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ON THE FORMATION OF THE ASTEROIDS
J. G. H/LLS
Recently Anders (1965) has shown that the smaller asteroids are overabundant relative to their initial numbers because of the collisional fragmentation of larger asteroids. Allowing for this fragmentation, he has derived a reconstructed initial radius distribution. Although the reconstructed distribution only comprises a little more than 100 asteroids and consequently has a large statistical uncertainty as well as the uncertainty introduced by the reconstruction process, Anders finds that when displayed on a log-log plot, the distribution can be represented fairly well by a gaussian curve with a peak near radius R = 30 km. This work has basically been substantiated by Hartmann and Hartmann (1968), although Hartmann (1968) notes that a gaussian distribution underestimates the observed number of more massive asteroids. This suggests that the initial asteroid distribution function was probably broader than a gaussian function, but it still retains a distinct bell-like appearance.
In this paper we propose a simple model for the accretion of objects in the solar nebula that permits a straightforward calculation of their radius function. This fits the Anders reconstructed asteroid distribution, and it also predicts reasonably well the number of terrestrial planets. (The terrestrial planets are assumed to be asteroids that formed slightly earlier than their fellows and consequently captured most of the available solid material in the solar nebula. Subsequently the asteroids and terrestrial planets will collectively be called planetoids.) A number of consequences of our model are explored in this paper. A preliminary report on this theory has been published (Hills, 1970).
The calculation of the radius function of the planetoids requires some knowledge of the rate of formation of the seed bodies that initiated their accretion. From knowledge of other processes requiring the formation of seed bodies, it is probable that their formation was governed by a stochastic process. In this case, the rate of formation of the seed bodies was independent of time as long as the total mass accumulated in the planetoids remained much smaller than the amount of unaccreted material.
With the rate of formation of the seed bodies being independent of time, the number of planetoids with radii between R and R + AR is directly proportional to the time necessary for the radius of a planetoid to grow from R to R + AR. This requires a radius distribution function of the form
where the constant of proportionality No' is the number of seed bodies formed per unit time in the nebula, and dR/dt is determined by the accretion equation. (See, e.g., Hartmann, 1968.)
Here o is the sticking coefficient, pa is the space density of the accretable material, Pp is the planetoid density, and V is the average preencounter velocity of the accreted particles relative to the planetoid. The equation is simplified by introducing a characteristic radius,
R. is the radius at which the accretion cross section of a planetoid is twice its geometric cross section.
Making use of the accretion equation, the radius distribution function becomes
Integrating equation (5), we find that the number of planetoids with radii equal to or less than R is
Thus the number of planetoids is formally bound even if the radius of the largest one and the total mass of the system are not. This results from the accretion cross section of the largest object formally growing much faster than its mass, which allows it to grow to infinite mass in a finite time if enough material is present. In a real system, the number of planetoids is similarly not determined by the total mass of the system but by the ratio of the time necessary for the largest object in the system to acquire most of its mass to the average time between the production of the seed bodies. Most of the mass in a typical planetoid system will be accumulated in the first one or two largest bodies. In any actual system there is an upper limit Rmax to the radius of the largest planetoid, but if Rimax > Re, then N(*) - N(Rmax). If we mathematically allow R → • so that N(co) is the total number of planetoids formed, the normalized integrated radius function is
We note that Re is the median radius of the planetoids. To compare the theoretical radius distribution function with the Anders distribution we have to convert the former into one in units of ln R. This yields
This function is plotted in figure 1. It is a serpentine curve and looks quasi-gaussian about the peak at R = Re.
The theoretical curve fits the reconstructed radius distribution to within the statistical errors if Re - 15 km. This function is noticeably broader than a gaussian function. For the initial asteroid system, the radius of Ceres is Rimax. We note in passing, that if we normalized the theoretical radius function to the reconstructed asteroid radius distribution, any planetoids with R P Rmax predicted by the theoretical relation can only have a mathematical and no physical significance.
With R. = 15 km and p, =3.6 g/cm3 for the asteroids we find by equation (3) that V = 0.02 km/s. The peak of Anders' (1965) proposed empirical function is Re - 30 km with an error of about 50 percent. For R. = 30 km, V = 0.04 km/s. We take V = 2 x 10-2 to 4 x 10-2 km/s as the likely range of V. This V was presumably due to large-scale turbulent motion in the solar nebula.
We shall now investigate some consequences of our theoretical model to shed more light on the planetoid formation process and to better test the validity of the theory by producing a larger body of results to check against observable data.
TOTAL NUMBER OF PLANETOIDS
Assuming that V was constant throughout the solar nebula allows the calculation of the total number N(oo) of planetoids formed in the region of the solar nebula now occupied by the terrestrial planets (terrestrial band). The total mass of a system of planetoids in which the largest body has a mass Mmax is found by integrating equation (10). This gives
T 3 " " 1 + (R/R.)? On completing the integration and rearranging terms, we find
We note that N(oo) a Pp 1/6. This weak dependence of N(x) on Pp makes rather immaterial whether we use 9p = 3.6 g/cm3 as for chondrite meteorites or