Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 317 / 1

For a family in stellar homology, dimensionless radial profiles are fixed. Mass conservation and hydrostatic equilibrium then give the central scalingsThe ideal gas equation of state, with fixed mean molecular weight, consequently giveswhere fixed dimensional constants such as $G$ are suppressed in homology relations.

The proton–proton chain law $ϵ\propto \rho T^{7/2}$ gives the nuclear luminosityOn the other hand, radiative diffusion in a star givesUsing the Kramers opacity law ${\kappa }_c\propto {\rho }_c{T}_c^{-7/2}$,Thermal equilibrium requires $L_{\mathrm{nuc}}=L_{\mathrm{rad}}$. Their common mass factor cancels, leaving $R^{-13/2}\propto R^{-1/2}$, and hence

This model captures the gas-pressure support, pp-chain burning, and strongly mass-dependent luminosity of the lower main sequence, including the Sun approximately. Its fully radiative assumption is an idealization: the Sun has a convective envelope, and sufficiently low-mass red dwarfs become largely or fully convective, so real radii are not exactly constant.

Along the pp-chain homologous sequence, $R$ is constant and therefore $T_c\propto M$. Normalizing to the Sun gives the transition mass

For the CNO cycle law $ϵ\propto \rho T^{23/2}$, the nuclear scaling becomesThe opacity law is unchanged, so $L_{\mathrm{rad}}\propto M^{11/2}{R}^{-1/2}$. Equating them givesand then

The effective temperature follows from the Stefan–Boltzmann law $L=4\pi R^2\sigma T_e^4$. Thus the pp branch has $T_e\propto M^{11/8}$ and $L\propto T_e^4$, whereas the CNO branch hasOn a Hertzsprung-Russell diagram, both branches rise toward larger luminosity and, conventionally, leftward toward larger temperature. They join near $4M_{\odot }/3$; the CNO branch has the steeper $L$--$T_e$ logarithmic slope and lies to the cooler side of the extrapolated constant-radius branch at fixed luminosity.

If $X$ is the hydrogen mass fraction and $X_{\mathrm{CNO}}$ the combined carbon-nitrogen-oxygen mass fraction, the reaction-pair and catalyst abundances give approximatelyThe CNO nuclei are catalysts, so their abundance multiplies the CNO-cycle rate rather than being consumed by the completed cycle.

At fixed hydrogen abundance, the ratio of the two rates near the transition scales asReducing $X_{\mathrm{CNO}}$ by $256=2^8$ therefore requires twice the temperature to recover equality:The metal-poor stars remain on the same pp-chain homology sequence up to this point because the question holds opacity and mean molecular weight fixed. Since that sequence has $T_c\propto M$,

The low-CNO pp branch therefore overlaps the solar-composition pp branch and simply continues farther upward and leftward before turning onto a CNO-dominated branch. The CNO branch retains $R\propto M^{4/7}$ and $L\propto M^{73/14}$, but its normalization changes. Since $R^{14}\propto {ϵ}_0{M}^8$, reducing the CNO coefficient by 256 gives, at fixed mass,It is consequently somewhat smaller, hotter, and more luminous than the solar-CNO homology model of the same mass. The sketch should show its delayed break at $8M_{\odot }/3$ and its CNO branch displaced upward and leftward.