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ESTIMATE OF PARTICLE DENSITIES AND COLLISION
DOWALD. J. KESSLER
The present lack of exact information about the distributions of asteroids and asteroidal meteoroids causes the largest uncertainty in the description of the interplanetary meteoroid environment between the orbits of Mars and Jupiter. Observed asteroids are inferred to have diameters from a few kilometers to a few hundred kilometers. The presence of these larger bodies suggests the presence of smaller, unobservable bodies. When asteroids collide, fragments are produced that eventually collide with other fragments. Because of this continuous collision process, much smaller asteroids most probably exist. Such asteroidal meteoroids, if present in sufficient number, could pose considerable danger to spacecraft.
ASTEROID MASS DISTRIBUTION
Various methods have been used to predict the number of smaller bodies in the asteroid belt. These methods include estimates of the mass distribution that are inferred from lunar and Martian crater distributions (Baldwin, 1964; Hartmann, 1965, 1968; Marcus, 1966, 1968), meteorite finds (Brown, 1960; Hawkins, 1964), and theoretical and experimental studies of rock crushing laws (Dohnanyi, 1969; Hawkins, 1960; Piotrowski, 1953). The number of smaller asteroids can also be estimated by the trend set by the larger asteroids. Most analyses indicate that the number of asteroids of mass m and larger (the cumulative mass distribution) varies as mo, where o is a constant. In extrapolating from the distributions for larger asteroids, a greater value for a indicates a larger number of smaller asteroids.
Several difficulties arise from inferring asteroid influx rates from lunar and Martian crater counts:
(1) The ages of the impacted surfaces are unknown.
*The full text of this paper appears in NASA SP-8038, Oct. 1970.
(3) The surface features of craters are eroded (by smaller meteoroids on the Moon or by atmospheric wind on Mars), which causes the smaller craters to disappear more rapidly than the larger CraterS.
(4) A surface can become saturated so that larger craters will obliterate a significant number of smaller craters. When this occurs, the number of impacting particles cannot be determined, and the original distribution is difficult to determine. Saturation appears to have taken place on the surface of Mars and in the lunar highlands (Marcus, 1966, 1968).
If meteorites are assumed to be of asteroidal origin, problems still exist in relating a meteorite mass to its original mass, because of ablation and fragmentation. Hawkins (1964) deduced the mass distribution of stony and iron meteorites and predicted their cumulative mass distribution in space to vary as m-' and m-97, respectively. Brown (1960), on the other hand, found both stony and iron meteorites to vary as m+9.77. Several investigators have attempted to predict the number of smaller asteroids by theoretical and experimental studies of the effects of collisions between rocks. Piotrowski (1953) found that under certain restrictive conditions, erosion and breakup of asteroids would lead to a cumulative mass distribution that varies as m+2/9. Hawkins (1960), however, pointed out that as terrestrial rocks are crushed, the value of o increases and approaches - 1. Dohnanyi (1969) used experimental results of hypervelocity impacts to determine a rock crushing law for the asteroids and their debris. He found that a steady-state solution exists if the cumulative mass distribution varies as m-9.8%. Dohnanyi also pointed out that his results are consistent with the observed asteroids of Kuiper et al. (1958). Kessler (1969) used the individual orbits and absolute magnitudes of asteroids given in the 1967 Ephemeris volume to obtain the spatial density (number density) of asteroids as a function of absolute magnitude at various positions in space (every 0.1 AU between 1.0 and 4.5 AU, and every 45° of heliocentric longitude). The results for 1.5, 2.0, and 2.5 AU are shown in figures 1, 2, and 3, respectively. The upper line in each figure partially corrects for selection effects. Except for a flattening of the curve, which occurs around absolute magnitude 11 (particularly noticeable in fig. 2), the mass distributions given by Kessler (1969) are consistent with Dohnanyi's results. An upper limit to the spatial density of asteroids is given by Kessler (1968) by the requirement that the intensity of the counterglow is not exceeded by the total light reflected by the asteroids. The cumulative asteroid spatial density S at 2.5 AU (Kessler, 1969) is shown in figure 4, along with the various models that have been discussed. Spherical particles with a mass density of 3.5 g/cm3 and a geometric albedo of 0.1 were used. For a spatial density greater than 10-13 or 10-16 particles/m3, the probability of encounter must be considered for large spacecraft (i.e., a
Figure 1.-Average asteroid spatial density in the ecliptic plane at 1.5 AU.
Absolute magnitude, Mo
spacecraft with 500 m” of surface area exposed to the environment for 1 yr). Therefore, if the asteroid spatial density varies as mT' (as suggested by Hawkins, 1960, and shown in fig. 4), such a spacecraft would require protection against impacts by masses as large as 10° g. However, this mass distribution exceeds the upper limit given by Kessler (1968) for masses less than 10° g. If the asteroid mass distribution varies as m-2/3 (Piotrowski, 1953), the spacecraft would have to be protected against asteroidal meteoroids of only 10-8 g (compared with cometary meteoroids of 10-3 or 10-2 g). The asteroid mass distribution suggested by Dohnanyi (1969) (S - m-9.8%) gives an intermediate result that is consistent with the upper limit for asteroid
Figure 3.-Average asteroid spatial density in the ecliptic plane at 2.5 AU.
masses greater than 107*. When Dohnanyi's mass distribution is extrapolated from the larger asteroids of Kessler (1969) at R = 1.0 AU, the resulting flux comes within a few percent of the flux found by Whipple (1967) from the Apollo asteroids, and the flux of meteoroids estimated from meteorite finds by Hawkins (1964) and Brown (1960). Such a mass law also gives results consistent with Öpik's prediction (Whipple, 1967) that the ratio of the number of Mars-crossing to Earth-crossing asteroids (of the same size) should be 300.
Thus, this asteroid mass distribution will be adopted, and the assumption will be made that the mass distribution is independent of the distance from the Sun. The spatial density of asteroids with an absolute magnitude Mo less than 10 can then be expressed as
where S, is the number of asteroids per cubic meter of absolute magnitude Mo and brighter, and CR is a constant for each distance R from the Sun.
The irregularity in the size distribution between asteroids with absolute magnitudes of 10 and 12 suggests that for Mo X 12, S, in equation (1) should be reduced by a factor of 7.6, or
The value of CR is then evaluated for each distance from the Sun by fitting equation (1) or (2) to the larger asteroids given by Kessler (1969); selection