Edit: People are confusing [this concept](https://en.m.wikipedia.org/wiki/Innermost_stable_circular_orbit) with the [Roche limit](https://en.m.wikipedia.org/wiki/Roche_limit). Wikipedia isn’t very helpful in explaining why the innermost stable circular orbit exists, but it does lay out the basic concept. It’s the innermost point at which an object can orbit another object at all, not the point at which it begins breaking up from tidal forces.
In: 5
So, with GR we see that objects in elliptical orbits tend to hang around the center more than in newton’s version. They curve around at a tighter radius than would be expected causing a procession. If you get just the right eccentricity at a low enough energy, you can get an object to “process” multiple revolutions near the center before being flung back out. If your orbit was slightly more eccentric, then it’ll instead just keep curving inward and fall into the black hole.
So that’s an upper bound on the eccentricity for a stable orbit at a given energy. Any higher and the orbit will dip too low and it gets sucked in. The lower bound would be 0: a circular orbit at that energy. As you keep getting closer to the minimum stable circular orbit the maximum eccentricity keeps getting smaller, until at the minimum energy, only a circular orbit is stable. Any eccentricity at all would cause it to dip too low and get sucked in.
In theory, you could orbit lower than this if you had a control system that actively stabilized your orbit; if you start dipping too low fire the rockets to try to get out (but then you’ll be going too fast once you get back to your orbital height so you’ll need to fire the retro rockets to slow back down)
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