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Here T0 is the blackbody temperature of the planetoid in the absence of accretion. To evaluate T we need to know pay the density of accretable material in the solar nebula. We have calculated this by assuming that the material presently in the terrestrial planets and asteroids, Mtotal = 2M, was distributed uniformly in an annular sector of the solar nebula lying between 0.3 and 2 AU from the Sun. The turbulent velocity V determined the scale height of the dust above the plane of the nebula and consequently the density p, in the plane. For V = 0.02 km/s, we find that p, = 1.6 x 10-10 g/cm3; whereas for V= 0.04 km/s, p, - 0.8 x 10-10 gems To illustrate the dependence of Tupon R, in figure 3 we have plotted this relation for a case in which To = 300 K and planetoid densities pe are 3.6 g/cm3 and 5.5 g/cm3. We note that if TP 270, then T is practically independent of the particular To chosen. We further note from the figure that Venus and Earth with T^ 3 X 10° K were probably the only terrestrial planets that thoroughly melted. Mercury and Mars with T^' 1100 to 1800 K only partially melted. The Moon with T^- 600 to 1000 K probably did not melt from accretion, and Ceres with T^ 303 to 320 K was essentially accreted cold.
Figure 3.-Maximum temperatures attained by planetoids as the result of accretion. Curves are drawn for turbulent velocities of 0.02 and 0.04 km/s. The upper curve in each pair is for Pp = 5.5 g/cm3 and the lower is for Pp * 3.6 g/cm3.
TABLE II.-Time (years) of Growth in the Solar Nebula
-17 . . . . . . . . . . . . . . . . . . 2.3 x 103 2.3 x 103 3.0 x 103 3.0 x 103
Thus an Earth-type core is expected on Venus but probably not on any of the other terrestrial planets. Because of their low accretion temperatures, the asteroids can be expected to have preserved the chemical integrity of the material that they accreted. Thus future onsite inspections of asteroid fragments may yield valuable insight into the chemical and physical properties of the preplanetoid material and, consequently, insight into the chemical and thermal properties of the solar nebula during the time of planetoid formation.
TIME OF FORMATION
If the seed bodies were formed at a uniform rate in time, as has been assumed, the average number of planetoids with radii in some range R1 to R2 remained constant even in the presence of further accretion as long as the radius Rimax of the most massive planetoid in the system was greater than or equal to R2. Thus at any given time during the accretion of the planetoids, their radius distribution function was the same as given in table I up to radius Rmax. To find the time required for the radius of the largest planetoid in the
system to grow to R we integrate equation (4). This gives
We note that, unlike the radius distribution function, this depends on the sticking coefficient o and the space density pa of accretable material. Setting
Rmax = •, we see that formally a planet grows to infinite mass in a finite time
This is the characteristic time for forming a planetary system. We further note that a planetoid takes only twice as long to grow to R = • as to grow to R =Re.
Table II tabulates t as a function of Rimax for a = 1 and previously calculated pa values. The table shows that if V = 0.02 km/s, a planetoid only required about 8X 10° yr to increase its mass from that of Ceres to that of Earth. This suggests that if a stable seed body had formed about 8X 104 yr earlier in the asteroid belt, there might be a terrestrial planet there today. This small difference is less than 3 percent of the time required for a planetoid mass to grow to 1Ms.
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Hartmann, W. K., and Hartmann, A. C. 1968, Asteroid Collisions and Evolution of Asteroidal Mass Distribution and Meteoritic Flux. Icarus 8,361-381.
Hills, J. G. 1970, The Formation of the Terrestrial Planets. Bull. Amer. Astron. Soc. 2, 320.
Piotrowski, S. 1953, The Collisions of Asteroids. Acta Astron. Ser. A 5, 115-138.