Photon sphere
A photon sphere is a spherical region of space where gravity is strong enough that photons are forced to travel in orbits. The radius of the photon sphere, which is also the lower bound for any stable orbit, is:
(where r is the radius, in meters, G is the gravitational constant, M is the mass in kg, and c is the speed of light in vacuum) which is one and one-half times the Schwarzschild radius.
This equation entails that photon spheres can only exist in the space surrounding an extremely compact object (a black hole or possibly a neutron star).
As photons approach the event horizon of a black hole, those with the appropriate energy avoid being pulled into the core of a black hole by traveling in a nearly tangential direction known as an exit cone. A photon on the boundary of this cone does not possess the energy to escape the gravity well of the black hole. Instead, it orbits the black hole. These orbits are rarely stable in the long term.
The photon sphere is located farther from the center of a black hole than the event horizon and ergosphere. Within a photon sphere, it is possible to imagine a photon that begins at the back of your head, orbiting the black hole, only then to be intercepted by your eyes, allowing you to see the back of your head. For non-rotating black holes, the photon sphere is a sphere of radius 3/2 Rs, where Rs denotes the Schwarzschild radius (the radius of the event horizon) - see below for a derivation of this result. There are no stable free fall orbits that exist within or cross the photon sphere. Any free fall orbit that crosses it from the outside spirals into the black hole. Any orbit that crosses it from the inside escapes permanently. No unaccelerated orbit with a semi-major axis less than this distance is possible, but within the photon sphere, a constant acceleration will allow a spacecraft or probe to hover above the event horizon.
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