Suppose that the matter (stars, gas, and 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 y of the star is given by v = r √ G M/R³and therefore that the star's period T of revolution is T = 2π√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?