PHY312 - Homework 10
1. Consider a satellite in a circular orbit around a black hole at r-coordinate ro.
(a) What is the angular momentum per unit mass in terms of vshell ? (Hint: the
answer is not rovshell)
(b) The satellite does one orbit and returns to its initial position. This distance is
2πro. How much time does this take on a stationary shell clock (in terms of vshell)
?
(c) What is the time taken as measured by a clock on the satellite ? (Hint: this is
the proper time elapsed i.e this question is asking about time dilation)
(d) How much time would the orbit take as recorded by a “far-away” observer? (Hint:
use the relationship between dt and dtshell)
2. From the formula given in class for the radius of stable circular orbits show that the
smallest stable circular orbit occurs for r= 3rSwhere rSis the Schwarzschild radius.
There are no stable orbits closer to the black hole than this.
3. A satellite is fired at 900to the radial direction at r-coordinate r0=8GM
c2at a local
(shell) speed of v0=3c
4. What is the angular momentum per unit mass and energy
per unit mass of the satellite ? Will this satellite escape to great distance ? (Hint:
plot a picture of the effective potential for this satellite and draw on this picture the
satellite’s energy)
4. Write down an expression for the effective potential for photons in a Schwarzschild
geometry. Find the radius of unstable circular orbits in terms of GM/c2(the radius of
the photon sphere). If the impact parameter b=6GM
c2will the photon be captured by
the black hole ?