How to Find Celestial Objects by Starhopping
version 1.11 (February 2008)
by Edwin H. Kaufman, Jr.
Professor of Mathematics (retired), Central Michigan University
edwin.h.kaufman@cmich.edu

List of Contents

I.  Introduction

II. Recommended equipment and other resources

III.  Distances, directions, locations, and brightnesses in the sky

     A.  Distances between celestial objects
     
     B.  North, south, west, and east
     
     C.  The celestial coordinate system
     
     D.  How stars and other objects move across the sky
     
     E.  Magnitudes
     
     F.  Inverted and reverted images
     
     G.  The ecliptic

IV.  Getting ready

     A.  Checking your slow-motion controls
     
     B.  Installing and aligning your finderscope and (if you
         have one) star pointer
     
     C.  Computing magnifications and field widths;  focusing

V.  Selecting an object to observe

VI.  Creating your starhopping plan to reach your object

     A.  Selecting a starting point
     
     B.  Why star charts seem to have east and west reversed from
         terrestrial maps
     
     C.  Scouting
     
     D.  Selecting intermediate objects in your path
     
     E.  Measuring distances between objects in your path
     
          1.  for free movement
          
          2.  for taxicab movement
          
          3.  for turn counting

     F.  Recordkeeping
     
     G.  An example:  The Ring Nebula
     
VII.  Carrying out your plan

     A.  Setting up your telescope with a rough polar alignment
     
     B.  Finding your starting point
     
     C.  Moving from point to point
     
          1.  Free movement
          
          2.  Taxicab movement
              
          3.  Turn counting

     D.  The "High in the sky" problem
          
     E.  Changing magnification
     
     F.  What if you realize you are lost in space?
     
     G.  What if you think you are pointing at your object, but you
         can't see it?
     
     H.  What if your object will appear only briefly between
         obstructions?

     I.  How much effect can errors in polar alignment have?

VIII.  Time

     A.  Computing the time correction for your date and location
     
     B.  Determining the transit time of your object
     
     C.  Using the Big Dipper as a clock
     
IX.  Viewing conditions

     A.  Light pollution
     
     B.  The moon
     
     C.  Other problems
     
X.  Buying a telescope

     A.  Types of telescopes

     B.  Aperture
     
     C.  Beware the large-magnification-small-aperture trap
     
     D.  Portability
     
     E.  Where to buy a telescope and related accessories

XI.  Taking care of your equipment

XII. Some beautiful objects for small telescopes

1.  M31--The Andromeda Galaxy (with satellite galaxy M32)

2.  65 Psc--A nice close pair of nearly equal magnitude (and
    sometimes colorful) stars

3.  NGC457--The ET Cluster

4.  M33--The Pinwheel Galaxy

5.  Gamma Ari--A pair of bright, nearly identical blue-white suns
    traveling side-by-side across the sky

6.  Gamma And--A blazing orange star followed by a bright green
    star--Mirror, mirror, on the wall, is this the fairest double
    star of them all?

7.  NGC869, NGC884--The Double Cluster, with a pretty little
    orange-blue double star along the way

8.  M34--A large, bright open cluster containing many double stars

9.  The Pleiades (M45)--The Seven Sisters

10. Beta Cam, NGC1502, and IC342--A bright, wide white-deep blue
    double star, the Golden Harp Cluster, and a large but faint
    galaxy

11. Omicron2 Eri and w Eri--A triple star and a superb yellow-
    green double star

12. The Hyades, with nearby open cluster NGC1647

13. Three from the hare--An attractive (but close) yellow-blue
    double (ADS 3954), a fairly bright but hard to resolve
    globular cluster (M79), and a wide yellowish-bluish double
    (gamma Lep)

14. The Terrific Trio--Open clusters M38, M36, and M37

15. Orion's sword, including the Great Orion Nebula (M42)

16. Zeta Ori and sigma Ori--A triple star and a quintuple star

17. M35--A splendid large cluster full of bright stars

18. A march through Monoceros--8 Mon (double star), NGC2244 (open
    cluster), C49 (the Rosette Nebula), NGC2264 (the Christmas Tree
    Cluster), the Cone Nebula, and (for the stout-hearted) NGC2261
    (Hubble's Variable Nebula)

19. Beta Mon--An attractive triple star

20. M41--One of the finest open clusters

21. 12 Lyn--A blue-blue-orange triple star, with nearby double 19
    Lyn as a bonus

22. NGC2281--An attractive, compact open cluster that vaguely
    resembles a bat flying east

23. NGC2392--The Eskimo Nebula, with double star delta Gem along
    the way

24. Castor--A gorgeous pair of brilliant white stars

25. M47--A big, bright milkshake in the sky, complete with a straw

26. NGC2403--A large and beautiful (but faint) galaxy

27. M48--"A tremendously pleasing cluster, a perfect arrowhead of
    bright stars with a tight, off-axis core"

28. A Cancerian quartet--Iota1 Cnc (a beautiful orange-green
    double), iota2 Cnc (a somewhat difficult white-yellow double),
    M44 (the Beehive Cluster), and zeta Cnc (a difficult triple of
    yellow stars, but an easy double)

29. M82 and M81--Two good galaxies for the price of one

30. A sextet of double stars--Regulus (a white-blue double), Gamma
    Leo (a beautiful pair of bright yellow stars), 54 Leo (an
    excellent yellow-blue pair), Denebola (a really wide white-blue
    pair), tau Leo (a stunning wide yellow-blue pair), and xi UMa
    (a close and somewhat difficult pair of nearly equal magnitude
    white stars)

31. NGC3242 and N Hya--The Ghost of Jupiter Nebula and a pleasing
    pair of nearly identical yellowish stars

32. An eight-galaxy tour through the Virgo Galaxy Cluster--M84, M86,
    M87, M58, M60, M59, M85, and M49

33. M104--The Sombrero Galaxy (and nearby multiple star)

34. M64--The Black Eye Galaxy

35. The Horse and Rider (Mizar and Alcor), two nice double stars
    (kappa Boo and alpha CVn), the Sunflower Galaxy (M63), and the
    superb "hypnotic eye" galaxy (M94)

36. M51--The Whirlpool Galaxy

37. M83--A galaxy which is best for southern observers, plus a red
    long-term variable star (R Hya) and an unexpectedly nice little
    double star along the way

38. M3--A glorious globular cluster

39. A seven-star tour through Bootes and Corona Borealis--Pi Boo
    (an unusual blue-yellow double), xi Boo (a white-reddish
    double), epsilon Boo (a striking but somewhat difficult white-
    bluegreen double), delta Boo (a wide white-deep blue pair),
    mu2,1 Boo (an amazing white-white-blue-blue quadruple), zeta CrB
    (a white-bluewhite double), and sigma CrB (a white-bluewhite
    double)

40. M5--A bright globular cluster

41. A bright double star and four globular clusters--Beta Sco, M80,
    M4, M19, and M62

42. M13--A spectacular globular cluster

43. NGC6210--A small but bright and easy-to-find planetary nebula

44. Rasalgethi--A beautiful orange-white double

45. M92--A pretty, well-resolved little globular cluster

46. M6--The Butterfly Cluster

47. IC4665--A large, bright, sparse open cluster with a shape that
    a dog mignt appreciate

48. M23--A superb open cluster

49. Barnard's Star--The fastest star

50. NGC6543--The Cat's Eye Nebula

51. Nu Dra, 16,17 Dra, 39 Dra, psi Dra, and 41,40 Dra--A double star
    quadruple-header plus a triple star
    
52. The Fantastic Four plus one--The Lagoon Nebula, NGC 6530, the
    Trifid Nebula, and M11, plus NGC6520

53. NGC6572--A bright little planetary nebula

54. M17 (the Omega Nebula), M16 (the Eagle Nebula), and M25--A big
    horseshoe in the sky, a nice little open cluster surrounded by
    nebulosity, and an open cluster containing a nice symmetric
    letter M

55. M22--A huge and bright globular cluster

56. IC4756 and NGC6633--Two bright open clusters

57. Epsilon1, epsilon2 Lyr--The double double star

58. Beta Lyr--A white variable star surrounded by 3 deep blue stars

59. M11--The Wild Duck Cluster

60. M57--The Ring Nebula

61. Albireo--A splendid orange-blue double

62. NGC6826--The Blinking Nebula

63. M27--The Dumbbell Nebula

64. 31,30 Cyg--A dazzling and wide orange-white-blue triple star

65. Omicron Cap--The chameleon double star?

66. 52 Cyg and NGC6960--An orange-blue double star and part of the
    Veil Nebula

67. NGC7009--The Saturn Nebula

68. 61 Cyg--A pretty little pair of orange dwarfs

69. M39--A big, bright Christmas tree in the sky

70. M15--A large and bright globular cluster

71. M2--One of the brightest globular clusters in the sky

72. NGC7209--A pretty open cluster contained (mostly) in an
    isosceles trapezoid

73. Delta Cep, mu Cep, and IC1396--A variable double star,
    Herschel's Garnet Star, and a large nebula containing an open
    cluster which contains a nice variable triple star
    
74. M52--A nice open cluster with many faint stars, with a
    quintuple star and The Bubble Nebula (NGC7635) along the way

75. NGC7662--The Blue Snowball Nebula

XIII.  Index of definitions, facts, and cautions

XIV.  Appendix A:  The constellations--abbreviations, names, and
       what they're supposed to be

XV.  Appendix B (optional):  A primer for spherical trigonometry


Chapter I.  Introduction


     This document discusses how celestial objects, such as
galaxies and multiple star systems, can be found by starhopping.
Starhopping means moving from object to object until the object
you want is reached.  
     There is also some general information in here about the
night sky and about telescopes, mainly in chapters III, IV, V,
VIII, IX, X, and XI.  People familiar with this information can
skim or skip it.  It is put in so that people just starting out
observing can find most of the information they need collected in
this one document.  Much of this has been borrowed from other
publications such as the ones mentioned in the next chapter, but
much of it is done in more detail here so that little prior
knowledge is assumed.  The main thing one needs to read this
document are an inkling of the concepts of angle, degree, circle,
and sphere.  There are a few more mathematical parts to intrigue
those who may be interested, but they are clearly marked (generally
by being placed inside square brackets) as being things that can be
skipped.
     The heart of this document consists of things that I have
never seen written down anywhere (which is not to say that they may
not have been written somewhere without my knowing about it);
these are mainly things I learned by experience over many nights of
observing.  Most of these things are in chapters VI and VII, with a
collection of beautifully objects in chapter XII that were found
with a 9.25 inch telescope (unless otherwise noted) in rather
light-polluted skies (although many of them would also look
beautiful in a smaller telescope, say a 5 inch telescope).  My
purpose in writing this document is to share these things with you,
in order that your viewing experience may be enhanced.  As a
secondary goal I would like to show that many beautiful things can
be seen even in the presence of fairly severe light pollution.  At
my location (which is my back yard), even in the better parts of
the sky one can barely see an object of magnitude 3 (namely the
star Albireo) with the naked eye, which is about 16 times brighter
than the faintest things one is supposed to be able to see with the
naked eye in excellent observing conditions; as far as seeing the
Milky Way with the naked eye from my back yard is concerned,
fuggedaboudit!
     There are many terms in this document;  most of them are
capitalized and defined where they first appear.  If you already
know some of these definitions, that is good, but it is not
necessary.  If you run across a term with which you are not familiar
that is not defined in here, you can probably find it in a
dictionary or encyclopedia.
     You may have heard of a kind of telescope called a go-to
telescope, which you can program to point to a desired object by
pushing some buttons.  If you have one of those, you may not need
to do much starhopping, although you may still find things of
value to you in this document.  If I may offer my humble opinion,
however, to me stargazing with a go-to telescope is like mountain
climbing with a helicopter;  you will be able to enjoy the view
from the top, but you will miss the fun and challenge in getting
there.  Also, you will not learn the night sky nearly as well as
you will if you employ starhopping.
     This document is written solely in plain text on a Macintosh
Powerbook using a text editor called BBedit.  In order to make it
possible for almost any word processor on almost any computer to
open this document and see the same thing I am seeing as I am
typing, I am foregoing drawing pictures and using certain symbols,
such as the symbol for degrees.
     I intend to keep this document on my website;  you can get it
by going to http://homepage.mac.com/edkaufman, scrolling to
"starhopping" if necessary, and using the "download" button to the
right of that file.  The document will be updated from time to
time to correct something or to add another object to the list in
Chapter XII;  when I do this, I will change the version number at
the start of the document.  For those who may prefer pdf format, an
identical copy of the document in pdf format will be kept on the
web site under the name starhopping.pdf;  pdf documents can be read
and printed using the free Adobe Acrobat Reader, which can be
downloaded from http://www.adobe.com.  If you have any comments or
corrections or suggestions, feel free to send them to me at the
e-mail address at the beginning of this document.
     Please feel free to make and distribute as many copies of this
document (or parts of it) as you wish without notifying or paying
me;  all I ask is that you acknowledge my authorship.
     I should make the usual statement that this material carries
no warranty of any kind.  I should also note that although several
companies are mentioned in here, I have no connection with any of
them, financial or otherwise.  I mention them only because I have
used their products or services and found them to be excellent;
there are many other companies not mentioned in here that would
also provide you with excellent products and services.
     I should warn you that what you will see with your telescope
will not look like a picture taken by the Hubble Space Telescope.
You will not see nearly as much detail as the Hubble does, and you
will not see nearly as much color either, except for a few stars.
For brilliant color in nebulas and things like that you generally
need a time-exposure photograph.  We will not discuss photography
in this document, but if you do want to photograph something, you
have to find it first, right?
     Some objects you will find fairly easily, and some objects you
will not find at all because your telescope is not big enough and/
or your viewing conditions are not good enough.  It is the purpose
of this document to bring you to a point where, with some practice,
you will be able to find anything which it is possible for you
to find, and when you can't find something, you will know
with a pretty fair degree of certainty that you were at least
looking in the right place.  You will discover that starhopping
has a substantial learning curve, but I hope this document will
shorten it for you.  To find certain objects you may need the
patience of a saint, the tenacity of a bulldog, and I dare say, the
heart of a lion!  When you find one of these difficult objects, you
will know that you have accomplished something that few other people
could do, and that, along with the final view, more than makes it
worthwhile.  Although some things you see may not seem too
impressive, you will also have some "Oh my Lord look at that"
experiences, often quite unexpectedly.
     There are many objects discussed in this document, mostly in
chapter XII.  These have starhopping paths given with them, which
have been field-tested and debugged already (although there is
always the possibility of a typo here or there).  Most of the
objects discussed in this document should be relatively easy for
you to find, with a few exceptions which are noted where they occur,
especially after you have had a little practice at it.  In the
beginning you will probably make many errors, such as mixing up
east and west, but you will get better as time goes on.  Although I
could have included a great many more objects, I have tried to
select ones that are especially beautiful or interesting in some
way, at least to me.
     As a final introductory comment, it might be good to point
out that one can read this document on a number of different
levels.  Some may be interested mostly in the list of beautiful
objects in chapter XII, some may be mainly interested in the
starhopping techniques, some may be interested in the introductory
material concerning observational astronomy (as a means to either
learn about this stuff or refresh one's memory of it), some may
be especially interested in the more mathematical parts (maybe even
including the proofs!), and some may find other tidbits of
interest.  It is certainly not required or expected that everyone
deeply study all of these things.  If even one person increases
his or her enjoyment of the night sky because of something they
read in this document, then I will feel that this project has been
a success.
     Clear skies, and good hunting!


Chapter II.  Recommended equipment and other resources


     This list consists mostly of things I have, since I can best
advise you about things I have used myself.  If you are missing
some of the items, some of the methods in this document may (or may
not) need to be modified.

1.  A telescope with a finderscope and equatorial mount.

Comment:  Essential.  The finderscope normally comes along with the
telescope.  My own observing has been done with a German equatorial
mount, but another kind of equatorial mount should work, although
the controls may be different than those described below.  If you
have a non-equatorial mount you will not be able to do a lot of the
telescope movements described in chapter VII below, but you can
still create a path using the ideas in chapter VI, then maybe
figure out how you should move your telescope.  An equatorial
mount, by the way, is one that can be aligned on the north
celestial pole (which is quite close to Polaris, the North Star),
so you can make separate movements toward and away from the pole,
and around the pole (more details are coming).  For more
information on mounts, see "Telescope basics" at the web site
http://www.celestron.com;  it would not be a bad idea to look over
the entire "Telescope basics" section while you are there.

2.  A pair of binoculars.

Comment:  Almost essential for the methods we will use.  This
doesn't have to be an expensive pair, but you need something to
help you scan the sky for some stars that are not bright enough for
you to see with your naked eye.

3.  The book "A Field Guide to the Stars and Planets", fourth
edition, by Jay M. Pasachoff, Peterson Field Guide series, Houghton
Mifflin, 2000.  ISBN 0-395-93432-x (cloth) or 0-395-93431-1
(paperback).

Comment:  Very highly recommended.  This little book contains a
wealth of information, including many useful star charts.  It is
recommended that you get it (about $30 hardbound or about $19
paperback) and read it straight through, perhaps skimming some
parts, but at least glancing at every page.  In the words of the
commercial, you'll be glad you did.

4.  The book "Star Atlas 2000.0" by Wil Tirion and Roger W. Sinott,
second edition, deluxe version (preferably), Sky Publishing
Corporation and Cambridge University Press, 1998.
ISBN 0-933346-87-5.

Comment:  Highly recommended.  This is a collection of large star
charts;  it includes a useful transparent overlay to help you
measure distances in the sky.  In order to use our procedures to
create your own starhopping paths, it is essential that you have
either this reference or the previous one or something else that
has the rectangular star charts of the type found in these
references.  When you are just learning about starhopping the star
charts in the previous reference should work well, but if you really
get serious about this, the star charts in this reference work
better since they are larger and include more stars (and that
overlay can be quite helpful).  Note, however, that if you want to
postpone the decision to acquire some charts, there are many
starhopping paths in this document that you can try out before you
start creating your own, and you won't need the charts for that.

5.  A chair to sit in while you are observing.

Comment:  Essential.  Without it you will not be able to keep your
head steady for long, and you will get tired.

6.  A small ruler (say 15 centimeters long), preferably marked in
millimeters (mm).

Comment:  Essential for creating your own starhopping paths.  A
larger ruler would work, but it would be awkward.

7.  At least two eyepieces, one with long focal length (say 25-40
mm) for searching and one with shorter focal length (say 8-20 mm)
for serious observing.

Comment:  Highly recommended.  One long focal length eyepiece and a
2x Barlow lens would fill the bill;  the Barlow halves the focal
length of any eyepiece with which it is used, and thus halves your
field width but doubles your magnification.  If you have only the
one eyepiece that came with your telescope and no Barlow, it might
be best if you do some observing first before buying another
eyepiece, although you would probably not regret getting a
high-quality Barlow.  There are many eyepieces you can buy, of
varying price and quality.  If you do buy another, make sure you
get the right barrel size to fit your telescope (refer to your
instruction manual;  one and a fourth inches is pretty standard).
You might also be wary of having too little eye relief (the
distance you have to hold your eye from the eyepiece);  5 mm may be
skimpy, especially if you wear glasses while observing.  Note that
an eyepiece with focal length too small may be of little use to you
since the image quality will be poor if the magnification is too
great (see Chapter X below or "Telescope basics" at
http://www.celestron.com for more details).

8.  A right ascension (or "single axis") drive motor that fits your
telescope.

Comment:  Recommended.  This will usually not help you find things,
but it will make it easier to observe them when you do find them
since it will keep them from drifting out of your field of vision
as the Earth turns.  You will find this especially desirable for
observing planets.  We will not specifically address finding planets
in this document, but some of the skills you learn here will be
helpful for that.  If you want to get a dual axis drive motor
(which might be the only kind available for your telescope), that
is fine too.

9.  A star pointer for your telescope.

Comment:  Recommended.  This is a zero-magnification sighting
device with a lighted red bulb that makes it easy to bring objects
that are visible with your naked eye into the field of your
finderscope.  If you do not have too much light pollution at your
observing site this will be especially helpful since then there
will be lots of naked-eye objects.

10.  A night vision flashlight.

Comment:  Recommended.  This will allow you to read your notes
outside at night withoug spoiling your night vision for 15 minutes
or so.  Putting red cellophane over the end of an ordinary
flashlight works too, but Celestron has a night vision flashlight
with two red LED's that is excellent.

11.  The two-page monthly sky map which is available for free
download at http://www.skymaps.com

Comment:  Recommended.  This contains a full-sky chart and a list
of objects suitable for viewing.  This can be quite useful, and
hey, the price is right.

12.  The monthly "Sky Calendar" from Michigan State University.

Comment:  Recommended.  To get this for a year, send your address
and $10 (currently) to Sky Calendar, Abrams Planetarium, Michigan
State University, East Lansing, MI 48824.  This contains a full-sky
chart plus a day-by-day calendar of celestial events (eclipses,
meteor showers, comets, etc.).  It has some information you may
be hard-pressed to find elsewhere, such as the locations of
Uranus and (sometimes) Neptune.  By the way, a subscription to this
makes a nice gift for kids.

13.  The book "Sky Atlas 2000.0 Companion" by Robert A. Strong and
Roger W. Sinott, second edition, Sky Publishing Corporation and
Cambridge University Press, 2000.  ISBN 0-933346-95-6.

Comment:  Recommended.  This contains descriptions and coordinates
for the star clusters, nebulae, and galaxies shown in reference 4
above, and may help you decide whether an object is worth looking
for.  This reference does not say anything about double stars,
triple stars, etc., but there is lots of information about these in
reference 3 above.  Be cautious about one thing, however:  When
they say something is "bright", this is in the eye of the beholder,
the telescope of the beholder, and the viewing conditions of the
beholder.

14.  A pack of 4x6 (or larger) index cards.

Comment:  Recommended.  This is for record-keeping, but if you want
to start out just writing stuff on a piece of paper, that is ok.
Once you have a lot of objects the cards will be quite convenient
since you can put one object on each card and pull out the ones
you want to look for on any given night.


Chapter III.  Distances, directions, locations, and brightnesses
in the sky


A.  Distances in the sky

     If you are already familiar with these concepts, you may want
to skim or skip this entire chapter, although there may be some
things in here you have not seen.
     When you look at the night sky, you will observe that some
objects appear to be close to each other, while others seem to be
far apart.  Note that just because two objects appear to be close
together in the sky does not necessarily mean that they are actually
close together in space; one of them could be much farther from the
Earth than the other.  We will not be concerned with the actual
distances in space between objects, but rather with how far they
appear to be apart from our vantage point, and we will measure
these distances in degrees.  For any two objects (or positions) in
the sky, imagine taking two sticks, touching them together at one
end, and pointing the other ends at the objects.  We then define
the DISTANCE BETWEEN THE OBJECTS to be the degree measure of the
angle where the sticks come together.  For example, the pointer
stars in the Big Dipper (see section VIA) are about 5 1/2
("five and a half") degrees apart.  You may recall that 90 degrees
make a right angle, 180 degrees make a straight angle, and 360
degrees make a full circle.
     Degrees are divided into smaller units, as follows.  60
ARCMINUTES make one degree, and 60 ARCSECONDS make one arcminute.
These units are sometimes called "minutes" and "seconds"
respectively, but we will usually call them arcminutes and
arcseconds to avoid confusion with time units.  Notation:  5' means
5 arcminutes, and 5" means five arcseconds.


B.  North, south, west, and east

     Now consider any object in the sky.  We need to have a way of
specifying directions from this object.  Here is how it is done.
First we imagine that all celestial objects are attached to a huge
sphere surrounding the Earth, called the CELESTIAL SPHERE.  If we
extend the Earth's axis northward until it hits the celestial
sphere, the point where it hits is called the NORTH CELESTIAL POLE.
The north celestial pole is only about 3/4 degree from the bright
star Polaris, the North Star, and for most practical purposes we
can use Polaris in place of the north celestial pole.  Likewise,
the point where the southward extension of the Earth's axis hits
the celestial sphere is called the SOUTH CELESTIAL POLE.  If we
pass a plane through the center of the Earth perpendiclar to the
Earth's axis, the circle where it hits the celestial sphere is
called the CELESTIAL EQUATOR;  one can think of the celestial
equator as being the Earth's equator projected outward onto the
celestial sphere.
     To give a precise and correct definition of north, south,
west, and east in the sky we need two more definitions.  A GREAT
CIRCLE on the celestial sphere is a circle formed when a plane
through the Earth's center hits the celestial sphere.  These are
called great circles because they are the biggest possible circles
on the celestial sphere.  We already know one great circle, namely
the celestial equator.  If a great circle contains the celestial
poles, it is called a POLAR GREAT CIRCLE (note that if a great
circle contains one of the celestial poles, it must contain the
other one also.)
     Now for any object in the sky which is not at one of the
celestial poles, there is one and only one polar great circle which
contains the object.  
NORTH is defined to be the direction one would go in the sky in
order to follow this polar great circle toward the north celestial
pole.
SOUTH is defined to be the opposite of north;  that is, it is the
direction one would go in the sky in order to follow this polar
great circle away from the north celestial pole (and thus toward
the south celestial pole).
WEST is defined to be the direction perpendicular to north and
south which would drag the part of the polar celestial great circle
between the north celestial pole and the object counterclockwise
(as seen from the Earth) around the north celestial pole (assuming
it is attached to the north celestial pole and is free to move
otherwise).  Another way of thinking of this is that if a little
man with his feet on the celestial sphere and his head down toward
you were to walk from the north celestial pole to the object
in question and then make a quarter-turn to his left, he would be
facing west.
EAST is defined to be the opposite of west;  that is, it is
perpendicular to north and south, and if one were to move east from
the object one would drag the part of the polar great circle
described above clockwise around the north celestial pole.
     These notions of north, south, west, and east coincide with
our usual notions of north, south, west, and east for objects that
are on your MERIDIAN, which is the north-south path in the sky
between the north celestial pole and the south point on your
horizon.  For objects not on your meridian, the definitions of
north, south, west, and east we have just given do not correspond
to our usual notions of north, south, west, and east.  It would be
a good mental exercise for you to check this, first thinking of an
object on your meridian just above the south point on your horizon,
then thinking of an object which is far from your meridian.  To
avoid confusing these concepts, from now on we will refer to our
previous notions of north, south, west, and east as TERRESTRIAL
north, south, west, and east.
     Now a simplified way of looking at this, which works well for
objects which are less than 90 degrees from the north celestial
pole (we say such objects are north of the celestial equator), is
that north means toward the north celestial pole, south means away
from the north celestial pole, west means perpendicular to north
and south in a direction that would take the object counterclockwise
around the north celestial pole, and east means perpendicular to
north and south in a direction that would take the object clockwise
around the north celestial pole.  Unfortunately, this simpler way
of looking at things can lead to problems for objects which are more
than 90 degrees away from the north celestial pole (that is, south
of the celestial equator).  For example, consider an object which
is just above your terrestrial southwest horizon.  If the object
were to move north, one would think that its path would have to be
tilted to the left with respect to your horizon in order to move
toward the north celestial pole, but in actuality the path would be
tilted to the right!  You can check this for yourself as follows.
Face terrestrial west, point your right arm and index finger toward
Polaris, and point your left arm and index finger toward an
imaginary object just above your terrestrial southwest horizon.
You are pointing at two points which are (approximately) on the
polar great circle that contains the object.  Since the celestial
sphere is so large that we usually consider the Earth to be just a
point, you can think of yourself as being at the center of the
Earth, and thus you are at the center of this polar great circle.
Now swing your left arm to your right arm, continuing to point
along this circle of which you are the center, then swing your left
arm back to where it was.  Your left index finger is pointing out a
north-south path in the sky.  Do this several times, so that you
have this path firmly in mind.  Now drop your arms and turn a
little to your left so that you are facing your object in the
terrestrial southwest, and you will see that the path you traced
several times in the sky tilts to your right!
     Now face terrestrial west again, point your right arm and
index finger toward Polaris again, and point your left arm and
index finger in the opposite direction to where your right arm and
index finger are pointing.  Your left arm and index finger will now
be pointing at the south celestial pole, although you will not be
able to see it, since it will be below your horizon.  For objects
south of the celestial equator, thinking of north as being away
from the south celestial pole works better than thinking of north
as being toward the north celestial pole.
     You may well ask, does thinking of these directions in a
somewhat complex but correct way instead of always thinking of
north as being toward the north celestial pole, actually
make any practical difference in your viewing?  The answer is
"Sometimes".  For example, if you want to view an object that
you know will only be briefly visible to you when it passes through
a gap in some trees a little terrestrial west of your terrestrial
south, and you know there is another object directly north of your
object that will be high enough to be seen above the trees, you
want to be able to use the higher object to judge where your object
is (even though it is behind the trees) so you can estimate when it
will brifly pop into the gap.  One is reminded of the words of
Albert Einstein when he said, "Everything should be made as simple
as possible, but not simpler."
     Oh by the way, north is often abbreviated as N, south is often
abbreviated as S, west is often abbreviated as W, and east is often
abbreviated as E.  We can also make combinations of these letters;
for example, NE means halfway between north and east, and NNE means
halfway between north and northeast.
     In case you are wondering why we don't just apply our notions
of terrestrial north, south, west, and east to the sky instead of
defining new directions, one reason is that these old notions of the
direction of one object from another change as the objects move in
the sky.  In contrast, with our new definitions the direction of
one object from another never changes;  for example, if star B is
north of star A at any particular time, then star B will always be
north of star A.  (The preceding statement ignores tiny effects
due to such things as the fact that the Earth slowly wobbles like
a top;  we will always ignore such things in this document.)


C.  The celestial coordinate system

     The location of any point on the Earth can be specified by
giving two coodinates, namely its latitude and longitude.
Similarly, any point in the sky can be specified by giving two
coordinates, namely its declination and right ascension.  Here's
how it works.
     First, we lay out a distance scale on the celestial equator
by dividing it into 24 equal-length pieces by means of points
which are numbered 0 to 23, moving eastward.  The zero point
happens to be a certain point in the constellation Pisces the fish;
we will say a little more about this point in section IIIG.  The
units of measurement are called hours, with each hour being
subdivided into 60 minutes, and each minute being subdivided into
60 seconds.  Now consider any point in the sky except the two
celestial poles.  We know there is one and only one polar great
circle that contains this point.  Consider that half of this great
circle that starts at the north celestial pole, goes through the
given point, and ends at the south celestial pole.  This half-circle
is called a LINE OF EQUAL RIGHT ASCENSION.  Then the RIGHT
ASCENSION of the given point is defined to be the number
corresponding to the point where this half-circle hits the
celestial equator, and the DECLINATION of the given point is
defined to be the distance (in degrees) between these two points,
with this number taken to be negative if the point is south of the
celestial equator.  Finally, we define the declination of the north
celestial pole to be 90 degrees and say that its right ascension is
undefined, and we define the declination of the south celestial
pole to be -90 degrees and say that its right ascension is
undefined.
     Here are some examples.  The declination of a point on the
celestial equator is 0 degrees, the declination of a point halfway
between the celestial equator and the north celestial pole is 45
degrees, and the declination of a point two-thirds of the way from
the celestial equator to the south celestial pole is -60 degrees.
The bright star Vega in the constellation Lyra has right ascension
18 hours, 36.9 minutes, and has declination 38 degrees, 46
arcminutes.  These numbers for Vega (and for most other objects we
will consider) are rounded, not exact.  We abbreviate this
information by saying that the coordinates of Vega are
RA 18h 36.9m, Dec 38 degrees 46' (or (18h 36.9m, 38 degrees 46')).
(There is a nice symbol for degrees, namely a small raised circle,
but since I want this document to be plain text and thus portable,
I will write out the word "degrees".)  This means that if we
consider the point on the celestial equator that is closest to Vega
(which is where the line of equal right ascension through Vega hits
the celestial equator), then Vega is 38 degrees 46' north of that
point, and that  point is 18h 36.9m east of the fixed point in
Pisces mentioned earlier.
     Now you may be asking, why do we measure right ascension in
time units instead of in degrees?  Good question!  Here is the
answer.  Note first that two points on the celestial equator
whose right ascensions differ by 1 hour are 15 degrees apart (since
360/24 = 15).  If two objects have the same nonzero declination
however and are 1 hour apart, then the distance between them is
less than 15 degrees, since the lines of equal right ascension get
closer together as you move away from the celestial equator.  Thus
using degrees for right ascension would lead to confusion.  Another
reason is that if you have two objects of equal declination, the
difference in their right ascensions will tell you approximately
how long it will take the object to the east to move to the position
occupied by the object to the west.  For example, if you have your
telescope pointed at an object with RA 10h 58m, Dec 23 degrees, and
you are interested in an object with RA 12h 27m, Dec 23 degrees,
then you will to wait about an hour and 29 minutes for your desired
object to come into view (since (12h 27m) - (10h 58m) = (11h 87m) -
(10h 58m) = (1h 29m)).
     Why did we say "about" an hour and 29 minutes?  It is because
there is a small discrepancy in these calculations;  this is
usually too small to bother us, but I mention it here since I know
you want to know the truth.  If for example you pointed your
telescope at an object and waited 24 hours, the object would have
gone past the position your telescope is pointing about 4 minutes
ago!  We have all been told that the Earth rotates on its axis
every 24 hours, but this is not quite right.  If you don't believe
that, suppose the Earth actually did rotate on its axis exactly once
every 24 hours, and consider what happens if you start at noon on
a sunny day and let exactly 180 24-hour days pass.  Then your
position on the Earth will still be facing the same direction in
space that it was at the start, but the Earth will have moved
around to the other side of the sun, so now the sun will be on
the other side of the Earth from where you are, and it will be
pitch black at noon!  This is unacceptable.  To avoid this, the
24-hour day was defined so that in 24 hours the Earth will turn a
little more than one rotation.  (To picture this, imagine looking
down on our solar system, so that the earth will be revolving
coounterclockwise around the sun and will be rotating
counterclockwise about its axis.)
     To get a figure for the time required for the Earth to make
one complete rotation on its axis, we note that the Earth takes
about 365.2422 24-hour days to make one complete revolution around
the sun, which is 365.2422 x 24 hours.  The actual number of turns
on its axis in this time will be one more than the number of 24-hour
days since otherwise we will eventually have problems like those
in the previous paragraph.  Thus the time for the Earth to make one
complete turn on its axis is about (365.2422 x 24)/366.2422 hours,
which is about 23.93447 hours, which is about 23 hours and 56.068
minutes.


D.  How stars and other objects move across the sky

     Because the Earth is rotating on its axis from terrestrial
west to terrestrial east, celestial objects (including the sun)
appear to move across the sky east to west.  This looks the same
as if the Earth were standing still, and the celestial sphere were
rotating east to west about the extended axis of the Earth.  Thus
celestial objects will appear to revolve about the extended axis
of the Earth.  An object will reach its highest point in the sky
(that is, the point that is closest to your ZENITH, which is the
point straight up from your location) when it crosses your meridian.
To better describe what this looks like, we consider three cases.
As usual, we assume you are a northern-hemisphere observer, that is,
you are observing from a place that is terrestrial north from the
Earth's equator.
Case 1.  If an object is north of the celestial equator (that is,
it has positive declination), it will appear to move in a
counterclockwise circle about the north celestial pole (even though
part of that circle may be below your horizon, and thus invisible
to you).  This will be especially obvious if the object is
so close to the north celestial pole that it never sets.
Case 2.  If an object is on the celestial equator (that is, it has
zero declination), it will rise due terrestrial east, climb to
its highest point directly south of your zenith, then descend and
set due terrestrial west.  You can convince yourself of this as
follows.  Point your right index finger toward Polaris, lay your
left index finger across your right index finger at right angles,
horizontal, and pointing right, then rotate your hands
counterclockwise as far as you can, keeping your right index finger
pointing at Polaris and and your left index finger at right angles
with your right index finger.  Your left index finger will then
point out a good approximation of the path followed by an object
on the celestial equator, and you will see that the path is as we
described it above.
Case 3.  If an object is south of the celestial equator (that is it
has negative declination), then it will rise somewhere terrestrial
south of terrestrial due east, climb until it reaches its highest
point due south of your zenith, then descend and set somewhere
terrestrial south of terrestrial due west.  Now this last case may
be confusing if you are trying to think of the object as revolving
about the north celestial pole;  remember that what the object is
actually doing is revolving about the extended axis of the Earth.
Another way to think of this is that the object is revolving
clockwise about the south celestial pole (which you cannot see
because it is below your horizon).
     We have just discussed how an object (or group of objects)
moves across the sky;  it is sometimes also useful to know how a
group of stars will change its orientation as it moves.  Let us
consider an isosceles triangle which points north;  for example, it
could consist of two stars one degree apart with the same
declination and a third star to the north which is three degrees
from each of the other two stars.  First suppose this triangle is
well north of the celestial equator, and we start watching it when
it is to the upper right of the north celestial pole (from your
point of view as you face terrestrial north).  It will then point
to your lower left, but as it moves it will point straight down
when it gets directly above the north celestial pole, and eventually
it will point to your lower right.  Thus if you think of a point
inside this triangle, the triangle will slowly rotate
counterclockwise around this point.  Now suppose the triangle is
well south of the celestial equator, and you are facing south.  The
object will first appear to your left, when it will be pointing to
your upper left.  As it moves it will point straight up when it gets
directly above the terrestrial south point on your horizon, and
eventually it will point to your upper right.  Thus it will appear
to rotate clockwise around any point inside it.
     We already know that an object will be at its highest in the
sky when it is on your meridian, but sometimes we also want to
know just how high it will be above the horizon then, so we can
judge in advance whether it will get high enough to clear trees
and other obstacles that may be in the way.  To do this we need
the following fact:

Fact 1:  For someone who is observing from a location on or north
of the equator of the Earth, the distance between the north
celestial pole and the point due terrestrial north on the horizon
equals the observer's latitude.

     Oh, so you want to know why this is true?  Good for you;  an
inquiring mind is a great asset.  We consider three cases:  The
observer is on the equator of the Earth, the observer is at the
north pole of the Earth, or the observer is somewhere in between.
     In the case where the observer is on the equator of the Earth,
the north celestial pole will be at the due terrestrial north point
on the horizon, so its distance from this point will be 0 degrees,
which equals the observer's latitude.  So far so good.
     In the case where the observer is at the north pole of the
Earth, the north celestial pole will be straight above his head,
so its distance from every point on the horizon will be 90 degrees,
which equals the latitude of the observer.  Still so far so good.
     The case where the observer is in between is more difficult.
We will give this proof inside square brackets to indicate that you
can skip it if you want to rest your mind.
[Suppose the observer is in between the equator of the Earth and
the north pole of the Earth, so his (north) latitude is greater than
0 degrees but less than 90 degrees.  Draw a circle to represent a
side view of the Earth, with the north pole at the top.  Mark a
point on the upper left quarter-circle of your circle, and label
it Y to represent you.  Label the center of the circle (and of the
Earth) C, and draw the line segment CY.  Mark the point directly
below Y and directly left of C with the letter F (since this is the
foot of a perpendicular you have dropped to the plane which
contains the equator of the Earth).  Draw the line segments YF and
CF, and you will have a right triangle YCF.  Now angle YCF equals
your latitude by definition of latitude, and since the angles of a
triangle always add up to 180 degrees, angle CYF equals 90 degrees
minus angle YCF, that is, 90 degrees minus your latitude.  Now
extend line segment YF upward and mark a point P on the extension
above Y.  Since the ray (or line if you like) YP is parallel to the
axis of the Earth, it points at the north celestial pole, at least
within our ability to measure, since the radius of the Earth is
tiny compared to the distance to the north celestial pole.  Now
draw a ray from Y perpendicular to line segment CY, extending
upward and to the right, and mark a point Q on this ray.  This ray
will be tangent to the circle (and thus tangent to the Earth) and
will point to the terrestrial north point on your horizon.  Thus
angle PYQ will give the distance between the north celestial pole
and the terrestrial north point on your horizon.  Now, it only
remains to show that angle PYQ equals angle YCF.  To do this, note
that angle PYQ + angle QYC + angle CYF = 180 degrees, since these
three angles form a straight angle.  Filling in what we know in
this equation, we get
angle PYQ + 90 degrees + (90 degrees - angle YCF) = 180 degrees.
Subtracting 180 degrees from both sides of this equation and adding
angle YCF to both sides gives angle PYQ = angle YCF, and the proof
is complete.  If you successfully made it through this proof, give
yourself one gold star!]
     Now let's consider three examples.  Suppose you are at
latitude 40 degrees (north) and you want to observe an object whose
declination is -30 degrees.  Your best chance may be when it is on
your meridian, since the object will be highest in the sky then.
(In chapter VIIIB we will learn how to find out what time this will
happen.)  At that time a half-circle can be drawn from the
terrestrial north point on your horizon, through your zenith, then
through the object, and finally to the terrestrial south point on
your horizon.  By Fact 1, moving from the terrestrial north point
on your horizon to the north celestial pole will take 40 degrees.
Continuing to the celestial equator will take 90 more degrees, and
continuing from there to your object will take 30 more degrees,
for a total of 160 degrees (since 40 + 90 + 30 = 160).  Since going
a full 180 degrees would take you to the horizon, your object will
be 20 degrees above the terrestrial south point on the horizon
(since 180 - 160 = 20).  That's not very high, being only 2/9 of a
right angle.
     For the second example, if we repeat the above calculation for
an object with declination -50 degrees, we will see that at its
highest your object will be on your horizon, and trying to see it
is hopeless (unless you can jump very high).
     For the third example, again suppose your latitude is 40
degrees, and the declination of your object is 80 degrees.  Then
when the object is on your meridian (and thus at its highest point
in your sky), it will take 40 degrees to go from the terrestrial
north point on your horizon to the north celestial pole, then 10
more degrees to reach your object (since the north celestial pole
has declination 90 degrees, and 90 - 80 = 10), so your object will
be 50 degrees above the north terrestrial point on your horizon
(since 40 + 10 = 50).


E.  Magnitudes

     You have no doubt observed that some heavenly bodies are much
brighter than others.  The brightness of an object is specified by
giving its MAGNITUDE, where the higher the magnitude, the dimmer
the object.  Part of the definition of magnitude is that for each
increase of the magnitude number by 1, the brightness diminishes
by a factor of the fifth root of 100 (which is about 2.5).  Thus an
object with magnitude 3.7 is about 2.5 times dimmer than an object
with magnitude 2.7, an object with magnitude 7.7 is 100 times
dimmer than an object with magnitude 2.7 (since 7.7 - 2.7 = 5, and
the fifth power of the fifth root of 100 is 100), and an object
with magnitude 1.8 is about 6.25 times brighter than an object with
magnitude 3.8 (since 3.8 - 1.8 = 2, and the fifth root of 100 to
the second power is about 2.5 to the second power, which is 6.25;
actually, using the more accurate value 2.5119 for the fifth root
of 100 gives the more accurate value 6.31, but you won't notice the
difference between 6.25 and 6.31 observationally).  As a final
example, an object with magnitude -0.2 is about 4.37 times as bright
as an object with magnitude 1.4 since 1.4 - (-0.2) = 1.6 and the
fifth root of 100 to the power 1.6 is about 4.37.  As an
approximate base point, we can use the fact that bright star Vega
in the constellation Lyra has magnitude approximately 0 (to be more
precise, the magnitude of Vega is about 0.03).
     One has to be careful when comparing the magnitudes of
different kinds of objects, however.  A person with excellent
eyesight can just manage to see a star with magnitude 6 if the
viewing conditions are excellent, but the same person might not be
able to see a galaxy with magnitude 6, since the galaxy is spread
out, and any point on it would be dimmer than magnitude 6.
     We note in passing that sometimes people define magnitudes
differently when they are doing photography with filters, but we
will only consider visual magnitudes, not photographic magnitudes.
     Now you may be curious as to where the fifth root of 100 comes
from.  It seems that once when magnitudes were specified only
informally, someone noticed that stars that differed by five
orders of magnitude differed in brightness by a factor of about
100, and they had the bright idea of redefining the magnitude
concept so that stars differing in brightness by a factor of 100
would have magnitudes that differed by exactly 5;  to make this
work, and to have brightness depend exponentially on magnitude, a
difference of 1 in magnitude had to correspond to a factor of the
fifth root of 100 in brightness.  [Although you don't need to
remember this, the equation relating brightness to magnitude looks
like this:  Brightness = (a constant) x ((1/the fifth root of 100)
to the power magnitude).  Knowing what the constant is would not be
all that helpful to us, but even without knowing that, the
interested reader who knows his or her laws of exponents can use
the equation to verify the examples in the previous paragraph.]


F.  Inverted and reverted images

     Whe you view something through a normal finderscope, the image
is INVERTED, that is, flipped both up and down, and left and right.
Another way to think of this is that any pair of points is flipped
180 degrees;  thus if star A is actually to the upper right of star
B in the sky, then when viewed through the finderscope star A will
be to the lower left of star B.  If a group of stars in the sky
looks like a capital L, through the finderscope the crossbar will
be at the top instead of the bottom, and will be pointing left
instead of right.
     It gets worse.  From here on we will assume that you are using
a star diagonal with your telescope, which is a tube with a right
angle bend (and a mirror or prism inside) that is used to allow you
to have a more comfortable viewing position.  Through the telescope
(with star diagonal) the image will be REVERTED;  this means that
if your star diagonal is pointing so that the plane which contains
both the center of the telescope and the center of the star
diagonal is perpendicular to the ground, then the image will be
flipped right and left but not up and down.  Thus the L in the
previous example will have the crossbar still on the bottom, but
pointing left.  If your star diagonal is oriented differently than
described above, then the view changes;  in all cases, the image is
flipped over the plane which contains the center of the telescope
and the center of the star diagonal.
     In addition, angles are reversed with the star diagonal, so
if you think of a little man walking with his feet on the L and his
head down toward you, from the top of the L to the end of the
crossbar, then he must turn left at the crook, but in the reverted
image he must turn right at the crook.  Angles are not reversed in
the inverted image of the finderscope, as you should check by
thinking about the L and the little man.
     In a rectangular star chart like the ones in reference II-4,
or the atlas charts in reference II-3, north is up, west is right,
south is down, and east is left.  Thus, starting with west, the
directions cycle clockwise as follows:  west, south, east,  north,
and back to west.  Because angles are reversed in the star diagonal,
the clockwise cycling is west, north, east, south, and back to west.
This is the normal cycling in a terrestrial map! (think California,
Minnesota, New York, Mississippi, and back to California).  Thus
once you know which way west is, you can use this cycling to
determine the other directions.  But west is easy to determine in
your telescope;  that is the direction all objects will drift if
you watch them for a few seconds.  Thus for example suppose you are
looking at a star, and the path you created says that there is a
check star to the northeast of your star.  Suppose further that
things are drifting straight down.  Then west is down, the cycling
says that north is to the left (since Minnesota is a quarter turn
clockwise from California on a map of the U.S., and a quarter turn
clockwise from down is left), and east is up, so northeast is to
the upper left.  Thus if you move the slow motion controls to put
your star to the lower right of your field of vision, then your
check star should appear to the upper left of your field of vision
(assuming that the distance between your star and the check star is
less than the width of your field of vision).
     We should note that there is another way to determine the
various directions in your telescope and star diagonal;  for
example, if you want to know which way is north, just turn your
declination slow-motion knob in the direction that moves your
telescope north, and north will be the opposite of the directions
that the stars move.  As another example, if you want to know which
direction NNW is in your telescope image, you can turn the Dec
slow-motion knob a little to make the telescope move N, then since
W is counterclockwise a quarter turn from N (since California is a
quarter turn clockwise from Minnesota on a map of the U.S.), NNW
will be one-fourth of a quarter turn counterclockwise from the
direction the telescope moved.
     If dealing with inverted and reverted image sounds like a
pain, there is a way around it.  One can avoid inverted
finderscope images by buying a correct-image right-angle
finderscope;  this also helps alleviate the "High in the sky"
problem discussed in section VIID below.  This is not a bad idea.
Cooncerning reverted telescopic images, there is a device called an
amici prism you can buy to avoid that, but with the starhopping
method we are using, reverted images are not much of a problem,
and the amici prism has the drawback that it can reduce your field
of view with low-power eyepieces, thus making searching for objects
harder.  My recommendations, which are not shared by everyone, are
to first try out your telescope with the normal finderscope that
comes with it, and buy the correct-image right-angle finderscope
if twisting yourself into the shape of a pretzel to look up through
your finderscope is too much of a pain (maybe literally!), but
don't bother with the amici prism.


G.  The ecliptic

     Although we will not need to consider the ecliptic after this
section, knowing about it does provide some useful information, and
it dovetails nicely with something we did earlier.  The ECLIPTIC is
defined to be the intersection of the celestial sphere with the
plane which contains the sun and the orbit of the Earth.  One could
also define it to be the path followed by the sun through the
heavens (even when the sun is below the horizon).  Since the orbits
of the moon and all the planets (except Pluto) are close to being
in the same plane, the moon and planets (except Pluto)
approximately follow the ecliptic across the sky, with some
variation up and down.  Unlike the celestial equator, the ecliptic
changes its position in your sky;  for a northern-hemisphere
observer, it is low in the south in summer (or below the horizon if
you live far to the terrestrial north) and higher in the winter.
By the way, why do we call it the ecliptic?  This probably has
something to do with the fact that if the moon is close enough to
the ecliptic when it is full, then we will have an eclipse of the
moon, and if the moon is close enough to the ecliptic when it is
new, then we will have an eclipse of the sun.
     Because of the tilt of the Earth's axis relative to the plane
which contains the Earth's orbit and the sun, the ecliptic is
tilted about 23 1/2 degrees relative to the celestial equator.
These two great circles on the celestial sphere meet at two places,
on opposite sides of the celestial sphere.  One of these is the
zero point for right ascension mentioned in section IIIC earlier.
This point is called the VERNAL EQUINOX.  The other point on the
celestial sphere where the ecliptic and the celestial equator meet,
which is directly opposite from the vernal equinox, is called the
AUTUMNAL EQUINOX.
     The term "autumnal equinox" also refers to the moment in late
September when the plane which contains the Earth's center and
equator hits the center of the sun;  this marks the end of Summer
and the start of Fall. This time and date change a little from year
to year due to leap years, and the fact that the time it takes the
Earth to make a complete revolution around the sun is not an
integer number of 24-hour days (it is about 365.2422 days, as
mentioned earlier).  At the time of the autumnal equinox, if your
Earthly longitude is evenly divisible by 15, then the right
ascension of any object on your meridian will equal your (24 hour)
local standard time.  We will have more to say about time in
chapter VIII.


IV.  Getting ready


A.  Checking your slow-motion controls

     Once you have set up your telescope and mount (the mount being
the thing that connects the telescope to the tripod), locate the
RA clamp and slow-motion knob, and the Dec clamp and slow-motion
knob.  As always, RA means right ascension and Dec means
declination;  the clamps might be called something else, for
example, locking levers.  We will now determine how far one
complete turn of either knob will take us.
     For declination, locate the declination scale (marked in
degrees) and pointer.  Turn the knob four full turns in either
direction, and see how many degrees the telescope moves.

CAUTION 1:  Never force anything!  With some telescopes, a
slow-motion knob may turn only so far;  if so, do the test by
turning the knob in the opposite direction.  Even if the knob will
turn, if you encounter resistance, parts of your equipment could be
hitting against each other, and trying to force the knob to turn
against resistance could cause damage.

CAUTION 2:  A slow-motion knob may not work correctly unless the
corresponding clamp is engaged, so engage the clamp before turning
the knob.

Note:  There should be a protrusion of some type on the knob or
its shaft that you can see and feel (if not, make a raised mark on
the knob with tape or something);  you will have completed one turn
when the protrusion returns to its original position.  Did the
telescope move about 10 degrees?  If so, one turn of the declination
slow-motion knob produces a movement of about 2 1/2 degrees.  I
think this is standard, but if you get something quite different,
you can compute the number TDec of turns to move 2 1/2 degrees in
declination as follows:  TDec = (the number of turns it takes to
move 10 degrees)/4. In the rest of this document we will assume
TDec = 1;  if TDec differs from 1 by more than about 1/8 (that is,
TDec is less than 7/8 or greater than 1 1/8), to improve accuracy
you may want to multiply all the north-south turn numbers later on
by TDec.
     For right ascension, locate the RA scale (marked in hours and
minutes) and pointer and turn the RA slow-motion knob 6 complete
turns in either direction, observing the same cautions as above.
If necessary, make a raised mark on the knob with tape or
something.  Did the telescope move about 1 hour (that is, 60
minutes)?  If so, then one turn of the RA slow-motion knob produces
a movement of about 10 minutes.  I think this is also standard, but
if you get something quite different, you can compute the number
TRA of turns to move 10 minutes as follows:
TRA = (the number of turns it takes to move 1 hour)/6.
In the rest of this document we will assume TRA = 1;  if TRA
differs from 1 by more than about 1/8, to improve accuracy you may
want to multiply all east-west turn numbers later on by TRA.
     You may wonder if there is any consistency between Dec turns
and RA turns.  Well, yes, there is;  in both cases, 144 turns would
produce a full circle. (Please don't actually try to move a full
circle, since parts of your equipment may bang into each other,
possibly causing damage.)  Let's check this.  For Dec,
(144 turns) x (2 1/2 degrees per turn) = 360 degrees = a full
circle.  For RA,
(144 turns) x (10 minutes per turn) = 1440 minutes = 24 hours = a
full circle.
     Now align your telesope with the front end of the mount, and
while looking directly at the end of your RA slow motion knob, by
turning the knob back and forth, determine which direction of
turning (clockwise or counterclockwise) causes your telelscope to
rotate counterclockwise about the fixed arm on the mount.  This is
the direction of turning that will cause your telescope to move
west while you are observing.  Try to remember this direction of
turning;  it will never change for your telescope unless you remove
the RA slow-motion knob and install it somewhere else (which is
possible with some telescopes).  We cannot do a similar test for
the Dec slow-motion knob and have the result remain constant, since
the same direction of turning of that knob will sometimes cause
your telescope to move north, and will sometimes cause it to move
south.  To verify this, imagine turning the Dec slow-motion knob so
that your telescope aim approaches the north celestial pole;  your
telescope will be moving north, but if you keep on going until you
pass the north celestial pole, then your telescope will be moving
south!  When actually observing, you can determine which direction
of rotation of the Dec slow-motion knob corresponds to north by
giving the knob a little twist one way and then back, and seeing
which way moves the far end of the telescope roughly toward the
north celestial pole.  Do not take your hand off the knob when
doing this test, since you want to be able to get back to the
object you were looking at.


B.  Installing and aligning your finderscope and (if you have
    one) star pointer

     Do this according to the instructions that came with these
things.  The finderscope should be aligned with sufficient
accuracy that when an object (say a bright star) is in the
crosshairs, with your longest-focal-length eyepiece the object will
be as close as possible to the center of the field of vision of
your telescope, preferably no more than half the radius of this
field away from the center.  If you have a star pointer, try to
align it so that when an object is centered in the star pointer,
the object is as close as possible to the center of the field of
vision of your finderscope, preferably no more than half the
radius of this field away from the center.  By the way, if it is
too dark to see the crosshairs in your finderscope, you should
re-align it sometime when there is enough light in the sky to see
the crosshairs, so that you can get an accurate alignment.

CAUTION 3.  If you have a star pointer, always turn if off
immediately after it has served its purpose in locating the object
that you pointed it at.  This is because the little red light will
not be obvious when you are not looking directly down the star
pointer, and it is fiendishly easy to forget that it is on, with
the result that you run down your battery and also wear out your
red LED faster.

     Your finderscope and star pointer can come out of alignment
after a while, especially if they are bumped or a little screw
comes loose.  If this happens, you will probably notice it quickly,
and can re-align them.


C.  Computing magnifications and field widths;  focusing.

     It will be useful for you to compute in advance the
magnifications and field widths for your telescope when used with
your various eyepieces.  Here are the formulas:

magnification = (focal length of telescope)/(focal length of
                eyepiece)

field width = (apparent field of eyepiece)/magnification

The focal length of your telescope will be stated in your
instruction manual.  The focal length of an eyepiece will be
printed on the eyepiece.  If you do not know the apparent field of
an eyepiece, it will probably be on the manufacturer's web site,
or you can call them and ask them.
     It may be useful for you to select a sequence of eyepieces
(possibly including a Barlow) that you will usually follow while
observing.  You will start with low magnification to get a wide
field for searching, then go to higher magnifications for
observing.  This sequence need not be set in stone;  you can change
it from time to time.  Then the first time you are observing with
this sequence, find a star that is easily visible at your lowest
magnification (but not extremely bright since an extremely bright
star may be hard to focus on accurately) and focus it using the
first eyepiece in your sequence.  Then go to the next eyepiece in
your sequence, and record how many quarter-turns of the focusing
knob (and in what direction) it takes to re-focus the object.
Continue through your sequence.  Why quarter-turns?  Well, they are
easy to estimate by feel;  just rotate your hand through a right
angle.  Why do this at all?  Well, when you switch to a new eyepiece
you may find yourself looking at a blank field, especially in a
light-polluted area, and then it is hard to refocus.  By knowing
about how many quarter-turns it takes to refocus you can probably
at least get the focus good enough to see something, and then it is
easy to finish the refocusing by sight.  Incidentally, because of
the effect of gravity on the internal parts of your telescope, you
can often get a shaper focus if your last little adjustment is
counterclockwise.  In difficult cases you may need to turn the
focusing knob back and forth through the best focus spot several
times before you are able to get the focus as sharp as possible,
given your equiment and the viewing conditions.
     To give you an example of these things, I will now give you a
table that I use, although yours will be different if you are using
a different telescope and/or different eyepieces.  The main
telescope I used is a 9.25 inch aperture Celestron
Schmidt-Cassegrain (non-computerized) with focal length 2350 mm.
My sequence consists of a 40 mm Celestron Plossl eyepiece with
apparent field 46 degrees, a 25 mm Celestron Plossl eyepiece with
apparent field 50 degrees, an 8.8 mm Meade ultra wide eyepiece with
apparent field 84 degrees, and the Meade used with a Celestron
deluxe 2x Barlow.  In the table below, "x" means "times", the
fields are in degrees, and the focusing changes are in quarter
turns of the focusing knob.  Some of the magnifications and fields
are rounded off.

eyepiece     magnification     field     focusing change

40 mm        58.75x            .78
                                         1 1/2 clockwise
25 mm        94x               .53
                                         1 1/4 clockwise
8.8 mm       267x              .31
                                         2 counterclockwise
8.8 mm +     534x              .16
Barlow

     Now it is a rule of thumb that your results will not be very
good if your magnification is more than about 50 or 60 times your
aperture (in inches), which in this case would be 462.5x or
555x;  if viewing conditions are poor, you may need to go even
lower than that in magnification in order to get a reasonably
sharp image.  In my experience, at 58.75x the image is sharp but
many faint objects cannot be seen;  at 94x the image is still
sharp and more objects can be seen;  at 267x the image may be a
little fuzzier but is still good; while at 534x the image is still
fuzzier and not too stable (it tends to fluctuate and jump around
due to air currents in the atmosphere).  The 267x usually gives
me the image with which I am happiest, but sometimes the 534x is
needed, for example, when I am trying to split (that is, see both
members of) a close double star.
     At times where noted below I also used an 11 inch aperture
Celestron Schmidt-Cassegrain (non-computerized) telescope with
focal length 2800 mm.  The table for this telescope looks the same
as the table above, except that the magnifications are
approximately 70x, 112x, 318x, and 636x, while the field widths are
approximately (in degrees) .66, .45, .26, and .13.
     It is also useful to know the field width of your binoculars
and finderscope.  The magnification and aperture (diameter of the
big lens at the end) are given in the description of the device;
for example, 7x50 binoculars have magnification 7x and aperture
50 mm.  If you do not know the field width, you can estimate it by
looking at one of the following pairs of stars, estimating what
fraction of your field is covered by the two given stars, (2/3 or
whatever), then dividing that into the given approximate separation.
The location of these pairs of stars can be found in section VIA
below; at least one of the pairs will probably be visible at any
given time:

Pointer stars in the bowl of the Big Dipper:  About 5 1/2 degrees

The two farthest-apart stars in the parallelogram just SE of Vega
(these are zeta2,1 Lyr and gamma Lyr;  see also section VIG
below):  About 6 degrees

The middle star in the W of Cassiopeia and the star at the end of
the W to the west:  About 6 degrees

Castor and Pollux in Gemini:  About 4 1/2 degrees


Chapter V.  Selecting an object to observe


     Up to now most of what we have said is stuff you could find in
various books, although we have often gone into greater detail than
they do.  In this chapter and the next two chapters there is a lot
of stuff not found in books because it comes from my own
personal experience.  I hope it helps you!
     In selecting an object to observe, if you are a beginner you
might want to start with the moon, Venus, Mars, Jupiter, or Saturn
(if they are up when you want to observe).  There are many objects
in chapter XII of this document along with paths to them and other
discussion.  References  II-11 and II-12 are good sources of ideas.
     You can also use reference II-3 as follows;  this is how I
selected most of the objects in chapter XII of this document.
First, use the monthly sky maps in chapter 3 of reference II-3 to
get an idea of what part of the sky will be visible when you want
to observe; references II-11 and II-12 would be good for this also.
Then look at the atlas keys on pages 224-227 of reference II-3 to
see which atlas charts correspond to a part of your sky that is
visible when you want to observe.  (You may want to bookmark or
paperclip pages 224-227 because you will refer to them often.)
Then go to the atlas charts in chapter 7 of reference II-3 and read
the descriptions.  There will be a number of objects mentioned from
which you can select.  One of the double or triple star systems
they suggest might be good to start with since it will probably
be visible in your binoculars and thus relatively easy to find.  It
would be good to select one that says "wide pair" or "easily
observed" however, since some of them may not be splittable
unless you have a bigger telescope or better viewing conditions.
     If you select a galaxy or nebula or open star cluster or
globular star cluster you can probably find more information about
it in reference II-13 if you have that reference.  Be warned that
when they say "bright" in that reference they may have in mind a
bigger telescope or better viewing conditions than yours.  It
would be good to pay attention to the visual magnitude given in
reference II-13 (they call it mv);  in my experience, if this
number is 7 or lower the object is probably fairly easy to find,
but if this number is 10 or higher the object is probably going to
be quite difficult.  This number doesn't tell the whole story,
since two objects with the same number could vary a lot in
difficulty depending on location in the sky (things near the
horizon tend to be tough) and other factors.


Chapter VI.  Creating your starhopping plan


A.  Selecting a starting point

     Now that you have selected an object in the previous chapter,
it is time to decide how to find it.  Your starhopping plan will
consist of a sequence of objects and instructions on how to get
from one object to the next, with the last object being the one
you are looking for.  You will need a starting point for this
sequence.  The purpose of this section is to give you a list of
good objects to use for starting points.  A starting point
should normally be a bright star (or group of stars) that you can
easily find with your naked eye.  It would be nice if the starting
point were close to your final object, but that is not necessary;
what you need is something you can find easily.  It is also helpful
(but not absolutely necessary) if there is another star nearby that
you can see with your binoculars and finderscope;  this "check
star" will help you verify that you have the correct object in view
in your telescope.
     We now list five groups of bright stars that are easy to
recognize;  no matter when you observe, one or more of these should
be visible to you.  For each one, we also list the season when it
is usually easiest to find it in the evening.  You will eventually
become familiar with these, and you may find other things on your
own that are convenient starting points for you.

1.  The Big Dipper group (Spring, Summer, Fall)

     The Big Dipper is probably the best-known of all collections
of stars;  it is part of the constellation Ursa Major the big bear.
Our use of the term "Big Dipper group" will extend far beyond Ursa
Major, however.  The Big Dipper didn't get its name for nothing;
that's what it looks like.  It has four stars in its bowl and three
stars in its handle.  The two stars in the bowl which are farthest
from the handle are called the pointer stars because they point
roughly to Polaris, the North Star (magnitude 2.0), which is about
3/4 degree from the north celestial pole.  The pointer stars both
have right ascension approximately 11h, and the northern one is
about 28 degrees from Polaris.  The pointers are about 5 1/2 degrees
apart, so they make a good measuring stick for the width of field
for your binoculars and finderscope (see if you can see them both
at once with your binoculars and finderscope).
     If you follow the pointer stars in the opposite direction
about 46 degrees, bending a little east, you will come to Regulus
(magnitude 1.35) in the constellation Leo the lion.  This star can
be recognized because it is at the end of the handle of a sickle
which is about 15 degrees from bottom to top;  the top is roughly
north of Regulus, and opens west.
     Now if you go back to the Big Dipper you can follow the arc
of the handle about 31 degrees to Arcturus (magnitude -.004) in
the constellation Bootes the herdsman, then continue on another
approximately 32 degrees to Spica (magnitude 0.98) in the
constellation Virgo the virgin.  Finally, from Arcturus go about 20
degrees WNW to Alphekka (magnitude 2.2), which is in the
constellation Corona Borealis the northern crown.  Alphekka is in
the SSW part of a group of 6 stars about 7 degrees across that looks
like a crown (or cup) and opens NNE.

2.  The Summer Triangle (Summer, Fall)

     This consists of the three bright stars Vega (magnitude 0.03)
in the constellation Lyra the harp, Deneb (magnitude 1.25) in the
constellation Cygnus the swan, and Altair (magnitude 0.77) in the
constellation Aquila the eagle.  These stars form a large, nearly
isosceles triangle that points south.  Altair (the south vertex)
has declination about 9 degrees, Vega (the NW vertex) has
declination about 39 degrees, and Deneb (the NE vertex) has
declination about 45 degrees;  Vega and Deneb are about 24 degrees
apart.  Cygnus contains a group of 6 stars called the Northern
Cross;  Deneb is at the top of the cross, and the bottom of the
cross is inside the Summer Triangle.

3.  Antares (Summer)

     Antares is a bright orange star of magnitude 0.96 (sometimes
called "The rival of Mars" because of its color and brightness) in
the constellation Scorpius the scorpion.  If your location is
such that you have a good enough view to the terrestrial south, it
is easy to visualize the head of the scorpion NW of Antares with
the body stretching out to the SE and curling up as the stinger.
Antares, Arcturus, and Vega form a large, nearly isosceles triangle
with the angle at Arcturus being close to a right angle, and the
distance from Arcturus to Antares being about 56 degrees.  With
declination about -26 degrees this star is pretty far south for
northern-hemisphere viewers, but we mention it here because
Scorpius and the constellation Sagittarius the archer (with its
characteristic teapot-shaped group of stars about 27 degrees (or
about 2h) east of Antares) contain many beautiful objects.

4.  Cassiopeia (Fall, Winter)

     Cassiopeia is a constellation;  she was the wife of Cepheus
(another constellation), who was a king of Ethiopia.  The main part
of this constellation is a group of 5 brught stars in the shape of
a lazy W which opens roughly north;  the W is about 13 degrees
across and 8 degrees deep.  The westernmost star in the W has right
ascension about 0h and declination about 59 degrees.  Since the
right ascension of this star is within about 1h of the right
ascension of the pointers in the Big Dipper, plus 12h (recalling
that right ascension 0h is the same as right ascension 24h), if you
follow the line of the pointers about 30 degrees on beyond Polaris
you will come close to this star.  [Why is this star (which is beta
Cas) only about 30 degrees from Polaris instead of 90 degrees - 59
degrees = 31 degrees?  Recall that Polaris is about 3/4 degree away
from the north ceiestial pole;  its coordinates are (2h 31.1m, 89
degrees 15').  Using the celestial distance formula, which is given
in square-bracketed (i.e. skippable) material in section VII(I),
along with the coordinates (0h 9.2m, 59 degrees 5') for beta Cas
which can be found in appendix 2 in reference II-3 among other
places, gives distance about 30.24 degrees, which we round to 30
degrees.]

5.  The Orion group (Winter, Spring)

     The constellation Orion the hunter contains a group of seven
bright stars that is probably the second most recognized star group
(after the Big Dipper).  The outer four stars are nearly a
north-south oriented rectangle (except the east side is longer than
the west side) about 17 degrees high and 8 degrees wide.  The
remaining three stars are in the middle, and are called Orion's
belt; they form a near-line about 3 degrees long running roughly
SE-NW.  The two brighest stars among the seven are Rigel (magnitude
0.12), which is a blue-white star in the SW corner, and Betelgeuse
(magnitude 0.50), which is a reddish (or orangish) star in the NE
corner.
     If you follow Orion's belt SE about 22 degrees, you will come
to Sirius (magnitude -1.46) in the constellation Canis Major the
big dog, which appears brightest of all the stars (not counting the
sun).  If at Sirius you make a right-angle turn to the NE and go
about 26 degrees you will come to Procyon (magnitude 0.38) in the
constellation Canis Minor the little dog.  Now turn north and go
about 23 degrees to Pollux (magnitude 1.14) and Castor (magnitude
1.57) in the constellation Gemini the twins. These bright stars are
about 4 1/2 degrees apart, with Castor to the NNW of Pollux.  Now
follow the line through Pollux and Castor, bending a little right,
about 30 degrees to the bright star Capella (magnitude 0.08) in the
constellation Auriga the charioteer.  Then turn right about a right
angle and go about 31 degrees to the bright star Aldebaran
(magnitude 0.85) in Taurus the bull.  Aldebaran is in a V-shaped
star group called the Hyades, which is pretty in binoculars.
Finally, another approximately right-angle turn to the right and a
move of about 22 degrees will bring you back to Orion's belt.


B.  Why star charts seem to have east and west reversed from
    terrestrial maps

     Here we mention something that may be confusing about star
charts at first.  Here and in the rest of this document we are not
talking about the circular, full-sky sky maps found in references
II-11 and II-12 (for example), but rather the rectangular star
charts that each cover only a small part of the sky.  Imagine you
are outside, facing terrestrial south, and holding a map of the
United States, with your right hand near New York and your left
hand near California.  Now imagine lifting the map above your head,
as if you were going to compare the map to the sky.  Then the top
of the map, where the northern states are, will be closest to the
terrestrial northern horizon, the bottom of the map, where the
southern states are, will be closest to the terrestrial southern
horizon (so far so good), but the right side of the map, where the
eastern states are, will be closest to the terrestrial western
horizon, and the left side of the map, where the western states are,
will be closest to the terrestrial eastern horizon.  In other words,
east and west on the map will be in the wrong places!  To remedy
this, star charts are reversed east and west when compared with
ordinary terrestrial maps;  that is, west on a star chart is to the
right, and east on a star chart is on the left (more precisely,
going west on the star chart means following a line or circle of
equal declination to the right, and going east on the star chart
means following a line or circle of equal declination to the left;
there will be some such lines or circles on the chart to help you,
and you can mentally put in lines or circles between the ones shown
on the chart).  North and south are still normal, that is, north is
up and south is down (more precisely, going north on the star chart
means following a line of equal right ascension in a generally
upward direction, while going south on the star chart means
following a line of equal right ascension in a generally downward
direction;  there will be some such lines on the chart to help you,
and you can mentally put in lines between the ones shown on the
chart).


C.  Scouting

     Now we have a (tentative) starting point and a desired final
object, and we have to figure out how to get from one to the other.
Technniques for finding a path are laid out in the next section,
but if you have a hard time applying these in any particular
example, or if you just want to make the path-finding easier for
youself, you can do some scouting first.  To do this, go outside
with your binoculars at the time of day you will normally be
observing and sweep the sky in the general vicinity of your final
object.  Look for stars or groups of stars that are easily visible
in your binoculars, then look at your star chart(s) and try to
match up what you saw in the sky with what you see in the charts.
If necessary, look at the sky again, then the charts, then the sky,
then the charts, etc., until you have made some matches.  Try to
match something that is as close as possible to your object, and
broadly scan the sky between your starting point and the things
you matched, seeing if you can find a pathway of naked-eye visible
or binocular visible stars leading from one to the other.  Scouting
in this way will make things easier when you go back to the charts
and actually write down a path.


D.  Selecting intermediate objects in your path

     You now have a (tentative) starting point and a final object,
and you want to select a sequence of objects from one to the other.
Doing this is not an exact science; each of these object will
usually be a star, but sometimes it is convenient to use a group of
stars instead, as is done in the example of M104 in chapter XII.
Usually there are many possible paths, and it is not always clear
which ones will be easiest to follow when you are outside observing.  
Eventually you just have to try something, and if it doesn't work
well, you can go back to the charts and try again.  Here are some
guidelines to help you select intermediate objects:

1.  It is good to select stars that are brighter than other nearby
stars.  It can be a little risky to select as intermediate objects
stars that are among the dimmest shown on your charts (indicated
by the dots representing them being the smallest) because there
may be lots of stars just a little dimmer than these that did not
make it onto the charts, but you may see them with your telescope,
and it is easy to confuse them with the stars that are actually in
your chosen path.  These dim stars, however, are ok for check
stars (see the next item).

2.  It is good to select intermediate objects that have other stars
nearby that can serve as check stars;  when you think you are
looking at a star in your path, you can check yourself by looking
to see whether the check stars are where they should be.

3.  It is good to select intermediate objects that are not too far
from the previous and/or following objects in your path, especially
when the object you going to is not visible in your binoculars and
finderscope, since then you have to rely on the field of view of
your telescope, which will be much narrower than the fields of view
of your binoculars and finderscope.

4.  It is convenient to select intermediate objects that are either
nearly due north of, or nearly due south of, or nearly due west of,
or nearly due east of, the previous and/or following objects in
your path.  In moving between two such objects, one of the
slow-motion knobs will require very little turning (or no turning),
and this tends to reduce errors and improve accuracy.

     In most cases some of the goals above will clash with each
other, and you will have to make a choice.  A general procedure
that works well is to start with your final object and look for
the star on your chart that is closest to it (although if this
closest star is one of the dimmest ones on the chart, you may (or
may not) want to choose a different one, and just use the dim one
as a check star).  Then look for a star close to the one you just
selected as another intermediate object, and continue working your
way away from your final object until you reach a star that you
think will be easily visible in your binoculars and finderscope
(magnitude 5.5 or better should be pretty safe even in the presence
of some light pollution, but you will learn from experience what
magnitude works for your own viewing conditions).  From there you
can connect the object to your starting object, working either
forward or backwards, but now you can safely move longer distances
at each step because you will have your binoculars and finderscope
to help you.
     I know this may sound complex if you have not done it before,
but there are numerous examples in this document (especially in
chapter XII) to help you.  You may develop strategies of your own
that work better for you than the ones suggested in this document.
In the end, the answer to the question of how you get to a point
where you are really good at this is the same as the answer to the
question of how you get to Carnegie Hall:  "Practice, practice,
practice!"


E.  Measuring distances between objects in your path
     
1.  for free movement

     FREE MOVEMENT from one object to another means moving straight
from one to the other by the shortest possible path (in actuality,
you will be following the great circle on the celestial sphere
that contains both objects).  To determine the distance, on your
chart just measure the number of millimeters between the objects in
a straight line, then divide by the scale factor. The scale factor
is 2.9 mm per degree (remember, mm means millimeters) in most of the
charts in reference II-3, and is 8.2 mm per degree in most of the
charts in reference II-4, although in the former case 3 mm per
degree is usually close enough and in the latter case 8 mm per
degree is usually close enough.  If there is any question, measuring
the number of millimeters between two circles on your chart that
are marked as being 10 degrees apart in declination and then
dividing by 10 will give you the scale factor.  You will make
things easier for yourself if you express your answer as a mixed
number rather than as a decimal (for example, 3 5/8 degrees);  using
only denominators 2, 3, 4, 6, or 8 (whichever of these is the
largest you need) for the fractional part (if any) will usually
give you enough accuracy, and will also make things easier if you
decide to do turn counting, as described in section VIIC3 below.
If your measurement falls halfway between two fractions that are
close together (for example, halfway between 3/4 degree (which
equals 6/8 degree) and 7/8 degree, you could just pick either of
the fractions, or use the smaller one followed by a + or the larger
one followed by a -, or those of you who really like math could use
a smaller division (13/16 in this case).  Note that even if your
measuring were perfect there would usually be a little inaccuracy
in your results because the charts themselves are not perfectly
accurate, since they are attempts to portray a curved surface
(namely part of the celestial sphere) on a flat piece of paper.  For
the directions, one of the following is usually accurate enough:
N, NNE, NE, ENE, E, ESE, SE, SSE, S, SSW, SW, WSW, W, WNW, NW, NNW.

2.  for taxicab movement

     TAXICAB MOVEMENT is movement that is divided into two parts,
a north or south part and a west or east part.  This tends to be
easier to carry out accurately since in each part of the move the
telescope moves only in declination or only in right ascension, so
you only have to turn one slow-motion knob;  in each part of the
move there are only two directions to go, as opposed to an infinite
number of directions in free movement.  The main method of measuring
distances in degrees for each part of the move is the same as in
free movement, but in the north or south part of the move if you
are using reference II-4 you have another choice:  Just lay the
appropriate one of the three templates over your star chart, with
one of the circles (or lines) of equal declination on the overlay
on top of the corresponding circle (or line) on the chart, and
count boxes from a horizontal circle (or line) that hits one object
to a horizontal circle (or line) that hits the other object;  each
box represents 1/2 degree north or south.  The labels on the
overlay may help you cut down on the counting;  for example, if one
object lies on the 28 degree circle and the other lies on the line
between the 35 degree circle and the 36 degree circle, then the
north-south distance is 7 1/2 degrees.  If it is not the case that
both objects lie on one of the horizontal circles (or lines) on the
overlay, you can slide the overlay slightly up or down to put one
of the objects on a line, then look at where the other object
falls;  when doing this, make sure that one of the vertical lines
on the overlay matches one of the vertical lines on the chart to
avoid distortion caused by incorrect tilting of the overlay.  Note
that this method cannot be used directly to measure degrees east or
west since the boxes have differing widths horizontally, so you
will need to use your ruler to get the degrees east or west.
[Alternately, you could use your overlay to get the number of
minutes of right ascension between your objects, and to estimate
their declinations, then use Fact 2 in section VIE3 to get the
degrees.  Actually, if you know the coordinates of both objects,
for example if they are stars which are bright enough (namely they
have magnitude greater than 3.55) to have their coordinates listed
in appendix 2 of reference II-3, then the most accurate way to get
all the information for the move is to get the degrees north-south
and the minutes east-west by subtraction, then use Fact 2 along
with the declination of the object for the east-west move to get
the degrees east-west;  you don't need to do any measuring at all
for this!  You can also apply the celestial distance formula which
is given in the skippable part of section VII(I) to the coordinates
to get the distance between the objects if you want that.]
     This brings up another question, namely, if you are going to
do taxicab movement, does it matter whether you do the north-south
part of the path first or the east-west part of the path first?
Well, actually it does, especially if you are not going to do
turn counting, since the number of east-west degrees you measure
will usually be different depending on whether you measure them at
the north end or at the south end of your two-way path;  this is
because the lines of equal right ascension get closer together as
you move away from the celestial equator.  If you are going to do
turn counting as described in the next section, however, the number
of minutes you measure (or compute) will be the same at each end
of the north-south part of the path.  Doing the north-south
part first does have the following advantage:  If you move
north-south first and get that right while you are observing, then
even if you mess up the east-west part, the north-part will remain
correct (as long as you don't change it) and you can try scanning
in the east-west direction.  If on the other hand you move
east-west first while you are observing, even if you get that
right, if the north-south part that follows is not right, then you
will soon be off in both directions due to the westward drift of
the stars, and then you will be in trouble.  On the other hand,
there may be some advantage in general in doing the shorter part
first, since that is the part where more accuracy in the distance
moved is required, as described later.
     To make our lives simpler, let's follow this rule:  Do the
north-south part first unless you are not doing turn counting,
and the east-west part is a lot shorter than the north-south
part;  if those two conditions hold, then use your own judgement,
but if you choose to do the east-west part first then do your
record-keeping in such a way that you will remember that you have
done this (see section VIF below for more on record-keeping).
     By the way, to possibly avoid confusion (and hopefully not
cause it), we point out that the length of a move in degrees is
always the distance defined in section IIIA, that is, it is
measured along a great circle.  In particular, even though in what
we call an east or west move the starting and ending points for the
move have the same declination, we are not measuring the distance
along a path of equal declination in the sky, but rather along this
great circle.

3.  for turn counting

     Turn counting is a form of taxicab movement where you
determine the distance you move by counting the number of turns you
make with your slow-motion controls.  The measurements you do in the
north or south direction are done the same way as they are done in
the previous section.  In the west or east direction, however, you
need to determine the number of minutes of right ascension between
the two objects instead of the number of degrees.  If you are doing
this with reference II-4, you line up your overlay exactly as in
the previous section, then you can use the scales on your overlay
and/or count boxes horizontally;  each box will represent 5 minutes.
If you slide the overlay a little to put one of your objects under
one of the vertical lines on the overlay, make sure that one of the
lines or circles of equal declination matches the line or circle
with the same declination on the chart, to avoid distortion due to
incorrect tilting of the overlay or having the overlay too far
north or too far south.  It will be helpful to express the number
of minutes as a mixed number, with the fractional part (if any)
having denominator 2 (this should give adequate accuracy;  however,
you can use a + of a - or use a smaller division if it makes you
feel better).  If you are using reference II-3 or some other
reference that does not include an overlay, the same measuring
techniques we have described for north and south moves still work
for the degrees east and west, but to get the minutes for east and
west moves you will need to do a more complex kind of measuring as
described near the end of section VIG below, or else measure the
degrees and use the formula in Fact 2 below to get the minutes from
the measured degrees and from the declination of the positions
between which you are moving (these positions will have the same
declination since we will be moving east or west).  If you are not
going to do turn counting in the east or west direction, you will
not need to know the minutes.
     We will now present the formulas relating degrees, minutes,
and declinations, and then discuss their use;  we will not give the
proof since it uses spherical trigonometry, which is an area of
mathematics that has just about died out due to the advent of the
computer (but someone still needs to know it since someone needs to
tell the computer what to do, right?).  In case there may be one or
two readers of this document who know something about spherical
trigonometry, I will mention that the proof is based on one of
Napier's rules for right spherical triangles.  If you intend to use
these formulas and you have a programmable calculator, you may find
that programming them into your calculator will save you time 
later.  If anyone is actually interested in learning some spherical
trigonometry, I would recommend the book "Concise Spherical
Trigonometry with Applications" by Jacques Redway Hammond, Houghton
Mifflin, 1943.  You might find this book to be difficult reading
unless you have a solid math background.  You could probably have a
bookstore order it for you, but it has been out of print for a long
time, and it might cost you 150 dollars or more.  In 1943 it cost
only two dollars and 22 cents, so you could save some money if you
had a time machine.  If the rest of this section appears to you
that it involves doo-doo that is too deep, you can skip down to
section VIF.
     An appendix (Chapter XV) on sphereical trigonometry is
included at the end of this document for anyone who might be
interested in learning a little about the subject.  The appendix
contains some of the basic definitions and theorems of the subject,
along with their applications to the astronomical ideas in this
document.  The appendix assumes little beyond some knowledge of
basic plane trigonometry, but it is not for the faint of heart!

     The first formula gives minutes in terms of degrees and the
second formula gives degrees in terms of minutes;  in both cases
you need to know the declination also.

Fact 2:  Consider two objects or positions, neither of which is at
a celestial pole, which have the same declination dec.  Let min be
the number of minutes between the objects (measured east or west,
whichever is shorter) and let deg be the distance between the
objects as defined in section IIIA.  Then

min = 8invsin[sin(deg/2)/cos(dec)], and

deg = 2invsin[sin(min/8)cos(dec)].

Sin and cos are abbreviations for the trigonometric functions sine
and cosine respectively;  you can compute them using a scientific
calculator or a table or an appropriate computer application (most
computers include a calculator).  Invsin is an abbreviation for
inverse sine; the invsin of a number is defined to be the quantity
whose sine is the given number (it also must lie between -90 degrees
and 90 degrees, but that will happen automatically with our
formulas, so we don't need to worry about that part).  On
calculator keyboards, inverse sine is usually abbreviated asin or
arcsin, or sin with an exponent -1.  You need to check that your
calculator or program is working in degree mode;  do this by
verifying that your calculator or program gives cos(60 degrees) =
0.5 and invsin(0.5) = 30 degrees.  If you cannot get your calculator
or program into degree mode, you can still use the formulas in Fact
2, but you will need to multiply the things to which sin and cos
are applied by pi/180 before computing the sin or cos, and multiply
the final answer by 180/pi.  You can take pi to be approximately
3.1416 (if you want more accuracy you could take pi to be
approximately 3.14159265358979323846264338327950288419716939937510
582097494459230781640628620899862803482534211706798214808651
--just kidding;  don't do that, since that would really be
overkill).
     For example, if we want to make a movement of 5 degrees to the
east at declination 60 degrees, the number of minutes that will
take is
min = 8invsin[sin(5/2)/cos(60)] = 8invsin[sin(2.5)/cos(60)] =
8invsin[.0436193873653/.5] = 8invsin[.0872387747306] =
8(5.00477557203) = 40.038204045762 minutes, or about 40 minutes.
Note that you would not normally write down the long decimals, but
would just write down the "about 40 minutes" at the end.  Most of
the digits in the long decimals are garbage anyway since our
measurement accuracy will not be good enough to justify such
precision;  I wrote them down so you can follow along as the
numbers flash across your screen.  We should also note that in this
sort of computation most of the equals signs really mean
"approximately equal to" since the numbers are rounded, not exact.
     You can also try out the computation with some of the examples
in chapter XII if you wish.  Some of them will not come out exactly
since they were measured with the overlay in reference II-3, and as
already mentioned, star charts necessarily have some inaccuracy
in them.  As a more general example,  if two objects are on the
celestial equator, then they have declination 0, and the first
formula gives
min = 8invsin[sin(deg/2)/cos(0)] = 8invsin[sin(deg/2)/1]
= 8invsin[sin(deg/2)] = 8(deg/2) (since invsin reverses sin) =
4deg.  
This makes sense, for the following reasons.  The number of degrees
between two objects or positions (which are less than 180 degrees
apart) is determined by the distance (in miles or whatever)
between them as measured along the one and only great circle on the
celestial sphere that contains them.  The number of minutes between
the objects (or positions) is determined by the distance (in miles
or whatever) between the points where the lines of equal right
ascension through the objects hit the celestial equator, measured
along the celestial equator.  When the objects or positions are
actually on the celestial equator, however, the degrees and minutes
are being measured along the same great circle (namely the celestial
equator itself), and there are four times as many minutes (1440) as
degrees (360) on the celestial equator, so we should have
min = 4deg.  Now the fact that we have just shown that our formula
works for objects on the celestial equator does not prove that it
works for all objects (unless you know spherical trigonometry, you
will have to trust me on that one), but it certainly gives us more
confidence in it;  what we have just done is known as "special-
case checking", which is quite useful since it can often (but not
always!) root out errors in a formula, and doing this kind of
checking is a useful habit.  Still, I cannot leave this paragraph
without adding the caution that there have been many cases where
people became convinced that something was correct because it
seemed to work for the examples they tried, but they later found
out to their dismay that the thing failed for some example they
had not tried.  To be really sure of a fact or formula, you need a
proof!
     We have now added Appendix B to this document, in which Fact 2
is actually proved.
     As a final remark in this section, we note that either of the
formulas in Fact 2 can be derived from the other.  For example,
starting with the first formula, we divide both sides by 8, take
the sine of both sides, multiply both sides by cos(dec), take the
invsin of both sides, and multiply both sides by 2 to get the second
formula.


F.  Recordkeeping

     Now that we know how to create a path and measure distances,
we need a good way to write down this information.  The following
scheme has worked well for me.  Get a pack of 4x6 or larger index
cards, one card for each object.  Record all the information below
that is known to you:

First line:  Name of object, what it is (galaxy, double star,
etc.), magnitude, size (if relevant), separation (if relevant).

Second line:  Right ascension, declination, constellation.

Next lines:  Your starhopping path, as described below.

Next lines:  Information about the object from books such as
references II-3 or II-13, or from other places.

Next lines:  Your observations.  For each time you see the object
you may want to record the date, time, viewing conditions (good,
fair, half-moony, etc.), what eyepiece you used for your best view,
a description of what the object looked like, and possibly a drawing
(consisting of a circle for your field of view, the object, and
maybe other objects in the same field;  include an arrow near the
picture marked W to indicate which way things are drifting).

     You may eventually have several stacks of cards, one stack for
objects already viewed for example;  within each stack, it may be
a good idea to order your cards by increasing right ascension,
since this is roughly the order in which the objects will appear
as the days move on.
     Now we come to the question of how to write down your path.
One way is this:

1.  Start by writing down your starting point (for example, "SW
star in bowl of Big Dipper").

2.  Draw an arrow pointing to the right.

3.  If you are using free movement to get to the next object,
above the arrow write the distance and direction, for example,
"3 1/4 degrees SE".  To save space, use a small raised circle
instead of the word "degrees".
     If you are using taxicab movement, you can write the north or
south part of the move above the arrow and the west or east part
of the move below the arrow.  If you are going to do turn counting,
also include the minutes with the west or east move, for example
"2 degrees (8m) W".  Following the rule we laid down at the end of
section VIE2 above, we make one exception:  If we have measured
the east-west degrees before the north-south degrees, then put
the east-west part of the move above the arrow and the north-
south part of the move below the arrow.
     Even if you are going to do turn counting, it may be helpful
to include the degrees along with the minutes because degrees are
better for eyeballing;  2 degrees looks the same no matter where 
you are in the sky, but this is not true of 8m.  Sometimes you
might include directional information such as "make a left turn of
about 45 degrees", which would mean turn left about half as much as
if you were making a right-angle left turn.  As usual, "left" or
"right" in the sky is meant from the point of view of a little man
walking with his feet on the path and his head down toward you.

4.  Write down the object you are going to right of the arrow.  If
is a star (which it will normally be except for the final object),
it may have a number or letter or Greek letter on your chart;  if
so, you can use that (probably along with the name or an
abbreviation of the constellation).  If there is no number or
letter on the chart, you could say something like "magnitude 6.5
star".  You can estimate the magnitudes using a scale given with
the chart (or on your overlay, if you are using one);  getting them
exactly right is not critical.  You can also include other
information in parentheses after the name, for example, magnitude
(if not there already), check stars (if any), and possibly the
English name of the star (for example, "Yed Prior" or "Yed
Posterior".  I am not making this up;  there really are two stars
with those names.  They are in the constellation Ophiuchus, the
serpent-bearer.  Why do they have such similar names?  Maybe the
person who named them thought that two Yeds were better than one.)
By the way, the Greek alphabet is given in the Introduction in
reference II-4, and you could probably also find it in a good
encyclopedia or dictionary.  Abbreviations and names for all the
constellations are given in chapter XIII of this document.

5.  Now repeat steps 2, 3, and 4 until you reach your final object.
If you want to, you can write down several paths and see which
works better while you are observing.


G.  An example:  The Ring Nebula

     The Ring Nebula is also known as M57;  it is an attractive
object that looks like a smoke ring, and is one of the few objects
which is given three exclamation points in the NGC CATALOG.  (The
"NGC catalog" refers to the "New General Catalog", a vast
collection of celestial objects published by J. L. E. Dreyer in
1888.  Many objects are identified today by their NGC numbers.  109
of the more interesting objects are also identified by Messier (M)
numbers, from a list published by Charles Messier in 1784, which he
compiled so he would not be confused by them when he was searching
for comets;  a few objects have been added since Messier's time.)
The information that goes in the first two lines of the record card,
as described in the previous section, looks like this:

M57 (Ring Nebula)          mag. 8.8                size 76"
18h 53.6m                  33 degrees 2'           Lyra

I got this information from reference II-13.  Among the statements
about the Ring Nebula in reference II-13 are the following:  "Also
the Smoke Ring Nebula or Donut Nebula.  Bright, pretty large;
irregular ring structure, a very impressive and stunning object.
Small telescopes usually show it as a featureless patch or disk,
but a 4-inch brings out the annular shape."
     Now we need to create a path.  Using the coordinates above, or
looking up "Ring Nebula" in the index of reference II-3 and then
looking at atlas chart 18 in that reference, we see that Vega is
quite close to to the Ring Nebula, so this should make a good
starting point.  We will now go through the creation of the rest of
the path twice, first using reference II-4 and then using
reference II-3.  If you have both of these references, then
reference II-4 is better for path-creating because it has a larger
scale, shows more stars, and has an overlay to help you (although
reference II-3 has lots of general information).
     Using reference II-4, we first scan the chart key, and see
that chart 8 contains Vega and vicinity.  Turning to chart 8, and
making use of the symbol key at the top if necessary, we locate the
green symbol for M57 a little SSE from Vega.  Looking for the
closest star to M57, we find a small dot a little N (and slightly
W) of M57.  Comparing the size of the dot with the sizes of the
dots on the overlay, we see that this star has magnitude 8.0.
There is a bright star a little W (and slightly S) of the mag. 8.0
star marked with a funny-looking B, which is a Greek letter beta.
This star is also called Sheliak.  We might also observe that
Sheliak is the SW corner of a parallelogram of bright stars;
noticing this will make our work a little easier outside.  Sheliak
(or beta Lyr) is marked by a dot inside a circle, which means it is
a variable star;  we will say more about that in Chapter XII.  We
now have a star which should be easily visible in our binoculars
and finderscope;  we see that we can complete the path by going
from Vega a little SE to a star marked zeta2,1 (this is actually
two stars close together), and from there to Sheliak.
     Now we need to determine distances and directions from Vega to
zeta2,1, then to Sheliak, then to the mag. 8.0 star, then to M57.
If we want to use free movement from Vega to zeta2,1, which would
work fine here since these stars are close and bright, we can use
our ruler to measure about 16 mm (mm means millimeters) from the
center of Vega to the center of zeta2,1, and since the scale factor
on this chart is about 8 mm per degree, this gives a move of 2
degrees SE (south being parallel to the vertical lines and down,
east being parallel to the horizontal circles and left).
     We will also go from Vega to zeta2,1 using taxicab movement
and turn counting to illustrate these measurements.  To do this, we
note that the middle template on our overlay seems to match the
chart, and we can align it properly by laying the 40 degree circle
on the overlay on top of the 40 degree circle on the chart (the 40
degree circles are the easiest ones to use in this example since
they are close to the stars we are looking at).  Now we slide the
overlay up or down slightly, making sure one of the vertical lines
on the overlay falls on top of one of the vertical lines on the
chart, until one of the horizontal circles goes through the center
of Vega.  We now count boxes vertically from Vega to zeta2,1,
finding 2 boxes and about a third of another box.  Since each box
represents 1/2 degree north-south, we need to go 1 1/6 degrees south
from Vega to get to zeta2,1.
     Now we re-align the 40 degree circles and slide the chart a
little left or right (keeping the 40 degree circles aligned) until
a vertical line passes through the center of Vega.  Counting boxes
horizontally, we see that it takes about 1 3/5 boxes to go from
Vega to zeta2,1.  Since each box represents 5 minutes east-west,
this gives a move of 8m E.  To get the degrees east, we can now
measure westward from zeta2,1 to the vertical line that goes
through Vega, getting 12 mm, thus 12/8 or 1 1/2 degrees.  We can
also note the mag. 8.5 star 3/8 degree SSE of zeta2,1 as a check
star if we wish;  although it takes a little time to do this,
check stars can be quite useful sometimes.  Doing similar
measurements from zeta2,1 to Sheliak, from Sheliak to the mag. 8.0
star near M57, and from the mag. 8.0 star to M57, we get a path
that looks like the following.  Due to the limitations of this
plain-text document, the path will be described less concisely here
than it would be on your card.  Although for this example I recorded
both turn counting distances and free movement distances, in most
later examples we will record only turn counting distances unless
the two objects are quite close to each other, in which case we may
record the free movement distance only.

Vega, then

   1 1/6 degrees S, 1 1/2 degrees (8m) E (or 2 degrees SE) to

zeta2,1 Lyr (4.0;  NW star of parallelogram, with a mag. 8.5 star
3/8 degree SSE), then

   4 1/4 degrees S, 1 1/8 degrees (5 1/2m) E (or 4 1/2 degrees SSE)
   to

beta Lyr (variable 3.3-4.3;  SW star of parallelogram;  Sheliak,
with a mag. 8.0 star 1/4 degree W), then

   slightly N, 5/8 degree (3m) E to

a mag. 8.0 star, then,

   1/2- degree S, 1/8 degree (1/2m) E to

M57.

By the way, the magnitude 3.3-4.3 came from reference II-3;  from
the chart, the best we can do (by comparing the outer and inner
circles for Sheliak to the circles on the overlay) is to say
3.5-4.0, but that is close enough, since these magnitude estimates
do not need to be accurate.
     Now let's try to create a path from Vega to M57 using
reference II-3 instead of reference II-4.  We will get a similar
(but slightly less accurate) path, and we will have to work harder
to get the minutes for the east-west parts.  Looking at the chart
key on pp. 224-227 of reference II-3, we see that chart 18 contains
Vega and its vicinity.  Turning to chart 18, we again locate M57 SSE
from Vega.  We notice beta Lyr a little ENE of M57; this bright star
should be easily visible in our binoculars and finderscope, and we
also see that we can get there by going from Vega to zeta2, zeta1
Lyr (same as zeta2,1 Lyr, named slightly differently), then to beta
Lyr.  Thus our path will use the same stars as before, with the
exception of the mag. 8.0 star near M57 and the check stars, which
were too faint to make it onto this chart.  We could select other
check stars, but the stars in the path in this example are pretty
bright and pretty close, so we will probably be able to get along
without check stars.
     Now let's determine our distances and directions from Vega to
zeta2, zeta1 Lyr.  If we think we might want to do free movement,
which would work well here since these stars are bright and close
together, we measure the distance between the centers of the dots,
getting 5 mm;  since the scale factor on this chart is about 3 mm
per degree, this gives us 1 2/3 degree SE.
     Next we do the measurements for taxicab movement with turn
counting to illustrate these ideas.  To get the number of degrees
going south from Vega, we imagine a circle parallel to the 40 degree
circle that passes through zeta2, zeta1 Lyr, and measure from the
center of Vega down perpendicular to this imaginary circle.  We get
about 4 mm, which gives about 4/3 or 1 1/3 degrees.  We now measure
from where the line southward met the imaginary circle eastward to
zeta2, zeta1 Lyr, getting about 4 mm, which is about 1 1/3 degrees.
     Now, how do we get the minutes?  The more accurate way would
be to use Fact 2 in section E of this chapter.  From the chart we
measure 7 mm down from the 40 degree circle to zeta2, zeta1 Lyr,
which corresponds to about 2 1/3 degrees, so the declination of
zeta2, zeta1 Lyr is about 37 2/3 degrees.  we get
min = 8invsin[sin(deg/2)/cos(dec)] =
8invsin[sin(1.33333333333/2)/cos(37.66666666667)] =
8invsin[sin(.666666666665)/cos(37.66666666667)] =
8invsin[.0116352658014/.791579171057] =
8invsin[.0146988023774] =
8(.842209669218) =
6.73767735374,
or about 6 1/2 minutes when rounded to the nearest half minute.  As
before, we comment here that in practice you would not normally
write down all these long decimals (most of whose digits are garbage
anyway, since our measurements are not accurate enough to justify
so many decimal places), but we are doing it here so you can follow
along with your calculator if you wish.  (If you are using a
calculator different from my HP 48SX, your numbers might look
slightly different).  By the way, for the declination, if you just
eyeball it as 38 degrees (or 37), that would be accurate enough.
     But what if you don't wan't to go through this computation?  A
somewhat less accurate (but probably good enough) result for the
minutes can be obtained by measuring, as follows:  Place your ruler
so it hits zeta2, zeta1 Lyr and is roughly parallel to the 40
degree circle halfway between the nearest vertical hour lines (in
this example they are marked 18h and 19h), and measure the distance
between the two points where your ruler hits these hour lines.
Divide this into the number of east-west millimeters you measured
between Vega and zeta2, zeta1 Lyr, which roughly gives you the
fraction of 1 hour that the east-west distance between Vega and
zeta2, zeta1 Lyr represents, then multiply by 60 to get the minutes.
In this case we measure about 34 mm;  dividing into 4 mm gives 2/17
mm, and multiplying by 60 gives 120/17 minutes, that is, about 7
minutes (and we even gave our calculator a rest on this one,
although we could have used it).
     By the way, in case anyone is having a panic attack out there,
I should point out that doing this stuff is a lot faster than
explaining it.  If we do the same things for the rest of the path,
using Fact 2 to get the minutes, this is the final path we get:

Vega, then

   1 1/3 degrees S, 1 1/3 degrees (7m) E (or 1 2/3 degrees SE) to

zeta2, zeta1 Lyr (4; NW star of parallelogram), then

   4 degrees S, 1 degree (5m) E (or 4 1/2 degrees SSE) to

Beta Lyr (variable 3.3-4.3;  SW star of parallelogram), then

   1/3 degree S, 2/3 degree (3m) E (or 2/3 degree ESE) to

M57.

     It is worth noting again that the NW and SE stars of the
parallelogram mentioned above are about 6 degrees apart, so they
make a good test of the field of view of your binoculars and
finderscope.
     Incidentally, one can get a pretty good estimate of the
coordinates of celestial objects from these charts.  Let's try it
for M57.  If we are using reference II-4, we lay the overlay on
chart 8 with the 40 degree circles matching again (the other
circles should match too), and we also match the left and right
margins of the template with the vertical lines marked 19h and 18h
on the chart.  The declination can be read from the left side of
the template as about 33 degrees.  M57 appears to be about 6 1/2
minutes from the left side of the template (as can be seen from the
scale at the bottom of the template, or by box-counting), so it is
about 53 1/2 minutes from the right side of the template, giving a
right ascension of about 18h 53 1/2m, so we have good agreement
with the values given at the beginning of this section.  If you are
using chart 18 in reference II-3, measuring to M57 on a line
perpendicular to the 30 degree circle and 40 degree circle gives
9 mm (or about 3 degrees) from the 30 degree circle, so again we
get declination about 33 degrees.  For the right ascension, we have
two ways we can proceed.  We can measure the distance from M57 to
the point with the same declination on the closest vertical line
(which is the 19h line), getting about 4 mm, or about 1 1/3 degree.
Using the formula in Fact 2 gives about 6 1/2 minutes, so
subtracting from 19h we again get 18h 53 1/2m.  If we want to avoid
Fact 2, we measure the distance between the 19h and 18h lines along
a line that passes through M57 and is parallel to the 30 degree
circle halfway between the 19h line and the 18h line.  This
measurement is about 36 mm;  dividing this into the 4 mm we just
measured and multiplying by 60 gives (4/36)x60 = 60/9 = 20/3 =
6 2/3 minutes, resulting in a right ascension of 18h 53 1/3m,
which is not quite as good as the previous value we got, but is
still ok.
     Being able to determine the coordinates of objects is useful
for several reasons.  For one thing, if you want to determine the
north-south distance and the east-west distance between two objects
that are not on the same star chart, then you cannot measure these
directions directly with your ruler, but if you know the
coordinates of the objects, then you can determine the distances by
subtraction.  As an example to show how this works, we consider
moving from Alphekka to zeta Her, which occur in the description of
M13 in chapter XII.  We could measure these coordinates as in the
previous paragraph, but these stars are bright enough to be listed
in Appendix 2 of reference II-3 (pp. 514-526), which lists the
brightest stars.  Looking them up (using approximate right
ascensions from the star charts to more quickly find them in the
appendix), we get the coordinates of Alphekka as (15h 34.7m, 26
degrees 43') and the coordinates of zeta Her as (16h 41.2m, 31
degrees 36').  Subtracting the right ascension of the starting
object from the right ascension of the target object, we get
1h 6.5m, or 66.5m.  (Note:  The fact that this came out positive
says that we are moving east;  had it come out negative, we would
have been moving west, and we would have needed to take the absolute
value to get the distance.  Note also that had the distance or its
absolute value come out greater than 12h, we would have had to
subtract it from 24h to get the shorter of the two east-west
routes.)  Subtracting the declination of the starting object from
the declination of the target object, we get 4 degrees 53'.  (Note:
The fact that this came out positive says that we are moving north;
had it come out negative, we would have been moving south, and we
would have needed to take the absolute value to get the distance.)
     But you may say that something is missing here;  we don't have
the number of degrees of eastward movement.  To get this, we use
Fact 2 in section VIF.  Applying the second formula with min = 66.5
and dec = 31 36/60 degrees = 31.6 degrees, we get deg =
14.1462419847 degrees, or about 14 1/8 degrees.  Note that 4 degrees
53 minutes = 4 53/60 degrees, which is about 4.88 degrees, or about
4 7/8 degrees.  Thus to summarize the example, to go from Alphekka
to zeta Her you go (about) 4 7/8 degrees N, 14 1/8 degrees
(66 1/2m) E.



Chapter VII.  Carrying out your plan


A.  Setting up your telescope with a rough polar alignment

     We are now ready to go outside and find something!  First we
have to take the telescope outside.  For the sake of your lower
back, it is recommended that you take it out in pieces, as follows.
     First, take the tripod (probably with the mount attached, but
you may be able to take them out separately if the combination is
too heavy).  You will probably have an idea where the objects you
want to observe will be in the sky, so you can set up the tripod in
a spot where trees and buildings do not block the view.  You may
find that the tripod is an awkward thing to carry with the legs
unfolded.  I do not recommend unfolding and refolding the legs
every time you use the tripod because this may require attaching
and detaching the accessory tray, and is a pain, but here is one
way of carrying the tripod with the legs unfolded (with a German
equatorial mount) which works well for me.  Grasp the base of the
counterweight arm with one hand, put the other hand under the center
of the tripod, put whichever leg of the tripod is convenient between
your legs, lift the tripod a little off the ground, and walk like a
duck (but not too fast!).  (The comments in this paragraph were made
mainly for my 9.25 inch telescope, but for my 11 inch telescope I
would recommend detaching the mount and accessory tray from the
tripod and partially folding the legs of the tripod for
transportation since the tripod is heavier and the accesory tray is
easier to remove.  For that telescope the Telrad star pointer and
right-angle finderscope are also easy to remove before moving the
(heavier) optical tube.)
     Having gotten your tripod set down where you want it, aim the
north part of the tripod (it may be marked N, or you can aim with
the fixed arm on the mount) towards terrestrial north.  In doing so,
you are adjusting the AZIMUTH of the mount, that is, the
terrestrial direction along the horizon the mount is aimed, which
we want to be due terrestrial north.  You will have better accuracy
in this alignment if you back up a few steps to the terrestrial
south and look at the tripod, then re-adjust if necessary.  Then
adjust the lengths of the legs of your tripod, if necessary, so the
top of the tripod looks level.  If there is a bubble level in the
mount, you can use that to help make the top of the tripod level.
The leveling could throw off the azimuth of the mount slightly, so
you may need to re-adjust the azimuth of the mount a little.
     
CAUTION 4.  You should not attempt to adjust the length of the legs
of your tripod while a heavy telescope is on it unless you have
someone with you to make sure the telescope does not fall.  The
reason is that when you lift a leg to adjust it, at first it will
seem heavy, but as the center of gravity shifts, it will quickly
seem much lighter, with the result that you may not adjust the
force of your lift fast enough and may cause the telescope to fall
away from you.  Pulling back on the leg you are adjusting might not
save the day because then the other two legs might slip, resulting
possibly in a broken telescope and possibly even injury to you.  If
your telescope is fairly light, say a 5 inch or smaller telescope,
then it will probably be safe to adjust the legs with the telescope
on it if you are careful.

     Now adjust the ALTITUDE of the mount so the fixed arm on the
mount will be inclined upward at an angle equal to your latitude.
There should be a scale on the mount to help with this;  once you
have set the altitude of the mount, you should not need to change
it for later observations from the same location.  This should
point the fixed arm on the mount roughly toward the north celestial
pole; that is, you will have done a rough polar alignment.  This
polar alignment will probably not be accurate enough for
astrophotograhy, but it will usually be accurate enough for our
purposes.  If you want the polar alignment to be more accurate,
there will be instructions with your telescope on how to accomplish
this.  Note that accuracy in alignment becomes more important when
you are looking for objects near a celestial pole, as discussed in
section VII(I) below.  
     If your mount is a German equatorial mount you will have
counterweight(s) to put on;  do that next.  (If you have two
counterweights, to keep things in balance better throughout the
setup process it may be slightly better to attach the optical tube
after attaching the first counterweight, then attach the other
counterweight, but this is normally not critcal.)  Remember to put
the safety screw back on so the counterweights do not slip off.
     Now you are ready to attach the optical tube to the mount.
With some telescopes the tripod may tend to tip in the direction
the telescope is pointing while you are attaching the optical tube,
so it would be a good idea to brace one leg of the tripod with
your knee while you are attaching the tube.

CAUTION 5.  Do not let go of the optical tube until you are sure it
is securely attached to the mount!

     If the instructions that came with your telescope say you need
to balance the telescope, do so.  After you have done this once you
can save some time in the future by using a marking pen to put marks
on the counterweight arm and on the outside of the optical tube to
show where the counterweights and optical tube go.  You may need to
rebalance during a future viewing session if you find that the
telescope moves by itself when the declination and right ascension
clamps are released.  Balancing will not only prevent the
frustrating experience of losing your path due to unexpected
telescope movement, it will also make your drive motor work better
(if you have one).  Some telescopes do not need balancing, however;
refer to your instruction manual.
     By the way, with my 11 inch Schmidt-Cassegrain telescope the
complete setting up process is a little more involved than what is
described above;  see section XD for more details.


B.  Finding your starting point

     You should be able to see your starting point with your naked
eye;  now you need to get it into the field of view of your
finderscope.  After that, you can move it to the center of the
field of your finderscope, using the crosshairs in your finderscope
if you can see them;  this should put the object into the field of
view of your telescope with your longest-focal-length eyepiece (if
not, your finderscope may need to be realigned).  Then you can
lock the slow-motion clamps (if they are not already locked) and
use the slow-motion knobs to center your starting point in your
telescope field of view.  Remember that this object will slowly
drift westward out of your field, so you may need to use the RA
slow-motion knob to recenter it from time to time until you are
ready for the next step.  This is one of the beauties of a polar
alignment of an equatorial mount;  you only need to turn one of
the slow-motion knobs to recenter an object which has drifted, and
the direction you have to turn it will always be the same.
     If it is dark enough outside (i.e. not too much light
pollution) that you cannot see the crosshairs in your finderscope,
then it becomes more difficult to get your object close enough to
the center of your finderscope to be seen in the field of your
telescope.  With practice you may still be able to do this by
direct observation, but if not, you can try scanning as follows.
First get the object as close to the center of the finderscope as
you can, then twiddle the RA slow-motion knob back and forth a
little while watching for the object in the field of the telescope;
keep you hand on the RA knob so you can get back to where you
started if necessary.  If the object has not appeared in the field
of view of the telescope, turn the Dec slow-motion knob by an
amount that would change the aim by an amount less than the width
of the telescope field (for example, if your telescope field has
width 3/4 degree with the eyepiece you are using, a quarter turn of
the Dec slow-motion knob would change the aim by (1/4 turn)(2 1/2
degrees per turn) = 5/8 degree, which is less than 3/4 degree, so
this should work well).  Again twiddle the RA slow-motion knob, and
if you still can't see the object in the telescope field, turn the
Dec slow-motion knob the same amount in the other direction (to get
it back where it started) then the same amount again, and twiddle
the RA slow-motion knob once more.  If you still can't see the
object in the telescope field, then try the whole process over
again.  
     The hardest part of the process with your starting point is
getting it into the field of view of your finderscope.  To do this,
first release both the clamps, and gently move the telescope with
your hands to try to aim it at your object.  In doing this, it may
be a little better to hold onto the handle (if there is one) or the
star diagonal instead of the optical tube itself, since otherwise
the heat from your hand may disturb the thermal equilibrium
inside the telescope and temporarily degrade your image a little.
Here is where a star pointer can really help you if you have one,
since it is relatively easy to get the object in the center of the
star pointer, and then it should be in the field of the finderscope.
     If you are having trouble getting the object into the field of
your finderscope, you can try sweeping, as follows.  Engage the Dec
clamp and swing the telescope back and forth slowly near the object
while watching the optical tube.  Your goal here is to have the
aim of the telescope swing back and forth either definitely a little
south of the object or definitely a little north of the object;  if
you don't have that, release the Dec clamp, move the telescope a
little in declination, engage the Dec clamp, and swing the telescope
again.  Once you have achieved this (say, for example, the telescope
is swinging definitely a little south of the object, that is, away
from the north celestial pole), slowly swing the telescope again,
this time looking through the finderscope as you do it.  If you see
the object flash by, then you can back up a little (if necessary)
and it will be in the field of your finderscope.  If you do not see
the object flash by, turn the Dec slow-motion knob about a turn and
a half in the direction that will move your telescope aim to the
north (you may need to watch the telescope while you are doing this
to make sure you are moving it north, not south).  You will have
moved your aim a little less than 4 degrees north (since 1 1/2
turns times 2 1/2 degrees per turn equals 3 3/4 degrees);  since
your finderscope probably has field about 5 or 6 degrees, you should
not have overrun the object.  Now look through the finderscope
and slowly swing the telescope again, then repeat the changing
of the Dec and the swinging as often as necessary until you either
find the object (which should not take long) or you have obviously
gotten to the north of it.  In the unlikely event that the last
event happens, try working your way back to the south, this time
turning your Dec slow motion knob 1 turn south each time.  Now if
the object is nearly straight overhead it may be physically hard to
get your head in position to look through the finderscope;  see
section VIID below for suggestions if you are having that problem.
     We should note here that if there are other stars in your
path that can be seen with the naked eye, then it may save you
time to follow your path to one of these stars with your naked eye
and/or binoculars only, then treat that star as a new starting
point.
     If your eyepiece or star diagonal are in the way of your
finderscope, you can loosen a little hand screw on the star
diagonal, rotate the star diagonal, and retighten the screw.  You
can also do this to get a more comfortable viewing position.  You
should, however, take heed of this:

CAUTION 6.  If you adjust your star diagonal so that your eyepiece
is sticking straight out to one side (or close to that), be careful
that the star diagonal is secure enough that the weight of the
eyepiece does not cause the star diagonal to turn downward, which
could result in dumping the eyepiece on the ground (which is not
the best thing in the world for the eyepiece).  Be especially
careful if the eyepiece is a heavy one (such as the Meade
wide-angle eyepiece mentioned in section VIC above), and even more
so if the eyepiece is heavy and you are also using a Barlow.  It
may be necessary to point the star diagonal in a more upward
direction to prevent this.  By the way, thanks go to the people at
the former astronomyquest.com for this caution;  just being aware
of this possibility has helped me avoid dumping an eyepiece so far
(fingers crossed).


C.  Moving from point to point
     
1.  Free movement

     As defined earlier, free movement means moving straight from
one object to another, by the shortest possible path.  To actually
move the telescope, you can either release both clamps and move the
telescope by hand, or you can engage the clamps and turn the
slow-motion knobs.  If you are moving between two objects that can
both be seen with your finderscope and binoculars, you should first
observe the path with your binoculars, then look in the finderscope
while you are moving the telescope, remembering that because of the
inversion of the image caused by the finderscope, you have to go
in exactly the opposite of the direction that you observed with
your binoculars.  For example, if in your binoculars the path goes
mainly up but a little right, then in the finderscope the path
should go mainly down but a little left.  Usually in free movement
you keep going in your selected direction until either the new
object appears in your field of view (in which case you may want to
get it centered in your telescope field and check the locations of
your check stars, if you selected any), or else you have gone so
far that it is pretty clear that you have missed the object, in
which case you try to get back to the object you started from so you
can try again.
     You can use the field of the thing you are looking through
(finderscope or telescope) as a rough yardstick to judge how far you
have moved.  For example, if you are looking through the finderscope
and it has a field width of about 6 degrees, then when you have
moved far enough to move the beginning object from the center of the
field to the edge you have moved about 3 degrees;  every time an
object moves from one edge of the field through the center and on
to the other edge you have moved about 6 degrees.
     When you are moving between two objects that can both be seen
in your finderscope and binoculars you may be able to successfully
make a fairly long move with free movement, especially if you have
a check star or two to verify the target object.  If one or both
objects is too dim to be seen in your finderscope and binoculars,
however, you may run into trouble unless the path is quite short,
say no more than one or two diameters of your field of vision;  not
only do you have a much more restricted field of vision, but your
binoculars cannot help you.
     In general, free movement is the quickest way to get from one
object to another, but it is also the one most likely to fail.  One
thing you need to remember for any kind of movement from one object
to another in your path is that once you have successfully completed
a link and are thinking about your next move, your object will be
trying to drift out of your field of view, so you may need to
repeatedly re-center it.

2.  Taxicab movement

     As defined earlier, taxicab movement means movement that is
divided into two parts, a north-south part and an east-west part.
To carry this out, engage both clamps and do the first part of the
move by turning the appropriate slow-motion knob.  In this part of
your move you will need to be careful to move as close to the
correct distance as you can, either by using your field as a
yardstick (as described above) or by turn counting (as described
below).  The second part of the move can then be either done with
the other slow-motion knob, or if the object to which you are going
will be easily visible in the field you are using, you can release
the second clamp (keeping the first one engaged) and move the
telescope by hand.  Remember that west and east will always be the
same (clockwise or counterclockwise) with the RA slow-motion knob,
but north or south could vary;  to figure those out, turn the Dec
knob a little (without letting go of it) and then back, and watch
to see whether the far end of the telescope moves generally
towards or away from the north celestial pole.

     Here is an example to illustrate why taxicab movement is safer
than free movement, even if you are not doing turn counting.  When
I was looking for galaxies M82 and M81 in Ursa Major, part of my
path involved the following free movement:  "Go from delta UMa (the
NE star in the bowl of the Big Dipper) through alpha UMa (the NW
star in the bowl of the Big Dipper), then continue on the same line
about 10 degrees to 23 UMa (magnitude 3.5)".  This would have been
a nice quick move if it had worked;  unfortunately, my direction
was a little off, and I instead arrived at 29 UMa (magnitude 3.5),
which is about 4 5/8 degrees SSE of 23 UMa.  Thinking that I was
looking at 23 UMa, I then tried the next step in my path (to
24 UMa), but this star was nowhere to be found!  I tried several
times with the same results.  The next night I used taxicab
movement by first going 1 1/3 degree north (which was pretty easy
to judge in the finderscope), then 10 1/2 degrees west (which I did
by moving the telescope by hand), and successfully found 23 UMa.
This time I was sure I had it because I had noted two check stars
as follows on my card:  "23 UMa is the E vertex of a nearly
equilateral triangle, the other two stars having mags. 4.5 and 5.0,
with side lengths about 2 1/2 degrees".  Thus I learned two lessons
from this experience, namely that taxicab movement is more accurate
than free movement (even though it is not as fast), and check stars
can be quite valuable (had I looked for those check stars the
previous night when I was at 29 UMa, I would have known I was not
at the right star, and I could probably have made my way to 23 UMa
with the help of my binoculars).  (In the discussion of M82 and M81
in chapter XII, we use turn counting, which is really an enhanced
form of taxicab movement, and do things a little more precisely than
discussed here.)

3.  turn counting

     To use turn counting, we need to know how far to turn the
slow-motion knobs to move a desired distance.  Here are the
facts:

a.  To get the number of turns of the Dec slow-motion knob needed
to move a certain number of degrees north or south, multiply the
number of degrees by 24/60.  (If TDec, as defined in section IVA
earlier is not close to 1, also multiply by TDec whenever you
determine the number of north-south turns, even though we will not
mention this again.)

b.  To get the number of turns of the RA slow-motion knob needed to
move a certain number of minutes east or west, multiply the number
of minutes by 6/60.  (If TRA, as defined in section IVA earlier is
not close to 1, also multiply by TRA, even though we will not
mention this again.)

     Let us check these results.  If you want to move 2 1/2 degrees
north or south, 2 1/2 times 24/60 gives 60/60, which equals 1, in
agreement with our assumption in section IVA that 1 turn of the Dec
slow-motion knob produces a movement of 2 1/2 degrees north or
south.  If you want to move 10 minutes east or west, 10 times 6/60
gives 60/60, which equals 1, in agreement with our assumption in
section IVA that one turn of the RA slow-motion knob produces a
movement of 10 minutes east or west.
     Now you are probably wondering why we set this up with a
denominator of 60.  Think of either slow-motion knob as sitting on a
clockface, with the bump on the knob or shaft pointing to 12:00.  A
clockface is divided into 60 minutes (which we will from now on
call "CLOCKFACE MINUTES" so we don't confuse them with RA minutes
or arcminutes or ordinary time minutes).  We can use this fact to
help us judge the correct amount of turning of the knobs.  For
example, if we wanted to move 1 degree north, that would be 24/60
of a turn;  if north is clockwise with your Dec slow-motion knob at
this time, it is probably most accurate to first figure out where
24 clockface minutes would be on your imaginary clockface (it would
be just short of the imaginary number 5 on the clockface), then put
a thumb or finger at that spot on the knob, and turn the knob with
little movements of your fingers until the bump reaches the place
where you put your thumb or finger.  An even more accurate, although
somewhat more awkward, way would be to put a finger of your other
hand lightly where you want the bump to end up, then turn the knob
until the bump reaches that finger.  Actually, if a turn will
involve one or more full revolutions of the knob plus (or minus) a
partial revolution, I find it convenient to place a finger of one
hand at the bump, do the full revolutions without moving the finger,
then move the finger to do the partial revolution.  If a
counterclockwise turn is required, you need to imagine the
clockface being numbered counterclockwise instead of clockwise, then
do the same thing.  As another example, if you want to move 35
minutes in RA, multiplying 35 times 6/60 gives 210/60, and
converting to a mixed number gives 3 30/60.  Note that we could
simplify the arithmetic by breaking 35 into 30 + 5; 30 is 3x10 so
that gives you 3 turns, and 5 times 6/60 gives you 30/60.  With a
little practice you may be able to do the arithmetic of computing
the number of turns in your head while you are observing, but if you
are not comfortable doing this, you can do the arithmetic in advance
and write the results on your card.
     This might be a good place to mention a problem that occurs
with some telescopes, namely, the slow-motion knobs will only turn
so far in each direction and no farther.  Remember Caution 1:
Never force anything (you may break the knob or something more
expensive if you force it).  To be steaming along counting turns
and have a knob suddenly refuse to turn further is a revolting
development (or revolving development?).  What you can do is turn
the knob back beyond where you started from at least a little
further than the number of clockface minutes you had to go when the
knob stalled, then disengage the clamp and move the telescope by
hand back to the place you started from  and then try the turn
counting again;  you should have enough room this time.  If you
were in the second half of a taxicab movement when the knob
stalled and there is nothing visible at the turning point of the
taxicab movement to tell you when you are there, you may have to
go back to the beginning of the taxicab movement and use the
object there to aid you in "uncoiling" the balky slow-motion knob.
     The great advantage of turn counting is that you are actually
measuring out the distance you want to go, instead of roughly
approximating it using the width of your field, or depending on
seeing something in your field that will tell you that you have
arrived.  With practice, you can do turn counting fairly accurately,
at least for short moves;  you may even be able to do it fairly
accurately for surprisingly long moves.  You just have to try it
and see how it works;  as time goes on, you will get better and
better at it.  My personal experience is that I am far less likely
to blunder if I use turn counting;  with practice, if your path is
constructed well enough, turn counting will almost always take you
where you want to go.


D.  The "High in the sky" problem

     Use of the finderscope becomes physically difficult for
objects that are close to being straight overhead.  For such
objects you need to get your eye close to the finderscope, then
look almost straight up, which can be hard to do.  In addition,
when you have to tilt your head a lot in order to look through the
finderscope, your sense of direction tends to get fouled up;  for
example, if you are looking at an object that is low in the sky,
then it is pretty clear which way up and right are, but this is not
so clear when you tilt your head to look up.  These problems also
exist for your star pointer (if you have one) but are not nearly so
difficult there since you have more freedom as to where to place
your eye and you have a much wider field of vision.
     So, what do we do about this?  One approach is to try to
maneuver your body so you can look through the finderscope;  for
example, you can put a pad on the ground to sit or kneel on while
looking up through the finderscope, or you can lean sideways or
backwards in your chair.  If you do this, be careful to not hurt
yourself, or you may have a stiff neck (or worse) in the morning.
     Another approach is to try to minimize the use of the
finderscope for objects that are nearly straight overhead.  In the
case of your starting object or any other object that you can see
with your naked eye, here is a place where a star pointer can be
really useful.  If you don't have a star pointer, you may find
sweeping for the initial object as described in section VIIA above
to be easier than trying to aim the finderscope at it.  If there
is a naked-eye object in your path that is lower in the sky than
your starting object, you could locate it by following the part of
the path up to it with your binoculars only, then using that as
your starting point for using the telescope.  Another approach
would be to select a new starting point less high in the sky,
use your star charts to determine the number of degrees north-south
and the number of minutes east-west to get from this new starting
point to the old one, then use turn counting to get to your
original starting point.  You may still need to briefly look
through your finderscope to center the original starting point well
enough to get it into the field of your telescope, but that will be
easier than having to take a prolonged look through the finderscope
while searching for your original starting point.  If you have
gotten started but are having trouble moving from point to point in
your path because of the high in the sky problem, you may want to
modify your path so as to avoid free movement and rely more
heavily on turn counting.  It may be possible to eliminate use of
the finderscope entirely by choosing a new path with only short
links, and preferably at least one check star for each point in the
path.
     There is another thing you could do, and that is do your
observing for this particular object at a different time when it is
not so high in sky.  There is one unfortunate aspect to this plan,
however, and that is that the seeing conditions nearly straight
overhead are often better than they are low in the sky since there
is often less light pollution straight overhead, and light from
objects that are nearly straignt overhead has less air to pass
through and thus usually suffers less atmospheric distortion (as
well as distortion caused by heated air rising from buildings and
things like that).
     There is yet one more thing you can do if you find the "High
in the sky" problem to be too burdensome, and that is to buy a
right-angle finderscope.  If you do this, by all means put out a
few extra dollars and get the correct-image kind, since then you
will not have to deal with inverted images.  I dealt with the "High
in the sky" problem for a long time before getting a correct-image
right-angle finderscope, and when I finally bought one, I was
really glad I did.  You will still need to look upwards to use your
star pointer (if you have one) but this is much easier than trying
to use a normal finderscope this way since you have a much wider
field with the star pointer and you also have much more freedom as
to where you put your eye.


E.  Changing magnification

     Once you have reached your final object, if you have a Barlow
lens or more than one eyepiece, you will want to try higher
magnifications, using the sequence you set up in section IVC above.
If you can see your final object, center it and go to the next
eyepiece (or insert the current eyepiece into your Barlow and use
the combination as a new eyepiece).  Do the number of quarter-turns
in the table you set up in section IVC to get an approximate focus,
then sharpen the focus visually.  If your object is a galaxy or
other extended object, you may get a sharper focus by looking at a
nearby star (moving a little east or west to find one if necessary;
moving a little north or south is also possible but increases the
risk of losing track of your object).  Remember that you can often
get a sharper focus if your last little adjustment of the focusing
knob is counterclockwise.  If you reach a point in your sequence
where the object looks worse than it did at the previous step, then
back up to the previous eyepiece.
     But what if you can't see your final object?  First try
wiggling one of the slow-motion knobs back and forth a little,
since the movement may make it easier to pick out your object.
Then try going a little way at least down your sequence since
an object that was not visible at low power may become visible at
higher power.  Remember that the object will be drifting westward
(unless you are using a drive motor, which would not be a bad idea
at this stage), so you may have to adjust your telescope
frequently, using nearby stars to keep track of where you are.  If
you still cannot see your final object, then see the next two
sections for further suggestions.

     
F.  What if you realize you are lost in space?

     Being lost in space means not being able to find something in
your path.  One example of this was already given in section VIC2
above.  If you are attempting to move to an object that is bright
enough to be visible in your binocluars and finderscope, you can
attempt to get back onto your path by pattern-matching, that is,
by moving your telescope aroung a little until you can see two or
move stars in your finderscope, then trying to find this same
pattern with your binoculars (remembering that what you see in the
finderscope and what you see in your binocluars will be inverted
with respect to each other--see section IIIF above).  If you are
successful at this then see which way you have to move in your
binoculars to get from the matched pattern to an object on your
path, then move the opposite way using the finderscope.
     Pattern-matching sometimes works, but often it is surprisingly
hard to carry out, and you may well have to go back to your
starting point and begin again.  This time do things a little more
slowly and carefully, paying special attention to any check stars.
Also be careful that you have not mixed up north and south, or
east and west, somewhere along your path;  it is fiendishly easy to
make that mistake!  If you get lost at the same point, you can try
some scanning to try to latch onto the missing object, perhaps
using a technique similar to that discussed near the end of the
discussion of M92 in chapter XII.
     If you are still getting lost, there may be something wrong
with your path.  Go back to your chart and remeasure everything,
being especially careful not to mix up east and west.  If you were
doing free motion around the place you got lost, you may want
to re-measure for taxicab motion, preferably with turn counting.
You may also want to add more check stars, and even add more stars
to your path so you will be taking shorter steps.
     You may even spot a different path that would be better.  As
an example of this, the first time I tried to find NGC2403 I tried
to approach it from the south and got lost in a maze of stars.
Looking again at chart 1 in reference II-4, I noticed a little
bow of stars (with two stars in an east-west line and a third star
between and a little below them) about 1/2 degree across;  this
looked like something that would fit easily into the field of my
telescope with the longest focal length eyepiece, and would be easy
to recognize.  About 3/4 degree below the middle of the bow was
another star, and NGC2403 was about 3/4 degree southeast of that
star.  On the other hand, there was a brighter star a degree or so
north of the bow, and a pair of bright stars a few degrees west of
that star;  a ways south of the pair was a group of stars I had
already found before getting lost.  Thus it appeared I had the
makings of a good path, and indeed, the next night I was able to
follow this path to NGC2403.  This path may seem a little bizarre
since it goes north and then circles back to the south, but there
is no law that says that you have to choose the shortest or most
direct path;  you just need a path you can follow.  By the way,
was it worth all this work to find this thing?  Absolutely!  In
fact, NGC 2403 is one of my favorites among the things I have found
so far, although it is quite faint, and you need relatively good
viewing conditions to make much out of it.  Here then is the path,
along with some other information, essentially in the format
described in section VIF.

NGC 2403 (spiral galaxy)       mag. 8.5        size 23'x11'
7h 36.9m                 65 degrees 36'        Camelopardalis

Capella, then

   1 degree S, 7 5/8 degrees (43m) E to

beta Aur (2.0, with a mag. 6.5 star 3/8- degree SSE), then

   9 3/8 degrees N to

delta Aur (3.5, with a mag. 7.5 star 1/8 degree NNE and a mag. 6.0
star 1/4 degree N), then

   4 3/4 degrees N, 2 5/8 degrees (20m) E (or 5 1/2 degrees NNE) to

2 Lyn (4.5, with 37 Lyn (5.5) 1 1/4 degrees W, 40 Lyn (5.5) 1 1/8
degrees NNW, 5 Lyn (5.0) 1 1/8 degrees SE, and a mag. 7.5 star 3/8
degree WSW), then

   1/2- degree N, 3 3/8 degrees (27m) E to

12 Lyn (4.5, with a mag. 7.0 star 1/4- degree W), then

   7/8 degrees (7m) E to

14 Lyn (5.5) (with 15 Lyn (4.5) 1 1/4 degrees SSE, so 15 Lyn, 14
Lyn, and 12 Lyn make a triangle that is readily recognizable;
there is also a mag. 7.5 star 1/4- degree SW of 14 Lyn), then

   9 1/2 degrees N, 1/8- degree (1m) E to

43 Cam (5.0, with 42 Cam (5.0) 1 3/8 degrees SSW, and a mag. 4.5
star 3 degrees WNW of 43 Cam, forming a right triangle with right
angle at 43 Cam), then

   1/2- degree S, 3 1/3 degrees (37m) E (or 3 1/3 degrees ESE) to

a mag. 5.5 star (with 42 Cam and 43 Cam this makes a triangle with
angle slightly greater than 90 degrees at 43 Cam), then

   1 1/2 degrees S to

a bow of three stars of mags. 7.5-8.0-8.0 from E to W, with two
stars on an E-W line about 1/2 degree apart and a third between
them and a little S, then from the middle star of the bow,

   2/3 degree S to

a mag. 7.0 star with a mag. 6.5 star 1/2- degree WNW and a mag. 7.5
star 1/8 degree NNE of the mag. 6.5 star, then

   1/2 degree S, 1/2 degree (5m) E (or 3/4 degree SE) to

the center of NGC2403 (with a mag. 8.5 star 1/4 degree S, a mag.
7.5 star 1/2 degree S, and a mag. 8.5 star at the NW edge of
NGC2403;  the two stars to the S point at the SE edge of NGC2403,
about 1/8 degree farther, which is aligned SE-NW;  NGC2403 is about
1/3 degree long and about 1/6 degree wide.  There are 3 stars on an
E-W line that cut across NGC2403).

     This object is given two exclamation points in the NGC catalog.
It is quite large, so you can miss it if you are looking for
something smaller.  I first saw this around January 2002 with a
5 inch telescope, then saw it a couple of times later with a 9.25
inch telescope.  It was impressive even in the 5 inch (even though
I could barely see it) and was better with the larger telescope,
even though it was not in the most light-pollution-free part of
my sky.  With the 9.25 inch telescope, I could just make out most
of the galaxy itself at 267x.  Some day I want to look at this when
I am away from city lights;  more than most objects in this
document, the view of NGC2403 seems to be sensitive to viewing
conditions.  By the way, the star 12 Lyn which occurs in the path
to NGC2403 is a nice triple star;  it is one of the objects
discussed in chapter XII.

     
G.  What if you think you are pointing at your object, but you
    can't see it?

     If you think you are pointing at your object but you can't see
it, even after wiggling the slow-motion controls and trying
different magnifications, then one of three
things has happened:

1.  The object is not in your telescope field, or

2.  The object is in your telescope field, but you will not be able
to spot and recognize it unless you know exactly where in the field
it is, or

3.  The object is in your telescope field, but even if you know
exactly where in the field it is, you will not be able to spot and
recognize it without better viewing conditions or a bigger
telescope.

     It will be our aim to modify our path, if necessary, so that
we can be reasonably certain that the object is in the field, and
to use other objects in the field to locate exactly where the
object is in the field, if possible.  If we still can't see it,
then about the only hope will be to try again when viewing
conditions are better, or make the conditions better by transporing
the telescope away from light pollution.  For tips on transporting
your telescope, see section IXA later.
     To be able to verify with reasonable certainly that an unseen
object is in the field, you may need to find one or (preferably)
more check stars for the next-to-last object in your path.  Once
you have pretty much verified that you have found that object, if
it is close enough to the final object, then you should be able to,
with reasonable certainty, move your telescope from that object to
a place where the final object lies, even if you cannot see the
final object.  Then using objects on your chart that are within the
same field, you may be able to tie down its location using a
triangle or intersecting lines or something like that.  You may
need to be creative.
     We will illustrate the process with M97 (the Owl Nebula,
magnitude 9.9) and M108 (a magnitude 10 galaxy).  The chart being
used is chart 2 in reference II-4.  These objects are quite close
together, and the paths to them start out the same way, as follows:

Beta UMa (SW star in bowl of Big Dipper), then

   1 1/3 degrees S, slightly E to

a mag. 7.5 star with a mag. 8.0 star 1/2 degree W, then

   1/8 degree S, 1 1/2 degrees (10m) E to

a mag. 6.5 star, with a pair of mag. 7.0 stars 3/4 degree S and a
little W;  these stars are on a nearly E-W line and are 1/4 degree
apart, with a mag. 8.0 star 1/8 degree NW of the western one.

     Having three check stars for the mag. 6.5 star in an
easily recognizable pattern makes it almost certain that we have
the correct mag. 6.5 star if we can see the check stars.  Now let's
deal with M97.  Formerly I had just said to go 1/4 degree ENE from
the mag. 6.5 star to get to M97, but although I was sure it was in
the field, I could not see it.  Then on the chart I noticed a mag.
8.5 star about 1/4 degree NNW of the mag. 6.5 star, with M97 and
these two stars forming an isosceles right triangle with right
angle at the mag. 6.5 star.  This information tied down the
location of M97 exactly, and after jiggling the slow-motion
controls back and forth for a while I was able to see (I think) a
faint patch of light at that spot.
     Now let's deal with M108, which turned out to be a little
easier to see than M97.  On the chart, one could see that a line
drawn through the mag. 6.5 star and the mag. 8.5 star nearly hits
the W edge of M108.  There is also a mag. 7.5 star 3/8 degree NE of
the mag. 8.5 star, and if one draws a line through the mag. 8.5 star
and the mag. 7.5 star, and then draws a line through the mag. 7.5
star perpendicular to this line, then this perpendicular line hits
M108. Now we have two lines that (nearly) intersect at M108, and
this ties down its location almost exactly.  After some more
wiggling of the knobs I was able to see a faint patch of light
there.  Success!
     There is another technique for locating faint objects that
utilizes stars that are too faint to be on your star chart.  These
stars will serve as markers and also as focus stars.  Here is how
it works.  First, move to the next-to-last object in your path as
usual.  This object needs to be quite close to the final object.
Then do the north or south part of the final move, using the field
of your telescope as a yardstick.  This is more accurate than
turn counting for short moves.  For example, suppose your field
width is about 3/4 degree;  then the radius of your field is half
that, namely about 3/8 degree.  Suppose you want to move 1/3 degree
south.  Dividing 1/3 by the field radius gives (1/3)/(3/8) =
(1/3)(8/3) = 8/9.  Turn the Dec slow motion to move the telescope
south, stopping when the next-to-last object in the path has moved
from the center of your field 8/9 of the way to the edge.  If you
needed to move 1/2 degree instead of 1/3 degree, again dividing by
the radius gives (1/2)/(3/8) = (1/2)(8/3) = 8/6 = 4/3.  You can
turn the Dec slow-motion knob with four small equal-length jumps,
so that after three of them the next-to-last object in the path
has moved to the edge of the field, then one more jump of the same
length as the other three will take your telescope where it needs
to go.  Now do the east or west part of the move in a similar
manner;  this one may be trickier because the next-to-last object
in the path does not start in the center of the field, but
hopefully there will be some object in the field that will help you
judge the distance.  (Comments about the east-west move:  (1) If
the east-west move is a lot longer than the north-south move, you
might want to do it before the north-south move, when you still
have the next-to-last object in the path at the center of the field
to help you.  (2)  For the east-west move we are dealing with
degrees, not minutes, except (3) if the east-west move is to the
east, you have the option of not doing it at all, but rather
waiting the number of minutes you measured (or computed) for the
final object to drift into the center of the field).  Note that in
these moves you will probably need to use magnification low enough
so that both the north-south and east-west parts of the move are
shorter than the diameter of the field (shorter than the radius is
even better) to allow for accurate moves.  By the way, it is
probably better not to use a wide-angle eyepiece when using the
field as a yardstick since with such an eyepiece you need to "peek
around the edge" to see the full field, and this can give you a
deceptive idea of your field width.
     Now that you think you have the final object in the center of
your field, look for any stars in the field, as close to the center
as possible.  These stars will help you mark the location where you
think the final object is, so you can do whatever adjustments are
needed to keep the object from drifting away from the center of the
field, and after any other moves you make you will be able to
re-center the (as yet unseen) object.  Now try wiggling the field
back and forth a little to see if that enables you to detect the
object.  You can also try averted vision, that is, look a little
away from where you think the object is to bring the object to a
part of your retina which is more sensitive to faint light.  If you
don't see the object, increase your magnification;  if this takes
your marker stars out of your (shrunken) field, you may have to
move to one of them to refocus, then move back.  Your marker stars
will still guide you here, but remember that when you increase the
magnification, the visual distance between objects increases
proportionately, so that (for example) if you go from 267x to
534x, objects will appear twice as far apart.  Then try wiggling
the field again, and continue in this way until you either detect
the object or reach your highest magnification.
     Although I had been using the field as a yardstick for quite
a while, the first time I tried the complete technique in the
previous paragraph for a faint object, I was astonished to be able
to find a magnitude 11.3 galaxy (namely NGC7640).  If you want to
try this technique for this galaxy, I should mention that your
task could be easier or harder than mine, depending on your
telescope and viewing conditions.  With my 9.25 inch telescope and
moderate to heavy light pollution, I was just barely at 267x (and
a little better at 534x) able to see a faint oval glow in the right
place, about the right size (10.7'x2.5'), and with the right
orientation (long axis north-south).  There was a star or two that
seemed to be embedded in the glow, but these were probably
foreground objects, not part of the galaxy.  Note that the more
information you have about an object, the better chance you have of
finding it.  To get to this object, first follow the path for the
Blue Snowball Nebula in Chapter XII as far as 10 And, then continue
as follows:

10 And (6.0), then

   3/8 degree SW to

9 And (6.0), then

   1/8 degree S, 1 1/8 degrees (6m) E to

a mag. 6.5 star, with a mag. 6.5 star 1/2 degree S and a mag. 8.5
star to the west of these stars, forming an equilateral triangle,
then from the west (mag. 8.5) star of this triangle,

   1/2 degree S, slightly W to

 NGC7640.
 
     Now some of you may be aching to ask, "If an object is so
faint, why bother with it?"  There are a number of possible
answers to this question;  here are two of them.  First, if you can
find the object, then if you later have another chance to find it
under better viewing conditions, having found it once will make it
easier to find the second time, when it may be more impressive.
The second answer is the same as the answer to the question "Why
climb a mountain?", namely, "Because it's there!"

     
H.  What if your object will appear only briefly between
    obstructions?

     If your object will appear only brifly between obstructions,
then you have less freedom in choosing a path because some stars
close to the object will be hidden by the obstructions.  Timing
also becomes more important since the object will be visible only
briefly, so you will need to compute the transit time of the object
as in section VIIIB below, and from that you can estimate when it
will be visible.  Normally this problem occurs when the object you
want is far to the south, and you have obstructions between you and
your terrestrial southern horizon.  In this case you would normally
want your path to approach your final object from the north, so
that the stars in the path will be visible above the obstructions,
at least for a while.  
     There are a couple of things you can try.  If you will have a
clear view between the obstructions, you can try to get your
telescope pointed at a point in the gap which the object will cross,
then wait for it.  You can judge the time you will have to wait by
using the fact that the waiting time will be about the right
ascension of the object minus the right ascension of the point at
which you point your telescope at the moment you point it there.  
     If you do not have a nice clear spot ahead of the object where
you can point your telescope, but only have an area of mixed tree
leaves and little holes between them (or something like that), there
is something else you can try if you have a drive motor for your
telescope.  Try to get your telescope pointed as directly at your
object as you can, even though you cannot see the object yet.  This
has to be done pretty accurately since you may not be able to pick
out the object well with your finderscope even when it appears, and
you may have to depend on the telescope only.  Then engage your
drive motor and turn it on so your telescope will track the unseen
object.  Then watch through the telescope until the object appears.
Even if it is still blurred by the leaves, you can then try to
center it.  North-south adjustments can be made with the
declination slow-motion knob, but you cannot use the right
ascension slow-motion knob since your right ascension drive motor
will be running.  If the object is too far west in your field, use
an extra-speed button on your control so the telescope can catch
up to the object.  If the object is too far east in your field,
use the stop button on your control so the object can catch up to
the telescope;  it is not a good idea to reverse the motor to get
back to the object since the motor gears will probably have a
little play in them (which is necessary to avoid putting too much
strain on the motor) so when you switch back to forward motion the
telescope will not move for a few seconds, resulting in the object
getting too far to the west in your field.  Once you have centered
the object, you can change magnification as desired whenever you
can get a good enough look.  When you have reached the desired
magnification, you can watch the object until it gets behind the
thick obstructions and is gone for the night.
     We now consider xi Lup as an example.  This is a double star
with right ascension 15h 57m and declination -34 degrees.  Here is
a possible path;  since when I viewed it the next-to-last object in
the path was behind trees until it reached a V-shaped gap shortly
before xi Lup appeared among leaves lower down, both of the
strategies in the previous paragraph were used.

Antares, then

   7/8 degree N, 1 7/8 degrees (8 1/2m) W to

sigma Sco (3.0, with a mag. 6.0 star 1 1/2 degrees WNW), then

   3 degrees N, 4 7/8 degrees (21m) W to

delta Sco (2.0, with a mag. 7.5 star 1/2- degree E and a mag. 7.0
star 3/4 degree NNE;  delta Sco is slightly N of a line through
Antares and sigma Sco), then

   3 1/2 degrees S, 3/8 degree (1 1/2m) W to

pi Sco (2.5, with 4 Sco (4.5) 3/4 degree W and a mag. 5.0 star
1 degree ENE), then

   3 1/8 degrees S, 3/8+ degree (2m) W to

rho Sco (4.0, with a mag. 8.0 star 1/4 degree W), then

   4 3/4 degrees S to

xi Lup.

     What I did was start by getting pi Sco into my telescope field
(the trees were actually helpful for this part, since if there is a
tree close to an object you want you can use it as a landmark in
your finderscope), then when rho Sco appeared in the V-shaped gap I
went to it, looked for the check star to be sure I had rho Sco,
centered rho Sco, then went 4 3/4 degrees S (which was 1 54/60 turns
with the declination slow motion knob;  this is the most critical
step, and needs to be done very carefully).  Then I engaged and
started the motor, waited until a flash of light appeared in the
telescope, centered it, noticed happily that even at low
magnification it looked like a double star, changed magnification
twice (re-centering as needed), then watched it until it
disappeared.  It was a nice little double star, but the involved
procedure required to find it made it look even nicer.  It is
amazing that you can get a good image when you only have small
holes between leaves to look through.  Although I didn't do this,
when you have found a hard-to-find object like this you are
entitled to celebrate with your best Tarzan yell, unless of course
you have neighbors who would not understand.  Actually, in this case
a wolf howl might be better since the object is in the constellation
Lupus, the wolf.


I.  How much effect can errors in polar alignment have?

     Back in section VIIA we said that our rough polar alignment
should usually be accurate enough for our purposes.  One needs to
be careful, however, especially with objects near a celestial pole.
For example, one time when I was making a 30 minute move to the
east at declination about -25 degrees, the center of the field of
my telescope missed the target object by more than the radius of
the field, with the result that I sailed right by the object
without seeing it, latched onto a similar-looking object a little
farther on, and became lost.  This was foolish on my part;  had I
looked in the finderscope after making the three turns of the RA
slow-motion knob required to go 30 minutes, I would have seen the
target object and could have made the necessary adjustment to get
it into the field of the telescope.  Still, however, it is useful
for us to consider how much effect a polar alignment error can have.
     Another situation in which polar alignment errors sometimes
show up occurs when you are tracking an object across the sky,
either using a drive motor or by making little westward adjustments
as needed with the RA slow-motion knob.  When you are doing this
you may see the object drifting a little north or south.  This is
due to a polar alignment error.  For visual observing this is not
a big deal since you can make small adjustments with the Dec slow-
motion knob as needed, but for time-exposure astrophotography it
could really mess up a picture.
     In order for a polar alignment error to cause you to miss an
object, you need to be a little unlucky.  If the error was made in
a direction directly toward or directly away from the point halfway
between the objects, then it will not cause you to miss the object;
the polar alignment errors that are most likely to cause you to
miss an object are ones that are made in a direction roughly
perpendicular to the line going from the north celelstial pole to
the point halfway between the objects.  In the table below we give
the worst distances in degrees (rounded to the nearest .001 degree)
that the center of your field could miss the object you are moving
toward in a move to the east or west, in all cases assuming the
objects are 5 degrees apart, under various assumptions on the size
of the polar alignment error (p) (in degrees) and the declination
(dec) (in degrees) of the object from which you are moving.

   p     1      3      5
dec
0      .087   .262   .436
30     .101   .302   .503
60     .174   .523   .871
80     .502  1.506  2.509

     Now you may ask, what about moves to the north or south rather
than the east or west?  It turns out that these errors differ from
the values in the table by at most .002 degrees, so we did not
bother to tabulate them separately.  You may also ask, what about
negative declinations?  It turns out that these errors are exactly
the same as the ones in the table, using the absolute value of the
declination (for example, for values for declination -30 degrees,
look in the table for declination 30 degrees).  This fact about
the declinations is tied up with the fact that when you are lining
up the fixed arm of the telescope with a point close to the north
celestial pole, the other end of that arm is being lined up with a
point near the south celestial pole.
     We can make several observations about the table.  First,
the errors are worse for objects near a celestial pole than for
errors near the celestial equator.  Thus we need to be a little
more careful for objects near a celestial pole, for example, we
may need to make shorter hops.
     Second, the size of the worst miss is roughly (but not exactly)
proportional to the size of the polar alignment error, at least for
the values in the table;  for example, for declination 30 degrees,
5 times .101 is .505, which is close to .503.
     This brings up the question of whether the size of the worst
miss is also roughly (but not exactly) proportional to the distance
we move.  The answer is yes (if we don't move too far, the polar
alignment error is not too large, and we are not too close to a
celestial pole).  For example, in the case of a 5 degree polar
alignment error and declination 80 degrees, the worst miss for a
move of 10 degrees east or west (instead of 5 degrees) is about
4.998 degrees, which is close to twice 2.509.
     The next question is, what can we do about these errors?  One
approach is to just be a little careful, especially near the
celestial poles.  A second approach would be to try to make your
initial polar alignment more accurate, using the instructions that
came with your telescope.  Note that even if you line up the fixed
arm of your telescope perfectly on Polaris, you will still be
making an error of about 3/4 degree, since Polaris is that far from
the true north celestial pole.  A third approach would be to try to
correct your polar alignment while you are observing if the errors
you are getting are large enough to be bothersome.  The next
two paragraphs tell in which direction an alignment correction
needs to be made, using a "stick" analogy;  this analogy is not
perfect since the stick pierces the celestial sphere instead of
running along it, but it is good enough to tell you in what
direction to adjust.  After making a small adjustment, you can check
to see if the situation has improved sufficiently for your taste,
and if not, you can make a further adjustment, guided by the results
of the first adjustment.
     In this paragraph we consider adjustments for movements east
or west.  Imagine a (very long) stick attached to the point
(hopefully near the north celestial pole) on which the fixed arm of
your telescope is aligned.  Assume the other end of the stick is
attached to the point on the celestial sphere at the center of your
field of vision, which in turn lies on the object from which you
want to move.  As you move, the first end of the stick will stay
where it is, and the end attached to the center of your field of
vision will swing, with the stick keeping it a constant distance
from the point on which your telescope is polar aligned.  If the
center of your field of vision passes north of your target object,
then the first end of the stick is too far in the direction
opposite to which you moved, resulting in the stick pulling the
center of your field of vision above the target object;  you must
correct your polar alignment point in the same direction you moved.
If on the other hand the center of your field of vision passes south
of your target object, then the first end of the stick is too far
in the direction you moved, resulting in the stick pushing the
center of your field of vision below below the target object;  you
must correct your polar alignment point in the direction opposite
to that in which you moved.  For a more specific example, suppose
your two objects are near your meridian, and you are moving east.
If the center of your field of vision passes north of the second
object, then you need to adjust the azimuth of your mount a little
to the terrestrial east.  If on the other hand your two objects are
to the terrestrial east of the north celestial pole, you are moving
east, and the center of your field of vision passes north of the
second object, then you need to adjust the altitude of your mount 
a little upward.  If the objects are somewhere else, you may need
to adjust both the azimuth and altitude of your mount a little.
     In this paragraph we first consider adjustments for movements
from north to south.  Again assume the first end of the stick is
attached to the point (hopefully near the north celestial pole) on
which the fixed arm of your telescope is aligned, but now also
assume that the stick is also attached to your first object.  As
you move north to south, the center of your field of vision will be
moving down the stick.  If the center of your field of vision
misses your second object to one side or the other, then the first
end of the stick is too far in the opposite direction to the
direction of miss, creating a lever effect with fulcrum at the
first object, and you need to correct your polar alignment in
the same direction as the miss.  If on the other hand you are
moving from south to north, again assume the stick is attached at
your alignment point and at the first object.  This time the
situation is reversed since the fulcrum of the lever is below the
second object, and you need to correct your polar alignment in the
opposite direction to which the miss occurred.  In both this
paragraph and the previous one, don't forget the effects of inverted
and reverted images on your perception of what is happening!
     We already said that polar alignment errors that are directly
toward or directly away from the halfway between the objects will
not cause you to miss the second object. In the case of a move to
the east or west, however, they can change the number of minutes
that your telescope thinks are between the objects, which in turn
will affect the number of turns of the RA slow motion knob that are
needed to go from one object to the other.  Examples for
declinations of 0, 30, 60, and 80 degrees are given in the next
four paragraphs;  you may want to just skim these paragraphs
without worrying too much about the specific numbers given.
     Again assuming a move of length 5 degrees, if the objects have
declination 0 degrees, then the true number of minutes between them
is 20 minutes;  if a polar alignment error of 5 degrees either
directly toward or directly away from the halfway point between the
objects was made, then the telescope will think that there are
about 20.076 minutes between the objects.  Thus if you centered the
object and then ran your drive motor for 20.076 minutes, after that
time the telescope would be aimed at the second object, but it
would be .076 minutes later getting there than if the polar
alignment had been correct (and the path the telescope aim followed
will also bulge a little north or south of the correct path), which
would degrade the image from a 20-minute time exposure.  To fix that
problem, you would need to correct your polar alignment.
     If the declination is 30 degrees, then the true number of
minutes between the objects is about 23.096 minutes;  if a polar
alignment error of five degrees directly toward the halfway point
between the objects is made, then the telescope will think that
there are about 24.417 minutes between the objects, and if a polar
alignment error of 5 degrees directly away from the halfway point
between the objects is made, then the telescope will think that
there are about 22.070 minutes between the objects.
     If the declination is 60 degrees, then the true number of
minutes between the objects is about 40.038 minutes;  if a polar
alignment error of five degrees directly toward the halfway point
between the objects is made, then the telescope will think that
there are about 47.353 minutes between the objects, and if a polar
alignment error of 5 degrees directly away from the halfway point
between the objects is made, then the telescope will think that
there are about 34.906 minutes between the objects.
     If the declination is 80 degrees, then the true number of
minutes between the objects is about 116.385 minutes;  if a polar
alignment error of five degrees directly toward the halfway point
between the objects is made, then the telescope will think that
there are about 224.993 minutes between the objects, and if a polar
alignment error of 5 degrees directly away from the halfway point
between the objects is made, then the telescope will think that
there are about 78.174 minutes between the objects.
     If the declination is some negative number, and a polar
alignment error is made directly toward the halfway point between
the objects, then the number of minutes that the telescope will
think are between the objects will be the number of degrees it
would have thought were between the objects if the declination were
replaced by its absolute value and a polar alignment error was made
directly away from the halfway point between the objects, and vice
versa.
     As we can see from these examples, the situation gets worse as
we get closer to one of the celestial poles.
     It turns out that polar alignment errors do not affect the
length of a move to the north or south.  The reason is that when
you turn the Dec slow-motion knob a certain distance, this will
cause the aim of your telescope to move a certain number of degrees
along a great circle on the celestial sphere, and this number of
degrees is not affected by the direction the fixed arm of the
telescope is pointing, or by the starting aim of the telescope.
If, however, you turn the RA slow-motion knob a certain distance,
this will cause the aim of the telescope to move a certain distance
along a circle on the celestial sphere, but this circle will not
usually be a great circle (just imagine what happens if the
starting aim of the telescope is 10 degrees from the north celestial
pole).  This is why the situation is more complicated for east-west
moves than it is for north-south moves.
     For those of you who are clamoring to know how all these
numbers were computed, we now provide the formulas and some
discussion in square brackets;  those who are not interested can
skip this.  The first part of the skippable square-bracket
material concerns the formula for finding the distance between
two objects, given their right ascensions and declinations.  This
"celestial distance formula" will be needed for some of the rest of
this skippable section, and it has some interest in its own right.
It is surprising that, unlike the case of taxicab movement in a
plane, in taxicab movement on the celestial sphere it is not enough
to know the north-south distance and the east-west distance between
the objects;  to compute the correct distance between the objects,
you also need their actual declinations.  Now begins the skippable
part;  I admire your gumption if you go through this, because it
is not for the faint of heart.
[Suppose you know the coordinates (RA1,dec1) and (RA2,dec2) of two
objects.  Let RAdiff be the smallest nonnegative difference of RA1
and RA2 (in minutes).  To compute RAdiff, subtract the right
ascensions, take the absolute value, and if this number is greater
than 12 hours, subtract it from 24 hours;  then convert to minutes.
(For example, if the right ascensions are 23h and 2h, then the
absolute value of the difference is 21h, which is greater than 12h,
and 24h - 21h = 3h, which equals 180m.)  Then the formula for the
distance deg (in degrees) between the objects is

deg = invcos[sin(dec1)sin(dec2) + cos(dec1)cos(dec2)cos(RAdiff/4)].
  
In case anyone may be interested, the proof comes from the Law of
Cosines for Sides for spherical triangles.
     The inverse cosine (which we denote by invcos) of a number
which is between 0 and 1 is defined to be the angle between 0
degrees and 90 degrees whose cosine equals the given number.  You
may have to deal with some negative numbers;  here are the rules
which will allow you to do this:  The sine of a negative number
equals minus the sine of the absolute value of the number;  the
cosine of a negative number equals the cosine of the absolute value
of the number (no minus here!), and the inverse cosine of a number
between -1 and 0 equals 180 degrees minus the inverse cosine of the
absolute value of the number.  If you do the computations with a
calculator, it should take care of the minuses for you
automatically.
     Now let's do an example.  In section VIA above we said the
distance between Arcturus and Antares is about 56 degrees.  This
distance is difficult to measure with star charts since these stars
are far apart, and they are not both on the same chart.  We can
get the result with the formula we just gave, however.  The
coordinates of Arcturus are (14h 15.7m, 19 degrees 12'), and the
coordinates of Antares are (16h 29.5m, -26 degrees 26').
Subtracting the right ascensions gives 2 hours 13.8 minutes, so
RAdiff = 133.8m.  Thus our formula gives
deg = invcos[sin(19 degrees 12')sin(-26 degrees 26') +
cos(19 degrees 12')cos(-26 degrees 26')cos(133.8/4)].
Now converting the declinations into decimal form (so 19 degrees
12' = 19.2 degrees and -26 degrees 26 minutes = -26.433333333333
degrees) we get
deg = invcos[(.328866646739)(-.445156207887) +
(.944376370237)(.89545293041)(.83436710369)] =
invcos[.559181044324] = 56.0008195959 degrees, which is about
56 degrees.
     Now let us consider the formula for computing the (nearly)
worst miss d (in degrees) when you are moving east or west a
distance of deg degrees, you have made a polar alignment error of
p degrees, and the objects have declination dec.  Here and in the
rest of this square-bracketed section we will assume that the
objects are more than p degrees from both celestial poles, for
otherwise you might not make any progress at all in the direction
of your intended target, and you might even move directly away from
it!  We first use Fact 2 in section VIE to compute the number min
of minutes between the objects.  Next we compute the quantity
M = invcos[tan(p)tan(dec)].  Here tan denotes the tangent function
from trignometry.  The formula then becomes

d = invcos[cos(p)sin(dec) + sin(p)cos(dec)cos[M + min/8]]
- invcos[cos(p)sin(dec) + sin(p)cos(dec)cos[M - min/8]].

This formula does not give the exact value of the worst error, but
it should be very close;  in one case where I worked out a close
approximation to the worst error using a kind of mathematics
called numerical analysis, it was greater than the value given by
the formula by less than .00000001 degree.
     Now you may (or may not) ask, where did these formulas come
from?  Well, M is the number of degrees so that if the false north
celestial pole on which your telescope is actually aligned has
right ascension 4M minutes more than the average right ascension of
your two objects, and has declination 90 degrees - p, then the
line from the true north celestial pole to the false north celestial 
pole would be perpendicular to the line from the false north
celestial pole to the point with declination dec and rignt
ascension equal to the average right ascension of your two objects
(the formula for M is derived from one of Napier's rules for right
spherical triangles).  The formula for d then arises when we use
the celestial distance formula above to compute the difference in
the distances from the false north celestial pole to your two
objects.
     Now let us consider the formula for computing the worst miss d
(in degrees) when you are moving north or south a distance of deg
degrees, you have made a polar alignment error of p degrees, and
the declination of the object from which you are moving is dec.
Unlike the formula for east or west movements, the formula for
north or south movements gives the exact value for the worst error.
The formula is

d = invsin[sin(p)sin(deg)/cos(dec)].

     This formula comes from solving two right spherical triangles
simultaneously using Napier's rules.  The picture looks different
depending on whether we are moving north to south or south to
north, but the result is the same.  The idea is to choose the false
north celestial pole so that the line from the true north celestial
pole to the false north celestial pole is perpendicular to the line
from the false north celestial pole to the object from which you
are moving, which maximizes the angle between the second line
mentioned above and the line from the true north celestial pole to
the object from which you are moving.
     Finally, let us consider the computation of the number newmin
of mimutes the telescope thinks there are between two objects with
declination dec which are deg degrees apart, assuming a polar
alignment error of p degrees either directly toward or directly
away from the halfway point between the objects.  To do this, first
use Fact 2 in section VIA to compute the number of minutes min
that actually exist between the objects.  Then the formula is

newmin = 8invsin[(sin(deg/2))/sin[invcos[cos(p)sin(dec)
(+ or -) sin(p)cos(dec)cos(min/8)]]],

where in place of the (+ or -) you use + if the polar alignment
error is toward the points, and you use - if the polar alignment
error is away from the points.
     Here is where the formula comes from.  Note that the
declination of the false north celestial pole is 90 degrees - p.
For a polar alignment error toward the two points, the right
ascension of the false north celestial pole is just the average of
the right ascensions of the two points (with an appropriate
adjustment if the right ascensions of the two points fall on
opposite sides of the zero point of right ascension).  For a polar
alignment error away from the two points, the right ascension of
the false north celestial pole is the same as above, with 720
minutes added (since adding the 720 minutes reverses the direction
of the polar alignment error).  In either case, applying the
distance formula to the false north celestial pole and either of
the original two points, and using the trigonometric identities
cos(90 degrees - angle) = sin(angle), sin(90 degrees - angle) =
cos(angle), and (if necessary) cos(180 degrees + angle) =
-cos(angle), gives the distance between the false north celestial
pole and either of the points as
invcos[cos(p)sin(dec) (+ or -) sin(p)cos(dec)cos(min/8)].
The telescope will think that this distance is 90 degrees - the
declination of the two points, so the telescope will think that
the declination of the two points is 90 degrees - this distance.
Finally, applying Fact 2 using what the telescope thinks is the
correct declination and using the identity cos(90 degrees - angle)
= sin(angle) gives the formula above for newmin.
     If you made it through this (except for the references to
spherical trigonometry, which most of you will not have studied),
give yourself another well-earned gold star!]
     It is worth noting that most taxicab movements involve both a
N-S movement and an E-W movement.  Any errors in the N-S movement,
whether due to polar alignment errors or errors in the amount of
turning of the Dec slow-motion knob or whatever, may result in the
ending point of the N-S movement being off a little in declination
and/or right ascencion.  Since the ending point of the N-S movement
is also the starting point of the E-W movement, these errors in
the N-S movement will affect the result of the E-W movement also.
We will not consider the details here, but in spite of the fact
that our work on polar alignment errors does not tell the whole
story, it does tell a lot of the story, and should give us a better
idea of the results of polar alignment errors.  For an example that
uses our error table earlier in this section, see the discussion of
M92 in chapter XII.
     We close this chapter by noting that further hints for finding
objects in some difficult situations are given in some of the
examples in chapter XII.

          
Chapter VIII.  Time


A.  Computing the time correction for your date and location


     The main reason we want to do time calculations is to find the
approximate TRANSIT TIME of an object of interest, that is, the time
when it crosses our meridian, since it will be highest then and may
be easiest to observe around that time.  The fundamental equation
we will use in our calculations is this:

Fact 3.  R = T + C,
where R = the right ascension of an object,
T = the transit time of the object (in 24 hour local standard time),
and C is a correction.

     Our purpose in this section is to learn how to compute a good
approximation to C;  we will see how to use the equation in the
next section.  Note that whenever you are using Fact 3, if
you get a negative answer, then add 24, and if you get an answer
that is greater than 24, then subtract 24.
     The exact value of C, which we will not use and you do not
need to remember, is 24 hours times the length of time since the
last autumnal equinox, divided by the time it takes the Earth to
make one orbit around the sun.  There are two parts involved in our
approximation of C, one of which depends on the date, and the other
(and less important of which) depends on your longitude.
     The more accurate of our two methods for getting the part of
the approximation to C that depends on the date is to look it up in
Appendix 12 on pp. 544-545 of reference II-3 (or some other
reference).  If you don't have this table handy, you can get an
approximation which is slightly less accurate (but still good
enough for our purposes) as follows:  Count the number of months
since last September (so October would be 1, November would be 2,
etc.), and multiply by 2 hours.  Then if the day of the month is
greater than 21, add 4 minutes for each day past 21, and if the day
of the month is less than 21, subtract 4 minutes for each day
before 21.
     To get the part of C that depends on your longitude, you need
to know your longitude (you can consult an atlas) and the longitude
of the line of longitude in your time zone that is evenly divisible
by 15 (this will probably be about in the middle of your time
zone).  If you are terrestrial east of the line of longitude that
is evenly divisible by 15, for each degree that you are terrestrial
east, add 4 minutes.  If you are terrestrial west of the line of
longitude that is evenly divisible by 15, for each degree that you
are terrestrial west, subtract 4 minutes.  The nice thing about
this calculation is that you only have to do it once.  Now I can
hear some of you out there saying "Why do we do this?"  For you who
want to know why, the justification follows in square brackets;
those who don't care why can skip this.
[Justification:  First suppose you are one degree of longitude
terrestrial east of the line of longitude in your time zone that is
evenly divisible by 15.  Consider an object with right ascension R
that transits (that is, crosses your meridian) at 24-hour local
standard time T.  For an observer on the longitude line that is
evenly divisible by 15, the same object will not transit until
about 4 minutes later (since he or she is one degree west of you,
and the Earth will turn a little more than 360 degrees in 24 hours
(as we saw at the end of section IIIC), and 24 hours divided by 360
degrees equals 4 minutes).  Thus the T for that observer is about 4
minutes greater than your T (so your T is about 4 minutes less than
his or her T), but R is the same for both of you, so for the
equation R = T + C to be true for both of you, your C must be about
4 minutes greater than his or hers to compensate.  Now the other
main case would be the case where you are one degree terrestrial
west of the line of longitude in your time zone that is evenly
divisible by 15;  why don't you try to figure this one out for
yourself, since that will help you understand this.  If you can do
it, award yourself another gold star.]
     Now let's consider the example of determining the C for
someone living in Decatur, Illinois, on July 1, 2002.  We will
first compute the geographic part of C, then use this in
approximating C by both methods described above.  The longitude of
Decatur is about 89 degrees (west) longitude.  Decatur happens to
be in the U.S. Central time zone;  the line of longitude that lies
in this time zone and is evenly divisible by 15 is the 90 degree
line.  Thus we are one degree east of this line, since (west)
longitude increases as you go terrestrial west.  Thus the geographic
part of C is plus 4 minutes (since (1 degree) x (4 minutes per
degree) = 4 minutes).
     If we look up the date part in the table on pp. 544-545 of
reference II-3, we note that the table is for the year 2001, but
below the table it says we have to subtract 1 minute from the values
in the table to get the values for the year 2002.  To make things a
little easier in case we will want to do this many times, we
subtract the 1 minute from the geographic 4 minutes we were going
to add, so that to get values for 2002 at Decatur, Illinois, we will
always have to add 3 minutes to values in the table.  Now under
July 1 we find 18:36, meaning 18 hours, 36 minutes, and adding the
3 minutes gives C = 18 hours, 39 minutes (approximately).
     Now if we use our other method to compute the date part of C,
July is 10 months past September, and 2 hours times 10 is 20 hours.
July 1 is 20 days before July 21, and 4 minutes times 20 is 80
minutes (or 1 hour, 20 minutes), and subtracting 1 hour, 20 minutes
from 20 hours (noting that 20 hours is the same as 19 hours, 60
minutes after we borrow 60 mimutes) we get 18 hours, 40 minutes.
Finally, adding the geographic correction of 4 minutes gives C = 18
hours, 44 minutes.  Thus in this example the difference between the
value obtained using the table and our rough approximation is only
5 minutes, which is not very much.


B.  Determining the transit time of your object

     Now we can determine transit times using Fact 3, rewritten in
the form T = R - C.  Remember that we may have to add or subtract
24 hours as needed to get a result that is positive and less than
or equal to 24 hours.  We can then convert this 24-hour time to
the usual 12-hour time by subtracting 12 hours if necessary, and
if daylight time is in effect we will have to add 1 hour to get the
actual local time.
     As an example, suppose we want to know what time xi Lup
transits on July 1, 2002 in Decatur Illinois.  As noted in section
VIIH, the right ascension of xi Lup is 15h 57m (that is, 15 hours,
57 minutes).  Let's use the value of 18 hours, 39 minutes we
computed for C in the last section.  We compute
T = R - C = 15 hours, 57 minutes - (18 hours, 39 minutes) =
(-3 hours), 18 minutes;  adding 24 gives T = 21 hours, 18 minutes.
Subtracting 12 hours gives 9:18 p.m.  In July we are on daylight
time, so adding 1 hour gives the final result that on July 1, 2002
in Decatur, Illinois xi Lup transits at (about) 10:18 p.m. Central
Daylight Time.
     By the way, the reason we put the -3 hours in parentheses in
the above calculation is to indicate that the minus sign affects the
hours, but not the minutes;  we have negative 3 hours, but positive
18 minutes.  When we do not use parentheses in this way, the minus
sign is intended to control the entire quantity after it;  for
example, a declination of -27 degrees 14' means -27 degrees plus
-14 arcminutes, so the object would be 14 arcminutes farther south
than an object with declination -27 degrees.
     Now some of you may be asking, how can we determine what time
an object will rise and what time it will set?  This information is
usually not as useful as the transit time, and when you see the
answer you may wish you had not asked the question, but for those
who may be interested we will now put the answer and its
justification down in square brackets;  you may skip this if you
are not interested, since it gets sort of heavy.  We should observe
that the actual time you will see an object depends on more than
just the results below;  it depends on your topography, and it is
also affected a little by the atmospheric bending of light rays,
which may make an object visible even when it is a little below the
horizon.
[Suppose you want to observe an object which is neither at the
north celestial pole nor at the south celestial pole.  Assume also
that your latitude is greater than 0 degrees but less than 90
degrees, so you are terrestrial north of the Earth's equator but
not at the Earth's north pole.  Also suppose that you have already
computed the transit time T for your object.  To get the 24-hour
approximate local standard rising and setting times (if any), first
let dec be the declination of the object, let lat be your latitude,
and attempt to compute the quantity H (which stands for half-time)
as follows:
H = (invcos[tan(abs(dec))tan(lat)])/15.
Here abs(dec) means the absolute value of dec, and tan is an
abbreviation for the tangent function, which is a trigonometric
function.  If tan(abs(dec))tan(lat) is greater than 1, then H is
not defined since you can't take the inverse cosine of a number
which is greater than one (and your calculator will give an error
message or some weird answer like a pair of numbers), but if
tan(abs(dec))tan(lat) is less than or equal to 1, then H will be
defined and you can compute it.  Now there are four cases to
consider.

Case 1:  If dec is negative and H is not defined (because we tried
to compute the inverse cosine of a quantity which is bigger than 1),
then the object will never rise;  it will always be below the
horizon.
Case 2:  If dec is negative and H is defined, then the object will
rise at approximately time T - H (24-hour local standard time), and
the object will set at approximately time T + H.
Case 3:  If dec is greater than or equal to zero and H is not
defined, then the object will never set;  it will always be above
the horizon.
Case 4:  If dec is greater than or equal to 0 and H is defined, then
the object will rise at approximately time T - (12 - H) (24-hour
local standard time), and the object will set at approximately
time T + (12 - H).

     Now if you think that was hairy, wait until you see the
proof!  By the way, as will be seen in the proof, the reason for
the word "approximately" is that the Earth takes only about 23
hours and 56 minutes to make a full rotation on its axis (as noted
at the end of section IIIC).  We will not worry about this small
discrepancy.
     Let's try Case 2.  Thus we assume dec is negative and H is
defined;  we want to show that the object will rise at time T - H
and set at time T + H.
     Draw a circle to represent a side view of the celestial sphere,
mark the center as point E (for Earth), mark the top point of the
circle N (for north celestial pole), and draw a line from N through
E and on down to the bottom of the circle (this line is the extended
axis of the Earth).  Next mark a point P on the lower left part of
the circle to represent the position of the object when it is on
your meridian, draw the line segment EP (which will have length r,
the radius of the celestial sphere), and drop a perpendicular from P
to the extended axis of the Earth, marking the point where it hits
with the letter C.  We now have a right triangle PEC.
     We claim now that angle PEC equals 90 degrees - abs(dec).  To
show this, imagine a point Z on the circle directly left of E and
the resulting line ZE (but to avoid clutter don't put Z or ZE in
your picture if you don't have to;  just imagine them).  Then the
negative of angle ZEP is dec, by definition of declination.  Thus
angle ZEP equals abs(dec).  Also, angle ZEC is a right angle, so
angles ZEP and PEC must add up to 90 degrees, so we have shown that
angle PEC equals 90 degrees - abs(dec), as we claimed.
     Now we claim that PC equals rcos(abs(dec)), and EC equals
rsin(abs(dec)).  To verify the first of these two facts, note that
angle EPC must equal abs(dec), since the angles in a triangle must
add up to 180 degrees, and the other two angles are 90 degrees and
(90 degrees - abs(dec)).  Then by the definition of the cosine of
an angle in a right triangle as being the adjacent side divided by
the hypoteneuse, cos(abs(dec)) equals the cosine of angle EPC, which
equals PC/r, and multiplying the equation by r gives the first part
of the claim.  Likewise, by the definition of the sine of an angle
in a right triangle as being the opposite side divided by the
hypoteneuse, sin(abs(dec)) equals the sine of angle EPC, which
equals EC/r, and multiplying the equation by r gives the second
part of the claim.  We will squirrel these facts away until we need
them later.
     Now we bring you, the observer, into the proof.  Consider the
plane which is tangent to the Earth at your location, that is, it
goes through your location and just grazes the Earth there.  When
you look towards the horizon in any direction, you are looking along
this plane.  Draw a line that hits the line segment PC at a point Q,
passes through E, and hits the circle twice.  This line represents
an edge-on view of the plane;  the plane is perpendicular to your
paper.  Then...now wait a minute, someone wants to know how we know
the line hits the line segment PC--couldn't it pass left of P (so
that the plane passes above the object in the sky)?  If you noticed
that little discrepancy, you may have the blood of Einstein in your
veins.
     Here is how we know that the line hits the line segment PC.
Remember that we assumed that H is defined.  This says that
tan(abs(dec))tan(lat) is less than or equal to 1.  This in turn says
that tan(lat) is less than or equal to 1/tan(abs(dec)); but by a
trigonometric identity,
1/tan(abs(dec)) = cot(abs(dec)) (where cot is an abbreviation for
the cotangent function), and by another trigonometric identity, this
equals tan(90 degrees - abs(dec)), which we already know is the
tangent of angle PEC.  Summarizing the preceding sentence,
tan(lat) is less than or equal to the tangent of angle PEC, but
since the tangent of an angle gets bigger as the angle gets bigger
(at least for angles between 0 degrees and 90 degrees, like the ones
we are dealing with), this says that the latitude is less than or
equal to angle PEC.  Now the angle our new line makes with the north
half of the extended axis of the Earth is just the distance between
the north celestial pole and the terrestrial north point on your
horizon, which by Fact 1 in section IIID says this angle is just
your latitude.  But this says that angle QEC equals your latitude,
so angle QEC is less than or equal to angle PEC, and this implies
that the new line hits line segment PC, which we claimed.
     But this brings up an interesting point--suppose we were
looking at case 1 instead of case 2, so H was not defined.  Then
we can rerun the argument of the preceding paragraph, with every
"less than or equal to" replaced by "greater than", and this says
that the new line passes to the left of point P, so the object is
below the plane of your horizon even though the object is at its
highest point (since it is on your meridian), so the object will
always be below your horizon, and will never rise.  We have just
proved case 1!
     Now getting back to case 2, QEC is a right triangle.  By the
definition of the tangent of an angle in a  right triangle as being
the opposite side divided by the adjacent side, we have that
QC/EC equals the tangent of angle QEC, which is the tangent of the
latitude.  Therefore QC equals EC times the tangent of the
latitude.  Substituting in the value of EC that we determined four
paragraphs ago, we have QC = rsin(abs(dec))tan(lat).
     Now we are going to need another picture.  Since the object is
revolving around the extended axis of the Earth (or so it appears to
us), it is moving in a circle with center at C.  Imagine looking
down on the circle from the Earth, and draw this circle, marking C
at the center and P at the top;  also draw the line PC, and mark the
location of Q on this line.  Remember that the last line we drew in
the previous picture represents an edge-on view of a plane;  this
plane cuts the circle in a line segment that is perpendicular to
PC, so in the new picture draw this as a line through Q
perpendicular to PC, with its ends on the circle.  Mark the right
point where the line hits the circle with an S; this represents
the point on your horizon where the object sets.  Draw the line SC,
and note that SCQ is a right triangle with with right angle at Q.
By the definition of cosine, the cosine of angle SCQ equals QC/SC,
but since SC and PC are equal (being radii of the same circle),
this equals QC/PC. Now we fill in the values we have already
computed for QC and PC, cancel out the factors of r, and use a trig
identity that says that the tangent of an angle equals the sine of
the angle divided by the cosine of the angle.  The work looks like
this:
The cosine of angle SCQ = (rsin(abs(dec))tan(lat))/(rcos(abs(dec))
= tan(abs(dec))tan(lat).  Thus by definition of inverse cosine,
angle SCQ = invcos[tan(abs(dec))tan(lat)].
     But angle SCQ divided by 360 degrees equals the fraction of
the circle that is covered by the object in going from its high
point in the sky to its setting point, which approximately equals
the fraction of a 24-hour day that is takes the object to go from
its high point to its setting point.  (We said "approximately"
because, as noted at the end of section IIIC, the object takes only
about 23 hours and 56 minutes to make a complete circle.)  Thus the
approximate number of hours it takes the object to go from its high
point to its setting point is obtained by taking the number of
degrees in angle SCQ, dividing by 360, then multiplying by 24
hours;  this gives (invcos[tan(abs(dec))tan(lat)])/15, which is
precisely the quantity we have been calling H!  Now noting that it
takes the object the same amount of time to go from its rising point
to its high point as it takes it to go from its high point to its
setting point, we get T - H for the approximate rising time and
T + H for the approximate setting time, and this completes the
proof of Case 2.
     Now, what about cases 3 and 4?  For these, I will utilize a
procedure known politely in mathematical circles as "hand-waving",
and just say that the details are similar to those in cases 1 and 2;
the main difference is that whereas in case 2 the horizon plane cuts
off the top part of the object's orbit (the part where the object is
above the horizon), in Case 4 the horizon plane cuts off the bottom
part of the object's orbit (the part where the object is below the
horizon).  With H as half the time the object is below the horizon,
and with an approximate 24 hours to make a full circle, the
approximate half time the object is above the horizon is 12 - H, and
the results for Case 4 follow.  The arguments for Case 3 are similar
to those for Case 1.
     We will close this topic by asking and answering the
question, "What are the approximate rising and setting times for xi
Lup on July 1, 2002 in Decatur, Illinois?"  We have already computed
the transit time as T = 21 hours, 18 minutes.  Recall that the
declination for xi Lup is -34 degrees (so dec = -34), and we also
need the fact that the latitude of Decatur is about 40 degrees (so
lat = 40).  We will now compute H;  I will show the digits appearing
with my calculator so you can follow along, although all but about
the first two digits are garbage, since our given information is not
accurate enough to support more digits than that.
H = (invcos[tan(abs(dec))tan(lat)])/15
= (invcos[tan(abs(-34))tan(40)])/15
= (invcos[tan(34)tan(40)])/15
= (invcos[(.674508516842)(.839099631177)])/15
= (invcos[.565979847708])/15.
We pause to note that the number of which we want to take the
inverse cosine is not greater than 1, so H will be defined.
Continuing, we get
H = (55.5296386184)/15 = 3.70197590789.
Multiplying the fractional part of H by 60 minutes gives H to be
about 3 hours, 42 minutes.  T - H is then 21 hours, 18
minutes - (3 hours, 42 minutes) = 20 hours, 78 minutes - (3 hours,
42 minutes) = 17 hours, 36 minutes, so xi Lup rises at 6:36 p.m.
Central Daylight time.  Also, T + H = 21 hours, 18 minutes +
3 hours, 42 minutes = 24 hours, 60 minutes = 25 hours; since this
is greater than 24, we subtract 24 to get 1 hour.  Thus xi Lup sets
at 2:00 a.m. Central Daylight Time.  If you got through all this,
you get another gold star.]
     Although the main use for our equation R = T + C will be to
find transit times, there are other possible uses as well.  For
example, for a given time, date, and geographical location, we can
use the equation directly to determine R, which is the right
ascension of whatever objects are transiting at that time.  Another
application would be this:  If we have a certain time of day we
like to observe, and we know the right ascension of an object we
would like to study, we can find a date when the object will
approximately transit at that time.  For better accuracy, we would
also need to know our longitude, since that is involved in a minor
way in C.
     Here is an example to illustrate this.  Suppose we would like
to study the Lagoon Nebula at 10:00 local daylight time, and we
would like to know on what date(s) in 2003 it will approximately
transit at that time.  Suppose also we are viewing from Decatur,
Illinois, where the geographical component of C is 4 minutes, as
computed earlier.  Our approach will be to use the equation in the
form C = R - T, determine the date component of C, and get a date
from that.  From chapter XII (or reference II-13 or elsewhere) we
find that the right ascension of the Lagoon Nebula is 18h 3.8m, so
this will be the R in the equation.  10:00 local daylight time is
the same as 9:00 local standard time, and the corresponding 24-hour
local standard time is 21h.  We compute C = 18h 3.8m - 21h.  Seeing
that we will get a negative number, we see that we will have to add
24h at some stage, and it may be less confusing to add it to R
first, so we compute C = 42h 3.8m - 21h = 21h 3.8m.  Since C has
the 4 minute geographical part in it, we subtract this out to get
the date part of C, namely 20h 59.8m.  Now we have two ways to get
a date.  If we use the table on pp. 544-545 of reference II-3, we
look for an entry close to 20:59.8, after first mentally
subtracting 2 minutes from all entries to get the values for 2003
(as explained below the table);  we find the (modified) entry 21:00
for August 7, so the desired date is August 7, 2003.  If we do not
have the table handy we can get a slightly less accurate answer by
first noting that 2 goes into 20h 59.8m ten times, with 59.8
minutes left over.  Ten months after September is July.  Now 4 goes
into 59.8 approximately 15 times, so we have do add 15 days to July
21, getting July 36 (or, since July has 31 days, we subtract 31 to
get August 5).  Thus our answer this time is August 5, 2003, which
is only 2 days off from the previous answer.  Now if we change our
mind and decide to observe at midnight daylight time, viewing two
hours later moves the date about a month earlier, so we would
observe around July 7 (or July 5).  If we want to observe the
Lagoon Nebula before it transits, then we would move the viewing
time earlier by the number of hours before transit we want to
observe, or else move the viewing date earlier by about one month
for each two hours before transit we want to observe.

     
C.  Using the Big Dipper as a clock

     To use the Big Dipper as a clock, think of Polaris as being
the center of the clockface, and the pointer stars in the bowl of
the Big Dipper as being the hour hand.  Think of time 0 (or time
24 if you prefer) as occuring when the hour hand is straight up,
that is, when the pointer stars are transiting, and think of the
other times (1 to 23) being distributed evenly around the clockface
counterclockwise.  Thus, for example, time 18 occurs when the hour
hand is pointing straignt to the right of Polaris.  Thus this is a
24 hour clock, and it runs backwards because the pointer stars
circle around the north celestial pole counterclockwise.  We will
call the time read off this clock Big Dipper time.  The relationship
of Big Dipper time to ordinary time is given by the following:

Fact 4.  Local 24-hour standard time approximately equals
Big Dipper time + 11 - C, where C is the correction computed in
section A of this chapter.

     Why is this so?  Well, when Big Dipper time is 0, then the
pointer stars are transiting, and by Fact 3 of section A of this
chapter, the local 24-hour standard time approximately equals
11 - C since the right ascension of the pointer stars is close to
11h.  Thus Fact 4 is true at Big Dipper time 0.  But as local
24-hour standard time marches on, Big Dipper time marches on at
about the same rate, since after 24 hours the pointer stars have
completed one complete trip (plus a little more).  Thus the two
sides of the equation in Fact 4 are increasing at approximately the
same rate, so they remain approximately equal.
     Consider the following question:  An observer in Decatur,
Illinois, observes during the night of July 1-2, 2002 that the
pointer stars are pointing directly to the left of Polaris.  What
time is it (approximately)?  To answer this, recall that in section
A of this chapter we found by one of our methods that C is 18
hours, 39 minutes for July 1, 2002 in Decatur.  Even if it turns
out we have passed midnight and gone into July 2, the change in
C will be only about 4 minutes, and since we are getting only an
approximate time anyway, we may as well stick with 18 hours, 39
minutes for C.  Since the hour hand is pointing straight to the
left, the Big Dipper time is 6 hours.  Thus we can approximate the
approximate local 24-hour standard time as 6 + 11 - (18 hours, 39
minutes) = 17 hours - (18 hours, 39 minutes).  Now adding 24 hours
to the 17 hours to avoid a negative result and borrowing 60
minutes, we have 40 hours, 60 minutes - (18 hours, 39 minutes) =
22 hours, 21 minutes.  This gives 10:21 p.m. local standard time,
so it is about 11:21 p.m. local Daylight time.
     We close this chapter on time by mentioning several other
concepts of time, since you may see these mentioned in other
references, but we will not discuss them further since to do so
might be more confusing than helpful.  Your SIDEREAL TIME is just
the right ascension of any object or location that happens to be
on your meridian at the time.  UNIVERSAL TIME (abbreviated U.T.)
is the time at Greenwich, England;  it is also called GREENWICH
MEAN TIME, and is 24-hour time which is 5 hours ahead of (U.S.)
Eastern Standard Time.
     We also mention here the concept of JULIAN DAYS, which is
sometimes used in astronomy.  These are numbered consecutively from
noon on January 1, 4713 B.C.  The fact that a Julian day runs from
noon to noon instead of from midnight to midnight means that the
Julian day does not change during a night of observing.  There are
tables (for example appendix 11 in reference II-3) to help you
determine Julian days, but you can also compute them directly
without a table as follows.  The Julian day which starts at noon on
January 1, 2003 is 2,452,641.  For the Julian day starting at noon
on any date after January 1, 2003, you add to this number the number
of days from January 1, 2003 to your date, and for the Julian day
starting at noon on any date before January 1, 2003, you subtract
the number of days from your date to January 1, 2003.  For example,
the Julian day which begins at noon on March 17, 2004 (and ends at
noon on March 18, 2004) is 2,452,641 + 365 (to get to January 1,
2004) + 31 (to get to February 1, 2004) + 29 (to get to March 1,
2004;  remember that 2004 is a leap year) + 16 (to get to March 17,
2004) = 2,453,082.  One reason that Julian days are useful is that
you can get the number of days between two given Julian days by
subtraction, without having to worry about different calendars, or
the number of days in a month, or leap years.  This makes them
especially useful for studying variable stars;  for more
information on variable stars, you can check the website of the
American Association of Variable Star Observers, which is
http://www.aavso.org.

     
Chapter IX.  Viewing conditions


A.  Light pollution

     The term "Light pollution" refers to artificial light that
makes it difficult to see objects in a telescope, especially faint
objects.  Street lights and house lights are often the worst
culprits.  If you find your viewing experience diminished by light
pollution, there are several things you can do.  One is to take
your telescope to a location away from city lights.  If you do this,
try to avoid jostling the optical tube any more than necessary,
since mirrors and other internal parts can be jarred out of
alignment.  If that happens, you may be able to fix the problem by
touching up the collimation as described in your instruction
manual, but if not, then you may have to call the place you got the
telescope or the manufacturer for advice (and you will probably
need to arrange to have the telescope sent in for repair).  The
safest way to transport the optical tube may be to put it in the
box it came in, using the padding that came with it.  An easier (but
less effective) thing to try would be to time your viewing so that
the thing you want to observe is in a less light- polluted part of
your sky, that is, a part of the sky that looks darker.  You could
also try getting a light pollution reduction (LPR) filter;  I have
the type that screws onto an eyepiece, but I don't use it much.  It
is sometimes helpful, but it can create strange colors when you are
viewing bright objects, and it blocks out some light from the
object itself.
     Although high magnification can cause objects to appear
fuzzier and less distinct, as well as increasing problems with
jitter (that is, even touching the focus knob may cause an object
to rapidly vibrate for a few seconds), high magnification has at
least one advantage (beyond the obvious one of making extended
objects look bigger):  It can help reduce the effects of light
pollution.  When you increase magnification, although the eye
thinks it is still seeing the same size disk of space, this disk
contains less area of space, so it contains less total background
light, so the average background light over the disk is less, so
the sky looks darker.  Still, we should not get carried away;  many
times you get a much better view with lower magnification and a
wider field, and the wider field makes searching easier.
     As noted in Chapter I, one purpose of this document is to show
that it is possible to see many beautiful things even in the
presence of moderate (or worse) light pollution.

     
B.  The moon

     The moon is an interesting object to study, but its light can
make it difficult to see other objects, especially faint ones.  An
LPR filter is of little help for moonlight;  these filters are
designed to filter out artificial light.  Although you may be able
to get good views of planets and double stars in the presence of
moonlight, if you want to view faint objects about all you can do
is restrict your viewing times to times when the moon is not up, or
when it is only a thin crescent (which happens when it is close to
being new).  When the moon is full it rises about sunset and sets
about sunrise, so that is a particularly poor time to view.  Each
night thereafter, however, it rises about 48 minutes later, so
2 or 3 days after it is full you may have a brief time to do some
viewing before the moon comes up.  Note that if you were to look at
the moon at the same time (say 9:00 p.m.) every night, it would
seem to move counter to the sun, that is, it would get farther east
every night.  You can understand why this is so by imagining you
are looking down on our solar system and watching the Earth rotating
counterclockwise on its axis while the moon moves counterclockwise
around the Earth.  On pp. 350-353 of reference II-3 there are tables
that give all the dates for new, first quarter, full, and last
quarter moon through the year 2010.  

     
C.  Other problems

     Some other viewing problems are the following.

1.  The air may lack transparency due to clouds, haze, smoke, or
high humidity.  In some cases you may still be able to see bright
objects under these conditions, for example, thin, high cirrus
clouds may let quite a bit of light through.  There is not much
you can do about this except restrict your viewing to bright
objects and wait for a better night for faint objects.

2.  There could be air currents inside or very close to the optical
tube.  These could be caused by someone having recently touched the
optical tube with his warm hand or by people standing close to the
tube.  In these conditions there may be rapid changes in the image
you see in the telescope;  for example, the moon may shimmer and
stars may appear to be double when they are not.  To prevent this,
try to avoid touching the tube for about 20 or 30 minutes before
you use it (you can move it by holding the star diagonal, or the
handle if there is one), and don't let anyone stand in front of or
below the tube while you are observing.

3.  The lower atmosphere could be disturbed, most likely by heat
waves from the ground or buildings.  This can cause blurry images
and loss of contrast and fine detail.  To avoid this problem,
avoid viewing over buildings or asphalt parking lots;  broad
hilltops or open grassy fields are ideal.  Alternatively, you can
do your viewing in the early morning when the surroundings are
uniformly cool and the seeing conditions are usually better.

4.  The upper atmosphere could be disturbed, causing quick little
movements (although the image may still be sharp with fine detail
visible).  About all you can do about this is put up with it or
wait for a better night.

     Most of the things in this section were borrowed from the
user's manual for my Celestron G-9.25 telescope.  (Celestron has
changed the name of this telescope, but as of 7/2/04 they were
advertising only one non-computerized 9.25 inch Schmidt-Cassegrain
telescope, and it appears to be the same one.)  If I may add a
comment of my own, although perfect seeing conditions are wonderful,
they are pretty rare in some places, and if you always wait for
perfect conditions you may not get much viewing done.  Thus you may
want to try to view your object even if conditions are less than
ideal, and then if you think the object may be an impressive one,
you can try again some other time when the viewing conditions are
better.  At least, it will be easier to find it the second time
since you will have found it once before.  If you have practiced
the ideas in this document, you will probably have the ability to
find things under conditions that may make others throw up their
hands.

     
Chapter X.  Buying a telescope


A.  Types of telescopes

     The best advice I can give you if you are thinking about
buying a telescope is to go to http://www.celestron.com and read
the section on "Telescope basics".  It has lots of information
which is not biased towards Celestron products;  this includes a
thorough discussion of the three main types of telescopes:
Refracting telescopes, which use lenses and are what most people
think of when they hear the word "Telescope", reflecting
telescopes, which use mirrors instead of lenses, and Schmidt-
Cassegrain (or related) telescopes, in which the light comes in
the far end, is reflected from a mirror at the near end back to a
smaller mirror at the far end, and then is reflected back to an
opening in the near end.  The only telescopes of any size I have
used are Schmidt-Cassegrain telescopes, and I have been more than
satisfied with them, although the other types have their good
points too.
     For kids who may not be old enough to handle and appreciate a
serious telescope, it is possible to buy a smaller kid's-type
telescope for much less money.  For example, I used to have a
10-20-30-40x refractor with a small tripod, and I had a lot of fun
with it looking at the moon and bright planets.


B.  Aperture

     One of the most important features of a telescope is its
APERTURE, which is the diameter of the big lens at the front of
the telescope.  This is because the light-gathering power of the
telescope, which determines how much you can see, is proportional
to the square of the aperture.  This is because the light-gathering
power is proportional to the area of the big lens at the front, and
this area is proportional to the square of the radius (remember
A = pi times r squared?), and is thus also proportional to the
square of the diameter.  Thus, for example, a 6 inch telescope has
nine times the light-gathering power of a 2 inch telescope (since
(6/2) squared equals 9).
     
C.  Beware the large-magnification-small-aperture trap

     Sometimes you will see telescopes advertised as having high
magnification, but when you look at them, they have small aperture.
A good rule of thumb is that magnifications greater than 50 or 60
times the number of inches in the aperture are not useful (unless
you like looking at indistinct fuzzy blobs), and even those
magnifications are not too useful unless the viewing conditions are
excellent.  About 20-35 times the aperture would be pretty good for
a variety of conditions.  Thus if someone wants to sell you a
telescope with an advertised high magnification, you would be well
advised to not even consider it until you find out what the
aperture is.

     
D.  Portability

     Portability is a critical factor since you will need to take
your telescope outside to observe and bring it back in afterwards,
and you do not want to hurt your back or some other part of your
anatomy in the process.  Thus you should find out how much a
telescope weighs before buying it, but remember that you can
usually alleviate the weight problem by moving the telescope in
pieces as described in section VIIA above.  The tripod is the most
awkward piece to move because it is spread out (and folding it
every time is usually too much of a hassle if it can be easily
avoided, but note the end of this section), but the optical tube
is the most critical part.  The optical tube for my non-computerized
Celestron Schmidt-Cassegrain 9.25 inch telescope weighs about 20
pounds, but fortunately it has a handle.  The optical tube for my
Celestron non-computerized Schmidt-Cassegrain 11 inch telescope
weighs about 27 1/2 pounds, and the optical tube for my Celestron
non-computerized Schmidt-Cassegrain 5 inch telescope is much
lighter.
     As a general rule of thumb in buying a telescope, once you
have decided what kind of telescope you want, you should probably
buy the largest aperture you can subject to conditions of expense
and portability.
     There is something more to this portablility question than
whether you will be able to take your telescope out and set it up,
namely, will the hassle involved make you less likely to use the
telescope?  For my 11 inch Schmidt-Cassegrain telescope, the
procedure I have settled on is this:  (1)  Take out the tripod,
spread the legs, and aim it roughly north (using a metal piece on
it as a guide).  (2)  Take out the mount, install it, and adjust
the legs of the tripod to level the mount (using the bubble level
in the mount).  Also correct the northward alignment of the fixed
arm if necessary.  (3)  Take out the accessory tray and one
counterweight.  Install the accessory tray.  (4)  Take out the
other two counterweights and install them.  (5)  Take out the
optical tube and install it.  Install the other counterweight and
replace the counterweight safety screw.  (6)  Take out the
right-angle finderscope and star pointer, and install them (except
if the sun is still up, wait until later if you have a Telrad star
pointer to install it to avoid any chance of pointing the star
pointer at the sun and damaging it).  (7)  Take out the binoculars,
night vision flashlight, extra eyepieces, barlow lens, and index
cards with the objects you will be observing (along with the drive
motor power pack and controller if you will be using them).
     Now this process takes 20-30 minutes, with the reverse process
taking somewhat less time when you are done observing.  This time
and effort does not bother me one whit, but it might seem like a
lot of trouble to someone who is less dedicated, especially
someone who is just starting out, such as a teenager.  Someone
just learning to stargaze may greatly prefer the process I used
with my 5 inch Schmidt-Cassegrain telescope, which I left set up
all the time, namely (1)  Take out the tripod, mount, optical
tube, finderscope, and star pointer together, aim the fixed arm to
the north, and level the tripod.  (2)  Take out the binoculars,
night vision flashlight, extra eyepiece, barlow, and index cards.
The point here is that if due to the reduced setup hassle, the 5
inch telescope is going to be used more often than the 11 inch
telescope, then in that sense the 5 inch telescope is more
powerful than the 11 inch telescope (besides being less expensive).
One should also consider whether the one who will be using the
telescope will be able to set it up himself or herself, since if he
or she can, then he or she will be more likely to consider the
telescope to be HIS or HERS, and thus will be more likely to use
it.  On the other hand, if the user of the telescope becomes
really serious about stargazing than he or she may want a larger
telescope later, so if you start with a smaller one you may end up
buying two telescopes.  Still, even a 5 inch telescope is a fine
instrument that can give a lifetime of pleasure to someone who
takes good care or it;  the difference in what you can see between
a five inch telescope and binoculars is far greater (it seems to
me) than the difference between an 11 inch telescope and a 5 inch
telescope.  These are judgments to be made by the purchaser of the
telescope, keeping in mind who will be using it.

     
E.  Where to buy a telescope and related accessories

     If there is a store near you that specializes in astronomical
equipment, that would be an ideal place to go telescope shopping.
That way you can examine telescopes before buying one, and you can
talk to people who know about telescopes.  My second choice would
be to try a reputable web site that specializes in selling
astronomical equipment.  Telescope manufacturers often have lists
of astronomical retailers of both the brick-and-mortar and online
varieties;  see, for example, http://www.celestron.com or http://
www.meade.com.  There are also publications, such as Sky and
Telescope magazine, that have useful advice and reviews.  The web
site I have used to purchase telescopes and accessories was at
http://www.astronomyquest.com, and I have been completely satisfied
with their products and services.  You also used to be able to call
them on the phone and they would be glad to talk to you, but I am
sorry to report that they have probably gone out of business, since
their phone has been disconnected and the web site no longer sells
stuff.  I do not have any personal experience with retailers other
than the former astronomyquest.com and so cannot comment on them
(except that I did buy references II-4 and II-13 through the
website in reference II-11;  that worked out well).


Chapter XI.  Taking care of your equipment


     The best advice I can give you about taking care of your
equipment is to read and follow the instructions in your
instruction manual.  The second best advice is to remember that
your telescope is a not a toy, it is a fine instrument;  if you
treat it well it should give you many years of good service.  The
third best advice is to contact the place you got the telescope
if you have questions that are not answered by the instruction
manual.  They will be glad to talk to you, in part because they
no doubt hope they can sell you more stuff later.  The fourth best
advice is to not get too excited about a little dust on the big
lens of your telescope.  These lenses seem to be dust magnets;
sometimes I think that I could get rid of all the dust in my house
by taking off the lens cap of the telescope, because then all the
dust would immediately go to the lens (well, not actually, but
sometimes it seems that way).  One time I shone my flashlight on
the big lens outside at night and was startled at the dust I saw;
when I asked someone about this, they said "How often do you look at
a flashlight through the telescope?"  The point is well taken;  you
will not actually see the dust since your telescope will be focused
far away from it.  Still, if dust (or smears or the like) get
excessive, your viewing will be enhanced if you clean the lens.
Again, follow the instructions in your user's manual, remembering
that "Gentle" is the key word.  After all, dust is temporary, but a
scratch is forever (or at least until you send the telescope in for
repair).  If you follow the instructions carefully, you should be
fine.  By the way, the instructions often say to gently remove dust
with a soft brush (brushing from the center out, not in circles);
this should be a very soft brush.  One good brush for this is part
of something called a lenspen, which Celestron sells for about a
dollar.
     By the way, sometimes when you are observing your big lens
may become fogged, which will drastically degrade your image.  The
instructions for one of my telescopes says that if this happens,
and you want to continue observing, you should point your
telescope downward and wait for the fog to dissipate (but be
careful that you don't bang parts of the mount together or even
dump the optical tube).  Since they didn't recommend trying to
remove the fog with a cloth or something, I gather that that would
be a bad idea, possibly causing smearing or even scratching of the
lens.  What I usually do is put the lens cap on for safety, bring
the optical tube inside, take the lens cap off, and wait for the
fog to dissipate.  Then I put the lens cap back on and either take
the optical tube back outside or give up for the night.
     If your telescope is a Schmidt-Cassegrain, it may need
collimating once in a while;  this corrects the tilt of the
secondary mirror, and is most likely to be needed if the telescope
has been bumped hard or dropped (which is a major no-no!).  I don't
know whether other types of telescopes need or can benefit from a
similar procedure, but if yours does, there will be a section in
your instruction manual on how to do it.  In September of 2005 I
first worked up the nerve to try this with my 11 inch
Schmidt-Cassegrain after I noticed that a defocused bright star
image was significantly nonsymmetrical.  This takes quite a bit of
patience, and it is easier if there are two people doing it (one to
look through the telescope while the other works the collimation
screws).  The result in my case was considerably improved viewing.
One thing that was quite noticable was that before collimation, I
could not focus some stars down to a point;  there were always two
images, which I thought was a problem with my eyes.  Collimation
fixed the problem, and also made it a lot easier for me to split
certain double stars.  I would like to mention one caution,
however:  Don't do an Arnold Schwarzenegger on the collimation
screws!  By this I mean, if you want to tighten one of the screws
(by turning it clockwise), and it really does not want to turn this
way, then do not force it too much (since you could strip the head
of the screw), but rather loosen the other two screws a little
instead, which should have a similar effect.  You don't want the
screws to be too loose either, but that would be a lesser evil.
After you have collimated your telescope you will probably need to
realign your finderscope since the collimating will have changed
the aim of the telescope slightly.  Finally, as one of my
instruction books said, don't be intimidated out of doing a
collimation if needed, since it can pay considerable dividends.  If
I can do it, so can you!


Chapter XII.  Some beautiful objects for small telescopes

     I viewed all these objects (except where otherwise noted) 
using a Celestron Schmidt-Cassegrain 9.25 inch telescope in the
presence of moderate to heavy light pollution (as described in
Chapter I).  I also viewed some of them using a Celestron
Schmidt-Cassegrain 5 inch telescope;  although they looked better
in the 9.25 inch telescope (and even better in an 11 inch
telescope, which I acquired in July 2005), many of them also looked
good in the 5 inch telescope.  Two notable exceptions are the
triple stars 12 Lyn and beta Mon;  I could see all three components
with the 9.25 inch telescope, but I could only see two components
with the 5 inch telescope (since two of the three components were
smeared together into one).  Still, if your telescope is smaller
than a 9.25 inch, do not despair;  it may well be that your viewing
conditions are better than mine, and in any event there are many
beautiful things in this list for you.  Nearly all the objects
presented in this chapter were easily visible with my telescope
under my viewing conditions, except that when several nearby
objects are listed under one heading, sometimes some of them were
harder to find, as indicated in the discussion of the objects;
they were put in partly because they were close to other objects in
the group that were easier to find.
     The format in which the information is presented is basically
that in section VIF above, except that the limitations of this
plain-text document make it necessary to use much more space to
describe the paths than you would need on your cards.  The objects
are presented in order of increasing right ascension, except that
when several objects are grouped under one heading, then under that
heading the objects are discussed in the order suggested for
viewing them.  The starting objects for the paths are either
objects discussed in section VIA, or else they are objects which
were found in specified other paths in the list.
     Let's add five more definitions to aid in describing some of
these objects:
A GALAXY is an enormous group of billions of stars, plus dust and
gas.
A GLOBULAR CLUSTER is a spherical group of several thousand old
stars.
An OPEN CLUSTER is a group of tens or hundreds of relatively young
stars.
A PLANETARY NEBULA is the remnant of a shell of gas blown off by a
star.  There are other types of nebulas too, all of which involve
gas;  in most of the nebulas in this list the gas will be
illuminated, although there also are so-called dark nebulas, where
unilluminated gas blocks the light from behind.
The POSITION ANGLE of a double star is the number of degrees you
would have to rotate counterclockwise in the sky from north in
order to point from the brighter component to the dimmer one.
Because of the reversal of angles due to reverted images, when you
are looking through your telescope you will have to rotate
clockwise.  For example, if the dimmer star is due west of the
brighter star, the position angle is 270 degrees.  Position angles
for double stars range from 0 to 360 degrees.  The position angle
of a galaxy is the number of degrees you would have to rotate
counterclockwise from north (clockwise through your telescope) to
first point along the long axis of the galaxy;  position angles
for galaxies range from 0 to 180 degrees.
     Note that for the double stars listed, for the ones I could
find in Appendix 6 of reference II-3 magnitudes for both
components are given, along with the separation (distance between
the components), but for the ones not in that table only the
combined magnitude (as discerned from the size of the dot in the
chart in reference II-4) is given, and for these latter doubles the
coordinates were measured from the chart, and so are not quite as
accurate as the coordinates for the doubles listed in the table.
     A variety of path construction types are illustrated in these
examples so you can see what kinds of paths you might like to create
and use.


1. M31--The Andromeda Galaxy (with satellite galaxy M32)

M31 (Andromeda Galaxy)    mag. 3.4   size 3.2 degrees x 1.0 degrees
0h 42.7m              41 degrees 16'              Andromeda

M32 (galaxy)              mag. 8.2   size 7.6' x 5.8'
0h 42.7m              40 degrees 52'              Andromeda

Beta Cas (W star of letter W in Cassiopeia), then

   30 degrees S, 1/4- degree (1m) W to

alpha And (2.0, with a mag. 6.0 star 3/8 degree W;  NE corner of
the Great Square of Pegasus;  note also that alpha And can be
quickly identified with binoculars or finderscope by checking for
the flat bow pointing roughly E consisting of epsilon And (4.0)
6 1/2 degrees E of alpha, delta And (3.0) 1 3/4 degrees N of
epsilon, and pi And (4.0) 3 degrees NNW of epsilon), then

   6 1/2 degrees N, 12 1/2 degrees (61 1/2m) E to

beta And (2.0, a bright orange (in telescopes) star with a mags.
8.0-7.5 ENE-WSW pair 3/4 degree W), then

   2 7/8 degrees N, 2 1/2 degrees (13m) W to

mu And (3.5, with a mag. 6.5 star 3/4 degree W), then

   2 1/2 degrees N, 1 1/3 degrees (7m) W to

nu And (4.5, with a mag. 7.5 star 1/8- degree W and a mag. 8.0 star
just NW of the mag. 7.5 star), then

   slightly N, 1 1/3 degrees (7m) W to

the center of M31.

After looking at M31, proceed as follows:

Center of M31, then

   3/8+ degree S to
   
M32 (with a mag. 7.0 star 1/4- degree SSW and a mag. 8.5 star 1/2
degree SSE).

     Viewed 10/5/02, 8-10:30 p.m.  Conditions excellent.  I watched
this thing starting when it was in the terrestrial northeast, which
is the most light-polluted part of my sky, until it climbed nearly
overhead.  The big, bright, glowing, oval-shaped center was visible
the whole time, and after a while some of the fainter glow arranged
NNE-SSW around the center became visible.  58.75x was good for
sweeping back and forth across the galaxy;  as usual, movement
makes faint things easier to detect.  94x was good too, and even
267x showed some things (like glimpses of a bright spot at the
center of the center, and a boundary of the fainter glow in some
places).  This is truly a magnificent object, and it certainly
deserves the three exclamation points that it is awarded in the NGC
catalog.
     M32 is much smaller than M31, but it is a good little galaxy,
with a bright fat-star-like center and a faint glow around the
center.  It is easy to find, even in the worst of the light
pollution.  On the star chart this appeared inside M31, but my
conditions were not good enough to see any of M31 that far out.
This galaxy received one exclamation point in the NGC catalog.
     There is another satellite galaxy nearby, namely M110, but
this is much fainter and harder to find than M32, which is why I
did not give it equal billing with M31 and M32.  What I could see
of it just looked like a small faint glowing spot.  It is 1/2-
degree N, 3/8+ degree W of the center of M31.  On the star chart it
is just outside M31.
     Incidentally, those of you with a very sharp memory may recall
that we said earlier (in section VIG) that there are 109 Messier
objects;  we get up to M110 because two objects in Messier's list
were later discovered to be the same object.


2. 65 Psc--A nice close pair of nearly equal magnitude (and
   sometimes colorful) stars

65 Psc (double star)      mags. 6.3, 6.3     separation 4.4"
0h 49.4m              27 degrees 43'            Pisces

Beta Cas (W star of letter W in Cassiopeia), then

   30 degrees S, 1/4- degree (1m) W to

alpha And (2.0, with a mag. 6.0 star 3/8 degree W;  NE corner of
the Great Square of Pegasus;  note also that alpha And can be
quickly identified with binoculars or finderscope by checking for
the flat bow pointing roughly E consisting of epsilon And (4.0)
6 1/2 degrees E of alpha, delta And (3.0) 1 3/4 degrees N of
epsilon, and pi And (4.0) 3 degrees NNW of epsilon), then

   1 2/3 degrees N, 6 2/3 degrees (31m) E to

delta And (3.0, orange, with a mag. 7.5 star 3/8 degree NE), then

   1 1/2 degrees S, 1/4- degree (1m) W to

epsilon And (4.0), then

   1 1/2+ degrees S, 2 1/2 degrees (11 1/2m) E to

65 Psc (5.0, with a mag. 7.5 star 1/2 degree E).

     Viewed 9/30/05, 7:55 p.m.  Conditions very good.  This is a
nice close pair of roughly equal magnitude stars, with a
(sometimes) yellowish star just SE of a (sometimes) bluish star.
The colors seemed to fluctuate some, depending in part on the
magnification used, sometimes appearing white.  With an 11 inch
Schmmidt-Cassegrain telescope at 318x this pair was quite
attractive.  It also splits at 112x, and even (barely) at 70x.
     I viewed this double again at 7:25 p.m. on 10/17/05 under
conditions that were good except that a one-day-past-full moon was
getting high enough to have an effect.  This time the yellowish
star looked more orangish, but again there was fluctuation in the
colors.  I would be interested in knowing what colors other people
see.
   
   
3. NGC457--The ET Cluster

NGC457 (open cluster)          mag. 6.4             size 12'
1h 19.1m               58 degrees 20'           Cassiopeia

Delta Cas (2.5, star in the letter W in Cassiopeia just E of center,
with a mag. 8.0 star 1/4- degree E and a NNE-SSW pair of mag. 7.5
stars 1/2 degree E), then

   2 degrees S, 3/4 degree (5 1/2m) W to

phi Cas (5.0, with a mag. 6.5 star 3/8 degree E and a mag. 8.5 star
1/4 degree E, making a bow pointing N), then

   1/8 degree WNW to

NGC457.

     Viewed 9/4/02, 10:30 p.m.  Conditions very good.  This is an
attractive, bright open cluster.  Reference II-13 says this cluster
"Resembles an owl's face or a stick figure with two very bright
eyes".  To me, it looks like a pteranodon.  He is flying SSE;  his
head is turned to the left (when viewed from above), with the
brightest star being the tip of his beak.  He is probably looking
for those people who dared to say that he resembles an owl's face
or a stick figure.


4. M33--The Pinwheel Galaxy

M33 (galaxy)           mag. 5.7              size 70'x41'
1h 33.9m           30 degrees 39'            Triangulum

Beta And (2.0, a bright orange (in telescopes) star with a mags.
8.0-7.5 ENE-WSW pair 3/4 degree W;  see gamma And for a path to
this star), then

   5 1/2 degrees S, 3/8 degree (2m) E to

tau Psc (4.5, with a mag. 6.5 star 1/4 degree E, a mag. 7.0 star
3/8 degree SSE of the mag. 6.5 star, and a mag. 5.5 star 1 3/8
degrees N), then

   1 3/8 degrees S, 2 1/8 degrees (9 1/2m) E to

91 Psc (5.5, with a mag. 8.5 double just N and a mag. 8.0 star 1/4
degree ENE), then

   1/8 degree S, 7/8 degree (4m) E to

a mag. 7.0 star with mag. 8.0 stars 1/8 degree SSW and 1/4 degree
NNW, then

   1 1/4 degrees N, 1/2 degree (2+ m) E to

a mag. 7.5 star with a mag. 8.0 star 1/2 degree ESE and a mag. 7.5
star 1 degree ESE, then

   3/4+ degree N, 1/4- degree (1m) E to

a mag. 7.0 star with mag. 8.0 stars 1/4 degree W, 3/8 degree NNE,
and 1/2 degree NNE, then

   slightly N, 3/4 degree (3 1/2m) E to

a mag. 8.0 star with mag. 8.0 stars 1/2 degree ESE and 3/4 degree
ENE, then

   3/8 degree E to

the center of M33 (note that the mag. 8.0 star just before M33 is
just outside M33, but its two check stars appear inside M33;  note
also that if you move another 3/8 degree E past the center of M33
you will pass out the other side of M33 and see darker sky and some
stars, including a small triangle of stars).

     Viewed 11/23/02, 6:00 p.m.  Conditions excellent.  This is a
huge ghostly white elliptical shape.  58.75x gave a good view,
although one couldn't fit it all in even with that wide field.
94x gave a good view too, but higher powers gave a less appealing
view because their fields were too narrow.  Although the average
surface brightness is low because the brightness is spread out
over such a large area, and I couldn't see the spiral arms (but
could imagine them), this is an impressive object if your viewing
conditions are not too bad.  The most striking thing about it was
the sharp contrast at a boundary between the milky white of the
galaxy (with few stars) and the darker neighboring sky (with many
stars).  I spent quite a while scanning back and forth across the
galaxy in both the E-W and N-S directions.  A couple of times I
scanned too fast and too far, became lost, and found myself back at
91 Psc (which is fortunately easy to recognize because of the mag.
8.5 double star nearby), so I had to follow the part of the path
from 91 Psc back to to the galaxy again.  Incidentally, this is
not the only time I have noticed how far you can wander from where
you want to be if you are just sort of thrashing around.
     I actually looked for M33 on three different nights.  The
first night it was low in the heavily light-polluted terrestrial
eastern part of my sky, and I did not see it at all.  The second
night conditions were better and I could faintly see it, but I was
not impressed, and thought I would not put it into this list.
Fortunately I tried once more on a clear moonless night when M33
was high in the sky and well away from the worst light pollution,
and I was glad I did not give up on this object.  Large, faint
objects like this are often more sensitive to viewing conditions
than other objects.  The first two nights were not wasted, however,
since I got used to the path (and even added a few more steps in
one shaky part) and also got used to the general shape of the
object, which set the stage for the successful third night.  Thus,
don't be afraid to try to find an object under poor conditions, but
if you are not satisfied, try again when the conditions are better.
     By the way, M33 was given one exclamation point in the NGC
catalog.


5. Gamma Ari--A pair of bright, nearly identical blue-white suns
   traveling side-by-side across the sky

Gamma Ari (double star)      mags. 4.6, 4.6     separation 7.5"
1h 53.6m              19 degrees 18'            Aries

Gamma And (2.0, see gamma And for the path to this star), then

   7 1/3 degrees S, 1 1/8 degrees (5 1/2m) E to

beta Tri (3.0;  this is the NW vertex of the skinny SW-pointing
isosceles triangle in Triangulum, with the other two vertices being
gamma Tri (4.0) 2 degrees SE and alpha Tri (3.5) 6 1/2 degrees SW
of beta), then

   11 1/2 degrees S, 1/2 degree (2+ m) W to

alpha Ari (2.0, with kappa Ari(5.0) 7/8 degree S and a mag. 8.0
star 1/2 degree NNE), then

   2 2/3 degrees S, 2 7/8 degrees (12 1/2m) W to

beta Ari (2.5, with a mag. 6.0 star 1 1/4 degrees E and a mag. 7.0
star 7/8 degree WSW), then

   1 1/2 degrees S, 1/4 degree (1m) W to

gamma Ari (3.5, with a mag. 8.0 star just E;  note that alpha Ari,
beta Ari, and gamma Ari form an easy-to-recognize broken line).

     Viewed 10/5/02, 8:00 p.m.  Conditions poor--moon just past
half.  This can be split with 58.75x, but it looks better at 94x,
and it looks really attractive at 267x.


6. Gamma And--A blazing orange star followed by a bright green
   star--Mirror, mirror, on the wall, is this the fairest double
   star of them all?

Gamma And (double star)       mags. 2.3, 5.0     separation 9.7"
2h 3.9m                42 degrees 20'            Andromeda

Beta Cas (W star of letter W in Cassiopeia), then

   30 degrees S, 1/4- degree (1m) W to

alpha And (2.0, with a mag. 6.0 star 3/8 degree W;  NE corner of
the Great Square of Pegasus;  note also that alpha And can be
quickly identified with binoculars or finderscope by checking for
the flat bow pointing roughly E consisting of epsilon And (4.0)
6 1/2 degrees E of alpha, delta And (3.0) 1 3/4 degrees N of
epsilon, and pi And (4.0) 3 degrees NNW of epsilon), then

   6 1/2 degrees N, 12 1/2 degrees (61 1/2m) E to

beta And (2.0, a bright orange (in telescopes) star with a mags.
8.0-7.5 ENE-WSW pair 3/4 degree W), then

   6 2/3 degrees N, 10 degrees (54m) E to

gamma And (2.0).

     Viewed 9/25/02, 9:00 p.m.  Conditions very good.  Even 58.75x
splits this, it looks even better at 94x, and at 267x it is
absolutely gorgeous!  The orange star is spectacular even by
itself, and it is followed acrosss the sky by a pretty bright green
star that is slightly N of straight E from it.  My personal answer
to the question posed in the title for this object is "Yes!".


7. NGC869, NGC884--The Double Cluster, with a pretty little
   orange-blue double star along the way

Eta Per (double star)      mags. 3.8, 8.5        separation 28.4"
2h 50.7m                55 degrees 54'           Perseus

NGC869 (open cluster)       mag. 5.3             size 30'
2h 19.0m                57 degrees 9'            Perseus

NGC884 (open cluster)       mag. 6.1             size 30'
2h 22.4m                57 degrees 7'            Perseus

Follow the line from gamma Cas (middle star of letter W in
Cassiopeia) through delta Cas (letter W star just E of center) on
12 degrees to eta Per (3.5, with gamma Per (3.0) another 3 1/8
degrees along basically the same line and alpha Per (2.0) another
4 3/4 degrees further on that line;  there is a mag. 7.0 star 1/2
degree SW of eta Per and a mag. 8.0 star 1/8 degree N of the mag.
7.0 star;  note also that eta Per, gamma Per, and tau Per (4.0, 
1 3/4 degrees WSW of gamma Per) form an isosceles triangle pointing
NNW (with eta Per at the NNW tip) which is easy to spot with
binoculars), then after viewing eta Per as discussed below,
continue as follows:

Eta Per (3.5), then

   5/6 degree S, 1 1/8- degree (7 1/2m) W to

11 Per (5.5, with a mag. 7.5 star 1/2- degree NNW), then

   1/2- degree NNW to

a mag. 7.5 star, then

   1 5/8 degrees (11 1/2m) W to

an E-W bow 1/2 degree long with mags. 7.0-7.0-6.5 pointing N, then
from the W star of the bow,

   1/3 degree N, 1 degree (7m) W to

9 Per (5.5, with a mag. 7.0-8.5 N-S pair 1/4 degree W), then

   1 1/4 degrees N to

the center of NGC884, then

   3/8+ degree (3m) W to

the center of NGC869.

     Viewed 9/24/02, 9:00 p.m.  Conditions excellent.  Eta Per is
a neat little orange-blue double, with the orange star much
brighter than the blue star.  This can be split at 58.75x, but it
looks better at 94x and 267x.  The two clusters are both quite
nice, with the one to the W maybe having a better center.  Even at
94x one can get the centers of both clusters into the same field,
which is a pretty sight.  This is a case where the whole is greater
than the sum of its parts.  Each cluster received one exclamation
point in the NGC catalog.


8. M34--A large, bright open cluster containing many double stars

M34 (open cluster)           mag. 5.2             size 34'
2h 42.0m             42 degrees 47'               Perseus

Gamma And (2.0, see path for gamma And), then

   5/6 degree S, 3/8 degree (2m) E to

a mag. 7.0 star, then

   1/8 degree S, 3 1/8 degrees (17m) E to

a mags. 7.5-6.0 N-S pair with a mag. 8.5 star 1/4 degree S, then
from the S (mag. 6.0) member of the pair,

   3/4 degree N, 1/4 degree (1 1/2m) E to

a mag. 7.0 star, then

   slightly S, 2 3/8 degrees (13m) E to

a mag. 7.5 star with a mag. 7.0 star 3/8 degree N and a mag. 8.5
star 3/8+ degree W, forming an isosceles triangle pointing W, then
from the N star of the triangle,

   1/3 degree N, 1 degree (5 1/2m) E to

the center of M34.

     Viewed 11/1/02, 6:00 p.m.  Conditions excellent, except at
this time M34 was in the NE part of my sky, which is the most
light-polluted part.  M34 almost fills the field at 58.75x, and
94x does not get quite all of it, but 94x gives a good view of the
central part where most of the stars are concentrated.  There are
quite a few scattered bright stars here, and many of them are
paired off (so maybe this should be called the "Dance Club
Cluster").  I counted five fairly close doubles, with about that
many that were a little wider.  There seemed to be a faint glow
behind the cluster.  This is an attractive cluster, with stars
bright enough compared to the vicinity that the cluster jumps out
at you when you first bring it into view.


9. The Pleiades (M45)--The Seven Sisters

M45 (The Pleiades)          mag. 1.2           size 1.8 degrees
3h 47.0m              24 degrees 7'            Taurus

Aldebaran, then

   7 5/8 degrees N, 11 1/8 degrees (49m) W to

The Pleiades.

     Viewed Thanksgiving 2002, 9:45 p.m.  Conditions excellent.
This is a truly stunning object!  It looks good in binoculars too,
but it is more striking with a telescope at low power, even though
you won't get it all in the field at once with a telescope.  The six
brightest stars form a small dipper opening NE, with the (short)
handle pointing E;  the far end of the bowl actually consists of
two stars.  These six stars are so bright in a telescope at 58.75x
that they just about knock your eyes out.  The brightest one, mag.
2.9 eta Tau, is where the handle and bowl come together.  There are
at least 100 stars in this cluster, and it is fun to explore by
sweeping through the cluster at 58.75x or 94x.  This cluster is
mentioned in the Bible in Job 38:31--here is the quote:  "Canst
thou bind the sweet influences of Pleiades, or loose the bonds of
Orion?"


10. Beta Cam, NGC1502, and IC342--A bright, wide white-deep blue
    double star, the Golden Harp Cluster, and a large but faint
    galaxy

Beta Cam (double star)          mag. 4.0
5h 3m            60 degrees 25'              Camelopardalis

NGC1502 (open cluster)          mag. 5.7     size 7'
4h 7.7m          62 degrees 20'              Camelopardalis

IC342 (galaxy)                  mag. 8.4     size 21'
3h 46.8m         68 degrees 6'               Camelopardalis

Capella, then

   1/2 degree S, 7 5/8 degrees (43m) E to

beta Aur (2.0, with a mag. 6.5 star 3/8- degree SSE), then

   9 3/8 degree N to

delta Aur (3.5, with a mag. 7.5 star 1/8 degree NNE and a mag. 6.0
star 1/4 degree N), then

   6 1/6 degrees N, 6 3/4 degrees (56m) W to

beta Cam (4.0, with a mag. 8.0 star 3/8- degree SSE and mag. 6.0
stars 5/8 degree N and 7/8 degree NNE;  beta Cam is discussed
further below), then

   6- degrees N, 7/8 degree (9m) W to

alpha Cam (4.0, with mag. 7.5 stars 1/8 degree N and 1/2 degree
WNW, a mag. 7.0 star 3/8 degree S, and a mag. 8.0 star 1/4 degree
NW), then

   5/6 degree S, 6 1/3 degrees (64m) W to

a mag. 4.5 star with a mag. 7.0 star 3/8 degree E, then

   2 1/4 degree S to

a mag. 6.0 star;  this star forms a line with a mag. 5.0 star 7/8
degree ESE and a mag. 5.0 star 3/8 degree WNW, and the mag. 6.0
star forms a north-pointing bow with the mag. 5.0 star 3/8 degree
WNW and another mag. 5.0 star 1/2 degree WSW of the last-mentioned
star, so we have a four-star combination which is readily visible
in binoculars or finderscope, then from the mag. 5.0 star to the
ESE,

   3/4 degree S, 1 1/8 degree (10m) E to

the center of NGC1502 (with a mag. 7.0 star 1/4 degree S, 1/8
degree W of the center of NGC1502).

Now after observing NGC1502 as discussed below, if you want a more
challenging assignment you can do this:

First, backtrack (or start over) to get to the mag. 4.5 star with
the mag. 7.0 star 3/8 degree E mentioned above, then

   1 2/3 degrees N to

a mag. 7.0 star with a mag. 6.0 star 3/8 degree W and a mag. 7.5
star 1/2 degree NNE, forming an obtuse triangle, then

   1/2 degree NNE to

the NNE (7.5) star in the triangle mentioned in the previous step,
then

   1- degree N, slightly W to

a mag. 6.5 star with a mag. 7.0 star 5/8- degree N and a mag. 8.0
star 1/8+ degree WNW of the mag. 7.0 star, then

   3/8+ degree S, 1/2- degree (5m) W to

the center of IC342.

     Viewed 11/23/02, 5:45 p.m.  Conditions excellent.  Beta Cam is 
an attractive double star because it is fairly wide (you can easily
split it with 58.75x but it looks better at 94x), and both
components are fairly bright;  in particular, the deep blue
component is brighter than most stars of this color.
     NGC1502 is an attractive small cluster with lots of bright
stars (as well as fainter ones).  The brightest and most striking of
these was SZ Cam, which is near the center;  this is a mags.
6.5-8.0 variable double star, and when I looked at it there were
two nearly identical bright white components, close together.
Since SZ Cam is a variable star, it may look different when you look
at it.  The cluster as a whole looks to me like a crossbow aiming
west.  There are a couple of stars a little north of the line of
aim of the crossbow, which may represent the target;  if so, the
target seems to have escaped (yea!).  Reference II-3 says that
NGC1502 is "...readily visible through small telescopes.  Larger
telescopes show it to be tightly packed with many stars of differing
magnitudes.  On nights with good seeing, the cluster is visible
even through binoclars."  I could not see it with binoculars, but
through the telescope I could verify the "tightly packed" part.
     While you are in the neighborhood, if you are up for a much
more difficult challenge, you can try for NGC1501, which is a faint
(magnitude 11.5) planetary nebula of size 52".  It should look like
a faint patch of light if you can see it;  I could not see it even
though I am almost certain I had it at the center of my field.  To
get to it, from the center of NGC1502 go 1 1/2 degrees S, slightly
E to a mag. 7.5 star, then slightly N, 3/16 degree W to NGC1501.
In the last move we said 3/16 degree W rather than 1/8+ degree W or
1/4- degree W to gain a little precision;  note that if (for
example) your field has width about 1/2 degree, then the radius of
the field will be about 1/4 degree, and since (3/16)/(1/4) =
(3/16)(4/1) = 3/4, if at the last step you center the mag. 7.5 star
then move west so that the mag. 7.5 star moves from the center of
the field to a point 3/4 of the way to the edge of the field, and
then you move a tad north, NGC1501 should be very close to the
center of your field.  What is "a tad"?  Well, it is no more than
1/16 degree (had it been greater than 1/16 degree but less than 1/8
degree we would have said "1/8- degree N" instead of "slightly N"
(assuming our ability to judge such things that closely)).
     IC342 is a large round galaxy, but its average surface
brightness is quite low, so don't feel bad if you can't spot it.
At 58.75x I could (faintly but clearly) see a right triangle in the
field with longer leg about 1/6 degree long extending from the
right angle vertex to the SSW, with the shorter leg about half that
long extending from the right angle vertex to the ESE.  At 94x I
could see some light near the two endpoints of the shorter leg.  At
267x I could see the boundary of the galaxy slightly to the SSW of
these two vertices, so these stars were just outside the galaxy.
Proceeding slowly and carefully, I could move the telescope so as
to pretty well trace the boundary all the way around;  the third
vertex of the triangle clearly appeared inside the galaxy, and
might even be the center of the galaxy instead of a star (although
I'm not ready to bet either way on that).  The galaxy is about 1/3
degree across, so at 94x it will stretch over about 2/3 of a 1/2
degree field.  I looked for IC342 for several nights (with gradually
improving viewing conditions) before I was sure I had it.  IC342
looks like a smaller, rounder, fainter version of M33.  Although
this is not an easy object to see, it is quite rewarding if you do.
By the way, with declination 68 degrees 6' this is getting pretty
far north, so we are getting into a region where polar alignment
errors can have larger effects as discussed in section VII(I), so we
need to be careful, and pay attention to our check stars.


11. Omicron2 Eri and w Eri--A triple star and a superb yellow-
    green double star

Omicron2 Eri (triple star)       mag. 4.5
4h 15.5m                 -7 degrees 40'            Eridanus

w Eri (double star)              mag. 4.5
3h 54m                   -2 degrees 50'            Eridanus

Rigel, then

   1/2 degree N, 14 5/8 degrees (59m) W to

Omicron2 Eri (4.5, with Omicron1 Eri (4.0) 1 1/4 degrees NW, 37
Eri (5.5) 3/8 degree W of Omicron1 Eri).

After observing Omicron2 Eri as discussed below, do the following:

Omicron2 Eri, then

   4 2/3 degrees N, 5 1/4 degrees (21m) W to

w Eri (4.5, with a mag. 7.5 star 3/4 degree SW and a mag. 8.5 star
1/4- degree N of the mag. 7.5 star).

(Note:  Omicron2 Eri is also known as 40 Eri, and w Eri is also
known as 32 Eri or ADS 2850.)

     Viewed 1/10/03 at 6:30 p.m.  Conditions good except for a
bright half moon.  The three components of Omicron2 Eri are also
known as 40 Eri A (mag. 4.5), a whitish star, 40 Eri B, a ninth
magnitude white dwarf star which is the easiest white dwarf star to
study with a small telescope, and 40 Eri C, a small red star which
is one of the least massive stars known.  40 Eri B is about 84" E
of 40 Eri A;  both of these stars can be seen at 58.75x.  40 Eri C
is slightly N of 40 Eri B (I think, at least at 267x I could see
something there which kept blinking on ond off due to atmospheric
turbulence, but it kept reappearing in the same spot, so it is
probably the right star).  Note that the angle from 40 Eri A to 40
Eri B then to 40 Eri C is a little less than 90 degrees.  40 Eri
C is so faint that I could not detect any color.
     W Eri is an exceptionally fine double star, and is my second
favorite double star (after gamma And).  The brighter component is
yellowish, and the other component is a slightly fainter green
star that is a little NNW of the yellowish star.  This pair can be
split at 94x but it looks better at 267x.
     In passing we mention that Rigel is also a double star, with
the fainter component (mag. 6.7) being about 400 times fainter than
the brighter component (mag. 0.2) and about 9.4" S of the brighter
component.  The brightness of the brighter component makes the
fainter component somewhat hard to find, but I could clearly see it
at 267x.  The brighter component is bluish white, while the fainter
component looked white to me.


12. The Hyades, with nearby open cluster NGC1647

The Hyades (open cluster)       mag. 0.5        size 5.5 degrees
4h 27.0m               15 degrees 50'           Taurus

NGC1647 (open cluster)          mag. 6.4        size 44'
4h 46.0m               19 degrees 4'            Taurus

Aldebaran appears to be at the E end of the S arm of the V-shaped
Hyades cluster;  the V points WSW, with arms about 4 degrees long.

After perusing the Hyades (see discussion below), you can do this:

Aldebaran, then

   2 3/8 degrees N, 3 3/4 degrees (16m) E to

97 Tau (5.0, with a mag. 7.0 star 5/8 degree W and a mag. 7.5 star
5/8 degree ENE), then

   slightly S, 5/8 degree (2 1/2m) W to

a mag. 7.0 star, then

   5/8 degree (2 1/2m) W to

a mags. 6.0-7.5 SSE-NNW pair of stars, then from the SSE (6.0)
star, 

   3/8 degree N to

the center of NGC1647.

     Viewed Thanksgiving 2002, 9:15 p.m.  Conditions excellent.
The Hyades is actually more beautiful with binoculars than with a
telescope, but you can explore it with the telescope at low power,
using your finderscope to guide your exploration.  The star
Aldebaran (mag. 0.9) is a brilliant orange and is a treat by itself
with a telescope.  This star is not actually part of the cluster,
since it is much closer to Earth than the stars of the cluster.
The Hyades were mentioned in the works of Homer, Virgil, and other
early authors.
     NGC1647 is a round and pretty cluster that just fits into the
field at 58.75x.  It also looks good at 94x, although it doesn't
quite all fit into the field at one time at that power.  This
attractive cluster has lots of bright stars (at least, they look
bright through the telescope).


13. Three from the hare--An attractive (but close) yellow-blue
    double (ADS 3954), a fairly bright but hard to resolve
    globular cluster (M79), and a wide yellowish-bluish double
    (gamma Lep)

ADS 3954 (double star)      mags.  5.4, 6.6    separation 3.2"
5h 22m          -24 degrees 40'                Lepus

M79 (globular cluster)      mag. 8             size 6'
5h 24.5m        -24 degrees 33'                Lepus

Gamma Lep (double star)     mags. 3.7, 6.3     separation 96"
5h 44.5m        -22 degrees 27'                Lepus

Saiph (2.0, SE corner of Orion, with a mag. 6.0 star 1 degree S),
then

   5 1/6 degrees S, 1/8 degree (1m) W to

zeta Lep (3.5, with eta Lep (3.5) 2 3/8 degrees ENE, a mag. 5.5 star
3/4 degree NE, and a mag. 7.5 star 3/8 degree S), then

   3 degrees S, 3 1/3 degrees (14m) W to

alpha Lep (2.5, with a mag. 8.5 star 1/8 degree E, a mag. 8.5 star
1/8 degree W, and a mag. 8.0 star 5/8 degree SSE), then

   2 5/6 degrees S, 1 degree (4 1/2m) W to

beta Lep (2.5, yellow, with 10 Lep (5.5) 3/4 degree E, a mag. 6.0
star 5/8 degree S, a mag. 8.0 star 3/8 degree SE, and a mag. 8.0
star 1/8 degree WSW), then

   1 2/3 degree S to

a mag. 7.0 star with a mag. 7.5 star 1/2 degree SE, then

   1 5/6 degrees S, 1/8 degree (1/2m) E to

a mag. 7.0 star with a mag. 8.5 star 1/2 degree W and a mag. 8.0
star 3/4 degree SSW, then

   1/2 degree S, 1 5/8 degrees (7m) W to

ADS 3954 (5.0, with a mag. 7.0 star 5/8 degree WSW, a mag. 7.5 star
1/4+ degree WSW of the mag. 7.0 star, and a mag. 8.0 star 3/8 degree
W of the mag. 7.5 star).

Then after looking at ADS 3954 as described below, do this:

ADS 3954, then

   1/4 degree N, 1/2+ degree (2 1/2m) E to

M79.

Then after looking at M79 as described below, return to beta Lep
and do this:

Beta Lep, then

   1 2/3 degrees S, 3 2/3+ degrees (16m) E to

gamma Lep (3.5, with 12 Lep (6.0) 1/2 degree W and a mag. 8.0 star
1/8 degree SE).

     Viewed 2/14/06, starting at 7:30 p.m.  Conditions pretty good.
These objects were viewed using an 11 inch Schmidt-Cassegrain
telescope.  At 318x, ADS 3954 is an unexpectedly attractive double
consisting of a bright yellow star with a fainter blue star
following directly to the E.  I could also barely split this pair
at 112x;  the colors were obvious even at this lower power.
     M79 is a nice little globular which jumps out at you even at
low power.  It looks like a bright little round fuzzy patch of
light.  It is one of the few globular clusters visible in the
winter.
     Gamma Lep is a nice wide yellowish-bluish double, with the
fainter bluish component to the N.


14. The Terrific Trio--Open clusters M38, M36, and M37

M38 (open cluster)          mag. 6.4           size 20'
5h 28.7m         35 degrees 50'                Auriga

M36 (open cluster)          mag. 6             size 12'
5h 36.5m         34 degrees 8'                 Auriga

M37 (open cluster)          mag. 5.6           size 23'
5h 52.4m         32 degrees 33'                Auriga

Capella, then

   4 3/4 degrees S, 2- degrees (10m) W to

eta Aur (3.5, with zeta Aur (4.0) 3/4 degree W), then

   8+ degrees S, 2 degrees (9 1/2m) W to

iota Aur (3.0, with a mag. 7.0 star 1/2 degree SW), then

   1/6 degree N, 4 3/8 degrees (21+ m) E to

16 Aur (4.5, with 17 Aur (5.5) 3/8 degree NNE and a mag. 7.0 star
1/4 degree W), then

   1/2+ degree N, 3/8 degree (2- m) E to

19 Aur (5.0, with 18 Aur (6.5) 1/8 degree W and 17 Aur (5.5) 1/4
degree SW), then

   1/2 degree N, 1 5/8 degree (8m) E to

phi Aur (5.5, with a mag. 8.0 star just N, a mag. 6.0 star 1/4-
degree SW, and a mag. 6.5 star 1/4 degree NE), then

   1 degree N, 1/8 degree (1/2m) W to

a mag. 6.0 star with a mag. 8.5 star 1/4 degree NW, then

   3/8 degree N, 3/8 degree (2- m) E to

the center of M38.

Then after looking at M38 as discussed below, to get to M36 you can
backtrack to phi Aur (see above) or follow the path above to phi
Aur, then

   1/4 degree NE to

a mag. 6.5 star, then

   1+ degree (5m) E to

a mag. 6.5 star with a mag. 8.5 star 1/8 degree SE and mag. 8.0
stars 3/8 degree NW and 3/8 degree SW, then

   1/2 degree S, 1/8 degree (1/2m) E to

a mag. 8.0 star, then

   slightly S, 1/2 degree (2 1/2m) E to

the center of M36.

Then after looking at M36 as discussed below, to get to M37 you can
proceed as follows:

Center of M36, then

   1 7/8 degrees S, 3/4+ degree (4- m) W to

chi Aur (5.0, with a mag. 6.5 star 1/8 degree NE, a mag. 7.5 star
1/8 degree S, and another mag. 7.5 star 1/4 degree S of the mag.
7.5 star), then

   1/4 degree S, 1 5/8 degrees (8m) E to

a mag. 6.5 star with a mag. 8.0 star 1/8 degree W and a mag. 6.0
star 1/2 degree S, then

   1/4 degree N, 2 1/4 degrees (11m) E to

a mag. 6.5 star with a mag. 7.0 star 1/2 degree SW and a mag. 6.0
star 7/8 degree SE, then

   1/2- degree N, 1/8+ degree (1- m) E to

the center of M37.

[Alternately, to get to chi Aur in the above path without having
to find M36 first, follow the path for M38 to 16 Aur, then go
1 1/6 degrees S, 3 degrees (14 1/2m) E.]

     Viewed 4/14/02, 8:30 p.m.  Conditions hazy.  94x gives a good
view of each cluster, but M36 and M37 (which are smaller than M37)
also look good at 267x;  267x for M38 is interesting, but you don't
get the whole picture at that magnification.  M38 contains a
beautiful cross!  M36 is sparser and has brighter stars than M38 or
M37.  M37 has lots of little stars.  Reference II-13 waxes poetic
about M37, saying it is "A magnificent object, the whole field
being strewed as it were with sparkling gold dust....The brightest
star is mag. 9.2, a topaz jewel surrounded by a pear-shaped cluster
of scintillating diamonds."  Although these clusters are all quite
different from each other, each is beautiful in its own way.


15. Orion's sword, including the Great Orion Nebula (M42)

Iota Ori (double star)       mag. 2.5
5h 35.5m             -5 degrees 55'             Orion

Theta1 Ori (quadruple star)  mags. 5.0, 6.7, 6.7, 7.9
5h 35.3m             -5 degrees 23'             Orion

M42 (Great Orion Nebula)     mag. 4             size 70'x60'
5h 35.4m             -5 degrees 27'             Orion

M43 (nebula)                 mag. 9             size 19'x15'
5h 35.6m             -5 degrees 16'             Orion

NGC1977 (nebula)                                size 19'x9'
5h 35.5m             -4 degrees 52'             Orion

NGC1981 (open cluster)       mag. 4.6           size 25'
5h 35.2m             -4 degrees 26'             Orion

The middle star of Orion's belt, then

   4 3/4 degrees S, 1/4- degree (1m) W to

iota Ori (2.5, with a mag. 5.5 star 3/8 degree ESE, a mag. 6.0 star
1/2 degree E, and an E-W mags. 6.5-7.0 pair 3/8 degree NNE).

After looking at iota Ori as discussed below, go 1/2+ degree N and
slightly W to

theta1 Ori (5.0, with a mag. 7.0 star 1/8 degree NNE and a mag. 8.0
star 1/8+ degree N, forming a right triangle, and an E-W line of 3
stars just SE).

M42 is a bright hazy area with theta1 Ori in its NE part.

After looking at theta1 Ori and M42, go 1/8 degree NNE from theta1
Ori to a mag. 7.0 star (which is one of the check stars given above
for theta1 Ori);  this star is roughly in the center of the faint
hazy patch M43.

After looking at M43, from the mag. 7.0 star at its center go

   1/2- degree N to

a mag. 7.5 star which is in the middle of a NNE-pointing bow with
45 Ori (5.0) just ESE and 42 Ori (5.0) just W;  the bow is
contained in NGC1977, which is roughly rectangular E-W with 42 Ori
near its center.

After looking at NGC1977, from 42 Ori go

   3/8 degree N to

the center of NGC1981.

     Viewed starting at 6:15 p.m. on 2/20/03.  Conditions were
good, and got better as time went on.  Iota Ori is a double star
consisting of a bright white component and a much fainter blue
component to the S and a little E.  I could barely split this at
58.75x, it was better at 94x, and it was quite attractive at 267x.
     Theta1 Ori is actually a sextuple star, but two of the
components are of magnitude 11 and hard to see, so for most of us
it is visually a quadruple star.  This is a trememdously pleasing
multiple star!  It is also called the Trapezium.  The four main
components are roughly in the form of a rectangle, with the
brightest to the S and the faintest to the N;  the faint one looked
sort of reddish, while the others were white.  I was able to see
all four components at 94x, but they were much prettier at 267x.
     By the way, to be more precise I could say that the four main
components were roughly oblong, meaning rectangular but not square.
In central Illinois there are towns called Normal and Oblong;  my
aunt Myra once saw an announcement in the paper that said "Normal
man marries Oblong woman".  I also thank my aunt for finding the
Biblical reference in the description of the Pleiades elsewhere in
our list of objects.
     Theta1 Ori is in the NE part of the Great Orion Nebula (M42);
the part I could see was shockingly bright compared to most nebulas
in our list.  It was visible even at 58.75x, better at 94x, and
really bright at 267x.  There is a dark cone just NE of theta1 Ori.
The part of the nebula I could see looked vaguely like a headless
bird flying ENE.  The combination of theta1 Ori and M42 has to be
one of the prettiest things in the sky.  In time exposures M42 and
M43 look red, while NGC1977, NGC1975, and NGC1973 (see below) look
blue, but you are likely to see only white visually;  it takes the
colors a while to build up on the film.  M42 is one of the few
objects to receive three exclamation points in the NGC catalog.
     All I could see of M43 was a roundish hazy patch surrounding
the mag. 7.0 star we use to find it (and which provides the energy
to illuminate it, just as theta1 Ori provides the energy to
illuminate M42).  M43 received one exclamation point in the NGC
catalog.
     NGC1977 is faint compared to M42; it can be seen as a hazy
patch in places, especially around 42 Ori.  NGC1977 received two
exclamation points in the NGC catalog.  It may be worth mentioning
that nebulas NGC1975 and NGC1973, wihch lie just N of NGC1977, are
similar to NGC1977, but are smaller and fainter.
     NGC1981 is an open cluster consisting of a dozen or so fairly
bright stars.  It is not as spectacular as most of the other open
clusters in our list, but it makes a good place to end our
south-to-north tour of Orion's sword.


16. Zeta Ori and sigma Ori--A triple star and a quintuple star

Zeta Ori (triple star)      mags. 2, 4, 10
5h 40.8m            -1 degree 57'                Orion

Sigma Ori (quintuple star)  mags. 4, 6, 7, 8, 10
5h 38.8m           -2 degrees 36'                Orion

Zeta Ori (1.5) is the one of the three bright stars in Orion's
belt that is farthest to the SE;  there is a mag. 8.0 star 1/4-
degree W.

After observing zeta Ori as described below, do the following:

Zeta Ori, then

   2/3 degree S, 1/2 degree (2m) W to

sigma Ori (3.5, with a mag. 7.5 double 1/8- degree NW, a mag. 8.0
star 1/8- degree SE, and a mag. 8.5 star 1/8+ degree NE).

     Viewed 1/10/03, 7:20 p.m.  Conditions good except for a bright
half moon.  The mag. 4 component of zeta Ori is just S of the mag.
2 component, and this pair is close enough that it is somewhat
difficult to split, althogh I could just do it at 267x and could do
it a little more convincingly at 534x.  The mag. 10 component
(assuming that I was looking at the right star;  there was another
faint star to the E of the bright pair) is a good bit farther away,
to the N, and can be viewed nicely at 267x.  These components are
supposed to be bluish-white, but they just looked whitish to me.
     Although sigma Ori has at least 5 components, it is most
unlikely you will see all five, since the two brightest components
are only 1/4" apart, so this bright pair looks like a single star
in most telescopes.  At 58.75x the bright pair and the mag. 7
component to the NE are clearly visible, and at 94x the mag. 8
component just E of the bright pair comes clearly into view.  At
267x the mag. 10 component WSW of the can be seen, so this star
with the bright pair and the mag. 8 component form a bow pointing
roughly N.  Sigma Ori is one of the easiest multiple star systems
for seeing more than two components.


17. M35--A splendid large cluster full of bright stars

M35 (open cluster)          mag. 5.1            size 28'
6h 8.9m              24 degrees 20'             Gemini

Pollux, then

   11 5/8 degrees S, 16 1/4 degrees (67 1/2m) W to

Alhena (2.0, with a mag. 6.0 star 7/8 degree E and a mags.
6.5-7.0-7.0 S-pointing isosceles triangle 3/8 degree tall 3/8
degrees NW), then

   6 1/8 degrees N, 3 1/2 degrees (15m) W to

mu Gem (3.0, orange, with nu Gem (4.0) 2 3/4 degrees SSE and a mag.
7.5 star 5/8 degree E), then

   1 7/8 degrees (8m) W to

eta Gem (3.5-4.0 variable, orange, with a mag. 7.0 star 3/4 degree
S and a mag. 7.0 star 3/4- degree SE), then

   5/8 degree N, 1 1/8 degree (5m) W to

3 Gem (6.0, with a mag. 6.0 star 1- degree S and 4 Gem (7.0) 1/4
degree SE), then

   1 1/6 degree N, 3/8 degree (1 1/2+ m) E to

5 Gem (6.0), then

   1/2 degree (2 1/2m) W to

a mag. 7.5 star, with a mag. 8.0 star 1/8 degree SW;  both these
stars are inside M35, with the mag. 8.0 star slightly NE of the
center of M35.

     Viewed 4/1/03, 7:10 p.m.  Conditions good.  M35 is a really
nice large cluster full of bright stars, with lots of fainter ones
too.  This is a good one!


18. A march through Monoceros--8 Mon (double star), NGC2244 (open
    cluster), C49 (the Rosette Nebula), NGC2264 (the Christmas Tree
    Cluster), the Cone Nebula, and (for the stout-hearted) NGC2261
    (Hubble's Variable Nebula)

8 Mon (double star)          mag. 4.5           separation 13"
6h 24m                  4 degrees 35'           Monoceros

NGC2244 (open cluster)       mag. 4.8           size 23'
6h 32.0m                4 degrees 55'           Monoceros

C49 (The Rosette Nebula)                        size 80'x60'
6h 32.3m                5 degrees 3'            Monoceros

NGC2264 (the Christmas Tree Cluster)  mag. 3.9  size 35'x15'
6h 41.1m                9 degrees 53'           Monoceros

NGC2261 (Hubble's Variable Nebula)              size 2'x1'
6h 39.2m                8 degrees 44'           Monoceros

Betelgeuse, then

   2 5/6 degrees S, 7 degrees (28+ m) E to

8 Mon (4.5, with mag. 7.0 stars 3/8 degree NE and 1/2 degree SSW,
a mag. 7.5 star 1/2 degree SSE, and mag. 8.0 stars 1/4 degree ESE
and 1/4 degree WSW).

After looking at 8 Mon as described below, do the following:

8 Mon (4.5), then

   3/8 degree NE to

a mag. 7.0 star with a mag. 8.0 star 1/2- degree E, then

   1/2- degree E to

a mag. 8.0 star with a mag. 6.5 star 3/8 degree NNE, then

   1 3/8 degrees (5 1/2m) E to

12 Mon (6.0);  this star is in the SE part of NGC2244.

NGC2244 is roughly in the middle of the roundish Rosette Nebula.

After looking at NGC2244 and the Rosette Nebula as described below,
do the following:

12 Mon (6.0, see above), then

   2 1/2 degrees N, 1/8 degree (1/2m) E to

13 Mon (4.5, with a mag. 7.5 star 1/4- degree E and a N-S pair of
mag. 8.0 stars 1/8 degree E of the mag. 7.5 star), then

   2 5/8 degree N, 2 1/8 degrees (8 1/2m) E to

15 Mon (4.5, with a mag. 7.5 star 1/8 degree ESE);  15 Mon is
roughly in the middle of NGC2264.

After looking at NGC2264 as described below, do the following:

15 Mon (4.5, see above), then

   3/8+ degree S to

a mag. 7.5 star with a faint companion just W and a still fainter
companion just N of that;  the Cone Nebula is a small dark cone
pointing N with its tip just S of the mag. 7.5 star.

After looking at NGC2264 and the Cone Nebula as described below,
if you are brave and/or ambitious try this:

The mag. 7.5 star described near the tip of the Cone Nebula (see
above), then

   1/2 degree S to

a mag. 6.5 star with a mag. 8.5 star 3/8 degree ESE (the mag. 6.5
star is a little N-S blue-orange double, which showed up well at
267x and could also be split at 94x), then

   1/4 degree S, 1/2- degree (2m) W to

NGC2261.

     Viewed starting 3/7/03, 6:45 p.m.  Conditions good in spite
of some moon interference.  8 Mon shows as a bright white star with
a fainter blue companion a little to the NE.  It can be split at
58.75x, but 94x looks better and 267x looks better still.
     NGC2264 is a neat little cluster.  The main part looks like a
top view of an old rectangular-winged biplane flying ENE, with 12
Mon (see above) in the right wing.  If you look carefully a little
S of the tail you may see a much fainter and smaller biplane that
appears to be on a collision course with the larger biplane.  Could
this be Snoopy and the Red Baron?  94x gives a good view, but you
will need to change your aim a little to see the second biplane.
     The Rosette Nebula looks like a faint glow around NGC2244.
With 94x or 267x you may be able to trace part of the boundary.
This nebula might be spectacular away from light pollution.
     The Christmas Tree Cluster is ok, but it was hard for me to
see the tree.  There may be several ways to look at it.  There is
a glow around this cluster too, and the Cone Nebula is a dark cone
in the glow, caused by unilluminated gas in front of the glowing
gas.  The Cone Nebula is an example of a so-called dark nebula.
     NGC2261 is a small but interesting fan-shaped nebula pointing
S.  It has been compared to a comet's tail.  It collects light from
the irregular variable star R Mon, which ranges from tenth to
thirteenth magnitude, so you may need to be lucky enough to be
looking at it at the right time to be able to see it.  The shape of
the nebula sometimes seems to change in a matter of weeks.  I did
not observe this nebula at the same time as the other objects in our
march through Monoceros because reference II-3 said it was hard to
observe with a small telescope, but since the other two nebulas in
this bunch are supposed to be hard to find, and our techniques were
sufficiently effective to allow me to find them, I decided to come
back and take a shot at it.  This attempt occurred on 3/14/03 at
7:15 p.m. under poor conditions--the moon was a little past half
full, and was so bright that I could (barely) read my object cards
by its light.  The mag. 6.5 star which is the last object in the
path before NGC2261 turned out, as noted above, to be a double;
this was a pleasant surprise.  Making the move 1/4 degree S, 1/2-
degree (2m) W at 58.75x, I could not see NGC2261, although there
were several stars in the vicinity.  Thus I tried to make the move
again with maximum accuracy, the idea being to mark where NGC2261
should be by a nearby star, then use that star both as a marker and
as a star to observe while focusing the telescope at higher
powers.  At 58.75x my field width is about 3/4 degree, so the
field radius is half that, namely about 3/8 degree.  The movement
1/4 degree S is about 2/3 of the field radius (since (1/4)/(3/8)
= (1/4)(8/3) = 8/12 = 2/3).  Thus moving S so that the mag. 6.5
star moved 2/3 of the way from the center of the field toward the
north edge gave an accurate move of 1/4 degree S.  Now trying to
make the move 1/2- degree W all at once might work, but a different
approach offered a chance to be a little more accurate, namely
first move W just far enough to bring the desired location into
the W edge of the field;  this can be done while keeping the mag.
6.5 star in the field, which is why it may be more accurate.  The
distance from the W edge of the field to the center is about 3/8
degree, and 1/2 - 3/8 = 1/8, and 1/8 degree is about 1/3 of the
field radius (since (1/8)/(3/8) = 1/3), so I moved the telescope W
just far enough so that the (imaginary) N-S line through the mag.
6.5 star moved 1/3 of the way from the center of the field to the E
edge.  Fortunately a faint star then appeared a little inside the W
edge of the field;  centering this showed it to be the E star of an
obtuse isosceles triangle pointing E, with the NW vertex of the
triangle being a little N-S double.  Now I could keep the E vertex
centered while going to 94x, then 267x, then 534x.  At 267x a small
fan-shaped blob of light appeared just W of this star, well inside
the isosceles triangle mentioned above.  Success!  534x made it
look twice as big.  I think I could see a faint star inside, perhaps
the elusive R Mon.  Now as mentioned above R Mon and NGC2261 are
variable, so you may have a harder time if you are unlucky enough
to look at the wrong time, but on the other side of the coin,
viewing conditions were poor when I looked due to the moon, and you
now have a road map to pinpoint NGC2261 pretty exactly, so if your
viewing conditions are better than mine, you may have a good shot
at finding this.  If you don't find it, don't feel bad because it
is a difficult object to find, but at least you tried, and I
congratulate you for that!


19. Beta Mon--An attractive triple star

Beta Mon (triple star) mags. 4.7, 5.2, 6.1 separations 7.3" and 2.8"
6h 28.8m            -7 degrees 2'          Monoceros

Saiph (2.0, the SE corner star of Orion, with mag. 7.5 stars 1/8+
degree N and 1/4 degree SW), then

   1/8 degree N, 2 7/8 degrees (11 1/2m) E to

2 Mon (5.0, with 1 Mon (6.5) 1/4- degree N), then

   2 1/2+ degrees N, 7 1/4 degrees (29 1/2m) E to

beta Mon (4.0, with a mag. 7.0 star 1/8- degree NNW, a mag. 7.5
star 3/8 degree SSE, and a mag. 6.0 star 3/4 degree SW).

[Alternate path:  A little less than halfway from Orion's belt to
Sirius, about 2 degrees above this line, find gamma Mon (4.0),
then from gamma Mon go about 3 1/2 degrees almost parallel to the
belt-Sirius line but up a little to beta Mon (4.0).]

     Viewed 3/6/03, 7:15 p.m.  Conditions excellent.  At 58.75x
this looked like a nice NW-SE double star with close
equal-magnitude white components.  At 94x one could barely see a
fainter white star just E of the SE member of the pair, closer to
that star than that star is to the NW member of the pair, and at
267x all three were clearly visible, forming an attractive arc
shape.  According to reference II-11, a telescope with aperture at
least 8 inches is needed to see all three stars;  I could see all
three clearly with a 9.25 inch telescope, but could see only two
stars with a 5 inch telescope.


20. M41--One of the finest open clusters

M41 (open cluster)            mag. 4.5           size 38'
6h 46.0m              -20 degrees 44'            Canis Major

Sirius, then

   4 degrees S, 1/4 degree (1m) E to

The center of M41.

     Viewed 3/6/03, 6:30 p.m.  Conditions excellent.  This large
and bright cluster is beautiful at 58.75x or 94x or 267x, although
with the larger magnifications you will not see it all at once.  Of
the three magnifications, 94x gave the best view for me.  There is
a bright orange star (mag. 6.9) near the center.  There are about
two dozen bright stars and many fainter ones.  This cluster can
also be seen with binoculars, and possibly even with the naked
eye under better viewing conditions than mine.


21. 12 Lyn--A blue-blue-orange triple star, with nearby double 19
    Lyn as a bonus

12 Lyn (triple star)         mag. 4.5
6h 46m            59 degrees 25'                Lynx

19 Lyn (double star)         mag. 5.5
7h 23m            55 degrees 15'                Lynx

Capella, then

   1 degrees S, 7 5/8 degrees (43m) E to

beta Aur (2.0, with a mag. 6.5 star 3/8- degree SSE), then

   9 3/8 degrees N to

delta Aur (3.5, with a mag. 7.5 star 1/8 degree NNE and a mag. 6.0
star 1/4 degree N), then

   4 3/4 degrees N, 2 5/8 degrees (20m) E (or 5 1/2 degrees NNE) to

2 Lyn (4.5, with 37 Lyn (5.5) 1 1/4 degrees W, 40 Lyn (5.5) 1 1/8
degrees NNW, and 5 Lyn (5.0) 1 1/8 degrees SE, and a mag. 7.0 star
3/8 degree WSW), then

   1/2- degree N, 3 3/8 degrees (27m) E to

12 Lyn (4.5, with 14 Lyn (5.5) 7/8 degrees E and 15 Lyn (4.5) 1 1/4
degrees SSE of 14 Lyn, so 15 Lyn, 14 Lyn, and 12 Lyn make a
triangle that is readily recognizable;  there is also a mag. 7.5
star 1/4- degree W of 12 Lyn).

After viewing 12 Lyn as discussed below, you can get to 19 Lyn by
following the line from 12 Lyn through 15 Lyn to the SE about three
times as far as the distance between 12 Lyn and 15 Lyn, or to be
more precise, you can go to 15 Lyn, then do this:

15 Lyn (4.5), then

   3 1/6 degrees S, 3 2/3 degrees (25 1/2m) E to

19 Lyn (5.5, with a mag. 7.5 star 1/8- degree N).

     Viewed 3/21/02, 7:45 p.m.  Conditions ok except for half moon
nearby.  Two close bright blue stars in an E-W line SE of a fainter
orange star.  Beautiful!  It took 267x to split this into 3 parts,
and 534x split it more easily.
     19 Lyn is a cute little double, with a bluish component and a
slightly fainter orangish component just to the NW.  19 Lyn can be
split at 58.75x, but it looks better at 94x and 267x.  As with
omicron Cap, which is elsewhere in this list, the colors tend to
fluctuate.
     By the way, 12 Lyn is part of the path to NGC2403 as discussed
in section VIIF, so while you are here you can go on to look at
this galaxy if you want.


22. NGC2281--An attractive, compact open cluster that vaguely
    resembles a bat flying east

NGC2281 (open cluster)       mag. 5.4            size 14.0'
6h 49.3m             41 degrees 4'               Auriga

Capella, then

   1 degree S, 7 5/8 degrees (43m) E to

beta Aur (2.0, with a mag. 6.5 star 3/8 degree SSE and pi Aur(4.5)
1 degree N), then

   2 1/2- degrees S, 7 1/4 degrees (39 1/2m) E to

psi2 Aur (5.0, orange, with a mag. 7.5 star 1/2 degree S and a
mags. 7.0-7.5 NNW-SSE close pair 1/2 degree S of the mag. 7.5
star), then

   2/3- degree S, 2 1/8 degrees (11 1/2m) E to

psi7 Aur (5.0, orangey, with a mag. 8.0 star 1/4+ degree SE), then

   1/2 degree S, 3/8+ degree (2 1/2m) W to

a mag. 7.5 star with a mag. 8.5 star 1/4+ degree WSW, then

   3/8 degree S to

the center of NGC2281.

     Viewed 4/12/06, 9:00 p.m.  Conditions only fair due to "A bad
moon on the rise" (Creedence Clearwater Revival).  This is an
attractive, fairly small cluster that looks ok at 70x, much better
at 112x (where the bat shape shows up best) and nifty at 318x (all
using an 11 inch Schmidt-Cassegrain telescope).  Near the center
there is a nice rhombus with the long diagonal NE-SW.  The NE and
(faint) SE stars of this rhombus appear to be double.  A little
ENE of the rhombus there is a nice equilateral triangle, with
another (apparent) double star a little further ENE.  Most of the
stars in this cluster are fairly bright.  Can you see the bat?


23. NGC2392--The Eskimo Nebula, with double star delta Gem along
    the way

Delta Gem (double star)      mags. 3.6, 8.2      separation 5.8"
7h 20.1m              21 degrees 59'             Gemini

NGC2392 (The Eskimo Nebula)  mag. 9.2            size 19.5"
7h 29.2m              20 degrees 55'             Gemini

Pollux, then

   3 2/3 degrees S, 1/4- degree (1m) W to

kappa Gem (3.5, yellow, with a mag. 7.0 star 1/4- degree SW and a
mag. 8.5 star 1/8 degree ESE), then

   2 1/3 degrees S, 5 5/8 degrees (24m) W to

delta Gem (3.5, yellow, with a mag. 8.0 star 1/8+ degree W).

After observing delta Gem as discussed below, do this:

Delta Gem, then

   1/2 degree S, 1 7/8 degree (8m) E to

63 Gem (5.5, the middle of a S-pointing bow along with a mag. 7.0
star 1/4- degree NE and a mag. 6.5 star 1/4 degree NW), then

   5/8 degree S, 5/8 degree (2 1/2m) E to

a mag. 7.5 star which is in the center of a S-pointing letter Y,
and with a mags. 8.0-7.5 ESE-WNW pair 3/8 degree SSE, then

   1/6- degree N, 1/4 degree (1m) W (or just follow the NW-pointing
arm of the letter Y a distance equal to the length of the arm) to

a mag. 8.5 star, then

   slightly S to

NGC2392.

     Viewed 4/11/03, 8:15 p.m.  Conditions poor due to a bright
1/2+ moon.  Delta Gem consists of a bright white star with a much
fainter bluish companion to the SW.  To split this well required
267x, but at 94x I could barely split it when I knew where the
components were.  This is not a spectacular double star, but it is
ok.
     NGC2392 (also known as the Clown Face Nebula) is a really nice
planetary nebula, especially when one considers that it was easy to
find even under poor viewing conditions.  I could clearly see a
blob even at 58.75x, and at 94x I think I could even see a little
bluish color.  267x and 534x also gave interesting views.  The mag.
10.5 central star was clearly visible at 267x and 534x, and at 534x
I thought I could even catch glimpses of some other little starlike
features in the blob.  This nebula, like many planetary nebulas,
has a tendency to blink on and off;  it is helpful to use averted
vision occasionally (that is, focus your gaze a little away from
the nebula to bring the image to a part of your retina which is
more sensitive to faint light).


24. Castor--A gorgeous pair of brilliant white stars

Castor (multiple star)      mags. 1.9, 3.0        separation 3.9"
7h 34.6m              31 degrees 53'              Gemini

Castor is the NNW member of the Castor-Pollux pair;  it is slightly
fainter than Pollux.

     Viewed 3/14/03, 8:00 p.m.  Conditions poor--the moon was a
little past half full, and was so bright that I could (barely) read
by it.  It took 267x to split this, since the glare of these bright
stars was so great that they ran together at lower powers.  This is
a stunningly beautiful pair!  The WSW member of the pair is the
brighter of the two.  There is supposed to be a third fainter 
component nearby, but I could not see it.  Each of the three stars
is actually a spectroscopic double (that is, a double as
determined by the light coming from the stars using a spectroscope,
but too close to determine visually), so Castor is actually a
sextuple star, but we don't need to worry about that.  Just enjoy
the view of the two components you can see!


25. M47--A big, bright milkshake in the sky, complete with a straw

M47 (open cluster)            mag. 4.4               size 30'
7h 36.6m                 -14 degrees 30'             Puppis

Sirius, then

   1 1/8 degrees N, 4 1/2 degrees (19m) E to

gamma CMa (4.5, with a mag. 7.0 star 3/8 degree SSW and a mag. 8.5
star 1/4+ degree NNE), then

   slightly N, 3 degrees (12 1/2m) E to

a mag. 5.5 star, with a mag. 7.5 star 3/8 degree WNW, then

   1+ degree N, 4 1/4 degrees (17 1/2m) E to

an orange mag. 5.0 star with a mag. 6.0 star 1/4- degree NNW, then

   3/4 degree (3m) E to

the center of M47.

     Viewed 3/15/03, 7:00 p.m.  Conditions poor due to a bright 3/4
moon.   This is a neat cluster that really looks like a large
milkshake cup opening N with a straw sticking up out of it.  Many
of the stars are bright, including a nice double star in the base
of the straw.  94x gives a good view.


26. NGC2403--A large and beautiful (but faint) galaxy

See section VIIF for details.


27. M48--"A tremendously pleasing cluster, a perfect arrowhead of
    bright stars with a tight, off-axis core"

M48 (open cluster)            mag. 5.8            size 50'
8h 13.8m             -5 degrees 48'               Hydra

Procyon, then

   8 1/4 degrees S, 7 1/4 degrees (29m) E to

zeta Mon (4.5, with 28 Mon (4.5) 2 1/2 degrees NW, 27 Mon (4.5)
2 3/8 degrees S of 28 Mon, forming a near-equilateral triangle,
and with a mag. 7.0 star 1/2 degree SSW of zeta Mon and a mag. 7.5
star 3/4 degree NW of zeta Mon), then

   4 3/4 degrees S, 3/4 degree (3m) E to

a mag. 5.5 star with a mag. 7.5 star 5/8 degree NNE, a mag. 8.5
star 1/4 degree SSE of the mag. 7.5 star, and a mag. 8.0 star 3/8
degree W of the mag. 7.5 star), then

   1 degree N, 1/8 degree (1/2m) W to

a mag. 6.5 star with a mag. 7.5 star 1/4 degree W, then

   1/6 degree N, 5/8 degree (2 1/2m) E to

a mag. 7.0 star, then

   3/4 degree N to

the center of M48.

     Viewed 3/22/03, 7:00 p.m.  Conditions excellent.  The quote in
the heading for this object, which comes from reference II-13, tells
the story.  I would add that this object looks interesting at
58.75x, 94x, and 267x, with most of the cluster visible at once at
58.75x.  One can make an arrowhead out of this in a couple of
different ways, depending on whether you prefer a broad arrowhead
pointing E or a narrower arrowhead pointng NW.


28. A Cancerian quartet--Iota1 Cnc (a beautiful orange-green
    double), iota2 Cnc (a somewhat difficult white-yellow double),
    M44 (the Beehive Cluster), and zeta Cnc (a difficult triple of
    yellow stars, but an easy double)

Iota1 Cnc (double star)        mags. 5.3, 6.6    separation 30.4"
8h 46.7m                28 degrees 46'           Cancer

Iota2 Cnc (multiple star)      mag. 5.5
8h 54m                  30 degrees 40'           Cancer

M44 (the Beehive Cluster)      mag. 3.1          size 1.7 degrees
8h 40.1m                19 degrees 59'           Cancer

Zeta Cnc (triple star)         mag. 4.0
8h 12m                  17 degrees 40'           Cancer

Pollux, then

   2/3 degree N, 13 3/8 degrees (61m) E to

iota1 Cnc (4.0, with a mag. 6.5 star 5/8 degree S, a mag. 7.5 star
3/8 degree SSW, and a mag. 8.5 star 1/8 degree ESE of the mag.
7.5 star).

After looking at iota1 Cnc as discussed below, do this:

Iota1 Cnc (4.0), then

   1 5/6 degrees N, 1 5/8 degrees (7 1/2m) E to

iota2 Cnc (5.5, with 61 Cnc (6.0) 7/8 degree ESE, a mag. 8.0 star
3/8 degree S, a mag. 7.5 star 3/8 degree SW of the mag. 8.0 star,
and a mag. 7.0 star 1/4 degree WSW of the mag. 7.5 star).

After looking at iota2 Cnc as discussed below, backtrack to iota1
Cnc, then do this:

Iota1 Cnc, then

   7 1/3 degrees S, 7/8- degree (3 1/2m) W to

gamma Cnc (4.5, with a mag. 8.0 star 5/8 degree SSE.  Gamma Cnc is
one of the four stars that forms the body of Cancer the crab;  the
others are delta Cnc (3.5) 3 1/3 degree S, eta Cnc (5.0) 2 3/4
degrees WSW, and theta Cnc (5.0) 2 3/8 degree S of eta Cnc), then

   1 3/4 degree S, 3/4 degree (3m) W to

the center of M44 (note that M44 is inside the body of Cancer, on
the E side).

After gazing spellbound at M44 as discussed below, backtrack to
gamma Cnc, then do this:

Gamma Cnc (4.0, NE star of body), then

   1+ degree S, 2 1/2 degrees (10 1/2m) W to

eta Cnc (5.0, NW star of body, with a mag. 8.5 star 1/4 degree S),
then

   2 1/3+ degrees S, 1/4 degree (1m) W to

theta Cnc (5.0, SW star of body, with a mag. 8.5 star 1/4 degree
NNW), then

   1/3+ degree S, 4 5/8 degrees (19 1/2m) W to

zeta Cnc (4.0, with a mag. 6.5 star 1/2 degree E and a mag. 7.5
star 3/8 degree ESE of the mag. 7.5 star).

     Viewed 3/24/03, 7:00 p.m.  Conditions good.  Iota1 Cnc is a
beautiful wide orange-green pair, with the green component NW of
the brighter orange component.  It can be easily split at 58.75x,
but it looks better at 94x and 267x.  According to reference II-3,
this is "outstanding in even a small telescope because of the
beautiful color contrast between its components".
     Iota2 Cnc is actually a triple star, but I could only see two
of the components, and I could barely see both of those at 267x
(it was a little easier at 534x).  At 58.75x and 94x this looks
like a fat star (fat because there are two stars there).  It is
rather attractive at 534x, with a white star NW of a yellow star.
     M44, the Beehive Cluster, is one of the most attractive groups
of stars in the heavens.  It is also huge, having almost three
times the diameter of the full moon, so you won't get it all in
view at once with your telescope.  With binoculars it really does
resemble a beehive.  My favorite view of it is at 94x;  this shows
quite a number of bright stars, many of them having interesting
colors.  Galileo said "I have noticed 36 stars".  As you pan back
and forth across this cluster, look for the attractive small
equilateral triangle and isosceles triangle (pointing S) near the
center.  Enjoy!
     Zeta Cnc is an attractive and somewhat unusual triple star,
partly because of the yellow color of all three components, and
partly because all three are lined up E-W.  58.75x, 94x, and 267x
show two yellow stars, with the brighter one to the W.  At 534x I
could barely split the one to the W into two stars, but this seems
to be about at the limit for my 9.25 inch telescope (with my
viewing conditions), so don't feel bad if you can only see two
components.  Even so, it makes a nice yellow-yellow double.


29. M82 and M81--Two good galaxies for the price of one

M82 (galaxy)           mag. 8.4           size 11.2'x4.6'
9h 55.8m            69 degrees 41'        Ursa Major

M81 (galaxy)           mag. 6.8           size 26'x14'
9h 55.6m            69 degrees 4'         Ursa Major

Alpha UMa (NW bowl star of Big Dipper, with a mag. 6.5 star 1/8-
degree SW), then

   1 1/3 degrees N, 10 1/2 degrees (92 1/2m) W to

23 UMa (4.5, with a mag. 8.0 star 3/8 degree SE and a mag. 7.5 star
5/8 degree SE;  23 UMa is the E vertex of a nearly equilateral
triangle with side lengths about 2 1/2 degrees, and the other two
stars having mags. 4.5 and 5.0), then

   6 3/4 degrees N, 1/4 degree (3m) E to

24 UMa (4.5, with mag. 7.0 stars 1/4 degree SE and 1/2 degree NE;
there is a small bright isosceles triangle of mag. 5.0 stars 3 1/2
degrees SW of 24 UMa), then

   5/8 degree S, 2/3 degree (7 1/2m) E to

a mag. 5.5 star with a mag. 8.0 star 1/4+ degree WNW and a mag. 7.5
star 3/4 degree SSW), then

   5/8 degree N, 7/8+ degree (11m) E to

a mag. 8.0 star with mag. 8.5 stars 1/4 degree NE and 3/8 degree
NW, then

   1/6 degree S, 1/4 degree (3m) E to

M82 (there are 3 stars in a NW-SE line, with M82 about in the
middle;  M82 is oriented roughly E-W).

After observing M82 as described below, from the center of M82 go
5/8 degree S to the center of M81;  M81 is oriented roughly N-S.

     Viewed 4/1/03, 8:15 p.m.  Conditions good.  M82 is a nice,
pencil-thin galaxy.  It is clearly visible even at 58.75x, but
94x and 267x give better views.
     M81 is much larger than M82, but all I could see of it was a
bright roundish fuzzy blob at its center.  It can be seen at 58.75x
(and in fact with this magnification it is possible to get both M82
and the center of M81 into the same field), but 94x and 267x give
better views.  Of the two galaxies I like M82 better;  even though
its overall brightness is much less than that of M81, it is
sufficiently compact that its shape can be seen better.


30. A sextet of double stars--Regulus (a white-blue double), Gamma
    Leo (a beautiful pair of bright yellow stars), 54 Leo (an
    excellent yellow-blue pair), Denebola (a really wide white-blue
    pair), tau Leo (a stunning wide yellow-blue pair), and xi UMa
    (a close and somewhat difficult pair of nearly equal magnitude
    white stars)
    
Regulus (double star)         mag. 1.35
10h 8.2m            11 degrees 58'               Leo

Gamma Leo (double star)       mags. 2.2, 3.5     separation 4.4"
10h 19.9m           19 degrees 51'               Leo

54 Leo (double star)          mags. 4.5, 6.4     separation 6.5"
10h 55.6m           24 degrees 45'               Leo

Denebola (visual double star) mag. 1.9
11h 49.1m           14 degrees 35'               Leo

Tau Leo (double star)         mags. 5, 8         separation 90"
11h 28m              2 degrees 50'               Leo

Xi UMa (double star)          mags. 4.3, 4.8     separation 1.8"
11h 18.2m           31 degrees 32'               Ursa Major

Regluds is one of our starting points discussed in section IVA.
After viewing it as discussed below, do this:

Regulus, then

   7 5/6 degrees N, 2 7/8 degrees (12m) E to

gamma Leo (with 40 Leo (5.0) 3/8 degree S;  gamma Leo is the third
star in the sickle counting Regulus as the first star, and is the
second brightest star in the sickle).

After viewing gamma Leo as discussed below, do this:

Gamma Leo, then

   2/3 degree N, 12 2/3 degrees (54m) E to

delta Leo (2.5, with a mags. 7.5-7.0 ESE-WNW pair 1/2 degree S),
then

   1/3 degree S, 2 3/4 degrees (12- m) W to

60 Leo (4.5, with mag. 8.0 stars 5/8 degree S and 5/8 degree N),
then

   4 5/8 degrees N, 1 1/2 degrees (6 1/2m) W to

54 Leo (4.5, with a NE-SW pair of mag. 7.5 stars 3/4 degree SW;
delta Leo, 60 Leo, and 54 Leo form a slightly obtuse triangle).

After viewing 54 Leo as discussed below, backtrack to delta Leo,
then go 6- degrees S, 8 1/2 degrees (35m) E to Denebola (2.0, the
tail of the lion, with a mag. 6.0 star 1/4 degree SSW).

After viewing Denebola as discussed below, do this:

Denebola, then

   4+ degrees S, 6 1/8 degrees (25m) W to

iota Leo (4.0, with a mag. 5.5 star 1- degree NNE), then

   4 1/2 degrees S, 3/4 degree (3m) W to

sigma Leo (4.0, with a mag. 6.5 star 5/8 degree N and a mag. 7.0
star 1/2 degree E of the mag. 6.5 star), then

   3 1/8 degrees S, 1 3/4 degrees (7m) E to

tau Leo (5.0, with a mag. 7.5 star 1/8 degree ESE and a mag. 6.0
star 3/8 degree WNW).

After viewing tau Leo as discussed below, backtrack to delta Leo
and do this:

Delta Leo, then

   11 1/8 degrees N, 7/8 degree (4m) E to

xi Uma (3.5, with nu UMa (3.5) 1 5/8 degrees N, and mag. 8.0 stars
5/8 degree W and 1/2 degree ENE).

     Regulus and xi UMa were viewed 4/9/02, 8:15 p.m., conditions
ok except for a bright half moon.  The other four double stars were
viewed 4/1/03, 8:45 p.m., conditions good except starting to get a
little foggy.  Regulus consists of a bright white star with a much
fainter blue star to the NW.  94x gives a good view.  According to
reference II-3 this pair can sometimes be viewed with binoclulars,
but I think you would need excellent conditions to see the faint
blue star that way.
     Gamma Leo is a really nice pair of bright yellow stars (among
the yellowest I have seen) with the brighter one to the NW.  Because
of the glare from these bright stars it took 267x to separate them
well.  The glare made the stars look like a pair of bright yellow
pincushions (or porcupines) traveling across the sky.
     54 Leo is an attractive double with a bright yellow star to
the NW and a fainter blue star to the SE.  It can be split nicely
with 267x.
     Denebola is called a visual double because the two components
are not close to one another in space;  they just happen to appear
close due to their alignment with Earth.  This is a really wide
pair with a bright white star to the N and a fainter blue star to
the S.  You can split it easily at 58.75x but it looks better at
94x.
     Tau Leo is a wide pair with a bright yellow star to the N and
a fainter (but still pretty bright) blue star to the S.  It can be
split easily at 58.75x, but it looks better at 94x.  This is a
beautiful pair, one of the best double stars I have seen.
     Xi UMa is a nearly E-W pair of bright white stars of nearly
equal magnitude (the one to the W is actually slightly S of W).  I
could barely split this at 267x, but 534x split it much more
easily.  These stars are close, and getting closer, so don't feel
bad if you can't split it.  Although you won't be able to see this
visually, according to reference II-3 each component is actually a
double star.  In one double the components are separated by only
half the distance between the Earth and the sun (this is called
1/2 AU, or 1/2 astronomical unit);  these two orbit each other
every 1.8 years.  In the other double the stars are separated by
only the distance from the Earth to the moon, and orbit each other
every four days.
     For those of you who remember the TV comercial that said
"You'll wonder where the yellow went", all you need to do is look
to Leo, and wonder no more!


31. NGC3242 and N Hya--The Ghost of Jupiter Nebula and a pleasing
    pair of nearly identical yellowish stars

NGC3242 (planetary nebula)         mag. 7.8           size 25"
10h 24.8m               -18 degrees 38'             Hydra

N Hya (double star)                mags. 5.8, 5.9
11h 32m                 -29 degrees 20'             Hydra

Regulus, then

   12 1/6 degrees S, 1/8 degree (1/2m) W to

alpha Sex (4.5, with a mag. 7.0 star 1 1/4 degrees N and a second
mag. 7.0 star 3/8- degree NE of the first mag. 7.0 star), then

   12 degrees S, 3/4 degree (3m) E to

lambda Hya (3.5, with a mag. 6.0 star 1/4+ degree NNW, a mag. 5.5
star 1/2 degree S, and nu2 Hya (4.5) 1 1/2 degree WSW), then

   4 1/2 degrees S, 3 3/4 degrees (16m) E to

mu Hya (4.0, with a mag. 7.5 star 5/8 degree SSE, mag. 8.5 stars
1/4- degree N and 3/4 degree SSW, and nu Hya (3.0) 2/3 degree N
and 5 3/4 degrees E), then

   5/8 degree SSE to

a mag. 7.5 star, then

   7/8 degree S to

a mags. 7.5-8.0 E-W pair, then from the mag. 7.5 star

   3/8 degree S, 5/8 (2 1/2m) degree W to

NGC3242.

After viewing NGC3242 as discussed below, do this:

NGC3242, then

   3/8 degree N, 8 3/8 degrees (35- m) E to

alpha Crt (4.0, with a mag. 8.0 star 1/2 degree W, a mag. 7.5 star
3/4 degree NNW, and nu Hya (3.0) 3 1/4 degrees NW), then

   9 degrees S, 1 1/4 degrees (5 1/2m) E to

chi1 Hya (5.0, with chi2 Hya (5.5) 1/8+ degree E, a mag. 7/5 star
1/2 degree WSW, and a mag. 6.0 star 3/4 degree NW), then

   3/4 degree S, 3/4 degree (3m) E to

a mag. 5.5 star, with a mag. 8.5 star 1/4 degree SE and a mag. 7.5
star 3/8 degree SSE, then

   1/4 degree S, 2 5/8 degrees (12m) E to

a mag. 6.5 star with mag. 7.0 stars 1/4- degree NW and 1/4 degree
SW, forming a right triangle, and a mag. 7.5 star 1/4+ degree W of
the SW star in the triangle, then

   1/3 degree N, 2 1/8 degree (9 1/2m) E to

a mag. 6.5 star with a mag. 8.5 star 3/8 degree NNW, then

   1 1/4 degrees S, 2/3 degree (3m) E to

N Hya (5.0, with a mag. 7.5 star 1/8 degree NNE, a mag. 8.0 star
1/2+ degree NNW, a mag. 5.0 star 1 7/8 degree S, and xi Hya (3.5)
3/4 degree S of the mag. 5.0 star;  N Hya and the last two stars
form a line which is easily recognizable if it is not too far south
for you to see it).

     Viewed 4/22/03, 8:30 p.m.  Conditions excellent at the start,
but getting a little foggy by the time I found N Hya.  NGC3242 is
A bright, fairly large planetary nebula that looks like a round,
whitish ball, with maybe a little bluish tinge.  It seems to
flicker on and off sometimes.  This is one of the few cases where
the LPR (light pollution reduction) filter may have helped a little
at 94x.  This nebula is easily recognizable even at 58.75, but
looks better with the higher powers.  At 534x it looked to me like
a small ghostly white full moon.  There are faint stars about 1/12
degree SE and a little S, forming a right triangle, which serve as
good stars to focus on when changing magnifications.  NGC3242 is
one of the better planetary nebulas for viewing, even though it is
pretty far south.  Incidentally, this nebula is somtimes called the
"CBS Eye Nebula".
     N Hya is a pleasant little double star with two nearly
identical yellowish components arranged SW-NE.  It can be split at
58.75x, but it looks better at 94x, and it looks best at 267x.
     Since N Hya is far enough south that you may have tree
problems with it, with it appearing only briefly in a gap between
trees, a few more words on the path may be in order.  The path was
chosen to come in basically from the N and W because that often
lessens gap problems.  One way of approaching this star is to go to
alpha Crt, follow it until it is above the gap, move 9 degrees S
(with 3 24/60 turns of your Dec slow-motion knob), then instead of
going 1 1/4 degree E, just wait 5 1/2 minutes for chi1 Hya to come
into view, look for the check stars (by the way, the pair
chi1 Hya - chi2 Hya is quite conspicuous in binoculars), center
chi1 Hya, then move 3/4 degree S and wait 3 minutes for the mag. 5.5
star to appear, and continue in this way.  If you find yourself
drifting out of the gap you may need to adjust yourself back in
with your RA slow-motion knob, remembering that for each 6
clockface minutes (that is, 1/10 turn) you turn the knob W you will
have to wait an extra minute for the next object, and for every 6
clockface minutes (that is, 1/10 turn) you turn the knob E you will
have to wait one minute less for the next object.  In my case there
was a tall tree overhanging the gap which would have swallowed up
alpha Crt if I had waited for it to get over the gap, so after
finding alpha Crt I went to chi1 Hya right away, then to the next
star in the path, then followed this star across the sky with
little westward adjustments with the RA slow-motion knob until it
got over the gap (and under the tall tree), then did the rest of
the path by adjusting the Dec slow-motion knob appropriately and
waiting for the next object.  If your view of the terrestrial
southern horizon is clear all the way down to the declination of N
Hya, then you can follow the path directly as given earlier without
waiting.


32. An eight-galaxy tour through the Virgo Galaxy Cluster--M84, M86,
    M87, M58, M60, M59, M85, and M49

M84 (galaxy)              mag. 9.3           size 5.0'x4.4'
12h 25.1m            12 degrees 53'          Virgo

M86 (galaxy)              mag. 9.2           size 7.4'x5.5'
12h 26.2m            12 degrees 57'          Virgo

M87 (galaxy)              mag. 8.6           size 7.2'x6.8'
12h 30.8m            12 degrees 24'          Virgo

M58 (galaxy)              mag. 9.8           size 5.4'x4.4'
12h 37.7m            11 degrees 49'          Virgo

M60 (galaxy)              mag. 8.8           size 7.2'x6.2'
12h 43.7m            11 degrees 33'          Virgo

M59 (galaxy)              mag. 9.8           size 5.1'x3.4'
12h 42.0m            11 degrees 39'          Virgo

M85 (galaxy)              mag. 9.2           size 7.1'x5.2'
12h 25.4m            18 degrees 11'          Coma Bernices

M49 (galaxy)              mag. 8.4           size 8.9'x7.4'
12h 29.8m            8 degrees 0'            Virgo

Arcturus, then

   1 5/8 degrees S, 15 2/3 degrees (65 1/2m) W to

alpha Com (4.5, with a mag. 8.0 star 1/8 degree W, a mag. 7.5 star
5/8- degree E, and a mag. 6.0 star 3/4 degree S), then

   6 1/2+ degrees S, 2- degrees (8m) W to

epsilon Vir (3.0, with a mag. 7.5 star 1/2 degree NE and a mag. 8.5
star 1/4 degree NW), then

   1/2 degree N, 1 5/8 degree (7m) W to

a mag. 7.0 star with a mag. 9.0 star 1/8+ degree W and 41 Vir (6.0)
1 degree NNW, then

   1/2 degree N, 2 degrees (8+ m) W to

34 Vir (6.0, with a mag. 9.5 star just N and a mag. 7.0 star 3/8+
degree ENE;  be careful not to confuse the second check star with
34 Vir, as the check star also has a faint star (mag. 10.0) N of
it, but a little farther away from it than the mag. 9.5 star is
from 34 Vir), then

   4 2/3 degrees (19m) W to

the center of an isosceles triangle of mag. 8.0 stars pointing S;
the S star is about 1/3 degree from the other two, which are about
1/4 degree apart. This triangle, which will be used in finding the
galaxies, will be referred to in the rest of this item as "The
triangle".  Then from the center of "The triangle",

   2/3 degree (3- m) W to

a mag. 9.0 star with a mag. 10.0 star just WNW, then

   1 degree N, slightly W to

M84 (with a mag. 10.0 star 1/8 degree W).

M86 is slightly N and 1/4 degree (1+ m) E of M84.

After viewing M84 and M86 as discussed below, backtrack to "The
triangle", then do this:

NE star of "The triangle", then

   3/4 degree (3m) E to

a mag. 8.0 star, then

   3/8 degree N, 1/4- degree (1- m) W to

a mag. 8.5 star, then

   1/8- degree S to

M87.

After viewing M87 as discussed below, backtrack to "The triangle"
and do this:

S star of "The triangle", then

   2 1/8 degrees (9m) E to

a mag. 8.0 star with a mag. 9.0 star 1/3 degree N, then

   1/8 degree (1/2m) E to

M58.

After viewing M58 as discussed below, do this:

Last star in path before M58 (mag. 8.0), then

   slightly S, 1 3/8+ degrees (6- m) E to

a mag. 9.5 star with a mag. 10.0 star 1/3 degree W, then

   1/4 degree S, 1/8+ degree (1/2+ m) E to

M60.

M59 is 1/3+ degree WNW of M60;  it is also 1/8+ degree SSW of the
mag. 10.0 star mentioned in the path to M60.  M60, the mag. 9.5
star mentioned in the path to M60, and M59 form a near-isosceles
right triangle with right angle at the mag. 9.5 star.

After viewing M60 and M59 as discussed below, backtrack to "The
triangle" and do this:

NW star of triangle, then

   5 5/8 degrees N, 1 2/3 degrees (7m) W to

11 Com (5.0, with a mag. 8.0 star 1/3 degree W, a mag. 10.0 star
1/8- degree SSE, and a N-S pair of mag. 9.5 stars 1/2- degree E),
then

   slightly N, 1- degree (4m) E to

a mag. 8.5 star with a mag. 7.0 star 2/3+ degree E, and mag. 9.5
stars 1/4- degree WSW and 1/4 degree SSE, then

   1/3 degree N, 1/8+ degree (1/2+ m) E to

M85.

After viewing M85 as discussed below, backtrack to "The triangle"
and do this:

S star in "The triangle", then

   3 1/4- degrees S, 1/8+ degree (1/2+ m) W to

a mag. 6.5 star with a mag. 8.5 star 1/3- degree N and a mag. 8.0
star 5/8 degree S, then

   5/8 degree S, slightly W to

a mag. 8.0 star with a mag. 8.0 star 1/4 degree WSW and a mag. 9.5
star 1/6 degree ESE, then

   5/8- degree (2 1/2- m) E to

M49 (with a mag. 6.0 star 5/8- degree SE).

     The first four galaxies were viewed 4/26/03 starting at 9:30
p.m. under excellent conditions.  M84 appears as a faint hazy
patch which can be seen at 58.75x, 94x. 267x, and 534x, but it
looks best at 267x.  The mag. 10.0 star 1/8 degree W is a good
focus star.
     M86 is also a faint hazy patch which is a little fainter than
M84.  Both of these galaxies can be gotten into the same field at
94x, but 267x gives the best view of M86.
     M87 is one of the brightest of the bunch.  It is another faint
hazy patch which is readily visible at all four magnifications,
with 267x giving the best view.  By the way, this galaxy is also
known as the Virgo A radio source;  it emits powerful radio and
x-ray emissions, and many astronomers believe there is a giant
block hole at its center which contains millions of times more mass
than our sun.  This information comes from reference II-3, along
with the following quote:  "One of the brightest and most
interesting galaxies in the Virgo Cluster, ... a huge peculiar
galaxy."
     M58 is another faint fuzzy blob, much fainter than M84 and
M86.  I could barely see it at all four magnifications, so this
galaxy may make a good test of your telescope and viewing
conditions.
     The next two galaxies were viewed 4/27/03 starting at 8:45
p.m. under excellent conditions.  M60 is a faint fuzzy blob
similar to M84, easily visible at all four magnifications.
     M59 is fainter than M60, but it is easily seen at 58.75x, 94x,
and 267x (and barely seen at 534x;  high power often makes faint
diffuse objects hard to see).  One can get both M60 and M59 into
the same field at 94x.  The mag. 10.0 star mentioned in the path
to M59, a fainter star just E of this star, and M59 form a right
triangle with right angle at the last-mentioned star.
     M85 was viewed 4/28/03 at 8:55 p.m. under less favorable
conditions;  there seemed to be some high thin clouds at the start
and the big telescope lens was starting to get a little foggy at
the end.  Still, this faint fuzzy patch was clearly visible at
94x and 267x.  There is a faint star just SE of M85 that helps with
location and focusing.
     M49 was viewed 4/30/03 at 9:15 p.m. under fair-to-poor
conditions (due to high humidity).  It was a faint fuzzy blob with
a brighter center;  this galaxy seemed bigger than the others in
this bunch, and was among the brightest of them.  It could be seen
at all four magnifications, but was best at 267x.
     In case you are wondering why some fainter stars than usual
are mentioned in the path and discussion for these galaxies, it is
because I was using a more detailed chart (chart B) in reference
II-4 that showed more stars.  It should be pointed out that the
galaxies mentioned here are only a small sample from the many
galaxies in the Virgo Galaxy Cluster.
     


33. M104--The Sombrero Galaxy (and nearby multiple star)

M104 (Sombrero Galaxy)       mag. 8.3        size 8.9'x4.4'
12h 40m           -11 degrees 37'            Virgo

Spica, then

   about 10 degrees W to

the south-central part of an isosceles trapezoid with long base
extending 4 2/3 degrees NE-SW, short base to the NW 2 degrees
long, slanted sides 4 degrees long;  stars and magnitudes clockwise
from S are a mag. 5.0 star, psi Vir (4.5), chi Vir (4.5), and
21 Vir (5.5).  This trapezoid is easy to pick out with binoculars
since the four stars are brighter than any other stars in the
immediate vicinity.  Then from the S star of the trapezoid, which
has a mag. 7.0 star 3/4 degree E and a mag. 6.5 star 1 1/8 degrees
S,

   1 1/2- degrees N, 3/8 degrees (1 1/2m) W to

M104 (with a mag. 7.5 multiple star 3/8 degree WNW and a mag. 6.5
star 1 degree ESE).

     Viewed 6/18/03, 9:30 p.m.  Conditions excellent.  This is a
good galaxy;  it looks like a blob (elongated E-W) with a bright
center.  It can be seen at 58.75x, 94x, 267x, and 534x, with the
last two of these giving the best view.  On my chart there is a
mag. 7.5 star marked 3/8 degree WNW of M104 with a line through it
to indicate a multiple star;  actually there are four bright
components there, showing an interesting color contrast, in a shape
that a golfer may recognize.  At higher power a fifth component
can be seen.  There is a faint star about 1/16 degree WSW of M104
with another faint star about 1/16 degree WSW of that, making a
line pointing to M104.  M104 received one exclamation point in the
NGC catalog.
     This example illustrates the use of a group of stars as an
intermediate object in a path rather than just a single star.  If
conditions are such that the isosceles trapezoid is not readily
visible, you can get directly from Spica to the S star of the
trapezoid by going 1 7/8 degrees S, 10 3/4 degrees (44m) W.


34. M64--The Black Eye Galaxy

M64 (Black Eye Galaxy)         mag. 8.5         size 9.3'x5.4'
12h 56.7m             21 degrees 41'            Coma Bernices

Arcturus, then

   1 5/8 degrees S, 15 2/3 degrees (65 1/2m) W to

alpha Com (4.5, with a mag. 8.0 star 1/8 degree W, a mag. 7.5 star
5/8- degree E, and a mag. 6.0 star 3/4 degree S), then

   1/8 degrees S, 2 5/8 degrees (11m) W to

36 Com (5.0, with 38 Com (6.0) 5/8 degree ESE and a mag. 8.0 star
3/8 degree NNW), then

   3 7/8 degrees N, 1 3/8 degrees (6- m) W to

35 Com (5.0;  this is a double star with a bright white component
and a much fainter blue component a little ESE, perhaps needing
94x to see the blue component), then

   1/2- degree N, 7/8 degree (4- m) E to

M64.

     Viewed 4/10/03, 10:00 p.m.  Conditions poor due to a bright
half moon.  I was able to spot M64 easily even at 58.75x;  94x,
267x, and 534x all gave interesting views.  There is a faint star
about 1/16 degree SE of M64 that is useful for focusing and for
keeping track of the location of M64 as you change magnification.
What I could see of M64 looked like a faint roundish blob with a
bright center that looked like a fat star.  This is a pretty good
galaxy, especially considering that it was easy to find even with a
bright half moon.  It gets its name from a dark dust lane that is
(according to reference II-13) "obvious in photographs but visually
difficult in small telescopes".  I didn't see the dust lane, but
the galaxy was good anyway.  This galaxy received one exclamation
point in the NGC catalog.


35. The Horse and Rider (Mizar and Alcor), two nice double stars
    (kappa Boo and alpha CVn), the Sunflower Galaxy (M63), and the
    superb "hypnotic eye" galaxy (M94)

Mizar (double star)        mags. 2.2, 3.9       separation 14.4"
13h 24.0m             54 degrees 56'            Ursa Major

Kappa Boo (double star)    mag. 4.5
14h 14m               51 degrees 45'            Bootes

Alpha CVn (double star)    mags. 2.9, 5.5       separation 19.3"
12h 56.0m             38 degrees 19'            Canes Venatici

M63 (Sunflower Galaxy)     mag. 8.6             size 12.3'x7.6'
13h 15.8m             42 degrees 2'             Canes Venatici

M94 (galaxy)               mag. 8.1             size 11.0'x9.1'
12h 50.9m             41 degrees 7'             Canes Venatici

Mizar is the middle star in the handle of the Big Dipper.
Alcor (4.0) (otherwise known as 80 UMa) is 12' E of Mizar.

After viewing Mizar and Alcor as discussed below, use your
finderscope to move to the star at the end of the handle of the
Big Dipper (which is called eta UMa), then do this:

Eta UMa (1.9), then

   2 1/2 degrees N, 4 degrees (26m) E to

kappa Boo (4.5, with iota Boo (5.0) 5/8 degree SSE and theta
Boo (4.0) 1 7/8 degrees E, forming a triangle which is easily
visible in binoculars or finderscope).

After viewing kappa Boo as described below, backtrack to eta UMa,
then go 11 degrees S, 10 1/8 degrees (51 1/2m) W to alpha CVn (2.5).

After viewing alpha CVn as discussed below, do this:

Alpha CVn (2.5), then

   2 1/4 degrees N, 4 1/8 degrees (22m) E to

20 CVn (4.5, with a mag. 8.0 1/8 degree just ESE, 19 CVn (6.0) 1/2
degree NW, 23 CVn (5.5) 3/4 degree SE, and a mag. 5.0 star 7/8
degree SW), then

   1/2 degree NW to

19 CVn (6.0), then

   1 1/6 degree N to

M63 (with a mag. 6.5 star 5/8 degree N and 1/2 degree (3m) W;  you
will probably not need this star, but you may find it helpful to
look for a faint star with a fainter star just SW, and M63 just SE
of the faint star--it may require 94x to see the fainter star, but
you can change to 94x while you are looking at 19 CVn if you wish).

After viewing M63 as discussed below, backtrack to alpha CVn and
do this:

Alpha CVn (2.5), then

   3 degrees N, 4 1/8 degrees (22m) W to

beta CVn (4.0, with a mag. 6.5 star 1 1/4 degrees W), then

   1/2 degree S, 1- degree (5m) E to

9 CVn (6.5, with a mag. 8.0 star 1/8 degree NNE and a mag. 7.5 star
5/8 degree WSW;  9 CVn is nearly on the line connecing alpha CVn
and beta CVn), then

   1/2- degree N, 1- degree (5m) E to

a mag. 8.0 star, surrounded by 4 fainter stars, with a mag. 7.5
star 3/4 degree SSW, then

   1/8 degree S, 1 3/8 degrees (7 1/2m) E to

M94 (with a mag. 8.0 star 1 degree S and 1/4 degree E).

     Viewed 4/11/03, 9:00 p.m.  Conditions poor due to bright 1/2+
moon.  Many people know Mizar and Alcor by the name given to them
by American Indians (and probably other people too), namely the
horse (Mizar) and rider (Alcor).  People with good vision can see
this pair with the naked eye under good enough viewing conditions,
and in fact it is said that American Indians used this as a test of
vision for the young braves, asking them where the rider was in
relationship to the horse.  It is not as well known that Mizar is
a gorgeous double star by itself, consisting of a pair of white
stars, with the brighter one to the NNW.  This pair can be split
at 58.75x, but it looks better at 94x, and it looks better still at
267x.  In actuality, both of these components are also double
stars, but this cannot be determined visually with a telescope;
the motions of the sub-components show up when the light from these
stars is analyzed with a spectroscope.
     Kappa Boo is an attractive double, with a bright yellowish
white component and a fainter (but also fairly bright) dark blue
component to the SW.  This can be split at 58.75x, but it looks
better at 94x, and it looks really nice at 267x.  By the way, the
name of the constellation Bootes the hersdman should have an umlaut
(pair of dots) above the second letter o, but I can't make umlauts
in this plain text document, so I left it out.
     Alpha CVn (also known as Cor Caroli) is a nice double with
a bright white component and a fainter (but not dim) bluish
component to the SW.  This can be split at 58.75x, but it looks
better at 94x, and it looks better still at 267x.
     The remaining two objects were viewed 4/27/03, starting at
11:00 p.m. under excellent conditions.  M63 is a pretty good
galaxy;  it is not terribly bright, but it does look bigger than
most of our faint galaxy blobs.
     M94 is a superb galaxy!  It is one of the best of the "small"
galaxies in our list.  The reason I called it the "hypnotic eye"
galaxy is a sentence in reference II-13 that says "Seems to peer
back at you like a hypnotic eye."  (OK, I know that is not a true
sentence because it lacks an expressed subject, but if you noticed
that, then I applaud your English skills.)  Reference II-13 also
says "Easy to find and see in the littlest of telescopes."  At
58.75x M94 looked like a fat fuzzy star, it was better at 94x, and
at 267x and 534x one could see a bright central blob and a fainter
surrounding halo (like the iris and white part of an eye).  There
was an occasional blinking effect.  You may note good focus stars
1/8 degree N and 1/16 degree W of M94.


36. M51--The Whirlpool Galaxy

M51 (Whirlpool Galaxy)          mag. 8.1         size 11.0'x7.8'
13h 29.9m            47 degrees 12'             Ursa Major

Eta UMa (1.9, end of handle of Big Dipper), then

   2/3 degree S, 2 1/8 degrees (13m) W to

24 CVn (4.5, with a mag. 6.5 star 5/8 degree NNE and a mag. 8.0 star
1/4 degree WNW), then

   1 1/4 degrees S, 1/4- degree (1 1/2m) W to

a mag. 7.5 star with a mag. 7.0 star 3/4 degree W, then

   1/2 degree S, 1/4- degree (1 1/2m) W to

a mag. 7.0 star, then

   3/8 degree (2m) W to

M51.

     Viewed 4/27/03, 10:45 p.m.  Conditions excellent.  This galaxy
is excellent, and pretty bright.  It consists of galaxies NGC5194
and, adjacent to this and just to the NNW, NGC5195.  I could barely
see them at 58.75x, but 94x was better and 267x was better yet;  I
could get both galaxies into the same field at 267x.  This is one
of the few objects that is given three exclamation points in the
NGC catalog.


37. M83--A galaxy which is best for southern observers, plus a red
    long-term variable star (R Hya) and an unexpectedly nice little
    double star along the way

R Hya (variable star)       mag. 3-11
13h 30m               -23 degrees 20'             Hydra

double star                 mag. 5.5
13h 37m               -26 degrees 30'             Hydra

M83 (galaxy)                mag. 7.6              size 11.2'x10.2'
13h 37.0m             -29 degrees 52'             Hydra

Spica, then

   12 degrees S, 1 1/2- degrees (6 1/2- m) W to

gamma Hya (3.0, with mag. 7.0 stars 3/4 degree SE, 3/4 degree NE,
and 7/8 degree ENE, and a mags. 8.0-5.0-6.5 threesome 2 1/4 degrees
W).

Now if you want to check out R Hya, do this:

Gamma Hya, then

   1/6 degree S, 2 1/2 degrees (11m) E to

R Hya (3-11 red variable, with a mag. 7.0 star 3/8 degree S and
1/8 degree (1/2m) E, a mags. 8.5-7.5 E-W pair 1- degree NNW, and
with a mags. 8.5-8.5 SE-NW pair 1/4 degree W of the E-W pair).

After looking at R Hya as discussed below, backtrack to gamma Hya
and do this:

Gamma Hya, then

   3 1/3 degrees S, 4 degrees (18m) E to

a mag. 5.5 double star, with a mag. 7.5 star 1 1/8- degrees E, a
mag. 7.0 star 7/8 degrees W, and a mag. 8.5 star 1/2+ degree SE.

The mag. 5.5 double star mentioned above is the one discussed below;
after looking at it, do this:

The mag. 5.5 double, then

   3+ degrees S, 3/8+ degree (2m) E to

a mag. 5.5 star with a mag. 7.0 star 1/4 degree SSW and a mag. 8.0
star 1/2 degree ESE, then

   1/4 degree SSW to

the mag. 7.0 star above, then

   slightly S, 5/16 degree (1 1/2m) W to

M83.

     Viewed 5/20/03, 11:00 p.m.  Conditions pretty good except for
several yard lights in neighbors' yards to the terrestrial south of
mine.  According to reference II-3, R Hya is a long-term variable
star which varies in magnitude from 3 to 11 over a period of 390
days, and is noted for its deep red color;  it is the third
long-term variable star discovered, and is one of the easiest to
observe.  To me it looked reddish or orangey.  It looked to be not
as bright as the mag. 7.0 check star to its SE, but when I looked
at it again 11 days later it looked a little more than twice as
bright as the check star, which would translate to about a magnitude
brighter, so I would estimate its brightness at about 6 on May 31.
On June 18 under excellent conditions it seemed brighter still;  it
was now visible with binoculars and finderscope, and although it
was not as bright as the mag. 5.0 check star which is part of the
threesome 2 1/4 degrees W of gamma Hya, it was almost as bright.
There is a faint star about 1/8 degree NW of R Hya and a SW-NE pair
of faint stars about 1/16 degree apart about 1/8 degree NE of R Hya;
these stars (along with the color) can help to verify that you are
really looking at R Hya even when it is dim.
     The mag. 5.5 double star consists of a blue-white star with a
fainter yellowish or orangey star to the S.  It can be split at
58.75x, but it looks better at 94x and 267x.  This is a pretty nice
one!
     I could just see M83 at 58.75x, 94x, and 267x as a faint patch
of light.  At 94x the addition of an LPR filter may have helped
slightly.  There was a faint star about 1/12 degree S of M83 and
another faint star about 1/12 degree W of this star, forming a
nearly isosceles triangle with the angle at the first faint star
being a little less than 90 degrees.  These faint stars are useful
for marking the location of M83 and for focusing.  Reference II-13
refers to M83 as "A delightful spiral for small telescopes", and
reference II-3 mentions that it is one of the 10 largest galaxies
and one of the 25 brightest.  This galaxy received two exclamation
point in the NGC catalog.  Reference II-3 also mentioned, however,
that M83 is hard for northern observers to see because it is very
low in the southern sky, in a field rich with star clouds and dust.
As described above it did not look all that impressive to me, but I
expect it could be quite attractive to viewers located farther
terrestrial south, and without the consarned yard lights (pardon my
French).  By the way, for me this galaxy appeared only for about a
half hour in a V-shaped gap between trees, so I had to wait a while
after looking at the double star discussed above and just follow
the double star until the rest of the path (including M83)
trundled into the gap.


38. M3--A glorious globular cluster

M3 (globular cluster)          mag. 6.4         size 16'
13h 42.2m            28 degrees 23'             Canes Venatici

Arcturus, then

   8 1/3 degrees N, 4 1/4 degrees (19m) W to

9 Boo (5.0, with a mag. 8.0 star 3/4- degree W and 11 Boo (6.0)
1 degree E), then

   3/8+ degree N, 1 3/4 degrees (8m) W to

a mag. 7.0 star with a mag. 8.0 star 1/2 degree W, then

   slightly N, 1/2 degree (2m) W to

the mag. 8.0 star above, then

   1/8- degree N, 1 3/8- degrees (6- m) W to

a mag. 6.5 star with a mag. 7.0 star 7/8 degree ESE, then

   1/2 degree NE to

M3 (with a mag. 8.5 star 1/8 degree SSE).

     Viewed 4/25/03, 8:30 p.m.  Conditions excellent.  At 58.75x
M3 appears as a bright ball of light, at 94x it appears as a
bigger ball of light, at 267x the ball of light gets bigger still
and one sees many individual stars around the edges and inside the
ball, and at 534x the ball is yet bigger and more individual stars
are seen.  My favorite view is at 267x, since the image is sharper
than at 534x, but the view is breathtaking at either of these two
higher powers.  This object contains about 45,000 stars (but don't
try to count all of them).  M3 deservedly received two exclamation
points in the NGC catalog.  Since this is the first nice globular
cluster to appear in the Spring, perhaps we should call it the
Robin Cluster.


39. A seven-star tour through Bootes and Corona Borealis--Pi Boo
    (an unusual blue-yellow double), xi Boo (a white-reddish
    double), epsilon Boo (a striking but somewhat difficult white-
    bluegreen double), delta Boo (a wide white-deep blue pair),
    mu2,1 Boo (an amazing white-white-blue-blue quadruple), zeta CrB
    (a white-bluewhite double), and sigma CrB (a white-bluewhite
    double)

Pi Boo (double star)       mags. 4.8, 5.8     separation 5.5"
14h 40.7m          16 degrees 25'             Bootes

Xi Boo (double star)       mags. 4.7, 6.9     separation 6.6"
14h 51.3m          19 degrees 6'              Bootes

Epsilon Boo (double star)  mags. 2.6, 4.7     separation 2.6"
14h 44.9m          27 degrees 4'              Bootes

Delta Boo (double star)    mag. 3.5
15h 15.5m            33 degrees 19'             Bootes

Mu2,1 Boo (double star)    mag. 4.0
15h 24.5m          37 degrees 25'             Bootes

Zeta CrB (double star)     mags. 5.0, 5.9     separation 6.3"
15h 39.9m          36 degrees 38'             Corona Borealis

Sigma CrB (double star)    mag. 5.0
16h 14.7m            33 degrees 52'             Corona Borealis

Arcturus, then

   2 5/6 degrees S, 6 degrees (25m) E to

pi Boo (4.5, with omicron Boo (4.5) 1 1/4 degrees ENE).

After viewing pi Boo as discussed below, do this:

Pi Boo (4.5), then

   1/2 degree N, 1 1/8- degrees (4 1/2m) E to

omicron Boo (4.5, with a mag. 7.5 star 1/4 degree SE), then

   2 1/8 degrees N, 1 1/2 degrees (6+ m) E to

xi Boo (4.5, with mag. 8.0 stars just E and 3/8- degree E, and a
mag. 6.0 star 1/2 degree E).

After viewing xi Boo as discussed below, go 8 degrees N, 1 3/8
degrees (6 1/2- m) W to epsilon Boo (2.0, with a mag. 8.0 star 1/2
degree NNE and W Boo (4.5-5.5 variable) 5/8 degree SSW).

After viewing epsilon Boo as discussed below, go 6 1/4 degrees N,
6 1/2 degrees (30 1/2m) E to delta Boo (3.5, with a mag. 8.0 star
1/4 degree NNW and a mag. 6.5 star 5/8 degree ENE).

After viewing delta Boo as discussed below, go 4 1/8 degrees N,
1 3/4 degrees (9m) E to mu2,1 Boo (4.0, with a mag. 8.5 star 1/2
degree E).

After viewing mu2,1 Boo as discussed below, go 3/4 degree S, 3
degrees (15m) E to zeta CrB (4.5, with mag. 7.0 stars 1/2- degree
NNE and 1/2- degree SSW).

After viewing zeta CrB as discussed below, do this:

Zeta CrB (4.5), then

   1 degree S, 2 3/8 degrees (12- m) E to

kappa CrB (5.0), then

   1 5/6 degrees S, 4 7/8 degrees (23 1/2m) E to

sigma CrB (5.0, with a mag. 8.0 star 7/8- degree NW, a mag. 6.5
star 7/8 degree SW, and the N-S pair nu1 CrB (5.0), nu2 CrB (5.0)
1 5/8 degrees E).

     Viewed 5/2/03, 9:30 p.m.  Conditions wonderful--a cold front
had passed through the previous night, leaving the air cooler,
drier, and clearer, which allowed better views of many of these
objects than when I looked at them earlier.  Pi Boo is a striking
blue-yellow double, which is unusual because the blue component,
which is to the W of the yellow component, is the brighter of the
two.  Although I could not split this at 58.75x, I could detect the
colors;  94x splits it, and 267x is better.  Pretty!  The colors
did occasionally fluctuate a little.
     Xi Boo consists of a bright white component with a fainter
(sort of) reddish component to the WNW.  I could barely split it at
58.75x, but 94x was better and 267x was best.
     I could just barely split epsilon Boo at 267x, and 534x was
better.  It consists of a bright white (or orangey) component with
a fainter (but not dim) and attractive blue-green component a
little W of due N from the brighter component.  Reference II-11
says this pair is difficult to split, and at times in the past
under poorer conditions I could not split it (since the brighter
component is so bright that its glare sometimes drowns out the
other component), and recently I was able to only barely split it at
534x.  At that time I checked the position angle, which was
fortunately given in appendix 6 in reference II-3; it was 344
degrees, which fitted with my observation, and confirmed that I had
actually split it.  This striking double illustrates that viewing
conditions can make a difference even in viewing double stars.  By
the way, don't feel bad if you can't split this, since even with a
9.25 inch telescope I couldn't split it at first.  Knowing where
the blue-green component is with respect to the white one may help
you.
     Delta Boo is a wide, fairly attractive bright white-fainter
deep blue pair, with the blue component to the ENE of the white
one.  This splits easily at 58.75x but it looks better at 94x.
     Mu2,1 Boo (otherwise known as 51 Boo or Alkalurops) was a
suprise which provided two "Oh my Lord look at that" experiences.
I had looked at this star before at 94x, and since it split easily
into a nice bright yellowwhite-fainter bluish pair at 94x and even
at 58.75x, I didn't try higher magnifications, but this night for
some reason I tried 267x and was startled to see that the bluish
component appeared to actually be a pair of bluish stars (knowing
this, I was even barely able to see the pair at 94x later).  534x
comfirmed this pair, and at that magnification I was startled again
to see a little white component just S of the bright white star!
Knowing this, I was able to later barely split this pair at 267x.
All four components are lined up N-S.
     Zeta CrB consists of a bright white component with a somewhat
fainter blue-white component to the WNW.  It is attractive at
58.75x, 94x, and 267x.
     Sigma CrB is a pretty bright white-not quite as bright blue
pair, with the blue component to the SW of the white component.
It can be split at 58.75x, but 94x is better and 267x is best.
     All seven stars in our tour of Bootes and Corona Borealis are
pretty good, but I consider two of them (pi Boo and epsilon Boo) to
be "star" stars, with mu2,1 Boo being a "superstar" star!


40. M5--A bright globular cluster

M5 (globular cluster)         mag. 5.8          size 21'
15h 18.6m            2 degrees 5'               Serpens

First, follow the line from beta CrB (3.5, crown star NW of
Alphekka) through Alphekka, bending a little left, about 9 degrees
SSE to a "triangle plus one":  gamma Ser (3.5) and beta Ser (3.5)
are 2 5/8 degrees apart, with beta W of gamma, kappa Ser (4.0) is
about 3 degrees from each, to the N, and iota Ser (4.5) is 2 1/3
degrees NW of kappa, about on a gamma-kappa line.

   Then follow a kappa-beta line, bending a little W, 5 3/8
   degrees SSW to

delta Ser (3.5), then

   bend about 45 degrees right and go 4 2/3 degrees SSE to

alpha Ser (2.5, with lambda Ser (4.5) 1 1/8 degree NNE), then

   2 1/2 degrees further, slightly right of a delta-alpha line, to

epsilon Ser (3.5), then

   follow the perpendicular bisector of the alpha-epsilon line
   segment about 7 1/2 degrees WSW to

another "triangle plus one":  10 Ser (5.0), 5 Ser (5.0), and 6 Ser
(5.5) form a near-isosceles triangle pointing ENE with 10 Ser at
ENE, 5 Ser 2 1/4 degrees W of 10 Ser, and 6 Ser 2 1/4 degrees SW of
10 Ser, 5 Ser and 6 Ser being 1 1/8 degrees apart, while 4 Ser (5.5)
is 1 3/8 degrees WSW of 6 Ser.  Then, from 5 Ser

   3/8 degree NNW to

M5 (which can be seen as a fuzzy ball in binoculars even in poor
viewing conditions).

     Viewed 6/4/02, 11:30 p.m.  Conditions ok.  A magnificent ball
of light, although its individual stars are not seen separately as
well as those of M13 are.  267x would give a good view, and 534x
probably would too (I was using an extra attachment that gave 336x).
This object is given two exclamation points in the NGC catalog.


41. A bright double star and four globular clusters--Beta Sco, M80,
    M4, M19, and M62

Beta Sco (double star)        mags. 2.6, 4.8     separation 13.7"
16h 5.4m             -19 degrees 48'             Scorpius

M80 (globular cluster)        mag. 7.2           size 8'
16h 17.0m            -22 degrees 59'             Scorpius

M4 (globular cluster)         mag. 5.9           size 34'
16h 23.6m            -26 degrees 32'             Scorpius

M19 (globular cluster)        mag. 7.2           size 14'
17h 2.6m             -26 degrees 16'             Ophiuchus

M62 (globular cluster)        mag. 6.6           size 10'
17h 1.2m             -30 degrees 7'              Ophiuchus

Antares, then

   7/8 degree N, 1 7/8 degrees (8 1/2m) W to

sigma Sco (3.0, with a mag. 6.0 star 1 1/2 degrees WNW), then

   3 degrees N, 4 7/8 degrees (21m) W to

delta Sco (2.0, with a mag. 7.5 star 1/2- degree E and a mag. 7.0
star 3/4 degree NNE;  delta Sco is slightly N of a line through
Antares and sigma Sco), then

   2 5/6 degrees N, 1 1/4- degrees (5m) E to

beta Sco (2.5, with the SSE-NNW pair omega2 Sco (5.0) - omega1
Sco (4.5) 1 degree S, a mag. 7.5 star 1/8 degree NNE, and a mag.
8.0 star 3/8 degree W).

After observing beta Sco as discussed below, return to sigma Sco
and do this:

Sigma Sco (3.0), then

   1 1/2- degrees N, 1/8 degree (1/2m) W to

omicron Sco (4.5, with a mag. 8.5 star 1/4 degree SSW and a mag.
7.0 star 1/2 degree NNE), then

   1/2- degree N, 1/8 degree (1/2m) E to

a mag. 7.0 star, then

   1/2- degree N, 3/4- degree (3m) W to

a mag. 7.5 star, then

   1/3 degree N, 3/8 degree (1 1/2+ m) W to

M80 (with a mag. 8.5 star 1/6 degree S, 1/8 degree (1- m) W).

After viewing M80 as discussed below, return to Antares and do this:

Antares, then

   slightly S, 1 3/8 degrees (6m) W to

M4 (with a mag. 8.0 star 1/2 degree E, a mag. 8.5 star 3/8 degree
SE, and a mag. 7.5 star 5/8 degree SSE).

After viewing M4 as discussed below, return to Antares and do this:

Antares, then

   1 3/8 degrees N, 1/4 degree (1m) E to

22 Sco (5.0, with a mag. 7.5 star just SW and a mag. 8.0 star 1/8+
degree NW), then

   6 3/4 degrees (30m) E to

a mag. 6.0 star, with 26 Oph (6.0) 1/8 degree NNE and a NNE-SSW
pair of mag. 8.5 stars 1/4+ degree apart slightly S and 3/8 degree
E), then

   1/2+ degree S, 7/8 degree (4m) E to

28 Oph (6.5, with 31 Oph (6.5) 1/4- degree ENE and a mag. 8.0 star
1/4 degree NE of 31 Oph), then

   1/2+ degree S, 3/8- degree (1 1/2m) W to

M19 (with a mag. 8.0 star 3/8 degree S and 1/4 degree (1m) W).

After viewing M19 as discussed below, return to 28 Oph and do this:

28 Oph (6.5), then

   3- degrees S, 1/8- degree (1/2m) E to

a mag. 6.5 star with a mag. 8.0 star 3/8 degree NNW and a mag. 7.5
star 1/2 degree SE, then

   3/4 degree (3 1/2m) W to

a mag. 7.5 star with a mag. 8.0 star 1/8 degree N and a mag. 7.5
star 1/8 degree SW, forming an obtuse isosceles triangle,
then

   1 1/8 degree S to

a mag. 7.0 star;  this star is a sort of nice wide E-W double with
nearly identical whitish components, easily splittable at low
power, then

   1/2- degree S to

M62 (with a mag. 8.5 star 1/4 degree SW).

     The first three objects were viewed 6/28/03, 10:00 p.m.
Conditions excellent.  We first note that Antares is itself quite
interesting, partly because of its brightness and its striking
orange color.  Also, it is actually a double star, with an emerald
green companion 2.9" away;  I have never been able to split this,
but at high power I can sometimes see flashes of green around the
bright orange main star.
     The remaining objects are presented mainly west-to east since
they may be pretty far south in your sky, and if we went the other
way an object might disappear behind trees before we got to it.
Beta Sco is a nice double with a bright white component and a
somewhat fainter white (or maybe a little bluish) component to the
NNE;  it is splittable at 58.75x, but looks nicer at 94x and 267x.
     M80 is an nice bright white little ball of light;  it can be
seen at 58.75x, 94x, 267x, and 534x, but I like the view with 267x
best.  I could not see individual stars in it.
     M4 is much larger and looser than M80;  in spite of the fact
that its total brightness is greater than that of M80, M4 is harder
to see because it is much more spread out.  It can be seen at all
four powers mentioned for M80, but I like the view with 267x best.
There appears to be a N-S vertical bar of stars at the center.  One
can see many stars around the center, in fact, M4 at times looks
almost like an open cluster with faint stars.  This is an unusual
globular cluster.
     M19 was viewed 6/27/03, 11:15 p.m., conditions excellent.
M19 is fairly easy to see at 58.75x, 94x, and 267x, with 267x being
best.  It looks like a fuzzy white ball, and at 267x I could see a
few individual stars (I think).
     M62 was viewed 7/31/02, 10:00 p.m., under fair conditions.  It
looks like an attractive white ball, a little brighter than M19;  I
could not see individual stars.  It is easy to see at 58.75x, 94x,
and 267x, with 267x probably being best.


42. M13--A spectacular globular cluster

M13 (globular cluster)       mag. 5.9            size 20'
16h 41.7m              36 degrees 28'            Hercules

First, follow the line from beta CrB (3.5, crown star NW of
Alphekka) through gamma CrB (4.0, crown star E of Alphekka),
bending slightly right, 5 2/3 degrees SE to

pi Ser (4.5, with a mag. 7.5 star 1/8 degree E), then 

   continue 5 7/8 degrees on the same line to

gamma Her (3.5, with beta Her (2.5) 3 degrees NE, a mag. 7.5 star
1/2 degree NE, and a mag. 7.0 star 3/4 degree E), then

   on a gamma Her - beta Her line, except make a right turn of
   about 30 degrees, go 10 1/3 degrees to

zeta Her (2.5, with a mag. 8.0 star 3/8- degree E and a mag. 7.0
star 3/4 degree S), then

   7 1/3 degrees N, 3/8 degree (2m) E to

eta Her (3.5, with a mag. 8.5 star 5/8- degree ENE and a mag. 8.0
star 1/4 degree N of the mag. 8.5 star), then

   2 1/2 degrees S to

a mag. 7.0 star with a mag. 7.5 star 1/2 degree SW, then

   1/4 degree (1 1/2m) W to

the center of M13.

[Shortcut to zeta Her:

Alphekka, then

   4 7/8 degrees N, 14 1/8 degrees (66 1/2m) E to

zeta Her (2.5, with a mag. 8.0 star 3/8- degree E and a mag. 7.0
star 3/4 degree S).]

     Viewed 6/8/02, 11:00 p.m.  Conditions good.  A magnificent
cluster, with many stars visible.  Some believe this is the best
globular cluster north of the celestial equator, and I would not
argue with that.  267x gives a good view, and M13 also looks good at
lower magnifications.  This object received two exclamation points
in the NGC catalog.


43. NGC6210--A small but bright and easy-to-find planetary nebula

NGC6210 (planetary nebula)       mag. 8.8           size 16"
16h 44.5m                 23 degrees 49'            Hercules

Beta Her (2.5, with a mag. 5.5 star 3/4 degree NNE and s Her (5.0)
1 degree S;  see M13 for the path to beta Her), then

   3 1/3 degrees N, 10 1/8 degrees (45- m) E to

delta Her (3.0, with 70 Her (5.0) 1 3/8 degrees ESE and a mag. 5.0
star 1 3/8 degrees ENE, forming an isosceles triangle pointing W,
and with a mag. 7.0 star 1/2 degree WNW), then

   1/6 degree S, 5 1/3 degree (23 1/2m) W to

51 Her (5.0, with a mag. 7.0 star 3/4 degree N and 56 Her (5.5)
3/4+ degree ENE of the mag. 7.0 star), then

   1 1/8 degree S, 1 2/3- degree (7 1/2m) W to

a mag. 6.5 star, then

   3/8- degree NE to

a mag. 7.5 star, then

   1/8+ degree NW to

NGC6210;  the last two stars in the path and NGC6210 form a right
triangle.

     Viewed 6/14/03, 10:30 p.m.  Conditions fair, but sort of
humid.  There is a faint star about 1/16 degree NE of NGC6210
which is useful as a marker and as a focus star.  At 58.75x,
NGC6210 looked like a star.  At 94x it looked like a fat star, or
as stated in reference II-13, "Exactly like a star out of focus,"
and one could see some of the blinking on-and-off behavior that
often occurs with planetary nebulas.  At 267x one could see a disk
of light, and at 534x one could see a larger disk.  Many nebulas
look pretty diffuse at 534x because of low average surface
brightness, but NGC6210 was bright enough to hold up relatively
well in this respect.  One is supposed to be able to see a faint
ring enclosing the bright disk with larger telescopes, but all I
could see was the disk (which was not bad).


44. Rasalgethi--A beautiful orange-white double

Alpha Her (Rasalgethi--double star)  mags. 3.5, 5.4 separation 4.6"
17h 14.6m             14 degrees 24'                Hercules

Roughly on a line from Alphekka through beta Her (2.5) (see M13
description for the way to beta Her), go 17 5/6 degrees ESE to

alpha Oph (2.0, Rasalhague, with a mag. 7.5 star 1/4 degree W, 56
Oph (7.0) 5/8 degree N, and 54 Oph (6.0) 1/8+ degree W of 56 Oph),
then

   backtrack 5 1/4 degrees WNW on that line to

alpha Her (3.5, Rasalgethi).

[Alternate path:

Vega, then

   26 1/4 degrees S, 15 1/8 degrees (62m) W to

alpha Oph (2.0, Rasalhague), then

   5 1/4 degrees WNW to

alpha Her (3.5, Rasalgethi).]

     Viewed 6/17/02, 10:30 p.m.  Conditions ok except for half
moon.  The first thing you notice, even with low power, is the
bright orange color.  Then at higher power the orange color gets
even more stunning, and you notice the not-quite-as-bright white
star just E of the orange star, following it across the sky like a
caboose following a locomotive.  Rasalgethi barely splits at 94x,
but it looks much better at 267x.


45. M92--A pretty, well-resolved little globular cluster

M92 (globular cluster)        mag. 6.5           size 14'
17h 17.1m             43 degrees 8'              Hercules

Eta Her (3.5, see M13), then

   2 1/6 degrees S, 6 3/8 degrees (32m) E to

pi Her (3.0), then

   3/4 degree NE to

69 Her (4.5, with rho Her (4.0) 1 1/4 degrees E;  note that pi,
69, and rho form a bow pointing N), then

   5 5/6 degrees N, slightly W to

M92.

     Viewed 9/21/02, 8:30 p.m.  Full moon, but otherwise conditions
ok;  at this time the moon was still blocked by trees, which
helped.  M92 was an easily visible glowing ball at 58.75x and 94x,
and many individual stars were visible at 267x.  This cluster is
much better resolved (meaning more individual stars are visible)
than many clusters I have seen.  It is a nice little cluster!
     Now some of you may wonder about the wisdom of traveling so
far at the last step in the path when the target object (namely
M92) is not visible with binoculars or finderscope (at least with
my sky conditions).  We had some trouble with this while following
the path to The Blue Snowball Nebula.  Here, however, we have three
things in our favor:  First, the target object is instantly
recognizable when it comes into the field of view of the telescope.
Second, the move consists of (essentially) only one part, not two,
which tends to reduce errors.  Third, the object from which we are
moving is not terribly close to a celestial pole (the declination
of 69 Her is about 37 degrees), so we can hope that any polar
alignment error we made will not cause too much trouble.  To
consider this last point a little more closely, since the field
width of my telescope at 58.75x is a little more than 3/4 degree,
we should not miss M92 entirely if the amount by which the center
of the field misses M92 is no more than 3/8 degree, or .375 degree.
Looking at the table in section I of chapter VII, using the row for
declination 30 degrees (which is not too far from 37 degrees), we
see that the worst miss with a polar alignment error of 3 degrees
would be about .302 degree, which is less than .375 degree (and we
would have to be unlucky for our polar alignment error of 3 degrees
to be in a direction which would produce this worst miss).  In fact,
I used this path on two different nights and had no problems;  M92
didn't appear exactly in the center of the field, but it was a long
way from the edge.  By the way, if you do your polar alignment as
described in the instructions that came with your telescope (that
is, align the optical tube with the fixed arm on the mount, then
adjust this fixed arm using the azimuth and altitude controls (not
the Dec and RA controls) to get Polaris into the center of your
telescope field) instead of using our rough alignment procedure as
described in section VIIA, your error in alignment will probably
not be more than a degree or so.
     But what if you do miss M92 entirely?  There are several
things you can try.  First, you can twist the Dec slow motion knob
back and forth a little in case you didn't get the distance right.
If this fails you can try a scanning procedure similar to that in
section VIIB:  Turn the Dec knob until you are pretty sure you are
north (or south) of M92, turn the RA knob back and forth a little
(not letting go so you don't lose the starting position), change
the Dec knob to move closer (you hope) to M92, scan east-west
again, continuing until you hit M92 or are pretty sure you have
passed it in declination.  Note that while this is going on M92 is
drifting westward, so you may have to try to make a compensating
westward adjustment with the RA knob.  If this fails, you can go
back to 69 Her (or back to an earlier point in the path if you
can't find 69 Her) and try again.  If this fails, you can try to
redo (or adjust) your polar alignment and try again.  If this fails
too, you can go back to your chart and select some intermediate
stars on the way to M92 so that you will be taking shorter steps.
For example, there is a mag. 5.0 star a little to the W of the
direct path and about 2/3 of the way up;  I would probably try to
go from pi Her to this star (leaving 69 Her out) and then try to
go from the mag. 5.0 star to M92.  I will leave the measuring to
you if you want to try this.  One way or another, I have confidence
that you will find M92 eventually!  After all, since you have
decided to read this document, you must be pretty smart.


46. M6--The Butterfly Cluster

M6 (Butterfly Cluster)         mag. 4.2          size 15'
17h 40.1m             -32 degrees 13'            Scorpius

Antares, then

   1 3/8 degrees N, 1/4- degree (1m) E to

22 Sco (5.0, with a mag. 7.5 star just SW and a mag. 8.0 star 1/8+
degree NW), then

   1/8 degree N, 11 3/4 degrees (52m) E to

theta Oph (3.0, with a mag. 6.0 star 1/8 degree NNW and a mag. 7.5
star 1/4 degree ESE;  see The Fantastic Four for more about theta
Oph), then

   3 1/8 degrees S, 3/8- degree (1 1/2m) E to

43 Oph (5.5, with a N-S mags. 7.0-7.5 pair of stars 3/4 degree ESE;
36 Oph (4.0) is W of a theta-43 line, and is at the right angle in
a right triangle with those two stars), then

   1 3/4 degrees S, 7/8 degree (4m) E to

45 Oph (4.5, with a mag. 6.0 star 1/8+ degree NNE), then

   2 1/3 degrees S, 2 3/4 degrees (13m) E to

the center of M6.

(If 45 Oph is obscured by tree branches, one can do 43 Oph, then
4 1/6- degrees S, 3 5/8 degrees (17m) E to M6.)

     Viewed 7/27/02, 10:15 p.m.  Conditions ok.  Beautiful!  It
does look like a butterfly, flying east.  There is a bright gold
star in the front outside part of the left wing (thinking of
looking down on the butterfly).  This is one where I like to set
the drive motor to track it, then just look at it for a while.
At first I thought it looked like a moth (maybe even Mothra!) but
now I think it looks like a butterfly.  Besides, "The Butterfly
Cluster" sounds better than "The Moth Cluster".  This cluster is
far to the south, so it might be hard to find for northern
observers, but if you can avoid trees and such, at least it is
quite bright.  94x gives a good view.  It can also be seen with
binoculars, but it looks a lot better with a telescope.
     By the way, M7 is a larger and brighter open cluster than M6,
and its center lies 2 5/8 degrees S, 2 3/4 degrees (13 1/2m) E of
the center of M6.  I am not reporting details on this since it was
a little too far south for me to get a good look at, although I did
see a few bright stars through the trees.  If you have found M6 and
have some clearance below it, you might try moving your telescope
2 5/8 degrees south from the center of M6 and waiting about 11
minutes for parts of this large cluster to begin appearing (or you
could do the entire move described above rather than waiting).


47. IC4665--A large, bright, sparse open cluster with a shape that
    a dog mignt appreciate

IC4665 (open cluster)      mag. 4.2             size 40'
17h 46.3m               5 degrees 43'           Ophiuchus

Alpha Oph (2.5) (see Rasalgethi above for the path to alpha Oph),
then

   8 degrees S, 2 1/8 degrees (8 1/2m) E to

beta Oph (2.5, with gamma Oph (3.5) 2 1/4 degrees SSW, a mag. 8.0
star 1/4 degree SSE, and a mag. 7.0 star 1/2- degree WSW), then

   1 1/6 degrees N, 3/4 degree (3m) E to

the center of IC4665.

     Viewed 6/22/03, 9:30 p.m.  Conditions good.  At 58.75x I could
see about a dozen bright stars and some fainter ones in the same
field.  94x gives a better view of the individual stars but a
narrower field.  The bright stars form what looks to me like a fire
hydrant with the top to the SSW, and water shooting out the W side
and up. The bright stars are unusually bright for an open cluster,
so IC4665 should be visible to people with more light pollution
than I have.


48. M23--A superb open cluster

M23 (open cluster)         mag. 5.5             size 26'
17h 56.8m             -19 degrees 1'            Sagittarius

Alpha Oph (2.5) (see Rasalgethi above for the path to alpha Oph),
then

   8 degrees S, 2 1/8 degrees (8 1/2m) E to

beta Oph (2.5, with gamma Oph (3.5) 2 1/4 degrees SSW, a mag. 8.0
star 1/4 degree SSE, and a mag. 7.0 star 1/2- degree WSW), then

   1/6 degree S, 4 1/8 degree (17m) E to

66 Oph (5.0), then

   1 1/2 degrees S, slightly E to

67 Oph (4.0) (note that 66 Oph, 67 Oph, 68 Oph (4.5), and 70 Oph
(4.0) form a "triangle plus one" which is easy to spot with
binoculars, with 68 almost on a line with 66 and 67, 1 5/8 degrees
S of 67 and slightly E, while 70 is 1 1/4 degrees ESE of 67), then

   6 5/8 degrees S to

zeta Ser (5.0, with a mag. 8.0 star 1/4 degree S), then

   6+ degrees S, 3/8 degree (1 1/2m) W to

nu Oph (3.5, with a mag. 8.5 star 3/4+ degree SE), then

   6 degrees S, 5/8 degree (3m) W to

a mag. 6.0 star with a mag. 7.5 star 1/8+ degree NE, then

   3 degrees S, slightly W to

a mag. 6.5 star, then

   1/4 degree SE to

the center of M23.

     Viewed 7/23/02, 10:00 p.m.  Conditions crappy--one day before
full moon, with intermittent milky high clouds.  In spite of the
conditions, this was a very attractive cluster, with many bright
stars of approximately equal magnitude, pretty evenly distributed
in a ball that just about filled up the field at 94x.  The next-to-
last star in the path makes a nice counterpoint.  According to the
description in reference II-13, this is "A large open cluster that
appears prominent against a dark nebula", and the background
darkness does indeed enhance the attractiveness of this cluster.
     By the way, I apologize of anyone is offended by my use of a
certain word in this description;  I don't consider that to be a
swear word, since it comes from a man's name.  He was Thomas
Crapper, the Englishman who is generally supposed (incorrectly) to
have invented the flush toilet.
     Viewed again 8/8/02, 9:30 p.m.  Conditions good.  This was
still a most attractive cluster, but now with the better
conditions, at 267x I could see a faint glow behind the cluster,
split by some narrow dark lanes.


49. Barnard's Star--The fastest star

Barnard's Star            mag. 9.5
17h 58m in 2002      4 degrees 45' in 2002        Ophiuchus

Alpha Oph (2.5) (see Rasalgethi above for the path to alpha Oph),
then

   8 degrees S, 2 1/8 degrees (8 1/2m) E to

beta Oph (2.5, with gamma Oph (3.5) 2 1/4 degrees SSW, a mag. 8.0
star 1/4 degree SSE, and a mag. 7.0 star 1/2- degree WSW), then

   1/6 degree S, 4 1/8 degree (17m) E to

66 Oph (5.0, see the M23 path for more details about 66 Oph),
then

   1/8- degree N, 5/8 degree (2 1/2m) W to

a mag. 8.5 star (with a mag. 10.0 star 1/6 degree E, a mag. 8.0 star
1/4 degree W and slightly S, and a mag. 8.5 star 1/6 degree SSW of
the mag. 8.0 star), then

   1/4 degree N (in 2002) to

Barnard's star (9.5;  in 2002 there was a mag. 10.5 star 1/6 degree
SW of Barnard's star, and a mag. 10.0 star slightly N and 1/4-
degree E of Barnard's star.  Also, about 1/6 degree E of the mag.
10.0 star there is a "candy cane" about 1/4 degree long in the N-S
direction and about 1/8 degree wide in the E-W direction, with the
head to the S and opening E;  there are 6 stars in the candy cane,
all mag. 10.0 except for the one farthest N, which is mag. 8.0).

     Barnard's star is not particularly beautiful in the aesthetic
sense, but it is beautiful in the sense of being interesting;  it
changes its position against the background stars faster than any
other star (excluding the sun).  This change is still quite slow,
about 16 arcseconds per year to the north, but it is fast enough
that a patient amateur may have a chance to detect it from one
year to the next.  We give three ways to line up Barnard's star
with other nearby stars, with fainter stars required to be seen as
we go down the list, but with the chances of detecting the motion
greater as we go down the list.
Way 1:  In the path we noted that there is a mag. 10.0 star between
Barnard's star and the candy cane, about 1/4 degree from Barnard's
star and about 1/6 degree from the candy cane;  let's call this
Star i (for "intermediate star"), and let's number the stars in the
candy cane 1 through 6, starting with the one farthest to the east.
If you imagine a line through Barnard's Star and Star i, this line
if extended to the east in July 2002 would pass between stars 4 and
5 in the candy cane.  These stars are about 1/12 degree apart, 
which is 5 arcminutes apart, which is 300 arcseconds apart, which
is about 19 times the distance Barnard's star moves in one year, so
one would have to really be patient to be able to detect anything
using this method.  To make matters worse, the yearly movement of
the end of the line at the candy cane is only about 2/3 of the
yearly movement of Barnard's star, since (1/6)/(1/4) = 2/3.
Way 2:  There is a faint star which was almost directly E of
Barnard's star;  you may need to increase magnification (and you
may need good viewing conditions) to see it.  Let's call this Star
f (for "faint star").  With higher magnification (and thus narrower
field) things will drift out of your field faster, so it will be
helpful if you have a drive motor to freeze the objects you want in
your field, although it is possible to proceed without a motor.  In
July 2002 an imaginary line through Barnard's star and Star f would
just about hit Star i, so one could watch to see if this line moves
south of Star i as Barnard's star moves north.  Since Star f is
much closer to Barnard's Star than it is to Star i, there is a
lever effect working in our favor that will multiply the apparent
rate of motion of the line near Star i by approximately the ratio
of the distance of Star f to Star i divided by the distance of Star
f to Barnard's star.
     To try to quantify this we need to estimate the distance 
between Star f and Barnard's Star.  My method of doing this is
sort of rough;  any of you who are into astrophotography could
take pictures each year and compare them, and also do more
accurate measurements on the pictures.  Anyway, here's what I did.
The separation of the components of the double star Albireo (see
later) is given in reference II-3 as being 34.2 arcseconds.  I
looked at Albireo at 534x, put one of the components in the center
of the field, and noted that the other component was about 1/3 of
the way from the center to the edge (that's without "peeking
around the edge", which one can do with a wide-angle eyepiece).
Doing the same thing with the Barnard's Star-Star f pair, when one
was at the center of the field the other was about 1/2 of the way
to the edge.  Thus it appears that the distance between Barnard's
Star and Star f is about 3/2 the distance between the components of
Albireo, so it is about (3/2)(34.2 arcseconds) or 51.3 arcseconds.
Now these measurements are too crude to support that much
precision, but I think it is pretty safe to say that the distance
between Barnard's Star in July 2002 and Star f is about 1 arcminute,
give or take 1/2 arcminute.
     Now armed with this estimate, since Barnard's star and Star i
are about 1/4 degree apart (which is 15 arcminutes), then Star f
and Star i are about 14 arcminutes apart, so the rate of movement of
the end of the line near Star i should be about (14/1)x(16
arcseconds per year) = 224 arcseconds per year, which is about
3 3/4 arcminutes per year.  Now this is about 1/4 of the distance
between Barnard's Star and Star i, so this should be detectable
in July 2003.
Way 3:  If we look at Barnard's Star and Star f under fairly high
magnification, we may see an even fainter star south of Star f,
and about twice the distance from Star f that Star f is from
Barnard's star;  let's call this Star ff (for "faint faint star").
The angle between the lines passing through Barnard's Star and
Star f, and Star f and Star ff, looked like a right angle in July
2002.  Now by July 2003 Barnard's star will have moved about 16
arcseconds north on a line which is close to being parallel to the
line through Star ff and Star f.  We should be able to detect then
that the angle mentioned above is significantly greater than 90
degrees;  here is why.  If you consider a triangle with vertices at
Star f, the July 2002 position of Barnard's Star, and the July 2003
position of Barnard's Star, this will be (roughly) a right triangle
with right angle at the July 2002 position of Barnard's Star.  By
the definition of the tangent of an angle in a right triangle as
being the opposite side divided by the adjacent side, the tangent
of the angle at Star f in the triangle just described will be
(16 arcseconds)/(the distance from Barnard's star to Star f in
July 2002), which is about (16 arcseconds)/(51.3 arcseconds).
Computing this ratio and then computing the inverse tangent of the
result gives about 17 degrees.  Thus the angle between the line
through Star ff and Star f, and the line through Star f and
Barnard's Star, will have gone from about 90 degrees to about 107
degrees, and this should be noticeable.  Even if our estimate
of the distance of Barnard's Star in July 2002 to Star f were
pretty far off, say the correct distance were 1 1/2 arcminutes
instead of 51.3 arcseconds, a similar computation would give a
change in the angle from about 90 degrees to about 100 degrees,
which should still be noticeable.
     Pushing this a little further, consider the position of
Barnard's Star in July 2003.  If you consider the line which passes
through Barnard's star and is parallel to the line through Star f
and Star ff, and you drop a perpendicular from Star f to this line,
where it hits should be roughly the July 2002 position of Barnard's
Star, and the distance between these two positions of Barnard's
Star will be the distance that Barnard's Star moved in one year (or
about 16 arcseconds).  Note that this distance should be about 1/3
of the distance between Barnard's Star in July 2002 and Star f (and
thus about 1/6 of the distance between Star f and Star ff), since
16/51.3 is about 1/3.  Then by July 2004 Barnard's Star should have
moved the same distance in the same direction (which will be along
a line that is roughly parallel to the line through Star f and Star
ff).  Thus if you are looking at these stars some July, you can
tell what year it is by seeing how many of these hops down this
line Barnard's Star has made, so we have a sort of Barnard's
Calendar!  (I am speaking sort of with tongue in cheek here.)
     By the way, one reason that Barnard's Star appears to move so
fast (relatively speaking) is that if you count the Alpha Centauri
system as a single star and ignore our sun, then Barnard's Star is
the second closest star to the Earth.  Do you know what star is
third?  It happens to be a star called Wolf 359, which has
magnitude 13.5 in spite of its closeness to us, and is one of
the least luminous stars known.  This star is interesting to Star
Trek fans, because it is the location where the Federation armada
made its stand against, and was destroyed by, a Borg cube on "Star
Trek:  The Next Generation".
     Epilog03:  I looked at Barnard's star again on 7/12/03 under
poor conditions (moon one day before full), and although the changes
were hard to estimate, they seemed about right.  Hooray!
     Epilog05:  I looked at Barnard's star again on 7/29/05 at
10:40 p.m. under pretty good conditions, this time using an 11 inch
SCT (Schmidt-Cassegrain telescope) at 318x.  All the stars
discussed above could be seen clearly, although star ff seemed to
wink on and off, probably due to some atmospheric instability.
Referring to Way 1 above, the line through Barnard's star and
star i still passed between stars 4 and 5 of the candy cane, as
expected.  Referring to Way 2 above, the line through Barnard's
star and star f passed far south of star i, showing a considerable
change from 2002, again as expected.  Referring to Way 3 above, the
angle between the lines through star f and Barnard's star, and star
f and star ff, looked to be about 135 degrees, which would be about
right (recall that in 2002 this angle appeared to be about 90
degrees).  Note that if the path of movement of Barnard's star
through the sky were exactly parallel to the line through star f
and star ff, and the angle referred to above were exactly 90
degrees during the 2002 observation, and between the 2002 and 2005
observations Barnard's star moved exactly the distance between
Barnard's star at the 2005 observation and star f, then the angle
referred to above would be exactly 135 degrees (since the triangle
with vertices at the two observations of Barnard's star and at star
f would be an isosceles right triangle, so the angle in this
triangle at star f would be 45 degrees).  I must admit that the
distance moved by Barnard's star between the 2002 and 2005
observations looked a little shorter to me than half the distance
between stars f and ff, but the angle mentioned above worked out so
well that it appears that Barnard's star is doing what it is
supposed to do.  That is comforting somehow.

50. NGC6543--The Cat's Eye Nebula

NGC6543  (Cat's Eye Nebula)       mag. 8.1       size 20"
17h 58.6m             66 degrees 38'             Draco

Vega, then

   12 3/4 degrees N, 6 1/4 degrees (40 1/2m) W to

gamma Dra (2.5, SE star of Draco "lozenge", which represents the
head of the dragon, with a mag. 8.0 star 1/4 degree W), then

   5 1/3 degrees N, 3/8 degree (3- m) W to

xi Dra (4.0, NE star of lozenge, with a mag. 8.0 star 1/2 degree
S and a mag. 6.0 star 1 degree SSE), then

   7 1/2 degrees N, 2 1/4 degrees (20 1/2m) E to

36 Dra (5.0, with 42 Dra (5.0) 1 3/4 degrees NE, a mag. 7.5 star
3/8 degree N, and a mag. 8.0 star 3/8- degree ENE), then

   1/6 degree S, 3/4 degree (7 1/2m) W to

a mag. 7.0 star (with a mag. 8.0 star 3/8- degree S and a mag. 7.5
star 1/2 degree SW, forming an isosceles right triangle), then

   2 3/4 degrees N, 1/8 degree (1 1/2- m) W to

a mag. 7.0 star (with a mag. 8.0 star 1/2+ degree NNE), then

   1/3 degree S, 5/8 degree (6 1/2m) W to

NGC6543 (with a mag. 8.0 star 5/8 degree WSW on a line with the
previous star in the path and NGC6543).

   Viewed 7/7/02, 9:30 p.m.  Conditions ok.  It looks like a bright
white baseball, elongated a little N-S.  There is a sort of faint
star a little NW.  At 58.75x, NGC6543 could be easily mistaken for
a fat star, but the disk shape shows clearly at 94x, and the best
view for me was at 267x.  Unlike some planetary nebulae, this one
is bright enough to be clearly visible.  This is a good one!


51. Nu Dra, 16,17 Dra, 39 Dra, psi Dra, and 41,40 Dra--A double star
    quadruple-header plus a triple star

Nu Dra (double star)         mag. 4.0
17h 32m                55 degrees 10'           Draco

16,17 Dra (triple star)      mags. 5.4, 6.4, 5.5 seps. 3.4", 90"

39 Dra (double star)         mags. 5.1, 8.1     separation 3.8"
18h 24.0m              58 degrees 48'           Draco

Psi Dra (double star)        mag. 4.5
17h 42m                72 degrees 10'           Draco

41,40 Dra (double star)      mag. 5.0
18h 0m                 80 degrees 0'            Draco

Vega, then

   12 3/4 degrees N, 6 1/4 degrees (40 1/2m) W to

gamma Dra (2.5;  this is the SE star in the Draco "lozenge", which
represents the head of the dragon, with a mag. 8.0 star 1/4 degree
W), then

   5 1/3 degrees N, 3/8 degree (3m) W to

xi Dra (4.0;  this is the NE star of the lozenge, with a mag. 8.0
star 1/2 degree S and a mag. 6.0 star 1 degree SSE), then

   1 2/3 degrees S, 3 degrees (21m) W to

nu Dra (4.0;  this is the NW star of the lozenge, with the SW and
final star of the lozenge being beta Dra (3.0), which is 2 7/8
degrees SSW).

After looking at nu Dra (see discussion below), continue as
follows:

nu Dra (4.0, NW star of the lozenge), then

   2/3 degree S, 3 7/8 degrees (26 1/2m) W to

mu Dra (5.0, with mag. 8.0 stars 1/2 degree ESE and 1/2+ degree
WNW), then

   1 1/2+ degrees S, 4 1/4 degrees (29m) W to

16,17 Dra (4.5, with a mag. 7.5 star 3/4 degree S and a mag. 6.5
star 1/2 degree W of the mag. 7.5 star)

After looking at 16,17 Dra (see discussion below), backtrack to xi
Dra and proceed as follows:

Xi Dra (4.0, NE star of lozenge), then

   1/6 degree N, 5 1/3 degrees (39m) E to

45 Dra (5.0, with a mag. 7.0 star 3/4+ degree NNE), then

   1 3/4 degrees N, 1 1/8 degrees (9- m) W to

39 Dra (5.0, with a SE-NW pair with mags. 6.5-7.0 1 degree NNE).

After looking at 39 Dra (see discussion below), continue as
follows:

39 Dra (5.0), then

   12 1/2 degrees N, 1/4 degree (3m) W to

phi Dra (4.5, with chi Dra (3.5) 1 3/8 degrees N, a mag. 6.5 star
1/4 degree NW, and a mag. 7.5 star 1/2 degree NE), then

   1 3/8 degrees N to

chi Dra (3.5), then

   1/2 degree S, 3 degrees (39m) W to

psi Dra (4.5, with 34 Dra (5.5) 1 degree ESE).

After looking at psi Dra (see discussion below), continue as
follows.  Note that once we turn north from 34 Dra we will be
pushing deep into the "badlands" near the north celestial pole,
where polar alignment errors may have a much larger effect than
usual (see section VII(I) earlier).  If we had enough stars in this
region which are visible in binoculars and finderscope to allow us
to do free movements, we could do an end run around the polar
alignment errors that way, but lacking that, we reduce the effects
of these errors by taking baby steps.

Psi Dra (4.5), then

   1/6 degree S, 1 degree (13m) E to

34 Dra (5.5, with a mag. 7.5 star 1/2 degree SE), then

   1 1/6 degrees N, 1/8 degree (1 1/2m) W to

a mag. 8.0 star with a mag. 7.5 star 1/2 degree W, then

   1 1/2 degrees N, 1/8+ degree (3m) W to

a mag. 7.0 star with a mag. 8.0 star 1/2 degree W, then

   1/2+ degree N, 1/8+ degree (3m) E to

a mag. 6.5 star, then

   5/6 degree N, 1/8 degree (2m) W to

a mag. 7.5 star, then

   1 degree N, 1/8+ degree (3m) W to

35 Dra (5.0, with a mag. 7.5 star 1/8- degree NW), then

   1 1/3 degrees N, slightly E to

a mag. 6.5 star with a mag. 7.5 star 3/8+ degree NNE, then

   1 degree N, slightly W to

a mag. 7.5 star with a NW-SE mags. 7.5-8.5 pair 1/2 degree W, then

   2/3 degree N, 1/2- degree (11m) E to

41,40 Dra (5.0, with a mag. 7.5 star 1/8- degree SE and a mag. 8.5
star 1/4 degree S).

     All except 16,17 Dra were viewed 10/4/02, 8:15 p.m., using a
9.25 inch telescope.   Conditions were quite good except for my
having to play "tag" with some rapidly moving cumulus clouds.  A
cold front had just pushed through, which often gives colder,
drier, more stable air, and thus pretty good conditions, if you can
just avoid those cumuli.  16,17 Dra was viewed 7/19/05, 10:00 p.m.,
using an 11 inch telescope.  The moon, two days before full, was
not a big problem since it was behind trees, but conditions were
quite humid;  the big telescope lens was starting to fog up as I
was viewing 16,17 Dra.
     Nu dra is a pretty white-yellowish pair, and is also an
unusually wide pair.  This can be split easily at any power;  94x
gives a good view.
     16,17 Dra is a nice triple consisting of a wide pair of fairly
bright white stars on roughly a SSW-NNE line and a fainter white
star just ESE of the NNE member of the pair.  The close pair was
easily split at 318x, and 112x could split it also.  The star mu
Dra in the path just before 16,17 Dra is also a double star;  I
could just split this close pair (1.9") at 318x.
     39 Dra is a close pair with a white star and a much fainter
reddish star.  It took 267x to split it.  There is a blue star
nearby that adds to the overall attractiveness.
     Psi Dra is a nice yellowish-bluish pair;  94x gives a good
view.
     41,40 Dra is a pretty little yellow-blue pair with nearly
equal magnitudes.  94x gives a good view, but it is prettier at
267x.  I would say that this is the best double star of this bunch,
and worth venturing into the badlands near the north celestial pole
(where small errors in polar alignment can have big effects on
one's path-following) to see.


52. The Fantastic Four plus one--The Lagoon nebula, NGC 6530, the
    Trifid Nebula, and M21, plus NGC6520.

M8 (the Lagoon Nebula)        mag. 5.8          size 90'x39'
18h 3.8m             -24 degrees 23'            Sagittarius

NGC 6530 (open cluster)       mag. 4.6          size 15'
18h 4.8m             -24 degrees 20'            Sagittarius

M20 (the Trifid Nebula)       mag. 6.3          size 28'
18h 2.6m             -23 degrees 2'             Sagittarius

M21 (open cluster)            mag. 5.9          size 12'
18h 4.6m             -22 degrees 30'            Sagittarius

NGC6520 (open cluster)        mag. 7.6          size 6'
18h 3.4m             -27 degrees 54'            Sagittarius

Vega, then

   64 1/6 degrees S, 2 degrees (9m) E to

lambda Sgr (3.0;  top of teapot;  Kaus Borealis;  sigma Sgr (2.0)
is 6 1/4 degrees ESE, phi Sgr (3.0) is 1 degree S of a lambda-
sigma line, closer to sigma, and delta Sgr (3.0) is 4 2/3 degrees
SSW of lambda, with sigma-lambda-delta being a right triangle),
then

   1 1/8 degrees N, 5 3/4 degrees (25+ m) W to

7 Sgr (5.0, with 9 Sgr (5.5) 1/4 degree ESE, a mag. 8.0 star 1/4-
degree NNE of 9 Sgr, and the center of bright open cluster NGC 6530
1/4- degree E of 9 Sgr;  all these objects are inside M8).

[Here is an alternate path to M8, which may be better in case there
are tree problems since it comes in from the west, and so one can
spot 7 Sgr as soon as it emerges from the trees, with no need to
wait for lambda Sgr to emerge:

Antares, then

   1 3/8 degrees N, 1/4 degree (1m) E to

22 Sco (5.0, with a mag. 7.5 star just SW and a mag. 8.0 star 1/8+
degree NW), then

   1/8 degree N, 11 3/4 degrees (52m) E to

theta Oph (3.0, with a mag. 6.0 star 1/8 degree NNW and a mag. 7.5
star 1/4 degree ESE;  theta Oph forms a recognizable bow with the
next two stars in the path and with 36 Sgr (4.0) which is 2 1/4
degrees SW, so once you learn to recognize this pattern you can
start here even if Antares has disappeared behind trees), then

   1 1/4 degrees NE to

44 Oph (4.0, with a mag. 6.0 star 1/4 degree W), then

   1 1/4 degrees E to

51 Oph (4.5, with a mag. 7.5 star 1/8 degree NW), then

   slightly N, 6 1/2 degrees (28 1/2m) E to

4 Sgr (5.0, with 5 Sgr (6.5) 1/2 degree S and a mag. 6.5 star 5/8
degree W), then

   1/2- degree S, 3/4 degree (3+ m) E to

7 Sgr (5.0, with 9 Sgr (5.5) 1/4 degree ESE, a mag. 8.0 star 1/4-
degree NNE of 9 Sgr, and the center of bright open cluster NGC 6530
1/4- degree E of 9 Sgr;  all these objects are inside M8)].

After viewing M8 and NGC6530 as discussed below, from 7 Sgr go
1 1/4 degrees N and slightly W to the center of M20 (the Trifid
Nebula).

After viewing M20 as discussed below, note the bow of stars 1/2
degree across pointing N just N of M20 with mags. (E to W)
7.5-7.0-6.0, and go 3/8 degree N and slightly E from the E star of
this bow to get to M21.

After viewing M21 as discussed below, backtrack to 7 Sgr and do
this:

7 Sgr (5.0), then

   3 1/2 degrees S to

a mag. 6.5 star, with mag. 8.0 stars 3/8- degree NW and 1/4 degree
NNE of the first mag. 8.0 star, and a mag. 8.5 star 1/4 degree W of
the mag. 6.5 star;  also, slightly N and 1 1/4 degree (6m) E of the
mag. 6.5 star there is a 1/4 degree long NE-SW line of three mag.
7.0 stars, with a mag. 7.5 star 1/4 degree NW of the middle star of
the line, then

   1/8 degree ESE to

NGC 6520.

     All these objects except for NGC6520 were viewed with the 9.25
inch telescope on 8/6/02 at 10:00 p.m.  Conditions were excellent.
M8 is one of the best objects for seeing nebulosity (illuminated
gas), the brightest of which is just W of open cluster NGC6530.
This looks stunning in binoculars too.  We should note that for
seeing nebulosity, viewing conditions are more critical than for
most other things, so if you look at M8 and it doesn't look too
impressive, you might want to try again on a better night.
     There are two mag. 7.5 stars inside M20 1/8 degree apart in a
roughly N-S line.  The S one is a triple star, and lies at the
joining of three dark dust lanes.  You will need excellent
conditions to see the dust lanes.  It is easy to see two components
of the triple star, but seeing all three is much harder.  The
nebulosity in M20 is not as bright as in M8.  M20 also looks very
good in binoculars.
     M21 is a really nice open cluster with many bright stars and
the glow of fainter stars at the center.  After looking carefully
at this, give yourself a treat and drop back down to view M20 again,
then drop down to view M8 and NGC 6530 again!
     Both the Lagoon Nebula and the Trifid Nebula received three
exclamation points in the NGC catalog.
     NGC6520 was viewed with the 11 inch telescope on 7/29/05 at
10:00 p.m.  Conditions were pretty good.  At 70x this did not look
like much of anything.  It was somewhat better at 112x.  At 318x,
however, the view was spectacular!  The outer part of the object
looked a lot like a regular pentagon, with the base to the west,
except there were two stars at the top instead of one, and the top
was somewhat flattened.  Inside the pentagon near the top, centered
N-S, was a circle of stars, with a star in the center of the
circle.  This object showed more symmetry than any other open
cluster I can think of.
     By the way, the mag. 6.5 star just before NGC6520 in the path
to NGC6520 is (from our point of view) in the eastern part of the
small (4') roundish dark nebula B86, but I was unable to detect
this nebula.  I think it should look like a dark area in a brighter
background.  If your skies are darker than mine, you may have
better luck.


53. NGC6572--A bright little planetary nebula

NGC6572 (planetary nebula)     mag. 9.1            size 11"
18h 12.1m                 6 degrees 51'            Ophiuchus

Alpha Oph (2.5) (see Rasalgethi above for the path to alpha Oph),
then

   8 degrees S, 2 1/8 degrees (8 1/2m) E to

beta Oph (2.5, with gamma Oph (3.5) 2 1/4 degrees SSW, a mag. 8.0
star 1/4 degree SSE, and a mag. 7.0 star 1/2- degree WSW), then

   1/6 degree S, 4 1/8 degrees (17m) E to

66 Oph (5.0, see the M23 path for more details about 66 Oph), then

   2 degrees N to

an isosceles triangle consisting of a mag. 6.5 star with a mag.
7.0 star 3/8 degree NNW and a mag. 7.5 star 3/8 degree WNW, then
from the N star (7.0) of the triangle,

   1 3/4 degrees (7m) E to

a "snake" 1/2 degree long, consisting of a mag. 8.0 star, a mag.
7.0 star 1/8+ degree SW of the mag. 8.0 star, a mag. 7.5 star 1/8+
degree S of the mag. 7.0 star, and another mag. 7.5 star 1/8+
degree SW of the mag. 7.5 star, then from the NE star (mag. 8.0) of
the snake,

   slightly N, 1+ degree (4 1/2m) E to

NGC6572 (with a mag. 8.5 star 3/4+ degree (3+ m) E).

     Viewed 7/8/02, 11.00 p.m.  Conditions not great (high
humidity).  This looks like a small bright ball at 267x, like the
Cat's Eye Nebula but smaller.  It looks good at 94x also.  Just to
the E of NGC6572 are four stars in the shape of a rectangle, so that
these stars along with the nebula look like a view from above of a
semi rolling west on Interstate 80.  This is a good little nebula!


54. M17 (the Omega Nebula), M16 (the Eagle Nebula), and M25--A big
    horseshoe in the sky, a nice little open cluster surrounded by
    nebulosity, and an open cluster containing a nice symmetric
    letter M

M17 (Omega Nebula)         mag. 7            size 45'x37'
18h 20.8m          -16 degrees 11'           Sagittarius

M16 (Eagle Nebula)         mag. 6            size 34'x27'
18h 18.8m          -13 degrees 47'           Serpens

M25 (open cluster)         mag. 4.6          size 32'
18h 31.6m          -19 degrees 15'           Sagittarius

Alpha Oph (2.5) (see Rasalgethi above for the path to alpha Oph),
then

   8 degrees S, 2 1/8 degrees (8 1/2m) E to

beta Oph (2.5, with gamma Oph (3.5) 2 1/4 degrees SSW, a mag. 8.0
star 1/4 degree SSE, and a mag. 7.0 star 1/2- degree WSW), then

   1/6 degree S, 4 1/8 degree (17m) E to

66 Oph (5.0), then

   1 1/2 degrees S, slightly E to

67 Oph (4.0) (note that 66 Oph, 67 Oph, 68 Oph (4.5), and 70 Oph
(4.0) form a "triangle plus one" which is easy to spot with
binoculars, with 68 almost on a line with 66 and 67, 1 5/8 degrees
S of 67 and slightly E, while 70 is 1 1/4 degrees ESE of 67), then

   6 5/8 degrees S to

zeta Ser (5.0, with a mag. 8.0 star 1/4 degree S), then

   6+ degrees S, 3/8 degree (1 1/2m) W to

nu Oph (3.5, with a mag. 8.5 star 3/4+ degree SE), then

   6 degrees S, 5/8+ degree (3m) W to

a mag. 6.0 star with a mag. 7.5 star 1/8+ degree NE, then

   5 3/4 degrees (24m) E to

a mag. 5.5 star with a mag. 8.5 star 1/4 degree NW, then

   3/8+ degree SSE to

the center of M17.

After viewing M17 as discussed below, return to the mag. 5.5 star
in the path just before M17, then go 2 degrees N, 3/8 degree
(1 1/2m) W to the open cluster in M16.

After viewing M16 as discussed below, return to the mag. 5.5 star
in the path just before M17 and M16, then do this:

Mag. 5.5 star with a mag. 8.5 star 1/4 degree NW, then

   2 1/2 degrees S, 2 2/3 degrees (11 1/2m) E to

a mag. 5.0 star with a mag. 8.0 star 1/8 degree N, a mag. 7.0 star
1/2- degree E, and a mag. 5.5 star 3/8+ degree SW, then

   1/2 degree S to

a mag. 7.0 star with another mag. 7.0 star 1/4+ degree ESE and a
mag. 5.5 star 3/8 degree NW, then

   3/8 degree S, slightly E to

the center of M25.

   M17 was viewed 7/25/02, 10:30 p.m.  Conditions poor--1 day past
full moon, humid.  M17 (at least what I saw of it) looks like a
horseshoe opening E, which appears about 1/2 degree N to S.  The S
arm has two mag. 8.0 stars, with the W one double, so that it looks
like three stars almost lined up E to W.  The N arm has an open
cluster.  This is a superb object!  It rates three exclamation
points in the NGC catalog.  94x gave a good view.  M17 is also
known as the Horseshoe Nebula.
     M17 was viewed again 8/8/02, 9:45 p.m.  Conditions good.  This
time I also saw a distinct streak of nebulosity near the bend of
the horseshoe running NW-SE.
     M16 was viewed 8/18/03, 9:45 p.m., under excellent conditions.
The cluster looks a little like a compact Christmas tree with the
top to the NNW.  I could clearly see a glow around the cluster,
although I could not make out an eagle shape;  maybe you will have
better luck.  Moving a little S from the cluster shows up the
boundary of the glow nicely.  94x gives a a good view, but 267x and
534x give interesting views also.  M16 is also called the Star
Queen Nebula (for celestial chess players perhaps?).
     M25 was viewed 8/1/02 under ok (but humid) conditions.  94x
gives a good view of the cluster and the 5-star letter M (with the
top close to N) inside it.  The center star of the letter M looks
gold.  This is a pretty little cluster!


55. M22--A huge and bright globular cluster

M22 (globular cluster)           mag. 5.1          size 33'
18h 36.4m                   -23 degrees 54'        Sagittarius

Vega, then

   64 1/6 degrees S, 2 degrees (9m) E to

lambda Sgr (3.0;  top of teapot;  Kaus Borealis;  sigma Sgr (2.0)
is 6 1/4 degrees ESE, phi Sgr (3.0) is 1 degree S of a lambda-sigma
line, closer to sigma, and delta Sgr (3.0) is 4 2/3 degrees SSW of
lambda, with sigma-lambda-delta being a right triangle),
OR get to lambda Sgr by using the alternate path under the
Fantastic Four to get to 7 Sgr (5.0, inside M8) and then go 1
degree+ S and 5 2/3 degrees (25m) E,
OR once you have used lambda Sgr often enough you may be able to go
to it directly as your starting point, then

   1 3/8 degrees N, 1 3/8 degrees (6m) E to

24 Sgr (5.5, with a mag. 7.0 star 1/8+ degree SW and 25 Sgr (6.5)
1/4 degree SSE), then

   1/8 degree N, 5/8 degree (2 1/2m) E to

the center of M22.

     Viewed 7/6/02, 10:45 p.m.  Conditions excellent.  It's so big!
After viewing small faint globular clusters, you are about knocked
out of your chair when this big bright thing comes into view.  This
must be the Arnold Schwarzenegger of globular clusters.  It is a
large glowing ball of stars.  94x and 267x give good views.  This
object received two exclamation points in the NGC catalog.


56. IC4756 and NGC6633--Two bright open clusters

IC4756 (open cluster)        mag. 4.6             size 50'
18h 39.0m             5 degrees 27'               Ophiuchus

NGC6633 (open cluster)       mag. 4.6             size 26'
18h 27.7m             6 degrees 34'               Ophiuchus

Alpha Oph (2.5) (see Rasalgethi above for the path to alpha Oph),
then

   8 degrees S, 2 1/8 degrees (8 1/2m) E to

beta Oph (2.5, with gamma Oph (3.5) 2 1/4 degrees SSW, a mag. 8.0
star 1/4 degree SSE, and a mag. 7.0 star 1/2- degree WSW), then

   1/4 degree S, 4 1/8 degree (17m) E to

66 Oph (5.0, see the M23 path for more details about 66 Oph),
then

   1/4 degree S, 14- degrees (56m) E to

theta Ser (4.0;  this is a nice double star, splittable at 58.75x,
consisting of a WNW-ESE pair of nearly identical white stars), then

   1 1/4 degrees N, 2 5/8 degrees (10 1/2m) W to

a mag. 6.0 star with a mag. 8.0 star 3/8+ degree SSE, then

   slightly S, 1 5/8 degrees (6 1/2m) W to

the center of IC4756;  there is a mag. 6.5 star in the SE part.

After observing IC4756 as discussed below, do this:

Center of IC4756, then

   1 1/6 degrees N, 1/2+ degree (2+ m) W to

a mag. 5.5 star with a mag. 7.5 star 1/4 degree SSE, then

   slightly S, 2 1/4- degrees (9m) W to

the center of NGC6633;  there is a mag. 5.5 star 3/8 degree SSE of
the center of NGC6633.

     Viewed 7/13/02, 11:00 p.m.  Conditions good.  IC4756 is an
attractive large cluster.  It filled the field with bright stars
at 94x, although it will not all fit into the field at this power.
     NCG6633 is smaller than IC4756 but contains an attractive
bunch of bright stars.  94x gives a good view.


57. Epsilon1, epsilon2 Lyr--The double double star

Epsilon1, epsilon2 Lyr (double double star)    mag. 4.5
18h 44m               39 degrees 40'               Lyra

Vega, then

   7/8 degree N, 1 3/8+ degree (7 1/2m) E (or 1 5/8+ degree NE) to

epsilon1, epsilon2 Lyr. (Note:  Epsilon1 is just N of epsilon2.
Also, zeta2,1 Lyr (4.0) is 2 degrees S of epsilon1, epsilon2 and 2
degrees SW of Vega, with Vega, epsilon1, epsilon2 and zeta2,1
forming a near-equilateral triangle.)

     Viewed 7/1/02, 9:40 p.m.  Conditions pretty good.  This
consists of four white, nearly equal-magnitude stars arranged in
a N-S pair to the N and an E-W pair to the S.  It looked good with
267x;  it is quite an attractive grouping, but fairly high power is
needed to split each of the pairs.


58. Beta Lyr--A white variable star surrounded by 3 deep blue stars

Beta Lyr (variable, multiple)     mag. 3.3-4.3
18h 50.1m             33 degrees 22'             Lyra

Vega, then

   1 1/6 degrees S, 1 1/2 degrees (8m) E (or 2 degrees SE) to

zeta2,1 Lyr (4.0;  NW star of parallelogram), then

   4 1/4 degrees S, 1 1/8 degrees (5 1/2m) E (or 4 1/2 degrees SSE)
   to

beta Lyr (variable 3.3-4.3;  SW star of parallelogram;  Sheliak).

     Viewed 7/1/02, 10:00 p.m., conditions pretty good.  I had
looked at this star with low magnification on the way to the Ring
Nebula (see section VIG), and noticed a blue star nearby.  This
night I decided to look at it with 267x, and was flabbergasted to
see two more blue stars nearby, so what we have is a triangle of
blue stars with a bright white star inside!  This was one of those
unexpected "Oh my Lord look at that" experiences mentioned in
Chapter I.
     There is something else interesting about this star, namely it
varies between magnitude 3.3 and magnitude 4.3 about every 12.94
days.  You can easily keep track of this with binoculars if you
cast a glance at the parallelogram near Vega each night;  when
beta Lyr is at its brightest it is about the same brightness as
the SE star of the parallelogram (gamma Lyr), and at its dimmest
(which is not terribly dim) it is about the same brightness as the
NW star of the parallelogram (zeta2,1 Lyr).


59. M11--The Wild Duck Cluster

M11 (Wild Duck Cluster)       mag. 5.8           size 14'
18h 51.1m             -6 degrees 16'             Scutum

Alpha Oph (2.5) (see Rasalgethi above for the path to alpha Oph),
then

   8 degrees S, 2 1/8 degrees (8 1/2m) E to

beta Oph (2.5, with gamma Oph (3.5) 2 1/4 degrees SSW, a mag. 8.0
star 1/4 degree SSE, and a mag. 7.0 star 1/2- degree WSW), then

   1/6 degree S, 4 1/8 degree (17m) E to

66 Oph (5.0, see the M23 path for more details about 66 Oph), then

   7 1/3 degrees S, 5 1/8 degrees (21m) E to

eta Ser (3.5, with a mag. 8.0 star 1/4 degree S), then

   1 7/8 degrees S, 6 1/2 degrees (26m) E to

beta Sct (4.0, with a mag. 8.5 star 3/8 degree WNW; also, alpha Sct
(4.0) is 4 1/2 degrees SW of beta Sct, with eta Ser, beta Sct, and
alpha Sct forming a nearly isosceles triangle that points NW), then

   1 1/6 degrees S to

a mag. 7.0 star which is the W star in a bow consisting of this
star, a mag. 6.0 star 5/8 degree E of this star, and a mag. 6.5
star a little S of the line joining these two, closer to the one on
the east;  then from the east star (6.0) in this bow,

   3/8 degree S, 3/8 degree (1 1/2m) E to

the center of M11 (note:  M11 contains a mag. 8.0 star).

   Viewed 7/15/02, 10:25 p.m.  Conditions good.  This is a
beautiful open cluster!  If you use your imagination a little, you
can see a flight of wild ducks flying east.  There is a pair of
bright stars just S of the E end of the cluster;  the mag. 8.0
star mentioned in the last line of the path is in the S wing of
the flight.  M11 received two exclamation points in the NGC
catalog.
     By the way, did you happen to notice a bright star about 1/4
degree N and a little E of the W star in the bow described in the
path?  I did, and it surprised me because I did not expect anything
so bright there.  Was it a UFO? a killer asteroid? Superman?  Well,
no, it was R Sct, a long-term variable star that varies between
mag. 4.5 and mag. 8.2 over the course of 140 days.  This is one
you can watch with binoculars over time if you want.


60. M57--The Ring Nebula

See section VIG for details.


61. Albireo--A splendid orange-blue double

Beta Cyg (Albireo--double star)   mags. 3.2, 5.4   separation 34.2"
19h 30.7m            27 degrees 58'               Cygnus

Deneb, then

   5 degrees S, 3 2/3 degrees (19 1/2m) W (or 6 1/8 degrees SW) to

gamma Cyg (2.0, middle star of the Northern Cross), then

   5 1/6 degrees S, 5 1/3 degrees (26m) W (or 7 1/4 degrees SW) to

eta Cyg (4.0, next star in the Northern Cross, with mag. 7.5 stars
1/4- degree ESE and 3/8- degree SW), then

   7 1/6 degrees S, 5 5/8 degrees (25 1/2m) W (or 9 degrees SW,
   bending a little E from gamma-eta line) to

beta Cyg (3.0, Albireo--bottom star in the Northern Cross) (Note:
Albireo is a little SW of a Vega-Altair line, about 1/3+ of the way
from Vega to Altair).

[Alternate path:  Vega, then

   10 2/3 degrees S, 11 7/8 degrees (54m) E to
   
beta Cyg (3.0, Albireo).]

   Viewed 7/8/02, 9:30 p.m.  Conditions ok to poor.  This double
star consists of bright orange and blue components, widely
separated.  Reference II-3 says this is one of the prettiest
doubles, and I do not disagree.


62. NGC6826--The Blinking Nebula

NGC6826 (Blinking Nebula)         mag. 8.8           size 25"
19h 44.8m                 50 degrees 31'             Cygnus

Deneb, then

   1/6 degree S, 10 degrees (56 1/2m) W to

delta Cyg (3.0, NW star in the crossbar of the Northern Cross, with
two mag. 7.5 stars 1/8+ degree S and a mag. 5.0 star 7/8 degree
NW), then

   5 3/8 degrees N, 1/2 degree (3m) W to

16 Cyg (5.5, with a mag. 8.0 star 1/8+ degree SSW;  you will
recognize 16 Cyg even without the check star since it is a nice
double with two white components of roughly equal magnitude), then

   1/2 degree (3m) E to

NGC6826 (There is a mag. 8.0 star 1/4 degree N which could serve as
a check star, but you don't need it, since 16 Cyg (the previous
star in the path) is the best check star--it is easy to recognize
and is almost directly W of NGC6826 1/2 degree).

     Viewed 8/27/02, 9:00 p.m.  Conditions very good.  This thing
really does appear to blink;  I didn't really believe it until I
saw it, but I believe it now.  At 58.75x this object is small
enough that you may mistake it for a star;  you need higher power
than that to appreciate it.  At 94x if you stare right at it it
seems to fade away, but glance a little away from it and it
instantly flashes back looking like a whitish BB (as in BB gun).
Although sometimes objects (including this one) seem to wink off
and on due to atmospheric turbulence, this one does it on cue, so
there is something more than atmospheric turbulence at work here.
This behavior is probably due to a phenomenon known as averted
vision;  when you look a little bit away from a faint object you
can sometimes see it better since you are then using a part of
your retina that is more sensitive to faint light.  If so, it is
certainly the most extreme example of this phenomenon I have seen.
At 267x it looks like a bigger white ball, the blinking behavior
is still present, and you may be able to see the central star
(magnitude 10.4).  At 534x the central star shining in the center
of the ball is quite obvious;  this is a rarity, since with
planetary nebulas the central star is usually too faint to be seen
with a small telescope.  When you go for this object, schedule some
time to look at it, since you may want to play with it for a while!


63. M27--The Dumbbell Nebula

M27 (planetary nebula)          mag. 7.3          size 8'x4'
19h 59.6m             22 degrees 43'              Vulpecula

From Altair, go about 10 degrees N to find the constellation
Sagitta the arrow, which is about halfway between Altair and
Albireo.  The main part of this constellation consists of the
following five stars:  Gamma Sge (4.0), delta Sge (4.0), which is
2 7/8+ degrees WSW of gamma, alpha Sge (4.5), which is 1 7/8-
degrees WSW of delta, beta Sge (4.5), which is 1/2 degree SSE of
beta, and zeta Sge (5.0), which is 3/4 degree NNE of delta.  Once
you have found this rather distinctive looking little group of
stars, the first four of which resemble an arrow flying NNE,
proceed as follows:

Gamma Sge (4.0, tip of arrow), then

   3 5/8 degrees N, slightly E to

14 Vul (6.0, with a mag. 8.5 star 1/8 degree S), then

   3/8 degree S, 1/8 degree (1/2m) E to

M27 (Note:  There is a roughly N-S line of stars 1/2 degree long
with mags. 8.0, 7.5, and 8.0, with the middle one closer to the top
than the bottom, and M27 is 1/4 degree W of the middle star).

     Viewed 9/30/02, 7:45 p.m.  Conditions good.  This appeared as
a ghostly white, vaguely dumbbell-shaped (or hourglass-shaped)
object.  It is much larger than most of our planetary nebulas, and
also has less average brightness across its area, although its
total brightness is fairly large due to its size.  94x gave a good
view.


64. 31,30 Cyg--A dazzling and wide orange-white-blue triple star

31,30 Cyg (triple star)      mags. 3.8, 6.7, 4.8   seps. 107", 337"
20h 13.6m               46 degrees 44'             Cygnus

Deneb, then

   1 1/2 degrees N, 5 degrees (28m) W to

31,30 Cyg (3.5, with 32 Cyg (3.5) 1 degree NNE).

     Viewed 7/30/05, 9:45 p.m., under excellent conditions.  This
consists of a bright orangish star, with a not-quite-as-bright
white star about 6' to the NW and a fainter blue star about 2' to
the S.  This is a bright and attractive combination!  With an 11
inch SCT, 112x gave a good view.
     While you are in the neighborhood, check out 32 Cyg 1 degree
to the NNE, which is the check star in the path to 31,30 Cyg.  This
three-star combination features a blazingly bright orange star.


65. Omicron Cap--The chameleon double star?

Omicron Cap (double star)       mag. 5.5
20h 30m                -18 degrees 35'              Capricornus

Altair, then

   9 2/3 degrees S, 5 1/8 degrees (20 1/2m) E to

theta Aql (3.0, with 66 Aql (5.5) 1/2 degree WSW and 64 Aql (6.0)
3/4+ degree ENE), then

   11 2/3 degree S, 1 5/8 degrees (6 1/2m) E to

alpha2 Cap (3.5, with alpha1 Cap (4.0) 1/8 degree WNW and nu
Cap (4.5) 5/8 degree ESE, forming a 3-star line), then

   5/8 degree ESE to

nu Cap (4.5), then

   2 degrees S, slightly E to

beta1 Cap (with beta2 Cap (5.5) just W), then

   3 5/6 degrees S, 2 1/8 degrees (9m) E to

omicron Cap (5.5, with rho Cap (5.0) 7/8- degree NNW and pi Cap
(5.0) 3/4 degree NW, forming a nice near-isosceles triangle
pointing SSE that is easy to spot with binoculars).

     Viewed many times under varying conditions.  This is a nice
little double star with two components of nearly equal magnitudes
that looks good at 94x and at 267x.  When I first saw this in the
Fall of 2001 with a 5 inch telescope I thought it was a red-blue
pair, but now when I look at it with a 9.25 inch telescope the
components both usually look bluish-white, but one of them will
occasionally flash red.  It is not unusual for the colors of stars
to fluctuate some when the star is near the horizon (so the
starlight has to come through a lot of atmosphere and possible
turbulence), but the fact that we have two near-equal magnitude
components close together fluctuating differently makes the
fluctuations more dramatic.  What colors will you see?  By the way,
I first stumbled on this double star while looking for the planet
Neptune, which was lurking nearby;  it probably is still nearby
since it moves so slowly.


66. 52 Cyg and NGC6960--An orange-blue double star and part of the
    Veil Nebula

52 Cyg (double star)          mag. 4.5             sep. 6.5"
20h 45.5m               30 degrees 40'             Cygnus

NGC6960 (nebula)                                   size 70'x6'
20h 45.7m               30 degrees 43'             Cygnus

Deneb, then

   11 1/3 degrees S, 1 degree (5- m) E to

epsilon Cyg (2.5, SE end of the crossbar of the Northern Cross,
with a mag. 7.5 star 1/8+ degree E and T Cyg (5.5-6.0 variable)
3/8+ degree NNE), then

   3 1/3 degrees S, 1/8 degree (1/2m) W to

52 Cyg (4.5).

     Viewed 9/21/02, 9:00 p.m.  Conditions ok except for full moon
(which was partly tree-blocked at this time).  52 Cyg consists of a
bright orange star followed by a much fainter blue star.  267x was
needed to split it.  It is a nice little double, but what makes it
more interesting is that if you allow the star to drift westward,
about 1/3 degree E of the star you may see lots of little stars that
appear to be embedded in a narrow (essentially) N-S band of faintly
glowing gas.  This was visible at 267x and even at 94x, although
the background sky looked brighter at 94x (since a bigger chunk of
light pollution was taken in), obscuring the gas somewhat.  This
strip of gas is part of NGC 6960, which is the west segment of the
Veil Nebula, which is part of the Cygnus Loop.  The Cygnus Loop is
a faint broken ring about 2 1/2 degrees across;  it is the remnant
of a supernova that exploded more than 100,000 years ago.  NGC6960
received two exclamation points in the NGC catalog.
     I viewed 52 Cyg again on 9/2/05 at 8:30 p.m. under pretty good
conditions as part of a path to NGC 6940.  Even with an 11 inch SCT
(Schmidt-Cassegrain telescope) at 318x, the faint blue component
was very difficult to see, so do not be too disappointed if you do
not see it.  This component is slightly S of due E of the much
brighter orange component.
     I viewed 52 Cyg yet again on 9/12/05 at 8:30 p.m. under fairly
good conditions in spite of a half moon far to the south.  This
time I could see both components easily at 318x with the 11 inch
SCT.  Two things that helped considerably were that I had cleaned
the big lens in front and collimated the telescope a few nights
earlier.


67. NGC7009--The Saturn Nebula

NGC7009 (planetary nebula)      mag. 8.3          size 28"
21h 4.2m                 -11 degrees 22'          Aquarius

Altair, then

   2 1/2 degrees N, 10 3/8 degrees (42 1/2m) E to

epsilon Del (4.0, with a mag. 6.5 star 1/2 degree W and a mag. 8.0
star 1/4 degree E;  tail of dolphin), then

   1 1/8 degrees S, 9 1/8 degrees (37m) E to

gamma Equ (4.5, with 6 Equ (6.0) 1/8- degree S), then

   1/8 degree S, 1 degree (4m) E to

delta Equ (4.5), then

   1/8 degree S, 7 1/4 degrees (29 1/2m) E to

epsilon Peg (2.0, with a mag. 7.5 star 1/2 degree NW), then

   3 2/3 degrees S, 6 1/2 degrees (26m) E to

theta Peg (3.5, with a mags. 7.5-8.0 NNE-SSW pair 3/8 degree E and
nu Peg (4.5) 1 5/8 degrees SW), then

   6 1/4 degrees S, 4 5/8 degrees (18 1/2m) E to

zeta Aqr (3.5;  this is the center of a distinctive letter Y in
Aquarius, opening to the west, which is so easily recognizable that
you can quickly get to it with binoculars, then use its stars as
starting points to find other objects.  The other three stars in
the Y are eta Aqr (4.0), 1 5/8 degrees E;  gamma Aqr (4.0), 2 1/4
degrees SW;  and pi Aqr (5.0), 1 5/8 degrees NNW.  There is also a
mag. 7.5 star 3/8- degree NE of zeta Aqr and a mag. 8.0 star 1/4
degree SE of zeta Aqr.  Zeta Aqr is actually a double star, but
534x was required for me to barely split it into a N-S pair of
nearly identical white stars.). then

   1/3 degree S, 5 3/4 degrees (23m) W to

alpha Aqr (3.0, with a mag. 5.5 star 5/8 degree SSW), then

   5 1/4 degrees S, 8 1/2 degrees (34m) W to

beta Aqr (3.0, with a mag. 8.0 star 5/8 degree WNW and a mag. 8.5
star 1/4 degree N of the mag. 8.0 star), then

   5 5/6 degree S, 5 3/8 degree (22m) W to

nu Aqr (4.5, with a mag. 8.0 star 1/2 degree SE), then

   1 3/8 degrees (5 1/2m) W to

NGC7009 (with a mag. 8.5 star 3/8 degree NNE, a mags. 8.5-8.0 NE-SW
pair 3/8 degree NW, and a mag. 7.0 star 5/8 degree WSW).

[Alternate path to nu Aqr coming from the west, which may be an
advantage if your view to the south is partially blocked, since
the stars in the path will then break into the clear before the
final object does, giving you more time to find the final object:

Altair, then

   9 2/3 degrees S, 5 1/8 degrees (20 1/2m) E to

theta Aql (3.0, with 66 Aql (5.5) 1/2 degree WSW and 64 Aql (6.0)
3/4+ degree ENE), then

   11 2/3 degree S, 1 5/8 degree (6 1/2m) E to

alpha2 Cap (3.5, with alpha1 Cap (4.0) 1/8 degree WNW and nu Cap
(4.5) 5/8 degree ESE, forming a 3-star line), then

   1 1/6 degrees N, 12 5/8 degrees (51 1/2m) E to

nu Aqr (4.5, with a mag. 8.0 star 1/2 degree SE).]

     Viewed 9/5/02, 9:30 p.m.  Conditions very good.  A whitish
ball, maybe a little bluish, maybe flattened a little.  At 94x it
tends to blink on and off, but look away and it blinks on.  267x
gives a good view.  Quite bright and attractive!  The September
2002 issue of reference II-11 says "Requires 8-inch telescope to
see Saturn-like appendages."  Apparently it also requires better
viewing conditions than mine since I did not see any appendages
with a 9.25 inch telescope, but it was pretty even without them.
By the way, this object is one of the few objects to receive three
exclamation points in the NGC catalog.


68. 61 Cyg--A pretty little pair of orange dwarfs

61 Cyg (double star)        mags. 5.2, 6.0        separation 30.6"
21h 6.9m            38 degrees 45'                Cygnus

Deneb, then

   4 1/8 degrees S, 3 degrees (16m) E to

nu Cyg (4.0, with mag. 7.0 stars 1/4+ degree NE and 1/2 degree SE),
then

   3 1/8 degrees S, 3 1/2- degrees (17 1/2m) E to

tau Cyg (4.0, with sigma Cyg (4.5) 1 1/2 degrees NNE, a mag. 5.5
star 7/8+ degree E, and a mag. 7.5 star 1/4 degree SSW), then

   3/4 degree N, 1 1/2 degrees (8m) W to

61 Cyg (5.5-6.0 variable;  sigma, tau, and 61 form a near-isosceles
right triangle with right angle at tau).

     Viewed 9/21/02, 8:00 p.m.  Conditions ok except for full moon.
This is a pretty little pair of nearly equal magnitude orange
dwarfs.  94x gives a good view.  Incidentally, this is the first
star (except for the sun) whose distance from Earth was measured.
It was done using parallax;  that is, the difference in position of
61 Cyg relative to more distant background stars was noted for two
different positions of the Earth in its orbit around the sun, and
this allowed the distance of 61 Cyg from the Earth to be computed
by trigonometry (note that the closer to us a star is, the more its
position in the sky relative to the background stars will seem to
change as the Earth moves in its orbit).  61 Cyg is about 11.38
light years from Earth. 


69. M39--A big, bright Christmas tree in the sky

M39 (open cluster)            mag. 4.6              size 32'
21h 32.2m                48 degrees 26'             Cygnus

Deneb, then

   1 1/3 degrees S, 4 1/4 degrees (23 1/2m) E to

xi Cyg (4.0, with a mag. 8.0 star 3/8 degree SSW and a mag. 8.5 star
3/8 degree ESE), then

   1 2/3 degrees N, 5+ degrees (29m) E to

rho Cyg (4.0, with a mag. 6.0 star 1/4 degree NNW), then

   2 7/8 degrees N, 1/4 degree (1 1/2m) W to

the center of M39.

     Viewed 8/9/02, 10:30 p.m.  Conditions pretty good.  It looks
like a Christmas tree pointing north, with the top bent a little
east, as if the ornament on top was a little too heavy.  This will
just barely fit in the field at 94x, but it actually looks prettier
at 58.75x since you don't seem to be right on top of it then.  The
stars in the tree are quite bright and pretty spread out.


70. M15--A large and bright globular cluster

M15 (globular cluster)       mag. 6.4            size 18'
21h 30.0m            12 degrees 10'              Pegasus

Altair, then

   2 1/2 degrees N, 10 3/8 degrees (42 1/2m) E to

epsilon Del (4.0, with a mag. 6.5 star 1/2 degree W and a mag. 8.0
star 1/4 degree E;  tail of dolphin), then

   1 1/8 degrees S, 9 1/8 degrees (37m) E to

gamma Equ (4.5, with 6 Equ (6.0) 1/8- degree S), then

   1/8 degree S, 1 degree (4m) E to

delta Equ (4.5), then

   1 1/6 degrees N, 1 1/8- degrees (4 1/2m) E to

a mag. 6.0 star with a mag. 7.0 star 3/8 degree N, then

   1/8 degree S, 2 1/2 degrees (10m) E to

a mag. 6.0 star with mag. 7.0 stars 5/8 degree SW and 7/8 degree
ESE, then

   7/8 degree N, slightly E to

a mag. 7.5 star, then

   1/4 degree NE to

the center of M15 (with a mag. 6.0 star 1/4 degree E).

     Viewed 9/6/02, 9:00 p.m.  Conditions good.  A bright and large
cluster!  94x gives a good view, and 267x shows lots of stars.
Bright center.  A good one!  This object received one exclamation
point in the NGC catalog.


71. M2--One of the brightest globular clusters in the sky

M2 (globular cluster)        mag. 6.5            size 15'
21h 33.5m              -0 degrees 49'            Aquarius

Zeta Aqr (3.5, center of Y;  see NGC7009), then

   1/3 degree S, 5 3/4 degrees (23m) W to

alpha Aqr (3.0, with a mag. 5.5 star 5/8 degree SSW), then

   7 degrees (28m) W to

a mag. 6.5 star with 24 Aqr (7.0) 5/8 degree NE, then

   1/2- degree S, 1 degree (4m) E to

M2 (with a mags. 8.0-8.5 ESE-WNW pair 3/4 degree S and slightly W).

[Alternate path to the last star in the path (6.5) coming from the
west, which may be an advantage if your view to the south is
partially blocked, since the stars in the path will then break into
the clear before the final object does, giving you more time to
find the final object:  First use the alternate path given following
the path to NGC7009 to get to nu Aqr, then go 5 5/6 degrees N,
5 1/2 degrees (22m) E to get to beta Aqr (3.0, with a mag. 8.0 star
5/8 degree WNW and a mag. 8.5 star 1/4 degree N of the mag. 8.0
star), then go 5 1/6 degrees N, 1 1/2 degrees (6m) E to the mag.
6.5 star.]

     Viewed 9/5/02, 10:00 p.m.  Conditions very good.  This is a
big bright ball, brighter in the center.  It is one of the
brightest globular clusters in the sky according to reference II-3,
and I certainly believe that.  It looks fine at 94x, and at 267x I
could see individual stars around the edges.  This object was given
two exclamation points in the NGC catalog.


72. NGC7209--A pretty open cluster contained (mostly) in an
    isosceles trapezoid

NGC7209 (open cluster)          mag. 6.7           size 25'
22h 5.2m               46 degrees 30'              Lacerta

Deneb, then

   1 1/3 degrees S, 4 1/4 degrees (23 1/2m) E to

xi Cyg (4.0, with a mag. 8.0 star 3/8 degree SSW and a mag. 8.5
star 3/8 degree ENE), then

   1 2/3 degrees N, 5+ degrees (29m) E to

rho Cyg (4.0, with a mag. 6.0 star 1/4 degree NNW), then

   1/2- degree S, 5 2/3 degrees (32m) E to

the middle star of a west-pointing S-N bow 1/4 degree long with
stars of magnitudes 5.0, 6.5, 6.0, then from the N star of the bow

   1 1/2 degrees N, 1/8 degree (1m) W to

a mag. 6.0 star, with a mag. 7.0 star 1/2 degree N, then

   1/4 degree S to

the center of NGC7209.

     Viewed 9/4/02, 9:00 p.m.  Conditions good.  This is a round,
pretty cluster which is mostly contained in an isosceles trapezoid
of four stars with the short base facing west.  Most of the stars
in the cluster have about the same magnitude.  This was a surprise;
I did not expect to see something this attractive when I started
looking for this object!


73. Delta Cep, mu Cep, and IC1396--A variable double star,
    Herschel's Garnet Star, and a large nebula containing an open
    cluster which contains a nice variable triple star

Delta Cep (variable double star)  mag. 3.5-4.4
22h 29.2m             58 degrees 25'            Cepheus

Mu Cep (Herschel's Garnet Star)   mag. 4.0-5.0
21h 43.5m             58 degrees 50'            Cepheus

IC1396 (nebula)                      size 2.8 degrees x 2.3 degrees
21h 39.1m             57 degrees 30'            Cepheus

Beta Cas (W star of letter W in Cassiopeia), then

   1 degree S, 15 1/2 degrees (118 1/2m) W to

zeta Cep (3.5, with a mag. 7.5 star 1/4 degree S;  also, epsilon
Cep (4.0) is 1 1/4 degrees SSE of zeta, and these two stars along
with delta Cep (3.5-4.4) make a nearly isosceles triangle pointing
ENE with long sides of length about 2 1/2 degrees, and this triangle
is readily recognizable in binoculars and finderscope), then

   1/4 degree N, 2 3/8 degrees (18 1/2m) E to

delta Cep (3.5-4.4 variable, with a mag. 8.0 star 1/8 degree NNW
and a mag. 8.0 double star 1/8- degree W).

[Note:  You can also get to zeta Cep by starting at Deneb and going
12 7/8 degrees N, 11 3/4 degrees (89 1/2m) E.]

After viewing delta Cep as discussed below, return to zeta Cep, and
then go 2/3 degree N, 3 1/2 degrees (27 1/2m) W to mu Cep.

After viewing mu Cep as discussed below, go 1 1/3 degrees S, 5/8
degree (4 1/2m) W to a mag. 6.5-7.5 variable triple star, with a
mag. 7.5 double star 1/4 degree ENE;  the variable triple star is
in the middle of the open cluster, which is roundish and about 3/4
degree across, and the open cluster is a little S of the center of
IC1396 (which is sort of roundish), with mu Cep in the NE edge of
IC1396.

     Viewed 9/4/03, 8:15 p.m.  Conditions good except for a half
moon.  Delta Cep is a nice double star, with a relatively bright
white component and a not quite as bright blue component to the
south.  This double splits easily with 58.75x, but it looks even
better with 94x and 267x.  The claim to fame of delta Cep, however,
is that it is the prototype of a class of variable stars called
Cepheid variables.  These stars are important because once you have
determined the period (which is the amount of time between peaks of
brightness, and is about 5.37 days for delta Cep), you can compute
the intrinsic brightness of the star, which is the amount of light
the star puts out.  Then by comparing the intrinsic brightness with
the brightness observed from Earth, you can determine how far away
the star is.  The distances from Earth to many galaxies have been
computed by looking at Cepheid variables within them.  According to
reference II-3, you can easily detect the variation in brightness
of delta Cep by looking at it every night, although I have not done
this yet.
     Mu Cep is said to be one of the reddest stars visible to the
naked eye (under good conditions).  This is why it is called the
garnet star.  It actually looks more orange than red to me, but the
color is certainly striking.  Mu Cep is also a variable star.
     IC1396 is a large nebula, and I could detect its glow with
58.75x, 94x, and 267x (I didn't look at it with 534x).  I was able
to locate much of the boundary of the nebula by scanning back and
forth across it with 58.75x and 94x.  The open cluster inside
IC1396 is OK, and the variable triple star is quite nice.  It
consists of a relatively bright white component, a bluish
component a little to the NNW, and a reddish component a little to
the SE of the white component.  (The reddish component sometimes
looked more whitish or even bluish.)  The double star to the NNE
consists of two whitish components lined up SSW-NNE.  Both of these
multiple stars can be split at 58.75x, but they look better at 94x
and 267x.  The triple star, by the way, is also known as Struve
2816.


74. M52--A nice open cluster with many faint stars, with a
    quintuple star and The Bubble Nebula (NGC7635) along the way

quintuple star                  mag. 5.0
23h 30m            58 degrees 30'                Cassiopiea

NGC7635 (The Bubble Nebula)                      size 15'x8'
23h 20.7m          61 degrees 12'                Cassiopiea

M52 (open cluster)               mag. 6.9        size 12'
23h 24.2m          61 degrees 35'                Cassiopiea

Beta Cas (W star of letter W in Cassiopeia), then

   1/2 degree S, 2 7/8 degrees (22m) W to

tau Cas (5.0, with a mag. 6.5 star 1/2- degree NE), then

   1/8 degree S, 2 1/4 degrees (17m) W to

a mag. 5.0 star with mags. 8.0 and 8.5 stars 1/4- degree SE (the
mag. 5.0 star is the quintuple star;  see discussion below), then

   1 1/2+ degrees N, 7/8+ degree (7 1/2m) W to

a mag. 5.5 star (with a mag. 7.0 star 3/8 degree NNW and a mag. 6.5
star 1/4 degree WNW, forming a right triangle), then from the W
(mag. 6.5) star of the triangle,

   1- degree N, slightly W to

a mag. 7.0 star (this star is in the W edge of NGC7635 (the Bubble
Nebula);  see discussion below), then

   slightly S, 3/4 degree (6m) E to

a mag. 6.5 star, then

   3/8 degree N, slightly W to

a mag. 8.0 star, with a mags. 8.5-8.0 ESE-WNW pair 1/4 degree E,
then

   1/4 degree NW to

a mag. 8.5 star, which is in the SW edge of M52.

     Viewed 9/24/02, 8:30 p.m.  Conditions excellent.  At 58.75x
the quintuple star looks like a fairly nice double with a bright
white star and a fainter blue star a little WNW, and a hint of
something else;  at 94x a faint, close pair of white stars
oriented parallel to the first two appears a little NNE of the
white star, and at 267x a still fainter and closer white star
appears just S of the white star.  I don't know if this is a true
quintuple system, or if some of these stars are actually far apart
in space, but this is interesting in any event.
     Don't feel bad if you can't see the luminosity of the Bubble
Nebula;  it is quite faint.  Most of it seems to be contained in
a pentagon formed by the mag. 7.0 star in the W edge and a
diamond-shaped group of 4 stars just to the SE.  At 267x the mag.
7.0 star seems to be sitting smack on the boundary of the
nebulosity, which runs N-S.  Viewing conditions are critical here.
     Except for the mag. 8.5 star in the SW edge, the stars in M52
are fainter than in most open clusters in our list, but there are
many stars here, and at 267x this round cluster is really quite
pretty.  It can also be seen at 94x, but 267x is better.  The
cluster looks sort of like a globular cluster without the bright
central core.


75. NGC7662--The Blue Snowball Nebula

NGC7662 (planetary nebula)        mag. 8.3           size 17"
23h 25.9m                42 degrees 33'            Andromeda

Deneb, then

   1 1/3 degrees S, 4 1/4 degrees (23 1/2m) E to

xi Cyg (4.0, with a mag. 8.0 star 3/8 degree SSW and a mag. 8.5
star 3/8 degree ENE), then

   1 2/3 degrees N, 5+ degrees (29m) E to

rho Cyg (4.0, with a mag. 6.0 star 1/4 degree NNW), then

   1/2- degree S, 5 2/3 degrees (32m) E to

the middle star of a west-pointing S-N bow 1/4 degree long with
stars of magnitudes 5.0, 6.5, 6.0, then from the S star of the bow

   2/3 degree S, 6 1/8 degrees (34 1/2m) E to

11 Lac (4.5, with a mag. 7.0 star 1/4+ degree W and 6 Lac (4.5)
2 1/8 degrees SW), then

   2 degrees S, 4 degrees (21 1/2m) E to

omicron And (3.5, with a mag. 8.0 star 1/4 degree ESE and 2 And
(5.0) 1/2- degree NNE), then

   1/2 degree S, 3/4+ degree (4 1/2m) E to

a mag. 7.5 star with a mag. 8.0 star 1/3- degree S, then

   1/3 degree N, 1 1/3- degrees (7m) E to

a mag. 7.0 star with a mag. 8.0 star 3/8 degree WSW, then

   1 1/4 degrees (7m) E to

10 And (6.0, with a mag. 7.5 star 3/8- degree S and 9 And (6.0)
3/8 degree SW, forming an isosceles right triangle), then

   7/8 degree N, 1 3/8 degrees (7m) E to

13 And (6.0, with a mag. 8.5 star 1/4 degree NNE), then

   1/3 degree S, 1/8- degree (1/2m) W to

a mag. 8.0 star, with a mag. 8.0 star 3/8 degree S and a mags.
7.5, 7.5 ESE-WNW pair 1/2 degree ESE, then

   1/8 degree (1/2m) W to

NGC7662.

     Viewed 9/7/02, 11:15 p.m.  Conditions ok.  Wonderful!  I could
even see some blue coloration.  Even at 58.75x it was instantly
clear that I had found it.  It did a lot of blinking at 94x (even
on cue when I glanced away from it, although this was maybe not
as pronounced as in The Blinking Nebula).  267x and 534x also gave
good views.  The next-to-last object in the path is a good focus
star, that is, it makes a good object on which to sharpen the focus
when changing magnifications.  This nebula received three
exclamation points in the NGC catalog, and deservedly so.
     I took at least three attempts, starting at Deneb, to find
this object.  After becoming lost a time or two I went back to chart
9 of reference II-4 and remeasured everything (correcting one
error), then got lost again in trying to go directly from omicron
And to 10 And in one step.  The problem here was that under the
prevailing conditions I could not quite make out the pair of mag.
6.0 stars 10 And, 9 And with my finderscope, and the direct path to
10 And (which was 1/6 degree S, 3 1/3 degree (18m) E) was long
enough that I was not getting these stars into the narrower field
of the telescope.  That is when I added two fainter stars in
between, which allowed me to take steps short enough that I could
dispense with the finderscope and just use the telescope itself.
About halfway through the path I was feeling pretty good, when some
cumulus clouds rolled in and brought things to a halt.  About 10
minutes later they rolled out again, allowing me to successfully
reach NGC7662.  There seem to be at least three morals here:  (1)
When the next object in the path is too dim to be seen with your
binoculars and finderscope, you should take short steps, although
you can try a longer step first if you want, especially if the
thing you are going to will be readily recognizable if you hit it.
(2) Our methods really do work.  (3) Persistence pays!  By the way,
was it worth taking a couple of hours to find the Blue Snowball
Nebula? You betcha! 

     Well, that brings our document nearly to its end.  Let me
close (except for the appendex and index) by quoting a verse from
the "Rime of the Ancient Mariner" by Samuel Taylor Coleridge.  (Yes,
I know we spell it "Rhyme" today.)

"This Seraph-band, each waved his hand,
It was a heavenly sight!
They stood as signals to the land,
Each one a lovely light."


Chapter XIII.  Index of definitions, facts, and cautions

     This index gives the section in which various things are
defined or stated.  It does not give the page number since that
will depend on what program you use to view and print this
document.

thing                                  section

Altitude                               VIIA
Aperture                               XB
Arcminutes                             IIIA
Arcseconds                             IIIA
Autumnal equinox                       IIIG
Caution 1 (never force anything)       IVA
Caution 2 (slow motion knobs)          IVA
Caution 3 (star pointer)               IVB
Caution 4 (adjusting tripod legs)      VIIA
Caution 5 (letting go of optical tube) VIIA
Caution 6 (dumping your eyepiece)      VIIB
Celestial equator                      IIIB
Celestial sphere                       IIIB
Clockface minutes                      VIIC3
Declination                            IIIC
Distance between objects               IIIA
East                                   IIIB
Ecliptic                               IIIG
Fact 1 (altitude)                      IIID
Fact 2 (degrees and minutes)           VIE
Fact 3 (time)                          VIIIA
Fact 4 (Big Dipper time)               VIIIC
Free movement                          VIE1
Galaxy                                 XII
Globular cluster                       XII
Great circle                           IIIB
Greenwich Mean Time                    VIIIC
Inverted image                         IIIF
Julian days                            VIIIC
Line of equal right ascension          IIIC
Magnitude                              IIIE
Meridian                               IIIB
North                                  IIIB
North celestial pole                   IIIB
Open cluster                           XII
Planetary nebula                       XII
Polar great circle                     IIIB
Position angle                         XII
Reverted image                         IIIF
Right ascension                        IIIC
Sidereal Time                          VIIIC
South                                  IIIB
South celestial pole                   IIIB
Taxicab movement                       VIE2
Terrestrial                            IIIB
Transit time                           VIIIB
Turn counting                          VIE3
Universal Time (U.T.)                  VIIIC
Vernal equinox                         IIIG
West                                   IIIB
Zenith                                 IIID


Chapter XIV.  Appendix A:  The constellations--abbreviations,
              names, and what they're supposed to be

     This table is taken from reference II-3, with the columns for
the pronunciation ond genitive omitted.  As an example of the
genitive form, which is not used in this document, the genitive for
Monoceros is Monocerotis;  thus the star w in Monoceros is often
called w Monocerotis ("w of Monoceros"), but we will refer to it
simply as w Mon.

abbreviation    name                 what it's supposed to be

And             Andromeda            Andromeda
Ant             Antlia               pump
Aps             Apus                 bird of paradise, bee
Aqr             Aquarius             water bearer
Aql             Aquila               eagle
Ara             Ara                  altar
Ari             Aries                ram
Aur             Auriga               charioteer
Boo             Bootes               herdsman
Cae             Caelum               chisel
Cam             Camelopardalis       giraffe
Cnc             Cancer               crab
CVn             Canes Venatici       hunting dogs
CMa             Canis Major          big dog
CMi             Canis Minor          little dog
Cap             Capricornus          goat
Car             Carina               ship's keel
Cas             Cassiopeia           Cassiopeia
Cen             Centaurus            centaur
Cep             Cepheus              Cepheus
Cet             Cetus                whale
Cha             Chameleon            chameleon
Cir             Circinus             compass
Col             Columba              dove
Com             Coma Bernices        Bernice's hair
CrA             Corona Australis     southern crown
CrB             Corona Borealis      northern crown
Crv             Corvus               crow
Crt             Crater               cup
Cru             Crux                 southern cross
Cyg             Cygnus               swan
Del             Delphinus            dolphin
Dor             Dorado               swordfish
Dra             Draco                dragon
Equ             Equuleus             little horse
Eri             Eridanus             river Eridanus
For             Fornax               furnace
Gem             Gemini               twins
Gru             Grus                 goose
Her             Hercules             Hercules
Hor             Horologium           clock
Hya             Hydra                water snake
Hyi             Hydrus               water snake
Ind             Indus                Indian
Lac             Lacerta              lizard
Leo             Leo                  lion
LMi             Leo Minor            little lion
Lep             Lepus                hare
Lib             Libra                scales
Lup             Lupus                wolf
Lyn             Lynx                 lynx
Lyr             Lyra                 harp
Men             Mensa                table
Mic             Microscopium         microscope
Mon             Monoceros            unicorn
Mus             Musca                fly
Nor             Norma                level
Oct             Octans               octant
Oph             Ophiuchus            Ophiuchus
Ori             Orion                Orion
Pav             Pavo                 peacock
Peg             Pegasus              Pegasus
Per             Perseus              Perseus
Phe             Phoenix              Phoenix
Pic             Pictor               easel
Psc             Pisces               fish
PsA             Piscis Austrinus     southern fish
Pup             Puppis               ship's stern
Pyx             Pyxis                ship's compass
Ret             Reticulum            net
Sge             Sagitta              arrow
Sgr             Sagittarius          archer
Sco             Scorpius             scorpion
Scl             Sculptor             sculptor
Sct             Scutum               shield
Ser             Serpens              serpent
Sex             Sextans              sextant
Tau             Taurus               bull
Tel             Telescopium          telescope
Tri             Triangulum           triangle
TrA             Triangulum Australe  southern triangle
Tuc             Tucana               toucan
UMa             Ursa Major           big bear
UMi             Ursa Minor           little bear
Vel             Vela                 ship's sails
Vir             Virgo                virgin
Vol             Volans               flying fish
Vul             Vulpecula            little fox


Chapter XV.  Appendix B (optional):  A primer for spherical
             trigonometry

     In this optional appendix we give some of the definitions and
theorems of spherical trigonometry (without proofs) and then apply
them to derive one of the results in this document.  This appendix
is only for thise interested in learning a little spherical
trigonometry;  everyone else can skip it.  The only mathematical
background needed to read this appendix is a little basic plane
trigonometry.  The spherical trigonometry ideas here are adapted
from the book "Concise Spherical Trigonometry with Applications" by
Jacques Redway Hammond, Houghton Mifflin, 1943.
     Sometimes we will use the Earth as an example of a sphere;
when we do that, we will be making the slightly incorrect
assumption that the Earth is a perfect sphere.
     We start with the idea of the angle between two intersecting
planes.

Definition 1.  Consider two planes whose intersection is a line.
From any point on the line of intersection, draw two lines
perpendicular to the line of intersection, one lying in each of the
planes.  The four angles formed by these lines are called plane
angles of the planes, and the (measure of the) angle between the
planes is the measure of one of the four angles, where the context
will dictate which of the four angles is meant.

Comment 1.  Two planes are said to be perpendicular if the angle
between the planes is 90 degrees.

     Next we need to define great circles on a sphere, the
spherical angle between intersecting great circles, and the measure
of a spherical angle.

Definition 2.  Consider a sphere.  A circle on the sphere formed by
interecting the sphere with a plane which contains the center of
the sphere is called a great circle on the sphere.  If two great
circles are not identical but intersect at a point on the sphere
(in which case they must also intersect at the diametrically
opposite point on the sphere), any one of the four openings on the
sphere at the point of intersection and between the great circles
is called a spherical angle.  The measure of the spherical angle is
the measure of the angle between the two planes of the great
circles.  This is also the angle between the two tangent lines to
the sphere at the point of intersection of the great circles.

Comment 2.  Great circles are the largest possible ciecles that can
be drawn on a sphere, which is why they are called great circles.
Aren't mathematicians clever about how they name things?

     Now we define an arc of a great circle and its measure.

Definition 3.  An arc of a great circle is a continuous portion of
the great circle which is not the entire great circle.  The measure
(or length) of the arc is the measure (in degrees) of the angle at
the center of the sphere formed by drawing line segments from the
center of the sphere to the endpoints of the arc.

Example 1.  On the Earth, consider a great circle which contains
the north pole, and is thus perpendicular to the equator (which is
another great circle).  Consider a portion of the first great
circle from the north pole to the equator.  This portion is an arc
of a great circle, and its length is 90 degrees.  Note that this
length is given in degrees, not miles.

     Now we come to the definition of a spherical triangle, its
sides, its vertices, and its angles.

Definition 4.  Any three-sided, closed, curvilinear figure on a
sphere, bounded by three great circle arcs of length less than 180
degrees, is called a spherical triangle.  The three great circle
arcs are called sides of the spherical triangle.  The three points
where the sides join are called the vertices of the spherical
triangle.  The spherical angles between pairs of the great circle
arcs are called the angles of the spherical triangle.

Comment 3.  Our definition of spherical triangle implies that the
sides do not all intersect in the same pair of points, and each of
the three angles os the spherical triangle has measure less than
180 degrees.

Example 2.  On the Earth, select two points on the equator which
are less than 180 degrees apart, and draw great circle arcs from
each of the two points to the north pole.  Then these two great
circle arcs, along with the shorter portion of the equator between
the two selected points, are the sides of a spherical triangle.
Note that this spherical triangle has at least two angles equal to
90 degrees.

     We now collect some results about spherical triangles in a
theorem.  Some of these results are the same as the corresponding
results for plane triangles, and some are not.

Theorem 1.  Consider a spherical triangle with angles A, B, and C,
and opposite sides a, b, and c respectively (so a is opposite A,
etc.)

(a)  The sum of the lengths of any two sides is greater than the
length of the third side, e.g. a + b > c.

(b)  180 degrees < a + b + c < 360 degrees.

(c)  180 degrees < A + B + C < 540 degrees.  (Yes, unlike the case
for plane triangles, the sum of the angles of a spherical
triangle is always greater than 180 degrees.)

(d)  The area of a spherical triangle equals the area of the sphere
times (A + B + C - 180 degrees)/720 degrees.

(e)  Two angles of a spherical triangle are equal if and only if
the opposite sides are equal, e.g. A = B iff a = b.

(f)  If two angles (sides) of a spherical triangle are unequal,
then the opposite sides (angles) are unequal in the same order,
e.g. A > B iff a > b, and A < B iff a < b.

(g)  (Law of Sines for spherical triangles)  a/sinA = b/sinB =
c/sinC  (Note:  By our assumptions, each angle lies strictly
between 0 degrees and 180 degrees, so the denominators cannot equal
zero.)

(h)  (Law of Cosines for Sides for spherical triangles)
cosa = cosbcosc + sinbsinccosA, and similarly for cosb and cosc.
(In words, the cosine of any side equals the product of the cosines
of the other two sides, plus the product of the sines of the other
two sides times the cosine of the included angle.)

(i)  (Law of Cosines for Angles for spherical triangles)
cosA = -cosBcosC + sinBsinCcosa, and similarly for CosB and cosC.
(In words, the cosine of any angle equals minus the product of the
cosines of the other two angles, plus the product of the sines of
the other two angles times the cosine of the included side.)

     Now we consider the important special case of right spherical
triangles.

Definition 5.  A right spherical triangle is a spherical triangle
which has at least one right angle, that is, at least one 90 degree
angle.

     From Example 2 we saw that a right spherical triangle can have
more than one right angle.  The next theorem concerns such
triangles.

Theorem 2.

(a)  If a right spherical triangle has any side of length 90
degrees, then the triangle has at least two right angles.

(b)  If a right spherical has any angle equal to its opposite side,
then the triangle has at least two right angles.

(c)  If a spherical triangle has two or more right angles, then
every angle is equal to its opposite side.

     Now we discuss ten equations (called Napier's rules) for right
spherical triangles which have only one right angle.  Some of these
formulas would contain undefined terms if we tried to apply them to
triangles with more than one right angle, which is why we restrict
ourselves to the case of only one right angle.  Triangles with more
than one right angle are so special that they usually can be
handled directly.
     We present a memory trick for deriving Napier's rules which
allows us to boil the ten equations down into two equations.

1.  Draw a right spherical triangle which has its only right angle
at C.
2.  Label the other two angles A and B, and label the sides a
(opposite A), b (opposite B), and c (opposite C).
3.  Write the letters "co" before A, c, and B.
4.  Scratch out the letter C.

We now have a triangle with the five parts b, co A, co c, co B, and
a.  The effect of "co" is to flip a trig function to its
cofunction, so if x is A or c or B, then sin(co x) = cosx,
cos(co x) = sinx, and tan(co x) = cotx.  For any of the five parts
there are two parts that are adjacent in the triangle (remember
that C is gone), and the remaining two parts are said to be
opposite the part.  For example, b has adjacent parts co A and a,
and opposite parts c and B.  The original part (which is b in the
previous sentence) is called the middle part.  The two equations
which can then be used to generate all ten of Napier's rules are

The sIne of any mIddle part = the product of the cOsines of the
Opposite parts,

The sIne of any mIddle part = the product of the tAngents of the
Adjacent parts.

Note the use of capitalization to help us remember these equations.

Example 3.  Using a as the middle part gives

sina = cos(co A)cos(co c), so sina = sinAsinc,

sina = tanbtan(co B), so sina = tanbcotB.

These are the first two of Napier's rules;  using b, c, A, and B in
turn generates the remaining Napier's rules.  We now list all ten
rules:

(1)  sina = sinAsinc
(2)  sina = tanbcotB
(3)  sinb = sinBsinc
(4)  sinb = tanacotA
(5)  cosc = cosacosb
(6)  cosc = cotAcotB
(7)  cosA = cosasinB
(8)  cosA = tanbcotc
(9)  cosB = cosbsinA
(10) cosB = tanacotc

     We conclude this appendix by using our knowledge of spherical
trigonometry to givr a careful proof of Fact 2 in section VIE.
First we repeat the statement of Fact 2.

Fact 2:  Consider two objects or positions, neither of which is at
a celestial pole, which have the same declination dec.  Let min be
the number of minutes between the objects (measured east or west,
whichever is shorter) and let deg be the distance between the
objects as defined in section IIIA.  Then

min = 8invsin[sin(deg/2)/cos(dec)], and

deg = 2invsin[sin(min/8)cos(dec)].

Proof.  We will verify the equation

sin(min/8)cos(dec) = sin(deg/2),

and the two equations above will follow from this.

Draw great circle arcs from the north celestial pole N to the two
objects (or positions).

We claim that the number of degrees in the spherical angle at N is
min/4.

To see this, temporarily extend (or shrink) the two great circle
arcs, if necessary, so that they terminate on the celestial
equator.  From the definition of right ascension, these two great
circle arcs cut off min minutes on the celestial equator, which is
the fraction min/1440 of the entire celestial equator.  But this
fraction of the celestial equator is the same as the fraction
(number of degrees in the spherical angle at N)/360.  Equating
these two fractions snd multiplying both sides of the equation by
360 verifies the claim.

We now break the remainder of the proof into three cases, of which
the third case is the most interesting, although the first two
cases are needed too for a complete proof.

Case 1.  dec = 0

To prove this case, note that since the two objects (or positions)
lie on the celestial equator, which is a great circle, the distance
between them is measured along the celestial equator, and this
distance equals the number of degrees in the spherical angle at N.
since we have shown that this quantity equals min/4, the equation
we are trying to verify becomes

sin((min/4)/2)cos0 = sin(deg/2),

which is an identity.  Q.E.D. for case 1.

Case 2.  dec does not equal 0, and min = 720 (which is the maximum
possible).

In this case, the two great circle arcs lie on the same great
circle, which is a vertical great circle which passes through N
(and through the south celestial pole S).  Thus the distance
between these points is measured along this great circle.

If dec > 0, then we measure from one of the points up to N, then
back down to the other point, getting deg = 2(90 degrees - dec).
Then sin(deg/2) = sin(90 degrees - dec) = cos(dec).

If dec < 0, then we measure from one of the points down to S, then
back up to the other point, getting deg = 360 degrees -
2(90 degrees - dec).  Then sin(deg/2) = sin(180 degrees -
(90 degrees - dec)) = sin(90 degrees - dec) = cos(dec).

Thus in either case, the equation we are trying to verify becomes

sin((720 degrees)/8)cos(dec) = cos(dec), which is an identity.
Q.E.D. for case 2.

Case 3.  dec is not 0 and min < 720.

Draw the great circle arc through the two points.  Since
min < 720, this arc does not pass through N, and we have a
legitimate spherical triangle.

Now draw the great circle arc from N to the midpoint of the
previous arc.  This divides the original spherical triangle
into two spherical triangles, which are identical except for
their position (since corresponding sides are equal), and this
implies that corresponding angles are equal too (for example, just
use Theorem 1h)).  This proves that the last-drawn arc bisects
the original angle at N, and the angles at the bottom where that
arc terminates are right angles (since they are equal angles that
add up to 180 degrees).

Now consider the left-most of these two right spherical triangles.
The side opposite the right angle (at the lower right of this
triangle) has length 90 degrees - dec, which is not equal to 90
degrees since dec is not 0.  Thus by Theorem 2c, our triangle has
only one right angle, so we can apply Napier's rules.

We use our memory trick to derive the rule we need, with the
middle part being the bottom side (which has length deg/2), and one
of the opposite parts being co (the left side), that is,
co (90 degrees - dec), and the other opposite part being co (the
angle at the top), that is, co ((min/4)/2).  Then we have

sin(deg/2) = cos(co (90 degrees - dec))cos(co min/8).  Thus

sin(deg/2) = sin(90 degrees - dec)sin(min/8).  Thus

sin(deg/2) = cos(dec)sin(min/8),

which is the equation we were trying to verify.  Q.E.D. for Fact 2.

To use an old Midwestern expression, that proof was rough as a cob,
but I think it was sort of pretty too.

"That's all, folks!"--Mel Blanc