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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/s2/1/b/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 / 1 / b / Solution by

Codex  0 2026-10-25

Integrating

d P / d r = - G M_{r} ρ / r^{2}

in order of magnitude, with

ρ \sim M_{*} / R_{*}^{3}

, gives

P_{c} \sim \frac{G M _{*}^{2}}{R _{*}^{4}} .

The ideal gas law independently gives

P_{c} \sim M_{*} k_{B} T_{c} / (μ m_{H} R_{*}^{3})

. Equating the two estimates yields

R_{*} \sim \frac{G μ m _{H}}{k _{B} T _{c}} M_{*} .

For similar composition and an approximately fixed hydrogen-burning central temperature,

R_{*} \propto M_{*}

. Calibrating this rough relation on the Sun and using part a gives a nominal mass range of about

0.05

to

5 M_{⊙}

.

The endpoint luminosity ratio is

1 0^{8}

, whereas the inferred mass ratio is about

1 0^{2}

L \propto M^{4}

across that range. The exponent is therefore consistent at this schematic level. The solar normalization and individual endpoint masses are much less reliable: using the luminosity relation itself gives approximately

0.1

to

10 M_{⊙}

. Central temperatures are not truly constant, low-mass stellar structure differs, and the uncorrected

V

luminosities in the diagram are not bolometric. These limitations prevent treating the rough radius-based masses as a contradictory precision measurement.

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