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effects are minimal at the larger sizes. At R = 2.5 AU, C2.5 =-37.18. If a radial distribution is defined such that
f(R)= CR - C2.5 (3) the spatial density in the ecliptic plane for the smaller asteroids becomes log S. = 0.504Mo-38.06+f(R) (4)
where Mo X 12. The function f(R) was evaluated from equation (3) and is given in figure 5.
If the adopted values for asteroid mass density (3.5 g/cm3) and geometric albedo (0.1) are used,
where 10-9 s m s 101°, and S, is the number of asteroids per cubic meter near the ecliptic plane of mass m (in grams) or larger at distance R in astronomical units from the Sun. Equation (5) is limited to masses larger than 10-9 g for two reasons: (1) If the mass distribution is extrapolated to masses smaller than 10-9, the upper limit given by Kessler (1968) would be exceeded; and (2) solar radiation pressure and the Poynting-Robertson effect would
Figure 6.—Asteroid mass distribution compared to the cometary mass distribution near Earth.
probably cause the mass distribution between 10-1” and 10-6 g to change in a manner similar to the change in cometary meteoroids in the same mass range. Such a change can be approximated by limiting the distribution to asteroids of 107°g or larger. Use of a curved function similar to the function for cometary meteoroids would imply more knowledge of the asteroid mass distribution than is available. Equation (5) is compared with cometary meteoroids at 1 AU in figure 6.
LATITUDE AND LONGITUDE DEPENDENCE
The asteroid spatial density is a function of both heliocentric latitude and longitude. At a latitude of 16° from the ecliptic plane, the number of asteroids is reduced by about an order of magnitude from that given by equation (5). Between 1.5 and 2.4 AU, the spatial density is longitudinally dependent and reaches a maximum dependency at R=1.8 AU. At R=1.8 AU, the spatial density increases or decreases by one-half an order of magnitude in the direction of Jupiter's longitude of perihelion or the opposite direction, respectively. A complete discussion is contained in NASA SP-8038 (1970).
The asteroids given in the 1967 Ephemeris volume were used to compute the average velocity of asteroids relative to a spacecraft. The asteroid data were corrected for observational selection effects (Kuiper et al., 1958), and weighted according to the probability of collision (Wetherill, 1967) with the spacecraft, and the velocity of each asteroid relative to the spacecraft was computed. Velocity distributions are obtained at distance R from the Sun for a spacecraft whose velocity vector makes an angle 6 with a circular orbit in the same plane. The speed of the spacecraft is o, in units of the speed necessary to maintain a circular orbit of radius R around the Sun. The average asteroid velocity V, is then found from each distribution. The velocity parameter U, is introduced and is less dependent on distance from the Sun than is Va. The values of U, shown in figure 7 were computed for R = 2.5 AU, but can be used with fair accuracy for all distances by applying the relationship
where R is in astronomical units, V, is in meters per second, and U, is given in figure 7. A more detailed discussion, which introduces a slight R dependence of U, is contained in NASA SP-8038 (1970).
FLUX AND NUMBER OF IMPACTS
Flux on a randomly tumbling surface is related to spatial density and relative velocity by
where F, is the number of asteroid impacts per square meter per second. The total number of impacts on a spacecraft is given by
Figure 7.-Average relative velocity of asteroid particles at R = 2.5 AU where o is the ratio of the heliocentric spacecraft speed to the speed of a spacecraft in a circular orbit at the same distance from the Sun.
where A is the surface area of the spacecraft in square meters, t is the time of spacecraft exposure to the environment, and the limits of integration are the beginning and end times of the mission.
The model that describes the asteroid size distribution is anchored at the large-mass end by the observed asteroids and is assumed to vary as moo. The assumption of the mo" dependence and the uncertainty in the value of a probably constitute the largest uncertainties in the asteroidal-meteoroidenvironment model. An upper limit for the number of smaller asteroids is established by not allowing the number to be so great as to produce more reflected sunlight than is observed in the counterglow (Kessler, 1968). The value of a that was chosen for the model was not arbitrary, but (as was pointed out previously) was a value that agreed with the steady-state mass distribution based on collisions within the asteroid belt. The choice of a value for o is also consistent with a limitation in particle mass to 107* g and larger, again based on counterglow observations. A larger value for of than that selected for the model would lead to a particle cutoff at larger masses, to remain consistent with counterglow observations. Because no method exists for establishing the proper cutoff, the upper limit of the flux of particles with greater than a given mass is just the value that would not produce reflections exceeding that of the counterglow. The ratio of this upper limit to the model used (eq. (5)) is 26m.9.17 (m = 10−9 g). Thus, if the asteroid mass is limited to particles of 10 * g or larger, the flux predicted from the model is essentially at the upper limit, whereas if the asteroid mass is limited to particles 1 g or larger, the limit of uncertainty permits the actual flux to be higher than the flux predicted by the model by a factor of 26.
Additional uncertainty is introduced through the geometric albedo used in relating a particle distribution to the counterglow. Theoretically, geometric albedo may range from zero to infinity; however, the extremes of this range are never observed. A geometric albedo of 0.1 is used for estimating the mass of the visual asteroids in the asteroid-environment model. If the true albedo, however, is as low as 0.05, the ratio of the upper limit of the asteroid flux predicted by the model becomes 52mo, 17.
The lower limit of the meteoroid population in the asteroid belt is set by the cometary environment. If the cometary meteoroid size distribution shown in figure 6 is assumed to vary as R-3 (which is an extreme case), the ratio of the asteroid-model environment to the cometary environment in the heart of the asteroid belt is 4.7 x 103 mO-37 (m= 10−6). This expression implies that the meteoroid flux in the asteroid belt could range from a factor of almost 30 lower than predicted by the asteroid model at m = 10-6 g to a factor of nearly 5000 lower for m = 1 g.
IMPACTS ON A SPACECRAFT
If a typical trajectory through the asteroid belt (e.g., a mission to Jupiter) is integrated by using equations (8), (7), (6), and (5) and figures 5 and 7, the total number of asteroid impacts per square meter of spacecraft surface area is given by
The average asteroid velocity relative to the spacecraft would be approximately 15 km/s. As can be seen from equation (9), the probability is small that a spacecraft with a random trajectory through the asteroid belt will collide with any of the large observed asteroids. In fact, only one asteroid of 10°g or larger could be expected to pass within 107 km of the spacecraft. The real danger to spacecraft is from the much smaller asteroidal meteoroids. Equation (9) predicts an average of one asteroidal meteoroid impact of 10-6 g or larger for every square meter of spacecraft surface area.