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because they involve two additional angles for each cone necessary to specify its orientation. This misalined case has been reduced from the 15 original equations to 3 equations in 3 unknowns. Because of their complexity, further reduction appears impractical. Numerical solutions are obtained by computer iteration. Thus, independent of the amplitude of the signals detected by the individual optical systems, one can establish the three velocity components and the range of the body. Using this calculated range, the measured light intensity at the detector, and the known solar intensity, one can solve equation (1) for the product of the reflectivity and the cross-sectional area, and thus determine the mean radius of the body to an uncertainty of the square root of the reflectivity. Further, from the real time at which the event took place; the known position, velocity, and orientation of the vehicle from which the measurement was made; and the three velocity components of the body, the complete orbit of the body in the solar system can be determined. The A/MD, as designed for the Pioneer F and G missions, uses four 20 cm Cassegrainian telescopes, each coupled to a photomultiplier tube (S20 photocathode) as sensor. The four optics are arranged in a square array 22 cm on a side. The use of four telescopes yields an inherent redundancy because any three sensors yield all of the required data. If a particle exceeds the threshold in all four fields of view, one obtains four sets of solutions to equation set (2). An artist's sketch of the A/MD instrument on the Pioneer spacecraft is shown in figure 2. The instrument hardware (excluding spacecraft mounting panel) has a mass of 2.4 kg and an average power requirement of 2.0 W. The A/MD is a background-limited detector. The noise inherent in such a
detector is given by in - W2 ibqf (3)
where it is the total background current, q is the unit electrical charge, and f is the bandwidth of the circuitry
Because the Pioneer is a rotating spacecraft, the sky background viewed by the telescopes is continuously varying. The threshold in each telescope is designed to “follow” the background and high-frequency noise. Because the telescopes remain approximately alined, the background and, consequently, the thresholds of all four telescopes should remain approximately equal. The background is averaged through a comparatively long time constant circuit that has the effect of introducing a delay in the background response. The relative threshold for each telescope at any instant t is designed to be self-setting at a value of
The operating modes of the instrument with the values for K 1, K2, and the time constant At for background averaging are given in table I. A computer program that calculates the change in the relative threshold over a complete spacecraft rotation as a function of the orientation of the spacecraft spin axis has been written (Soberman and Neste, 1971). The program has a sky brightness map (Roach and Megill, 1961) built in. A typical output for the wideband instrument mode is shown in figure 3. Because the instrument is “triggered” by noise when the intensity exceeds the threshold value, a technique for reducing the number of false events must TABLE I.–Modes of Pioneer F and G Sisyphus Instrument
Operating Bandwidth f, K1 K2 Time mode kHz constant, S Wideband 500 2.0 | 1.1 0.047 Mediumband 160 2.0 | 1.1 .047 Narrowband 13 2.0 | 1.1 .047 High threshold (a) 2.0 | 1.2 .047 Calibration 13 2.0 | 1.1 to 1.2 .5
*For all three bandwidths.
be employed. This is accomplished by a simple threefold coincidence requirement that must be fulfilled before an event is considered legitimate and is recorded. This means that the transit time of a particle through the overlap region of three of the four fields of view must be greater than or equal to a predetermined minimum (3.2 pis). This criterion limits the number of recorded false events to approximately one per month for the wideband mode. In addition, the thresholds are designed to have hysteresis to reduce the probability of rejecting a legitimate signal. For example, if a target is passing through the field of view and noise or fluctuations in target intensity cause the level to drop below the threshold before coincidence is confirmed, the signal would be regarded as an error and rejected. However, if the threshold is reduced to a lower level after being exceeded, the probability of dropouts due
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Figure 3.—Computer-generated cyclic threshold variation for Pioneer F and GA/MD in the wide band mode. Galactic longitude = 333°; galactic latitude = 0°; bandwidth = 500 kHz. x indicates threshold; o indicates noise.
to noise is reduced. The rate at which the threshold is reduced and the level to which it is reduced do not affect noise rejection nor prevent a legitimate end of signal determination.
Because the threshold follows the background, a distant large asteroid moving with low relative angular velocity will, in several time constants of the averaging circuit, be considered as part of the background. When this happens, a false termination of the signal will occur. Another problem relating to the measurement of relatively slow-moving asteroids is the star exclusion circuit of the A/MD. To limit the amount of data telemetered, any “event” that recurs on successive vehicle rotations is ignored after the first measurement. Thus, if such a relatively slow-moving asteroid is detected, it can only be measured once in a given region of the sky. Operation in the “star exclusion disabled” mode, which would allow multiple measurements of the passage of slow-moving asteroids through the complete field of view, is incompatible with the present Pioneer A/MD telemetry assignment. Modifications of that assignment were considered too costly in view of the low probability of such measurements. (See below.)
ASTEROID BELT MODELS
Current models of the particle number density within the asteroid belt are usually expressed by a relationship of the form
where a is the particle radius and C and o. are constants. The major uncertainty, and point of controversy, between the various models lies in determining the value of os. Arguments for the various values of o are usually based on theoretical studies regarding the collision and subsequent fragmentation of particles within the belts. On the basis of such a grinding mechanism, Piotrowski (1953) argues that particles near a = 1 cm should follow an a = 3 law. However, Anders (1965) does not believe that the fragmentation history of the asteroids has progressed as far as does Piotrowski and favors a value of a more nearly equal to 2.
A more recent model of the asteroid distribution is that published by Dohnanyi (1969) in which he considers the evolution of a system of particles undergoing inelastic collisions and fragmentation.” He derives a theoretical density function for asteroids given by
where m is the particle mass. After performing the integration and expressing the number density in the form of equation (5), we obtain
where a is the particle radius in meters. Following Dohnanyi, a particle density of 3.5 × 103 kg/m3 was assumed in the conversion from mass to radius. Figure 4 presents some recent data on the cumulative asteroid distribution. The histograms are for the McDonald survey (Kuiper et al., 1958) and the more recent Palomar-Leiden survey (PLS) (van Houten et al., 1970). Distributions are shown for power-law exponents of -3 (Piotrowski), -2.5 (Dohnanyi), -2 (Anders), and the literal interpretation of the PLS of -1.75. Implicit in