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

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.