The diameter distribution of Earth-crossing asteroids

Abstract

The knowledge of the diameter distribution of Earth-Crossing Asteroids (ECAs) is important because it gives information about the evolutionary processes that have affected the asteroid population in the main belt as well as their ejection into Earth-crossing orbits. On a more practical sense it is also of interest to know the frequency of collisions with the earth of asteroids of different diameters.In the present investigation we briefly discuss the luminosity functions of van Houten et al. (1970)(hereafter VH) for main belt asteroids (MBAs) and of Shoemaker et al., 1990(hereafter SH) for earth-crossing asteroids (ECAs). The luminosity function of VH is well represented by an exponential N(⩽H) ∼ eαH in the interval 11.25 < H ⩽ 16.25, while that of SH is represented, in the same magnitude interval, by two exponentials with different values of the constant α. The exponential behavior of VH luminosity function is consistent with the mass distribution function derived theoretically by Dohnanyi (1969)for the fragments of a population of objects in collisional equilibrium. Specifically, Dohnanyi shows that the power law n(m)dm ∼ m−βdm holds in principle for masses down to sizes of centimeters, where the Poynting–Robertson effect depopulates the distribution.The apparent inconsistency between the distributions of VH and SH, and the expectations that follow from Dohnanyis work, led us to study the problem of the distribution of diameters of ECAs, taking advantage of the much larger number of currently known earth-crossers (457, as of February, 1998, in the WEB page of the Minor Planet Center, http://cfa-www.harvard.edu:80/ ∼graff/lists/Unusual.htm).In the first section of this paper we derive the luminosity functions of MBAs and of ECAs. For MBAs we use the data of VH in the absolute magnitude interval 11.25 < H ⩽ 16.25, and for ECAs we use the data given in the Web page of the Minor Planet Center, in the magnitude interval 12.0 < H ⩽ 15.5. We find that in both cases the constants α are very similar to the value 1.1513 predicted by Dohnanyis theory.In the second section we use the known values of the mean albedos to transform the magnitude distribution N(⩽H) into the equivalent diameter-frequency distribution n(d). This distribution is then transformed to a mass-frequency distribution n(m) by means of the relevant densities, assuming that the asteroids are spherical. Finally, we use the distribution of masses and the kinetic energy for a given encounter velocity v with the earth to obtain the frequency distribution of impact energies n(E).In the third section we find the expected frequency of impacts with the earth of objects larger than a given diameter d. We also exhibit the frequency of impacts of energies equal or larger than E. Finally, our frequencies are compared with those obtained using the distribution of SH. We find that for small asteroids (∼10 m), the frequency of impacts predicted from our distribution is an order of magnitude larger than the one obtained from the distribution of SH.It is to be noted, on the other hand, that our frequency distribution n(d) of diameters is reasonably consistent, within the uncertainties, with those of Rabinowitz et al., 1994and Menichella et al. (1996).