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distribution. Alfvén (1964a,b, 1969), on the other hand, discussed the origin of asteroids making the alternate assumption that asteroids in Hirayama families constitute original jetstreams. (Also see Kuiper, 1953.) In view, however, of the fact that next to nothing is known about the distribution of asteroids too faint to be observed, and much still remains to be learned about those cataloged, it appears worthwhile to employ statistical methods to improve our understanding of some of the gross properties of the population of asteroids. Ideally, one would like to combine the distribution of orbital elements for asteroids with their mass distribution in a complete statistical analysis. This difficult problem can be simplified by two methods:
(1) Studying the distribution of the masses of asteroids using an assumed spatial (and velocity) distribution
(2) Studying the asteroidal population by using precise spatial and velocity distributions combined with an assumed mass distribution (Wetherill, 1967).
This second method has its basis on Öpik's (1951, 1963, 1966) statistical treatment of the dispersal of stray objects by planetary (gravitational) perturbations, and in its most highly developed form has been applied to asteroids by Wetherill (1967). In this paper we shall limit our attention to method (1). Method (2) is, however, complementary to method (1), because a complete analysis would employ a combination of both methods; i.e., a joint mass, velocity, and position distribution. Method (1) is the physical and mathematical modeling of a population of objects that undergo mutual inelastic collisions. Such collisions take place with an assumed mean encounter velocity, and the larger of the colliding masses may completely shatter (catastrophic collision) or it may lose a modest fraction of its mass (erosive collision) depending on the relative size of the other colliding object. The result is a process by which the masses of individual objects in the population decrease with time because of erosion and by which some objects are violently destroyed from time to time. Redistribution of the comminuted debris produced during erosive and catastrophic collisions constitute a particle creation mechanism. A correct modeling of these processes would enable one to describe the evolution of the distribution of these colliding masses. Piotrowski (1953) has derived a mathematical expression for the rate at which asteroids disappear because of catastrophic collisions and the rate at which the number of asteroids in any given mass range changes because of the erosive reduction of their masses caused by the cratering collisions with relatively small objects. He did not include the particle creation resulting from fragmentation during collisions and his analysis therefore is restricted to cases in which the replenishment (i.e., feedback) of the population by comminuted fragments is insignificant.
Jones (1968) has studied the evolution of the mass distribution of asteroids using a more detailed model; the contribution of fragmentation was considered but later discarded because the size of the fragments produced during collisions was taken to be insignificantly small.
Dohnanyi (1969) (see also Dohnanyi, 1967a,b,c, 1970a) has discussed a model that includes the influence on the distribution of asteroidal masses of the following collisional processes:
(1) Disappearance of asteroids because of catastrophic breakup
(2) Reappearance of new asteroids from the fragments of catastrophically disrupted ones
(3) Progressive change in the number of asteroids in any given mass range caused by the gradual reduction of asteroidal masses by erosive cratering of small projectile particles
(4) Reappearance, as tiny asteroids, of secondary ejecta produced during erosive cratering
Numerical values for all parameters were taken from experiment and observation, wherever possible; and a particular solution of a simple power-law type was obtained, under the provision that the distribution could be assumed stationary.
The study was continued (Dohnanyi, 1970b), and it was found that the mass distribution of asteroids may indeed approach a stationary form, regardless of initial conditions, after a sufficiently long time period has elapsed. The uniqueness of the solution obtained in Dohnanyi (1969) was considered, and it was found to be the only analytic solution that can be expanded into a power series in m, for masses m far from the limiting masses of the distribution. An approximate solution for large asteroids was also obtained.
Hellyer (1970) has also examined this problem; he considered large asteroids and small ones separately. For small asteroids he studied the influence on the mass distribution of fragmentation and his treatment is comparable to that in Dohnanyi (1969, 1970b) except that it is less detailed but mathematically much shorter.
Because of their completeness compared with earlier work, we shall, in what follows, give a review of these studies (Dohnanyi, 1969, 1970b). Most of the earlier work can readily be discussed by comparing it with special cases of these studies.
McDonald Asteroidal Survey!
In their survey of asteroids at the McDonald Observatory (the McDonald survey (MDS)), Kuiper et al. (1958) obtained statistical data for the brighter
"Currently under revision; see van Houten in the “Discussion” following this paper.
asteroids up to a limiting apparent magnitude of 16. The observation covered the asteroid belt over all longitudes and a 40° width in latitude. The absolute photographic magnitudes of 1554 asteroids were obtained in half-magnitude intervals together with correction factors for estimating the true number of asteroids in each magnitude interval, based on the completeness of the survey. To estimate the masses of asteroids, we assume a geometric albedo of 0.2X 3* and material density of 3.5 x 103 kg/m3. The upper limit on the geometric albedo represents a completely white smooth surface and the lower limit corresponds to basalt. The nominal value of 0.2 is the mean of the estimated geometric albedos of the asteroids Ceres, Pallas, Juno, and Vesta. (See, e.g., Sharonov, 1964.) The result is
where m is the mass, in kilograms, of a spherical asteroid with absolute photographic magnitude g (i.e., relative photographic magnitude at a distance of 1 AU from both Earth and the Sun). A measure of the uncertainty due to albedo is indicated.
The observational material of MDS is presented in figure 1. Plotted in this figure are the cumulative numbers of observed asteroids (solid histogram) as well as the probable true number of asteroids (dashed line histogram) versus absolute photographic magnitude g, as given by MDS. The curve is complete up to g = 9.5; i.e., the observed number of these objects is believed to equal the true number. Above g × 9.5 the difference between the true and the observed number of asteroids, based on the completeness of the survey, has been tabulated in MDS (also see Kiang, 1962); the dashed line histogram in figure 1 is their mean value.
The solid curve in figure 1 is the cumulative number N(m) of asteroids larger than m
as a function of mass m (or g) obtained in Dohnanyi (1969). In that study, we took M. = 1.86 X 1029 kg corresponding to g = 4 and
where the numerical (normalization) factor is empirical and the numerical value of the exponent was theoretically obtained for relatively small (faint) asteroids. It can be seen that there is close agreement between theory and the statistical results of MDS.
Figure 1.-Cumulative number of asteroids having an absolute photographic magnitude g or smaller (i.e., mass m or greater), obtained by the MDS. Observed value = solid line histogram; probable value = dashed line histogram; earlier theory (Dohnanyi, 1969) = solid curve.
A series of observations of faint asteroids with limiting apparent magnitudes of less than 20 was made by van Houten et al. (1970) at Hale Observatories (Mount Palomar) (the Palomar-Leiden survey (PLS)). The angular area covered was only 18° by 12° and a compilation of the estimated number of faint asteroids as a function of absolute magnitude, in the range 11 sg s 17, was prepared. Whereas in the MDS results, the number of asteroids found is believed to be complete up to an absolute magnitude of about g = 9.5, in the case of the PLS results, the number of observed asteroids needs to be corrected for completeness for all values of g because of the smaller area covered (about 1 percent of the MDS area). Thus, to estimate the total number of faint asteroids in the entire asteroid belt as a function of absolute magnitude g, the PLS data have to be extrapolated over the large regions not covered by the survey.
The result is displayed in figure 2, a plot of the cumulative number of asteroids having an absolute magnitude g or greater (per half-magnitude intervals) obtained by MDS and PLS, as indicated. It can be seen that the two curves display the same trend, i.e., the shapes of the two distributions are identical, but that the MDS results are almost an order of magnitude higher than corresponding PLS results, and likewise for their respective extrapolations. It was pointed out in the PLS report that this discrepancy may be due to the method of estimating completeness factors in MDS. Because the true cause for this discrepancy has not yet been given, we shall avoid combining the results of MDS with those of PLS and will consider them separately.
In figure 3, we plot the cumulative number of asteroids from PLS as a function of absolute magnitude g and seek to represent the results by an empirical formula of the form
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Absolutt PhotocRAPHIC MAGNITUDE, 9
Figure 2.-Cumulative number of asteroids obtained by the MDS and the PLS. Solid line is the observed number; dashed line is the corrected number for completeness.