Local thermodynamic equilibrium and Kirchhoff's law

In LTE, translational velocities follow thermal distributions and excitation and ionization populations follow local Boltzmann and Saha relations at a common temperature, with the applicable particle statistics and chemical potentials. The state can vary with position. The radiation field need not equal the Planck function in every direction because photons connect regions with different temperatures and can escape through boundaries. LTE also does not require zero net radiative energy transport. Stellar interiors can carry an outward luminosity while their matter populations are close to local thermal equilibrium.

LTE is most plausible when relevant equilibration processes are fast compared with changes in the local gas and radiative perturbations of the populations. Low-density photoionized nebulae often require rate equations instead of a local Saha prescription. A Maxwellian electron distribution by itself does not prove that every atomic level or ion stage is in LTE. Conversely, an approximately thermal continuum can arise from one process without making all line populations thermal. Each diagnostic must state which part of the material distribution is assumed equilibrated.

In thermodynamic equilibrium a thermal emitter placed in a blackbody radiation bath neither gains nor loses photons through a given balanced process. The transfer equation then gives emissivity divided by net absorption equal to the Planck function. The same local relation can be used for thermal matter in LTE even when the actual incoming radiation differs from B. Spontaneous and stimulated transitions must be combined consistently; using a gross upward absorption coefficient while omitting stimulated emission gives the wrong source function.

If coherent scattering also contributes, its emissivity is proportional to the actual mean intensity rather than automatically to B. The total source function is consequently the weighted B and J combination derived previously. Large scattering optical depth alone does not ensure thermalization; enough true absorption and re-emission must occur to exchange energy with matter. In a varying atmosphere the emergent intensity samples a range of depths with different local temperatures. Replacing that entire contribution function by one blackbody requires an additional approximation, not merely the statement that matter is in LTE.

Kinetic temperature describes a particle velocity distribution. An excitation temperature parameterizes the ratio of two level populations. Brightness temperature is the temperature assigned to a measured intensity by a chosen spectral convention. Under LTE the excitation and kinetic temperatures agree, but a partially transparent thermal slab can still have a lower brightness temperature. A nonthermal or amplified source can have a brightness temperature with no direct interpretation as the gas kinetic temperature. Always specify whether the definition uses the full Planck inversion or the Rayleigh-Jeans form.

A useful limiting check is to hold the slab temperature fixed while decreasing optical depth. Its Rayleigh-Jeans brightness falls linearly with small depth, even though the particles have not cooled. An observed brightness decrease can therefore indicate less emitting material, a smaller beam filling factor, or an opacity change instead of a different kinetic temperature. Frequency-dependent opacity further changes which depths are sampled. Comparing continuum brightness with line ratios and independent geometry helps separate these possibilities, while recognizing that each inference carries its own population and transfer assumptions.