Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 317 / 3 / i / Solution

A fully convective monatomic perfect-gas star follows an

n = 3/2

adiabatic stellar polytrope. At the photosphere,

τ = 2/3

and hydrostatic equilibrium in optical depth gives

P_{ph} \sim \frac{g}{κ _{ph}} \propto \frac{M}{R ^{2}} T_{eff}^{- 19/2} .

For an ideal gas on an adiabat,

K = \frac{P}{ρ ^{5/3}} \propto T^{5/3} P^{- 2/3} .

Evaluating this at the photosphere gives

K \propto M^{- 2/3} R^{4/3} T_{eff}^{8} .

The

n = 3/2

polytropic mass-radius relation is

R \propto K M^{- 1/3}

, so

R \propto M^{3} T_{eff}^{- 24} .

Finally the Stefan–Boltzmann law gives

L = 4 π R^{2} σ T_{eff}^{4} \propto M^{6} T_{eff}^{- 44} .

This is the specified Hayashi track.

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