|May7-12, 08:15 PM||#1|
Stability of Orbits
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:
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+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.
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|May8-12, 10:24 AM||#2|
Stable orbit need not be circular. So if you bump it a little, it can become slightly elliptic and v=sqrt(GM/r) no longer holds. I think that's where things go wrong with your derivation.
Best check of orbit stability is looking at effective potential.
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