# Scattering Cross-Section from a Central Force

1. Nov 23, 2014

### CM Longhorns

1. The problem statement, all variables and given/known data
An object from space (like an asteroid) approaches Earth. A collision will occur if the scattering cross-section is less than π*Re2. If the distance of closest approach is much greater than Re, no collision would occur. Find an implicit expression for the cross-section in terms of g, vo, and Re. Show explicitly that the cross-section approaches the minimum value for large vo, and find the appropriate dimensionless parameter.

g: acceleration due to Earth's gravity
vo: velocity of incoming object

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

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

3. 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:
GMem ≈ g*m
Me + m ≈ Me ⇒ μ ≈ m
and approximating E by the kinetic energy T at very large r:
E ≈ ½mvo2
I arrive at
dσ/dΩ = g^2/4vo2*1/sin4(θ/2)

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

It seems like the differential cross section will approach Re for large vo, 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.

2. Nov 24, 2014

### Staff: Mentor

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.

3. Nov 24, 2014

### CM Longhorns

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

4. Nov 24, 2014

### Staff: Mentor

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

5. Nov 24, 2014

### CM Longhorns

Resolved. I derived it the second way. Thanks