Cross Section for Asteroid Impact

[ebjessee]

Homework Statement

In this problem, you will estimate the cross section for an earth-asteroid collision. In all that follows, assume that the Earth is fixed in space and that the radius of the asteroid is much less than the radius R of the earth. The mass of the Earth is M_e, and the mass of the asteroid is m. Use G for the universal gravitational constant.

I've already solved:
E_initial = (m/2)v²)
L_initial = bmv
E_at surface of Earth = ((v_f)²*m)/2 +(GmM_e)/R
L_at surface of Earth = (mv_fR
b_max² = R²+(2RGM_e)/v²

Where I get confused is the actual significance of b_max:

The collision cross section S represents the effective target area "seen" by the asteroid and is found by multiplying (bmax)² by π. If the asteroid comes into this area, it is guaranteed to collide with the earth.
A simple representation of the cross section is obtained when we write v in terms of ve, the escape speed from the surface of the earth. First, find an expression for v_e, and let v=Cv_e, where C is a constant of proportionality. Then combine this with your result for (bmax)² to write a simple-looking expression for S in terms of R and C.

Express the collision cross section in terms of R and C.

I know that a version of v_e is sqrt(2GM/R), but I have no idea how to go on from here.

 

[gneill]

Did you take your expression for v_e and form v = Cv_e? What did that do for your b_max² equation? Show your work.

 

[ebjessee]

Ok. We say that v=Cv_e
So b_max² = R² + (2RGM_e)/Cv_e²

The collision cross section S represents the effective target area "seen" by the asteroid and is found by multiplying (bmax)² by π

Then π bmax² = π R² + 2π RGM_e/Cv_e²

Distributing pi to both sides of the equations helps me see that I'm now looking at circular area equations, so that helps to but bmax into perspective. I plugged in v_e = sqrt(2Gm/R) which helped me reduce the previous equation to:

S = π R² +π R²M_e/Cm

That didn't work. I got a message the answer did not depend on m or M_e.

 

[ebjessee]

Nevermind- just forgot to square C and use the same M_e. Thanks!

 

[gneill]

You want to replace $v_e^2$ in your $b_{max}^2$ equation with your expression for escape velocity right away and cancel out what can be cancelled. I don't know how m snuck into your equations...