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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.

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.

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 constant radiative-diffusion coefficient and constant gravity give $T^4=T_0^4+4AP/g$. Thus ${\nabla }_{\mathrm{rad}}=AP/(g{T}^4)<1/4$ for $T_0>0$. A diatomic ideal gas has adiabatic temperature gradient $2/7$, so this idealized profile never becomes convective. A physical radiative-convective boundary requires additional opacity, flux, or thermodynamic variation.

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.

An exoplanet thermal phase curve measures infrared brightness throughout the orbit. Its amplitude constrains day-night heat redistribution, while its maximum's displacement from secondary eclipse constrains the offset of the hottest atmospheric region. Wavelength-dependent opacity makes different channels probe different pressures.

In an optically thick steady spherical stellar wind of constant mean molecular weight ${\mu }_{\mathrm{mol}}$, let $v$ be the outward speed and $a_g^2=k_{B}T/({\mu }_{\mathrm{mol}}{m}_u)$ the isothermal sound speed squared. Mass conservation gives ${\rho }^{\prime }/\rho =-2/r-v^{\prime }/v$, where a prime denotes differentiation with respect to radius $r$. Use $P_{\mathrm{gas}}=\rho a_g^2$ and radiative diffusion to write $P_{\mathrm{rad}}^{\prime }/\rho =-g\Gamma$, where $g=G{M}_r/r^2$ and $\Gamma =L_r/L_{\mathrm{Edd}}$ uses the local Eddington luminosity $L_{\mathrm{Edd}}=4\pi cG{M}_r/\kappa$ and opacity $\kappa$. Substituting into the momentum equation yields the displayed wind equation. Smooth sonic point crossing requires both its denominator and numerator to vanish. The radiation-pressure gradient already supplies radiative acceleration; adding that force separately would count it twice.

For an optically thick alpha disk dominated by gas pressure, vertical hydrostatic equilibrium gives $H\propto T^{1/2}/\Omega$ and $\rho \propto \Sigma \Omega T^{-1/2}$. The specified opacity therefore scales as $\kappa \propto {\Sigma }^{-3/2}{\Omega }^{-3/2}{T}^2$. Viscous heating scales as $Q^{+}\propto \alpha \Sigma T\Omega$, while radiative diffusion gives $Q^{-}\propto T^4/(\kappa \Sigma )\propto {\Sigma }^{1/2}{\Omega }^{3/2}{T}^2$. Their balance gives $T\propto \alpha {\Sigma }^{1/2}{\Omega }^{-1/2}$, hence $\nu \propto {\alpha }^2{\Sigma }^{1/2}{\Omega }^{-3/2}$. Fixed $\alpha$ and Keplerian rotation yield the displayed law.

The local thermal stability of an accretion disk follows because, at fixed surface density and radius, cooling has logarithmic temperature slope two while heating has slope one. The viscous stability of an accretion disk follows from ${\partial }_{\Sigma }(\nu \Sigma )>0$ on the thermal-equilibrium branch. Both conclusions apply within this pressure, opacity and transport closure.

For small positive radiation-to-gas ratio, constant opacity, fixed alpha and molecular constants, eliminate $\lambda$ between $\lambda \propto H\Omega \Sigma$ and $\lambda \Sigma \propto H^7{\Omega }^6$. This gives $H^6\propto {\Sigma }^2{\Omega }^{-5}$ and $\bar{\nu }\propto {\Sigma }^{2/3}{\Omega }^{-2/3}$. Keplerian rotation supplies the displayed radial power. Since $\bar{\nu }\Sigma \propto {\Sigma }^{5/3}$, its derivative with respect to surface density is positive and local viscous perturbations diffuse rather than amplify.

Let $\mathcal{R}=k/({\mu }_m{m}_H)$, take constant opacity $\kappa$, and prescribe the dynamic viscosity $\mu =\alpha p_g/\Omega =\alpha \mathcal{R}\rho T/\Omega$ in a Keplerian disk. The pressure supporting the disk may contain both gas pressure and radiation pressure. Combining viscous dissipation and radiative diffusion gives ${\partial }_m^2{T}^4+A\Omega T=0$ in the vertical mass coordinate of a disk, where $A=27\alpha \kappa \mathcal{R}/(16\sigma )$. Set $\zeta =2m/\Sigma$, $T=T_{\ast }t(\zeta )$ and $T_{\ast }^3=A\Omega {\Sigma }^2/4$. With a radiative-zero disk surface, $t$ satisfies $(t^4{)}^{\prime \prime }+t=0$, $t^{\prime }(0)=t(1)=0$. Writing $I_t={\int }_0^{1}td\zeta$, the density-weighted mean kinematic viscosity is $\bar{\nu }=\alpha \mathcal{R}{T}_{\ast }{I}_t/\Omega$, giving the displayed formula. This calculation does not assume gas-pressure domination: the closure depends on gas pressure while hydrostatic support can be mixed.