How heat crosses a star

Photons do not take a clear path to the surface

Deep inside a star, a photon does not normally travel freely to space. Radiation interacts repeatedly with electrons, ions and atoms. On larger scales, energy diffuses from hotter regions toward cooler ones. The diffusion calculation follows the net energy flow through the gas. Individual photons are repeatedly absorbed, re-emitted and scattered.

Opacity, $\kappa$ , measures how strongly matter blocks radiation per unit mass. A local mean free path is roughly the reciprocal of opacity times density. In a denser or more opaque layer, the energy travels only a short distance before the next interaction.

The diffusion approximation applies when conditions change gradually over many interactions. Near the photosphere, where radiation begins to escape, a stellar model needs an atmosphere boundary condition. A deep-interior diffusion formula should not be extended unchanged into freely escaping light.

\ell\sim\frac{1}{\kappa\rho}

The mean free path is

\ell

, density is

\rho

, and opacity has units of area per mass. Their combination gives a length.

Sources: Paxton et al., MESA IV, arXiv:1710.08424v1, energy equations and convective boundaries · Keszthelyi et al., arXiv:2209.06350v1, rotating stellar grids and physical prescriptions

A temperature gradient carries a luminosity

Let $F$ be outward radiative energy flow per unit area and $T$ the local temperature. The diffusion expression has a minus sign because temperature usually decreases outward while the energy flow is outward. The radiation constant is $a$ and the speed of light is $c$ .

A whole sphere at radius $r$ has surface area $4\pi r^2$ . Multiplying the flux by this area gives the luminosity $L(r)$ flowing through that layer. This is an interior luminosity, which can change from one layer to another when energy is produced or stored.

Combine radiative diffusion with spherical hydrostatic balance. The result tells us the temperature gradient that radiation would need in order to carry the specified luminosity. The pressure gradient from the previous chapter removes radius and density from the final local ratio.

F=-\frac{4acT^3}{3\kappa\rho}\frac{dT}{dr},\qquad L=4\pi r^2F

These expressions describe optically thick radiative diffusion in the spherical limit.

\nabla_{\mathrm{rad}}\equiv\left(\frac{d\ln T}{d\ln P}\right)_{\mathrm{radiation}}=\frac{3\kappa LP}{16\pi acGmT^4}

The dimensionless logarithmic gradient compares fractional changes in temperature and pressure.

P

is local pressure and

m

is enclosed mass. This is the gradient required if radiation carries the specified luminosity; the actual gradient can differ where convection operates.

Sources: Paxton et al., MESA IV, arXiv:1710.08424v1, energy equations and convective boundaries

Test a displaced parcel of gas

Now imagine lifting a small parcel slightly upward. It moves into lower pressure and expands. If it exchanges little heat during that short motion, the expansion cools it approximately adiabatically. Adiabatic means that heat exchange is small during the displacement, not that the parcel’s temperature remains constant.

The important comparison is with the surroundings. If the surrounding gas cools with height more rapidly than the parcel does, the rising parcel remains warmer and less dense. Buoyancy then pushes it farther upward. A downward displacement has the complementary effect, and convection can develop.

For uniform composition, compare the radiative gradient with the adiabatic gradient. A larger radiative gradient indicates that radiation alone would require an unstable stratification. The actual convective gradient is set by how effectively moving gas carries heat. A hot centre by itself is not evidence of convection: a stable radiative layer also has a temperature gradient.

\nabla_{\mathrm{rad}}>\nabla_{\mathrm{ad}}

This is the Schwarzschild instability criterion for a compositionally uniform layer under the stated parcel assumptions.

Sources: Paxton et al., MESA IV, arXiv:1710.08424v1, energy equations and convective boundaries

Where the familiar value of two fifths comes from

For an ideal gas with constant composition and ratio of specific heats $\gamma$ , an adiabatic change satisfies a power-law relation between pressure and density. Combine it with the ideal-gas relation to express temperature in terms of pressure.

For a monatomic ideal gas, $\gamma=5/3$ , giving an adiabatic logarithmic gradient of two fifths. This is a useful controlled example. Partial ionization, radiation pressure and electron degeneracy change the thermodynamics; actual stellar models obtain their gradients from an equation of state.

Suppose a local radiative gradient is $0.30$ and the ideal-gas adiabatic value is $0.40$ . Radiation can carry the specified luminosity without triggering this instability. If opacity doubles while the other local quantities are held fixed for the test, the required gradient doubles to $0.60$ . The same parcel test now predicts instability.

The star would subsequently readjust. Holding every other variable fixed was a way to diagnose the local response, not a calculation of a new equilibrium star.

P\propto\rho^\gamma,\quad T\propto P^{(\gamma-1)/\gamma},\quad\nabla_{\mathrm{ad}}=\frac{\gamma-1}{\gamma}=\frac25

The last equality uses the monatomic ideal-gas value only. A real MESA profile carries its own thermodynamic gradient.

Sources: Paxton et al., MESA IV, arXiv:1710.08424v1, energy equations and convective boundaries · Jermyn et al., MESA VI, arXiv:2208.03651v2, Section V energy consistency

Burning changes the buoyancy of the gas

Hydrogen burning leaves helium behind. At a given pressure and temperature, composition affects density through the average mass per freely moving particle. A parcel lifted out of a helium-rich interior can retain a denser composition than its new surroundings. That extra density can oppose its tendency to rise.

The Ledoux criterion includes this composition contribution. A sharp chemical boundary therefore matters for both mixing and stability. It is not enough to locate a sign change in a temperature-gradient curve without considering which side of the boundary supplies the material properties.

In one-dimensional evolution calculations, convection and mixing beyond its formal boundary need a prescription. Different choices can feed different amounts of hydrogen or helium into a burning core. They can consequently change a lifetime even when two models look similar from the outside. The native atlas keeps its prescriptions fixed; its saved composition curves show the resulting structure.

Sources: Paxton et al., MESA IV, arXiv:1710.08424v1, energy equations and convective boundaries · Keszthelyi et al., arXiv:2209.06350v1, rotating stellar grids and physical prescriptions · Anne Thoul and Morgan Chidester (2022), convective-boundary school solutions, Zenodo 7082536, CC BY 4.0

A composition comparison changes more than opacity

The Solar and SMC mixtures used here differ in opacity, helium content and individual metal abundances. Their saved interiors provide a useful comparison of complete stellar models. To isolate the effect of one opacity change, use the controlled local calculation above or compute a new matched evolution experiment. Keeping those two kinds of comparison distinct makes the temperature response easier to interpret.

Sources: Keszthelyi (2022), fixed reproduction package, Zenodo 7069766, CC BY 4.0 · Keszthelyi et al., arXiv:2209.06350v1, rotating stellar grids and physical prescriptions