Non-LTE Radiative Transfer

Deep inside a star, collisions are so frequent that every atomic level is populated exactly as the local temperature dictates: Boltzmann and Saha hold, and the source function equals the Planck function, $S_\nu = B_\nu(T)$ . This is Local Thermodynamic Equilibrium, and it lets a stellar atmosphere be solved layer by layer, each shell knowing only its own $T$ and density. But climb outward into the thin skin of the atmosphere — or into a chromosphere, a planetary nebula, or the recombining plasma of the early universe — and the assumption collapses: radiative processes compete with collisions, and the level populations are set not by the local temperature but by the ambient radiation field. This is Non-LTE radiative transfer, the physics that makes real stellar spectra interpretable. The difficulty is a two-way coupling LTE never faces: the populations $n_i$ obey statistical equilibrium, whose radiative rates depend on the mean intensity $J_\nu$ , but $J_\nu$ comes from the transfer equation, which needs the source function $S_\nu$ built from those same populations. Because $J_\nu$ depends on conditions everywhere along every ray and the emissivity depends nonlinearly on the populations, the system is coupled, nonlocal, and nonlinear. The organizing idea is photon escape: a photon trapped in an optically thick line core is scattered thousands of times and can leak out of the surface before a collision destroys it, draining energy collisions can no longer replace. Then the source function sinks below $B_\nu$ , lines develop deep dark cores, and the full Non-LTE apparatus — statistical equilibrium, the thermalization parameter, accelerated lambda iteration — is needed to predict what a spectrograph sees.

Key takeaways

Non-LTE abandons Boltzmann and Saha statistics at the local temperature; the level populations are fixed instead by the balance of radiative and collisional rates in statistical equilibrium.
The coupling is circular: the populations $n_i$ depend on the radiation field $J_\nu$ , while $J_\nu$ depends on the source function $S_\nu$ built from those same populations — a nonlinear, nonlocal problem.
The source function departs from the Planck function, $S_\nu \neq B_\nu(T)$ , whenever photon escape or scattering decouples matter from a Planckian field — low-density atmospheres and strong line cores.
For a two-level atom it collapses to $S_\nu = (1-\varepsilon)\,J_\nu + \varepsilon\,B_\nu$ , with the thermalization parameter $\varepsilon$ the probability an absorbed photon is destroyed by a collision rather than re-emitted.
A scattering surface sinks to $S_\nu(0) = \sqrt{\varepsilon}\,B_\nu$ (the √ε law), recovering $B_\nu$ only below $\tau_{\mathrm{th}} \sim \varepsilon^{-1/2}$ ; classical lambda iteration stalls here and accelerated lambda iteration cures it.
Non-LTE is essential for stellar abundances, chromospheric Ca II and Mg II lines, nebular forbidden-line diagnostics, and the recombination that shapes the microwave background.
Key idea: Level populations are set by rate balance between radiative and collisional processes, and the radiation field driving those rates is itself determined by the populations through the transfer equation; solving this self-consistently makes the source function depart from the Planck function.

Where this lives today

Non-LTE line formation is now routine infrastructure, decisive wherever spectra are used to weigh matter or read temperatures. In galactic archaeology, abundances of light and iron-peak elements in metal-poor stars carry Non-LTE corrections of 0.1–0.5 dex from overionization by a super-thermal ultraviolet field, and codes such as MULTI, RH, and Turbospectrum are applied to millions of Gaia, APOGEE, and 4MOST spectra. In solar physics, the chromospheric Ca II H and K, Mg II h and k, and H-alpha lines are pure Non-LTE phenomena, inverted by IRIS and the Daniel K. Inouye Solar Telescope against 3D radiation-MHD models. In nebular astrophysics, solvers such as PyNeb and Cloudy turn forbidden-line ratios into densities and temperatures. And in cosmology, recombination of the primordial plasma at redshift about 1100 is a Non-LTE problem — Saha equilibrium fails because Lyman-alpha and Lyman-continuum photons cannot escape fast enough — so codes like CosmoRec and HyRec compute the recombination history that sets the visibility function shaping the cosmic microwave background.

The physical picture

Follow a single photon born in the core of a strong resonance line, deep in a stellar atmosphere. It is absorbed and re-emitted again and again, a random walk in space and frequency, and at each scattering two fates compete. An inelastic collision can destroy it, converting its energy into thermal motion — this couples the radiation back to the local temperature. Or it can random-walk all the way to the surface and escape. Deep down, where the line is opaque across many collision mean free paths, destruction wins and the populations thermalize to LTE; near the surface, escape wins, the upper level is starved, and the source function falls below $B_\nu$ . The first figure makes this competition one picture.

Figure

A photon random-walks in a line core: at each scatter a collision destroys it with probability ε, otherwise it eventually escapes the surface. Only below the thermalization depth τ_th ~ ε^(−1/2) ≈ 100 (for ε ≈ 10⁻⁴) does the source function recover S_ν → B_ν; nearer the surface, escape drives S_ν below B_ν

The single most important number is the thermalization depth. Only below $\tau_{\mathrm{th}} \sim \varepsilon^{-1/2}$ (for complete redistribution) does the source function "forget" the surface and climb back to $B_\nu$ . Because $\varepsilon$ can be $10^{-4}$ or smaller for a strong line, Non-LTE cores remember the surface across a hundred optical depths — far deeper than the naive $\tau \sim 1$ intuition. The second figure plots the source function for a scattering line.

Figure

The two-level source function for a scattering line (ε = 10⁻⁴): S_ν rises from its surface value S(0) = √ε B_ν = 0.01 B_ν and reaches the Planck value B_ν only below the thermalization depth τ_th ~ ε^(−1/2) = 100 — the √ε law, origin of deep dark line cores

A useful bookkeeping device runs through everything below: the departure coefficients $b_i = n_i/n_i^\ast$ , the ratio of the actual population to its LTE value at the local $T_e, n_e$ . Every Non-LTE effect can be phrased as how the $b_i$ differ from unity and how that reshapes the opacity ( $\propto b_l$ ) and the line source function ( $\propto b_u/b_l$ ). The word "temperature" also fractures: kinetic $T_e$ , excitation, ionization, and radiation temperatures can all differ — LTE forces them equal, Non-LTE lets each go its own way.

The core equations

The transfer equation is unchanged from LTE — it is the source function that carries the new physics. Along a ray the intensity relaxes toward $S_\nu$ .

\frac{\mathrm{d} I_\nu}{\mathrm{d}\tau_\nu} = I_\nu - S_\nu, \qquad S_\nu \equiv \frac{j_\nu}{\kappa_\nu}, \qquad J_\nu = \frac{1}{4\pi}\oint I_\nu\, \mathrm{d}\Omega \qquad (1)

where: $I_\nu$ — specific intensity; $\tau_\nu$ — monochromatic optical depth along the ray; $S_\nu$ — source function, emissivity $j_\nu$ over extinction $\kappa_\nu$ ; $J_\nu$ — mean intensity, the isotropic average that drives the radiative rates. Reading: intensity is dragged toward $S_\nu$ . In LTE, $S_\nu = B_\nu(T)$ and we are done. In Non-LTE the populations that build $S_\nu$ are unknown, so we replace Boltzmann and Saha with statistical equilibrium — every level steady, gains balancing losses.

\sum_{j\neq i} n_j P_{ji} = n_i \sum_{j\neq i} P_{ij}, \qquad P_{ij} = R_{ij} + C_{ij} \qquad (2)

where: $n_i$ — number density in level $i$ ; $P_{ij}$ — total transition rate from $i$ to $j$ ; $R_{ij}$ — radiative rate (depends on $J_\nu$ ); $C_{ij}$ — collisional rate (depends on $n_e, T_e$ ). Reading: closed by $\sum_i n_i = n_{\mathrm{tot}}$ , this reduces to Boltzmann and Saha only when $C_{ij} \gg R_{ij}$ . The radiative rates are where transfer reaches back into the atom: a bound-bound transition is pumped upward by absorption and drained by spontaneous plus stimulated emission, weighted by the line profile.

R_{lu} = B_{lu}\!\int_0^\infty \!\phi_\nu J_\nu\, \mathrm{d}\nu, \qquad R_{ul} = A_{ul} + B_{ul}\!\int_0^\infty \!\phi_\nu J_\nu\, \mathrm{d}\nu \qquad (3)

where: $A_{ul}$ — spontaneous emission coefficient; $B_{lu}, B_{ul}$ — absorption and stimulated-emission coefficients; $\phi_\nu$ — normalized line profile; $J_\nu$ — mean intensity, the explicit coupling to transfer. Reading: carrying $J_\nu$ , the populations inherit the nonlocality of the field. The three Einstein coefficients are not independent; demanding consistency with thermodynamic equilibrium once fixes two relations that then hold universally, as properties of the atom.

g_l B_{lu} = g_u B_{ul}, \qquad A_{ul} = \frac{2h\nu^3}{c^2}\,B_{ul} \qquad (4)

where: $g_l, g_u$ — statistical weights of the levels; $\nu$ — line-center frequency ( $h, c$ standard). Reading: these collapse three coefficients into one, so $S_\nu$ depends only on a population ratio. Assembling the line emissivity and opacity from (3) and (4), the line source function depends only on the ratio $n_u/n_l$ — the frequency profile cancels.

S_\nu^{\,l} = \frac{2h\nu^3}{c^2}\left(\frac{g_u n_l}{g_l n_u} - 1\right)^{-1} \qquad (5)

where: $n_l, n_u$ — actual Non-LTE populations; $g_l, g_u$ — statistical weights. Reading: a Boltzmann ratio gives $B_\nu(T)$ ; any departure pushes $S_\nu$ off the Planck curve. For an atom with only two levels, statistical equilibrium solves in closed form and the source function collapses to a weighted average of field and Planck function — the most illuminating equation in the subject.

S_\nu = (1-\varepsilon)\,\bar J + \varepsilon\, B_\nu, \qquad \varepsilon = \frac{C_{ul}\left(1 - e^{-h\nu/kT}\right)}{A_{ul} + C_{ul}\left(1 - e^{-h\nu/kT}\right)} \qquad (6)

where: $\bar J = \int \phi_\nu J_\nu\,\mathrm{d}\nu$ — profile-averaged mean intensity; $\varepsilon$ — photon destruction probability (thermalization parameter); $C_{ul}$ — collisional de-excitation rate; $B_\nu$ — Planck function at $T_e$ . Reading: $\varepsilon$ dials between pure scattering ( $S_\nu \to \bar J$ ) and LTE ( $S_\nu \to B_\nu$ ); to leading order $\varepsilon \approx C_{ul}/(A_{ul} + C_{ul})$ . Substituting the formal solution of transfer for $\bar J$ turns (6) into a single integral equation for $S_\nu$ , governed by the lambda operator — the linear, nonlocal map returning $\bar J$ from $S_\nu$ .

\bar J = \Lambda_\nu[S_\nu], \qquad S_\nu = (1-\varepsilon)\,\Lambda_\nu[S_\nu] + \varepsilon\, B_\nu \qquad (7)

where: $\Lambda_\nu$ — the lambda operator, whose plane-parallel kernel is the first exponential integral $E_1$ . Reading: the whole Non-LTE line problem is this one equation; how you solve it sets ten versus ten thousand iterations. Solving (7) by direct substitution (lambda iteration) stalls fatally for small $\varepsilon$ . The cure is operator splitting: precondition with an approximate operator $\Lambda^\ast$ (usually the cheap-to-invert diagonal of $\Lambda$ ) and iterate the correction.

S_\nu^{\mathrm{new}} = \left[\,1 - (1-\varepsilon)\Lambda^\ast\,\right]^{-1}\!\left[\,(1-\varepsilon)(\Lambda - \Lambda^\ast)\,S_\nu^{\mathrm{old}} + \varepsilon\, B_\nu\,\right] \qquad (8)

where: $\Lambda^\ast$ — approximate lambda operator, usually the diagonal of $\Lambda$ . Reading: inverting $1 - (1-\varepsilon)\Lambda^\ast$ absorbs the near-unity eigenvalue, so it converges in order ten steps regardless of $\varepsilon$ . Finally, results are reported not as raw populations but as departure coefficients, which measure the deviation from LTE level by level and feed back into the opacity and source function.

b_i \equiv \frac{n_i}{n_i^\ast}, \qquad n_i^\ast = n_i^{\mathrm{Saha\text{-}Boltzmann}}(T_e, n_e) \qquad (9)

where: $b_i$ — departure coefficient of level $i$ (unity in LTE); $n_i^\ast$ — LTE population from Saha plus Boltzmann at the local $T_e, n_e$ . Reading: opacity scales with $b_l$ and $S_\nu^l/B_\nu$ with $b_u/b_l$ , so Non-LTE effects are encoded in how the $b_i$ stray from one.

Derivation AFrom statistical equilibrium to the two-level source function

The goal is to derive the central relation $S_\nu = (1-\varepsilon)\bar J + \varepsilon B_\nu$ from an honest two-level atom, seeing exactly where the thermalization parameter $\varepsilon$ comes from and why the Planck function reappears though we never assumed LTE. The route: balance the rates, solve for the population ratio, feed it into the line source function (5), and use collisional detailed balance to collect the algebra.

Step 1. Write statistical equilibrium for an atom with only levels $l$ and $u$ : every upward transition balances every downward one, $n_l(B_{lu}\bar J + C_{lu}) = n_u(A_{ul} + B_{ul}\bar J + C_{ul})$ . Why: with two levels the sum in (2) has a single term, collapsing to one solvable balance.

Step 2. Solve for the ratio, $\dfrac{n_u}{n_l} = \dfrac{B_{lu}\bar J + C_{lu}}{A_{ul} + B_{ul}\bar J + C_{ul}}$ , which carries $\bar J$ explicitly. Why: this is where nonlocality enters the atom — the populations depend on radiation from other layers, not the local temperature alone.

Step 3. Insert the ratio into $S_\nu^{\,l} = \dfrac{2h\nu^3}{c^2}\left(\dfrac{g_u n_l}{g_l n_u} - 1\right)^{-1}$ and eliminate the coefficients with $g_l B_{lu} = g_u B_{ul}$ , $A_{ul} = (2h\nu^3/c^2)B_{ul}$ . Why: (4) trades three coefficients for one, so $S_\nu$ reduces to $\bar J$ , $B_\nu$ , and the collision rates.

Step 4. Use collisional detailed balance, $\dfrac{C_{lu}}{C_{ul}} = \dfrac{g_u}{g_l}e^{-h\nu/kT_e}$ — where the Planck function at $T_e$ enters. Why: thermal electrons obey microscopic reversibility at $T_e$ , so collisions drive the populations toward Boltzmann and $S_\nu$ toward $B_\nu$ .

Step 5. Collect the algebra: it collapses to $S_\nu = (1-\varepsilon)\bar J + \varepsilon B_\nu$ with $\varepsilon = C_{ul}(1 - e^{-h\nu/kT})/[A_{ul} + C_{ul}(1 - e^{-h\nu/kT})] \approx C_{ul}/(A_{ul}+C_{ul})$ . Why: $\varepsilon$ is a genuine probability — the fraction of de-excitations that are collisional (destroying the photon) not radiative (recycling it).

Step 6. Limits: $C_{ul} \gg A_{ul}$ gives $\varepsilon \to 1$ , $S_\nu \to B_\nu$ (LTE); $C_{ul} \ll A_{ul}$ gives $\varepsilon \to 0$ , $S_\nu \to \bar J$ (pure scattering). Why: collisions couple matter to the local temperature, radiation decouples it, and $\varepsilon$ is the dial. *Verified: reproduces the two-level source function of Mihalas, Stellar Atmospheres Ch. 5–6; limits match Rybicki & Lightman Ch. 1.*

Go deeper · Derivation B · 1 minThe √ε law for the surface source function

Derivation BThe √ε law for the surface source function

The goal is to estimate how far below the Planck function the source function of a scattering line drops at the surface, and to show the answer is the non-intuitive $\sqrt{\varepsilon}$ rather than the naive $\varepsilon$ . Take a semi-infinite isothermal plane-parallel atmosphere with constant $\varepsilon$ and complete redistribution, replace the lambda operator by its diffusion approximation, and solve with the correct boundary conditions.

Step 1. Deep in, the lambda operator is approximated by the diffusion (Eddington) expansion $\bar J \approx S + \tfrac13\, \mathrm{d}^2 S/\mathrm{d}\tau^2$ . Why: at large optical depth radiation is nearly isotropic and the exact nonlocal operator reduces to a local differential one.

Step 2. Substitute into $S = (1-\varepsilon)\bar J + \varepsilon B$ to get $S = (1-\varepsilon)\big(S + \tfrac13 S''\big) + \varepsilon B$ , then rearrange to $\tfrac13 S'' = \dfrac{\varepsilon}{1-\varepsilon}(S - B)$ . Why: this turns the source relation into a linear second-order ODE whose small parameter is $\varepsilon$ .

Step 3. With $\delta S = S - B$ , this is $\delta S'' = 3\,\dfrac{\varepsilon}{1-\varepsilon}\,\delta S$ , decay length $\ell \approx (3\varepsilon)^{-1/2}$ in optical depth. Why: the length over which $S$ heals to $B$ is set by $\sqrt{\varepsilon}$ , not $\varepsilon$ — the first square root, and the origin of $\tau_{\mathrm{th}} \sim \varepsilon^{-1/2}$ .

Step 4. The bounded solution is $S(\tau) = B\big(1 - (1-\sqrt{\varepsilon})\,e^{-\sqrt{3\varepsilon}\,\tau}\big)$ , fixed by $S \to B$ as $\tau \to \infty$ and the scattering surface condition $S(0) \approx \sqrt{3}\,\bar J(0)$ . Why: thermalization at depth and free escape at the surface pin down the single decaying exponential and its amplitude.

Step 5. At the surface $S(0) = \sqrt{\varepsilon}\,B$ (up to an order-unity constant), so $\varepsilon = 10^{-4}$ starts at $10^{-2}\,B$ and recovers $B$ only below $\tau_{\mathrm{th}} \sim 100$ . Why: this is the √ε law made numerical, and why strong lines carve cores far darker and deeper than an optically-thin estimate predicts. *Verified: standard result; matches Mihalas Ch. 6 and Rutten's lecture notes for the two-level atom with complete redistribution. The curve of Step 4 is the one plotted in the second figure above.*

Go deeper · Derivation C · 1 minWhy direct lambda iteration fails and accelerated lambda iteration succeeds

Derivation CWhy direct lambda iteration fails and accelerated lambda iteration succeeds

The goal is to explain, with a concrete convergence estimate, why the obvious way to solve the integral equation (7) — plugging the old source function into the lambda operator to get a new one — is catastrophically slow for scattering lines, and why splitting the operator into a local part plus a correction restores fast convergence. We track how an error propagates through one iteration and show preconditioning removes the controlling eigenvalue.

Step 1. Classical iteration computes $S^{(n+1)} = (1-\varepsilon)\Lambda[S^{(n)}] + \varepsilon B$ ; subtracting the exact solution, the error obeys $\delta S^{(n+1)} = (1-\varepsilon)\,\Lambda[\delta S^{(n)}]$ . Why: convergence is governed by the spectral radius of $(1-\varepsilon)\Lambda$ .

Step 2. The lambda operator has eigenvalues near unity for the deepest modes (photons trapped over many optical depths), so for the slowest mode $\Lambda \to 1$ and the error shrinks by only $(1-\varepsilon)$ per step. Why: with $\varepsilon = 10^{-4}$ each step removes just $0.01\%$ of the error, needing of order $1/\varepsilon \sim 10^4$ iterations — impractical.

Step 3. Split $\Lambda = \Lambda^\ast + (\Lambda - \Lambda^\ast)$ , taking $\Lambda^\ast$ to be the diagonal — the stiff, local trapping, trivial to invert. Why: the near-unity eigenvalue lives in the local coupling, so isolating it into a cheap operator tames it.

Step 4. Iterate $[1 - (1-\varepsilon)\Lambda^\ast]\,S^{(n+1)} = (1-\varepsilon)(\Lambda - \Lambda^\ast)S^{(n)} + \varepsilon B$ ; the left inversion absorbs the stiff eigenvalue and the effective spectral radius drops far below one. Why: what remains is the smooth nonlocal correction, which converges geometrically.

Step 5. In practice accelerated lambda iteration converges in of order ten iterations independent of $\varepsilon$ , and with Ng acceleration and multilevel extensions is the workhorse of every modern Non-LTE code. Why: turning $10^4$ iterations into $10$ is the difference between intractable and routine. *Verified: convergence analysis follows Olson, Auer & Buchler (1986); operator-splitting logic in Hubeny & Mihalas (2014) Ch. 13.*

Limiting cases

The Non-LTE source function interpolates between thermal emission and scattering by how efficiently collisions destroy photons:

Case	Result	Meaning
Collision-dominated (LTE limit)	$\varepsilon \to 1 \Rightarrow S_\nu \to B_\nu(T_e)$	Frequent inelastic collisions enforce Boltzmann and Saha populations; Kirchhoff's law is recovered. Deep stellar interiors.
Scattering-dominated (extreme Non-LTE)	$\varepsilon \to 0 \Rightarrow S_\nu \to \bar J$	Pure resonance scattering; the source function is set entirely by the nonlocal radiation field, not the local temperature.
Semi-infinite atmosphere surface	$S_\nu(0) \approx \sqrt{\varepsilon}\, B_\nu$	The √ε law: an isothermal scattering surface sits at $\sqrt{\varepsilon}$ of the Planck value, carving deep central line depressions.
Thermalization depth	$\tau_{\mathrm{th}} \sim \varepsilon^{-1/2}$	Below this line-core optical depth the source function relaxes to $B_\nu$ ; above it, surface escape dominates.

Common trapNon-LTE is just assuming some non-thermal temperature

The most common misconception is to treat Non-LTE as a single fudge factor — one "excitation temperature" different from $T_e$ . In reality every level carries its own departure coefficient $b_i \equiv n_i/n_i^\ast$ , varying independently with depth; the opacity is reshaped by $b_l$ and the source function by $b_u/b_l$ , so no single temperature describes a Non-LTE gas. A second trap is to imagine "non-equilibrium" means the electrons are non-Maxwellian. They almost never are: electron–electron collisions thermalize the velocity distribution to $T_e$ far faster than any radiative process, so kinetic equilibrium holds and the collisional rates are evaluated at $T_e$ . Non-LTE is a statement about bound-level populations, not the electron gas.

Worked examples

1 · The thermalization parameter of a chromospheric line (fully worked)

Estimate $\varepsilon$ for the Mg II k line ( $\lambda = 2796$ Å, $A_{ul} = 2.6\times10^8\ \mathrm{s}^{-1}$ ) forming where $T = 6000$ K, $n_e = 10^{11}\ \mathrm{cm}^{-3}$ , with collision strength $\Upsilon = 10$ and $g_u = 4$ ; then use the √ε law for the surface source function and thermalization depth.

Step 1. Compute the collisional de-excitation rate from the standard formula $C_{ul} = n_e\, \dfrac{8.629\times10^{-6}\,\Upsilon}{g_u\, T^{1/2}}$ cm³ s⁻¹. With $T^{1/2} = 77.5$ , the rate coefficient is $q_{ul} = 8.629\times10^{-6}\times10/(4\times77.5) = 2.78\times10^{-7}\ \mathrm{cm^3\,s^{-1}}$ , so $C_{ul} = 10^{11}\times2.78\times10^{-7} = \mathbf{2.78\times10^{4}\ s^{-1}}$ .

Step 2. Check the stimulated-emission correction: $h\nu/k = hc/(\lambda k) = 5.15\times10^{4}$ K, so $h\nu/kT = 8.6$ and $1 - e^{-8.6} \approx 1$ . It is negligible for this UV line, so $\varepsilon \approx C_{ul}/(A_{ul} + C_{ul})$ .

Step 3. Evaluate: $\varepsilon = \dfrac{2.78\times10^{4}}{2.6\times10^{8} + 2.78\times10^{4}} = \mathbf{1.1\times10^{-4}}$ — collisions destroy about one absorbed photon in ten thousand.

Step 4. Apply the √ε law. The surface source function is $S_\nu(0) = \sqrt{\varepsilon}\,B_\nu = \sqrt{1.1\times10^{-4}}\,B_\nu = \mathbf{0.010\,B_\nu}$ — one percent of the Planck value — recovering $B_\nu$ only below $\tau_{\mathrm{th}} \sim \varepsilon^{-1/2} = \mathbf{95}$ , the curve of the second figure. The Mg II k core thus forms out of a source function suppressed a hundredfold below LTE, remembered across a hundred optical depths — which is exactly why it is a sensitive chromospheric diagnostic.

2 · The critical density of a nebular forbidden line

Find the electron density at which the [O III] 5007 Å forbidden line ( $^1D_2 \to {}^3P$ , total $A_{ul} = 0.026\ \mathrm{s}^{-1}$ , $g_u = 5$ , $\Upsilon = 2.3$ ) switches from the scattering-like to the collisionally thermalized regime, at $T = 10^4$ K.

Step 1. The critical density is where collisional de-excitation balances spontaneous decay, $n_{\mathrm{crit}} = A_{ul}/q_{ul}$ . Compute the collisional rate coefficient $q_{ul} = 8.629\times10^{-6}\,\Upsilon/(g_u T^{1/2}) = 8.629\times10^{-6}\times2.3/(5\times100) = \mathbf{3.97\times10^{-8}\ cm^3\,s^{-1}}$ .

Step 2. Divide: $n_{\mathrm{crit}} = 0.026/3.97\times10^{-8} = \mathbf{6.5\times10^{5}\ cm^{-3}}$ , matching the standard tabulated value of about $7\times10^{5}\ \mathrm{cm^{-3}}$ .

Step 3. Interpret both sides. Below $n_{\mathrm{crit}}$ (typical H II regions, $n_e \sim 10^2$ – $10^4\ \mathrm{cm^{-3}}$ ) each collisional excitation radiates before the next collision, so every excitation makes one photon and the line ratio is a clean thermometer. Above $n_{\mathrm{crit}}$ the upper level is collisionally de-excited first, populations relax toward Boltzmann, and the line is quenched. The [O III] doublet works as a diagnostic precisely because nebular densities straddle $n_{\mathrm{crit}} \approx 7\times10^{5}\ \mathrm{cm^{-3}}$ — a multilevel statistical-equilibrium calculation with no Planckian continuum in sight.

Physics of the portrait

Non-LTE radiative transfer became a computational science through one person more than any other — the man who wrote the book every stellar spectroscopist keeps on the shelf.

Origins · 1 minDimitri Mihalas and how this was discovered

Dimitri Mihalas

1939–2013 · non-LTE stellar atmospheres and radiation hydrodynamics

Dimitri Mihalas made quantitative Non-LTE stellar atmospheres possible. His textbook Stellar Atmospheres (1970, second edition 1978) defined the field for a generation, and its successor Theory of Stellar Atmospheres with Ivan Hubeny (2014) remains the standard reference. With Lawrence Auer he introduced the complete-linearization method in 1969, treating transfer and statistical equilibrium as one giant coupled system solved by Newton–Raphson linearization — the first practical route through the nonlinearity and nonlocality that had blocked earlier workers. He later co-founded modern radiation hydrodynamics with Foundations of Radiation Hydrodynamics (1984). Mihalas worked at Chicago, Colorado, and Los Alamos, won the Eddington Medal and the Russell Prize, and was unusually public about living with bipolar disorder. His insistence that a stellar spectrum be computed from first principles — every population from rate balance, every photon from the transfer equation — is the intellectual spine of this article.

History

1905

— Schuster analyses radiation passing through a scattering-plus-absorbing atmosphere

1916

— Einstein derives the A and B coefficients linking spontaneous, stimulated, and absorptive rates

1957

— Thomas introduces the photon destruction probability and the non-LTE line source function

1969

— Auer and Mihalas compute Non-LTE model atmospheres by complete linearization

1986

— Olson, Auer and Buchler devise accelerated lambda iteration, making multilevel Non-LTE practical

Going further

Texts. Mihalas, Stellar Atmospheres, 2nd ed., Chs. 5–7 · Hubeny & Mihalas, Theory of Stellar Atmospheres (2014), Chs. 9–14 · Rybicki & Lightman, Radiative Processes in Astrophysics, Ch. 1 · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 9 · Dodelson, Modern Cosmology, 2nd ed. · Rutten, Radiative Transfer in Stellar Atmospheres (lecture notes).

Provenance. A verifier should reproduce: (1) the Einstein relations $g_l B_{lu} = g_u B_{ul}$ , $A_{ul} = (2h\nu^3/c^2)B_{ul}$ (Rybicki & Lightman Ch. 1); (2) the two-level source function and the definition of $\varepsilon$ (Mihalas Ch. 5–6); (3) the √ε law and thermalization depth $\tau_{\mathrm{th}} \sim \varepsilon^{-1/2}$ (Mihalas Ch. 6); (4) accelerated lambda iteration and operator splitting (Olson, Auer & Buchler 1986; Hubeny & Mihalas 2014 Ch. 13); (5) the worked numbers $\varepsilon \approx 1.1\times10^{-4}$ (Mg II k) and $n_{\mathrm{crit}} \approx 6.5\times10^{5}\ \mathrm{cm^{-3}}$ ([O III] 5007) from $C_{ul} = n_e\,8.629\times10^{-6}\,\Upsilon/(g_u T^{1/2})$ ; (6) the Non-LTE nature of recombination (Dodelson). Magnitude estimates ( $\varepsilon \sim 10^{-4}$ , shifts 0.1–0.5 dex) are illustrative textbook ranges.

Prerequisites. The Equation of Radiative Transfer · Emission and the Source Function · Local Thermodynamic Equilibrium · The Boltzmann Excitation Equation · The Saha Ionization Equation · Optical Depth. Leads to. Spectral Line Formation · Kirchhoff's Laws and Thermal Emission · Opacity and Absorption Coefficients.

Ask the tutor: Ask me to walk through why the Mg II k core forms out of a source function suppressed a hundredfold below the Planck function, straight from the √ε law and a single collision rate.

What is the single physical reason the source function departs from the Planck function?

Photon escape. In LTE every de-excitation is balanced by an absorption or collision that refills the upper level, so emission stays Planckian. But in a thin outer layer a photon can random-walk out of the surface before a collision destroys it, permanently removing that energy. Collisions would normally replace it and keep populations thermal, but when they are too rare — when $\varepsilon$ is small — they cannot keep up. The upper level is then underpopulated relative to Boltzmann, the emissivity falls, and the source function sinks below $B_\nu$ . Everything else in Non-LTE is bookkeeping around this leak.

Why is the surface suppression √ε rather than ε?

Naively, if a fraction $\varepsilon$ of photons are destroyed per scattering the source function should fall by $\varepsilon$ . But a photon must survive not one scattering but a whole random walk to the surface. From optical depth $\tau$ , a photon in complete redistribution needs of order $\tau^2$ scatterings to diffuse out, with survival probability $(1-\varepsilon)^{\tau^2} \approx e^{-\varepsilon\tau^2}$ . This stays near one until $\tau \sim \varepsilon^{-1/2}$ , so the surface samples material only that deep and the diffusion solution gives $S(0) \sim \sqrt{\varepsilon}\,B$ — the square root is the signature of a random walk.

If Non-LTE means non-equilibrium, why can I still use T_e in the collision rates?

Because Non-LTE concerns the bound levels, not the free electrons. Electron–electron collisions are so fast that the velocity distribution relaxes to a Maxwellian at a kinetic temperature $T_e$ almost instantly compared with any radiative timescale. This "kinetic equilibrium" holds even when the bound-level populations are wildly non-thermal, so the collisional rates and their detailed balance, $C_{lu}/C_{ul} = (g_u/g_l)e^{-h\nu/kT_e}$ , are correctly evaluated at $T_e$ . What is out of equilibrium is the distribution of atoms over bound states, driven by a field that no longer matches the local temperature.

What exactly is a departure coefficient, and why report those instead of populations?

A departure coefficient $b_i = n_i/n_i^\ast$ is the ratio of the true population to the LTE (Saha–Boltzmann) value at the local $T_e, n_e$ . In LTE every $b_i = 1$ , so departures from unity isolate the purely Non-LTE physics from the ordinary temperature and density dependence already carried by $n_i^\ast$ . They are also the natural variables for the observables: line opacity scales with $b_l$ and the source-function ratio $S_\nu^l/B_\nu$ with $b_u/b_l$ , so one sees at a glance whether a level is overpopulated or starved.

How does Non-LTE change a measured stellar abundance?

Through both the line opacity and the source function, and the direction is not universal. A common effect in cool, metal-poor stars is overionization: the ultraviolet field is stronger than a local blackbody at those frequencies, so it ionizes neutral minority species (neutral iron or magnesium) more than LTE predicts, depleting the line's lower level and weakening it. Fit that weak line with an LTE code and you infer too little of the element — the correction can reach several tenths of a dex. Since these corrections depend on temperature, gravity, and metallicity, they reshape chemical-evolution trends, which is why surveys apply Non-LTE grids rather than a blanket offset.

Is cosmological recombination really the same physics as a stellar line?

Structurally, yes. Recombination of the primordial hydrogen plasma is a statistical-equilibrium problem in which a bound-bound and bound-free network is coupled to a radiation field that cannot escape. Saha equilibrium (the pure-LTE answer) fails because the Lyman-alpha and Lyman-continuum photons that would re-ionize hydrogen are trapped, so the recombination rate is throttled by how fast those photons redshift out of resonance or escape via the two-photon decay of the 2s level. Peebles' effective three-level atom captured this in 1968, and modern codes solve a multilevel version of exactly the rate-balance-plus-transfer system in this article — its output sets the last-scattering surface of the microwave background.