Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 37 / 1

Take homologous profiles and hold composition, hence mean molecular weight, fixed. The stellar mass conservation equation and stellar hydrostatic equation give ${\rho }_c\propto M/R^3$ and $P_c\propto G{M}^2/R^4$. With ideal gas pressure support, $T_c\propto M/R$. Integrating the specific nuclear rate through homologous profiles then givesThe radiative diffusion in a star equation gives $L_{\mathrm{rad}}\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. Here ${\kappa }_c\propto {\rho }_c^{1/2}{T}_c^{-5/2}\propto M^{-2}R$, soEquality of generated and transported luminosity implies $M^7{R}^{-8}\propto M^5{R}^{-1}$, whenceThis is the radiative homology with density-half inverse-five-halves opacity scaling.

The Stefan–Boltzmann law defines the effective temperature by $L=4\pi R^2{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4$. Therefore $T_{\mathrm{eff}}\propto M^{29/28}$. In theoretical Hertzsprung-Russell diagram coordinates, the slope isThe usual graphical convention puts increasing temperature to the left, so the line descends as one moves right. If the horizontal coordinate is $-\log T_{\mathrm{eff}}$, its plotted slope is $-132/29$.

A fully convective, monatomic ideal gas is approximately isentropic through its deep interior. Its adiabatic index is $5/3$, so $P=K{\rho }^{5/3}$ and $T\propto K{\rho }^{2/3}$. The $n=3/2$ stellar polytrope scaling, or direct substitution of the central homology scalings into $K=P_c/{\rho }_c^{5/3}$, gives $K\propto R{M}^{1/3}$.

Near the photosphere, hydrostatic balance expressed in optical depth gives $P_{\mathrm{ph}}\sim g/{\kappa }_{\mathrm{ph}}$, with $g\propto M/R^2$. Treat $T_{\mathrm{ph}}$ as proportional to $T_{\mathrm{eff}}$ and match to the convective adiabat, neglecting a varying surface entropy jump in this homology approximation. Since $\rho \propto P/T$ and ${\kappa }_{\mathrm{ph}}\propto \rho T^{10}\propto P{T}^9$,The adiabat gives $P\propto K^{-3/2}{T}^{5/2}$, so the surface condition yieldsThe Stefan–Boltzmann law consequently gives $L_{\mathrm{surf}}\propto M^{4/7}{R}^{16/7}$. The same specific nuclear law as before still gives $L_{\mathrm{nuc}}\propto M^7{R}^{-8}$. Equating these scalings produces $R^{72/7}\propto M^{45/7}$. Thus the fully convective homology with tenth-power surface opacity isThen $T_{\mathrm{eff}}\propto M^{3/16}$, and the theoretical HR slope isIts steepness expresses the relatively weak change of effective temperature with mass. The graph using $-\log T_{\mathrm{eff}}$ has the negative of this slope.

The radiative family is a stylized approximation to gas-pressure-supported main sequence stars with mostly radiative interiors and moderately temperature-sensitive hydrogen burning, such as the roughly solar-mass regime. Its opacity exponents are local approximations, not a universal composition-independent opacity law. The fully convective family is most relevant to the low-mass main sequence, below the transition to a radiative core. Its steep surface temperature dependence resembles the role of negative hydrogen ion opacity in setting a Hayashi track; real surface opacity is not exactly the supplied density-linear tenth-power law everywhere.

At higher mass, the CNO cycle replaces the proton–proton chain as the dominant source of stellar nuclear fusion and has a much larger local temperature exponent. This promotes a convective core. Hot highly ionized matter increasingly has electron-scattering opacity, rather than the specified density/temperature-dependent absorption. Radiation pressure becomes important in the equation of state, changing $T_c\propto M/R$ and eventually making the luminosity relation approach an Eddington luminosity scaling. Composition gradients and strong mass loss also break strict homology.

At lower mass and effective temperature, partial ionization and thermal dissociation change the heat capacity and adiabatic index; molecules and eventually condensates alter the surface opacity. Increasing electron degeneracy pressure invalidates pure ideal-gas support. Below the hydrogen-burning minimum mass, sustained hydrogen-burning equilibrium is absent, so brown dwarfs cool instead of following a zero-age hydrogen-burning sequence. These changes explain why neither derived slope should be extrapolated indefinitely.