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An uncertainty due to albedo of about half an order of magnitude is present, in addition to other uncertainties. The value of ree for the largest asteroids is on the order of 10° yr (fig. 8). It can be seen, from figure 8, that the lifetime of the six largest asteroids with masses m > 10% kg is about 4 X 10° yr or longer and therefore these may have survived since the time of their creation. The other asteroids have shorter lifetimes Tce and may therefore be collisional fragments.
Figure 8.-Double logarithmic plot of particle lifetimes in years as a function of particle masses in kilograms (or particle radii in meters).
Using a more detailed spatial and velocity distribution, Wetherill (1967) has calculated collisional probabilities and obtained values comparable to but smaller than the values that a randomly distributed asteroid population (particle-in-a-box) would imply. He also estimated tee for a 1 m diameter object for a number of assumed mass distributions. He considered (Wetherill, 1967, table 7) population indexes in the range o = 5/3 to o = 1.8; corresponding values of ree were then computed for T' = 10°, 10°, and 10°. Because these population indexes are lower than the steady-state value of a = 1 1/6, the values in Dohnanyi (1969) for tec are correspondingly shorter. The difference is about an order of magnitude in Tee.
The lifetime with respect to erosion (i.e., erosive reduction of the particle mass) can be obtained when the expression for m (eq. (23)) is integrated. The result is, from Dohnanyi (1969),
where the erosive lifetime re of an object was taken to be the time required to erode it to one-half its initial radius and where a = 1 1/6 was used. The logarithmic term is significant for masses approaching the value T'u, as can be seen from figure 8, Te becoming infinitely long for masses m s T'u. This happens because erosion stops for these small particles and all collisions they experience are catastrophic. We also plot, in figure 8, the particle lifetimes with respect to the Poynting-Robertson effect (Robertson, 1936) TPR and the lower limit of the lifetime of small objects T1 due to the influence of cometary meteoroids and cosmic rays estimated by Whipple (1967). Here the definition of T is similar to that of Te; i.e., it is the time for erosion of an object to one-half its radius. TPR is the time required for an object to traverse radially one-half of the asteroidal belt, because of the Poynting-Robertson effect. It can be seen, from the figure, that catastrophic collisions dominate the lifetime of the particles greater than about 10-3 kg (or 1 mm in radius). Smaller particles may be subject to erosion by cometary particles to an extent that this mechanism dominates.
Using a stochastic model of asteroidal collisions, their mass distribution has been estimated. The results individually agree with the observed distribution of bright asteroids (MDS) and faint asteroids (PLS). After correction for completeness, the MDS and PLS distributions are similar in form but differ from each other by a numerical factor. Until this difficulty is resolved, some uncertainty remains in the precise form of the distribution of bright asteroids. Subject to this reservation, we may conclude that the mass distribution of most asteroids has reached (i.e., relaxed into) a stationary form that is independent of the original distribution and is a power-law function with index ~ 1 1/6 for faint asteroids.
The influence of catastrophic collisions dominates the evolution of the population; erosion plays a minor part. The influence of the PoyntingRobertson effect becomes dominant, however, for particles with masses of 10-19 g or smaller.
Whereas the particle lifetimes, erosion rates, collision probabilities, and other derived quantities of physical interest are expected to be self-consistent, uncertainties in the albedo of asteroids and in other parameters introduce an appreciable systematic error; the numerical values of these quantities should therefore be regarded as order of magnitude approximations.
Thanks are due to D. E. Gault for important suggestions and to T. Gehrels for helpful discussions.
Alfvén, H. 1964a, On the Origin of Asteroids. Icarus 3, 52-56.
Marcus, A. H. 1965, Positive Stable Laws and the Mass Distribution of Planetesimals.
VAN HOUTEN: I wish to comment on figure 2 of Dohnanyi's paper. In this figure, the cumulative number of asteroids, as a function of absolute magnitude, is shown for MDS and PLS. In the overlapping part, the MDS values are approximately 10 times as large as the PLS values. This discrepancy could be traced to the following causes:
(1) The correction factors for incompleteness in table 15 of MDS in group III (3.0 < a < 3.5) are incorrect; the correct values are given in table D-I.
(2) Dohnanyi apparently used table 5 of PLS for the computation of his cumulative numbers of PLS asteroids. But to this table should be added the objects that were too bright for measurement in the iris photometer; these are five in total.
TABLE D-I.—MDS Correction Factors
3.0 < a 3 3.5 2.0 < a. 33.5 g Wobs Amin Wmax Nobs Wmin ^max 9.75. . . . . . . . . . 39 39 40 114 114 115 10.25. . . . . . . . . . 64 65 69 150 151 155 10.75. . . . . . . . . . 93 121 143 180 208 231 11.25. . . . . . . . . . 78 137 169 184 246 288 11.75. . . . . . . . . . 77 220 320 168 334 453 12.25. . . . . . . . . . 45 250 450 150 4.18 661 12.75. . . . . . . . . . 17 255 472 138 568 913
After these corrections, and using the average of Nmax and Nmin for the MDS value as Dohnanyi did, the comparison between MDS and PLS becomes as given in table D-II.
The MDS values are still about twice as large as the PLS values, after these corrections. But the comparison is based on only 12 objects in the PLS. The statistical uncertainty of this number is such that maybe not too much importance should be attached to this difference.
TABLE D-II.—Comparison Between MDS and PLS
Number of asteroids g < – MDS PLS 11.25. . . . . . . . . . 961 505 11.75. . . . . . . . . . 1355 600 12.25. . . . . . . . . . 1895 912 12.75. . . . . . . ...] 2635 1704
DOHNANYI: If the five objects, too bright for photometry in PLS, are included, the resulting change in figure 2 is not significant for the purposes of my present study. A least-squares fit for asteroids with g » 11 gives a new population index o' = 1.815, which does not significantly differ from the previous value of a = 1.839 that I have obtained earlier. The maximum and minimum probable number of asteroids was estimated in MDS. No such quantitative estimate is given in PLS even though large correction factors affecting every asteroid observed in PLS were employed in extrapolating the relatively small sample of PLS to the rest of the asteroid belt. Large nonlinear corrections that have been applied are particularly visible when comparing figures 9 and 11 in PLS. The maximum in distribution of inclinations (fig. 9, PLS) is near 3° whereas the average inclination for cataloged asteroids is about 10° (Watson, 1956). The simple method based on several assumptions for estimating the completeness factors due to the inclination cutoff in PLS may be especially vulnerable to systematic error. The large correction factors employed for the small number of high-inclination orbits may be subject to noise because the number of these asteroids probably fluctuates in time. (See Nairn, 1966.) Assuming that the PLS results are free from the type of difficulties that led van Houten to revise the MDS data, uncertainties in PLS data do still exist. Without a quantitative estimate of these uncertainties it might be arbitrary and misleading to connect the PLS results with those of the MDS without further comment at this time. VAN HOUTEN: Nmax and Nmin in table 15 of MDS should not be regarded as the maximum and minimum probable number of asteroids. They are simply numbers derived from two different extrapolations of the log N(mo) relation. They should be compared with the PLS to see whether any of them approximates the real numbers. This comparison yields the results given in table D-III (cf. table 14 of MDS). It is seen that at po - 18 even the use of equation (7) results in values that are too small. Fortunately the incompleteness corrections of the MDS were not extended to such faint magnitudes. Their lower limit is po = 17, and here the PLS value is indeed between equations (6) and (7) (from which Nmin and Nmax were derived, respectively). A check is possible on the correctness of the correction factors for the declination cutoff, used in the PLS. The corrected values should reproduce the same distribution of asteroids with respect to the ecliptic as was found in the MDS. (See fig. D-1.) It was shown in the PLS that one strip of the MDS yielded 1.90 times as many asteroids as a PLS strip for the same limiting magnitude; moreover it was found in the MDS that 10 percent of the