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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 coecient 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
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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.
 Fall '09
 dion

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