2.1 Equations of Stellar Structure

Fundamental Equations

Stellar structure equations are the coupled differential equations that describe how pressure, mass, luminosity, and temperature vary with radius inside a star. Solving them simultaneously gives you a complete model of a star's interior, from core to surface. These four equations, combined with an equation of state and opacity data, form a boundary value problem whose solutions predict stellar luminosities, lifetimes, and evolutionary tracks.

Hydrostatic Equilibrium and Mass Conservation

Hydrostatic equilibrium describes the balance between the inward pull of gravity and the outward push of the pressure gradient at every point inside a star. If this balance breaks, the star either collapses or expands on a dynamical (free-fall) timescale.

$\frac{dP}{dr} = -\frac{G M_r \rho}{r^2}$

where $P$ is pressure, $r$ is radial distance from the center, $G$ is the gravitational constant, $M_r$ is the mass enclosed within radius $r$, and $\rho$ is the local density. The negative sign tells you pressure decreases outward, which makes sense: the deepest layers bear the weight of everything above them.

Mass conservation (the continuity equation) relates the enclosed mass to the local density:

$\frac{dM_r}{dr} = 4\pi r^2 \rho$

This says that as you move outward by $dr$, the additional mass enclosed is just the density times the volume of a thin spherical shell. Together with hydrostatic equilibrium, it lets you reconstruct the full mass and pressure profiles once you know $\rho(r)$.

Energy Conservation and Equation of State

Energy conservation (the luminosity equation) tracks how luminosity builds up through the star:

$\frac{dL_r}{dr} = 4\pi r^2 \rho \epsilon$

Here $L_r$ is the luminosity at radius $r$ and $\epsilon$ is the energy generation rate per unit mass. In the core, $\epsilon$ is dominated by nuclear reactions (pp-chain, CNO cycle). Outside the nuclear burning region, $\epsilon \approx 0$ and $L_r$ stays roughly constant. During contraction phases, gravitational energy release (the Kelvin-Helmholtz mechanism) also contributes to $\epsilon$.

The equation of state (EOS) closes the system by relating thermodynamic variables. For an ideal, non-degenerate gas:

$P = \frac{\rho k_B T}{\mu m_H}$

where $k_B$ is Boltzmann's constant, $T$ is temperature, $\mu$ is the mean molecular weight, and $m_H$ is the hydrogen atom mass. This is adequate for most main-sequence stellar interiors, but the EOS must be modified in several regimes:

• Radiation pressure becomes significant in massive stars ($\gtrsim 10\,M_\odot$), adding a term $P_{\text{rad}} = \frac{1}{3}aT^4$
• Electron degeneracy pressure dominates in white dwarfs and the cores of evolved low-mass stars, where the Pauli exclusion principle supports the star independent of temperature
• Partial ionization zones near the surface alter $\mu$ and the adiabatic exponent, affecting convective stability

Stellar Structure

Pressure and Temperature Gradients

The pressure gradient is set directly by hydrostatic equilibrium. Steeper gradients correspond to regions of stronger gravitational compression, typically deep in the core where $M_r/r^2$ is large.

The temperature gradient depends on how energy is transported. In radiative zones, photons carry the energy outward, and the gradient is:

$\frac{dT}{dr} = -\frac{3}{4ac} \frac{\kappa \rho L_r}{4\pi r^2 T^3}$

where $\kappa$ is the opacity, $a$ is the radiation density constant ($a = 4\sigma/c$), and $c$ is the speed of light. Notice that high opacity or high luminosity steepens the radiative gradient, which can trigger convection.

In convective zones, the actual temperature gradient is nearly equal to the adiabatic gradient:

$\nabla_{\text{ad}} = \left(\frac{\partial \ln T}{\partial \ln P}\right)_s$

For a fully ionized ideal gas, $\nabla_{\text{ad}} = 2/5$. Convection is so efficient at transporting energy that the temperature profile barely departs from adiabatic, except in superadiabatic surface layers.

Density Profile and Stellar Layers

Density decreases from center to surface due to gravitational stratification. To give you a sense of scale for a solar-type main-sequence star:

• Core: $\rho \sim 1.5 \times 10^5$ kg/m$^3$ (about 150 g/cm$^3$ for the Sun)
• Outer envelope: $\rho$ can drop below $10^{-4}$ kg/m$^3$ near the photosphere

A star's interior divides into layers defined by the dominant energy transport mechanism and nuclear processes:

• Core: site of nuclear burning; highest $T$, $\rho$, and $P$
• Radiative zone: energy carried by photon diffusion; stable against convection
• Convective zone: energy carried by bulk fluid motions; present where the Schwarzschild criterion is satisfied
• Photosphere: the optically thin surface from which photons escape

The boundaries between these layers are not arbitrary. They shift based on opacity, composition gradients (e.g., hydrogen exhaustion in the core), and the local temperature gradient. In a $1\,M_\odot$ star, the outer convective zone extends inward from the surface, while in a $10\,M_\odot$ star, a convective core sits beneath an extended radiative envelope. This reversal is driven by the strong temperature sensitivity of the CNO cycle ($\epsilon \propto T^{16}$), which concentrates energy generation and steepens the core temperature gradient.

Energy Transport and Opacity

Radiative and Convective Energy Transport

Radiative transport moves energy outward through repeated photon absorption and re-emission. Because the mean free path of a photon in a stellar interior is very short (on the order of centimeters in the solar core), this process is really photon diffusion. The radiative diffusion approximation gives the luminosity as:

$L_r = -\frac{4\pi r^2}{3\kappa\rho} \frac{4acT^3}{1} \frac{dT}{dr}$

This is just a rearrangement of the radiative temperature gradient equation above. Radiative transport dominates wherever the material is hot and highly ionized, keeping opacity relatively low.

Convective transport takes over when the radiative temperature gradient becomes steeper than the adiabatic gradient. The Schwarzschild criterion for convective instability is:

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

where $\nabla_{\text{rad}} = \frac{3}{16\pi acG}\frac{\kappa L_r P}{M_r T^4}$ is the temperature gradient that would be required if all energy were carried radiatively. When this exceeds $\nabla_{\text{ad}}$, a displaced parcel of gas is buoyant and keeps rising, so convection sets in.

Convection dominates in:

• Outer envelopes of cool stars ($\lesssim 1.5\,M_\odot$), where partial ionization of H and He raises $\kappa$ and steepens $\nabla_{\text{rad}}$
• Cores of massive stars ($\gtrsim 1.5\,M_\odot$), where the CNO cycle's extreme temperature sensitivity produces a steep luminosity gradient

Conduction is generally negligible in normal stellar interiors but becomes the dominant transport mechanism in degenerate matter (white dwarf interiors), where electrons have long mean free paths.

Opacity Sources and Rosseland Mean Opacity

Opacity ($\kappa$) quantifies how effectively stellar material absorbs or scatters radiation. It's one of the most important inputs to stellar models because it controls both the radiative temperature gradient and the onset of convection.

The main opacity sources in stellar interiors:

• Bound-free absorption (photoionization): a photon ionizes an atom. Dominant at intermediate temperatures ($\sim 10^4$–$10^5$ K) where partially ionized species are abundant.
• Free-free absorption (inverse bremsstrahlung): a free electron absorbs a photon while passing near an ion. Scales roughly as $\kappa_{\text{ff}} \propto \rho T^{-7/2}$ (Kramers' opacity law).
• Electron scattering (Thomson scattering): photons scatter off free electrons. Nearly independent of frequency and temperature; dominates in hot, fully ionized interiors of massive stars. $\kappa_{\text{es}} \approx 0.2(1+X)$ cm$^2$/g, where $X$ is the hydrogen mass fraction.
• Bound-bound absorption (line opacity): significant near the surface and in partial ionization zones, but less important in deep interiors.

Since opacity varies strongly with photon frequency, stellar structure calculations use the Rosseland mean opacity, a weighted harmonic mean:

$\frac{1}{\kappa_R} = \frac{\int_0^\infty \frac{1}{\kappa_\nu} \frac{\partial B_\nu}{\partial T}\, d\nu}{\int_0^\infty \frac{\partial B_\nu}{\partial T}\, d\nu}$

Here $\kappa_\nu$ is the monochromatic opacity and $B_\nu$ is the Planck function. The weighting by $\partial B_\nu / \partial T$ means the Rosseland mean is most sensitive to frequencies near the peak of the radiation field's temperature derivative, which is where most of the energy flux occurs. Because it's a harmonic mean, the Rosseland opacity is biased toward frequency windows of low opacity, reflecting the fact that radiation preferentially leaks through the most transparent channels.

In practice, opacity tables (such as the OPAL tables from Livermore) are pre-computed as functions of $T$, $\rho$, and composition, then interpolated during numerical stellar model calculations.