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 / S26 / 2 / b

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- Solution b

Solution

Use characteristic stellar values in the supplied radiative temperature gradient:

T_{c} / R \sim κ ρ L / (R^{2} T_{c}^{3})

. Hence

L \propto T_{c}^{4} R / (κ ρ)

. With Kramers opacity,

κ = κ_{0} ρ T_{c}^{- 3.5}

,

L \propto κ_{0}^{- 1} \frac{T _{c}^{7.5} R}{ρ ^{2}} \propto κ_{0}^{- 1} \frac{T _{c}^{7.5} R ^{7}}{M _{s}^{2}} .

Hydrostatic support by an ideal gas gives

T_{c} \propto μ M_{s} / R

, or

R \propto μ M_{s} / T_{c}

. Substitution then yields

L \propto κ_{0}^{- 1} μ^{7} M_{s}^{5} T_{c}^{1/2} .

The result assumes homologous density and temperature profiles, comparable structural constants, gas-pressure support and radiative transport throughout. It is not simultaneously an electron-scattering-opacity result; replacing Kramers opacity changes the exponents. Convection, degenerate cores or a different opacity regime limit quantitative application to real evolved helium stars.

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