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Presume there is a satellite orbiting the Earth in an orbit that follows a closed path around the planet (that is, escape orbits are not permitted here). As I understand it, there are two possibilities, ignoring the massive timescales that this might require:

  1. The orbit can decay due to the action of the atmosphere
  2. The orbit can expand until the satellite eventually leaves orbit. As I understand it this happens with larger bodies, like moons, around planets, but not satellites. I also could be wrong here.

Considering the $\langle x, y, z \rangle$ coordinates of the spacecraft in a suitable coordinate frame and their associated derivatives, in case 1 above the orbit would be stable in the sense that it decays to the origin of the state space, if the Earth were a point mass centered at the origin. However, it isn't. Would such an orbital decay still be Lyapunov stable?

Is case 2 possible? In case 2 the orbit would be unstable in the sense of Lyapunov and just about everything else, correct?

Is there a 3rd case in which the orbits are actually stable in the sense that the coordinates in the state space are bounded and never zero?

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  • $\begingroup$ Concerning Mechanism #2: Moons can migrate outwards due to their tidal interactions with their planet, but this process is usually self-limiting, stopping once the planet becomes tidally locked with the moon. An example where this has happened is the Pluto-Charon system. (Not sure how/whether this applies to gas giants, though.) $\endgroup$ Commented Jul 23, 2020 at 19:56
  • $\begingroup$ Also, mechanism #3: inspiral due to gravitational wave emission. $\endgroup$ Commented Jul 23, 2020 at 19:56

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