Question

3. The dark matter halos which form in numerical simulations have density profiles, a(r), which are well approximated by a Navarro Frenk White (NFW) profile Ts Ts where rs, is the scale radius, the radius at which d ln ρ/dln r Show that in this case the rotation curve, v(r), has the form: -2. 1/2 s +r

Answer 1

Form the problem given data p(r) = p_0/(r/r s) (1 + r/rs)^2 now we have to use passion equation rightarrow nabla^2 Phi (r) = 4 pi a P(a) at a = R here theta (r) = -4 pi G P_0 r^2_s/R r_n (1 + R/r^3) with sin theta (R) = 0 R rightarrow a 4 lin theta (R) = 4 pi GM P_0 r^2_s \r\n we have the minimum velocity of escape v_r (r) = [- 2 theta (R)^1/2] v_e (R) = [4 pi G P_0 r^2_s/R ln (1 + R/r)^1/2] above problem graph diagram rightarrow v(R) = (R partial differential theta/partial differential R)^1/2