The encounter of the probe with the target body is likely to be the most challenging phase of the mission. As discussed in Section 6 typical encounter velocities for candidate target NEOs range from 7.4 km/s to 30.6 km/s, with a mean of 15.5 km/s. Rather than design the probe to cope with this full range of encounter velocities, it was decided to assume a peak encounter velocity of 20 km/s. Of the 14 forthcoming predicted close encounters modelled earlier, 12 fall within this range. Not only are encounters at a higher velocity than this relatively rare, but a newly-discovered object moving at such a high speed is unlikely to be found with adequate warning time to mount even a rapid-response mission. Nonetheless, 20 km/s is still a very high-speed flyby to cope with; in more day-to-day terms it equates to 45,000 mph, or fast enough to fly the distance from London to New York in under 5 minutes.

This last comparison indicates why the encounter is so challenging. The probe is within a few thousand km of the target for a matter of minutes at most, whilst for a flyby at the sort of ranges usually associated with high-resolution remote sensing (a few hundred km) the probe is near the target for only a few seconds. During such a rapid close flyby the instantaneous field of view (IFOV) of the probe's sensors will need to slew rapidly to maintain lock on the target. Such slewing can be achieved either by rotating the entire spacecraft or by using some form of scan platform or periscope to track the target.

It is thus evident that there are two sets of options for the target encounter, both of which have a major impact on the mission design. Firstly, the probe can carry our either a distant or close flyby. This decision has direct implications for payload sizing and affects probe configuration and attitude control. Secondly, the probe can either slew itself to track the target or can use some form of scan mechanism. This decision has direct implications for payload sizing, probe configuration and attitude control.

Flyby Distance. The flyby distance (i.e. the minimum distance between the probe and the target) affects two main design drivers: the size of the payload and the sensor IFOV slew rate.

• Payload Size The larger the minimum distance at which the target can be imaged is, the larger the payload optics need to be to give sufficient spatial resolution to meet the imaging objectives. Two factors come into play: the limiting resolution of the payload is determined by its imaging aperture (although sensor design also has to be taken into account), and the long focal lengths needed to obtain high-resolution images at long range can lead to very 'slow' optical systems with long exposure times. This second point is particularly important during a target flyby, as it can lead to image blurring.
• Sensor IFOV Slew For a given encounter velocity, the slew and slew rate required to keep the target within the sensor IFOV will depend on the flyby distance. For a probe moving at relative velocity v towards a flyby at distance d_f the slew angle theta at a time t before encounter is given by

  (8-1)

  whilst the slew rate is given by

  (8-2)

  We can use this to make a rough distinction between a 'close' and distant' flyby. A close flyby is one in which the probe is within a factor of 3 of its flyby distance for a period in the order of seconds, whilst during a distant flyby a probe is within a factor of 3 of its flyby distance for at least a few minutes. Taking 30 s and 10 min as close and distant flyby encounter times, then for a 20 km/s encounter velocity a close flyby involves a flyby distance of 106 km and a distant flyby involves a flyby distance of 2120 km. Example plots of slew and slew rate for 20 km/s flybys at approximately these values (i.e.100 km and 1000 km) are shown at Fig 8.1a-d.

The variation of slew and slew rate with time affects the design of whichever method is used to provide the slew.

Slew Method The sensor IFOV can be slewed in a number of ways:

• Spacecraft Body Slew The sensor IFOV can be fixed with respect to the spacecraft body and the entire spacecraft can be slewed to track the target. This technique avoids the use of any form of scan mechanism and allows the payload size to be limited only by the size of the spacecraft body. It has the disadvantages of requiring the whole spacecraft to be rapidly but accurately repointed, placing great demands on the attitude control system. It also precludes normal communications during the encounter unless a fully articulated Earth-tracking antenna is used.
• Payload on Scan Platform An approach adopted on many space probes (e.g. Voyager and Galileo) is to mount the imaging instruments on an articulated 'scan platform' that can be driven to track the target. The effort required to scan the platform is much less than that needed to slew the entire spacecraft, whilst the use of such a platform allows the spacecraft to keep a fixed antenna pointed at Earth. However, the size and mass of the payload are limited by the platform's capacity and the use of a complex scan mechanism is required*.
• Fixed Payload with Scanned IFOV Rather than scan the entire payload a mirror/periscope arrangement can be used to scan its IFOV. This approach is often used on Earth-imaging satellites, e.g. SPOT and the scanning IR imager on METEOSAT. The payload size is constrained mainly by the spacecraft body, and although a scan mechanism is required it only has to drive a relatively low-mass mirror rather than the entire payload. Depending of the encounter geometry, a probe with a scanned IFOV may be able to keep a repointable antenna pointed at Earth without having to drive it during the flyby.

[*Indeed, Voyager 2 suffered a scan platform malfunction during its 1981 Saturn encounter.]

As described in Section 9, a payload with an imaging telescope of 0.2 m aperture and mass of approximately 20 kg was selected to give the required resolution of a few metres at distances of the order of a few hundred km. This size was a compromise between resolution and light-gathering power on the one hand and payload size and mass on the other. To achieve a pixel resolution of 2 m (i.e. 500 pixels across a 1 km diameter target) with this aperture implies a minimum imaging distance of at most 700 km. However, a flyby at exactly this distance would result in the probe attempting to take its highest resolution images at the moment of maximum slew rate. Furthermore, if the probe is approaching the target from Sunwards (as is desirable for early tracking) then at closest approach the target illumination could be 50% or less. It is therefore prudent to ensure that the probe reaches minimum acceptable imaging distance some time before closest encounter, so as to reduce the slew rate and improve target illumination. To do this the probe must fly considerably closer to the target than 700 km. By the criteria established above, the probe will thus need to carry out a close flyby mission.

How close should a close flyby be? The closer the flyby is to the target, the smaller the slew angle and slew rate are at the point of minimum acceptable imaging distance. However, whilst this eases tracking in one respect it makes it more difficult in another, as becomes apparent when the encounter geometry is considered. If the slewing element (probe body, scan platform or scan mirror) is aligned such that the target's velocity vector relative to the probe lies in its plane of rotation then target tracking can be achieved simply by slewing in this place, as shown in Fig 8.2. However, if this plane is not aligned perfectly with the target velocity vector then during the slew the sensor IFOV will deviate above or below the target track. Given the necessarily long focal length and narrow IFOV of the payload optics, this could easily lead to the target leaving the IFOV during the slew.

Close Flyby Tracking. Given that the probe can be equipped with star trackers that can accurately measure its orientation such misalignment is unlikely to be due to a pointing error. Instead it would be due to the inherent uncertainty in the state vectors (position and velocity) of the target and probe. As stated in Section 5, range and range rate measurements of the target can accurately be carried out using radar, whilst the same technique can be applied to the probe via a coherent transponder. As quoted earlier, Ostro [Gerhels94] describes how such tracking allows the trajectory of the centre of mass of a NEO to be located to within 100 m with 3 days of tracking data. Assuming that the tracking of both target and probe is an order of magnitude worse than this, i.e. accurate to 1 km (still no mean feat at 7.5 x 10⁷ km) then the mean error between target and probe positions will be larger than this by a factor of 1.4**. This means that the uncertainty in target position relative to the probe is slightly larger than the nominal diameter of the target. With an open-loop guidance system (sensor IFOV aimed at predicted target position) this means that during the final stages of imaging the sensor IFOV could drift off the target above or below the target track, even if it had been maintaining tracking along the target track (Fig 8.3). Some form of on-board guidance is thus needed to correct this error should it arise.

[**I.e. 2^0.5, as there are 2 independent errors. In fact the errors may well not be independent, especially if the same equipment is used for both measurements. In this case the mutual position uncertainty would be closer to 1 km.]

We can see how this might be achieved by considering the sensor's view of the target as it approaches the encounter. For a close flyby the target will initially be close to the projection of the probe's velocity vector, i.e. the point relative to the background stars towards which the probe is heading. As the probe nears the target it will start to drift away from this point at a rate determined by Eqn 8-2. If the probe is correctly aligned so that the target velocity vector lies in the sensor slew plane then the target image will drift through the sensor IFOV parallel to the slew plane (Fig 8.4a). If, however, the probe is aligned so that its slew plane is tilted relative to this then the target will drift through the sensor IFOV at an angle to the slew plane (Fig 8.4b). By tracking the motion of the target relative to the slew plane this angle can be measured and corrected by rolling the spacecraft*** by the opposite amount.

[***Following the convention that 'roll' is rotation around the velocity vector relative to the parent or target body.]

This form of guidance requires that the probe's camera can detect and track the target at sufficient range to be used for such navigation. To confirm that this was the case the brightness of an example target was calculated as a function of distance.

Brightness of astronomical objects is quoted as magnitude. Magnitude was originally an empirical scale where the brightest stars were of magnitude 1 and the faintest of magnitude 6. In modern times the magnitude scale has been rationalized to a logarithmic scale where a change of +5 magnitudes equates to a 100-fold decrease in brightness. Faint telescopic objects have magnitudes greater than 6 (large telescopes can image distant galaxies of Mag +25) whilst very bright objects can have negative magnitudes (e.g. the full Moon is Mag -12.7).

Five factors affect the brightness of an object such as an asteroid: size, albedo, distance to the object, distance from the object to the Sun and phase of the object. For the purposes of an initial calculation it is assumed that the asteroid is 1 AU from the Sun and that the phase of the object is 50%. The former assumption is valid for an encounter within 0.05 AU of the Earth as the asteroid will be within 0.93 and 1.07 AU from the Sun, so its illumination by the Sun will be within 15% of the notional 1 AU value. It is also reasonable to assume that phase will be about 50% as an encounter with an illumination less than this during approach would be unfavourable for imaging.

The easiest way to calculate the brightness of an asteroid is to do so relative to a body of known size, distance and albedo. The full Moon's brightness is Mag -12.7 for a diameter of 3476 km, a mean distance of 384,400 km and an albedo of 0.07 [Weigert76]. Given the diameters of the asteroid and the Moon (d_Moon, d_ast), their distances (r_Moon, r_asr), albedos (A_moon, A_ast) and the asteroid's phase p then its magnitude M_ast can be calculated in terms of the Moon's magnitude M_Moon as

(8-3)

The factor within the outer bracket is the asteroid's brightness, which is proportional to the square of the ratio of distances, the square of the ratio of diameters, the ratio of albedos and the phase of the asteroid. Taking logs to base 10 and multiplying by 2.512 converts brightness to magnitudes.

Fig 8.5 shows the brightness of a 1 km diameter asteroid of albedo 0.04 (typical for a dark C-type asteroid [Kowal96]) for ranges from 10 to 1 million km. Even at a range of 200,000 km - some 3 hours before closest encounter at a closing speed of 20 km/s - the asteroid is at Mag +5. Although faint by human standards, this is easily within the capacity of a CCD camera / telescope combination, especially as long exposures can be used at this point as there is no need to avoid motion blur. It thus appears that even a small and dark asteroid can be tracked far enough in advance to allow the probe to set up its orientation properly for slewing to track it.

As for the tracking slew itself, there are a number of ways it could be carried out. For a probe equipped with a scan platform or periscope, the slew is relatively simple and relies only on the accuracy with which the platform or periscope mirror can be driven. However, for a probe with fixed payload that tracks the target by rotating the entire body then achieving an accurate slew is a much more difficult task. The probe could slew at a smoothly-varying rate to match the slew rate of the target as shown on Fig 8.1. However, this requires continuous fine application of torque, and while this may be possible with a yaw reaction wheel it would be difficult with an attitude control system using thrusters alone. Rather than commit the probe to the use of a reaction wheel it was decided to stay on the side of probe simplicity and work towards an attitude control strategy that could be carried out using thrusters alone, i.e. via discrete torque impulses.

An extreme application of this would be to step the probe through a series of slew angles to track the target. However, given the narrow sensor IFOV required to image a small target this would need a very large number of steps and would become impractical at close approach. Instead an intermediate approach was adopted of using a series of successive slews to track the target. The aim of this approach is to approximate the slew angle curve by a series of straight-line segments (i.e. constant slew rate) with each segment constrained not to deviate from the actual slew angle by more than half the sensor IFOV.

To verify that a probe with body-pointed payload was practical a successive-slew flyby was modelled for a flyby with a closest approach of 100 km and a sensor IFOV of 3 milliradians (about 0.2°). This IFOV was arrived at after a series of design tradeoffs between slew tracking performance and imaging payload design (see Section 9 for details of the latter). It views an area 2 km across, twice the size of a small NEO target, at a distance of 665km, allowing high-resolution imaging of the entire target before the point of maximum slew rate. The imaging sequence was assumed to start 10,000 km before closest encounter. The resulting series of slews is shown in Fig 8.6a-g.

As is seen in Fig 8.6a, the first slew is actually not a slew at all; the sensor IFOV is aimed so that the target drifts from one side of the IFOV to the other. When it does so a slew is set up (Fig 8.6b) such that the target seems to drift back to the other side of the IFOV and then across again. The appropriate slew rate was found by using a root-finding algorithm to obtain a slew that deviated no more than IFOV/2 from the actual asteroid slew angle. This process was repeated with five more slews (Fig 8.6 c-g).

As the probe closes on the asteroid both the slew rate and its rate of change increase so that the seventh slew covers only the portion of the approach from 660km to 500 km before closest approach. At typical encounter velocities this would take only a few seconds so closer slews were not modelled.

The successive-slew approach thus allows the target to be tracked to within the limits of a very narrow sensor IFOV whilst using only discrete torque impulses as could be provided by a thruster-based attitude control system.