Consider a galaxy consisting of a very large number N of identical stars, each of mass...

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Consider a galaxy consisting of a very large number N of identical stars, each of mass m, distributed uniformly as a very thin disk with radius R. The galaxy also has a concentric spherical halo of dark matter, with a finite but very large radius and uniform mass density p. The stars only interact gravitationally with the dark matter and each other. (a) If the star orbits are perfectly circular, find the speed v and the angular momentum (about the center of the galaxy) for each star, as a function of the star's distance from the center r and N,m, R, p, and G (Newton's constant). (b) Now consider star orbits that can be non-circular. The equation of motion of r for each star with angular momentum is the same as that of an equivalent 1-dimensional problem. For that problem, find the effective potential as a function of r. Your answer may depend on N,m, R, p, G, and l. (c) Consider a star with a nearly circular orbit r(t) = 0 + (t), where ro is constant and e(t) is very small. Find an expression for the frequency of oscillations for . Consider a galaxy consisting of a very large number N of identical stars, each of mass m, distributed uniformly as a very thin disk with radius R. The galaxy also has a concentric spherical halo of dark matter, with a finite but very large radius and uniform mass density p. The stars only interact gravitationally with the dark matter and each other. (a) If the star orbits are perfectly circular, find the speed v and the angular momentum (about the center of the galaxy) for each star, as a function of the star's distance from the center r and N,m, R, p, and G (Newton's constant). (b) Now consider star orbits that can be non-circular. The equation of motion of r for each star with angular momentum is the same as that of an equivalent 1-dimensional problem. For that problem, find the effective potential as a function of r. Your answer may depend on N,m, R, p, G, and l. (c) Consider a star with a nearly circular orbit r(t) = 0 + (t), where ro is constant and e(t) is very small. Find an expression for the frequency of oscillations for .