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
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 Fall '09
 dion
 Force, m1+ m2, Moon M2

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