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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.

The abundance of chemical elements heavier than helium in astronomical material. A metal mass fraction measures their combined mass; spectroscopic abundance ratios are other measures of metallicity and need not be numerically identical to that fraction.

After complete ionization, each hydrogen nucleus supplies one ion and one Electron, while each helium-4 nucleus supplies one ion and two Electrons. Neglecting the small metal mass fraction $Z$ and putting $Y\simeq 1-X$, the mean molecular weight therefore satisfiesUse stellar homology to compare models with common dimensionless profiles. The stellar mass conservation equation, stellar hydrostatic equation and ideal gas law giveThe radiative diffusion in a star equation then gives $L_{\mathrm{rad}}\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. Inserting the specified opacity yields $L_{\mathrm{rad}}\propto R{T}_c^7/(Z{\rho }_c^2)\propto {\mu }^7{M}^5/Z$. Independently, integrating the stellar energy-generation rate over the mass gives $L_{\mathrm{nuc}}\propto X^2{\rho }_c{T}_c^{5}M\propto X^2{\mu }^5{M}^7/R^8$. Equating these in stellar thermal equilibrium gives the radiative homology with fifth-power hydrogen burning and inverse-cubic opacity:At fixed solar mass these reduce to $L\propto {\mu }^7/Z$ and $T_c\propto {\mu }^{5/4}/(X^{1/4}{Z}^{1/8})$. The stellar homology approximation, rather than merely the local power laws, is what permits these relations between whole stars.

Equal luminosities imply ${\mu }_2/{\mu }_1=(Z_2/Z_1{)}^{1/7}=2^{-1/7}$. Thusso $X_2=0.8$ to one significant figure. Retaining the unrounded hydrogen mass fraction when comparing central temperatures givesThe first model has the higher central temperature.

For homogeneous hydrogen burning at fixed $M,Z$, fuel conservation gives $M{E}_0\dot{X}=-L$. Since $d\mu /dX=-5{\mu }^2/4$ and $L=L_0(\mu /{\mu }_0{)}^7$, the homogeneous fuel-depletion luminosity feedback obeysIntegration with the initial mean molecular weight ${\mu }_0$ givesThis idealized stellar evolution law stops when $X=0$; it does not predict a physically attainable infinite luminosity. Indeed $\mu =4/3$ at fuel exhaustion, giving $t_H=M{E}_0[1-(3{\mu }_0/4{)}^8]/(10{\mu }_0{L}_0)$, strictly before the formal singularity.

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 $L\propto {\mu }^7$.