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An absorption line is a deficit of observed spectral specific intensity relative to the surrounding continuum. In a photosphere with temperature falling outward, increased line opacity makes photons emerge from cooler layers than the continuum. For an absorption-dominated LTE source, the Planck function then supplies a smaller line specific intensity. A foreground layer with source below the incident continuum gives another direct transfer description.

An atmospheric pressure-temperature profile specifies temperature as a function of pressure through an exoplanet atmosphere. Pressure normally increases inward under hydrostatic equilibrium, so it is a useful vertical coordinate even when the absolute altitude is uncertain. In a radiative equilibrium region, heating and cooling by radiation determine the profile; where that profile would be unstable, convection can carry energy and bring the gradient close to an adiabatic temperature gradient.

The profile explains why a molecular band can appear in absorption or as emission lines. Greater opacity makes the emerging light sample a higher layer. If that layer is colder, the band is dimmer; an atmospheric thermal inversion can reverse this behaviour. An atmospheric retrieval therefore needs both abundances and the profile, rather than identifying composition from band depth alone.

An exoplanet transmission spectrum measures a wavelength-dependent transit depth. In a simple opaque-disk geometry with uniform stellar brightness, an effective planetary radius $R_p$ gives $D=(R_p/R_{\ast }{)}^2$, where $R_{\ast }$ is the stellar radius. If stronger atmospheric absorption increases that radius by $\Delta R=N_{H}H$, define $H$ as the atmospheric scale height and $N_H$ as the number of scale heights separating the two spectral levels. The additional occulted annulus givesThe last approximation requires $N_{H}H\ll R_p$. For instance, $R_p/R_{\ast }=0.1$ and $N_{H}H/R_p=0.05$ give exact $\Delta D=0.001025$ and leading estimate $0.001$; these are stipulated geometric ratios.

Since $H=k_{B}T/(\mu m_{H}g)$ in the thin isothermal atmosphere model, larger temperature, smaller mean molecular weight or weaker gravity can increase the signal. The factor $N_H$ is set by the atmospheric opacity, composition and cloud structure; it is not a universal constant. Limb darkening, stellar heterogeneity and instrumental effects also affect the observed transit depth.^[ref]

Consider a hypothetical star whose emitted specific intensity is exactly that of a uniform $5000\mathrm{K}$ blackbody. Assume transparent propagation with negligible Doppler shift and interstellar extinction, so geometric dilution multiplies every spectral interval by the same factor. The stipulated temperature givesUsing the converted frequency peak as though it maximized $B_{\lambda }$ would return $b/(c/{\nu }_{\max })\simeq 2841.26\mathrm{K}$, a substantial error caused solely by mixing spectral conventions.

Now stipulate a photodetector with constant quantum efficiency $\eta$ over the relevant wavelengths, collecting equal narrow wavelength bins. For fixed accepted projected area times solid angle $G$, with dilution included in $G$, its expected photon count rate density isThis follows by dividing the received power by the photon energy $hc/\lambda$. It peaks where $x=4(1-e^{-x})$, at $x=3.9206903949\dots$, giving $0.73394057\mu \mathrm{m}$ for the same model star. Constant quantum efficiency is not constant energy spectral responsivity: the latter includes the wavelength-dependent photon energy conversion. For example, $G=10^{-12}{\mathrm{m}}^2\mathrm{sr}$, $\eta =0.5$ and a $1\mathrm{nm}$ bin centered at $0.600\mu \mathrm{m}$ give approximately $1.93\times 10^{10}$ detected photons per second in the narrow-bin approximation. These are calculated model values, not observations of a named star.

A more general linear detector channel has outputwhere $S_{i,0}$ is the background reading and $R_i$ includes spectral selection, optical transmission and detector spectral responsivity. A raw channel maximum need not locate either Planck function maximum. Recovering a temperature requires an appropriate calibrated response and an emission model. For a real stellar atmosphere, opacity and radiative transfer can reshape the spectrum; interstellar extinction can reshape it again. A single peak does not establish a unique physical temperature without those assumptions.

A circumstellar habitable zone is an orbital region in which a specified planetary and atmospheric model can maintain surface liquid water. A simple radiative scaling gives distance $a\propto \sqrt{L_{\star }/S_{\mathrm{eff}}}$, where $L_{\star }$ is stellar luminosity and $S_{\mathrm{eff}}$ the incident irradiance at a chosen climate boundary. The required irradiance depends on the stellar spectrum and atmospheric opacity.

Model inner limits involve water loss or a runaway greenhouse; a maximum greenhouse outer habitable-zone limit is a common outer criterion. Published one-dimensional estimates explicitly depend on water and carbon dioxide absorption and on their treatment of clouds.^[ref] The zone is therefore a conditional climate calculation, rather than a guarantee that each planet within it is habitable.

For gas-pressure-supported stellar homology with specific nuclear heating $ϵ={ϵ}_0\rho T^{13}$ and opacity $\kappa ={\kappa }_0\rho T^{-3}$, the dimensionless radiative coefficient is proportional to $L{\mu }^{-7}{M}^{-5}$. The nuclear coefficient is proportional to ${\mu }^{13}{M}^{15}/(L{R}^{16})$. Holding both and the dimensionless profiles fixed gives the displayed scalings. The Stefan–Boltzmann law then gives $T_e\propto {\mu }^{25/16}{M}^{15/16}$ and $L\propto {\mu }^{-4/3}{T}_e^{16/3}$. The coefficients, pressure regime and composition assumptions must be held as specified; these relations are not universal laws for massive stars.

A collision or transient molecular pair can possess a dipole even when the isolated molecules do not. The resulting opacity can be important in a dense molecular hydrogen atmosphere. Pair absorption per unit length scales approximately as the product of the two partner number densities, so its mass absorption coefficient is generally density-dependent. It supplies continuum absorption between ordinary spectral lines.

For homologous, chemically uniform, gas-pressure-supported radiative stars with specific energy generation $ϵ\propto X\rho T^{13}$ and composition-dependent constant opacity, stellar homology gives $L_{\mathrm{nuc}}\propto X{\mu }^{13}{M}^{15}{R}^{-16}$ and $L_{\mathrm{rad}}\propto {\mu }^4{M}^3/\kappa$. Equating them gives the displayed scalings. Hydrogen depletion changes both mean molecular weight and electron-scattering opacity; holding composition fixed gives $R\propto M^{3/4}$ and $L\propto M^3$. The power-law burning approximation applies only while hydrogen fuel remains.

For constant positive $\lambda =p_r/p_g$ and molecular gas constant $\mathcal{R}=k/({\mu }_m{m}_p)$, the gas–radiation equation of state gives $\rho =4\sigma T^3/(3c\lambda \mathcal{R})$. Vertical hydrostatic equilibrium in harmonic central gravity therefore gives $T=T_0(1-z^2/H^2)$ with $T_0={\Omega }^2{H}^2/[8(1+\lambda )\mathcal{R}]$. The mass density and pressure powers are three and four. Integrating both faces yields $\Sigma =(32/35){\rho }_{0}H$ and $\int pdz=\Sigma {\Omega }^2{H}^2/9$. With constant opacity, radiative diffusion produces a linear flux and height-independent dynamic viscosity $\mu =4c\lambda /[9\kappa (1+\lambda )]$. This is an interior formal profile; its low-density surface requires a photospheric treatment beyond diffusion.

For radiative, gas-pressure-supported homologous stars with constant opacity, write $\rho =Mb/(4\pi R^3)$, $P=G{M}^{2}p/(4\pi R^4)$ and $T=\mu GMp/(\mathcal{R}Rb)$. The radiative diffusion in a star equation contains the dimensionless coefficient $D=3{\kappa }_0{\mathcal{R}}^{4}L/(64{\pi }^{2}ac{\mu }^4{G}^4{M}^3)$. A common fixed profile fixes $D$, so $L\propto M^3$ and the luminosity of a fixed-mass member is constant. Gravitational energy generation in a homologously contracting ideal-gas star gives $\dot{R}=-2EL{R}^2/(3G{M}^2)$ for a positive profile constant $E$. Thus $R^{-1}=R_0^{-1}+2ELt/(3G{M}^2)$. For CNO cycle heating $ϵ\propto \rho T^{16}$, the stellar nuclear fusion luminosity scales as $M^{18}{R}^{-19}$. Equating this with $M^3$ gives $R_{\mathrm{MS}}\propto M^{15/19}$. Neglecting $R_0^{-1}$, the arrival time is $t_{\mathrm{MS}}\propto M^{-34/19}$, hence a coeval cluster's newly arriving mass scales as $t^{-19/34}$. All comparisons hold stellar composition, opacity normalization and dimensionless profiles fixed.

A negligible-mass radiative envelope surrounding a core of mass $M_c$, carrying constant luminosity and having constant opacity, obeys $dP/dT=16\pi acG{M}_c{T}^3/(3{\kappa }_{0}L)$. Neglecting the outer pressure integration constant gives $\Lambda =4\pi acG{M}_c/(3{\kappa }_{0}L)$. The gas pressure fraction $\beta$ is constant because $P_{\mathrm{rad}}=a{T}^4/3$, and $\rho =\mu \beta \Lambda T^3/\mathcal{R}$. Hydrostatic balance gives $T=T_s+\mu \beta G{M}_c(1/r-1/R_s)/(4\mathcal{R})$; dropping this outer-boundary constant gives the displayed deep-envelope approximation. A positive gas-pressure fraction requires $L<4\pi cG{M}_c/{\kappa }_0$.

Assume a thin disk has height-independent opacity ${\kappa }_{\mathrm{op}}$ and radiation-to-gas pressure ratio $\beta$, with vertical gravity $-{\Omega }_z^{2}z$. Hydrostatic equilibrium and radiative diffusion implyBalancing its derivative against viscous heating $\rho \nu q^2{\Omega }^2$ fixes the dynamic viscosityIt is independent of height under these assumptions. For a spherically symmetric potential, ${\Omega }_z=\Omega$. Integrating $\rho \nu$ through the full surface-to-surface thickness $2H$ gives $\bar{\nu }\Sigma =2H\rho \nu$, which determines the thickness from the radial accretion rate solution.

Combining $F=c{\Omega }^{2}z/\kappa$ with viscous heating $dF/dz=(9/4)\mu {\Omega }^2$ gives a height-independent dynamic viscosity $\mu =4c/(9\kappa )$ under constant opacity. This exact limiting relation differs from an approximate vertically averaged alpha disk closure.

For a fixed-mass, fully convective monatomic gas protostar, write $P=K{T}^{5/2}$ and $K=K_0{\mu }^{-5/2}{M}^{-1/2}{R}^{-3/2}$. Matching the photosphere to $\kappa P=2g/3$ with $\kappa ={\kappa }_0\rho T^4$ gives $T_e^8=2G\mathcal{R}{\mu }^4{M}^{2}R/(3{\kappa }_0{K}_0^2)$. The Stefan–Boltzmann law gives $L=\pi ac[2G\mathcal{R}/(3{\kappa }_0){]}^{1/2}{K}_0^{-1}{\mu }^{2}M{R}^{5/2}$. The stellar virial theorem gives total energy $-3G{M}^2/(7R)$, so $L=-3G{M}^2\dot{R}/(7R^2)$. Integrating proves $C=(49\pi ac/6)[2\mathcal{R}/(3{\kappa }_{0}G){]}^{1/2}{K}_0^{-1}$. This holds before stellar nuclear fusion, changing opacity or electron degeneracy pressure invalidates the assumed contraction law.

A deep irradiated layer can remain stable against convection when its thermal opacity and outward radiative flux are constant. Specify a plane-parallel atmosphere in hydrostatic equilibrium, with constant gravity $g>0$, positive constant Rosseland mean opacity ${\kappa }_R$ and upward radiative flux $F>0$. At depths where incident stellar light has already been absorbed, radiative diffusion givesConsequentlywhere $T_0>0$ is the extrapolated boundary temperature of this simplified diffusion solution. The radiative temperature gradient obeysFor a homogeneous diatomic ideal gas with fixed rotational degrees of freedom, $\gamma =7/5$ gives adiabatic temperature gradient ${\nabla }_{\mathrm{ad}}=2/7>1/4$. The Schwarzschild criterion therefore predicts stability throughout this model. Variable opacity, changing thermodynamics or varying radiative flux can change that conclusion. The diffusion approximation also needs large optical depth; it does not establish stability of the entire optically thin irradiated atmosphere.

Gravitational binding is necessary but not sufficient for gas capture. A hydrostatic envelope contracts by losing energy on a Kelvin-Helmholtz scale $t_{\mathrm{KH}}\sim |E|/L$. Solid accretion heats it, while radiative transport depends on opacity and convection carries energy when the radiative gradient is unstable. The disk supply and outer tidal/thermal boundaries also matter.

A model-dependent threshold beyond which a cooling gaseous envelope cannot remain on a slowly growing low-mass branch around a solid core. Opacity, solid accretion heating, ambient disk conditions and envelope composition affect the threshold. It is not a universal constant assigning one exact mass to every gas-giant progenitor.

Day-night heat redistribution transports absorbed stellar energy from an illuminated hemisphere to a dark one through atmospheric motion or an ocean. A simple comparison uses the atmospheric advection time, $L/U$, and the radiative relaxation time in a planetary atmosphere. Fast transport relative to cooling allows gas to retain heat while crossing the planet, warming the nightside and reducing the day-night contrast.

An exoplanet thermal phase curve measures the resulting variation in brightness as different hemispheres rotate into view. HD 189733 b provides an observed example. The inference remains wavelength-dependent because different opacities sample different pressures. A small brightness contrast can therefore constrain redistribution at the sampled levels without proving that every level has the same temperature.

Delayed cooling preserves a young giant planet's large radius by reducing the escape of internal energy. Enhanced atmospheric opacity, an irradiation-maintained outer radiative blanket, and composition gradients that inhibit convection can slow Kelvin-Helmholtz contraction. These mechanisms retain existing heat rather than supplying a new deep heat source.

A brightness temperature equates a measured intensity with a Planck function; it does not determine its contribution pressure. In hydrostatic equilibrium, ${\tau }_{\lambda }(P)={\int }_0^P{\kappa }_{\lambda }d{P}^{\prime }/g$. Estimating $P\sim g/{\kappa }_{\lambda }$ therefore needs gravity and opacity, as well as a model of the contribution function. Different pressures can produce the same band brightness.