Photon Sphere

Radio emission from the accretion disk surrounding the supermassive black hole M87* (captured 2017, computed 2019) as imaged by the Event Horizon Telescope. The photon sphere lies within the dark shadow (which has a radius of approximately 2.6 times the Schwarzschild radius.

A Schwarschild radius is the distance from a Black hole's singularity (infinitely dense center of a black hole) to it's Event horizon (outermost part of a black hole). We know that light is influenced by the gravity. So when the photons come near the black holes, some of them fall into it, while some are not as close to the black hole so they are not consumed by it but their trajectory changes due to the immense gravity of the black hole. The third condition is utterly different, the photon as it comes near the black hole at a particular distance which is 3/2 or 1.5 Schwarschild radius, it literally starts to orbit the black hole. 

Now, the equation below shows that Photon Spheres can only form in space around objects which are extremely sense or compact and have an ultra-strong gravitational pull i.e.(Black holes or some very dense Neutron Stars.)

where G is the gravitational constant, M is the black-hole mass, and c is the speed of light in vacuum and r_s is the Schwarzschild radius (the radius of the event horizon).

For a better understanding we have an example.

Imagine you are at the point of Photon Sphere.

As we know that at 1.5 Schwarschild radius the light orbits the black hole. So you can actually see your head at that point as the light has orbited the black hole and reached you by circling around the black hole. 

One more important fact is that if there is a Photon Sphere around the an active black hole, the Accretion Disk of that black hole will always exist at a distance more than 1.5 Schwarschild radius because if anything except photons orbit at 1.5 Schwarschild radius will fall into the black hole because they are not at the speed of light. And if anything including light get closer than 1.5 Schwarschild radius to a black hole will fall into black hole.

Now, let's see how to find the distance of 1 Schwarschild radius from the center of black hole.

If a black hole is having a mass of 5M ☉ then the method of calculating it's Schwarschild radius will be (5km)×(5)= 25km. Hence, the Schwarschild radius of the black hole in of 25km.

Now, to find where will be the Photon Sphere of the black hole with 5M ☉ we will use this formula.

(3/2)(5km)(5)= 37.5km. Hence, the Photon Sphere will be 37.5 km far from the Singularity of the black hole..