Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 317 / 2

Use stellar homology: dimensionless profiles and composition are fixed, so proportionality constants do not change along the model sequence. Scaling mass conservation and the stellar hydrostatic equation giveswhere the last relation uses the ideal gas equation of state with fixed mean molecular weight. The specific stellar energy-generation rate is $ϵ\propto \rho T^{\eta }$, soBy stellar radiative diffusion, $T_{\ast }/R\sim {\kappa }_{\ast }{\rho }_{\ast }L/R^2{T}_{\ast }^3$, henceEquate the generated and transported luminosity in thermal equilibrium. The homology relations for radiative stars giveThis division presumes a nonzero denominator. If it vanishes, the scaling balance instead imposes a condition on $M$ or leaves the radius scaling undetermined, depending on the numerator. For later use, the corresponding mass-luminosity relation exponent is

For PP chain burning and the Kramers' opacity law, insert $\eta =4$, $\lambda =1$, and $\nu =-7/2$ into the homology relations for radiative stars:Thus the radiative homology with proton-proton burning and Kramers opacity isTo obtain the Hertzsprung-Russell diagram slope, apply the Stefan–Boltzmann law $L=4\pi R^2\sigma T_{\mathrm{eff}}^4$:Equivalently $T_{\mathrm{eff}}\propto L^{69/284}$. The Hertzsprung-Russell slope of a stellar homology sequence refers here to logarithmic luminosity versus logarithmic effective temperature; a conventional diagram reverses its horizontal axis, so the line descends toward the right.

For fixed composition, electron-scattering opacity is effectively independent of mass density and temperature, so $\lambda =\nu =0$. With the specified CNO cycle exponent $\eta =16$, the homology relations for radiative stars giveThe radiative homology with CNO burning and electron scattering is thereforeThe Stefan–Boltzmann law now givesEquivalently $T_{\mathrm{eff}}\propto L^{9/76}$. These slopes are illustrated with arbitrary separate reference-model normalizations:

The proton–proton/Kramers model applies to the lower-mass regime; the CNO/electron-scattering model applies to the higher-mass regime. Higher-mass cores are hotter, making the more temperature-sensitive CNO cycle efficient, and highly ionized material favors electron-scattering opacity. These are the stated idealized radiative, gas-pressure-supported sequences: a fully convective low-mass star or a radiation-pressure-dominated massive star needs a different structural model. The intersection of independently normalized lines in the figure does not specify a physical crossover mass.