Whats the difference between the Photon Sphere and the Marginally bound orbit? Whats the difference between the Photon Sphere and the Marginally bound orbit?
Why does photon sphere have a radius of 1.5Rs, while the Marginally bound orbit has a radius of 2Rs?
 A: Both the light ring (photon orbit) and the marginally bound circular orbit, are cases of unstable circular orbits. The light ring is a null orbit, i.e. the orbit followed by a massless particle (such as a photon). Any massive particle would need infinite energy to follow this orbit.
The marginally bound circular orbit on the other hand is defined as the unstable circular orbit whose energy is equal to the rest mass of the particle. This means that if the orbit is perturbed towards infinity, the particle would reach infinity with zero velocity. (A particle escaping the light ring on the other hand would reach infinity travelling at the speed of light.)
So, why is the radius of the light ring smaller than that of the marginally bound circular orbit? One way to see this is to note that the energy of unstable circular orbits increases as the radius of the orbit decreases. Since infinity is bigger than a finite energy, the marginally bound circular orbit must lie outside the light ring. (Alternatively we could have noted that the light ring is the smallest possible circular orbit.)
The actual values (2Rs and 1.5Rs) for these orbits are simply how these work out for a Schwarzschild black hole. If one were to add spin or charge to the black hole the radii of these orbits would change. However, it would still be true that the marginally bound circular orbit lies outside the light ring.
