ay45c4-page34 - D a θ r1 R Moon sub-lunar point Earth GM2...

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Unformatted text preview: D a θ r1 R Moon sub-lunar point Earth GM2 jr r2 0 CoM r2j = = GM2 = GM2 2 + a2 2Ra cos )1=2 D (R   GM2 1 + a2 2 a cos  1=2 : R 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 : =R R 2 R2 R (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. θ a 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  : 2 2 M1 + M2 M1 + M2 ...
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