Solving Angular Velocity of Particle Skimming a Planet

[treynolds147]

Homework Statement

A particle travels in a parabolic orbit in a planet's gravitational field and skims the surface at its closest approach. The planet has mass density ρ. Relative to the center of the planet, what is the angular velocity of the particle as it skims the surface?

Homework Equations

$L=mr^{2}\omega$
$r_{min}=\frac{L^{2}}{m\gamma(1+\epsilon)}$, where ε is the eccentricity of the orbit, and γ = GMm.

The Attempt at a Solution

Okay, I used the knowledge that the minimum radius for a parabolic orbit is d/2, d being the focal length of the parabola. Because the particle is skimming the surface of the planet at this point, the radius of the planet must also be d/2. I equated that to the expression of r_min, substituting in the expression for L and using d/2 as r. I ended up getting
$\omega=4\sqrt{\frac{MG}{d^{3}}}$.
The units of this come out properly, but I have this nagging feeling that I did something wrong along the way. Can someone point out my error?

 

[gneill]

One problem occurs to me: What's d? I mean, its value in terms of the variables given in the problem statement?

You'll want to find an expression for the angular velocity that uses the variables given. You can work with the density to find the mass in terms of the radius. Add G to the mix for an expression for μ...