 Solution

ID: past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2021/6/a/solution

For a spherical mass distribution, the shell theorem says the circular gravitational acceleration is

GM (< R) / R^{2}

. Balancing it against the centripetal acceleration

V_{c}^{2} / R

gives

M (< R) = \frac{R V _{c}^{2} ( R )}{G} .

R = 16 kpc = 4 r_{0}

V_{c} = 200 (1 - e^{- 4}) = 196.337 km s^{- 1}

. With

R = 4.93708 \times 1 0^{20} m

M (< 16 kpc) = 2.8515 \times 1 0^{41} kg = 1.4340 \times 1 0^{11} M_{⊙} .

This is the total dynamically inferred enclosed mass, including any dark matter; the galaxy rotation curve alone does not separate its components.

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