So I saw a video on youtube: http://youtu.be/uhS8K4gFu4s And so I thought I'd try to understand the whole stable orbit thing. So first you have a simple energy equation of E = K + U. K = (1/2)mv^2 (kinetic energy) U = -GMm/r^2 (gravitational energy) r = radius of orbit v = velocity of orbiting object M = mass of center object m = mass of orbiting object And we want it to be a stable orbit in the first place, so we have: v=sqrt(GM/r) Now, let's say we slightly bump the object in orbit. So r -> r + s, where s is much less than r. (also w is much less than v) So we can adjust the stable orbit equation: v=sqrt(GM/r) v+w = sqrt(GM/(r+s)) approximate for small distances: v+w = sqrt(GM/r)(1-s/(2r)) So then, you can subtract the original equation out and have: w = -sqrt(GM/r)s/(2r) This makes sense directionally, if you bump the orbiting object inwards, the velocity with increase. So now, back to the energy equation: E = K + U Since it's a stable orbit being bumped only slightly, you expect it to be able to eventually return to the same state, so E can't change. So initially you have: E = (1/2)mv^2 - GMm/r^2 Then apply the bump: E' = ((1/2)mv^2 )(1+2w/v) - (GMm/r^2 )(1-s/r) This then reduces to the form: E' = E + mvw + (GMm/r^2 )s Since E' must equal E, the two extra terms should add to zero: E' = E + (mv)sqrt(GM/r)(-s/(2r)) + (GMm/r^2 )s E' = E + (-1/2)(GMm/r^2 )s + (GMm/r^2 )s So I am doing something wrong. I have a missing factor of two somewhere, but I don't know why. If you guys can help, it will be appreciated. Thanks.