Shouldn't all moon orbits be inherently unstable? A hypothetical simulation starts with a planet called Earth in a perfectly circular orbit around a star called Sol. There is also a moon called Luna in a perfectly circular orbit around Earth.

Question:
Can Luna's orbit remain perfectly circular or does it become elliptical over time? At what point does it become stable? Never?

Thinking:
The distance between Sol and Luna is not constant, it fluctuates as Luna orbits Earth. Therefore the pull sunward also fluctuates. Shouldn't this cause all moon orbits to eventually deteriorate into longer and longer elliptical orbits until they collide with the planet they orbit?
The answer to your simple question is that while a circular orbit could remain in the absence of any other gravitational fields, in reality you'll have an ellipse and it can be stable from the beginning, remember stable has a wide range of values.

If you allow drag through tidal forces etc then the orbit will change, eg our Moon is slowly getting further away from us, but we still consider this stable over realistic timeframes.

Additionally, the actual orbit will track out a sort of helix as the planet travels round the sun.