gravity - Can a photon be made to orbit a known (or undiscovered theoretical) body?

Can a photon through some process be made to orbit a celestial or any other object?

Two follow-up questions.

1. 
   

   Can this orbit be described as the photon crossing its own path.

   
   


2. 
   

   Will this wave-function be effected by positive interference. To the effect of increasing frequency?

   
   

Answer

In short, yes. But there are 2 caveats for the orbit for a massless particle around a spherical body. Both can be seen from the following plot, borrowed form "Spacetime and Geometry" by Carroll pg.211:

i) There is only one possible orbit for $r = 3 GM = \frac{3 R_{s}}{2}$ where $R_s$ is the Schwarzschild radius. This orbital radius could very well be within the radius of the body being orbited, hence the particle may not even be able to get this trajectory in the first place.

ii) The orbit is unstable.

EDIT: As for the updated, additional questions:

First of all, the above analysis was done for a classical particle, so there is no notion of waves or interference there. In this sense, the photon will cross its own path the same way my dog crosses his own path when he runs in a circle.

If you really want to treat the photon quantum mechanically, I can only take a stab at it under some idealized circumstances. In one idealization we could take the photon state is very localized (high probability of detecting it in small range, virtually zero everywhere else). We can send this photon once around in the orbit - in this case it will act a lot like a classical particle in that it won't interfere at all with the 'tail' of the wavefunction from the prior pass. (That we can set up states like this is merely an intuitive notion on my part since I know we can set up single photon states and there is little chance of detecting them at say, Alpha Centauri, so localized states of a single photon seem to be possible). Now for the other idealization, we could take a state that isn't very localized at all, say a plane wave directed tangentially to the orbit. Now that the wave function isn't localized it's going to be able to 'feel' all around the space and I think its going to want to 'fall off' the trajectory where it can be at a state of lower energy. That is, over a long period of time its going to sense the instability and have a very small probability of being detected at along the radius at $ 3 G M$ rendering the question sort of moot. One thing you could do is stand on one side of the body being orbited and shoot a photon towards it and have your buddy on the other side detect it, which is pretty much the double slit experiment and will result in interference (the 2 paths are going left or right of the body being orbited). (I am more posting this part of my answer as a guess to see what other people think and less out of certainty).

I want to emphasize point (ii) above - something that is unstable is a bit like having a needle balanced on its tip - sure you can do it for a split second but the slightest movement will make it fall over so really no matter how many special circumstances you want to invent or questions you want to ask about the photon orbiting its just not going to stay there very long.