ay45c4-page32 - of the other body For the Earth-Moon...

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Unformatted text preview: of the other body. For the Earth-Moon system (e.g.) this causes the surface of the oceans (and to some extent the solid Earth as well) to be deformed into the familiar tidal shape: least gravitational attraction greatest gravitational attraction Earth Moon tidal deformation produced Our goal is to understand the above picture quantitatively. The surface of the ocean is an equipotential, because if it were not | that is, if there were a potential gradient along the surface | there would be a force causing the water to ow \downhill" until it lled up the potential \valley." But is it a purely gravitational potential we must consider? No, because the \stationary frame" in which things can come to equilibrium is the rotating frame in which the Earth and Moon are xed. (We are here assuming a circular orbit. Things would be more complicated if the orbit were signi cantly eccentric.) So there is also a centrifugal force and corresponding centrifugal potential. If the orbital angular velocity is !, with !2 = G(M1 + M2) ; R3 then the centrifugal acceleration at a position r with respect to the CoM is !  (!  r), so the centrifugal potential (whose negative gradient is the 1 acceleration) is 2 j!  rj2 or, in the orbital plane, 1 !2r2. The total potential 2 in the rotating frame is thus 1 1 2  = jrGMr j jrGMr j 2 !2r2 : 1 2 91 ...
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