Photon sphere

M87* By Event Horizon Telescope

I come across the photon sphere (or last photon orbit) which is another radius around a Schwarzschild black hole at a distance ##3R_s/2##. It is the closest distance for a stable orbit and light would orbit there in an exact circle. I would like to calculate it using Carroll conventions. ##R_s=2GM## is the Schwarzschild radius.

There is a Wikipedia article on it. The proof it gives has some peculiarities. I followed it avoiding those. It is one page.
Here: Commentary 5.4 Photon Sphere.pdf 

The famous picture is of M87* the black hole in our neighbour galaxy Messier 87. The black central sphere has a radius of ##2.6R_s## so the photon sphere is well inside that black bit and this is not a good illustration of a photon sphere!