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There are 4 configurations to consider. A satellite orbiting a planet that:

  • Has a prograde orbit and larger orbital period than its planet's rotational period (example: The Moon)

  • Has a prograde orbit and smaller orbital period than its planet's rotational period (example: Phobos)

  • Has a retrograde orbit and larger orbital period than its planet's rotational period (example: Triton)

  • Has a retrograde orbit and smaller orbital period than its planet's rotational period (example: N/A)

For the 1st example, the planet's rotation will slow down and the satellite will receed into a higher orbit. This will keep happening until both the planet and satellite are tidally locked to each other. (Tidal acceleration)

For the 2nd example, the planet's rotation will speed up and the satellite will spiral into the planet. (Tidal deceleration)

For the 3rd example, the satellite will spiral into the planet but the planet's rotation will also slow down. Angular momentum is conserved because the orbit is retrograde. (Tidal deceleration)

What would happen in the 4th example? Is it a "negative times negative gives a positive" situation and we refer to the 1st example?

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In the first case, the planet's rotation will carry the tidal bulge ahead of the satellite, this will tend to accelerate the satellite, causing its orbit to recede. In the other cases, the planet's rotation will pull the bulge behind the satellite, causing the orbit to decay. It doesn't matter if the period is greater or smaller in the case of a retrograde orbit. The bulge is behind the satellite, and so the orbit will decay.

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  • $\begingroup$ And what about the planet itself? Will its rotation speed up or slow down? $\endgroup$ – user267545 Aug 4 '20 at 13:03
  • $\begingroup$ The planet will be slowed down. Eventually it could be stopped and then the rotation reversed. The moon and the planet will evolve towards tidal locking. $\endgroup$ – James K Aug 4 '20 at 13:14

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