Stellar structure

Builds on Hydrostatic equilibrium Ordinary differential equation Schwarzschild criterion for convection

Definition

Consider a static, spherically symmetric, non-rotating star of mass $M$ and radius $R$ . Let $r \in [0, R]$ be the distance from the centre. Let $m(r)$ be the mass inside radius $r$ , $\rho(r)$ the density, $P(r)$ the pressure and $T(r)$ the temperature. Let $L(r)$ be the energy per unit time that flows outward through the sphere of radius $r$ . Let the material functions $P(\rho, T)$ (equation of state), $\kappa(\rho, T)$ (opacity) and $\varepsilon(\rho, T)$ (energy generation per unit mass) be given for a fixed chemical composition. Let $G$ be the gravitational constant, $c$ the speed of light and $a$ the radiation constant. A stellar structure model is a solution of the system

$\dfrac{dm}{dr} = 4 \pi r^2 \rho$ (mass conservation);
$\dfrac{dP}{dr} = -\dfrac{G m \rho}{r^2}$ (hydrostatic equilibrium);
$\dfrac{dL}{dr} = 4 \pi r^2 \rho \varepsilon$ (energy conservation);
$\dfrac{dT}{dr} = -\dfrac{3 \kappa \rho L}{16 \pi a c r^2 T^3}$ where energy is carried by radiation, and $\dfrac{dT}{dr} = \left(1 - \dfrac{1}{\gamma}\right) \dfrac{T}{P} \dfrac{dP}{dr}$ where it is carried by convection, with $\gamma$ the adiabatic index;

with the boundary conditions $m(0) = 0$ , $L(0) = 0$ , $m(R) = M$ , and $P(R) = 0$ , $T(R) = 0$ in the simplest form.

Remarks

Example: the uniform-density model with $\rho = \rho_0$ satisfies clauses 1 and 2 with $m(r) = \frac{4}{3} \pi \rho_0 r^3$ and $P(r) = \frac{2}{3} \pi G \rho_0^2 (R^2 - r^2)$ . Indeed, $dP/dr = -\frac{4}{3} \pi G \rho_0^2 r = -G m \rho_0 / r^2$ , and $P(R) = 0$ . Non-example: a ball with uniform density and uniform pressure $P_0 > 0$ fails clause 2, since $dP/dr = 0$ while $G m \rho / r^2 > 0$ for $r > 0$ . Which form of clause 4 applies at a given radius is decided by the Schwarzschild criterion for convection. More realistic models replace $P(R) = T(R) = 0$ by conditions from a stellar atmosphere, and add time derivatives to follow the evolution of the composition. Clause 4 for radiation is often written with $\sigma = ac/4$ , the Stefan–Boltzmann constant, as $dT/dr = -3 \kappa \rho L / (64 \pi \sigma r^2 T^3)$ .

Reading path

To understand this page from the ground up, read these 88 pages in order. The foundations come first.

Brunt–Väisälä frequency concept
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Euclidean space object
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Local maximum definition
Ordinary differential equation concept
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Poisson's equation concept
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Fréchet derivative definition
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Function of several variables concept
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Fermat's theorem on stationary points statement
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Intermediate value theorem statement
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Boundedness theorem statement
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C^k function characterised by continuous derivatives statement
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An iterate of the Picard operator is a contraction statement
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Newton's law of universal gravitation statement
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Schwarzschild criterion for convection statement
Stellar structure definition

Tag	`71YMD86GX7`
URI	`m2://71YMD86GX7`
Kind	definition
Writing	W0 stub
MSC	85A15
Defines	Stellar structure
Uses	Hydrostatic equilibrium, Ordinary differential equation, Schwarzschild criterion for convection
Changed	3 November 2026

Stellar structure

Definition

Remarks

Reading path

Further reading

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