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

A fully convective monatomic perfect-gas star follows an $n=3/2$ adiabatic stellar polytrope. At the photosphere, $\tau =2/3$ and hydrostatic equilibrium in optical depth givesFor an ideal gas on an adiabat,Evaluating this at the photosphere givesThe $n=3/2$ polytropic mass-radius relation is $R\propto K{M}^{-1/3}$, soFinally the Stefan–Boltzmann law givesThis is the specified Hayashi track.

Equate the accretion luminosity to the surface luminosity:Using $R\propto M^3{T}_{\mathrm{eff}}^{-24}$ from part i givesSubstitution into $L\propto M^6{T}_{\mathrm{eff}}^{-44}$ yields the accreting Hayashi track

The specific deuterium-burning energy-generation rate is $ϵ={ϵ}_0\rho T^{15}$. By stellar homology,soSince $L_{\mathrm{acc}}\propto M\dot{M}/R$,Use $R\propto M^3{T}_{\mathrm{eff}}^{-24}$ and $T_{\mathrm{eff}}^{68}\propto M^8/\dot{M}$ from part ii. Then

The two track relations also giveConsequentlyEquality occurs atThe preceding formulae imply $L_{\mathrm{nuc}}>L_{\mathrm{acc}}$ for $L<L_0$, not $L>L_0$. Thus the inequality printed in the question is reversed relative to its requested intermediate scaling; the exponent $964/43$ is consistent.