What would be required to cause photons to orbit an object?
Well certainly electromagnetism can't interact with it because well... photons carry electromagnetism. Don't know about the strong or weak nuclear forces though.
George Jones has already given the answer, if only you'd realise it, in post #7.Seems like the gravity to hold a photon in orbit around a black hole should be known? Is it?
Right. Photons cannot hold elliptical orbits.Because a regular mass can change speed. If it moves closer to the other object, the gravitational force will increase but its speed will also increase, so it won't end up crashing into the massive body. Photons can't increase or decrease speed, so if a photon is not at the ideal radius, then it won't get into a circular orbit because the attraction to the massive body is increased at a radius smaller than the circular orbit radius.
Why would a photon's orbit be any less stable than a regular mass orbiting a regular body?
As an aside, all orbits of objects of mass decay under GR
.I'm not sure what you mean.
Anything that makes the path of least time in the form of a circle. I wonder if that's even possible since then it becomes essentially infinite length.