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

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Past exam of the astrophysics course of the University of Oxford / 2021 / 6 / c / Solution by

Codex  0 2026-10-25

At large radius

V_{c} \to V_{0}

, so the spherical enclosed mass is

M (< r) ≃ V_{0}^{2} r / G

. A spherical mass density satisfies

d M / d r = 4 π r^{2} ρ (r)

, and hence

ρ (r) ≃ \frac{V _{0}^{2}}{4 π G r ^{2}} \propto r^{- 2} .

Equivalently, a local power law

ρ \propto r^{- p}

with enclosed mass dominated by that extended profile has

M \propto r^{3 - p}

and

V_{c}^{2} \propto r^{2 - p}

. A flat galaxy rotation curve requires

p = 2

.

Differentiating the exact mass profile verifies the limit and the finite central density:

ρ (r) = \frac{V _{0}^{2}}{4 π G r ^{2}} [(1 - e^{- x})^{2} + 2 x (1 - e^{- x}) e^{- x}], x = r / r_{0} .

x \to 0

, the bracket is

3 x^{2} + O (x^{3})

, so

ρ (0) = 3 V_{0}^{2} / (4 π G r_{0}^{2})

x \to \infty

, it tends to one, recovering the

r^{- 2}

tail. Extending the flat galaxy rotation curve to infinity makes total mass diverge linearly; a real finite halo must eventually depart from this idealization.

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