Complete Hill Sphere Derivation: Learn the Equation and its Derivation Process

[Tefeari]

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.

 

[Chris Hillman]

Hill's Equations and Hill Sphere

See Section 3.13 of Carl D. Murray and Stanley F. Dermott, Solar System Dynamics, Cambridge University Press, 1999.

 

[Tefeari]

Yes, thank you. However, that is the one I was talking about from Amazon.com's "See what's inside" feature. It shows me a derivation but does so while referring to equations found elsewhere in the book that arn't visible without me buying it.

If there is another place you could refer me to (especially one that doesn't require money) I would greatly appreciate it.

 

[Chris Hillman]

Can't you get it from your local university library?

BTW, this shows why anyone mentioning a book should take the time to say which book

 

[Tefeari]

I'm sorry, my University's library system does not have it (thanks for the idea though, I just checked). There isn't another library system I can easily access at the moment. If you could suggest another option that would be great, especially one that doesn't require buying or obtaining a particular textbook.

I'm suprised it's not somewhere on the internet, I was hoping there was some place I hadn't checked, or that it would be simple enough for someone to just put the derivation up here. Heck, I'll add it to wikipedia if someone could help me out with this.

 

[pervect]

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.