 Solution

ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2022/s02/2/b/solution

Write

m (r)

d P / d r = - G m ρ / r^{2}

. If radiation pressure dominates, compare it directly with the radiative diffusion result for

d P_{r} / d r

\frac{κ ρ L}{4 π r ^{2} c} ≃ \frac{G m ρ}{r ^{2}}, L ≃ \frac{4 π G c}{κ _{th}} M \propto M .

This is the limiting Eddington luminosity scaling, assuming fixed composition and electron-scattering opacity.

ρ \sim M / R_{*}^{3}

P \sim G M^{2} / R_{*}^{4}

. The ideal gas law then gives

T \sim \frac{μ m _{H}}{k _{B}} \frac{GM}{R _{*}} .

Estimating

∣ d T / d r ∣ \sim T / R_{*}

L \sim \frac{a c}{κ} \frac{R _{*} T ^{4}}{ρ} \propto \frac{μ ^{4} M ^{3}}{κ} .

Thus at fixed

μ

L \propto M^{3}

. These are stellar homology estimates for radiative stars, not exact relations valid for arbitrary composition and internal transport.

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