Orbits of a photon that can't quite cross the photon sphere of a BH

[pervect]

First, some background. Sorry, it's a bit long.

I've been considering the orbits of photons around a black hole. First some background. My reading indicates that massive objects have two conserved orbital parmeters, a constant energy-at-infinity per unit mass, $\tilde{E}$ and a similar angular momentum $\tilde{L}$. Unlike massive objects, photons only have one conserved parameter, which is basically the ratio of the photon's angular momentum to it's energy-at-infinity (or equivalently up to a factor of c, the ratio of the photon's angular momentum to it's linear momentum as stated by MTW).

The orbital equation of the photon basically states that b in the formula below, where b is the above "impact factor".

$$\left(\frac{1}{r^2} \frac{dr}{d\phi} \right) ^2 + \frac{1-2M/r}{r^2}=1/b^2$$

It can be useful to substitue u=1/r in this equation, as the chain rule makes $du/d \phi$ equal to $(1/r^2) (dr/d \phi).$ I won't be using that here.

The quantity $\frac{1-2M/r}{r^2}$ can be interpreted as an effective potential. This effective potential has a peak at the photon sphere, r=3M.

This background material is from MTW's gravitation, by the way, section $25.6

Onto the question. I am concluding from this, and some popularizations, that any photon that makes a close encounter to the photon sphere, whether said photon is rising from below the photon sphere, or falling from above the photon sphere, but in either case not quite crossing it, can orbit the black hole several times.

The argument is basically that $\frac{dr}{d\phi}$ will be very small as the photon in the 1d "effective potential" model hangs near the photon sphere, giving the photon enough time to orbit the black hole as many times as one desires (given enough fine tuning of the impact parameter).

If it is correct, objects can potentially have many images - in the usual case, we are far away from the event horizion. I gather these are not ordinary "Einstein rings", but have been observed as "photon rings" by the Event Horizon telescope.

Hopefully I don't have to go into the idea of "effective potential". If I were writing an explanation, I'd write more, but since I'm looking for some feedback that I'm on the right track here and not misleading people, it's better to omit them.

 

[Ibix]

That matches my understanding, yes. An Einstein ring in the usual sense is a different phenomenon in the weak field.

Minor point - I don't think the EHT has resolved photon rings. They are clearly visible in the simulations used to generate Gargantua in the film Interstellar, however. Note that both Sag A* and Gargantua are Kerr holes, not Schwarzschild, with $a$ of either ~0.3 or ~1 for Gargantua (modelling vs story) and between <0.1 and 0.93$\pm$0.15 for Sag A* (according to Wikipedia's summary of the literature).