Stellar structure

Stellar structure follows from mass conservation, hydrostatic equilibrium, energy generation, and energy transport together with an equation of state and opacity law.

The temperature at a star's centre. In gas-pressure-supported stellar homology, the stellar hydrostatic equation and ideal gas law give $T_c\propto \mu GM/(\mathcal{R}R)$, with a coefficient set by the dimensionless density and pressure profiles. It controls the strongly temperature-dependent stellar energy-generation rate.

A stellar core is the central region distinguished from the surrounding envelope by its composition, energy generation or thermodynamic state. A core boundary at radius $R_c$ encloses mass $M_c=m(R_c)=4\pi {\int }_0^{R_c}\rho (r)r^{2}dr$. In a hydrogen-shell-burning red giant, a helium-rich core can be inert while a narrow surrounding layer supplies the luminosity. Core mass and core radius are separate variables: the relation between them must come from a structural model, such as a polytropic mass-radius relation, rather than from the word core alone.

Uniform solid-body rotation adds the centrifugal potential $-{\Omega }^2{\varpi }^2/2$ to the gravitational potential. In the corotating frame, hydrostatic equilibrium becomes $\nabla P=-\rho \nabla \varphi$. The Poisson equation gives ${\nabla }^2\varphi =4\pi G\rho -2{\Omega }^2$. On regular connected equipotential surfaces, $P$ is constant; taking the curl shows that $\rho$ is constant too. Uniform composition and an invertible equation of state then make $T$ constant on each such surface. These statements do not require $|\nabla \varphi |$ to be constant there.

This approximation represents the gravitational potential by a central point mass while retaining the centrifugal potential. A surface through polar radius $R_p$ satisfies $R_p/r+q(r/R_p{)}^2{\sin }^2\theta /2=1$, where $q={\Omega }^2{R}_p^3/(GM)$. The inner closed surface is regular below $q=8/27$; at the limiting value its equatorial radius is $3R_p/2$ and equatorial effective gravity vanishes. The approximation is useful for illustrating oblateness and gravity darkening, rather than modeling the density distribution of every rotating star.

The divergence theorem and Poisson equation give ${\int }_S\nabla \varphi \cdot d\mathbf{S}=4\pi Gm-2{\Omega }^{2}V$ on a bounding equipotential surface. Equipotential luminosity conservation with stellar circulation then fixes $f=L/(4\pi Gm-2{\Omega }^{2}V)$. Combining $dT/d\varphi =-f/\chi$, $dP/d\varphi =-\rho$ and radiative conductivity gives the displayed logarithmic temperature-pressure gradient. Both appearances of mass mean the enclosed mass $m$; only on the outer boundary can both be replaced by the total mass.

Slow circulation in planes containing the rotation axis can transport entropy through a rotating star while the leading mechanical balance remains nearly hydrostatic equilibrium. Its steady mass flux satisfies the continuity equation. It is distinct from assuming an exactly motionless radiative interior, and from turbulent convection.

The thermal imbalance of a slowly rotating radiative star can drive stellar meridional circulation. Its fractional centrifugal distortion is ${ϵ}_{\Omega }\sim {\Omega }^2{R}^3/(GM)$, so balancing an order-${ϵ}_{\Omega }$ fraction of the luminosity by thermal advection estimates $t_{\mathrm{ES}}\sim t_{\mathrm{KH}}/{ϵ}_{\Omega }$, where $t_{\mathrm{KH}}\sim G{M}^2/(RL)$. Stable composition gradients can further impede the circulation. This radiative thermal estimate is different from rapid torque-driven circulation in a convective envelope.

For steady stellar meridional circulation, $T\nabla s=\nabla (h+\varphi )$ and $\nabla \cdot (\rho \mathbf{v})=0$. Thus $\rho \mathbf{v}\cdot T\nabla s=\nabla \cdot [\rho \mathbf{v}(h+\varphi )]$. In the leading uniformly rotating stellar hydrostatic equilibrium, $h+\varphi$ is constant on an equipotential surface; its total advected flux vanishes because the net mass flux vanishes. Integrating the energy equation then identifies the outward luminosity with enclosed nuclear energy generation, even though local heat advection is nonzero.

On a regular equipotential surface, $f,f^{\prime },\rho$ and ${\nabla }^2\varphi$ are constant, but $|\nabla \varphi {|}^2$ generally varies. Consequently a flux $\mathbf{F}=f(\varphi )\nabla \varphi$ cannot ordinarily satisfy local radiative equilibrium $\nabla \cdot \mathbf{F}=\rho ϵ$ at all latitudes. Special cases such as $f^{\prime }=0$ or spherical symmetry evade this obstruction. Stellar meridional circulation supplies heat advection in the leading hydrostatic treatment.

Gravity darkening is the variation of surface brightness and effective temperature with effective gravity over a rotating or tidally distorted star. Lower equatorial gravity generally corresponds to cooler equatorial regions. The radiative gravity-darkening law applies to the idealized uniformly rotating, radiatively transporting, composition-uniform case; its exponent is not universal for every stellar envelope.

In uniformly rotating stellar hydrostatic equilibrium, composition-uniform thermodynamic quantities are functions of the total potential. Radiative conductivity gives $\mathbf{F}=f(\varphi )\nabla \varphi$, with $f=-\chi dT/d\varphi$. On one surface, $f$ is constant, so the outward flux is $F_n=fg$. The Stefan–Boltzmann law therefore gives $T_e\propto g^{1/4}$. This is a surface-flux law: the material temperature constant on an interior equipotential is not the latitude-dependent emergent effective temperature.

A uniform-density spherical hydrostatic equilibrium model with zero surface pressure has $P(r)=2\pi G{\rho }^2(R^2-r^2)/3$, $M=4\pi \rho R^3/3$ and gravitational energy $-3G{M}^2/(5R)$. A perfect gas can have this mechanical structure with varying temperature and entropy; the imposed equilibrium density does not require incompressible perturbations.

An imposed spherical profile $\rho ={\rho }_c(1-r/R)$ gives $M=\pi {\rho }_c{R}^3/3$ in hydrostatic equilibrium. For an ideal gas with zero surface pressure, $P=P_c(1-x{)}^2(1+2x-9x^2/5)$ and $T=T_c(1-x)(1+2x-9x^2/5)$, where $x=r/R$, $P_c=5G{M}^2/(4\pi R^4)$ and $T_c=5GM/(12\mathcal{R}R)$. The density has a central cusp and temperature initially rises outward, so positive outward radiative diffusion with regular positive nuclear heating cannot satisfy this imposed profile near the centre.

For a slowly evolving spherical star, the Lagrangian energy equation separates nuclear rest-mass release, stellar neutrino energy loss, thermal storage and pressure-volume work. Specific internal energy and nuclear energy conventions must avoid counting nuclear mass changes or escaping reaction-neutrino energy twice. At fixed composition the gravothermal stellar energy generation is $-TDS/Dt$. Energy transport processes redistribute heat rather than creating it.

Escaping neutrinos remove energy produced by nuclear reactions or drawn from thermal reservoirs. Pair annihilation, plasma excitations, photoneutrino emission and electron-ion bremsstrahlung provide thermal losses. This reduces photon luminosity or increases the required nuclear and gravothermal supply. At sufficiently extreme collapse densities, neutrino transport must replace a free-escape loss approximation.

This term accounts for thermal storage and compression or expansion in the stellar energy balance equation. At fixed composition the first law of thermodynamics makes it $-TDS/Dt$. Kelvin-Helmholtz contraction can supply positive luminosity, while thermal adjustment may locally absorb energy. It is not simply the full change in gravitational potential energy: the stellar virial theorem also requires changes in internal energy.

The gas-pressure fraction describes the relative support by an ideal gas and radiation pressure. For monatomic gas plus equilibrium radiation, it determines the adiabatic exponents of a monatomic gas-radiation mixture. Its value at the unperturbed state may be specified, but it generally changes during an isentropic parcel displacement.

For a radiative envelope, $d{P}_{\mathrm{rad}}/dP=\kappa L_r/(4\pi cGm)$. Exactly uniform stellar gas-pressure fraction instead requires this derivative to be $1-\beta$. Constant opacity therefore requires $L_r$ to increase in proportion to enclosed mass. A positive-density finite-mass envelope with no local energy generation has constant $L_r$ and cannot satisfy all these conditions pointwise. The Eddington-model convective-core mass fraction uses constant $\beta$ as a global approximation, rather than an exact detailed envelope constraint.

In stellar structure, stellar thermal equilibrium means that internal heating balances the surface luminosity and any energy loss through neutrinos, with negligible secular storage of thermal energy on the timescale considered. A star can satisfy this balance while its interior is hotter than its surface and heat flows outward. It is therefore different from equal-temperature thermal equilibrium between systems. Neglecting neutrino losses, the balance is $L_{\mathrm{surface}}=L_{\mathrm{heat}}$. In a slowly mass-losing donor star, a stellar thermal equilibrium radius sequence applies only when thermal adjustment is fast compared with the mass-loss timescale; rapid loss instead requires the adiabatic stellar radius response exponent.

A temperature versus density diagram separates classical, degenerate, nonrelativistic and relativistic stellar-matter limits. The Electron degeneracy boundary is $T\sim T_F(\rho )$, with slope $2/3$ at low density and $1/3$ at high density. Thermal relativity is controlled by $k_{B}T\sim m_e{c}^2$, while relativity of degenerate Electrons is controlled by $p_F\sim m_{e}c$. Photon pressure and thermal electron-positron pairs add further crossovers; all depend on composition and are gradual, not thermodynamic phase boundaries.

For one ionic species of number density $n_i$, the ion-sphere radius is $a_i=(3/(4\pi n_i){)}^{1/3}$. The ionic Coulomb coupling parameter compares the interaction energy from Coulomb's law at that separation with thermal energy $k_{B}T$. At fixed composition, $\Gamma \propto {\rho }^{1/3}/T$. Values near one mark significant corrections to the ideal ion gas; strong coupling can lead to an ordered solid. Screening, mixtures and quantum motion modify detailed transitions, so this estimate is not a universal crystallization boundary. It is separate from the electron degeneracy criterion based on the electron Fermi temperature.

Equating radiation pressure to cold electron degeneracy pressure gives $T=[3P_e(\rho )/a_{\mathrm{rad}}{]}^{1/4}$. Because the two Electron pressure limits scale as ${\rho }^{5/3}$ and ${\rho }^{4/3}$, this boundary has logarithmic slopes $5/12$ and $1/3$, respectively. Above it, photon pressure can dominate even when the net Electrons remain degenerate; degeneracy of one component and dominance of the total pressure are different criteria. Thermal pairs modify the high-temperature pressure comparison.

For a nondegenerate fully ionized gas, equating $P_{\mathrm{gas}}=\rho k_{B}T/({\mu }_{\mathrm{mol}}{m}_u)$ with $P_{\gamma }=a_{\mathrm{rad}}{T}^4/3$ gives $T^3=3k_B\rho /(a_{\mathrm{rad}}{\mu }_{\mathrm{mol}}{m}_u)$. Radiation dominates at higher temperature for fixed density. A degenerate Electron gas requires its degeneracy pressure instead of extrapolating this ideal-gas formula.

In a dilute Electron gas, thermal kinetic energies become relativistic near $k_{B}T=m_e{c}^2$, corresponding to $T_{\ast }\simeq 5.93\times 10^9\mathrm{K}$. This is a crossover in the momentum distribution, not an abrupt temperature transition. Degenerate Electrons can instead be relativistic at much lower temperature if their Fermi momentum exceeds $m_{e}c$.

A stellar composition profile gives chemical mass fractions against radius or enclosed mass. Flat portions can indicate mixing in a convective core or convective envelope, while burning shells and moving convective boundaries leave gradients and discontinuities.

The stellar structure equations describe spherical hydrostatic stars through radial conservation of mass, force balance, energy generation, and energy transport. An equation of state, opacity, energy-generation law, and boundary conditions close the system.

In spherical stellar structure, a shell of radius $r$, thickness $dr$ and mass density $\rho$ has mass $dm=4\pi r^2\rho dr$. Integrating outward gives the enclosed mass, $m(r)=4\pi {\int }_0^r\rho (s)s^{2}ds$.

The stellar adiabatic exponents describe pressure and temperature responses to an isentropic density change:

For fixed-composition monatomic ideal gas plus equilibrium radiation pressure, specific internal energy is $u=3\mathcal{R}T/2+a_{\mathrm{r}}{T}^4/\rho$. An isentropic process satisfies $du=Pd\rho /{\rho }^2$, giving ${\Gamma }_3-1=(8-6\beta )/(24-21\beta )$ andThe stellar gas-pressure fraction must be allowed to change along the adiabat. The pure-radiation and pure-gas limits give ${\Gamma }_1={\Gamma }_2=4/3$ and $5/3$, respectively. Partial ionization or different gas heat capacities change these formulas.

For fixed-composition monatomic ideal gas plus equilibrium blackbody radiation, let $\mathcal{R}=k/(\mu H)$ and $\beta =P_g/P$. The mass-specific heats are $c_P=\mathcal{R}(32-24\beta -3{\beta }^2)/(2{\beta }^2)$ and $c_V=\mathcal{R}(24-21\beta )/(2\beta )$. The specific-heat ratio is ${\Gamma }_1/\beta$, not generally any one of the stellar adiabatic exponents. Differentiate at constant total pressure rather than at constant $\beta$.

For a spherically symmetric density $\rho (r)$, the mass enclosed inside radius $r$ is

For a static spherically symmetric Newtonian star, balancing the inward gravitational force against the pressure gradient gives

A stellar polytrope obeys $P=K{\rho }^{1+1/n}$ for constant $K$ and polytropic index $n$. Its dimensionless density profile satisfies the Lane-Emden equation.

The regular $n=5$ Lane-Emden equation solution has no finite first zero. Its Lane-Emden mass formula gives $m(\xi )=4\pi {\rho }_c{\alpha }^3{\xi }^3/[3(1+{\xi }^2/3{)}^{3/2}]$, so its infinite-radius total mass is $4\pi \sqrt{3}{\rho }_c{\alpha }^3$. Its enclosed mean mass density is ${\rho }_c(1+{\xi }^2/3{)}^{-3/2}$ and tends to zero. A finite externally confined core instead requires a specified truncation pressure.

For a gas-pressure-supported polytrope of index five with uniform mean molecular weight, $T/T_c=\theta$. A specific stellar energy-generation rate $ϵ={ϵ}_0\rho T^q$ therefore integrates to $4\pi {ϵ}_0{\alpha }^3{\rho }_c^2{T}_c^q{\int }_0^{\infty }{\xi }^2{\theta }^{10+q}d\xi$. Substituting $u=\xi /\sqrt{3+{\xi }^2}$ gives the displayed formula, convergent for $q>-7$. At $q=11$ its dimensionless coefficient is approximately $1.02942$. The scale factor is a volume, ${\alpha }^3$, rather than an area.

For a spherical stellar polytrope, the axial moment of inertia is $I=(8\pi /3){\int }_0^R\rho (r)r^{4}dr$. With Lane-Emden variables for a stellar polytrope, first zero ${\xi }_1$ and Lane-Emden surface mass constant ${\omega }_n$, its dimensionless form isA polytrope of index zero gives $2/5$, while a polytrope of index one gives $(2/3)(1-6/{\pi }^2)$. The latter is smaller because more of the mass lies near the center. The scalar second mass moment $\int r^{2}dm$ is $3I/2$, not the axial moment of inertia.

The regular $n=0$ solution of the Lane-Emden equation is $\theta =1-{\xi }^2/6$, with first zero $\sqrt{6}$. Its interior density is constant and its pressure varies as $\theta$. This is the structural incompressible limit; $P=K{\rho }^{1+1/n}$ itself is singular at $n=0$, and does not assign the perturbative stellar adiabatic exponent of a thermally stratified ideal-gas model.

For $p=K{\rho }^{1+1/n}$, the Lane-Emden mass formula gives $M\propto {\rho }_c^{(3-n)/(2n)}$ at fixed $K$, while $M/R\propto {\rho }_c^{1/n}$. For $3<n<5$, the Lane-Emden equation has a finite-radius solution, yet $M\to \infty$ as ${\rho }_c\to 0$. The relativistic Tolman–Oppenheimer–Volkoff equation approaches this limit since $p_c/{\rho }_c\to 0$. Monotone pressure and density alone therefore do not impose a finite stellar maximum mass.

For a spherical stellar polytrope of finite radius and zero surface pressure, the specific enthalpy is $(n+1)P/\rho$. Hydrostatic equilibrium gives $h+\Phi =-GM/R$, hence $(n+1)\int PdV=-G{M}^2/R-2\Omega$. Eliminating the pressure integral with the stellar virial theorem proves the displayed formula. For a gas adiabatic stellar polytrope, $n=1/(\gamma -1)$ and $U=G{M}^2/[(5\gamma -6)R]$; a general polytropic index need not have this relation to the specific-heat ratio.

Writingreduces Newtonian hydrostatic balance toThe stellar surface is the first zero ${\xi }_1$ of $\theta$, and $R=\alpha {\xi }_1$.

A stellar polytrope with polytropic index one has $\nabla \Phi =-2K\nabla \rho$ in hydrostatic equilibrium. Combining this with the Poisson equation for Newtonian gravity gives the Helmholtz equationThe spherical solution regular at the origin is $\rho (r)={\rho }_c\sin (kr)/(kr)$. Its first zero gives $R=\pi /k=\sqrt{\pi K/(2G)}$, independent of central mass density, and $\bar{\rho }/{\rho }_c=3/{\pi }^2$.

The interior Helmholtz equation for a polytrope of index one also admits a positive solution vanishing on the faces of a cube:Its central value is ${\rho }_c$ and its average is $\bar{\rho }=8{\rho }_c/{\pi }^3$. Together with $\Phi =C-2K\rho$, it solves the interior hydrostatic equilibrium and Poisson equation for Newtonian gravity. A local interior solution need not match the external field of its own mass.

For the cubic polytropic interior, $\nabla \rho$ vanishes at each vertex, so the locally defined potential $\Phi =C-2K\rho$ predicts zero gravitational acceleration there. But at the vertex $(0,0,0)$, the actual self-gravitational acceleration of the positive mass distribution iswhose three components are strictly positive. The fields cannot match continuously. Thus the interior Helmholtz equation and zero density on the faces do not construct an isolated static cubic star. External stresses or an external gravitational field would be needed to realize such a boundary-value construction.

The sine-product cubic polytropic interior has zero mass density gradient at every corner. Its hydrostatic potential $C-2K\rho$ therefore has zero gradient there. In contrast, at a corner of a cube containing a positive mass density, every interior mass element contributes a gravitational-potential gradient pointing into the same coordinate octant. Each component of the gradient is strictly nonzero. This directly proves failure of global isolated gravitational matching, even though the interior hydrostatic and Poisson equations hold.

The Eddington standard model is a chemically homogeneous $n=3$ stellar polytrope with constant gas-pressure fraction $\beta$. It approximates massive radiative stars in which gas and radiation both contribute appreciably to pressure.

The polytropic index $n$ is defined by $P=K{\rho }^{1+1/n}$. A monatomic convective ideal gas has $n=3/2$, while a radiation-supported Eddington model has $n=3$.

An adiabatic stellar polytrope has $P=K{\rho }^{\gamma }$ with constant specific entropy, so $n=1/(\gamma -1)$. For a monatomic perfect gas, $\gamma =5/3$ and $n=3/2$.

Eliminating central density from the Lane-Emden scalings givesfor $n\ne 3$. In particular, $R\propto K{M}^{-1/3}$ for $n=3/2$.

With $Y_n=-{\xi }_1^2{\theta }^{\prime }({\xi }_1)$, the exact rearranged relation iswhereFor $n=3$, the radius dependence disappears and the mass is proportional to $K^{3/2}$.

For a finite regular polytrope with fixed index, the scale ${\alpha }^2\propto K{\rho }_c^{(1-n)/n}$ implies $M^{n-1}{R}^{3-n}\propto K^n$. Thus, for $n\ne 3$, $R\propto K^{n/(3-n)}{M}^{(1-n)/(3-n)}$. For $n=3/2$, $R\propto K{M}^{-1/3}$. Fixed composition makes the cold electron degeneracy pressure coefficient approximately fixed, but a convective ideal gas has $K$ controlled by entropy. A stellar sequence with changing entropy therefore need not follow the fixed-$K$ negative slope.

If $\rho ={\rho }_c{\theta }^n$, $r=\alpha \xi$, and ${\xi }_1$ is the first zero of the Lane-Emden function, then a spherical polytrope has

For a Lane-Emden function whose first zero is ${\xi }_1$,

For a chemically uniform $n=3$ gas-plus-radiation polytrope with constant gas-pressure fraction $\beta$, the mass obeysIt follows from the $n=3$ mass scaling $M\propto K^{3/2}$ and the equation of state.

Homologous stellar models have the same dimensionless radial profiles after mass, radius, density, pressure, and temperature are scaled by characteristic values.

For a monatomic ideal gas fully convective star, $P=K{\rho }^{5/3}$ and $K\propto R{M}^{1/3}$. Matching its adiabat to a photosphere with $P\sim g/\kappa$ and $\kappa \propto \rho T^{10}$ gives $T_{\mathrm{eff}}^{14}\propto M^{2}R$, and hence surface luminosity $L\propto M^{4/7}{R}^{16/7}$. Specific heating $ϵ\propto \rho T^5$ gives $L\propto M^7{R}^{-8}$. Equality yields the displayed laws and $T_{\mathrm{eff}}\propto M^{3/16}$, so the theoretical HR slope is $32/3$. Fixed composition, efficient convection and the surface matching approximation are essential.

A mass-preserving stellar homology with scale $x$ has mass density proportional to $x^{-3}$. For an adiabatic process with constant adiabatic index $\gamma$, its internal energy scales as $x^{-3(\gamma -1)}$ and its Newtonian gravitational potential energy as $x^{-1}$. Writing $a=3(\gamma -1)$, stationarity at $x=1$ requires $W_0=-a{U}_0$, and the second scale derivative is $a(a-1)U_0$. Thus this particular displacement is restoring for $\gamma >4/3$, destabilizing for $1<\gamma <4/3$, and marginal at $\gamma =4/3$. This is a test of homologous radial stability; it does not by itself test every stellar displacement.

For radiative, gas-pressure-supported homologous stars with constant opacity, write $\rho =Mb/(4\pi R^3)$, $P=G{M}^{2}p/(4\pi R^4)$ and $T=\mu GMp/(\mathcal{R}Rb)$. The radiative diffusion in a star equation contains the dimensionless coefficient $D=3{\kappa }_0{\mathcal{R}}^{4}L/(64{\pi }^{2}ac{\mu }^4{G}^4{M}^3)$. A common fixed profile fixes $D$, so $L\propto M^3$ and the luminosity of a fixed-mass member is constant. Gravitational energy generation in a homologously contracting ideal-gas star gives $\dot{R}=-2EL{R}^2/(3G{M}^2)$ for a positive profile constant $E$. Thus $R^{-1}=R_0^{-1}+2ELt/(3G{M}^2)$. For CNO cycle heating $ϵ\propto \rho T^{16}$, the stellar nuclear fusion luminosity scales as $M^{18}{R}^{-19}$. Equating this with $M^3$ gives $R_{\mathrm{MS}}\propto M^{15/19}$. Neglecting $R_0^{-1}$, the arrival time is $t_{\mathrm{MS}}\propto M^{-34/19}$, hence a coeval cluster's newly arriving mass scales as $t^{-19/34}$. All comparisons hold stellar composition, opacity normalization and dimensionless profiles fixed.

In stellar homology, a fixed dimensionless density profile gives ${\rho }_c=C_{\rho }M/R^3$ through the mass integral. Integrating the stellar hydrostatic equation through the same profile gives $P_c=C_{P}G{M}^2/R^4$ when surface pressure is negligible. The constants depend on the shape, rather than the mass and radius. For an ideal gas, $T_c=(C_P/C_{\rho })\mu GM/(\mathcal{R}R)$.

During a spherical stellar homology contraction at fixed mass and angular momentum, a constant inertia coefficient gives $I=\alpha M{R}^2$ and solid-body rotation gives $\Omega =J/I\propto R^{-2}$. A fixed fluid label $s=r/R$ retains its enclosed mass and polar angle, so centrifugal acceleration scales as $R^{-3}$ and gravity as $R^{-2}$. Their ratio increases as $R^{-1}$. This assumes spin conservation, not continuous synchronization to an external orbit.

In the spherical approximation, equatorial centrifugal acceleration balances surface gravity when ${\Omega }_c^2{R}_c=GM/R_c^2$. For $I=\alpha M{R}^2$ and conserved angular momentum $J$, this gives the displayed critical radius. Real rotating gas can deform, lose matter and exchange torques before a spherical model remains accurate at this limit.

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.