 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/a/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 / a / Solution by

Codex  0 2026-10-25

At this small redshift,

D ≃ cz / H_{0} ≃ 20.0

Mpc. The outer angular radius gives

R_{out} ≃ D θ ≃ 21.3

kpc. Approximating the dynamical mass by a spherical equivalent,

M_{dyn} (< R_{out}) ≃ \frac{V _{rot}^{2} R _{out}}{G} ≃ 4.9 \times 1 0^{11} M_{⊙} .

The supplied Tully-Fisher relation gives

M_{V} ≃ - 22.49

, and hence

L_{V} ≃ 9.9 \times 1 0^{10} L_{V, ⊙}

. Treating the population as solar-like gives the estimate

Υ_{V} ≃ 1 M_{⊙} / L_{V, ⊙}

, so

M_{luminous} ≃ 1.0 \times 1 0^{11} M_{⊙} .

The dynamical estimate exceeds that by about a factor of five; a massive extended dark-matter component is the natural interpretation. Solar-like spectra do not exactly fix the population mass-to-light ratio, and a disk is not spherical, but order-unity corrections do not remove this large outer mass excess under the stated estimate.

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