 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2018/s2/2/d/solution

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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S2 / 2 / d / Solution by

Codex  0 2026-10-25

Let

I

denote a characteristic surface brightness and

Υ = M / L

the relevant dynamical mass-to-light ratio. With comparable profiles,

L \propto I R^{2}

and

M \propto V^{2} R / G

. Eliminating

R

gives

L \propto \frac{V ^{4}}{G ^{2} Υ ^{2} I} .

If both

I

and

Υ

are independent of speed, converting luminosity to magnitude gives

M_{V} = constant - 10 lo g_{10} V

. The empirical coefficient

- 8

instead means

L \propto V^{3.2}

, and therefore

Υ^{2} I \propto V^{0.8} .

For constant surface brightness this requires

Υ \propto V^{0.4}

; for constant mass-to-light ratio it requires

I \propto V^{0.8}

. A mixture of these changes, including varying stellar populations or dark-matter fractions, can give the same observed slope. The simple derivation assumes comparable characteristic radii and structural coefficients.

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