---
title: The Equations of Stellar Structure
draft: false
module: Stellar Structure
moduleNumber: 4
lessonNumber: 2
order: 402
summary: >
  A static star is described by four coupled first-order differential equations
  in the interior mass or radius: mass conservation, hydrostatic equilibrium,
  energy generation, and energy transport. Closed with an equation of state,
  opacity, and reaction rates, and subject to central and surface boundary
  conditions, they determine the structure uniquely from mass and composition,
  the Vogt-Russell theorem. Energy moves by radiation until the temperature
  gradient exceeds the Schwarzschild limit, where convection takes over.
topics: [Stellar Structure]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 10 — The Interiors of Stars; §10.4 Energy Transport, §10.5 Stellar Model Building"
  - book: Maoz
    ref: "Ch. 3 — Stellar Physics"
---

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 $\d r$ at radius $r$ has volume $4\pi
r^2\,\d r$ and contains mass $\d m = 4\pi r^2 \rho\,\d r$, so

$$
\frac{\d m}{\d r} = 4\pi r^2 \rho.
$$

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

$$
\frac{\d P}{\d r} = -\frac{G m \rho}{r^2}.
$$

**Energy generation.** Let $\epsilon$ 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:

$$
\frac{\d L}{\d r} = 4\pi r^2 \rho\, \epsilon.
$$

In the core $\epsilon$ is large and $L(r)$ rises steeply; outside the
energy-generating region $\epsilon \approx 0$ 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

$$
\frac{\d T}{\d r} = -\frac{3}{4 a c}\,\frac{\kappa \rho}{T^3}\,\frac{L(r)}{4\pi r^2},
$$

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 $\rho$,
$\kappa$, $\epsilon$. 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 $\epsilon = \epsilon(\rho, T, X_i)$ — all
functions of the local pressure, temperature, and composition $X_i$. With those, the
system closes to four equations in four unknowns.

$$
% caption: The four structure equations couple through the constitutive relations:
% each derivative feeds the microphysics that sets the next.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\tikzset{blk/.style={draw=black, minimum width=2.7cm, minimum height=0.95cm, align=center}}
\tikzset{mic/.style={draw=black, fill=black!6, minimum width=2.4cm, minimum height=0.85cm, align=center}}
\node[blk] (mass) at (0,3) {mass\\dm/dr};
\node[blk] (hse) at (3.6,3) {hydrostatic\\dP/dr};
\node[blk] (lum) at (0,1.3) {energy gen\\dL/dr};
\node[blk] (temp) at (3.6,1.3) {transport\\dT/dr};
\node[mic] (eos) at (7.4,3) {equation\\of state};
\node[mic] (opa) at (7.4,1.75) {opacity};
\node[mic] (rate) at (7.4,0.5) {reaction\\rates};
\draw[->, black] (hse) -- (eos);
\draw[->, black] (temp) -- (opa);
\draw[->, black] (lum) -- (rate);
\draw[->, black] (eos) to[out=200,in=20] (mass);
\draw[->, black] (opa) to[out=200,in=10] (temp);
\draw[->, black] (rate) to[out=200,in=350] (lum);
\draw[<->, black] (mass) -- (hse);
\draw[<->, black] (mass) -- (lum);
\draw[<->, black] (hse) -- (temp);
\end{tikzpicture}
$$

## 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:

$$
m(0) = 0, \qquad L(0) = 0.
$$

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

$$
P(R) \to 0, \qquad T(R) \to 0,
$$

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.[^co-model]

> **Theorem (Vogt-Russell).** The equilibrium structure and the subsequent
> evolution of a star are determined uniquely by its total mass and the
> distribution of chemical composition throughout its interior.

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,

$$
F_{\text{rad}} = -\frac{c}{3\kappa\rho}\,\frac{\d (a T^4)}{\d r}
             = -\frac{4 a c\, T^3}{3\kappa\rho}\,\frac{\d T}{\d r},
$$

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,

$$
\frac{\d T}{\d r} = -\frac{3}{4 a c}\,\frac{\kappa\rho}{T^3}\,\frac{L(r)}{4\pi r^2}.
$$

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.[^co-transport]

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

$$
\nabla \equiv \frac{\d \ln T}{\d \ln P},
$$

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,

$$
\nabla_{\text{rad}} = \frac{3}{16\pi a c G}\,\frac{\kappa L P}{m T^4}.
$$

## 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 $\d r$, 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

$$
\nabla_{\text{ad}} = \frac{\gamma - 1}{\gamma} = \frac{2}{5} = 0.4,
\qquad \gamma = \tfrac{5}{3}.
$$

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

$$
\nabla_{\text{rad}} > \nabla_{\text{ad}}.
$$

Convection is triggered by anything that inflates $\nabla_{\text{rad}} = \frac{3}{16\pi
a c G}\,\frac{\kappa L P}{m T^4}$: 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.

$$
% caption: The Schwarzschild criterion partitions the interior: convection where the
% radiative gradient exceeds the adiabatic value, radiation elsewhere.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% Sun: radiative core, convective envelope
\draw[black] (0,0) circle (2.0);
\fill[acc!10] (0,0) circle (2.0);
\fill[black!8] (0,0) circle (1.35);
\draw[black] (0,0) circle (1.35);
\node[black, anchor=center] at (0,0) {radiative};
\node[acc, anchor=center] at (0,1.68) {convective};
\node[black, anchor=north] at (0,-2.15) {Sun-like: core radiative};
% Massive star: convective core, radiative envelope
\begin{scope}[xshift=6cm]
\draw[black] (0,0) circle (2.0);
\fill[black!8] (0,0) circle (2.0);
\fill[acc!14] (0,0) circle (0.95);
\draw[acc!55] (0,0) circle (0.95);
\node[acc, anchor=center] at (0,0) {convective};
\node[black, anchor=center] at (0,1.55) {radiative};
\node[black, anchor=north] at (0,-2.15) {massive: core convective};
\end{scope}
\end{tikzpicture}
$$

## 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 $H_P = -P/(\d P/\d r) = P /(\rho g)$
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.

$$
% caption: A convective blob rises one mixing length, cooling adiabatically; the
% actual gradient exceeds the adiabatic gradient only where convection is inefficient.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% axes: gradient vs depth
\draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {depth into star};
\draw[->, black] (0,0) -- (6.6,0) node[right, black!70] {gradient};
% adiabatic gradient (vertical reference)
\draw[very thick] (2.4,0.2) -- (2.4,3.9);
\node[anchor=south, rotate=90] at (2.15,2.0) {adiabatic gradient};
% actual gradient: near-adiabatic deep, super-adiabatic near surface
\draw[black, very thick, densely dashed] (2.5,0.2) .. controls (2.6,1.3) and (3.5,2.6) .. (5.6,3.9);
\node[black, anchor=west] at (4.4,3.4) {actual gradient};
% blob rising
\fill[acc!25] (5.9,1.0) circle (5pt);
\draw[->, acc!70, thick] (5.9,1.25) -- (5.9,2.1);
\node[black, anchor=west] at (6.0,1.6) {rising blob};
\node[black, anchor=west] at (0.15,0.55) {deep: near-adiabatic};
\end{tikzpicture}
$$

## 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.

```algorithm
Input: total mass M, composition profile X(m), microphysics EOS, kappa, epsilon
Guess: central Pc, Tc and surface R, L
repeat
  integrate dm, dP, dL, dT outward from center with (Pc, Tc)
  integrate the same equations inward from surface with (R, L)
  evaluate the mismatch in P, T, L, m at the fitting mass
  apply a Newton-Raphson correction to (Pc, Tc, R, L)
until mismatch < tolerance
Output: run of P(m), T(m), rho(m), L(m), r(m)
```

For evolutionary sequences, the converged static model is advanced one timestep by
updating the composition from the local reaction rates, $\d X_i/\d t = f_i(\rho, T,
X_j)$, 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.

$$
% caption: Interior run of a solar model: pressure, temperature, and density fall
% steeply outward while mass and luminosity saturate within the inner core.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {radius fraction};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {fraction of central or total};
\foreach \x/\lab in {0/0,2/0.25,4/0.5,6/0.75,8/1.0}
  \draw[black] (\x,0.06) -- (\x,-0.06) node[below, black] {\lab};
% pressure/temp/density: high at center, falling
\draw[acc, very thick] (0.1,4.0) .. controls (1.6,3.2) and (2.6,1.2) .. (4.5,0.5)
  .. controls (6.0,0.2) and (7.0,0.1) .. (8.0,0.05);
\node[acc, anchor=south west] at (1.5,3.2) {pressure, density};
% temperature falls more gently
\draw[black, very thick, densely dashed] (0.1,3.6) .. controls (2.2,2.6) and (4.0,1.2) .. (8.0,0.2);
\node[black, anchor=west] at (4.3,1.0) {temperature};
% luminosity: rises and saturates quickly
\draw[very thick, densely dotted] (0.1,0.1) .. controls (1.2,3.2) and (2.2,3.85) .. (4.0,3.95)
  -- (8.0,3.95);
\node[anchor=south] at (5.2,3.6) {luminosity};
% mass: rises and saturates more slowly
\draw[black, very thick] (0.1,0.05) .. controls (2.0,1.8) and (4.0,3.4) .. (8.0,3.9);
\node[black, anchor=north west] at (5.6,3.0) {mass};
\end{tikzpicture}
$$

## Summary

The four coupled structure equations — mass conservation $\d m/\d r = 4\pi r^2\rho$,
hydrostatic equilibrium $\d P/\d r = -Gm\rho/r^2$, energy generation $\d L/\d r =
4\pi r^2 \rho\epsilon$, and energy transport $\d T/\d r$ — 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](/astrophysics-cosmology/stellar-structure/the-equation-of-state-and-polytropes)
that closes the system, and the polytropic models that solve it analytically, are the
subject of the next lesson.

[^co-transport]: Carroll & Ostlie, §10.4 — Energy Transport: the radiative diffusion equation, the radiative and adiabatic gradients, and the Schwarzschild criterion for convection.
[^co-model]: Carroll & Ostlie, §10.5 — Stellar Model Building: the coupled structure equations, boundary conditions, the Vogt-Russell theorem, and numerical integration.