Orbits of Particle Around a Black Hole using Effective Potential

[omegas]

Homework Statement

The possible orbit of a particle moving around a black hole can be described using the effective potential U_GR(r) (in effect, potential energy per unit mass):

U_GR(r) = -GM/r + l²/2m²r² - R_sl²/2m²r³

where the symbols have their usual meaning and in particular R_s = 2GM / c² is the Schwarzschild radius for the black hole. We then use the energy balance equation

K = 1/2 mr² + mU_GR = constant

to illustrate the main features of the possible orbits.

(a) Show that the function U_GR has a minimum and maximum value at r = r_- and r = r₊ respectively and determine expressions for these qualities.
(b) Briefly describe the possible orbits.

Homework Equations

Listed above

The Attempt at a Solution

Please help.

 

[chrisk]

Begin by plotting the effective potential with respect to r and look for turning points.