Schwarzschild black hole orbits spacecraft

Question

A spacecraft is in a circular orbit around a schwarzschild black hole of mass M.

a) find expressions for the orbital period as a function of r determined by an observer at rest at infinity and an astronaut on the spacecraft

b) Another spacecraft is hovering with fixed θ and Φ coordinatees at a point on the spacecraft orbit. What is the orbital period of the first spacecraft from the pov of the astronaut on this craft?

Answer 1

Hi Lena,

 

Gravity is gravity is gravity (as long as you are outside the black hole).  

 

Part a)

 

In a circular orbit the gravitational force causes the centripetal acceleration; additionally v =2πr/T.  One can then write

 

GMm/r² = mv²/r and substituting in for v  find that  T² = (2π)²r³/(GM)     (one of Kepler's laws)  

 

This will certainly work for our observer at infinity.

 

(sorry for the interrupt, I need to take a break here)