Unformatted text preview: D
a θ r1 R Moon sub-lunar point Earth GM2 jr r2 0
CoM r2j =
= GM2 =
2 + a2 2Ra cos )1=2
GM2 1 + a2 2 a cos 1=2 :
R2 R We need to do the binomial expansion to second order to consistently pick
up all terms of order a2=R2 . This gives
GM2 1 + a cos + 1 (3 cos2 1) a2 + O a 3 :
(If you know about multipoles, you will see the Legendre polynomials in
cos lurking here.) Note that the term (a=R) cos actually varies linearly in
the z direction from M1 to M2, since z = a cos , so the gradient of this part
of the potential is a constant force.
z Now, for the centrifugal potential term, we again apply the law of cosines,
1 !2r2 = 1 !2 M2 R
2 + a2 2 M2 R a cos :
M1 + M2
M1 + M2 ...
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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