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One group of asteroids that does not fall under this limitation is the Mars-crossing asteroids (Anders, 1964). Traversing the main part of the asteroid belt during every revolution, they suffer impacts from time to time. The debris, being ejected with low velocities, will move in similar, Mars-crossing orbits, but with slightly different periods. Consequently it soon spreads out in a toroid along the orbit of the parent body. Secular perturbations further disperse the fragments; and in a steady state, some of them are always in orbits intersecting the orbit of Mars. Close encounters with Mars reorient the velocity vectors of the debris (Arnold, 1965; Öpik, 1951), leading to Earth-crossing orbits in a fraction of cases. Thus Mars-crossing asteroids can serve as a source of meteorites.

The number of Mars asteroids, 34 according to the 1964 Ephemeris volume, is somewhat larger than the apparent number of meteorite parent bodies. However, 21 of these, comprising 98 percent of the mass and 92 percent of the cross section, belong to four Hirayama families (Anders, 1964). Thus the number of original Mars asteroids is indeed of the same order as the number of meteorite parent bodies.

Let us see how the picture changes when we make more optimistic assumptions about the escape of meteorites from the asteroid belt. Some asteroids of high inclination that are not Mars crossing at present will periodically become so, owing to secular perturbations (e.g., Pallas; Smith, 1964). Data are not available for most asteroids of interest, and would, moreover, not be quite appropriate for their debris, which moves in slightly different orbits. To obtain some sort of upper limit on the number of asteroids that can contribute meteorites, let us calculate U, the velocity relative to a hypothetical circular orbit at the same semimajor axis a (Öpik, 1951):

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The components of U in the x, y, and z directions are

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Minimum perihelion qmin is reached when U is reoriented such that U, and i = 0, causing e to reach emax. At each value of a, only debris from asteroids with U greater than some minimum value has a chance of reaching the orbit of Mars, and then only in the doubly favorable case that perturbations cause U. to approach zero and cause the node to occur at or near minimum perihelion.

Whether such reorientation can actually take place is completely uncertain. One must appeal to unknown or ill-understood effects; e.g., higher order terms in secular perturbations or commensurabilities with Jupiter. However, it may be significant that the asteroids with highest eccentricities (719, 887, 1036, 6344) occur near the 1/3 commensurability, a = 2.50 AU. Perhaps a substantial reorientation of U takes place when a is close to a major commensurability. Because all Mars-crossing asteroid families have a within 0.2 AU of the 1/3, 1/4, or 2/5 commensurabilities, some fraction of their ejecta will have commensurable orbits. They will certainly be subject to strong Jupiter perturbations, perhaps of the required kind. [Note added after colloquium: There now exists some support for these speculations. Williams' has found, using his new theory of secular perturbations, that several resonance surfaces exist in a, e', sini' space in the inner asteroid belt. Any object in the vicinity of these resonances will experience very large oscillations in e and i, which of course favor attainment of a Mars-crossing orbit. Interestingly, each of the high-velocity asteroid families in figure 1 adjoins one or more of these resonance surfaces. A significant fraction of their collisional debris thus will be thrown into these resonance regions, where the postulated reorientation of U may take place. Williams' resonances may thus be the long-sought factor permitting change of highly inclined to highly eccentric orbits.] The distribution of U is illustrated in figure 1 for the inner half of the asteroid belt. The minimum value of U required to reach q = 1.700 AU (a value part way between the present and maximum aphelion of Mars) is indicated by the solid line. Asteroids lying above the line are potential sources of Mars-crossing debris under the above, optimistic assumptions. Nearly one-tenth of all asteroids have amin less than 1.700 AU. Yet the number of potential meteorite parent bodies has not increased greatly. Many of the newly added asteroids are less eccentric but otherwise bona fide members of Mars-crossing families 5, 29, 30, and 31.” Others fall within the boundaries of these families on a U versus a plot, but have e' and i' outside the family limits. They are either interlopers or former members whose elements were changed by Mars encounters. Only three new families appear on this graph. The cluster at 1.9 AU, though quite disperse on an a, e, i' plot, is fairly compact on a U, a plot. Nine additional members of this cluster were discovered by van Houten et al. (1970), who referred to it as the “Hungaria group.” Family 28, including 2 Pallas, becomes marginally Mars crossing every 10° yr (Smith, 1964) but its velocity is high enough to give a potential qinin as small as 1.10 AU if i = 0. In fact, family 30 may have been derived from family 28 by a partial reorientation of U. Family 17 is a very marginal case, but has been included for the sake of completeness.

*See p. 177.

*The status of family 31 is in some jeopardy. Anders (1965) suggested that it might be related to the Flora families, 6 to 9. Arnold (1969) has questioned its reality on the grounds that it constitutes only a twofold enhancement of asteroid density in a, e, i space over the general “background.” Van Houten et al. (1970) have assigned number 31 to another family, not being aware of the previous assignment.

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Figure 1.-The curve represents the critical condition for achieving a Mars-crossing orbit. Asteroids lying above this line either have Mars-crossing
orbits or are potentially capable of ejecting debris into such orbits if secular perturbations reorient Uso as to make i small and allow the node to

occur at perihelion. Most of these actually or potentially Mars-crossing asteroids are members of six numbered Hirayama families or the Hungaria group at 1.9 AU.

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Only a few nonfamily asteroids fall above the q = 1.700 AU line, most of them by very marginal amounts. The only sizable objects are 18 Melpomene at 2.296 AU (possibly related to family 31), 6 Hebe at 2.426 AU, 247 Eukrate at 2.741 AU, and 148 Gallia at 2.771 AU. A list of actual or potential Mars crossers is given in table III. Original radii were reconstructed according to Anders (1965), using data for the first 1651 numbered asteroids from the 1964 Ephemeris volume. A geometric albedo of 0.12, as for Ceres, was assumed, resulting in the magnitude-radius relationship

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We see that the reconstructed family asteroids (table III) are only about one-half as large or one-eighth as massive as the meteorite parent bodies (table I). This is not surprising, because the family asteroids now visible represent but a fraction of the original population. If we ascribe the discrepancy to loss of 7/8 of the members by dispersal or deflection to terrestrial space, the combined half-life for these processes must be on the order of 1 to 2 aeons, depending on the age of the family. This is roughly consistent with the half-lives found in Monte Carlo calculations, and with Dohnanyi's (1969) estimated lifetimes for collisional destruction of asteroids 10 to 20 km in radius. The half-lives may be longer if the albedo of the asteroids is smaller than assumed. A value of 0.065, as for Phobos (Smith, 1970), would reduce the mean discrepancy in mass to a factor of ~3 and would lengthen the half-life accordingly.

We note in passing that the two most massive objects in table III, 2 Pallas and 6 Hebe, are not extensively broken up, and hence probably cannot serve as

TABLE III.-Actual and Potential Mars-Crossing Asteroids

Number | Representative al, q. I amin." U Ro, Family of member AU AU AU km

members Hungaria 23 1235 Schorria | 1.910 1.617 | 1.060 |0.457 b11 31 9 1204 Renzia 2.264 1.596 || 1.592 .300 || C71 5 Phocaea 34 1310 Villigera |2.392 1.538 | 1.219 .507 | by 3 30 Aethra 4 1036 Ganymed 2.658 1.216 .910 .703 || b28 29 Desiderata 7 1134 Kepler 2.683 1.432 || 1.289 .539 b53 17 11 36 Atalante 2.749 1.924 || 1.576 437 || bg5 28 Pallas 6 2 Pallas 2.772 2.123 | 1.103 .635 | 245 Hebe 1 6 Hebe 2.426 1.934 || 1.646 .326 || 116

*For zero inclination.

*Highly fragmented.

*Including 18 Melpomene; without this asteroid, Ro would be 5 km and the classification changed to highly fragmented.

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sources of iron meteorites. Most or all irons in each group apparently came from the deep interiors of their parent bodies (Fricker et al., 1970). But if the dispersal-deflection half-life is as short as 1 to 2 aeons, some older families may have been decimated, fragmented, and dispersed beyond recognition. Their remains are presumably hidden in the nonfamily background in figure 1. Thus far we have relied entirely on planetary perturbations to extract meteorites from the asteroid belt, neglecting the effect of ejection velocity. Actually, it appears that ejection velocity was an important factor in at least one case, the group III irons. Jaeger and Lipschutz (1967) have noted that group III irons, without exception, are shocked to >13 GN/m2 (130 kb). often to >75 GN/m2 (750 kb). Shock pressures of this order correspond to free-surface velocities of 1 to 3 km/s; and the fact that no lightly shocked members are found in group III (in contrast to other groups) suggests that high shock pressures, and the concomitant acceleration, were essential to the escape of these meteorites from the asteroid belt. Jaeger and Lipschutz propose that the parent body of group III was a ring asteroid not crossing the orbit of Mars. Only its high-velocity ejecta had a chance of achieving a Mars-crossing orbit, the essential prerequisite for deflection into terrestrial space. Various objections have been raised to Mars asteroids as the principal source of meteorites. Öpik (1965, 1968) has pointed out that their mean lives for deflection into Earth-crossing orbits, 109 to 1010 yr, are far longer than the radiation ages of stony meteorites, 2 x 10° to 6 x 107 yr. He maintains that such short capture times are completely unattainable for debris from Mars asteroids. However, it is important to make a distinction between mean capture times for a large population and actual capture times for individual objects. On the Mars asteroid model, the radiation age T is the sum of two intervals; tı, from ejection of the meteorite from its parent body to deflection into an Earth-crossing orbit, and t2, from achievement of the first Earth-crossing orbit to actual capture by Earth. Both ti and t2 are exponentially distributed about the mean lives for the two processes, T1 and r2, and, because the most probable values of ti and t2 in an exponential distribution are zero, small values of T are not at all inconsistent with large values of Ti and 12. One can also prove this by recognizing that the toroidal debris stream associated with each Mars asteroid is analogous to a meteor stream. Typically, orbits of planet and stream intersect for a few centuries during each 10° yr oscillation in e and i If the stream is continuous, some objects will be captured or deflected during each revolution as the planet crosses the stream. Because the distribution of radiation ages along the stream is random, the objects deflected will include some very young ones. Arnold (1965) and Wetherill (1967, 1968a) objected to a Mars asteroid origin mainly on the grounds that it would give a preponderance of long ages, in the range 10° to 10° yr. In principle, such long ages could be suppressed by collisional destruction of meteorites, but a careful analysis of the problem seemed to show that the density of dust and rubble in the asteroid belt was too

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