The Equations of Stellar Structure

The previous lesson established one of the four equations that a static star must satisfy at every radius. This lesson assembles the complete set: mass conservation, hydrostatic equilibrium, energy generation, and energy transport. These four first-order ordinary differential equations, together with the microphysics that closes them (the equation of state, the opacity, and the nuclear reaction rates) and with boundary conditions at the center and the surface, determine the run of pressure, temperature, density, mass, and luminosity through the star. The Vogt-Russell theorem states that this solution is unique once the total mass and the composition are fixed. The transport equation carries the most physics: it is radiative where the gas is transparent enough and convective where the radiative gradient would be too steep, a switch governed by the Schwarzschild criterion.

The four structure equations

Take the interior mass $m$ or the radius $r$ as the independent variable and write the four dependent quantities as functions of it. Each equation states how one quantity changes across a shell.

Mass conservation. A shell of thickness $\mathbf{d}r$ at radius $r$ has volume $4\pi r^2\mathbf{d}r$ and contains mass $\mathbf{d}m=4\pi r^2\rho \mathbf{d}r$, so

Hydrostatic equilibrium. The pressure gradient supports the weight of each shell, as derived in the previous lesson:

Energy generation. Let $ϵ$ be the nuclear energy generated per unit mass per unit time (plus any gravitational or neutrino contribution). The luminosity $L(r)$, the net energy per unit time crossing the sphere of radius $r$, grows outward by the energy produced in each shell:

In the core $ϵ$ is large and $L(r)$ rises steeply; outside the energy-generating region $ϵ\approx 0$ and $L(r)$ is constant at the surface value $L$.

Energy transport. The temperature gradient needed to carry the luminosity outward depends on the transport mechanism. Where radiation carries the flux (the next section), the gradient is

with $\kappa$ the Rosseland-mean opacity, $a$ the radiation constant, and $c$ the speed of light. Where convection carries the flux, the gradient is close to the adiabatic value (the last two sections).

These four equations contain seven quantities: $m$, $P$, $L$, $T$, and also $\rho$, $\kappa$, $ϵ$. The last three are supplied by the constitutive relations — the equation of state $\rho =\rho (P,T,X_i)$, the opacity $\kappa =\kappa (\rho ,T,X_i)$, and the energy-generation rate $ϵ=ϵ(\rho ,T,X_i)$ — all functions of the local pressure, temperature, and composition $X_i$. With those, the system closes to four equations in four unknowns.

Boundary conditions and the Vogt-Russell theorem

Four first-order equations require four boundary conditions, split between the two ends of the star. At the center, $r=0$, both the enclosed mass and the enclosed luminosity vanish:

At the surface, $r=R$, the pressure and temperature drop to values negligible compared with the interior. The simplest closure sets

though realistic models match onto a stellar-atmosphere solution where $T(R)\approx T_{\text{eff}}$ and the photospheric pressure is finite. The total radius $R$ is itself an eigenvalue, adjusted so that the outward integration from the center and the inward integration from the surface meet.

The composition enters through the constitutive relations. Given the total mass $M$ and the run of composition $X_i(m)$, the four equations plus four boundary conditions have a unique solution.¹

The theorem is the reason a star's place on the Hertzsprung-Russell diagram is set by mass and composition rather than by history: two stars of the same mass and the same composition profile have the same radius, luminosity, and internal structure. Evolution is the slow change of the composition profile $X_i(m)$ as nuclear burning converts one element into another, with the structure re-solving to a new equilibrium at each step. The theorem holds for the idealized problem; it can fail in the presence of multiple solutions or where the history of mixing matters, but it organizes the subject.

The radiative temperature gradient

Inside a star, radiation diffuses outward through an opaque medium. Photons are absorbed and re-emitted over a mean free path $\ell =1/(\kappa \rho )$ far shorter than the stellar radius, so the transport is a slow diffusion of radiative energy down the temperature gradient, not free streaming. The radiative flux follows a diffusion law analogous to heat conduction,

where $a{T}^4$ is the radiation energy density and $c/(3\kappa \rho )$ is the photon diffusion coefficient. Setting $F_{\text{rad}}=L(r)/(4\pi r^2)$ and solving for the gradient reproduces the transport equation quoted above,

The magnitude of the required gradient rises with the opacity $\kappa$, the density, and the local flux $L(r)/(4\pi r^2)$, and falls steeply with temperature. Where the opacity is high or the flux is large, radiation demands a steep temperature drop to carry the luminosity. Beyond a critical steepness the gas can no longer transport the flux by radiation alone, and convection sets in.²

It is convenient to express the steepness as the dimensionless logarithmic gradient

the fractional change in temperature per fractional change in pressure. The value it would take if all the flux were carried by radiation is the radiative gradient${\nabla }_{\text{rad}}$; combining the transport equation with hydrostatic equilibrium,

The Schwarzschild criterion for convection

Whether a region transports energy by radiation or by convection is decided by a stability argument. Displace a small blob of gas upward by $\mathbf{d}r$, letting it expand adiabatically to stay in pressure balance with its new surroundings. If the blob ends up denser than the surrounding gas, it sinks back and the region is stable, so radiation carries the flux. If the blob ends up less dense, buoyancy pushes it further up, the displacement grows, and the region is convectively unstable.

The density comparison reduces to a comparison of temperature gradients. The blob's interior follows the adiabatic gradient ${\nabla }_{\text{ad}}$; the surroundings follow the actual gradient $\nabla$. For a fully ionized ideal monatomic gas the adiabatic gradient is

The blob rises unstably when the surrounding temperature falls faster with height than the blob's own adiabatic cooling, i.e. when the actual gradient is steeper than the adiabatic one. Since the actual gradient equals the radiative one wherever radiation would carry the whole flux, the Schwarzschild criterion for convective instability is

Convection is triggered by anything that inflates ${\nabla }_{\text{rad}}=\frac{3}{16\pi acG}\frac{\kappa LP}{m{T}^4}$: a large opacity $\kappa$ (cool stellar envelopes, where $H^{-}$ and bound-free opacity are enormous) or a large local flux $L/m$ (concentrated CNO-cycle cores in massive stars). The Sun has a radiative interior and a convective outer envelope; a massive main-sequence star has a convective core and a radiative envelope; a low-mass red dwarf is convective throughout.

Mixing-length theory in outline

The Schwarzschild criterion says where convection occurs but not how much flux it carries or how far the actual gradient exceeds the adiabatic one. In the deep interior convection is extraordinarily efficient: the blobs carry the flux with only a negligible super-adiabatic excess, so $\nabla \approx {\nabla }_{\text{ad}}$ to high accuracy and the temperature gradient is fixed without further detail. Near the surface, where the density is low and the blobs radiate away their heat before travelling far, convection is inefficient and $\nabla$ lies well above ${\nabla }_{\text{ad}}$; the exact value requires a model of the turbulence.

Mixing-length theory supplies the simplest such model. A convective blob is assumed to rise a characteristic distance, the mixing length $\ell =\alpha H_P$, before dissolving and depositing its heat, where $H_P=-P/(\mathbf{d}P/\mathbf{d}r)=P/(\rho g)$ is the pressure scale height and $\alpha$ is a free parameter of order unity, calibrated so that a solar model reproduces the observed radius. Given $\ell$, the theory estimates the convective velocity from the buoyancy work, the convective flux from the heat carried per blob, and the super-adiabatic excess $\nabla -{\nabla }_{\text{ad}}$ needed to carry the required flux. The result is crude — real convection is turbulent and three-dimensional — but it captures the surface layers well enough for evolutionary models and is the standard closure. The parameter $\alpha$ absorbs the theory's ignorance and is transferred, once calibrated on the Sun, to other stars.

Numerical integration of a stellar model

The equations cannot be integrated straight through from one end, because a small error in the guessed central conditions grows explosively toward the surface and a small error in the surface conditions grows inward. The standard method is the Henyey relaxation scheme: discretize the four equations on a grid of mass shells, guess a trial solution, and iterate a Newton-Raphson correction over the whole grid simultaneously until all four equations and both boundary conditions are satisfied to tolerance. An alternative, the shooting method, integrates outward from the center and inward from the surface with trial values of the free parameters (central pressure and temperature, total radius and luminosity) and adjusts them until the two integrations match at an intermediate fitting point.

For evolutionary sequences, the converged static model is advanced one timestep by updating the composition from the local reaction rates, $\mathbf{d}{X}_i/\mathbf{d}t=f_i(\rho ,T,X_j)$, and the structure is re-relaxed at the new composition. A solar model built this way and integrated to the solar age reproduces the observed radius, luminosity, and, through helioseismology, the internal sound-speed profile. The characteristic result is a run of pressure, temperature, and density that all fall by many orders of magnitude from center to surface, while the mass and luminosity rise from zero to their surface values, with the luminosity nearly complete within the innermost quarter of the radius where energy generation is concentrated.

Summary

The four coupled structure equations — mass conservation $\mathbf{d}m/\mathbf{d}r=4\pi r^2\rho$, hydrostatic equilibrium $\mathbf{d}P/\mathbf{d}r=-Gm\rho /r^2$, energy generation $\mathbf{d}L/\mathbf{d}r=4\pi r^2\rho ϵ$, and energy transport $\mathbf{d}T/\mathbf{d}r$ — close on the equation of state, opacity, and reaction rates, and with central and surface boundary conditions determine the structure uniquely from mass and composition (Vogt-Russell). Transport is radiative, following a photon-diffusion law, until the radiative gradient exceeds the adiabatic gradient (the Schwarzschild criterion ${\nabla }_{\text{rad}}>{\nabla }_{\text{ad}}$), where convection takes over and mixing-length theory supplies the gradient in the inefficient surface layers. The equation of state that closes the system, and the polytropic models that solve it analytically, are the subject of the next lesson.