Local Thermodynamic Equilibrium

Every quantitative statement an astronomer makes about the interior or atmosphere of a star begins with a quiet act of faith: that a tiny parcel of gas, buried inside an object manifestly not in equilibrium — hot in the core, cool at the surface, hemorrhaging radiation into space — can nonetheless be assigned a single temperature that fixes all of its microscopic distributions. This is local thermodynamic equilibrium (LTE), the load-bearing wall of classical stellar astrophysics. In strict equilibrium — a closed cavity at one temperature — every distribution is nailed down by $T$ alone: speeds follow Maxwell–Boltzmann, levels follow Boltzmann, ionization follows Saha, and the radiation is the Planck spectrum $B_\nu(T)$ . Real stars have gradients and leak light, so strict equilibrium never holds; LTE is the pragmatic rescue. The crucial and often-misunderstood point is that LTE is a claim about matter, not radiation. It asserts that collisions are frequent enough to thermalize the local velocities, excitation, and ionization to a single kinetic temperature $T(\mathbf{r})$ faster than radiative processes can pull them away — even while the radiation streaming through the parcel is anisotropic and non-Planckian (it must be, or no net energy would flow). The payoff is Kirchhoff's law $S_\nu = j_\nu/\alpha_\nu = B_\nu(T)$ : the source function becomes a known function of the local temperature, and the transfer equation closes. Deep in a stellar interior LTE is essentially exact; in tenuous chromospheres, coronae, and nebulae it fails, and the full non-LTE rate equations are needed.

Key takeaways

Local thermodynamic equilibrium (LTE) assumes the matter at each point — velocities, excitation, ionization — obeys the equilibrium distributions (Maxwell–Boltzmann, Boltzmann, Saha) at one local temperature $T(\mathbf{r})$ , enforced by collisions.
Its central consequence is Kirchhoff's law $S_\nu = B_\nu(T)$ : the source function equals the Planck function, closing the transfer equation throughout a stellar atmosphere.
LTE does not require the radiation field to be Planckian; the intensity $I_\nu$ can be strongly non-blackbody. Only the source function is fixed.
It holds when collision rates dominate radiative rates ( $C_{ul} \gg A_{ul} + B_{ul}\bar J_\nu$ ) and gradients are small over a mean free path — excellent in stellar interiors, broken in coronae, nebulae, and the diffuse ISM.
LTE is what licenses the Saha and Boltzmann equations that turn a stellar spectrum into a temperature, gravity, and abundances.
Key idea: LTE means the matter distributions (Maxwell–Boltzmann, Boltzmann, Saha) take their equilibrium forms at a single local temperature $T(\mathbf{r})$ — enforced by collisions — even though the radiation field need not be Planckian; this yields $S_\nu=B_\nu(T)$ and closes the transfer equation.

Where this lives today

LTE remains the default engine of quantitative stellar astrophysics. Model-atmosphere grids — ATLAS, MARCS, and their descendants — assume LTE to close the transfer equation, and from them come the synthetic spectra fitted to millions of stars by surveys such as APOGEE, GALAH, and Gaia's RVS; when those pipelines report a star's temperature, gravity, and iron abundance, they are at leading order inverting the Saha–Boltzmann–Planck machinery of LTE. The OPAL and OP Rosseland-mean opacities behind every stellar-evolution calculation are likewise computed for LTE populations. The frontier is the correction of LTE where it fails: NLTE codes routinely revise iron, oxygen, and light-element abundances by a few tenths of a dex, corrections that matter for the ages of the oldest stars. And the same competition reappears in cosmology as the freeze-out condition $\Gamma \gtrsim H$ : each decoupling — neutrinos, the light elements at nucleosynthesis, the CMB at recombination — is a cosmic parcel falling out of local equilibrium.

The physical picture

Picture a small parcel of gas bombarded so relentlessly by collisions that its atoms retain no memory of how they were last excited or de-excited by a photon. Each collision reshuffles energy, and after enough of them the parcel relaxes to whatever the local kinetic temperature demands — Maxwell–Boltzmann speeds, Boltzmann level ratios, Saha ionization. This "collisional amnesia" is the essence of LTE. The medium as a whole is out of equilibrium, hot inside and cool outside, but zoom in far enough and each parcel looks like a blackbody cavity at its own temperature — provided it is smaller than the scale over which $T$ changes yet larger than a few mean free paths. LTE is local equilibrium threaded onto a global gradient.

Figure

A gas parcel in LTE: frequent collisions lock the velocity, excitation, and ionization distributions to the single local temperature T(r), so the emission and absorption coefficients obey S_ν = j_ν/α_ν = B_ν(T) — even though the radiation streaming through is anisotropic and non-Planckian, I_ν ≠ B_ν

A subtle point worth internalizing: LTE fixes the source function, not the intensity. The radiation coursing through the parcel is anisotropic and carries the net outward flux; LTE only ties the local emission and absorption coefficients together by $B_\nu$ at $T(\mathbf{r})$ . Whether LTE holds is decided by a race between collisions and radiation, won or lost with density and temperature. The clearest fingerprint is ionization: run the Saha equation for hydrogen up in temperature and it hands you a steep S-curve — neutral below, ionized above. The second figure plots it and marks the Sun's photosphere, where only about one hydrogen atom in ten thousand is ionized.

Figure

Saha ionization of hydrogen versus temperature at photospheric electron density: the ionized fraction climbs steeply from essentially zero near 6000 K to about one half near 9000 K. The Sun's photosphere at T = 5772 K sits far to the left with ionized fraction ≈ 10⁻⁴, yet its Balmer lines still form

The core equations

The engine of LTE is Kirchhoff's law: in equilibrium the ratio of emission to absorption coefficient is a universal function of temperature and frequency — the Planck function. Under LTE it holds locally, so the source function is known once the local temperature is.

S_\nu = \frac{j_\nu}{\alpha_\nu} = B_\nu(T) \qquad (1)

where: $S_\nu$ — source function at frequency $\nu$ ; $j_\nu$ — emission coefficient (emissivity); $\alpha_\nu$ — absorption coefficient per unit length; $B_\nu(T)$ — Planck function at local temperature $T$ . Reading it: matter emits and absorbs in a fixed ratio set by $T$ , whatever the radiation is doing. The Planck function is the blackbody cavity spectrum — the target both the source function (always, under LTE) and the intensity (only at large optical depth) approach.

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

where: $h$ — Planck constant; $c$ — speed of light; $k_B$ — Boltzmann constant; $\nu$ — frequency; $T$ — local temperature. Reading it: a single number $T$ specifies the entire equilibrium spectrum. The matter side of LTE is three equilibrium distributions. The first is the Maxwell–Boltzmann law for particle speeds, which fixes the collision rates.

f(v)\,\mathrm{d}v = 4\pi n\left(\frac{m}{2\pi k_B T}\right)^{3/2} e^{-m v^2 / 2k_B T}\, v^2\,\mathrm{d}v \qquad (3)

where: $f(v)$ — number density of particles with speed near $v$ ; $n$ — total particle number density; $m$ — particle mass; $v$ — particle speed. The same $T$ that sets $B_\nu$ sets the speeds driving the collisions that enforce LTE. The second is the Boltzmann excitation formula, the population ratio of two bound levels.

\frac{n_b}{n_a} = \frac{g_b}{g_a}\, e^{-(E_b - E_a)/k_B T} \qquad (4)

where: $n_a, n_b$ — populations of the lower level $a$ and upper level $b$ ; $g_a, g_b$ — statistical weights (degeneracies); $E_b-E_a$ — excitation energy. Reading it: upper levels are exponentially rarer, filling appreciably only at high $T$ . The third is the Saha equation, balancing ionization against recombination to set adjacent-stage ratios.

\frac{n_{i+1}}{n_i}\, n_e = \frac{2 Z_{i+1}}{Z_i}\left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} e^{-\chi_i / k_B T} \qquad (5)

where: $n_i, n_{i+1}$ — number densities of ionization stages $i$ and $i{+}1$ ; $n_e$ — electron number density; $Z_i, Z_{i+1}$ — partition functions; $m_e$ — electron mass; $\chi_i$ — ionization energy from stage $i$ . Reading it: ionization is set by temperature and, through $n_e$ , by density — a high electron density suppresses it by favouring recombination. Whether these forms apply is decided by the collision-dominance criterion: LTE requires the collisional rate out of a level to swamp the radiative rates.

C_{ul} \gg A_{ul} + B_{ul}\bar{J}_\nu \qquad (6)

where: $C_{ul}$ — collisional de-excitation rate (rate coefficient times $n_e$ ); $A_{ul}$ — Einstein spontaneous emission coefficient; $B_{ul}$ — stimulated emission coefficient; $\bar J_\nu$ — mean intensity over the line profile. Reading it: when collisions win, the populations forget the radiation and thermalize. A second, geometric requirement: the photon mean free path must be short compared with the scale over which $T$ changes, so a parcel sees a single temperature.

\ell_\nu = \frac{1}{\alpha_\nu} \ll H_T \equiv \left|\frac{\mathrm{d}\ln T}{\mathrm{d}s}\right|^{-1} \qquad (7)

where: $\ell_\nu$ — photon mean free path; $\alpha_\nu$ — absorption coefficient; $H_T$ — temperature scale height; $s$ — path length. Reading it: the parcel must be optically thick to itself yet nearly isothermal across a mean free path. Granting all this, the transfer equation closes: with $S_\nu$ replaced by the known $B_\nu(T)$ , it is an ODE along each ray.

\frac{\mathrm{d}I_\nu}{\mathrm{d}\tau_\nu} = I_\nu - S_\nu = I_\nu - B_\nu(T) \qquad (8)

where: $\tau_\nu$ — optical depth, $\mathrm{d}\tau_\nu=\alpha_\nu\,\mathrm{d}s$ ; $I_\nu$ — specific intensity; $S_\nu$ — source function, $=B_\nu$ in LTE. Reading it: the intensity relaxes toward $B_\nu(T)$ over one optical depth — deep layers glow as blackbodies, the surface does not.

Derivation AWhy the source function equals the Planck function under LTE

The goal here is to show that the most useful consequence of LTE — that the source function equals the Planck function, $S_\nu = B_\nu(T)$ — is not a fresh assumption but a theorem, provable two independent ways: from the stationarity of the radiation field in a closed cavity, and from the collisional detailed balance that fixes the level populations. We build both, then localize the result so it holds point by point in a medium whose radiation field is not Planckian — the whole content of "local" in LTE.

Step 1. In strict equilibrium the radiation in a cavity is uniform and isotropic, $I_\nu = B_\nu(T)$ , so along any ray $\mathrm{d}I_\nu/\mathrm{d}s = j_\nu - \alpha_\nu I_\nu = 0$ . Why: a uniform field has no gradient, so emission and absorption must balance at every frequency.

Step 2. Solving gives $j_\nu = \alpha_\nu B_\nu(T)$ , hence $S_\nu \equiv j_\nu/\alpha_\nu = B_\nu(T)$ . This is Kirchhoff's law: the ratio of emissivity to opacity is a universal function of $T$ and $\nu$ . Why: the cavity result cannot depend on what the walls are made of, so the ratio must be a material-independent thermodynamic identity.

Step 3. Now the microphysics. For a two-level atom the rate equation reads $n_l(C_{lu} + B_{lu}\bar J_\nu) = n_u(C_{ul} + A_{ul} + B_{ul}\bar J_\nu)$ , balancing upward against downward transitions. Why: in steady state each level's population is constant, so rate in equals rate out.

Step 4. In the collision-dominated limit $C_{ul}, C_{lu} \gg$ the radiative terms, the radiative pieces drop and $n_l C_{lu} \approx n_u C_{ul}$ . The collision coefficients obey detailed balance at the kinetic temperature, $C_{lu}/C_{ul} = (g_u/g_l)\,e^{-(E_u-E_l)/k_B T}$ , because the colliding electrons are Maxwellian at $T$ . Why: collisions are mediated by thermal electrons, so they know only the kinetic temperature — not the radiation field.

Step 5. Therefore $n_u/n_l = (g_u/g_l)\,e^{-(E_u-E_l)/k_B T}$ : the populations are exactly Boltzmann at $T$ . Since $j_\nu$ and $\alpha_\nu$ are built from them, their ratio is forced back to $B_\nu(T)$ , and localizing gives $S_\nu(\mathbf{r}) = B_\nu\big(T(\mathbf{r})\big)$ even where $I_\nu \neq B_\nu$ . Why: the matter carries the equilibrium relation locally, so $S_\nu$ is Planckian pointwise regardless of the radiation streaming past.

Go deeper · Derivation B · 1 minThe Eddington–Barbier relation and why the emergent intensity is not Planckian

Derivation BThe Eddington–Barbier relation and why the emergent intensity is not Planckian

The aim of this derivation is to make precise the most common confusion about LTE: that if $S_\nu = B_\nu(T)$ everywhere, the light leaving the star must be a blackbody. It is not. We solve the transfer equation for a semi-infinite plane-parallel atmosphere, linearize the source function in optical depth, and show that the emergent intensity samples the Planck function at optical depth of order unity — so only the deepest layers radiate as blackbodies while the surface does not.

Step 1. The formal solution of equation (8) for a ray emerging at angle $\theta$ , with $\mu = \cos\theta$ , is $I_\nu(0,\mu) = \int_0^\infty S_\nu(t)\,e^{-t/\mu}\,\mathrm{d}t/\mu$ , where $t$ is vertical optical depth and $S_\nu = B_\nu$ . Why: each layer contributes its source function attenuated by the optical depth to the surface — the exponential is a photon's survival probability.

Step 2. Expand to first order about the surface, $S_\nu(t) \approx a_\nu + b_\nu t$ , valid when $B_\nu(T(t))$ varies slowly over the layer dominating the emergent light. Why: the emergent intensity is weighted toward $t\sim\mu$ , so only the local slope of $S_\nu$ there matters.

Step 3. Evaluate using $\int_0^\infty e^{-t/\mu}\,\mathrm{d}t/\mu = 1$ and $\int_0^\infty t\,e^{-t/\mu}\,\mathrm{d}t/\mu = \mu$ , giving $I_\nu(0,\mu) = a_\nu + b_\nu\mu = S_\nu(t=\mu)$ . Why: the two moments of the exponential kernel pick out value and slope, collapsing the integral to a point.

Step 4. Hence $I_\nu(0,\mu) \approx B_\nu\big(T(\tau_\nu = \mu)\big)$ : the emergent intensity equals the Planck function at the layer where $\tau_\nu \approx \mu \sim 1$ . Only for $\tau_\nu \gg 1$ does $I_\nu \to B_\nu(T)$ ; near the surface the intensity is not Planckian even though LTE holds locally. Why: we see one optical depth in, so different frequencies with different opacities show different depths and hence different brightness — which is what carves absorption lines.

Go deeper · Derivation C · 1 minThe two-level source function and the thermalization parameter

Derivation CThe two-level source function and the thermalization parameter

The purpose here is to write down, for a two-level atom, the exact line source function as a weighted average of the Planck function and the local mean intensity, and to identify the dimensionless parameter measuring how close the gas is to LTE. This one expression holds both the LTE limit and its breakdown, explaining why low-density gas — where scattering beats collisions — falls out of LTE and needs non-local treatment.

Step 1. The line source function is spontaneous emission over net absorption, $S_\nu = n_u A_{ul}/(n_l B_{lu} - n_u B_{ul})$ . Why: source function is emissivity over opacity, and for a line both are built from the same two populations.

Step 2. Insert the Einstein relations $g_l B_{lu} = g_u B_{ul}$ and $A_{ul} = (2h\nu^3/c^2)B_{ul}$ to get $S_\nu = (2h\nu^3/c^2)\big[(n_l g_u)/(n_u g_l) - 1\big]^{-1}$ . Why: the Einstein relations encode equilibrium into the atomic constants, forcing $S_\nu\to B_\nu$ whenever the populations are Boltzmann.

Step 3. Solve the statistical-equilibrium equation for $n_u/n_l$ keeping both terms, and define the thermalization parameter $\epsilon = C_{ul}(1-e^{-h\nu/k_B T})/[A_{ul} + C_{ul}(1-e^{-h\nu/k_B T})]$ . Why: $\epsilon$ is the probability an excitation is quenched by a thermalizing collision rather than scattered — the fraction of "true absorption".

Step 4. The result collapses to $S_\nu = (1-\epsilon)\,\bar J_\nu + \epsilon\, B_\nu(T)$ , a linear interpolation between the radiation field and the Planck function. Why: a photon is re-thermalized with probability $\epsilon$ (giving $B_\nu$ ) or scattered with probability $1-\epsilon$ (giving $\bar J_\nu$ ); $S_\nu$ is the weighted mean.

Step 5. In the collision-dominated limit $C_{ul}\gg A_{ul}$ , $\epsilon\to1$ and $S_\nu\to B_\nu(T)$ — precisely LTE. In tenuous gas $\epsilon\ll1$ , so $S_\nu\approx\bar J_\nu$ : populations are set by the non-local radiation field and the problem is irreducibly non-local (NLTE). Why: $\epsilon$ is the single knob that turns LTE into NLTE, and it is small exactly where the density is low.

Limiting cases

Case	Result	Meaning
Deep stellar interior, large optical depth	$\tau_\nu \gg 1,\ \ I_\nu \to B_\nu(T)$	Photon mean free path tiny, radiation nearly isotropic and Planckian; LTE excellent and even the intensity approaches $B_\nu$ .
Collision-dominated regime	$C_{ul} \gg A_{ul} + B_{ul}\bar J_\nu$	Collisional rates swamp radiative rates, so populations follow Boltzmann and Saha at $T$ ; LTE holds even where $\tau_\nu\sim1$ .
Full thermodynamic equilibrium	$\nabla T = 0,\ \text{closed cavity}$	LTE plus vanishing gradients and no leakage recovers strict TE: matter and radiation both Planckian everywhere.
Tenuous, optically thin gas (NLTE)	$C_{ul} \lesssim A_{ul},\ \ \tau_\nu \ll 1$	Radiative rates dominate; populations set by the non-local radiation field. LTE fails — chromospheres, coronae, nebulae, and the ISM need full rate equations.

Common trapLTE does not mean the radiation field is a blackbody

The most persistent error is to assume that under LTE the specific intensity equals the Planck function, $I_\nu = B_\nu(T)$ . It does not. LTE is a condition on matter: collisions force the velocity, excitation, and ionization distributions to their equilibrium forms at the local kinetic temperature. What this fixes is the source function $S_\nu = B_\nu(T)$ — the ratio of emission to absorption — not the intensity. The radiation field $I_\nu$ can be, and generally is, anisotropic and non-Planckian; it must be, because a Planckian isotropic field carries no net flux and a star would not shine. Only deep down, where $\tau_\nu \gg 1$ , does the intensity relax to $B_\nu(T)$ . Equating $I_\nu$ with $B_\nu$ near the surface erases every spectral line — lines form because $I_\nu \neq B_\nu$ at $\tau_\nu \sim 1$ .

Worked examples

1 · Hydrogen in the solar photosphere — Boltzmann, then Saha (fully worked)

Take the solar photosphere at $T = 5772$ K. Hydrogen's n = 2 to n = 1 excitation energy is $E_{21} = 10.2$ eV with weights $g_1 = 2,\ g_2 = 8$ ; the ionization energy is $\chi = 13.6$ eV. Estimate the fraction of hydrogen in the n = 2 level (which absorbs the Balmer lines) and the fraction ionized, using a representative photospheric $n_e \approx 2\times10^{19}\ \mathrm{m^{-3}}$ .

Step 1. Thermal energy: $k_B T = (8.617\times10^{-5}\ \mathrm{eV/K})(5772\ \mathrm{K}) = 0.497$ eV — tiny compared with both 10.2 eV and 13.6 eV, the first hint that excited and ionized hydrogen will be rare.

Step 2. Boltzmann ratio for n = 2: $n_2/n_1 = (g_2/g_1)\,e^{-E_{21}/k_B T} = 4\,e^{-20.5}$ . With $e^{-20.5} = 1.24\times10^{-9}$ , $n_2/n_1 = \mathbf{5.0\times10^{-9}}$ — five hydrogen atoms in a billion in the Balmer-absorbing level.

Step 3. Saha ratio. The degeneracy factor is $2Z_{II}/Z_I = 2(1)/2 = 1$ , and $(2\pi m_e k_B T/h^2)^{3/2} = 1.06\times10^{27}\ \mathrm{m^{-3}}$ here. With $\chi/k_B T = 27.3$ , so $e^{-27.3} = 1.33\times10^{-12}$ , we get $(n_{II}/n_I)\,n_e = (1.06\times10^{27})(1.33\times10^{-12}) = 1.41\times10^{15}\ \mathrm{m^{-3}}$ .

Step 4. Divide by the electron density: $n_{II}/n_I = 1.41\times10^{15}/2\times10^{19} = \mathbf{7\times10^{-5}}$ . Only about one hydrogen atom in fourteen thousand is ionized — hydrogen is overwhelmingly neutral in the photosphere.

Step 5. Put the two together: the fraction of all hydrogen able to absorb a Balmer photon is $(n_2/n_1)\times(\text{neutral fraction}) \approx 5\times10^{-9}$ , since nearly all hydrogen is neutral. Yet the Balmer lines are prominent in the solar spectrum — proof of how few absorbers are needed. Running Saha and Boltzmann across temperature explains the spectral sequence: Balmer strength peaks near $T\approx9000$ – $10\,000$ K (A stars), where $n=2$ is populated but hydrogen is not yet fully ionized, and collapses in hotter and cooler stars. Key results: n₂/n₁ ≈ 5×10⁻⁹, ionized fraction ≈ 7×10⁻⁵, and only ~5 hydrogen atoms in 10⁹ can absorb a Balmer photon.

2 · Where does LTE hold? The critical density of a forbidden line

The [O III] λ5007 line has $A_{ul} \approx 0.02\ \mathrm{s^{-1}}$ and a collisional de-excitation coefficient $q_{ul} \approx 3\times10^{-8}\ \mathrm{cm^3\,s^{-1}}$ at $T\sim10^4$ K. Find the critical electron density where collisions and radiative decay balance, and decide whether the upper level is in LTE in (a) a stellar photosphere and (b) a planetary nebula.

Step 1. The critical density is where the collisional de-excitation rate $n_e q_{ul}$ equals the spontaneous rate $A_{ul}$ : $n_\mathrm{crit} = A_{ul}/q_{ul} = 0.02/(3\times10^{-8}) = \mathbf{7\times10^{5}\ cm^{-3}}$ . Above it, collisions win and the level thermalizes to Boltzmann (LTE); below it, radiative decay wins and the radiation field sets the populations (NLTE).

Step 2. A stellar photosphere has $n_e \sim 10^{13}\ \mathrm{cm^{-3}}$ , ten million times $n_\mathrm{crit}$ . The upper level is collisionally de-excited far faster than it can radiate, so it sits at its LTE population — and, tellingly, the forbidden line is collisionally quenched and does not appear.

Step 3. A planetary nebula has $n_e \sim 10^{3}$ – $10^{4}\ \mathrm{cm^{-3}}$ , far below $n_\mathrm{crit}$ . Almost every collisional excitation is followed by radiative decay, so the level is out of LTE and the forbidden line shines — the famous nebular "nebulium" lines no nineteenth-century lab could reproduce.

Step 4. The lesson generalizes: LTE for a transition holds when $n_e \gg n_\mathrm{crit}$ , and $n_\mathrm{crit}$ ranges from $\sim10^2\ \mathrm{cm^{-3}}$ for forbidden lines to $\gtrsim10^{15}\ \mathrm{cm^{-3}}$ for strong permitted lines. Key result: with $n_\mathrm{crit}\approx7\times10^{5}\ cm^{-3}$ , the level is in LTE in a photosphere but not in a nebula.

Physics of the portrait

The idea that a single temperature governs stellar matter has many parents — Kirchhoff, Boltzmann, Planck, Schwarzschild — but the one who made it a working tool, letting a spectrum be read as a thermometer, was Meghnad Saha.

Origins · 1 minMeghnad Saha and how this was discovered

Meghnad Saha

1893–1956 · thermal ionization and the physics of stellar spectra

Meghnad Saha, born in a Bengali village to a shopkeeper of modest means, became a founder of modern astrophysics with a single idea. In 1920 he wrote down the Saha ionization equation, treating ionization as a chemical reaction between an ion, an electron, and a bound atom in thermal equilibrium. It showed that the bewildering variety of stellar spectra — hydrogen lines here, ionized calcium there, molecular bands elsewhere — reflected not different compositions but overwhelmingly temperature and pressure acting on nearly the same mix of elements. This unlocked the Harvard spectral sequence (OBAFGKM) as a temperature ordering and made LTE quantitative: given $T$ and $n_e$ , one could compute the ionization and excitation state and predict which lines appear. Cecilia Payne applied Saha's and Boltzmann's equations in her 1925 thesis to show stars are overwhelmingly hydrogen and helium — a result so at odds with the prejudices of the day that she was pressured to disown it before it was vindicated. Saha went on to found the research culture of independent India, but his name is anchored to the equation that let astronomers take a star's temperature from its light.

History

1859

— Kirchhoff states the law relating emission and absorption and coins the blackbody idea

1900

— Planck derives the blackbody spectrum B_ν(T)

1906

— Schwarzschild formulates radiative equilibrium in a stellar atmosphere

1920

— Saha publishes the thermal ionization equation, making stellar spectra a thermometer

1925

— Payne applies Saha and Boltzmann to show stars are overwhelmingly hydrogen

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §8.1 (Saha and Boltzmann), §9.1–9.5 (transfer, source function, Eddington–Barbier), §10.1–10.4 (model atmospheres) · Rybicki & Lightman, Radiative Processes in Astrophysics, Ch. 1 (Kirchhoff's law, source function) · Mihalas, Stellar Atmospheres, 2nd ed., Ch. 5 (LTE, two-level atom, thermalization) · Osterbrock & Ferland, Astrophysics of Gaseous Nebulae, Ch. 3 (critical density, forbidden lines) · Dodelson, Modern Cosmology (equilibrium and freeze-out) · Mo, van den Bosch & White, Galaxy Formation and Evolution (cosmic thermal history). Key numerical checks: at $T=5772$ K, $k_BT=0.497$ eV gives $n_2/n_1 = 4\,e^{-20.5} = 5.0\times10^{-9}$ ; the Saha factor $(2\pi m_e k_B T/h^2)^{3/2}=1.06\times10^{27}\ \mathrm{m^{-3}}$ and $e^{-27.3}=1.33\times10^{-12}$ give $n_{II}n_e/n_I=1.41\times10^{15}\ \mathrm{m^{-3}}$ , hence at $n_e=2\times10^{19}\ \mathrm{m^{-3}}$ an ionized fraction $\approx7\times10^{-5}$ ; the critical density $A_{ul}/q_{ul}=0.02/(3\times10^{-8})\approx7\times10^{5}\ \mathrm{cm^{-3}}$ . Physical constants are CODATA values; nebular rate coefficients are representative.

Prerequisites. Blackbody Radiation and the Planck Law · The Equation of Radiative Transfer · Emission and the Source Function · Specific Intensity and Radiative Flux. Leads to. Kirchhoff's Laws and Thermal Emission · The Boltzmann Excitation Equation · The Saha Ionization Equation · Spectral Line Formation · Opacity and Absorption Coefficients.

Ask the tutor: Ask me to run the Saha and Boltzmann equations across the spectral sequence to see exactly why Balmer lines peak in A stars while hydrogen stays neutral in the Sun.

Does LTE require the radiation field to be a blackbody?

No — this is the most important thing to get right. LTE is a statement about matter: collisions force the velocity, excitation, and ionization distributions to their equilibrium forms at one local temperature. The radiation field can be strongly anisotropic and non-Planckian, and must be, because an isotropic blackbody field carries zero net flux and the star would not radiate. What LTE fixes is the source function $S_\nu = B_\nu(T)$ — the ratio of emission to absorption — not the intensity $I_\nu$ , which approaches $B_\nu$ only deep inside where the optical depth is enormous.

If the source function is Planckian everywhere, why isn't the emergent spectrum a perfect blackbody?

Because you see only about one optical depth into the star, and that depth differs by frequency. The Eddington–Barbier relation says the emergent intensity at $\nu$ equals the Planck function where $\tau_\nu \approx 1$ . At a line frequency the opacity is high, so $\tau_\nu=1$ is reached higher up in cooler gas, and the line looks dark against the brighter continuum formed deeper and hotter. Absorption lines exist precisely because $S_\nu=B_\nu$ is evaluated at different temperatures for different frequencies — the spectrum is a temperature map, not a single blackbody.

What actually makes LTE break down?

Two things, usually together: low density and low optical depth. LTE needs collisions to win the race against radiation, $C_{ul}\gg A_{ul}+B_{ul}\bar J_\nu$ . Since the collisional rate scales with $n_e$ while spontaneous emission does not, thin gas lets radiation control the populations — the threshold is the critical density $n_\mathrm{crit}=A_{ul}/q_{ul}$ , below which a level is in NLTE. LTE also needs $T$ nearly constant over a mean free path, which fails near a surface where photons escape freely. Chromospheres, coronae, winds, and nebulae violate both.

How is LTE related to the freeze-out of species in the early universe?

The logic is identical, with expansion playing the role of radiative loss. In the hot early universe a species stays in equilibrium as long as its interaction rate exceeds the Hubble rate, $\Gamma \gg H$ — the cosmic analogue of "collisions beat radiation". When expansion drops $\Gamma$ below $H$ , the species freezes out, its abundance locked in. Neutrino decoupling, nucleosynthesis, and recombination are all freeze-outs — each a parcel of the universe falling out of local equilibrium exactly as stellar matter does when collisions can no longer keep pace.

Why can Saha's equation be so sensitive that the Sun is only 10⁻⁴ ionized but an A star is half ionized?

Because ionization depends on temperature through an exponential, $e^{-\chi/k_B T}$ , and $\chi=13.6$ eV for hydrogen dwarfs $k_B T\sim0.5$ eV in the photosphere. A modest rise in $T$ shrinks that exponent dramatically: 5772 K to 9000 K changes $e^{-\chi/k_B T}$ by many orders of magnitude, sweeping hydrogen from almost neutral to about half ionized. Density enters too, through $n_e$ , because a denser gas recombines faster and stays more neutral. This exponential sensitivity is exactly why a spectral line is such a precise thermometer.

Does LTE mean the electron and radiation temperatures are equal?

Under LTE the matter has a single kinetic temperature — electrons and ions share it because collisions are frequent — and the source function is Planckian at that same $T$ . But the radiation field need not have a temperature at all: it is generally not a blackbody. Deep inside a star it is so nearly Planckian that a radiation temperature is meaningful and equals the matter temperature; near the surface it departs from Planckian and the notion loses meaning, even while the matter stays in LTE.