Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 10; Problem 10.9.
Suppose we are unfortunate enough to be orbiting a supermassive black hole with a mass of solar masses. For the sun, , so for the black hole, . The radius of the orbit is . For comparison, the average earth-sun distance is , so this orbit is about 10 times closer to the black hole than Earth is to the sun. A rather frightening prospect.
From its radius, we can work out the energy and angular momentum per unit mass. From the formula for angular momentum as measured by the orbiting object:
The energy is
The period of the orbit as measured by the spacecraft can be found from