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Ay45c4-page40 - 4.3.3 Roche stability limit for satellites Let us now apply the previous result for the tidal potential(slight change in notion 1 3

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Unformatted text preview: 4.3.3 Roche stability limit for satellites Let us now apply the previous result for the tidal potential (slight change in notion), 1 3 (r; ) = GM1 2 Gr3 3M2 cos2  + M1 + constant r R to the case where M1 is a small satellite orbiting close to a large parent (star or planet) M2, so M1  M2. Satellite: M2 B r M1 A z axis R It turns out that if R is too small, the satellite is torn apart by tidal forces. To see this, let's calculate the gravitational acceleration at points A and B in the gure. By symmetry it must be along the z axis, so   d GM 1 Gz2 (3M ) gz = rz  = dz jzj 2 R3 2   GM1 z + z 3GM2 = Gz M1 + 3M2 : = jzj3 R3 jzj3 R3 This is a restoring force only if the coecient of z is negative. Otherwise the force is away from the center of the satellite and the satellite is torn apart, starting at its surface. Putting jzj = r and considering a satellite of mean density , so 4 M1 = 3 r3 99 ...
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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.

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