# Wolfgang Tichy

## Orbits around Black Holes

The orbit of a test particle around a black hole of mass M and spin S
does not depend on the particle's mass. Rather it is
determined by the three constants of the motion E, Lz and C.
Here E is the total energy per rest mass, Lz is the z-component
of the angular momentum per rest mass and C is the Carter constant.
In order to understand these constants let's look at the Newtonian limit.
In this limit and using units where G=c=1,
we would have
E = 1 + (1/2) [ (dr/dt)^2 + L^2/r^2 ] - M/r ,

where r is the distance form the center and where

L^2 = Lz^2 + C

is the total angular momentum.

Below are examples of geodesic orbits around a black hole. The black
hole is located at the center of the coordinate system.

A Newtonian orbit (just for comparison)

Orbit around non-rotating (or Schwarzschild) black hole

Another Orbit around non-rotating (or Schwarzschild) black hole

Equatorial orbit around rotating (or Kerr)
black hole

Generic orbit around rotating (or Kerr) black
hole