In order to estimate the likely nature of an encounter between the probe and a NEO a number of close encounters with known NEOs were modelled to provide a set of sample missions. A list of all currently-predicted NEO close encounters for the next several decades was obtained from the IAU Minor Planet Center web site; from this every encounter within 0.07 AU (slightly larger than the planned maximum encounter range) for the next 10 years was extracted (see Table 5.1). Each encounter was then modelled in detail in order to determine its geometry.

Encounter Modelling. For each encounter it was assumed that the probe would intercept the target at the point of closest approach whilst travelling radially outwards from Earth at a notional velocity of 5 km/s (see Section 7). To locate the intercept point and determine the encounter relative velocity and angle the asteroid's position and velocity vectors at the time of closest approach were then needed. This was found by making use of the Jet Propulsion Laboratory's 'Horizons' on-line ephemeris generation system. 'Horizons' allows users to generate an ephemeris covering a large number of possible parameters for any body or pair of bodies in the solar system. For example, it can be used to generate a list of position and velocity vectors of an asteroid relative to Earth on an hourly basis around the time of predicted closest encounter. The output can be referred to one of several co-ordinate systems (ecliptic plane is most useful for this calculation) and presented in various sets of units. An example output run from 'Horizons' is shown at Appendix 1 to this section.

Using the range output the time of closest approach was confirmed and the position and velocity vector for that time noted. The position vector relative to Earth then gave the direction (and hence velocity vector, normalized to 5 km/s) of the probe. With position and velocity vectors for both asteroid and probe the circumstances of the encounter could be modelled.

Fig 6.1 shows the encounter geometry, projected onto the plane of the ecliptic. The vector difference between the asteroid and probe velocity vectors gives the relative encounter speed. Also of interest though are the two angles illustrated in more detail in Fig 6.2.

Figure 6.1. Asteroid Encounter Geometry

Figure 6.2. Asteroid Encounter Geometry - Detail

The Probe/Sun Angle is the angle between the probe flight path and the Sun. This is important for probe configurations where the high-gain antenna is fixed, as it determines the Sun's aspect relative to the probe body assuming that the probe cruises with its antenna in Earth-pointing mode. The Asteroid/Sun Angle (also known as the Phase Angle indicates the illumination of the asteroid as the probe approaches it. A low value indicates that the asteroid approaches almost fully illuminated, 90° that it is half-full during approach and values approaching 180° that the asteroid is coming out of the Sun relative to the probe. Although the probe flightpath can usually be adjusted to ensure that the encounter is over the Sunward side of the asteroid a high Asteroid/Sun angle should be avoided as it makes acquisition of the target for tracking purposes difficult. Once the asteroid's position and velocity vectors are known then basic vector algebra can be used to find these angles via the probe's velocity vector and the Sun's direction vector from the Earth (calculated as a function of date).

The flyby parameters for the encounters listed at Table 5.1 are given at Table 6.1.

From Table 6.1 it is apparent that not all encounters are suitable for flyby missions. The encounters with 1994 PM and 1996 GT both feature very high relative velocities, whilst six encounters (1996 EG, 1996 FG₃, Mithra, Nereus, 1991VK and Hathor) all involve poor asteroid/Sun angles. Note that the encounter with Nereus, whilst very favourable in relative velocity terms, involves the target being poorly illuminated. The encounters with 1992 SK, Golevka, Minos, Toutatis, 1992 BF and 1992 UY₄ are, however, all favourable. Thus 6 out of 14 encounters over the next 10 years are potentially suitable for probe flyby missions. Assuming that this is a representative sample, some 40% of all close NEO encounters should be suitable candidates for such flyby missions.

Fine-Tuning Encounters. The above encounters were all modelled by assuming that the probe was sent to intercept the asteroid at its point of closest approach to Earth. If that approach is significantly closer than the 0.05 AU baseline range of the probe then there is scope for bringing forward or delaying the flyby to make its parameters more favourable. Delaying the flyby in particular ensures that the probe is 'chasing' the asteroid and so reduces the encounter velocity, although the need to reach a more distant intercept point may mean that the probe actually has to be launched earlier than for a minimum-distance encounter, despite the later flyby date. Nonetheless such a move can have beneficial effects; for example, calculations show that intercepting Toutatis at a range of 0.05 AU on 7 Oct 2004, rather than at 0.01 AU on 29 Sep 2004, has the following effects:

• Encounter velocity reduced from 12.04 km/s to 6.26 km/s.
• Asteroid/Sun angle reduced from 62.2° to 55.8°.

In the event of a newly-discovered NEO being a suitable mission target, trajectory analysis will thus be required to identify the optimum launch and encounter dates, subject of course to constraints on the former.

Appendix 1

Example output from JPL 'Horizons' Ephemeris Generator

(data truncated to fit page)