Radiative Transfer and the Transfer Equation

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 $\mathbf{d}s$ along the ray.

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

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

Combining the two, the intensity along the ray obeys

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

so that ${\tau }_{\nu }=\int {\kappa }_{\nu }\rho \mathbf{d}s$ 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 ${\kappa }_{\nu }\rho \mathbf{d}s=\mathbf{d}{\tau }_{\nu }$ puts it in a form with a single dimensionless independent variable,

The ratio

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.

The formal solution

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

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

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.¹

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

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.

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 $\mathbf{d}{I}_{\nu }/\mathbf{d}{\tau }_{\nu }=0$ 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,

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.²

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 $\mathbf{d}s=\mathbf{d}{\tau }_{\nu }/\mu$ larger for slanted rays, and the emergent intensity is

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

Inserting this and using ${\int }_0^{\infty }{e}^{-\tau /\mu }\mathbf{d}\tau /\mu =1$ and ${\int }_0^{\infty }\tau e^{-\tau /\mu }\mathbf{d}\tau /\mu =\mu$, the emergent intensity is

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.³

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.

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, $S(\tau )=a(1+\frac{3}{2}\tau )$, giving

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.⁴

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.