Suppose that the matter (stars, gas, dust) of a particular galaxy, of total mass M, is distributed uniformly throughout a sphere of radius R.A star of mass m is revolving about the center of the galaxy in a circular orbit of radius r < R.
(a) Show that the orbital speed v of the star is given by
v = r (GM/R3,
And therefore that the star's period T of revolution is
T = 2((R3/GM,
Independent of r. Ignore any resistive forces.
(b) Next suppose that the galaxy's mass is concentrated near the galactic center, within a sphere of radius less than r. What expression then gives the star's orbital period?