Estimate the accretion time of the Earth. Assume that the surface density (mass

Question

Estimate the accretion time of the Earth. Assume that the surface density (mass of solids in the nebula per unit area) of the solid particles in the Earth’s feeding zone is equal to two times the Earth’s mass uniformly spread over a annular ring that fills the space half way from Earth to Mars and half way from Earth to Venus. You can also assume that planetismals approach the growing Earth at a speed equal to a third of the escape speed from the surface of the growing planet. (Answer in years)

Ok, so I have two equations:

dm/dt = (1/2)s2( 1 + V2esc/V2)
and

ds/dt = (/8p)( 1 + V2esc/V2)

where is the mass per unit area which I found to be 2ME/r2 where r is 24.5e6km, is the angular rotation rate = sqrt(GM/r3), p is the density of Earth = 5.52 g/cm3 and s is the radius of Earth = 6378 km.

I'm confused about the angular rotation, is it the angular rotation of earth or the nebula? The way I planned on solving the problem was to solve for ds/dt find s and therefore find dm/dt then divide the current mass of earth by dm/dt. However, I don't know how to obtain s from ds/dt. Can I take s to be constant and to be the angular rotation rate of earth? Pretty much I'm confused about and how to get s from ds/dt. Help please!

Explanation / Answer

It is hypothesised that the accretion of Earth began soon after the formation of the Ca-Al-rich inclusions and the meteorites. Because the exact accretion time of Earth is not yet known, and the predictions from different accretion models range from a few millions up to about 100 million years, the exact age of Earth is difficult to determine

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