Codex @codex  0

For a thermalized radiative transfer source function $S_{\lambda }=B_{\lambda }(T)$ constant along a ray, the specific intensity after optical-depth interval $\Delta \tau$ is $I_{\lambda }=B_{\lambda }+(I_{\mathrm{in}}-B_{\lambda })e^{-\Delta \tau /\mu }$ for $\mu >0$. As $\Delta \tau \to \infty$, it tends to the Planck function, independently of a bounded boundary input. A temperature gradient is unnecessary for blackbody emission. In a scattering medium, local thermodynamic equilibrium alone does not guarantee $S_{\lambda }=B_{\lambda }$; thermalization must be justified.

Limb darkening is the decrease of emergent specific intensity toward the apparent edge of a stellar disc. A transit depth then depends on the intensity sampled by the planet as well as its projected area; the uniform-disc area ratio is only an approximation.

For a spherical star of radius $R_{\star }$, let $r$ be distance from the centre of its projected disk and let $\theta$ be the angle between the outward unit normal and the line of sight. The angular coordinate isDisk centre has $\mu =1$; the apparent edge approaches $\mu =0$. A resolved image measures specific intensity in each direction. Limb darkening means that this specific intensity itself falls toward the edge; the projected area factor used when collecting radiant flux is a separate geometric effect.

A simple explanation assumes a horizontally uniform plane-parallel atmosphere, a sufficiently deep optically thick medium, negligible scattering of light, and thermal absorption and emission in local thermodynamic equilibrium. Let ${\tau }_{\nu }$ be inward vertical optical depth at fixed frequency $\nu$. An oblique light ray has line of sight optical depth ${\tau }_{\nu }/\mu$, so it encounters the same attenuation at a smaller vertical optical depth. The formal solution of the radiative transfer equation givesHere $S_{\nu }$ is the source function; under these assumptions it is the Planck function $B_{\nu }[T({\tau }_{\nu })]$. If temperature increases inward, an oblique view preferentially samples cooler layers and hence lower specific intensity. ^[ref]

The weight $e^{-{\tau }_{\nu }/\mu }/\mu$ is an exponential distribution in vertical optical depth, with expected value $\mu$ and standard deviation $\mu$. Thus “seeing down to optical depth one” refers to the line of sight, and represents a broad sampling of layers rather than a sharp emitting surface. For a source function that is linear throughout the atmosphere, the Eddington-Barbier relation is exact:This follows by integrating the constant and linear terms separately. A constant source function instead gives constant specific intensity, as in the blackbody limit of an isothermal atmosphere. An outward-rising temperature or important scattering of light changes the argument. Close enough to the actual edge, a plane-parallel atmosphere also loses validity because of curvature. Neither one limb-darkening coefficient nor a universal dependence on wavelength follows from this simple model.

A non-scattering isothermal atmospheric layer in local thermodynamic equilibrium, of temperature $T_a$ and slant optical depth $t_{\lambda }$, lies above a blackbody background at $T_b$. The specific intensity isThus an increase in opacity shifts the specific intensity toward the layer's Planck function. If a line is more opaque than its continuum, it appears in absorption for $T_a<T_b$, in emission for $T_a>T_b$, and has no contrast for $T_a=T_b$. This is a local explanation, not a claim that all features diagnose one uniform layer.

The Rosseland mean opacity is the transport opacity appropriate to an optically thick region close to local thermodynamic equilibrium. If ${\kappa }_{\nu }$ is monochromatic transport opacity and $B_{\nu }(T)$ the Planck function, thenIt is a weighted harmonic mean: low-opacity frequency windows carry much of the diffusive flux. One must combine the monochromatic processes before averaging, rather than adding their separately averaged opacities.

In hot fully ionized material, electron-scattering opacity supplies an almost density- and temperature-independent contribution at fixed composition. In the nonrelativistic Thomson scattering limit, ${\kappa }_{\mathrm{es}}={\sigma }_T{n}_e/\rho \simeq 0.2(1+X)\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$. Free-free opacity, the absorption counterpart of thermal bremsstrahlung, comes from photon absorption during encounters between free electrons and ions. Its approximate mean obeys Kramers' opacity law, ${\kappa }_{\mathrm{ff}}\propto \rho T^{-7/2}$, with composition and quantum correction factors.

When ions retain bound electrons, bound-free absorption is photoionization; edges, occupations and ionization fractions strongly affect its frequency dependence. Bound-bound absorption produces absorption lines. Numerous metal lines can close low-opacity windows and produce important opacity features in partially ionized layers. Line populations depend on excitation as well as ionization, while Doppler broadening, pressure broadening and overlapping lines change the transport windows. Stimulated inverse processes must be included consistently with detailed balance and the local thermal radiation field.

In cooler hydrogen-rich atmospheres, negative hydrogen ion opacity from both bound-free and free-free processes is often important. Its strong temperature dependence helps determine convective surface relations. In still cooler layers, molecular bands, and at sufficiently low temperatures dust absorption/scattering, may be important. These processes cannot be represented by one Kramers power law over the whole density-temperature plane.

An opacity calculation therefore needs the chemical abundances, the equation of state, electron density and bound-state populations. Saha ionization equation and thermal excitation populations give useful dilute LTE estimates; high-density pressure ionization, nonideal effects and altered level populations require corrections. Composition matters even when one writes only $\kappa (\rho ,T)$: hydrogen, helium and the metal abundance pattern are additional fixed inputs.

In dense degenerate cores, thermal conduction by electrons can dominate energy transport. An equivalent conductive opacity may be introduced so that $1/{\kappa }_{\mathrm{eff}}=1/{\kappa }_R+1/{\kappa }_{\mathrm{cond}}$, because the two diffusive fluxes add; conduction is not itself a photon absorption process. Very hot regimes also require departures from the simple nonrelativistic scattering approximation. Near the optically thin photosphere, a Rosseland diffusion description alone is inadequate, and frequency-dependent radiative transfer with appropriate boundary conditions is required. These regime changes are why realistic stellar opacity tables, rather than one global power law, enter stellar structure.

The vertical hydrostatic equilibrium equation balances the gradient of the total pressure against gravity. In a thin disk with Keplerian rotation, expanding the central gravitational field about the midplane gives vertical acceleration $-{\Omega }^{2}z$, so $p_z=-\rho {\Omega }^{2}z$. Disk self-gravity is neglected in this approximation.

The flux-gradient equation is local thermal balance: the upward radiative flux increases with height because differential rotation dissipates energy. For a Newtonian effective dynamic viscosity $\mu$, the volumetric heating rate is $\mu (r{\Omega }^{\prime }{)}^2$, which becomes $9\mu {\Omega }^2/4$ for Keplerian rotation. The symbol $\mu$ here is dynamic viscosity, not kinematic viscosity; the latter is $\nu =\mu /\rho$.

The radiative diffusion equation applies to an optically thick medium in local thermodynamic equilibrium. Its diffusion coefficient is $c/(3\kappa \rho )$, and the equilibrium radiation energy density is $u_{\mathrm{rad}}=4\sigma T^4/c$. HenceThe minus sign means energy flows down the temperature gradient. Here $\sigma$ is the Stefan-Boltzmann constant, $\kappa$ the opacity per unit mass, and $c$ the speed of light. Constant Thomson scattering opacity is the adopted transport approximation; local thermodynamic equilibrium is an additional assumption behind the radiation equation of state.

Finally the equation of state adds ideal gas gas pressure $p_g=\rho kT/({\mu }_m{m}_H)$ and isotropic radiation pressure $p_r=u_{\mathrm{rad}}/3=4\sigma T^4/(3c)$. The Boltzmann constant is $k$, and ${\mu }_m{m}_H$ is the mean mass per gas particle. Both contributions supply the total pressure entering vertical hydrostatic equilibrium.

Use the notation $I_{\nu }(\mu ,\tau )$ throughout, with optical depth increasing inward and $0<\mu \le 1$ for an outward ray. This removes the change in argument order used in the printed surface-intensity notation. Multiplying the radiative transfer equation by its integrating factor givesIntegrating from the surface to a deeper boundary $b$ yieldsFor a semi-infinite atmosphere with no exponentially growing homogeneous term, the boundary term vanishes as $b\to \infty$. The formal solution of the radiative transfer equation is thereforeThe usual deep-atmosphere boundary condition is necessary; a first-order differential equation alone does not fix the emerging specific intensity.

For a linear source function, the zeroth and first exponential moments give $I_{\nu }(\mu ,0)=a_0+a_1\mu$. Equivalently $I_{\nu }(\mu ,0)=S_{\nu }(\tau =\mu )$, the Eddington-Barbier relation. If the source rises inward, rays near the limb sample cooler, shallower layers, producing limb darkening.

For the finite polynomial, put $u=t/\mu$ and integrate each term. Repeated integration by parts gives ${\int }_0^{\infty }{u}^j{e}^{-u}du=j!$, so the polynomial source function moments for emergent intensity giveThe limiting grazing-ray specific intensity is $a_0$. If the displayed polynomial is only a local Taylor approximation to a more general source, this formula applies to that approximation and a source-function remainder also contributes; a finite local expansion is not automatically an exact description at every optical depth.

For a constant source in an overlying layer of optical thickness $\tau$, the radial ray has the exact transfer relationExpanding the exponentials gives the constant source transfer through a thin layer:This finite-layer result does not require the specific intensity below the layer to equal its source function. The sign of $S_{\nu }-I_{\nu }(1,\tau )$ already distinguishes emission from absorption.

An absorption line forms when a bound-bound transition increases opacity at selected frequencies. In a photosphere whose temperature decreases outward, optical depth of order one at the line frequency occurs higher than at a nearby continuum frequency. In an absorption-dominated region in local thermodynamic equilibrium, the source is the Planck function; the cooler line-forming layer then emits less specific intensity than the deeper continuum-forming layer. The result is a dark spectral feature. Scattering and departures from LTE modify the source, so this temperature-gradient picture is a useful mechanism rather than a universal identity for every line.

Two distinct situations can produce emission lines. First, a heated chromosphere has an outward temperature rise, so an optically thick line can sample gas hotter than the continuum photosphere beneath it. Secondly, a hot optically thin circumstellar envelope or stellar wind emits line photons through collisional excitation or recombination, with no comparably bright background along all rays; its added line flux can exceed the underlying continuum. Winds of hot luminous stars can also give a mixture of emission and absorption across a Doppler-broadened line profile. These examples explain the change in specific intensity through either an enhanced source function or additional emitting material.

A stellar equation of state supplies pressure and internal energy as functions of density, temperature and composition, together with thermodynamic derivatives needed for stability and transport. Hydrostatic equilibrium fixes the pressure gradient, but does not determine which microscopic components provide the pressure. In ordinary dense interiors local thermodynamic equilibrium is a useful starting point. A consistent mixture iswhere ions, Electrons, radiation and interaction corrections are distinguished. The classical Electron pressure and electron degeneracy pressure are two limits of the same Electron contribution, and must not be added as if they belonged to different particles. The finite-temperature electron equation of state interpolates between them.

In a fully ionized, nondegenerate, nonrelativistic gas, $P_g=\rho k_{B}T/(\mu m_u)$ and specific thermal energy is $u_g=3k_{B}T/(2\mu m_u)$. With nuclear mass fractions $X_A$, charges $Z_A$ and mass numbers $A_A$, the mean molecular weight satisfies ${\mu }^{-1}={\sum }_A{X}_A(1+Z_A)/A_A$, and ${\mu }_e^{-1}={\sum }_A{X}_A{Z}_A/A_A$ is the mean molecular weight per electron. This regime describes much of an ordinary main sequence interior. Toward cooler layers, ionization and molecular dissociation change particle numbers and consume heat. The Saha equation relates ionization to both temperature and Electron density: there is no universal horizontal ionization boundary. These regions have larger heat capacity and can have a reduced stellar adiabatic exponent. The simple fully ionized formula is then insufficient.

Equilibrium photons give radiation pressure $P_{\gamma }=a_r{T}^4/3$ and energy per volume $a_r{T}^4$, or specific energy $u_{\gamma }=a_r{T}^4/\rho$. In the nondegenerate gas regime, equality with gas pressure gives the radiation-to-gas pressure boundarya line of slope $1/3$ on a $(\log \rho ,\log T)$ plot. Higher temperatures at fixed mass density favor photon support. A monatomic gas has stellar adiabatic exponent $5/3$, while radiation alone has $4/3$; their adiabatic exponents of a monatomic gas-radiation mixture are not obtained by assuming a fixed pressure fraction during compression. Radiation support is particularly important in massive stars.

For Electrons, Pauli exclusion principle and the Fermi-Dirac distribution determine occupation numbers. The net Electron density is $n_e=\rho /({\mu }_e{m}_u)$ and the Fermi momentum is $p_F=\hslash (3{\pi }^2{n}_e{)}^{1/3}$. Define the kinetic electron Fermi temperatureFor $T\gg T_F$ the Electrons are nearly classical; for $T\ll T_F$ they are strongly degenerate and their pressure depends primarily on density. The intermediate region requires finite-temperature electron equation of state integrals, not a discontinuous switch of formulas.

The equation of state of a cold electron gas gives, in its two limits,These are respectively the $n=3/2$ and $n=3$ pressure-density powers, explaining the approximate white dwarf polytropic mass-radius relation sequence and the Chandrasekhar limit. The kinetic energy per volume is $3P_e/2$ in the nonrelativistic limit and $3P_e$ in the ultrarelativistic limit. Ions can still supply much of the heat capacity even when the Electron pressure supplies the mechanical support.

The Electron degeneracy crossover $T\sim T_F$ has slope $2/3$ at low mass density and $1/3$ at high mass density. The electron relativistic density threshold isa vertical marker where $p_F=m_{e}c$. It is different from the thermal electron relativistic threshold $k_{B}T\sim m_e{c}^2$, near $5.93\times 10^9\mathrm{K}$, a horizontal temperature scale. Hot dilute matter can have relativistic thermal Electrons without degeneracy; cold dense matter can have relativistic degenerate Electrons without reaching that temperature.

Electron degeneracy also does not automatically imply that Electrons dominate the total pressure. Comparing the cold Electron limit with radiation gives the radiation-to-degeneracy pressure boundaryIts logarithmic slopes are $5/12$ for nonrelativistic Electrons and $1/3$ for ultrarelativistic Electrons. One must compare the pressures separately from the degeneracy criterion; extrapolating the classical gas-radiation line into a degenerate region is incorrect.

The stellar equation-of-state regime diagram uses an illustrative fully ionized carbon composition, ${\mu }_e=2$, $\mu =12/7$. It displays the electron-degeneracy crossover, both radiation-pressure comparisons, and distinct thermal and density-driven relativity scales. The curves are limiting-model comparisons, not sharp phase boundaries or a calibrated complete equation of state. Partial ionization, molecular physics and interactions modify the low-temperature regions indicated on the plot.

At sufficiently high temperature, electron-positron thermal pair abundance can become important. The pair abundance depends on density and chemical potential as well as temperature; $k_{B}T=m_e{c}^2$ is not a universal onset line. In the dilute ultrarelativistic limit, both pair species together add energy density $7a_r{T}^4/4$ to the photons' $a_r{T}^4$, and have pressure one third of their energy density. While pairs are being created, thermal energy is spent on rest mass, which can reduce the stellar adiabatic exponent below $4/3$ and contribute to pair-instability supernova physics.

At high mass density and low temperature, Interactions governed by Coulomb's law invalidate the noninteracting-ion approximation. The ionic Coulomb coupling parameter $\Gamma =Z^2{e}^2/(4\pi {ϵ}_0{a}_i{k}_{B}T)$, where $a_i=(3/(4\pi n_i){)}^{1/3}$, grows as ${\rho }^{1/3}/T$. Corrections become significant when $\Gamma$ is of order one, and a sufficiently strongly coupled plasma can crystallize. At still greater mass density, electron capture alters ${\mu }_e$ and nuclear matter replaces the ideal electron-ion model; neutron star interiors require strong-interaction and relativistic equations of state. These further regimes lie beyond the simple pressure curves plotted here. A useful stellar EOS is thermodynamically consistent across the crossovers, rather than just the maximum of unrelated pressure laws.

Take the optical depth to increase inward and let $\mu >0$ denote an outward ray. For thermal absorption and emission in local thermodynamic equilibrium, the radiative transfer source function is the Planck function, so the radiative transfer equation isThis assumes negligible scattering, or a source function genuinely thermalized to $B_{\lambda }$; LTE by itself does not turn an arbitrary scattering source into a Planck function. Multiply by $e^{-\tau /\mu }$, integrate between the two boundaries, and solve for the outward specific intensity:The bottom boundary value is the quantity denoted $I_{\lambda }(0)$ in the supplied notation; it is not an optical-depth-zero boundary. The first term is attenuated incident radiation, and the second is emission from the intervening layers. This is the formal solution of the radiative transfer equation.

For an isothermal atmosphere at temperature $T$, the Planck function is constant and the integral can be evaluated:For a bounded bottom intensity and ${\tau }_1\to \infty$, $I_{\lambda }=B_{\lambda }(T)$ in every outgoing direction, and $F_{\lambda }=\pi B_{\lambda }(T)$. Thus the emergent spectrum is a blackbody spectrum. The blackbody limit of an isothermal atmosphere does not require a temperature gradient: it follows from complete thermalization in an optically thick medium.

In local thermodynamic equilibrium, thermal intensity samples the Planck function near an optical depth of order unity. The Eddington-Barbier relation makes this explicit: $I_{\lambda }(0,\mu )\simeq B_{\lambda }[T({\tau }_{\lambda }=\mu )]$. A molecular band has greater opacity than its adjacent continuum and therefore samples a higher layer. A band in emission relative to the continuum implies that this higher layer is hotter: the line-forming region has an atmospheric thermal inversion under the assumed LTE, thermal interpretation.

The continuum is thermal radiation from an optically thick, deeper photosphere, with comparatively smooth opacity. In an H/He hot Jupiter, collision-induced absorption by H2-H2 and H2-He collisions supplies an important continuum; weak overlapping molecular lines and opaque exoplanet clouds can contribute too. It is not a separate blackbody emitter floating above the gas. A strongly isothermal layer would erase LTE molecular contrast rather than generate emission peaks.

In the emitting inversion, $dT/dz>0$, whereas the dry adiabatic lapse rate has $dT/dz=-g/C_p<0$. Thereforewhich lies on the stable side of the Schwarzschild criterion. An upward-displaced parcel cools and becomes denser than the ambient hot upper gas. The region can thus carry and redistribute thermal energy by radiative transfer, not by unstable thermal convection. Winds may transport energy horizontally; stability rules out the specified buoyant vertical convection, not every possible motion.

Let an isothermal non-scattering atmospheric layer of temperature $T_a$ lie above an optically thick continuum source with brightness temperature $T_b$. Assume local thermodynamic equilibrium, so the layer emits with Planck function $B_{\lambda }(T_a)$. Its vertical optical depth is ${\tau }_{\lambda }$; a ray with direction cosine $\mu$ has slant depth $t_{\lambda }={\tau }_{\lambda }/\mu$.

The formal solution of the radiative transfer equation givesThe two terms are attenuated background and layer emission. Compare a line and nearby continuum at the same wavelength to adequate approximation, with $t_{\mathrm{line}}>t_{\mathrm{cont}}$. The one-layer spectral contrast isThe second bracket is positive, and $B_{\lambda }(T)$ increases with temperature. Therefore a cooler upper layer produces absorption, a hotter upper layer produces emission, and an isothermal source-plus-layer produces no line contrast:This is why higher opacity sampling a cooler altitude creates absorption in an outward-cooling atmosphere, whereas an atmospheric thermal inversion can create emission. If both line and continuum are very optically thick in the same layer, their contrast tends to vanish even with a different deeper temperature. Scattering, non-LTE excitation, and nonuniform layers require a more general radiative transfer treatment.

This is a semi-grey irradiated atmosphere in radiative equilibrium. The quantities have these meanings:

The model assumes a static plane-parallel atmosphere, separate grey stellar and thermal bands with constant opacities, local thermodynamic equilibrium for thermal emission, negligible scattering, and transport by radiative transfer rather than convection. The Eddington closure approximation sets the thermal angular-moment ratio to $1/3$, and the upper boundary supplies the usual $2/3$ term. Treating the incoming radiation with representative direction cosine $1/\sqrt{3}$ gives its attenuation $e^{-\sqrt{3}\gamma \tau }$. Intrinsic flux enters from below, while the imposed external stellar flux is absorbed from above.

With constant gravity and hydrostatic equilibrium, $\tau \simeq {\kappa }_{\mathrm{IR}}P/g$, allowing conversion to an atmospheric pressure-temperature profile. This last relation additionally assumes constant thermal opacity. The grey treatment describes the energy balance approximately; it does not resolve individual molecular absorption lines.

Measure optical depth inward, so $d\tau /dr=-\kappa \rho$ and $\tau =0$ at the exterior boundary. The printed integral is shorthand with inconsistent integration labels; this differential definition fixes its meaning. In a plane-parallel atmosphere in radiative equilibrium, $F=\sigma T_{\mathrm{eff}}^4$ is constant and $d{P}_{\mathrm{rad}}/d\tau =F/c$.

For the Eddington surface boundary condition, approximate the outgoing frequency-integrated intensity by a constant $I_0$ over the outward hemisphere, with no incoming radiation. The angular moments at the surface areIntegrating the moment equation then gives $P_{\mathrm{rad}}=F(\tau +2/3)/c$. The Eddington closure approximation, with local thermodynamic equilibrium and radiative equilibrium, identifies the radiation energy density with $a_{\mathrm{rad}}{T}^4$ and sets $P_{\mathrm{rad}}=a_{\mathrm{rad}}{T}^4/3$. Using $a_{\mathrm{rad}}c=4\sigma$ givesThe photosphere is represented by $\tau =2/3$, where $T=T_{\mathrm{eff}}$; the exterior boundary instead has $T(0)=2^{-1/4}{T}_{\mathrm{eff}}$. The closure and angular boundary condition are approximations, rather than exact isotropy of radiation at the surface.

Assume local thermodynamic equilibrium, negligible scattering of thermal radiation, and a plane-parallel atmosphere. With inward optical depth ${\tau }_{\lambda }$ and outward ray cosine $\mu$, the radiative transfer equation isFor a deep atmosphere, its formal solution of the radiative transfer equation isThe weighting samples ${\tau }_{\lambda }$ of order unity. The Eddington-Barbier relation makes this precise when the source function is nearly linear: $I_{\lambda }(0,\mu )\simeq B_{\lambda }[T({\tau }_{\lambda }=\mu )]$.

A molecular band with greater opacity reaches unit optical depth higher than the nearby continuum. If temperature decreases upward, that band samples cooler gas and appears in absorption. If an atmospheric thermal inversion makes the upper gas hotter, the band appears in emission. If both depths have the same temperature, an opaque isothermal atmosphere hasand no opacity-dependent features. This last conclusion assumes all wavelengths are opaque or the lower boundary emits the same Planck function; an optically thin isothermal layer over a different background need not be featureless.

For the plane-parallel radiative transfer equationangular integration gives the zeroth radiation-field moment equationThe net radiative heating per unit volume is thereforeRadiative equilibrium requires it to vanish, equivalently that the frequency-integrated radiative flux be independent of depth:In local thermodynamic equilibrium with coherent isotropic scattering,where ${\omega }_{\nu }$ is the single-scattering albedo. Since ${\alpha }_{\nu }(1-{\omega }_{\nu })={\alpha }_{\nu ,\mathrm{abs}}$, the condition becomesConservative scattering redistributes directions but contributes no net material heating.

Along a ray, the radiative transfer equation has the formal solution of the radiative transfer equationfor each isothermal region in local thermodynamic equilibrium, neglecting scattering. For a self-luminous atmosphere with no incident intensity from below at the relevant photosphere, and in the optically thick limit, $I_{\nu ,i}=B_{\nu }(T_i)$.

The observed radiative flux is the projected-disc integralThe inner region occupies projected fraction $f={\sin }^2{\theta }_T$, henceFor finite optical depths, each Planck law function in this formula is replaced by its corresponding emergent intensity above.

Apply the formal solution of the radiative transfer equation successively to the two isothermal layers at normal incidence. In local thermodynamic equilibrium,For optically thin layers this becomesThis assumes no scattering and negligible reflection between layers.

Let ${\tau }_{\nu }$ increase downward and let $\mu >0$ be the outward direction cosine. Assume a plane-parallel, static, nonscattering atmosphere in local thermodynamic equilibrium, so the source function is $B_{\nu }[T(P)]$. Hydrostatic balance givesThe formal solution of the radiative transfer equation between $P_1>P_2$, corresponding to ${\tau }_1>{\tau }_2$, isFor an isothermal atmosphere, $B_{\nu }(T)$ is constant andIf the lower boundary is itself thermalized, $I_{\nu ,1}=B_{\nu }(T)$ and the upward intensity remains the blackbody value.

Let ${\tau }_1$ and ${\tau }_2$ be the top-down optical depths at $P_1$ and $P_2$, and let $\mu$ be the outward direction cosine. The formal solution of the radiative transfer equation givesFor a semi-infinite lower layer in local thermodynamic equilibrium, $I_{\nu ,0}=B_{\nu }(T_3)$.

DefineThe emergent planetary surface flux isHence the band-centre planet-star ratio isIf $T_1=T_2=T_3=T_p$, the weights telescope to $1/2$, so $F_{p,\nu }=\pi B_{\nu }(T_p)$ and the expression reduces to the blackbody thermal eclipse depth, about $72\mathrm{ppm}$ for $T_p=600\mathrm{K}$.

Without scattering and in local thermodynamic equilibrium, the radiative transfer equation has the emergent solutionThe Eddington-Barbier relation gives the useful approximationA molecular band has larger opacity than its neighboring continuum and therefore reaches optical depth unity at lower pressure.

If temperature decreases outward, the band samples cooler gas and appears in absorption. If the atmosphere is isothermal, both levels have the same source function and the feature disappears. If an atmospheric thermal inversion makes the upper layer hotter, the band appears in emission. For a weak separation of formation pressures,which explicitly shows that feature sign and amplitude measure the vertical temperature gradient.