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Unformatted text preview: maintain a constant position in the rotating frame (that is, co-orbit with the
same period as the two main bodies at constant position relative to them)?
The condition for such an equilibrium is that there be no acceleration
at the chosen position, that is, the gradient of the potential (including
centrifugal term) must vanish, r = 0 :
Gradients vanish, in general, at extrema and at saddle points. In the contour
plot you can see that there are 3 saddlepoints co-linear with masses M1 and
M2. It is easy to see that there must always be these three (for any mass
ratio) as the following graph illustrates:
force due to M1
L3 L2 L1 A x B
due to M2 You can see that gravitational force and (negative) centrifugal force must
always cross at exactly three points, where Fgrav = Fcentrif or Fgrav + Fcentrif = 0 :
These three points, L1; L2; L3 are the rst three \Lagrange points" where
a test mass can orbit. However, since they are saddle points (see contour
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This note was uploaded on 12/29/2011 for the course AST 350 taught by Professor Dion during the Fall '09 term at SUNY Stony Brook.
- Fall '09