Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 317 / 2

For a homologous stellar sequence, mass conservation, hydrostatic equilibrium, and the ideal gas equation giveIntegrating the given stellar energy-generation rate over a fixed dimensionless profile givesTherefore

The radiative diffusion in a star equation gives the homology scalingUsing the central scalings from part (i) and the electron-scattering opacity $\kappa \propto 1+X$ yieldsThe radius cancels because an electron-scattering opacity is independent of density and temperature.

A fully convective monatomic ideal gas is an adiabatic stellar polytrope with index $n=3/2$. Its polytropic mass-radius relation givesAt the base of a thin grey atmosphere, the stellar surface boundary condition and electron-scattering opacity giveThe ideal-gas equation evaluated on the convective adiabat givesCombining these relations,The Stefan–Boltzmann law then gives

At fixed initial zero-metal composition, equating parts (i) and (ii) gives $R\propto M^{3/4}$ and $L\propto M^3$. Therefore $T_{\mathrm{eff}}\propto (L/R^2{)}^{1/4}\propto M^{3/8}$, and the fully radiative zero-age line on the Hertzsprung-Russell diagram has slope

For the convective sequence, equating parts (i) and (iii) at fixed composition givesIts slope is consequently

For a fully ionized zero-metal hydrogen-helium mixture with $Y=1-X$, the mean molecular weight obeysFor a radiative star of fixed mass,At the pure-hydrogen point $X=1$,andHenceAs hydrogen burns, $X$ decreases, so both $L$ and $T_{\mathrm{eff}}$ increase. The track moves upward and toward higher temperature from the zero-age main sequence, with initial slope $d\log L/d\log T_{\mathrm{eff}}=256/53\simeq 4.83$.