Scattering Cross-Section from a Central Force

[CM Longhorns]

Homework Statement

An object from space (like an asteroid) approaches Earth. A collision will occur if the scattering cross-section is less than π*R_e². If the distance of closest approach is much greater than R_e, no collision would occur. Find an implicit expression for the cross-section in terms of g, v_o, and R_e. Show explicitly that the cross-section approaches the minimum value for large v_o, and find the appropriate dimensionless parameter.

g: acceleration due to Earth's gravity
v_o: velocity of incoming object
R_e: radius of Earth

Homework Equations

Rutherford scattering formula:
dσ/dΩ = k²/16E²*1/sin⁴(θ/2)

Total energy:
E = ½μr'² - k/r + L²/2μr²

The Attempt at a Solution

I knew to use the Rutherford scattering formula because the scattering cross-section arises from an inverse square law (the gravitational force.) By doing some manipulation of the force constant:
GM_em ≈ g*m
M_e + m ≈ M_e ⇒ μ ≈ m
and approximating E by the kinetic energy T at very large r:
E ≈ ½mv_o²
I arrive at
dσ/dΩ = g^2/4v_o²*1/sin⁴(θ/2)

This is an expression including g, and v_o... but where can we relate Earth's radius, R_e?

It seems like the differential cross section will approach R_e for large v_o, but I can't seem to express that mathematically. I'll need help understanding that first part. I could then use hints for finding the impact parameter per the last part of the question.
I'm quite confused at this point.

 

[mfb]

The object will hit Earth if its minimal distance is below the radius of earth. For a fixed initial velocity, this corresponds to a certain impact parameter you can calculate.

 

[CM Longhorns]

Do I calculate the impact parameter in terms of v_o using equations of orbital motion?

 

[mfb]

What else?
If you don't find a direct formula, you can also derive it based on energy and angular momentum conservation.

 

[CM Longhorns]

Resolved. I derived it the second way. Thanks