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pe = 5.5 g/cm3 as for Earth or Mercury in calculating N(x). For the terrestrial band, Mtotal = 2.0M, (total mass of present terrestrial planets and the asteroids) and Mmax = 10Ms. With these values for the masses and V=0.02 km/s, N(x)=3508 for p, = 5.5 g/cm3 and 3269 for pp = 3.6 g/cm3; whereas for V = 0.04 km/s, N(x) = 1752 for pp = 5.5 g/cm3 and 1633 for pe = 3.6 g/cm3.
Table I shows the normalized planetoid mass distribution functions for pe = 3.6 g/cm3 and turbulent velocities of 0.02 and 0.04 km/s. Note that the mass distributions are almost the same for masses greater than 10-8M. One can show that the number of these more massive objects is similarly insensitive to pp. Presumably, except for about 100 initial asteroids and their fragments, the smaller planetoids have been accreted by the four remaining terrestrial planets and the Moon.
From the table we find that about 15 percent of the mass of the original planetoid system was in objects less massive than Mercury, which implies that about 15 percent of the mass of Earth and the other terrestrial planets was accreted as small planetoids, whereas the remaining 85 percent was accreted as subplanetoid bodies, primarily clumps of dust. About one-third the mass of the accreted planetoids was in objects having sublunar masses and two-thirds was in objects having masses between that of the Moon and Mercury.
We may be concerned that a collision between a large planetoid in the latter group and a terrestrial planet could cause their mutual destruction. A breakup is expected if the preencounter total kinetic energy of the two objects relative to their center of mass is greater than their combined gravitational binding energies; i.e., for planetoids of uniform density, breakup requires that
where u = (M1M2)/(M1 + M2), the reduced mass, and V is the relative velocity of the two objects prior to the encounter. An upper limit on V is probably its present value for interasteroidal collisions, 5 km/s (Piotrowski, 1953). After the terrestrial planets formed, their long-range gravitational perturbations increased V well above the 0.04 km/s maximum turbulent velocity in the solar nebula, but these forces were not likely to have had sufficient time before the accretion of most of the small planetoids to raise V much above its present interasteroidal value. Figure 2 shows the mass m of the smallest body required to cause the collisional breakup of a planetoid of mass M if V = 5 km/s and pe = 5.5 g/cm3 or 3.6 g/cm3. We note that planetoids more massive than 0.026M, for pp = 5.5 g/cm3 and 0.033M, for pp = 3.6 g/cm3 are safe against breakup in collisions with all objects equal to or less massive than themselves. As the critical mass is less than half that of Mercury (for pp " 5.5), it seems likely that all terrestrial planets were safe against collisional breakup.
V = 0.02 km/s V = 0.04 km/s
Figure 2.-In a collision between two planetoids having a preencounter relative velocity of 5 km/s, m is the minimum mass of the collision partner required to cause the breakup of a planetoid of mass M. We note that near the limit of M beyond which a planetoid is stable against breakup irrespective of m, m is double valued, which indicates that there is an upper as well as a lower limit on the m required for fragmentation. This results from the quadratic dependence of the gravitational potential energy on m and M.
Although a collision with a large fellow planetoid would not destroy a terrestrial planet, it would produce a drastic alteration in the direction of its rotational axis if the orbit of the planetoid did not lie in the equatorial plane of the planet." A simple calculation shows that even single collisions with planetoids of lunar mass would easily account for the magnitude of the deviation of the equators of the terrestrial planets from the planes of their orbits. From the theoretical radius distribution function (table I) we see that Earth is likely to have accreted several such bodies.
COLLISIONAL FRAGMENTATION OF THE PLANETOIDS
Their small number [N(x)^2X 103 to 3 X 10°] indicates that the accretion of primary planetoids only produced a very small fraction of the observed lunar and Martian craters. As we shall see, likely agents for the production of most craters are the collisional fragments of a few primary planetoids of approximately lunar mass. We note from figure 2 that lunar-sized objects can be broken apart by collisions with bodies an order of magnitude less massive than themselves, whereas objects the size of Ceres can be broken up in collisions with bodies less than two orders of magnitude less massive than themselves. This extreme fragility of the asteroids suggests an explanation for their failure to coalesce into one body. If an object of 0.03M, or greater had formed in the asteroid belt, we would likely see only one object today.
The average number of fragmentation collisions that occur among a group of n planetoids before they are accreted by the planets is
Here or is the total :cretion cross section of the terrestrial planets for the average planetoid and alj is the collision cross section for encounters between planetoids i and j. With V = 5 km/s, the collisional cross sections of sublunar planetoids are very nearly their physical cross sections. Thus
Because the radius of the average sublunar planetoid is small compared with the radius S, of a terrestrial planet, by conservation of energy and momentum we find
where W is the escape velocity from the ith planet. For V = 5 km/s, o, = 1.5 x 109 km2.
Table I indicates that there were 214 original planetoids in the mass range 0.001M, to 0.01Me. These have radii between 735 and 1580 km for pe = 3.6 g/cm3. For these objects
From equations (17) through (20) we find that N = 2.4. Thus it is highly likely that at least one fragmentation collision took place among the objects in this mass range with each collision causing the breakup of two objects. The observed fragments produced by asteroidal collisions have an integrated radius function of approximately the form (Hartmann and Hartmann, 1968)
where N(R) is the number of fragments with radii larger than R. The largest fragment, of radius Rmax, has usually about one-half the initial mass of the fragmented planetoid. For the objects we have considered, Rmax is typically 1000 km which implies the production of about 10° fragments with radii greater than 1 km. This is three orders of magnitude larger than the number of primary planetoids, and it is quite adequate to account for the number of large lunar craters. Because N(R) is very sensitive to Rmax, we can expect that only the first one or two largest fragmented planetoids produced a majority of all the fragments. This result suggests that although most of the integrated mass in planetoids and their fragments accreted by a terrestrial planet or the Moon was in the form of a handful of very large unfragmented primary objects, the vast majority of crater-forming bodies were fragments of a few primary planetoids with initial masses on the order of that of the Moon. If meteorites are fragments of planetoids that were formed in the vicinity of Earth rather than objects that have diffused in from the asteroid belt, we can expect most of them to be from a few primary objects with masses on the order of that of the Moon. Because the integrated cross-sectional area of the fragments of a planetoid is much larger than its initial cross-sectional area, one collisional fragmentation produces a chain reaction of further fragmentations. Thus, although the integrated mass in planetoids and their fragments that are being accreted by a planet per unit time decreases exponentially with time, an accelerating pace of further fragmentation actually causes the number of objects being accreted per unit time to increase. This is accompanied by a rapid decrease in the average mass of the individual fragments. The integrated accretion cross section on of the Jovian planets is about three orders of magnitude greater than that of the terrestrial planets. Consequently, if we apply the same arguments to the planetoids that formed in the region of the Jovian planets that we did to those in the terrestrial band, we find that no fragmentation collisions are likely to have occurred among these planetoids before they were accreted by the Jovian planets. Thus the surfaces of the satellites of the Jovian planets should not be scarred by the large number of impact craters that dominate the faces of the Moon and Mars; however, there may be some contamination in the case of the satellites of Jupiter due to the diffusion of fragments from the asteroid belt. It is hoped that this anticipated scarcity of craters can be tested by future space probes.
TEMPERATURE OF ACCRETION
It is desirable to know whether the temperature that a given planetoid attained during the course of accretion was sufficient to melt it and thereby allow the differentiation of a core of dense material. The minimum temperature maintained by a planetoid in the act of accreting material is one whereby the energy inflow due to accretion is just balanced by the loss of blackbody radiation, or