Opacity Sources and the Rosseland Mean

Opacity is the resistance a stellar gas offers to radiation. It sets the optical-depth scale of the previous lessons, and through it the rate at which energy diffuses out of a star. Four microphysical processes contribute, each with its own frequency and temperature dependence. Because a star transports energy at all frequencies at once, the many monochromatic opacities must be combined into a single mean, and the correct average is harmonic, dominated by the frequencies where the gas is most transparent. That mean fixes the temperature gradient and, where it grows too steep, triggers convection.

The four sources of opacity

The total opacity ${\kappa }_{\nu }$ is a sum of contributions from four distinct interactions between photons and matter.¹

• Bound-bound absorption (${\kappa }_{bb}$): an electron absorbs a photon and jumps between two bound levels. This produces the spectral lines of the previous lesson. It is sharply peaked in frequency and, summed over the forest of lines, contributes substantially to the mean opacity in cool, metal-rich gas.
• Bound-free absorption (${\kappa }_{bf}$): photoionization. A photon with energy above the ionization threshold ejects a bound electron into the continuum. The cross section jumps at each threshold frequency and falls as ${\nu }^{-3}$ above it, giving the characteristic sawtooth of ionization edges.
• Free-free absorption (${\kappa }_{ff}$): a free electron passing a nucleus absorbs a photon, the inverse of bremsstrahlung. It requires the third body to conserve momentum and operates at all frequencies, contributing everywhere but most in the hot, dense interior.
• Electron scattering (${\kappa }_{es}$): a photon scatters off a free electron by the Thomson process. It removes the photon from the beam without absorbing it, is independent of frequency, and dominates at high temperature where the gas is fully ionized and few bound electrons remain.

The Kramers law

The bound-free and free-free opacities share a temperature and density dependence, because both scale with the density of absorbers and with the population of photons able to interact. Averaged over frequency, both follow the Kramers opacity law,

The steep inverse temperature dependence means opacity falls rapidly as the gas heats: hotter gas is more transparent, because ionization removes bound electrons and the free-free interaction weakens as electrons move faster past ions. The linear density dependence reflects that both processes need a second particle, an ion or a nucleus, whose abundance scales with $\rho$.

Electron scattering has no such dependence. The Thomson cross section is a constant, so its opacity depends only on the number of free electrons per unit mass,

with $X$ the hydrogen mass fraction and ${\sigma }_T$ the Thomson cross section. It is independent of frequency, temperature, and density, so it forms a floor beneath the Kramers contribution. Where Kramers opacity has fallen below this floor, in the hot interior, electron scattering dominates and the opacity becomes nearly constant.

The negative hydrogen ion

In the cool photospheres of the Sun and less massive stars, temperatures near $5000$–$6000\text{K}$ are too low to keep hydrogen ionized, and the ordinary bound-free and free-free hydrogen opacities are weak. The dominant opacity is instead the negative hydrogen ion, ${\text{H}}^{-}$: a neutral hydrogen atom binding a second electron with an ionization energy of only $0.75\text{eV}$. The loosely held electron is photodetached by any photon with wavelength shorter than about $1.64\mu \text{m}$, so ${\text{H}}^{-}$ absorbs across the whole visible and near-infrared band.²

The extra electrons that form ${\text{H}}^{-}$ come from the ionization of trace metals with low ionization potentials, so the ${\text{H}}^{-}$ opacity depends on metal abundance even though the absorber is hydrogen. It is the reason the solar continuum forms where it does, and its threshold near $1.6\mu \text{m}$ imprints a minimum in the continuous opacity that makes the infrared photosphere the deepest layer visible in the Sun.

The Rosseland mean opacity

A star transports energy at every frequency simultaneously, so the structure equations need one representative opacity. The physically correct choice follows from how radiation diffuses through the deep interior, where the field is nearly isotropic and nearly Planckian. In that diffusion regime the monochromatic flux is

The key feature is the factor $1/{\kappa }_{\nu }$: the flux at each frequency is inversely proportional to the opacity there, so transparent frequencies carry the most energy. Integrating over frequency to get the total flux,

The Rosseland mean opacity ${\bar{\kappa }}_R$ is defined so that the total flux takes the same form with a single opacity,

It is a harmonic mean: the reciprocal of the opacity is averaged, not the opacity itself. The average therefore weights toward the frequencies of lowest opacity, the transparent windows, because those are the frequencies through which the flux leaks. A narrow transparent band lowers ${\bar{\kappa }}_R$ far more than a narrow opaque line raises it, since energy simply routes around the opaque line through the transparent continuum. The weighting function $\partial B_{\nu }/\partial T$ peaks somewhat blueward of the Planck peak and its total integral is $\partial B/\partial T=ac{T}^3/\pi$, from the Stefan-Boltzmann law.³

Opacity and the temperature gradient

Solving the diffusion relation for the temperature gradient gives the radiative temperature gradient a star must have to carry its luminosity by radiation alone,

with $L_r$ the luminosity crossing radius $r$. The gradient is proportional to the opacity: a more opaque region needs a steeper temperature drop to push the same flux through. Where the opacity is high enough that the required radiative gradient exceeds the adiabatic gradient a rising gas parcel would follow, the layer becomes unstable to convection, and energy is carried by bulk motion instead. This is the Schwarzschild criterion, and opacity is the switch that decides it.

The consequences track the opacity sources. In the cool outer envelope of the Sun, the surge of ${\text{H}}^{-}$ and bound-free opacity steepens the radiative gradient past adiabatic and drives the outer convection zone that produces granulation. In the hot interior, where electron scattering has flattened the opacity, the gradient is shallow and energy diffuses radiatively. The opacity computed in this lesson thus feeds directly into the stellar-structure equations of the next module, where it appears as the coefficient linking the luminosity to the temperature profile.

Radiation and matter, treated over these four lessons through intensity, transfer, line formation, and opacity, supply the boundary between what a star emits and what it is made of. The next module turns inward, using hydrostatic equilibrium and the opacity computed here to build the interior structure of a star.