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Data Reduction for Differential Photometry
These are notes on how to conduct differential photometry using the software available for students of Jeff Adkins at Deer Valley High School in Antioch, California. This procedure was painstakingly arrived at through the efforts of several students, most notably Tri Nguyen, Kyle Hornbeck, and Robert Johnson.
NOTE: Many software packages for measuring the magnitudes of stars in images exist, and many of them are more automatic or simple to use than the procedure described here. Nevertheless, I want my students to follow the procedure described here for two reasons. First, using the method described here gives definitive background on where the data comes from. There's nothing mysterious about how the computer generates the numbers. It could be done by hand if necessary. Second, this method generates standard error of the mean for the magnitudes determined, useful for generating error bars on the data points.
What you need:
A photograph of a variable star, AGN, or nova to be measured. The photo should be "calibrated" before beginning, so darks, flats, and bias frames should have already been applied to the photo prior to analysis described here.
A finder chart with standard stars. The standard stars must have been measured with the same filters as the one used to make the photograph. That might be the L filter from a LRGB, for example, or a V vilter from a UVBRI set.
Software: We usually use "Image Processing" from the HOU project because it will count brightness counts reliably, and Fathom for the curve fit. Graphical Analysis from Vernier Software or Excel can be used for the curve fit as well, but it is less intuitive.
Example:
As an example we are going to measure the magnitude of 4C 29.45, a blazar which is the centerpiece of our school's Spitzer space telescope project. 1. Obtain a finder chart. Finder charts for this object are located here.
Finder chart for 4C 29.45: go to http://gtn.sonoma.edu/participants/catalog/ and click on GTN 7.
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Here is the finder chart. Note the items with numbers. These are standard stars. On the same reference you will find the calibrated filtered magnitudes for the standard stars.
2. When you open the image with HOU's Image Processing software, the file will not open by double-clicking, or even by trying to use "open" fromt he file menu. It won't even show up in the list. In the "open files" dialog, you must select "All files" in the drop down that normally reads "Fits." Then the picture you are trying to use will appear on the list to open. When you try to open it, an unrecognized file type warning will appear. Ignore it and clear it, then the picture will open and be available for analysis.
Look at the photo you intend to analyze and try to identify the standard stars and the target. Here is a sample image with the target object indicated with an arrow.
3. Selecting HOU's image processing tool for measuring brightness counts (it looks like a little bull's eye) click on each target and write the numbers down that are shown.
Here is an example of the brightness counts for standard stars and the target. This image was filtered with an "L" filter, but we're going to pretend it's a "V" filter for the sake of this analysis. The software does an analysis of the brightness of the star compared to the nearby background, computing the full-width half-max diameter of the star, adds up all the brightness counts in that circle, then draws an annulus around the star and measures the background counts within in. The background is automatically subtracted from the star and the number is displayed.
4. Compare this to the finder chart above. From the finder chart reference page, make a table of magnitudes and brightness counts for the standard stars and the target.
Object V mag Brightness Count Standard 1 13.39 216120 Standard 13 15.36 35830 Standard 14 15.89 22758 Standard 15 16.6 11837 Target 4c29.45 unknown 9714 From this data you can see the target is a little dimmer than the dimmest standard--not the best position to be in, since you are not interpolating. However, it's not too much dimmer than the dimmest standard (15) so perhaps the measurements will hold up.
5. Compute the log (base 10) of the brightness counts. This is because magntitudes are logarithmic, and if we're going to compare magntiudes which are logarithmic to brightness counts which are linear, we have to "linearize" our data by making the brightness counts logarithmic as well.
Object V mag Brightness Count log(Brightness Count) Standard 1 13.39 216120 5.3347 Standard 13 15.36 35830 4.5542 Standard 14 15.89 22758 4.3571 Standard 15 16.6 11837 4.0732 Target 4c29.45 unknown 9714 3.9873 6. Open Fathom, start a new collection and create a table, and enter this data. Leave out the target data.
7. Start a new graph, and make logbright x and V -mag y.
8. Drag down a Model from the button area. Choose Simple Regression from the drop down menu. Then drag logbright to the predictor attribute, and v-mag to the response attribute. Click on the graph once, and from the graph menu choose "Least squares line."
9. In the model window, there will be a sentence that reads, "When logbright = 0, the predicted value of v is..."
Click on the 0, and enter the log of the brightness count of the target (in this example, 3.9873.)
The model will respond with something like this:
"When logbright = 3.9873, the predicted value for a future observation of v is 16.8194+/-0.0782161."
This is the official data point from this single image, and you have now "reduced" an image to a single value with an error estimate. The error is a 2-sigma standard error of measurement with a 95% confidence level. In this particular example the correlation coefficient of the regression line is better than 0.999, which is very very good.
When constructing a light curve, this represents ONE data point. The +/- values determine the placement of error bars above and below the data point.
Write this value down, record the image it came from, noting the date and time of the observation, save the file and move on to the next picture.
Note the slope of this line should be -2.51 for theoretical reasons. If the predicted slope from this model (including its estimated error) does not include 2.51, then there is probably a problem with the calibration of the image and you will have to discard the measurement if you cannot fix it. In the example shown, the predicted slope of the best fit line is 2.54737+/-0.065, which includes 2.51 in its range, so this is probably acceptable.
10. If you save the fathom file as a template, all you have to do is enter the new brightness counts and the log of the brightness count of the target for each new iteration.
Jeff Adkins, Director
astronomyteacher@mac.com
Cheryl Domenichelli, Assistant Director
cheryldomenichelli@antioch.k12.ca.us
4700 Lone Tree Way
Antioch, CA 94531
The ESPACE Academy is sponsored in part by a grant from the California Department of Education's Specialized Secondary Program.
