GRAVITATIONAL PRINCIPLES AND MATHEMATICS

Three cases of photon orbits near a gravitating body

The visual distortion that will be described here would be caused by gravitation in the Schwarzschild metric. Einstein's general relativity is not the only gravitational theory that admits the Schwarzschild metric as an exterior solution for a spherically symmetric, non-rotating gravitational field, but it is the preferred theory, and the theory that will be assumed implicitly here. The Schwarzschild metric is

Here ds is a metric measure of coordinate distance r, coordinate time t and coordinate angles theta and phi. The term R_S, the Schwarzschild radius, refers to the radius of a black hole event horizon, and c refers to the local speed of light. R_S is directly proportional to the mass that creates the metric through R_S = 2GM/c^2, where G is the gravitational constant and M is the mass interior to r.

For a photon, ds^2 = 0. Combining this with the conservation of angular momentum allows one to express the deflection angle phi of a photon moving in a gravitational field as

where b is a constant over the trajectory of the photon path, corresponding to a linear projected impact parameter of a photon at infinity for a photon that escapes. This impact parameter can be visualized by assuming that when the photon is far from the gravitating object it travels in a straight line; the impact parameter is the distance between the closest approach of the continuation of this straight line and the center of the gravitating object. Note that Delta phi is not the extra angle deflected by the lens but the total change in the phi angle between the observer and the source, emitted at radial coordinate r_emitted and observed at radial coordinate r_observed. This angle is measured with the lens at the vertex, and includes gravitational deflection. Therefore, for example, a source seen by an observer just over the limb of a lens which has only a small mass, and hence a negligible effect of the trajectory of the photon, has a Delta phi near pi.

An important radius is found from Eq. (2) when Delta phi diverges to infinity. Here a photon will circle the massive star at the photon sphere. The exact location of the photon sphere is R_P = 1.5 R_S. Note that a "normal" neutron star with a relatively weak external gravitational field does not have a photon sphere. Were it somewhat more compact, it would have a photon sphere, and were it even more compact, it would have an event horizon and be called a black hole. For black holes and the "ultracompact" neutron stars considered below, however, these circular photon orbits can exist.

Photons circling at the photon sphere are not in a stable orbit - any small perturbation will cause them to spiral either in or out. Photons emitted from infinity with impact parameters slightly greater than R_B = 3^(1.5) R_S / 2 will spiral around the compact star near the photon sphere and then spiral out. Photons emitted from infinity with impact parameters slightly less than R_B will spiral around near the photon sphere and then spiral in, eventually colliding with the neutron star surface or falling into the black hole. It is also possible for a photon to be emitted from a ultracompact neutron star surface, orbit near the photon sphere, and then spiral back in again impacting the surface. These describe, in general all of the distinct cases of photon orbit near an ultracompact neutron star. All shorter photon trajectories will lie on one of these paths.