The mass distributions of asteroids and comets are usefully described by power laws of form where N(> m) is the cumulative number of objects with mass greater than m, q is the exponent of the differential mass distribution dN/dm, and C is a constant related to mi, the largest object in the distribution. The differential form is useful for integations. The most important properties of the power law are that it is scale-free and that, for q < 2, most of the mass is in the largest objects. The latter ensures that the cumulative effects of impacts are always dominated by small-number statistics at all scales. The probable origin of the power law is a self-similar fragmentation cascade.
Reported values of q range from about 1.3 to 2.2, with the extreme values associated with comets. According to Safronov et al. (1986), q « 1.6 in a swarm of planetesimals dominated by coalescence; fragmentation increases q to ~1.8. Dohnanyi (1972) calculated that in a collisionally evolved distribution q ^ 11/6. Essentially the same result has also been obtained by Williams and Wetherill (1994) and by Tanaka et al. (1996). Asteroids with diameters between 100 and 250 km have q « 2 (Hughes, 1982; Donnison and Sugden, 1984). The best sample of a modern crater production population, that of Venus, is well matched by q = 1.8 (Zahnle, 1992; McKinnon et al., 1997).
Comets do not appear to obey a single power law. Familiar comets (110 km) seem to be described by q < 1.7 (Donnison, 1986; Hughes, 1988; Zahnle et al., 2003), but bigger and smaller comets are rarer than this. Weissman
(1990) suggests that q « 2.2 for long period comets bigger than 10 km. The Kuiper Belt, the probable source of the short period comets, seems to be telling the same story: Kuiper Belt Objects bigger than 50 km across apparently obey a q « 2.1 power law distribution (Gladman et al., 2001; Trujillo et al., 2001). In the other direction, Zahnle et al. (2003) argue from the discovery history of comets and from the paucity of small impact craters on the Galilean satellites that q « 1.3 for comets smaller than 0.5 km.
Lunar craters and basins of the late bombardment appear to demand low values of q. The consensus size distribution of lunar highland craters is N(> D) x D±b, with b = 1.8, fit to craters between 10 and 80 km diameters (Wilhelms, 1987, p. 257). Because these craters date from the end of the late heavy bombardment they are directly relevant. Crater-diameter scaling relations like Eq. (7.2) or (7.3) are then inverted to obtain q = 1 + 0.26b « 1.5 (Maher and Stevenson, 1988, Melosh and Vickery, 1989, Chyba, 1991). Chyba
(1991) includes slumping by using Eq. (7.3) to get q = 1.54.
The mass distributions of asteroids and comets are usefully described by power laws of form
The late large lunar basins are consistent with the crater distribution. Wilhelms lists 14 lunar basins from Nectaris to the present. (Lunar "basins" are defined as craters bigger than 300 km diameter.) The largest of these is Imbrium, for which we have estimated a nominal energy release of 2 x 1033 ergs (see above). The smallest, call it Schrodinger, is essentially a big crater; if treated as a crater its impact energy would be x 1031 ergs. For similar impact velocities, Eq. (7.5) implies that
This q is consistent with the record for smaller craters and consistent with a cometary source. The same argument applied to the 45+ basins postdating S. Pole Aitken gives the same result
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