---
title: Radiative Transfer and the Transfer Equation
draft: false
module: Radiation and Matter
moduleNumber: 3
lessonNumber: 2
order: 302
summary: >
  Along a ray, matter adds intensity through emission and removes it through
  absorption. Measuring path length in optical depth turns this into the transfer
  equation, whose formal solution superposes an attenuated background on the
  source function integrated along the line of sight. In local thermodynamic
  equilibrium the source function is the Planck function, and the Eddington-Barbier
  relation shows that the emergent intensity samples the source function at optical
  depth of order unity, explaining absorption lines and solar limb darkening.
topics: [Radiation and Matter]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 9 — Stellar Atmospheres; §9.2 The Transfer Equation, §9.3 The Emergent Spectrum"
  - book: Maoz
    ref: "Ch. 3 — Stellar Physics"
---

In empty space the specific intensity is constant along a ray. Matter breaks that
constancy: a gas both emits radiation into the beam and removes radiation from it.
The bookkeeping of these two processes along the line of sight is **radiative
transfer**. Reduced to its natural variable, the optical depth, it becomes a
single first-order differential equation whose solution is the emergent spectrum
of a star. The same equation, read backward, shows why some lines appear in
absorption and others in emission, and why the solar disk fades toward its edge.

## Emission and absorption coefficients

Two coefficients describe the interaction of radiation with matter over a path
element $\d s$ along the ray.

**Absorption** removes energy in proportion to the intensity present. Passing
through a slab of thickness $\d s$, the intensity changes by

$$
\d I_\nu = -\kappa_\nu \, \rho \, I_\nu \, \d s ,
$$

where $\rho$ is the mass density and $\kappa_\nu$ is the **opacity** (or mass
absorption coefficient), with units of $\text{m}^2\,\text{kg}^{-1}$. The product
$\kappa_\nu \rho$ has units of inverse length and is the fraction of intensity
removed per unit path. "Absorption" here includes scattering out of the beam;
both remove intensity from the ray in the observed direction.

**Emission** adds energy independent of the intensity already present. The
**emission coefficient** $j_\nu$ gives the energy added per unit volume, time,
frequency, and solid angle, so

$$
\d I_\nu = j_\nu \, \d s .
$$

Combining the two, the intensity along the ray obeys

$$
\d I_\nu = -\kappa_\nu \rho \, I_\nu \, \d s + j_\nu \, \d s .
$$

$$
% caption: Along a path element ds the beam loses intensity to absorption, in
% proportion to the intensity present, and gains intensity from emission by the
% material, independent of the incident beam.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% incoming beam
\draw[->, very thick] (-1.2,1.4) -- (0.6,1.4);
\node[anchor=south] at (-0.4,1.5) {in};
% slab
\draw[fill=acc!10, draw=acc, thick] (0.6,0.4) rectangle (3.0,2.4);
\node[black!70] at (1.8,1.4) {slab ds};
% outgoing beam
\draw[->, very thick] (3.0,1.4) -- (4.8,1.4);
\node[anchor=south] at (4.0,1.5) {out};
% absorption arrow down
\draw[->, black, thick] (1.2,0.4) -- (1.2,-0.6);
\node[black, anchor=north] at (1.2,-0.6) {absorbs};
% emission arrow up
\draw[->, black, thick] (2.4,2.4) -- (2.4,3.4);
\node[black, anchor=south] at (2.4,3.4) {emits};
\end{tikzpicture}
$$

## Optical depth and the source function

The natural measure of distance for radiation is not geometric length but the
number of mean free paths traversed. The **optical depth** $\tau_\nu$ increases
along the ray according to

$$
\d\tau_\nu = \kappa_\nu \, \rho \, \d s ,
$$

so that $\tau_\nu = \int \kappa_\nu \rho \, \d s$ counts the accumulated
absorption. A medium with $\tau_\nu \ll 1$ is **optically thin**: a typical photon
crosses it without interacting. A medium with $\tau_\nu \gg 1$ is **optically
thick**: a photon is absorbed or scattered many times, and only the outermost
layers are visible.

Dividing the combined intensity equation by $\kappa_\nu \rho \, \d s = \d\tau_\nu$
puts it in a form with a single dimensionless independent variable,

$$
\frac{\d I_\nu}{\d\tau_\nu} = -I_\nu + \frac{j_\nu}{\kappa_\nu \rho} \equiv -I_\nu + S_\nu .
$$

The ratio

$$
S_\nu = \frac{j_\nu}{\kappa_\nu \rho}
$$

is the **source function**, the emission per unit absorption. It has the same
units as intensity and sets the value toward which $I_\nu$ is driven: where
$I_\nu > S_\nu$ the derivative is negative and the intensity falls; where
$I_\nu < S_\nu$ it rises. The transfer equation says the beam relaxes toward the
local source function on a scale of one optical depth.

> **Definition (Source function).** $S_\nu = j_\nu / \kappa_\nu \rho$ is the local
> ratio of emission to absorption. The transfer equation
> $\d I_\nu / \d\tau_\nu = S_\nu - I_\nu$ drives the intensity toward $S_\nu$ over
> each unit of optical depth.

## The formal solution

The transfer equation is linear and first order, so it integrates directly.
Multiplying $\d I_\nu / \d\tau_\nu + I_\nu = S_\nu$ by the integrating factor
$e^{\tau_\nu}$ turns the left side into a total derivative,

$$
\frac{\d}{\d\tau_\nu}\!\left( I_\nu e^{\tau_\nu} \right) = S_\nu \, e^{\tau_\nu} .
$$

Integrating from $0$ to $\tau_\nu$ and dividing back by $e^{\tau_\nu}$,

$$
I_\nu(\tau_\nu) = I_\nu(0)\, e^{-\tau_\nu}
  + \int_0^{\tau_\nu} S_\nu(t)\, e^{-(\tau_\nu - t)} \, \d t .
$$

The two terms read directly. The first is the background intensity $I_\nu(0)$
attenuated by the factor $e^{-\tau_\nu}$, the survival probability across the
column. The second superposes the source function emitted at every depth $t$,
each contribution attenuated by $e^{-(\tau_\nu - t)}$ over the remaining path.
Radiation emitted deeper than a few optical depths is absorbed before it emerges,
so the integral is dominated by the layers within about one optical depth of the
surface.[^bob-transfer]

For a **uniform slab** with constant $S_\nu$ and no background, the integral gives

$$
I_\nu = S_\nu \left( 1 - e^{-\tau_\nu} \right) .
$$

Optically thin, $I_\nu \approx S_\nu \tau_\nu$: the emission grows in proportion
to the column. Optically thick, $I_\nu \to S_\nu$: the slab radiates its own source
function and the observer sees no deeper. This single expression already contains
the emission-versus-absorption distinction developed below.

$$
% caption: The observer sees into a stellar atmosphere to about optical depth
% one; radiation emitted deeper is reabsorbed before escaping, so the emergent
% intensity samples the layer where tau of order unity.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% atmosphere layers
\foreach \y/\op in {0/16, 0.55/22, 1.1/30, 1.65/40, 2.2/52} {
  \fill[black!\op] (0,\y) rectangle (7.4,\y+0.55);
}
\draw[black] (0,0) rectangle (7.4,2.75);
\node[black!70, anchor=south] at (3.7,2.8) {deep interior below};
\node[black!70, anchor=north] at (3.7,-0.05) {surface};
% observer
\draw[->, very thick] (3.7,-1.1) -- (3.7,0.0);
\node[anchor=north] at (3.7,-1.1) {to observer};
% tau=1 marker
\draw[acc, densely dashed] (0,0.55) -- (7.4,0.55);
\node[acc, anchor=west, fill=white, inner sep=1pt] at (5.6,0.55) {optical depth one};
% escaping photon
\draw[->, black, thick] (2.2,0.55) -- (2.2,-0.3);
\node[black, anchor=east] at (2.2,-0.1) {escapes};
% trapped photon
\draw[->, black, thick, densely dotted] (5.4,1.9) -- (5.4,1.35);
\node[black, anchor=west] at (5.5,1.7) {reabsorbed};
\end{tikzpicture}
$$

## Thermodynamic equilibrium and LTE

In strict **thermodynamic equilibrium** (TE), matter and radiation share one
temperature, the intensity is isotropic and equal to $B_\nu(T)$, and the transfer
equation forces $S_\nu = B_\nu(T)$: with $I_\nu = B_\nu$ constant, the derivative
$\d I_\nu / \d\tau_\nu = 0$ requires $S_\nu = I_\nu = B_\nu$. This is **Kirchhoff's
law** in the form $j_\nu = \kappa_\nu \rho \, B_\nu(T)$, tying emission to
absorption through the Planck function.

A stellar atmosphere is not in strict TE: it has a temperature gradient and a net
outward flux, or no radiation would escape. But over a region small compared with
the scale on which $T$ changes, collisions keep the level populations and velocity
distribution at their equilibrium values for the local temperature. This is
**local thermodynamic equilibrium** (LTE), and in it the source function keeps its
equilibrium form,

$$
S_\nu = B_\nu(T(\tau_\nu)) ,
$$

even though the intensity itself is neither isotropic nor Planckian. LTE is the
default assumption for computing stellar continua; it holds where collisions
dominate over radiative transitions and fails in low-density, radiation-dominated
regions such as the outermost photosphere and chromosphere.[^maoz-lte]

## The Eddington-Barbier relation

The emergent intensity leaving the surface of an atmosphere follows from the
formal solution with the observer at $\tau_\nu = 0$ looking inward along a ray at
angle $\theta$ to the vertical, $\mu = \cos\theta$. Measuring optical depth
vertically, the path element is $\d s = \d\tau_\nu / \mu$ larger for slanted rays,
and the emergent intensity is

$$
I_\nu(0, \mu) = \int_0^\infty S_\nu(\tau_\nu) \, e^{-\tau_\nu / \mu} \, \frac{\d\tau_\nu}{\mu} .
$$

Suppose the source function is approximately linear in optical depth over the
region that contributes,

$$
S_\nu(\tau_\nu) = a_\nu + b_\nu \, \tau_\nu .
$$

Inserting this and using $\int_0^\infty e^{-\tau/\mu}\,\d\tau/\mu = 1$ and
$\int_0^\infty \tau \, e^{-\tau/\mu}\,\d\tau/\mu = \mu$, the emergent intensity is

$$
I_\nu(0, \mu) = a_\nu + b_\nu \, \mu = S_\nu(\tau_\nu = \mu) .
$$

This is the **Eddington-Barbier relation**: the emergent intensity in a direction
$\mu$ equals the source function evaluated at the optical depth $\tau_\nu = \mu$.
Looking straight down ($\mu = 1$) one sees the source function at $\tau_\nu = 1$;
looking toward the limb ($\mu \to 0$) one sees the shallower, cooler layers at
$\tau_\nu \to 0$. The relation makes precise the earlier statement that the
spectrum is formed near optical depth unity.[^bob-eddbarbier]

## Absorption and emission lines

Whether a spectral feature appears in absorption or emission is decided by the
sign of the temperature gradient along the line of sight, through the source
function. In LTE $S_\nu = B_\nu(T)$, which increases with temperature. At the
frequency of a spectral line the opacity $\kappa_\nu$ is large, so a given optical
depth is reached higher in the atmosphere, where the temperature is lower; the
emergent intensity, sampling $S_\nu$ at $\tau_\nu = 1$, is therefore lower at line
center than in the neighbouring continuum. That intensity deficit is an
**absorption line**.

- **Absorption line**: temperature falls outward (the normal photospheric case).
  Line opacity pushes the $\tau_\nu = 1$ surface up into cooler gas, so
  $S_\nu$ and the emergent intensity drop at line center.
- **Emission line**: temperature rises outward, as in the chromosphere and
  corona, or a hot optically thin gas seen against a cold background. The line
  samples hotter gas than the continuum, so the intensity rises at line center.

$$
% caption: With a source function rising into the star, the strong line opacity
% samples shallow, cooler layers and cuts an absorption trough; a source function
% rising outward instead fills the line into emission.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% left panel: absorption
\begin{scope}
\draw[->, black] (0,0) -- (3.4,0) node[right, black!70] {wavelength};
\draw[->, black] (0,0) -- (0,2.8) node[above, black!70] {intensity};
\draw[very thick] (0.2,2.2) -- (1.2,2.2) .. controls (1.55,2.2) and (1.55,0.7) .. (1.7,0.7)
  .. controls (1.85,0.7) and (1.85,2.2) .. (2.2,2.2) -- (3.2,2.2);
\node[black!70, anchor=south] at (1.7,0.0) {temperature falls outward};
\node[anchor=south west] at (0.2,2.2) {absorption};
\end{scope}
% right panel: emission
\begin{scope}[shift={(5.0,0)}]
\draw[->, black] (0,0) -- (3.4,0) node[right, black!70] {wavelength};
\draw[->, black] (0,0) -- (0,2.8) node[above, black!70] {intensity};
\draw[very thick] (0.2,0.9) -- (1.2,0.9) .. controls (1.55,0.9) and (1.55,2.4) .. (1.7,2.4)
  .. controls (1.85,2.4) and (1.85,0.9) .. (2.2,0.9) -- (3.2,0.9);
\node[black!70, anchor=south] at (1.7,0.0) {temperature rises outward};
\node[anchor=north west] at (0.2,0.85) {emission};
\end{scope}
\end{tikzpicture}
$$

## Limb darkening

The Eddington-Barbier relation predicts a measurable consequence of the
photospheric temperature gradient: the solar disk is brighter at its center than
at its edge. The emergent intensity is $I(0, \mu) = S(\tau = \mu)$, so the center
of the disk ($\mu = 1$) samples $S$ at $\tau = 1$, deep and hot, while the limb
($\mu \to 0$) samples $S$ at $\tau \to 0$, shallow and cool. Since $S = B(T)$
increases with $T$, the limb is fainter.

For a grey atmosphere the source function in the Eddington approximation is linear,
$S(\tau) = a(1 + \tfrac{3}{2}\tau)$, giving

$$
\frac{I(0, \mu)}{I(0, 1)} = \frac{a\,(1 + \tfrac{3}{2}\mu)}{a\,(1 + \tfrac{3}{2})}
  = \frac{2 + 3\mu}{5} .
$$

At disk center the ratio is $1$; at the limb it drops to $2/5$, so the edge of the
Sun radiates about $40\%$ of the central intensity in the continuum. Measuring the
limb-darkening curve inverts the argument: its shape maps the run of the source
function, hence the temperature, with optical depth, making limb darkening a
direct probe of the outer temperature gradient.[^bob-limb]

$$
% caption: Continuum limb darkening across the solar disk. The ratio of emergent
% to central intensity follows the two-plus-three-mu over five law, falling to
% about two-fifths at the limb where cool shallow layers are sampled.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {cosine of viewing angle};
\draw[->, black] (0,0) -- (0,4.0) node[above, black!70] {relative intensity};
% axis ticks
\node[black, anchor=north] at (0,0) {limb};
\node[black, anchor=north] at (7.0,0) {center};
% baseline gridlines
\draw[black, densely dotted] (0,3.5) -- (7.0,3.5);
\node[black, anchor=east] at (-0.05,3.5) {1.0};
\draw[black, densely dotted] (0,1.4) -- (7.0,1.4);
\node[black, anchor=east] at (-0.05,1.4) {0.4};
% straight line from 0.4 to 1.0
\draw[acc, very thick] (0,1.4) -- (7.0,3.5);
\fill[acc] (0,1.4) circle (2pt);
\fill[acc] (7.0,3.5) circle (2pt);
\node[acc, anchor=north west] at (3.2,2.5) {linear limb law};
\end{tikzpicture}
$$

The transfer equation and its formal solution reduce the formation of a stellar
spectrum to the run of the source function with optical depth. What remains is to
compute the opacity $\kappa_\nu$ that sets the optical-depth scale. The next two
lessons treat the two pieces that determine it: the shapes of individual spectral
lines, and the microphysical processes that make a stellar gas absorb.

[^bob-transfer]: Carroll & Ostlie, §9.2 — the transfer equation, optical depth, the source function, and the formal solution.
[^maoz-lte]: Maoz, Ch. 3 — thermodynamic equilibrium, local thermodynamic equilibrium, and Kirchhoff's law relating emission and absorption.
[^bob-eddbarbier]: Carroll & Ostlie, §9.3 — the emergent spectrum and the Eddington-Barbier relation for a source function linear in optical depth.
[^bob-limb]: Carroll & Ostlie, §9.3 — limb darkening as a probe of the photospheric temperature gradient.