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accepting any of the three steeper values is the assumption of a systematic selection process in the surveys. How far any of these extrapolations can be extended is problematical. For a portion of the present analysis it is necessary to extrapolate beyond the point of any reasonable confidence. However, until the Pioneer spacecraft have actually penetrated the asteroid belts, these extrapolations form the only source from which anticipated data rates can be inferred. If we assume an average asteroid velocity of 15 km/s relative to the Pioneer spacecraft, a cumulative asteroidal flux can be calculated for each of the
Figure 5.-Asteroid and cometary meteoroid flux models at 2.5 AU.
distribution models. These are presented in figure 5 for the four models, together with a flux model for the meteoroid environment at 2.5 AU. The meteoroid environment was taken from NASA SP-8013 (1969), corrected for the Earth's focusing effect and reduced by an inverse-square dependence on the heliocentric distance.
Estimates of particle reflectivity are also open to speculation. For the present purposes, three values have been chosen. As pointed out above, the reflectivity used is equal to 3/2 the geometric albedo. The three values used are a reasonable lower limit of 0.07, a meteoroid value of 0.20, and the mean for the asteroids Ceres, Pallas, Juno, and Vesta (Gehrels, 1970), which is 0.3. This last was converted from a geometric albedo of 0.2 (the value used by Dohnanyi, 1969).
ASTEROID EVENT RATES
The anticipated A/MD event rates during the Pioneer transit of the asteroid belts can be determined for the various power-law models and particle reflectivities as shown below.
The area surrounding the field of view of the A/MD can be approximated as a truncated cone:
for small 6 (the half angle of the telescopes). In the wideband mode, it was shown in figure 3 that the mean sensitivity of the A/MD instrument is approximately 1.75 visual stellar magnitudes. This can be translated through equation (1) in terms of a range to radius relationship applicable at a heliocentric distance of 2.5 AU:
Substituting this into equation (8), the effective detector area as a function of particle radius can then be expressed as
and converted to a particle flux per unit area with the assumption of an average particle velocity v:
The event rate E is then determined from the integral
From this, we can obtain an event rate for each set of assumptions. Because the A/MD is oriented at an angle of 45° from the spacecraft (S/C) spin axis, its field of view will describe a 49° right cone in space of wall width approximately 8°. (See fig. 6.) Any particle that takes more than approximately 12 s (the vehicle rotation period) to pass through the 8° instrument field of view must be seen by the system (i.e., it cannot escape from the
45° 44° cone described during the vehicle rotation without being detected). Thus, for large bodies moving at low relative angular velocity, the effective field of view becomes 98°. As an example, for a minimum asteroid size (radius) of 10 m and a reflectivity of 0.2, the effective detection range is 650 km in the wide bandwidth mode. Event rates can thus be calculated for large asteroids using the effective cone area measured in square meters.
where amin is now equal to 10 m. The computations have been carried out for the combinations of o and r to yield the event rates shown in tables II and III.
From tables II and III and from the nature of the A/MD, it is apparent that for power-law distributions steeper than those with a = 2, the event rate would be dominated by small particles. Although larger bodies would tend to predominate for power-law distributions with a = 2 or less, it is unlikely that the A/MD would detect any asteroid events. In point of fact, if a. is much less than 2.5, it is possible that passage of the Pioneer spacecraft through the asteroid belt would not even be noted. In figure 7, we have plotted the anticipated cometary meteoroid event rate as a function of heliocentric distance. The values from table II are shown in the asteroid region. Included also is a maximum value based upon a zodiacal light interpretation (Kessler, TABLE II.-Asteroidal Events Per Day, Wideband Mode
Ox r 1.75 2.0 2.5 3.0 0.07 . . . . . . . . . 2.1 x 10-6 1.0 x 10-5 1.7 x 10-2 140 0.2 . . . . . . . . . . 6.0 x 10-6 2.9 x 10-5 4.8 x 10-2 400 0.3 . . . . . . . . . . 9.0 x 10-6 4.4 x 10-5 7.3 x 10-2 600
Angular velocity > 0.685 deg/s.
TABLE III.-Large Asteroid (a > 10 m) Events Per Day, Wideband Mode
1968). From this figure, one can see how the cometary flux might predominate during asteroidal passage. For small particles, the velocity determinations of the A/MD should, however, enable us to distinguish cometary from asteroidal material if both are noted. Assuming a uniform distribution in the asteroid belt, the probability of the A/MD detecting a single large asteroid (a P 10 m) during the entire passage can be calculated from table III. The results are presented in table IV. Although the extrapolation is more reasonable in this limited size region, it is obvious from table IV that the distribution slope still dictates the probability of detecting a large asteroid. The apparent magnitudes of 1735 asteroids that the Pioneer F spacecraft might encounter have been computed (NASA PT-204, 1970). For the vehicle trajectory ephemerides, which were used in that study, the brightest apparent magnitude anticipated is approximately 6.5. As shown above, the mean threshold for the wideband of the A/MD is about 1.75. Consequently, it is unlikely that any of the asteroids considered would be detected. However, as stated in the study, the results are “subject to a very large uncertainty.” One way of increasing the probability of large-body detection would be to narrow the bandwidth of the A/MD. In the narrowband mode, the mean threshold would be approximately 2.75 visual stellar magnitudes. (See fig. 8.) In this mode, linear velocity determination for small particles would be extremely poor at best. However, for large-body detection beyond the effective triangulation range of the Pioneer A/MD version, those consequences are