Emission and Source Function

Every photon that reaches a telescope was, at some point, both emitted and absorbed by matter — often many times over. Radiative transfer is the accounting of that traffic: how the specific intensity $I_\nu$ of a beam grows and fades as it crosses gas that adds light (spontaneous and stimulated emission) and removes it (absorption and scattering). Two microscopic coefficients carry the physics: the emission coefficient $j_\nu$ (energy poured into the beam per unit length) and the absorption coefficient $\alpha_\nu$ (the fraction taken back out). Each depends on density, composition, ionization, and level populations in complicated ways — yet the emergent radiation depends on them only through their ratio, the source function $S_\nu \equiv j_\nu/\alpha_\nu$ . That is the quiet miracle this article is about. The reason is structural. Measure distance not in centimetres but in optical depth $\tau_\nu$ , the number of mean free paths along the ray ( $\mathrm{d}\tau_\nu = \alpha_\nu\,\mathrm{d}s$ ). In that natural coordinate the transfer equation collapses to a single relaxation law, $\mathrm{d}I_\nu/\mathrm{d}\tau_\nu = S_\nu - I_\nu$ : brighter than $S_\nu$ the beam dims, fainter it brightens, and the fixed point is $I_\nu = S_\nu$ . Over a few optical depths the intensity forgets where it started and settles onto the local source function. In thermodynamic equilibrium the radiation is blackbody and unchanging, so the law forces $S_\nu = B_\nu(T)$ — Kirchhoff's law, $j_\nu = \alpha_\nu B_\nu(T)$ . When absorption gives way to scattering, $S_\nu$ becomes a mix of thermal $B_\nu$ and mean intensity $J_\nu$ , the radiation feeds itself, and the local ODE turns into a non-local integral equation governing spectral lines, limb darkening, and the escape of light from stars.

Key takeaways

The source function $S_\nu \equiv j_\nu/\alpha_\nu$ is the intensity toward which a beam relaxes; it carries all the microphysics the emergent light can ever see.
In optical-depth form the transfer equation is $\mathrm{d}I_\nu/\mathrm{d}\tau_\nu = S_\nu - I_\nu$ — a relaxation toward $S_\nu$ at one e-fold per unit optical depth, which is why only the ratio $j_\nu/\alpha_\nu$ matters.
In (local) thermodynamic equilibrium the source function is the Planck function, $S_\nu = B_\nu(T)$ — Kirchhoff's law; for the Sun at $5772\ \mathrm{K}$ , $500\ \mathrm{nm}$ this is $2.19\times10^{-8}\ \mathrm{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$ .
The formal solution makes the emergent intensity an $e^{-\tau}$ -weighted average of $S_\nu$ : we see to optical depth unity, and by Eddington-Barbier $I_\nu(0,\mu) = S_\nu(\tau_\nu=\mu)$ .
With scattering, $S_\nu = (1-\epsilon_\nu)J_\nu + \epsilon_\nu B_\nu$ couples the field to itself and only thermalizes to $B_\nu$ at depth $\tau_\nu \sim \epsilon_\nu^{-1/2}$ .
Key idea: the transfer equation in optical-depth form, $\mathrm{d}I_\nu/\mathrm{d}\tau_\nu = S_\nu - I_\nu$ , shows the specific intensity always relaxes toward the source function $S_\nu = j_\nu/\alpha_\nu$ ; in (local) thermodynamic equilibrium $S_\nu = B_\nu(T)$ .

The core insight

The transfer equation in optical-depth form, $\mathrm{d}I_\nu/\mathrm{d}\tau_\nu = S_\nu - I_\nu$ , shows that the specific intensity always relaxes toward the source function $S_\nu = j_\nu/\alpha_\nu$ ; in (local) thermodynamic equilibrium $S_\nu = B_\nu(T)$ .

Where this lives today

The source function is the working engine of quantitative spectroscopy. Every stellar-atmosphere code — from the classical LTE grids (ATLAS, MARCS) to the non-LTE machinery (TLUSTY, PHOENIX, RH) — is at heart a solver for $S_\nu$ versus depth, iterating equation (8) until $S_\nu$ and $J_\nu$ are consistent. Fitting synthetic spectra to observed line strengths is how elemental abundances are measured across the Galaxy, and the recognition that the solar oxygen and carbon lines form well out of LTE — with $S_\nu$ far below $B_\nu$ in their cores — drove the downward revision of the solar metallicity in the 2000s, reopening the "solar abundance problem" against helioseismology. Instruments now map solar limb darkening and center-to-limb line variations precisely enough to invert for the photospheric stratification — a direct descendant of Schwarzschild's 1906 calculation. The same formalism, with $S_\nu = J_\nu$ for electron scattering, governs hot-star winds, accretion-disk atmospheres, and supernova photospheres, while in the millimetre sky the Rayleigh-Jeans source function underlies the brightness-temperature calibration of ALMA and CMB experiments.

The physical picture

Picture a pencil of light threading a slab of gas. In each thin layer two things happen at once: some passing light is absorbed or scattered out of the beam in proportion to its brightness — the $-\alpha_\nu I_\nu$ term — and the gas adds its own emitted light, a fixed amount set by its state — the $+j_\nu$ term. Balance the two and there is a special brightness where addition cancels removal: $j_\nu = \alpha_\nu I_\nu$ , i.e. $I_\nu = j_\nu/\alpha_\nu = S_\nu$ . Brighter than $S_\nu$ , removal wins and the beam dims; fainter, emission wins and it brightens. The gas forever nudges the beam toward the single value $S_\nu$ , one e-fold of relaxation per unit optical depth $\tau_\nu$ .

Figure

The specific intensity relaxes toward the source function Sν: whatever the boundary value at τν = 0, after a few optical depths the beam forgets it and settles on Sν at one e-fold per unit optical depth, since dIν/dτν = Sν − Iν

Because absorption and re-emission are two sides of one coin — a good absorber is a good emitter, the content of Kirchhoff's law — it is the ratio $j_\nu/\alpha_\nu$ that fixes the outcome, never the coefficients separately. Double the density and both $j_\nu$ and $\alpha_\nu$ double; $S_\nu$ is untouched, and the slab merely reaches its equilibrium brightness in half the distance. This is why a spectrum can be predicted without knowing the absolute density: the emergent light reports on $S_\nu$ and on how much $\tau_\nu$ was crossed, and nothing else. In a real star we see down to about optical depth unity — deeper layers are hidden behind their own opacity, shallower ones are too thin to matter — so the emergent intensity is essentially $S_\nu$ at $\tau_\nu \approx 1$ . Because $S_\nu = B_\nu(T)$ in the photosphere and temperature rises inward, a slanted ray toward the limb reaches $\tau_\nu = 1$ while still high and cool, while a ray to disk centre reaches it deeper and hotter. The disk therefore darkens toward its edge, and the exact law that follows from a gray atmosphere is the punchline of the next figure.

Figure

Solar limb darkening from the Eddington-Barbier relation with a gray atmosphere: the emergent-intensity ratio Iμ/Icentre = (2 + 3μ)/5 falls linearly from 1 at disk centre μ = 1 to 0.40 at the limb μ = 0, because slanted rays reach τν = 1 in shallow, cooler layers

Pure scattering breaks this comfortable picture. A scattered photon keeps its energy but loses its memory of the local temperature: the gas re-emits it with a brightness set not by $B_\nu(T)$ but by the ambient mean intensity $J_\nu = \tfrac{1}{4\pi}\oint I_\nu\,\mathrm{d}\Omega$ . Now $S_\nu$ at one point depends on the radiation field everywhere, and the tidy local relaxation becomes a self-referential integral equation. Solving it — deciding how deep photons random-walk before they thermalize — is the central problem of stellar-atmosphere and line-formation theory, and everything below is built toward it.

The core equations

The emission coefficient measures how much energy the gas radiates into the beam per unit path length — a property of the matter alone, set by density, temperature, and level populations.

j_\nu = \frac{\mathrm{d}E}{\mathrm{d}V\,\mathrm{d}t\,\mathrm{d}\Omega\,\mathrm{d}\nu}, \qquad \mathrm{d}I_\nu = j_\nu\,\mathrm{d}s \qquad (1)

where: $j_\nu$ — emission coefficient (emissivity per unit solid angle), $\mathrm{erg\,s^{-1}\,cm^{-3}\,Hz^{-1}\,sr^{-1}}$ ; $\mathrm{d}E$ — energy radiated into $\mathrm{d}\Omega$ over band $\mathrm{d}\nu$ ; $\mathrm{d}s$ — path element; $I_\nu$ — specific intensity, $\mathrm{erg\,s^{-1}\,cm^{-2}\,Hz^{-1}\,sr^{-1}}$ . Reading it: over a step $\mathrm{d}s$ the gas adds $j_\nu\,\mathrm{d}s$ of intensity, independent of how bright the beam already is. The absorption coefficient measures the opposite process — light removed from the beam — and it is proportional to the intensity present, because you can only take away light that is there.

\mathrm{d}I_\nu = -\alpha_\nu I_\nu\,\mathrm{d}s \qquad (2)

where: $\alpha_\nu$ — linear absorption coefficient, $\mathrm{cm^{-1}}$ (bundles true absorption, stimulated emission as negative absorption, and scattering); $\alpha_\nu I_\nu$ — energy removed per unit path length. Reading it: the fractional loss $\mathrm{d}I_\nu/I_\nu = -\alpha_\nu\,\mathrm{d}s$ defines the mean free path $1/\alpha_\nu$ . Their ratio is the source function — the single combination the emergent light responds to, in units of intensity, not emissivity.

S_\nu \equiv \frac{j_\nu}{\alpha_\nu} \qquad (3)

where: $S_\nu$ — source function, same units as $I_\nu$ . Reading it: $S_\nu$ is the brightness at which emission into the beam exactly balances absorption out of it — the local equilibrium the intensity is driven toward. Adding the two processes gives the equation of radiative transfer in its raw, path-length form — loss plus gain per unit distance.

\frac{\mathrm{d}I_\nu}{\mathrm{d}s} = -\alpha_\nu I_\nu + j_\nu \qquad (4)

where: $-\alpha_\nu I_\nu$ — loss by absorption and scattering out of the beam; $+j_\nu$ — gain by emission and scattering into it. Reading it: everything reduces to a competition between a term that scales with $I_\nu$ and one that does not. Dividing by $\alpha_\nu$ and switching the ruler to optical depth turns equation (4) into a pure relaxation law — the form in which the whole theory becomes transparent.

\frac{\mathrm{d}I_\nu}{\mathrm{d}\tau_\nu} = S_\nu - I_\nu, \qquad \mathrm{d}\tau_\nu = \alpha_\nu\,\mathrm{d}s \qquad (5)

where: $\tau_\nu$ — optical depth, dimensionless, the number of mean free paths along the ray. Reading it: the sign of $S_\nu - I_\nu$ decides whether the beam brightens or dims, and the fixed point $I_\nu = S_\nu$ is approached at one e-fold per unit $\tau_\nu$ . In local thermodynamic equilibrium the matter populations follow Boltzmann and Saha statistics at the local temperature, which — as the first derivation shows — pins the source function to the Planck function. This is Kirchhoff's law for the source function.

S_\nu = B_\nu(T) = \frac{2h\nu^3}{c^2}\,\frac{1}{e^{h\nu/k_BT}-1} \qquad (6)

where: $B_\nu(T)$ — Planck function; $h$ — Planck constant; $k_B$ — Boltzmann constant; $c$ — speed of light; $T$ — local matter temperature. Reading it: in LTE the source function needs no knowledge of the radiation field — only the local temperature. Integrating the relaxation law along the ray gives the formal solution: the attenuated boundary value plus an exponentially weighted average of the source function.

I_\nu(\tau_\nu) = I_\nu(0)\,e^{-\tau_\nu} + \int_0^{\tau_\nu} S_\nu(t)\,e^{-(\tau_\nu - t)}\,\mathrm{d}t \qquad (7)

where: $I_\nu(0)$ — intensity at the boundary $\tau_\nu = 0$ ; $e^{-\tau_\nu}$ — attenuation of that boundary term; $t$ — dummy optical-depth variable. Reading it: layers within about one optical depth of the observer dominate; deeper ones are exponentially suppressed — the precise sense of "we see to $\tau_\nu \approx 1$ ." When scattering competes with true absorption, the source function is a weighted blend of the thermal Planck term and the mean intensity, set by the photon destruction probability.

S_\nu = \frac{\kappa_\nu B_\nu(T) + \sigma_\nu J_\nu}{\kappa_\nu + \sigma_\nu} = (1-\epsilon_\nu)\,J_\nu + \epsilon_\nu\,B_\nu(T) \qquad (8)

where: $\kappa_\nu$ — true (thermal) absorption coefficient; $\sigma_\nu$ — scattering coefficient; $J_\nu = \tfrac{1}{4\pi}\oint I_\nu\,\mathrm{d}\Omega$ — mean intensity; $\epsilon_\nu = \kappa_\nu/(\kappa_\nu+\sigma_\nu)$ — photon destruction probability per interaction. Reading it: $\epsilon_\nu \to 1$ recovers pure LTE ( $S_\nu = B_\nu$ ); $\epsilon_\nu \to 0$ gives pure scattering ( $S_\nu = J_\nu$ ) and the non-local problem.

Derivation AFrom the transfer equation to the relaxation form and Kirchhoff's law

The goal is to show that the raw path-form transfer equation, which mixes the two microphysical coefficients $j_\nu$ and $\alpha_\nu$ , collapses into a single relaxation law in which only their ratio survives, and then to read off why the source function must equal the Planck function in equilibrium. We change variables from geometric path to optical depth, identify the fixed point, and impose the conditions defining thermodynamic equilibrium.

Step 1. Start from $\mathrm{d}I_\nu/\mathrm{d}s = -\alpha_\nu I_\nu + j_\nu$ and define optical depth by $\mathrm{d}\tau_\nu = \alpha_\nu\,\mathrm{d}s$ , so that $\tau_\nu$ counts mean free paths and increases along the direction of propagation. Why: absorption sets the only natural length scale in the problem, the mean free path $1/\alpha_\nu$ ; measuring distance in those units is what removes the absolute magnitude of the coefficients from the equation.

Step 2. Divide the whole equation by $\alpha_\nu$ . The left side becomes $\alpha_\nu^{-1}\,\mathrm{d}I_\nu/\mathrm{d}s = \mathrm{d}I_\nu/\mathrm{d}\tau_\nu$ by the chain rule, and the right side becomes $-I_\nu + j_\nu/\alpha_\nu = -I_\nu + S_\nu$ . The result is $\mathrm{d}I_\nu/\mathrm{d}\tau_\nu = S_\nu - I_\nu$ . Why: after this single division the two coefficients appear only in the combination $j_\nu/\alpha_\nu$ , proving that the emergent radiation cannot depend on them any other way — the entire microphysics is compressed into $S_\nu$ .

Step 3. Read the fixed point. The right-hand side vanishes when $I_\nu = S_\nu$ ; for $I_\nu > S_\nu$ the derivative is negative (the beam dims) and for $I_\nu < S_\nu$ positive (it brightens). So $S_\nu$ is a stable attractor approached exponentially, $I_\nu - S_\nu \propto e^{-\tau_\nu}$ — one e-fold per unit optical depth. Why: this is the precise content of "the beam relaxes toward the source function," and what makes deep layers thermalize.

Step 4. Impose full thermodynamic equilibrium: the radiation is the isotropic blackbody field $I_\nu = B_\nu(T)$ and homogeneous, so $\mathrm{d}I_\nu/\mathrm{d}\tau_\nu = 0$ . The relaxation law then gives $0 = S_\nu - B_\nu(T)$ , hence $S_\nu = B_\nu(T)$ and $j_\nu = \alpha_\nu B_\nu(T)$ . Why: because $j_\nu$ and $\alpha_\nu$ are local matter properties, this equality persists in local thermodynamic equilibrium even when the radiation field is not blackbody — the generalization is Kirchhoff's law.

Go deeper · Derivation B · 1 minThe formal solution as an exponentially weighted average of the source function

Derivation BThe formal solution as an exponentially weighted average of the source function

The aim is to integrate the relaxation law exactly, turning the differential statement "the beam is pushed toward $S_\nu$ " into an explicit expression for the emergent intensity in terms of the run of $S_\nu$ along the ray. We treat it as a linear first-order ODE, multiply by an integrating factor that makes the left side a perfect derivative, integrate over the ray, and interpret the two surviving terms.

Step 1. Rewrite equation (5) as $\mathrm{d}I_\nu/\mathrm{d}\tau_\nu + I_\nu = S_\nu$ and multiply by the integrating factor $e^{\tau_\nu}$ , giving an exact derivative $\mathrm{d}(I_\nu e^{\tau_\nu})/\mathrm{d}\tau_\nu = S_\nu\,e^{\tau_\nu}$ . Why: the factor $e^{\tau_\nu}$ is chosen so the loss term $+I_\nu$ folds into the product rule, leaving a form that integrates in one stroke.

Step 2. Integrate from $t = 0$ to $t = \tau_\nu$ : $I_\nu(\tau_\nu)\,e^{\tau_\nu} - I_\nu(0) = \int_0^{\tau_\nu} S_\nu(t)\,e^{t}\,\mathrm{d}t$ . Why: the definite integral accumulates the source contribution from every layer along the ray, each weighted by where it sits.

Step 3. Multiply by $e^{-\tau_\nu}$ : $I_\nu(\tau_\nu) = I_\nu(0)\,e^{-\tau_\nu} + \int_0^{\tau_\nu} S_\nu(t)\,e^{-(\tau_\nu - t)}\,\mathrm{d}t$ . The first term is the boundary light attenuated by the full optical depth; the second is a running average of $S_\nu$ weighted by $e^{-(\tau_\nu-t)}$ , falling off one e-fold per unit depth back from the observer. Why: this makes precise "we see to optical depth unity" — layers a few mean free paths past $\tau_\nu = 1$ contribute exponentially little.

Step 4. Check the limits. For an optically thick isothermal slab in LTE ( $S_\nu = B_\nu$ , $\tau_\nu \to \infty$ ) the boundary term vanishes and the integral returns $B_\nu$ , so $I_\nu \to B_\nu(T)$ — blackbody. For a thin cloud ( $\tau_\nu \ll 1$ ) expanding gives $I_\nu \approx I_\nu(0) + \int j_\nu\,\mathrm{d}s$ — background plus raw emissivity. Why: one formula reproduces both the opaque blackbody star and the transparent nebula, confirming it at both extremes.

Go deeper · Derivation C · 1 minThe Eddington-Barbier relation and why we see to optical depth unity

Derivation CThe Eddington-Barbier relation and why we see to optical depth unity

The purpose is to make quantitative the claim that the emergent intensity is essentially the source function at optical depth of order one, and to derive the exact statement — the Eddington-Barbier relation — for a source function linear in depth. We write the formal solution for outgoing rays in a semi-infinite plane-parallel atmosphere, insert a linear $S_\nu(\tau_\nu)$ , and evaluate the two elementary exponential moments that appear.

Step 1. For an outgoing ray at inclination $\mu = \cos\theta$ with boundary at $\tau_\nu = 0$ , the formal solution gives $I_\nu(0,\mu) = \int_0^\infty S_\nu(\tau_\nu)\,e^{-\tau_\nu/\mu}\,\mathrm{d}\tau_\nu/\mu$ . Why: along a slanted ray the vertical optical depth is stretched by $1/\mu$ , so the weighting selecting visible layers is $e^{-\tau_\nu/\mu}$ , not $e^{-\tau_\nu}$ .

Step 2. Adopt $S_\nu(\tau_\nu) = a_\nu + b_\nu\,\tau_\nu$ — exact for a gray atmosphere in radiative equilibrium and a good local approximation generally. Why: a linear profile is the first non-trivial case, and it suffices because only the value and gradient of $S_\nu$ near the surface survive the weighting.

Step 3. Evaluate $\int_0^\infty e^{-\tau/\mu}\,\mathrm{d}\tau/\mu = 1$ and $\int_0^\infty \tau\,e^{-\tau/\mu}\,\mathrm{d}\tau/\mu = \mu$ , giving $I_\nu(0,\mu) = a_\nu + b_\nu\mu = S_\nu(\tau_\nu = \mu)$ . Why: the second moment picks $\tau_\nu = \mu$ as the formation depth — the emergent intensity is literally $S_\nu$ one slanted optical depth in.

Step 4. Read off limb darkening. Disk centre ( $\mu = 1$ ) shows $S_\nu(\tau_\nu = 1)$ , deep, hot, bright; the limb ( $\mu \to 0$ ) shows $S_\nu(\tau_\nu \to 0)$ , shallow, cool, faint. For the gray atmosphere $S_\nu \propto (\tau_\nu + 2/3)$ , so $I_\nu(0,\mu)/I_\nu(0,1) = (2 + 3\mu)/5 = 0.40$ at the limb. Why: the darkening of the solar disk toward its edge is a direct readout of $\mathrm{d}S_\nu/\mathrm{d}\tau_\nu > 0$ and an outward-decreasing temperature.

Limiting cases

The source function takes sharply different forms depending on whether photons are truly destroyed or merely redirected, and on how much optical depth the beam has crossed:

Case	Result	Meaning
Local thermodynamic equilibrium	$S_\nu = B_\nu(T)$	Pure thermal emission and absorption; populations set by the local temperature. Kirchhoff's law; the source function is purely local.
Pure coherent isotropic scattering	$S_\nu = J_\nu = \tfrac{1}{4\pi}\oint I_\nu\,\mathrm{d}\Omega$	Photons redirected, not destroyed; the source is the angle-averaged intensity, making the problem non-local and self-referential.
Mixed absorption and scattering	$S_\nu = (1-\epsilon_\nu)J_\nu + \epsilon_\nu B_\nu$	A weighted mean of thermal and scattered light; $\epsilon_\nu$ is the destruction probability per interaction.
Optically thick ( $\tau_\nu \gg 1$ ), homogeneous	$I_\nu \to S_\nu$	The beam forgets the boundary; in LTE the emergent light is a blackbody at the local temperature.
Optically thin ( $\tau_\nu \ll 1$ )	$I_\nu \approx I_\nu(0) + \int j_\nu\,\mathrm{d}s$	Background plus the raw emissivity integrated along the ray, independent of the equilibrium value of $S_\nu$ .
Scattering-dominated ( $\epsilon_\nu \ll 1$ )	$S_\nu(0) \approx \sqrt{\epsilon_\nu}\,B_\nu$ ; thermalizes at $\tau_\nu \sim \epsilon_\nu^{-1/2}$	Surface source function is depressed below Planck; photons must random-walk to the thermalization depth to reach $B_\nu$ .

Common trapConfusing the emission coefficient with the source function, or LTE with a blackbody radiation field

Two traps sit close together. First, $j_\nu$ and $S_\nu$ are not the same object: $j_\nu$ is an emissivity (energy per volume per time per frequency per solid angle) that scales with density, whereas $S_\nu = j_\nu/\alpha_\nu$ has the units of intensity and, in LTE, is density-independent — doubling the density leaves $S_\nu$ untouched while doubling $j_\nu$ . Reporting "the source function in $\mathrm{erg\,cm^{-3}}$ " is a units error. Second, $S_\nu = B_\nu(T)$ requires only local thermodynamic equilibrium of the matter, not full equilibrium of the radiation. Near any surface $I_\nu$ differs from $B_\nu$ — photons leak out, so $I_\nu < B_\nu$ there even though $S_\nu$ is exactly Planckian. Only at $\tau_\nu \gg 1$ does the radiation itself thermalize to $I_\nu \to B_\nu$ .

Worked examples

1 · Solar limb darkening from a gray atmosphere (fully worked)

Model the solar photosphere as a gray atmosphere in radiative equilibrium, for which $T^4(\tau) = \tfrac{3}{4}T_{\rm eff}^4(\tau + 2/3)$ with $T_{\rm eff} = 5772\ \mathrm{K}$ . Find the temperature at the layers that form disk-centre and limb radiation, and the ratio of limb to centre brightness.

Step 1. By Eddington-Barbier the continuum at inclination $\mu$ forms near $\tau = \mu$ , so disk centre ( $\mu = 1$ ) forms at $\tau = 1$ and the limb ( $\mu \to 0$ ) near $\tau \to 0$ . At $\tau = 1$ : $T = 5772\,(1.25)^{1/4} = \mathbf{6103\ K}$ . At $\tau = 2/3$ (where $T = T_{\rm eff}$ ): $\mathbf{5772\ K}$ . At the surface: $T = 5772\,(0.5)^{1/4} = \mathbf{4854\ K}$ .

Step 2. Since $S = B \propto T^4$ , the source function runs linearly in $\tau$ : $S(\tau) \propto (\tau + 2/3)$ . Eddington-Barbier $I(0,\mu) = S(\tau=\mu)$ then gives $I(0,\mu)/I(0,1) = (\mu + 2/3)/(5/3) = (2 + 3\mu)/5$ .

Step 3. At the limb, $\mu = 0$ : $I_{\rm limb}/I_{\rm centre} = 2/5 = \mathbf{0.40}$ — the disk edge is only $40\%$ as bright as its centre in the bolometric gray model, close to the observed solar darkening near $500\ \mathrm{nm}$ . The chain is complete: temperature falls outward, so $S_\nu$ falls toward the surface, so slanted rays escaping from shallow layers carry a lower source function. T(τ=1) = 6103 K, T(surface) = 4854 K, limb is 0.40 of centre.

2 · The value of the LTE source function in the Sun (fully worked)

Compute the solar photospheric source function at $\lambda = 500\ \mathrm{nm}$ assuming LTE at $T = 5772\ \mathrm{K}$ , i.e. evaluate $S_\nu = B_\nu(T)$ .

Step 1. Frequency: $\nu = c/\lambda = (2.998\times10^{8})/(500\times10^{-9}) = \mathbf{6.00\times10^{14}\ Hz}$ . Dimensionless exponent $x = h\nu/k_BT = (6.626\times10^{-34})(6.00\times10^{14})/[(1.381\times10^{-23})(5772)] = \mathbf{4.99}$ — just blueward of the Planck peak, the Wien-like regime where $e^{x} \gg 1$ .

Step 2. The prefactor $2h\nu^3/c^2 = 3.18\times10^{-6}\ \mathrm{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$ ; the occupation factor $1/(e^{x}-1) = 1/145.3 = 6.88\times10^{-3}$ .

Step 3. Multiply: $S_\nu = B_\nu = (3.18\times10^{-6})(6.88\times10^{-3}) = \mathbf{2.19\times10^{-8}\ W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$ . Because the deep photosphere is optically thick, the emergent continuum near $500\ \mathrm{nm}$ is essentially this Planck value at the $\tau \approx 1$ temperature — which is why the Sun looks so nearly blackbody. S_ν = B_ν(5772 K, 500 nm) = 2.19×10⁻⁸ W m⁻² Hz⁻¹ sr⁻¹, at x = 4.99.

3 · Thermalization depth in a scattering line

A resonance line has photon destruction probability $\epsilon_\nu = 10^{-4}$ per scattering. Estimate the depth at which $S_\nu$ thermalizes to $B_\nu$ , and the surface $S_\nu/B_\nu$ .

Step 1. A photon is truly destroyed only with probability $\epsilon_\nu$ per interaction, so it survives $\sim 1/\epsilon_\nu$ scatterings, random-walking a net optical depth $\sim\sqrt{1/\epsilon_\nu}$ before thermal contact: the thermalization depth is $\Lambda_\nu \sim \epsilon_\nu^{-1/2} = (10^{-4})^{-1/2} = \mathbf{100}$ .

Step 2. Above that depth the source function is starved of thermal photons; the square-root result for a semi-infinite scattering atmosphere gives the surface value $S_\nu(0)/B_\nu \approx \sqrt{\epsilon_\nu} = \mathbf{0.01}$ — the line core is dark not because the gas is cool but because scattering lets photons escape before they thermalize. Thermalization at τ ~ 100; surface S_ν ≈ 0.01 B_ν.

Physics of the portrait

The source function is a two-generation idea: Kirchhoff tied emission to absorption in the laboratory, and half a century later Schwarzschild lifted that relation into the atmosphere of a star and made it the pivot of radiative transfer.

Origins · 1 minKarl Schwarzschild and how this was discovered

Karl Schwarzschild

1873–1916 · radiative equilibrium and the birth of the source function

Karl Schwarzschild is remembered by most for the first exact solution of Einstein's field equations, found on the Russian front in 1915, months before his death. But a decade earlier he had already reshaped a different field. His 1906 paper on the solar atmosphere replaced the prevailing "adiabatic" model of the outer Sun with the hypothesis of radiative equilibrium — energy in the photosphere carried outward by radiation, not convection, with absorbed light balancing emitted light at each depth. To make it calculable he introduced, in effect, the modern source function and the two-stream description of the radiation field, and used them to predict solar limb darkening quantitatively — the first time the temperature stratification of a star was inferred from the shape of its disk. His framework, sharpened by Schuster's scattering-slab model, Eddington's approximation, and Milne's theory of line formation, became the backbone of stellar-atmosphere theory. Everything here — the relaxation law, the formal solution, Eddington-Barbier, the scattering source function — descends from his insight that a star's spectrum is set locally, layer by layer, by the ratio of what the gas emits to what it absorbs.

History

1859

— Kirchhoff states the law relating thermal emission and absorption, jν = αν Bν

1901

— Planck fixes the function Bν(T) that the LTE source function must equal

1905

— Schuster models radiative transfer through a scattering foggy atmosphere

1906

— Schwarzschild introduces radiative equilibrium and computes solar limb darkening

1950

— Chandrasekhar's Radiative Transfer systematizes the scattering source function

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., ch. 9 (The Interiors of Stars) and §9.2 (the transfer equation) · Rybicki & Lightman, Radiative Processes in Astrophysics, ch. 1 (§1.4 the transfer equation, §1.5 thermal radiation, §1.7–1.8 scattering and the source function) · Mihalas, Stellar Atmospheres, 2nd ed., ch. 1–2 (formal solution, Eddington-Barbier) and ch. 6–7 (non-LTE) · Chandrasekhar, Radiative Transfer (Dover reprint) · Hubeny & Mihalas, Theory of Stellar Atmospheres (modern non-LTE treatment) · Ryden, Introduction to Cosmology, 2nd ed. (thermal radiation and brightness temperature). Key results in this article: $S_\nu = j_\nu/\alpha_\nu$ and $\mathrm{d}I_\nu/\mathrm{d}\tau_\nu = S_\nu - I_\nu$ ; the formal solution follows from the standard integrating-factor method for a linear first-order ODE; Kirchhoff's law gives $S_\nu = B_\nu(T)$ , with $B_\nu(5772\,\mathrm{K},500\,\mathrm{nm}) = 2.19\times10^{-8}\ \mathrm{W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$ at $x = 4.99$ ; the Eddington-Barbier relation $I_\nu(0,\mu) = S_\nu(\tau_\nu = \mu)$ follows from $\int_0^\infty e^{-\tau/\mu}\mathrm{d}\tau/\mu = 1$ and $\int_0^\infty \tau\, e^{-\tau/\mu}\mathrm{d}\tau/\mu = \mu$ , giving the gray limb law $(2+3\mu)/5 = 0.40$ ; the scattering thermalization depth is $\tau \sim \epsilon^{-1/2}$ with surface value $S_\nu \approx \sqrt{\epsilon}\,B_\nu$ . The gray temperature structure $T^4 = \tfrac{3}{4}T_{\rm eff}^4(\tau + 2/3)$ is the standard Eddington result.

Prerequisites. Specific Intensity and Radiative Flux · The Equation of Radiative Transfer · Opacity and Absorption Coefficients · Blackbody Radiation and the Planck Law. Leads to. Kirchhoff's Laws and Thermal Emission · Local Thermodynamic Equilibrium · Spectral Line Formation · Optical Depth.

Ask the tutor: Ask me to walk you through why the Sun's limb is exactly 40% as bright as its centre, or to solve the scattering source function for any photon destruction probability you choose.

Why does only the ratio jν/αν matter, and not the two coefficients separately?

Because in optical depth ( $\mathrm{d}\tau_\nu = \alpha_\nu\,\mathrm{d}s$ ) the transfer equation reads $\mathrm{d}I_\nu/\mathrm{d}\tau_\nu = S_\nu - I_\nu$ , and the only place the microphysics enters is through $S_\nu = j_\nu/\alpha_\nu$ . The absolute sizes of $j_\nu$ and $\alpha_\nu$ set how many centimetres make one unit of optical depth — the thickness of the "skin" we see — but not the brightness the beam relaxes toward. Double the density and both coefficients double; the slab reaches the same equilibrium intensity in half the distance. This is why an emergent spectrum constrains $S_\nu$ and the optical-depth scale, but not the absolute emissivity.

Does S_ν = B_ν mean the star radiates as a perfect blackbody?

No — the most common misreading. $S_\nu = B_\nu(T)$ says the source function is Planckian, which follows from local thermodynamic equilibrium of the matter (Boltzmann and Saha populations at the local temperature). The emergent intensity $I_\nu$ is different: it is the $e^{-\tau}$ -weighted average of $S_\nu$ over depth, and near any surface photons escape, so $I_\nu < B_\nu$ . Only deep inside, at $\tau_\nu \gg 1$ , does the radiation itself thermalize to $I_\nu \to B_\nu$ . The Sun looks approximately blackbody because it is optically thick with a slowly varying temperature, not because $I_\nu$ is exactly Planck — the absorption lines and limb darkening are precisely the departures.

What is the difference between an absorption and an emission line, in source-function terms?

It comes down to the sign of the source-function gradient across the line-forming region. A line has extra opacity, so it forms higher than the neighbouring continuum. If $S_\nu$ decreases outward (temperature falling with height, as in a normal photosphere), the line samples a lower $S_\nu$ than the deeper continuum and appears in absorption. If a hot chromosphere or irradiated nebula makes $S_\nu$ rise outward, the line sits above a fainter background and appears in emission. The same gas can show a line in absorption against the disk and in emission off the limb — classic solar chromospheric behaviour — purely because the background and gradient change.

Why does scattering make the problem so much harder than absorption?

True absorption is local: a photon is destroyed, its energy thermalized, and the gas re-emits with brightness $B_\nu$ set only by the local temperature. Scattering conserves the photon but randomizes its direction, so the re-emitted brightness is the mean intensity $J_\nu$ — an average over the radiation arriving from the whole medium. Then $S_\nu$ at one point depends on $I_\nu$ everywhere, and $I_\nu$ everywhere depends on $S_\nu$ through the formal solution. The tidy local ODE becomes a coupled integral equation (the " $\Lambda$ -operator" problem) solved self-consistently by iteration. Physically, a photon random-walks many times before it escapes or is destroyed — which is why the answer is non-local.

What is the thermalization depth and why does it scale as ε^(−1/2)?

In a scattering-dominated medium a photon is truly destroyed only with probability $\epsilon$ per interaction, so it undergoes of order $1/\epsilon$ scatterings before thermal contact. A random walk of $N$ unit-optical-depth steps covers a net distance of order $\sqrt{N}$ , so the photon reaches a net optical depth $\sim \epsilon^{-1/2}$ before it thermalizes. Below that depth the source function has climbed to Planck; above it, photons leak out first and $S_\nu$ is depressed to a surface value $S_\nu(0) \approx \sqrt{\epsilon}\,B_\nu$ . For $\epsilon = 10^{-4}$ : thermalization only at $\tau \sim 100$ and a surface source function just $1\%$ of Planck — the reason strong resonance lines are deeply non-LTE.

Does the source function apply beyond stars — to nebulae or the CMB?

Yes; it is completely general. For an optically thin nebula ( $\tau_\nu \ll 1$ ) the emergent intensity reduces to $I_\nu(0) + \int j_\nu\,\mathrm{d}s$ , so bright emission lines directly measure the emissivity integrated along the line of sight — the emission measure, density, and abundances of the ionized gas — with the equilibrium value of $S_\nu$ irrelevant. In the microwave sky the Rayleigh-Jeans limit of the Planck source function underlies the brightness-temperature scale that calibrates ALMA and CMB experiments. Anywhere radiation crosses matter — stellar winds, accretion disks, the intergalactic medium — the same $S_\nu = j_\nu/\alpha_\nu$ and the same relaxation law govern what reaches the detector.