Homology relations for radiative stars

For chemically uniform, gas-pressure-supported stars with homologous dimensionless profiles, mass conservation and hydrostatic pressure support equation give ${\rho }_c\propto M/R^3$ and $T_c\propto M/R$. Suppose opacity scales as $\kappa \propto {\rho }^a{T}^b$ and specific nuclear energy generation as $ϵ\propto {\rho }^c{T}^d$. ThenThe second scaling follows from radiative diffusion in a star, $L\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. Equating the two giveswhen the denominator is nonzero. Fixed composition and self-similar profiles are essential; these are relations between idealized models rather than universal stellar laws.

For gas-pressure-supported stellar homology with $\kappa \propto {\rho }^{1/2}{T}^{-5/2}$ and specific nuclear heating $ϵ\propto \rho T^5$, the central scalings give $L_{\mathrm{nuc}}\propto M^7{R}^{-8}$ and radiative diffusion in a star gives $L_{\mathrm{rad}}\propto M^5{R}^{-1}$. Equating them yields the displayed laws. The Stefan–Boltzmann law gives $T_{\mathrm{eff}}\propto M^{29/28}$, hence $d\log L/d\log T_{\mathrm{eff}}=132/29$. Composition and dimensionless profiles are held fixed.

For a chemically uniform, gas-pressure-supported stellar homology with Kramers' opacity law and specific heating $ϵ={ϵ}_0\rho T^{\nu }$, radiative diffusion in a star gives $L\propto {\kappa }_0^{-1}{\mu }^{15/2}{M}^{11/2}{R}^{-1/2}$. Nuclear heating gives $L\propto {ϵ}_0{\mu }^{\nu }{M}^{\nu +2}{R}^{-\nu -3}$. Equating them yields $R^{\nu +5/2}\propto {ϵ}_0{\kappa }_0{\mu }^{\nu -15/2}{M}^{\nu -7/2}$. Thus $\nu =7/2$ gives a mass-independent radius, whereas $\nu =23/2$ gives $R\propto M^{4/7}$ and $L\propto M^{73/14}$. The coefficients and dimensionless profiles must be fixed; this is an idealized radiative model rather than a universal stellar law.

For gas-pressure-supported stellar homology with opacity $\kappa ={\kappa }_{0}Z\rho T^{-3}$ and stellar energy-generation rate $ϵ={ϵ}_0{X}^2\rho T^5$, radiative diffusion in a star gives $L\propto R{T}_c^7/(Z{\rho }_c^2)\propto {\mu }^7{M}^5/Z$. Integrating the heating rate instead gives $L\propto X^2{\mu }^5{M}^7/R^8$. Equality in stellar thermal equilibrium yields the displayed radius law and $T_c\propto {\mu }^{5/4}{M}^{3/4}{X}^{-1/4}{Z}^{-1/8}$. This fixes the dimensionless profiles and coefficients, and neglects radiation pressure; it is a model comparison rather than a universal law for all main sequence stars. At fixed mass and metal mass fraction, homogeneous hydrogen burning gives the homogeneous fuel-depletion luminosity feedback through $L\propto {\mu }^7$.

For a chemically uniform, ideal-gas-supported stellar homology sequence with nuclear energy generation $ϵ={ϵ}_0{X}^2\rho T^4$ and opacity $\kappa ={\kappa }_{0}Z(1+X)\rho T^{-4}$, radiative diffusion in a star fixes $LRZ(1+X)/({\mu }^8{M}^6)$ and nuclear heating fixes $X^2{\mu }^4{M}^6/(R^{7}L)$. Eliminating $L$ gives $R^6\propto X^{2}Z(1+X){\mu }^{-4}$, henceThe absence of $M$ in the radius law is a consequence of these particular power laws; it is not a universal relation for real stars. At fixed initial $X$ and $\mu$, the Stefan–Boltzmann law gives $T_e\propto Z^{-1/12}$ at fixed luminosity.

Let a fixed-mass, fully mixed stellar model evolve by hydrogen burning, with $L=C{M}^6{Z}^{-7/6}H(X)$ and $H(X)=\mu (X{)}^{26/3}{X}^{-1/3}(1+X{)}^{-7/6}$. Energy conservation gives $\dot{X}=-L/(M{E}_H)$. The time to reach $X=f{X}_0$ is therefore $t=(E_H/C)M^{-5}{Z}^{7/6}{\int }_{f{X}_0}^{X_0}H(X{)}^{-1}dX$. If $X_0,f$ and the function $\mu (X)$ are common to the population, the integral is constant and $M_{\mathrm{to}}\propto Z^{7/30}{t}^{-1/5}$. This includes the change of luminosity during fuel depletion rather than replacing the luminosity by its initial value.

For gas-pressure-supported stellar homology with specific nuclear heating $ϵ={ϵ}_0\rho T^{13}$ and opacity $\kappa ={\kappa }_0\rho T^{-3}$, the dimensionless radiative coefficient is proportional to $L{\mu }^{-7}{M}^{-5}$. The nuclear coefficient is proportional to ${\mu }^{13}{M}^{15}/(L{R}^{16})$. Holding both and the dimensionless profiles fixed gives the displayed scalings. The Stefan–Boltzmann law then gives $T_e\propto {\mu }^{25/16}{M}^{15/16}$ and $L\propto {\mu }^{-4/3}{T}_e^{16/3}$. The coefficients, pressure regime and composition assumptions must be held as specified; these relations are not universal laws for massive stars.

For a fixed-mass fully mixed hydrogen burning star with $\mu =4/(3+5X)$ whose radiative stellar homology gives $L\propto {\mu }^7$, put $s=3+5X$. The luminosity is $L=L_0(s_0/s{)}^7$ and fuel conservation gives $M{E}_H\dot{X}=-L$. Hence $d(s^8)/dt=-40L_0{s}_0^7/(M{E}_H)$, producing the displayed feedback law. This applies both to CNO homology with density-dependent inverse-cubic opacity and to radiative homology with fifth-power hydrogen burning and inverse-cubic opacity, at fixed metal mass fraction in the latter. Its physical domain ends at $X=0$, before the formal denominator vanishes. The idealization holds the coefficient normalizations fixed while stellar composition changes through the mean molecular weight and fuel abundance.

For homologous, chemically uniform, gas-pressure-supported radiative stars with specific energy generation $ϵ\propto X\rho T^{13}$ and composition-dependent constant opacity, stellar homology gives $L_{\mathrm{nuc}}\propto X{\mu }^{13}{M}^{15}{R}^{-16}$ and $L_{\mathrm{rad}}\propto {\mu }^4{M}^3/\kappa$. Equating them gives the displayed scalings. Hydrogen depletion changes both mean molecular weight and electron-scattering opacity; holding composition fixed gives $R\propto M^{3/4}$ and $L\propto M^3$. The power-law burning approximation applies only while hydrogen fuel remains.

With negligible heavy-element contribution to the mean molecular weight, $\mu =4/(3+5X)$ and $\kappa \propto 1+X$. At fixed mass, composition-dependent CNO homology gives $L\propto (1+X{)}^{-1}(3+5X{)}^{-4}$ and $R\propto [X(1+X){]}^{1/16}(3+5X{)}^{-9/16}$. The Stefan–Boltzmann law then gives $T_e\propto X^{-1/32}(1+X{)}^{-9/32}(3+5X{)}^{-23/32}$. From the formal $X=1$ reference, declining $X$ initially increases luminosity and effective temperature, unlike the usual redward main-sequence turn-off. A trace CNO catalyst abundance is implicit in the burning coefficient; an exactly metal-free pure-hydrogen star cannot burn through that cycle.

For a stellar homology sequence with $R\propto M^X$ and $L\propto M^Y$, the Stefan–Boltzmann law gives $T_{\mathrm{eff}}\propto M^{(Y-2X)/4}$. Eliminating mass gives the displayed Hertzsprung-Russell diagram slope when $Y\ne 2X$. The usual horizontal axis decreases in temperature toward the right, reversing the visual left-to-right sign. The relation is between logarithmic variables and depends on the specified structural assumptions.

With proton–proton chain energy generation $ϵ\propto \rho T^4$ and Kramers' opacity law $\kappa \propto \rho T^{-7/2}$, homology relations for radiative stars giveThe Hertzsprung-Russell diagram relation has $d\log L/d\log T_{\mathrm{eff}}=284/69$, with effective temperature increasing toward the left in the usual plotting convention.

Using the idealized CNO cycle law $ϵ\propto \rho T^{16}$ and constant electron-scattering opacity, homology relations for radiative stars giveThese retain ideal-gas pressure support; significant radiation pressure or convection changes the model.