Tefeari said:
I've been really looking but can't seem to find a complete derivation of the Hill/Roche Sphere equation: r[h] = a(m/3M)^(1/3)
http://en.wikipedia.org/wiki/Hill_sphere" and I havn't been able to find another across the entire internet (aside from one on Amazon.com's "See what's inside" book feature which wasn't finished).
I would really appreciate it if anyone could explain how it's derived or point me to a place that might do so.
Edit: Let me put this one first, it's probably the closest to what you want:
There's some interesting an insightful stuff on the "Interplanetary superhighway" (IPS) (in spite of the dippy sounding name) that takes the chaos theory POV, see for instance
http://www.cds.caltech.edu/~shane/papers/multiple_gravity_assists.pdf. This is more advanced than just the Hill sphere, but if you understand all (or most) of it, you'll understand the Hill sphere too.
Now for the rest of what I said:
If you happen to already know that the radius of the Hill sphere is given by the location of the Lagrange points, you can find the Lagrange point formulas at
http://www.physics.montana.edu/faculty/cornish/lagrange.pdf
If you don't know this, you need to know that the
Jacobi integral is a constant of motion for the restricted 3 body problem. If you want to demonstrate this, compute the Hamiltonian of the restricted 3-body problem (assuming you know what one is and how to compute it - don't want to be insulting but I don't know your background). See also the first link on the IPS.
Knowing that the Jacobi integral function is a constant of motion, if you pick a specific mass ratio and do some plots like those at
http://www.geocities.com/syzygy303/, you can demonstrate for specific mass ratio that there is a closed zero velocity surface that does not permit escape for sufficiently high values of J (low values of H, which is a negative number).
Note the similarity of these plots to those on pg 4 of the IPS paper. If you've got a good plotting program, you might want to generate plots like these for yourself.
A more general proof would require some insight into chaos theory. Unstable equilbrium points are the key here, just as they are for the classic inverted pendulum. And L1 and L2 are unstable equilbrium points.You might also want to look at the following PF threads:
https://www.physicsforums.com/showthread.php?t=198441
https://www.physicsforums.com/showpost.php?p=382232
(the discussion of the Hill sphere starts about post 16 in the second thread).
None of these is really a complete demonstration, but they might give you some insight.
The Interplanetary superhighway is a more advanced concept that's related to the original Hill sphere idea, basically the idea is that the unstable world tubes of bodies escaping from the Lagrange points can intersect. The abstract even mentions the Hill sphere.
