---
title: Opacity Sources and the Rosseland Mean
draft: false
module: Radiation and Matter
moduleNumber: 3
lessonNumber: 4
order: 304
summary: >
  Stellar opacity comes from four processes: bound-bound line absorption,
  bound-free photoionization, free-free absorption, and electron scattering. The
  bound-free and free-free terms follow a Kramers law, electron scattering sets a
  frequency-flat floor, and the negative hydrogen ion dominates cool photospheres.
  The Rosseland mean averages these harmonically, weighting transparent frequencies
  because they carry the flux, and its value fixes the radiative temperature
  gradient and decides where a star becomes convective.
topics: [Radiation and Matter]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 9 — Stellar Atmospheres; §9.2 Opacity; Ch. 10 — The Interiors of Stars; §10.3"
  - book: Maoz
    ref: "Ch. 3 — Stellar Physics"
---

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.[^bob-opacity]

- **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.

$$
% caption: The four opacity processes as electron transitions: a bound-bound jump
% between levels, a bound-free ejection into the continuum, a free-free absorption
% by an electron passing an ion, and photon scattering off a free electron.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% panel 1: bound-bound
\begin{scope}
\draw[black] (0,0) -- (1.4,0);
\draw[black] (0,1.0) -- (1.4,1.0);
\draw[->, very thick] (0.5,0) -- (0.5,1.0);
\node[black!70, anchor=north] at (0.7,-0.15) {bound-bound};
\end{scope}
% panel 2: bound-free
\begin{scope}[shift={(2.6,0)}]
\draw[black] (0,0) -- (1.4,0);
\draw[black, densely dashed] (0,1.4) -- (1.4,1.4);
\draw[->, very thick] (0.5,0) -- (0.5,1.6);
\node[anchor=south] at (0.5,1.6) {free};
\node[black!70, anchor=north] at (0.7,-0.15) {bound-free};
\end{scope}
% panel 3: free-free
\begin{scope}[shift={(5.2,0)}]
\fill[black] (0.7,0.2) circle (2.5pt);
\node[black, anchor=north] at (0.7,0.15) {ion};
\draw[very thick] (0.1,1.3) .. controls (0.5,0.9) and (0.9,0.9) .. (1.3,1.4);
\draw[->, thick] (1.3,1.4) -- (1.6,1.7);
\node[black!70, anchor=north] at (0.7,-0.35) {free-free};
\end{scope}
% panel 4: electron scattering
\begin{scope}[shift={(7.8,0)}]
\fill (0.7,0.7) circle (2.5pt);
\node[black, anchor=north] at (0.7,0.5) {electron};
\draw[->, thick] (0.0,1.3) -- (0.65,0.75);
\draw[->, thick] (0.75,0.65) -- (1.5,1.2);
\node[black!70, anchor=north] at (0.7,-0.35) {scattering};
\end{scope}
\end{tikzpicture}
$$

## 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**,

$$
\kappa_{\text{Kramers}} \propto \rho \, T^{-7/2} .
$$

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,

$$
\kappa_{es} = \frac{\sigma_T n_e}{\rho} = 0.02\,(1 + X)\ \text{m}^2\,\text{kg}^{-1} ,
$$

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.

$$
% caption: Opacity against temperature at fixed density. The Kramers term rises
% toward lower temperature and falls as T to the minus seven-halves toward higher
% temperature, meeting the flat electron-scattering floor in the hot interior.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {log temperature};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log opacity};
% electron scattering floor
\draw[black, thick, densely dashed] (0.4,1.2) -- (8.0,1.2);
\node[black, anchor=south east] at (8.0,1.2) {electron scattering};
% Kramers falling toward high T
\draw[acc, very thick] (2.2,4.2) .. controls (3.6,3.3) and (4.6,1.9) .. (5.4,1.25);
\node[acc, anchor=south west] at (3.4,3.1) {Kramers};
% low-T rise (bound-free / H-minus bump region)
\draw[acc, very thick, densely dotted] (0.6,1.9) .. controls (1.2,2.9) and (1.8,3.6) .. (2.2,4.2);
\node[black, anchor=east] at (1.5,3.2) {bound-free bump};
% combined floor continuation
\draw[acc, very thick] (5.4,1.25) -- (8.0,1.22);
\end{tikzpicture}
$$

## 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.[^maoz-hminus]

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.

$$
% caption: In a cool photosphere the negative hydrogen ion, a neutral atom holding
% a weakly bound extra electron, is photodetached by visible and near-infrared
% photons and dominates the continuous opacity below about six thousand kelvin.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% nucleus
\fill (1.4,1.6) circle (3pt);
\node[anchor=south] at (1.4,1.85) {proton};
% inner electron orbit
\draw[black] (1.4,1.6) circle (0.9);
\fill[black] (2.3,1.6) circle (2pt);
\node[black, anchor=west] at (2.35,1.6) {bound electron};
% weakly bound outer electron
\draw[black, densely dashed] (1.4,1.6) circle (1.7);
\fill[black] (1.4,3.3) circle (2pt);
\node[black, anchor=south] at (1.4,3.35) {weakly bound electron};
% incoming photon
\draw[->, acc, very thick] (4.6,3.3) -- (2.0,2.9);
\node[acc, anchor=west] at (4.6,3.3) {photon};
% ejected electron
\draw[->, black, thick] (1.4,3.3) -- (0.1,3.9);
\node[black, anchor=east] at (0.2,3.85) {detached};
\end{tikzpicture}
$$

## 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

$$
F_\nu = -\frac{4\pi}{3\, \kappa_\nu \rho} \, \frac{\partial B_\nu}{\partial T} \, \frac{\d T}{\d r} .
$$

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,

$$
F = -\frac{4\pi}{3\rho} \, \frac{\d T}{\d r} \int_0^\infty \frac{1}{\kappa_\nu} \, \frac{\partial B_\nu}{\partial T} \, \d\nu .
$$

The **Rosseland mean opacity** $\bar\kappa_R$ is defined so that the total flux
takes the same form with a single opacity,

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

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.[^bob-rosseland]

$$
% caption: The Rosseland weighting averages one over opacity, so a transparent
% window at moderate opacity contributes far more to the flux than an opaque line;
% the mean is pulled down toward the transparent frequencies.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.4,0) node[right, black!70] {frequency};
\draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {opacity};
% opacity curve with a transparent dip and an opaque line spike
\draw[very thick] (0.4,2.4) .. controls (1.6,2.3) and (2.2,0.6) .. (2.8,0.6)
  .. controls (3.4,0.6) and (3.6,2.2) .. (4.2,2.2)
  -- (4.6,2.2) -- (4.6,3.8) -- (4.9,3.8) -- (4.9,2.2)
  .. controls (5.6,2.2) and (6.4,2.4) .. (8.0,2.5);
\node[acc, anchor=north] at (2.8,0.55) {transparent window};
\node[black, anchor=west] at (4.95,3.6) {opaque line};
% flux weighting arrow to the window
\draw[->, acc, thick] (2.8,-0.7) -- (2.8,0.4);
\node[black, anchor=north] at (2.8,-0.7) {f\/lux escapes here};
\end{tikzpicture}
$$

## 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,

$$
\left(\frac{\d T}{\d r}\right)_{\text{rad}} = -\frac{3\, \bar\kappa_R \rho}{4\, a c\, T^3} \, \frac{L_r}{4\pi r^2} ,
$$

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.

$$
% caption: Where high opacity forces the radiative gradient steeper than the
% adiabatic gradient a rising parcel follows, the layer convects; where the
% radiative gradient stays shallow, energy diffuses as radiation.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% radiative zone
\begin{scope}
\draw[black] (0,0) rectangle (3.2,3.0);
\node[black!70, anchor=south] at (1.6,3.05) {low opacity};
\foreach \y in {0.5,1.1,1.7,2.3} \draw[->, thick] (0.6,\y) -- (2.6,\y);
\node[black] at (1.6,-0.35) {radiative};
\end{scope}
% convective zone
\begin{scope}[shift={(4.6,0)}]
\draw[black] (0,0) rectangle (3.2,3.0);
\node[black!70, anchor=south] at (1.6,3.05) {high opacity};
\draw[->, thick] (0.8,0.4) .. controls (0.4,1.5) .. (0.8,2.6);
\draw[->, thick] (1.6,2.6) .. controls (2.0,1.5) .. (1.6,0.4);
\draw[->, thick] (2.4,0.4) .. controls (2.0,1.5) .. (2.4,2.6);
\node[black] at (1.6,-0.35) {convective};
\end{scope}
\end{tikzpicture}
$$

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.

[^bob-opacity]: Carroll & Ostlie, §9.2 — the four sources of opacity, the Kramers law, electron scattering, and the definition of the Rosseland mean.
[^maoz-hminus]: Maoz, Ch. 3 — the negative hydrogen ion as the dominant continuous opacity in cool stellar photospheres.
[^bob-rosseland]: Carroll & Ostlie, §9.2 and §10.3 — the Rosseland mean as a harmonic, transparency-weighted average and its role in setting the radiative temperature gradient.