Question

Dark matter; C&O 24.22.

(a) Assume that the density of dark matter in our Galaxy is given by ρ(r) = ρ0/(1+(r/a) ^2) . Show that the amount of dark matter interior to a radius r is given by: Mr = 4πρ0a^2 [ r − a tan^−1 ( r/a )].

(b) If 5.4 × 10^11 M⊙ of dark matter is located within 50 kpc of the Galactic centre, determine ρ0 in units of M⊙ kpc−1 . Repeat your calculation if 1.9 × 10^12 M⊙ is located within 230 kpc of the Galactic centre. Assume that a = 2.8 kpc.

Answer 1

a)

Given,

Then the amount of dark matter interior to a radius is given by

  

  

Now put

Thus

and

Also the limit varies from to . Therefore,

  

  

  

  

Hence proved.

b)

Given

and

Using the expression

  

  

When and ,