The mass density of a spherical galaxy varies as K r over a large distance r from its center. In that region, a small star is in a circular orbit of radius R . Then the period of revolution, T depends on R as:

Step-by-step solution

M = ∫ ρ d V M = ∫ 0 R k r 4 π r 2 d r M = 4 π k ∫ 0 R r d r M = 4 π k R 2 2 = 2 π k R 2 F G = G M m R 2 = m ω 0 2 R ⇒    G 4 π k R 2 2 R 2 = ω 0 2 R    ⇒ ω 0 = 2 π k G R ∵ T = 2 π ω 0 = 2 π R 2 π k G = 2 π R k G ⇒ T 2 ∝ R