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Dark matter; C&O 24.22. (a) Assume that the density of dark matter in our Galaxy is given...

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Physics

Dark matter; C&O 24.22.

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

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

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