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ay45c4-page35 - M1 a Earth θ Moon M2 M2 R M1 M2 CoM So...

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Unformatted text preview: M1 a Earth θ Moon M2 M2 R M1+ M2 CoM So, collecting the three terms in the overall potential , and substituting for !2 (formula in 4.3),   GM1 GM2 1 + a cos  + 1 3 cos2  1 a2 = a RR 2 R2  M2 2 M2 1 G(M1 + M2) 2 + a2 2 2 R3 M1 + M2 R M1 + M2 Ra cos  : Look carefully and you will see that the term in (a=R) cos  exactly cancels out. This is not coincidence: it is because the constant force term is exactly canceled by the centrifugal force that keeps the bodies in a circular orbit. Also note that there are terms with no a or  dependence: Since these are just constants, they produce no gradients and can be ignored. So we get, combining the remaining terms, 2 (a; ) = GM1 1 Ga3 3M2 cos2  + M1 + constant: a 2R This is the local tidal potential near the Earth. To get a better feeling for its shape, let us expand a in terms of height h above the radius of the Earth (\mean sea level"):     2h +    ; a 1 = R 1 1 h +    : a = R + h ; a2 = R2 1 + R   R   94 ...
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