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Resonant Orbits Besides spin

# Resonant Orbits Besides spin - more details Loss of...

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Resonant Orbits Besides spin-orbit resonances, evolution of satellite orbits in a multi-satellite system can lead to orbit-orbit resonances between different satellite (or vetween satellites and ring particles for planets with rings). Thus Io, Ganymede and Europa have resonant orbital periods. We can understand this if we realise that the primary-satellite tidal torques will cause the satellite orbits to evolve gradually. Eventually they will evolve (randomly) until two satellites have resonance periods. Assuming for simplicity that one is in a circular orbit: Whenever the satellites come near to conjunction the outer satellite will get a "tug" in the same direction on every orbit. The overall effect of these increments - which will be in one direction on one side of the orbit and in the other direction on the other side - will be toe other side - will be to stabilise the outer satellite into a resonant period with the inner satellite. (See Lewis for
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Unformatted text preview: more details). Loss of satellite to a perturbinof satellite to a perturbing body If we have a satellite mass m, at a distance d from its primary mass M 1 , then a second large body of mass M 2 distance D from M 1 produces a perturbing acceleration on m of: A - B = [GM 2 /(D-d) 2 ] - GM 2 /D 2 or approximately 2GM 2 d/D 3 between orbiting body and primary (d << D). The gravitational aceleration b is given by: b = GM 1 /d 2 Thus, the instability limit, when the perturbation starts to be of the same order as the primary's effect, is given by: d = (M 1 /2M 2 ) 1/3 D as long as M 1 << M 2 . If M 1 >> M 2 we must use the exact relation: d 3 (2D-d) = (M 1 /M 2 )D 2 (D-d) 2 If we take the earth-moon system with the sun as perturber then d is about 1.7 10 6 km, four times the current orbital radius....
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