# Accretion Time of Earth

1. Oct 25, 2006

### Poop-Loops

My Astronomy homework has a problem asking to find the accretion time of Earth.

The forumula's the professor gave (since the book doens't have ANY) are:

$$\frac{dm}{dt}=\pi s^2\frac{\sigma \omega}{2}$$

And

$$\frac{ds}{dt}=\frac{\sigma \omega}{8 \rho} (1+ \frac{V_{esc}^2}{V_{relative}^2})$$

Where sigma = surface density of the accretion area (I'm guessing here, by the way), omega is the angular acceleration (which means radians/second around the sun, right?) and s is the radius of Earth.

I have no idea what rho is, probably the density of Earth.

Relative velocity = 1/3rd escape velocity, so those cancel. I know the accretion area is from halfway to Mars to half way to Venus from Earth and the total mass of that volume is 2 Earth masses.

My problem is: how do I calculate that? Since s is constantly changing, does that end up being a differential equation? Looks pretty ugly. Or do I assume that it is constant (say, the size of Earth), since it is so much smaller than the accretion area?

I have no idea what to do from here. If I can figure out s, I can then find dm/dt and then just divide Earth's current mass by that, right? So that shouldn't be a problem.