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An optically thick medium has optical depth $\tau \gg 1$ along the relevant path. Photons undergo many interactions before escaping. Radiative diffusion applies under suitable near-isotropy conditions, but a large scattering depth alone does not guarantee thermalization when true absorption is weak.

For ${\Omega }_m=1$ the preceding result reduces to $\kappa ={\Omega }_b{C}_H[(1+z_{\mathrm{re}}{)}^{3/2}-1]$. Using $C_H=0.032$ and ${\Omega }_b=0.05$ givesAt fixed $H_0$ and ${\Omega }_b$, a lower matter density lowers $H(z)$ relative to the flat dust value at positive redshift. The same Electron density then persists for a longer scattering time, increasing the optical depth. Hence the reionization redshift giving unit depth is lower in an open universe.

In the high-redshift matter regime the scaling is $1+z_{\mathrm{re}}\propto {\Omega }_m^{1/3}$ at fixed optical depth; for ${\Omega }_m=0.3$ this suggests a redshift near fifty rather than seventy. This comparison holds the baryon density fixed, not the baryon fraction ${\Omega }_b/{\Omega }_m$, and is not an estimate of the actual observed reionization epoch.

Efficient convection keeps the interior at nearly uniform specific entropy. For the fully ionized monatomic gas, the heat capacity ratio is $5/3$, so the adiabatic process gives $P=K_{\rho }{\rho }^{5/3}$, with $K_{\rho }$ spatially constant. The ideal gas relation $P=\mathcal{R}\rho T/\mu$ then impliesThe constant is spatial, not temporal: loss of specific entropy during contraction changes it. This is a stellar polytrope with polytropic index $n=3/2$. We neglect radiation pressure and take the thin photosphere as an atmospheric matching layer rather than a significant fraction of the stellar mass.

For a fixed dimensionless profile of the Lane-Emden equation, the stellar homology relations have the formwhere $C_{\rho },C_P$ are positive dimensionless structure constants. More explicitly, write $\rho (r)={\rho }_{c}f(x)$ and $m(r)=Mh(x)$ with $x=r/R$ and fixed dimensionless profiles. Mass conservation gives $M=4\pi {\rho }_c{R}^3{\int }_0^{1}f(x)x^{2}dx$. Integrating the hydrostatic pressure support equation, with negligible surface pressure on the interior scale, gives $P_c=(GM{\rho }_c/R){\int }_0^{1}h(x)f(x)x^{-2}dx$. Both integrals are fixed positive numbers, proving the two central scalings. Substituting these relations into $K=P_c/T_c^{5/2}$ givesThus the same polytropic index fixes $K_0$ throughout this homologous sequence, while the dimensional stellar polytrope constant changes.

At the photosphere, whose optical depth is $2/3$, identify the temperature with the effective temperature $T_e$. The ideal gas relation and the interior adiabat give ${\rho }_{\mathrm{ph}}=\mu K{T}_e^{3/2}/\mathcal{R}$. With the prescribed opacity and stellar surface boundary condition,Eliminating $K$ therefore provesUsing the Stefan–Boltzmann law, with $\sigma =ac/4$, gives the luminosity

The gravitational energy of a stellar polytrope is $\Omega =-6G{M}^2/(7R)$ for $n=3/2$. The stellar virial theorem for a monatomic gas gives $2U+\Omega =0$, and hence the total energy is $E=U+\Omega =-3G{M}^2/(7R)$. With no appreciable stellar nuclear fusion or accretion, energy conservation givesCombining the two expressions for $L$ yieldsMultiplication by $-7R^{-9/2}/2$ and integration proves the contraction law with density-linear fourth-power stellar opacity:When the accumulated contraction term dominates the initial-radius term, $R\propto {\mu }^{-4/7}{M}^{2/7}(t-t_0{)}^{-2/7}$. Substitution into the central ideal gas relation giveswhich has the stated $t^{2/7}$ behavior when the time origin is negligible. This late-time approximation still requires the assumed fully convective star, fixed composition and opacity law to remain applicable.

Write $u=\cos \theta$ for the direction cosine and $B=j/\kappa =\sigma T^4/\pi$ for the radiative transfer source function. Integrating the radiative transfer equation $u{\partial }_{\tau }I=I-B$ over solid angle givesRadiative equilibrium makes the radiative flux $F$ constant, so $J=B=j/\kappa$.

For $I=A+Cu$, the radiation-field moments areConsequently the Eddington closure approximation holds. Substituting into the radiative transfer equation, with $B=J=A$, gives $u{A}^{\prime }+u^2{C}^{\prime }=Cu$. Equality for every $u$ requires $A^{\prime }=C$ and $C^{\prime }=0$, with $C=3F/(4\pi )$. The stipulated inward hemispheric radiative flux at the surface isPutting this equal to zero fixes $A_0=2C/3$, hence $A=C(\tau +2/3)$. Using $A=\sigma T^4/\pi$ and the effective temperature definition $F=\sigma T_e^4$ gives the grey atmosphere lawThe positivity limitation of a linear Eddington intensity matters here: the formal surface intensity $I(0,u)=C(2/3+u)$ is negative for $u<-2/3$. Thus this angular ansatz gives the requested Eddington surface boundary condition for approximate moments; vanishing integrated inward flux is not an exact, nonnegative, pointwise no-incoming radiation field.

Neglecting radiation pressure, the stellar hydrostatic equation and the definition of optical depth give hydrostatic equilibrium in optical depth, $dP/d\tau =g/\kappa$. Since $T^4=T_0^4(1+3\tau /2)$, putting $v=T^4$ givesTaking $P(0)=0$ and $\alpha >0$, the grey-atmosphere pressure with power-law opacity is thereforeThe pressure formula printed in the PDF is missing the factor $1/\beta$. Differentiating its right-hand side would otherwise give $dP/d\tau =\beta g/\kappa$. For $\beta =0$ the correct limit is $P^{\alpha }=2\alpha g\log (T^4/T_0^4)/(3{\kappa }_0{T}_0^4)$.

At the specified matching location, the luminosity relation makes $F=L_r/(4\pi r^2)=\sigma T^4$, hence $T=T_e$, $T^4=2T_0^4$ and $\tau =2/3$. Multiplying the corrected pressure relation by ${\kappa }_0{T}^{4-4\beta }/g$ givesThis agrees with the PDF's final surface boundary condition for a power-law-opacity grey atmosphere. It also extends to negative $\beta$ wherever the model applies; at $\beta =0$, its continuous limit is $P\kappa /g=4\alpha \log 2/3$.

In the Maxwell-Boltzmann velocity distribution, each Cartesian component has mean-square velocity $k_B{T}_e/m_e$. Thus $⟨v^2⟩=3k_B{T}_e/m_e$. The original PDF calls this quantity an rms velocity, but the rms speed is its square root. Also, its $n_{\gamma }$ must mean the dimensionless mean occupation number $n$, since its equilibrium value is a Planck photon distribution; it is not the frequency-integrated photon number density.

Take $x=h\nu /(k_B{T}_{\gamma })$ at a fixed local radiation temperature. Because $x$ is proportional to frequency,The hot-electron Kompaneets equation therefore becomesFor traversal length $d\ell =cdt$, the Compton y parameter isIt measures electron pressure integrated along the line of sight, or the Thomson scattering optical depth weighted by $k_B{T}_e/(m_e{c}^2)$. In this hot-electron limit the mean fractional energy gain is approximately $4y$. The derivation assumes nonrelativistic electrons, Thomson-limit photon energies and $T_e\gg T_{\gamma }$. The original PDF uses $h\nu$, as required for ordinary frequency; the converted TeX incorrectly substitutes $\hslash \nu$.

Write $n=n_0+y\mathcal{L}{n}_0+O(y^2)$, with $n_0=(e^x-1{)}^{-1}$ and $\mathcal{L}=x^{-2}{\partial }_x(x^4{\partial }_x)$. Differentiating the Planck photon distribution givesDividing by $n_0$ proves the first-order thermal Sunyaev-Zeldovich effect:In the Rayleigh-Jeans law limit, $x{e}^x/(e^x-1)\to 1$ and $x\coth (x/2)\to 2$, so $\Delta n/n_0\to -2y$. In the Wien approximation, the prefactor approaches $x$ and the bracket approaches $x-4$, so $\Delta n/n_0\sim yx(x-4)\sim y{x}^2$. The last limit is an asymptotic description of the first-order coefficient: a perturbative prediction at large $x$ still requires $y{x}^2\ll 1$.

Hot electrons transfer energy to the cosmic microwave background, removing low-frequency photons and enhancing the high-frequency tail. The first-order distortion crosses zero where $x\coth (x/2)=4$, at $x\simeq 3.830$; for today's CMB this is near $217\mathrm{GHz}$. Photon conservation follows directly from the boundary termIn contrast, integration by parts gives $\int x^3\Delta ndx=4y\int x^3{n}_{0}dx$, so radiation gains a fraction $4y$ of its original energy. The thermal Sunyaev-Zeldovich effect is thus a spectral distortion, not simply a new blackbody temperature.

A galaxy cluster with bulk peculiar velocity along the line of sight additionally produces the kinetic Sunyaev-Zeldovich effect. Define $v_{\parallel }>0$ for recession and $\tau =\int n_e{\sigma }_{T}d\ell$. To first order in velocity and optical depth, $\Delta T/T=-\tau v_{\parallel }/c$: receding gas gives a decrement and approaching gas an increment. Its occupation number perturbation is $(\Delta T/T)x{e}^x/(e^x-1{)}^2$, the spectrum of a small blackbody temperature change. It remains nonzero at the thermal null and therefore complicates measurements near that frequency.

Typical weak Lyman-alpha forest absorbers have neutral hydrogen column densities roughly $10^{12}$–$10^{16}{\mathrm{cm}}^{-2}$, with stronger forest absorption extending towards $10^{17}{\mathrm{cm}}^{-2}$. Common Doppler broadening parameters are a few tens of ${\mathrm{km}\mathrm{s}}^{-1}$, consistent with gas at approximately $10^4$ K together with bulk velocity gradients. The thermal contribution isA line width is consequently a temperature upper bound if all other broadening is neglected, not a unique thermometer. The gas commonly lies in diffuse sheets and filaments at modest density contrast; only the strongest systems need substantial self-shielding or a galactic environment.

For optically thin gas illuminated by the cosmic ionizing background, photoionization equilibrium requireswhere the last approximation uses almost complete hydrogen ionization. For example, $n_H=10^{-5}{\mathrm{cm}}^{-3}$, $\alpha \simeq 4\times 10^{-13}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$ and ${\Gamma }_{\mathrm{HI}}=10^{-12}{\mathrm{s}}^{-1}$ give $x_{\mathrm{HI}}\simeq 4\times 10^{-6}$. A $100$ kpc path then has $N_{\mathrm{HI}}\sim 10^{13}{\mathrm{cm}}^{-2}$ despite its tiny neutral fraction. This illustrates the Lyman-alpha forest photoionization scaling: low density makes the two-body radiative recombination rate weak relative to the one-body photoionization rate, but a long path and a strong resonance still produce measurable absorption.

Independent information is important because a column density alone is not a volume density. Similar absorption in close pairs of quasars indicates substantial transverse coherence, while an ionizing optical depth $N_{\mathrm{HI}}{\sigma }_{912}\ll 1$ shows that these weak systems do not shield themselves from the background. The continuum-shielding threshold is $1/{\sigma }_{912}\simeq 1.6\times 10^{17}{\mathrm{cm}}^{-2}$. Together with the observed line widths and ionization balance, these facts favor extended, low-density, highly ionized gas over small cold neutral clouds. The physical interpretation and characteristic scales are consistent with Rauch's forest analysis and the measured column and line-width distributions.

Consider a small outward displacement $\delta r$ of a fluid element. It adjusts to the ambient pressure while retaining its specific entropy and stellar composition, so an adiabatic process gives $d\log {\rho }_{\mathrm{parcel}}=d\log P/\gamma$. Its excess mass density over the new surroundings is, to first order,Restoring buoyancy requires this bracket to be positive. With $\gamma =5/3$, the Schwarzschild criterion is thereforeEquality is neutral stability. For uniform mean molecular weight, the ideal gas law gives $d\log \rho =d\log P-d\log T$. Since hydrostatic equilibrium pressure decreases outward, the inequality is equivalent to $\nabla =d\log T/d\log P<{\nabla }_{\mathrm{ad}}=2/5$. Dividing the radiative diffusion in a star and hydrostatic equilibrium stellar structure equations givesThis is the stellar radiative temperature gradient test for stellar convective stability.

In the thin upper stellar atmosphere take $m\simeq M$, $L_r\simeq L$ and use gas pressure dominance. The prescribed opacity becomes $\kappa ={\kappa }_0\mu P{T}^{12}/\mathcal{R}$. Dividing the two structure equations then yieldsAt zero optical depth, the atmospheric boundary has $T_s^4=T_e^4/2$ and negligible gas pressure. Integrating from that boundary, with $T_s^{-8}=4T_e^{-8}$, givesTo locate the onset, put $y=T_e^8/T^8$. Logarithmic derivative gives $d\log P/d\log T=4y/(4-y)$, henceThis starts at zero at the outer boundary and increases inward. It first reaches $2/5$ atThe corresponding optical depth is ${\tau }_b=\frac{4}{3}(\sqrt{13/20}-\frac{1}{2})>0$, so the onset lies inside the stellar atmosphere. This is grey-atmosphere convection onset with thirteenth-power opacity.

Let the interior of the fully convective star have $P=K_T{T}^{5/2}$, with $K_T$ spatially constant. At its boundary $T_b/T_e$ is fixed, so the atmospheric pressure formula givesThe Stefan–Boltzmann law, $L=4\pi R^2\sigma T_e^4$ with $\sigma =ac/4$, consequently impliesThe ideal gas equation transforms the interior relation intoFor completeness, the mass-radius scaling follows by taking $\rho ={\rho }_c{\theta }^{3/2}$ and $r=\alpha \xi$ with ${\alpha }^2=5K_{\rho }{\rho }_c^{-1/3}/(8\pi G)$. Hydrostatic equilibrium and mass conservation reduce to the Lane-Emden equation $({\xi }^2{\theta }^{\prime }{)}^{\prime }=-{\xi }^2{\theta }^{3/2}$, with $\theta (0)=1$, ${\theta }^{\prime }(0)=0$. Its fixed dimensionless profile gives $R\propto \alpha$ and $M\propto {\alpha }^3{\rho }_c$. Eliminating ${\rho }_c$ gives $R\propto K_{\rho }{G}^{-1}{M}^{-1/3}$, or $K_{\rho }\propto G{M}^{1/3}R$. Thus $K_T^2\propto M^{-1}{R}^{-3}$ at fixed stellar composition. Equating this with the atmospheric expression gives $L^{17/4}\propto M^2{R}^{19/2}$, and henceThis Hayashi relation with thirteenth-power opacity concerns the interior of a fully convective star beneath a thin stellar atmosphere. The coefficient $K_T$ may vary between stars; spatially constant specific entropy does not mean equal specific entropy across the whole sequence.

Let $\Omega =-{\int }_0^{M}Gmdm/r$ be the gravitational energy. Multiply the stellar hydrostatic equation by $4\pi r^3$ and integrate through the star. Integration by parts and zero surface pressure givewhile its hydrostatic right-hand side is $-{\int }_0^{R}4\pi Gmr\rho dr=\Omega$. Hence the stellar virial theorem is

There is a genuine reference-sheet error to resolve. A monatomic ideal gas with adiabatic exponent $5/3$ has specific internal energy of an ideal gas $u=P/[\rho (5/3-1)]=3P/(2\rho )$, not the printed $3P/\rho$ on PDF page 2. The latter would give total internal energy $-\Omega$ and zero total energy, inconsistent with radiating contraction. The following derivation uses the consistent monatomic value.

A fully convective star is approximately isentropic and has an adiabatic stellar polytrope with index $3/2$. The given polytropic gravitational-energy formula yields $\Omega =-6G{M}^2/(7R)$. The virial relation gives total internal energy $U=-\Omega /2$, so total energy is $U+\Omega =-3G{M}^2/(7R)$. At constant mass with no nuclear heating, Kelvin-Helmholtz contraction obeys $d(U+\Omega )/ds=-L$, whenceHere $s$ denotes physical time.

In the plane-parallel atmosphere, the inward optical depth satisfies $d\tau =-\kappa \rho dr$, so hydrostatic equilibrium in optical depth gives $dP/d\tau =g/\kappa$. Neglecting radiation pressure, $\rho =\mu P/(\mathcal{R}T)$ and the specified opacity becomes $\kappa =({\kappa }_0\mu /\mathcal{R})P{T}^{16}$. ThereforeThe grey atmosphere relation and the boundary $P(0)=0$ integrate toTo apply the Schwarzschild criterion, put $w=2+3\tau$. Logarithmic differentiation givesThis increases inward, starting from zero. Marginal convection has ${\nabla }_{\mathrm{rad}}=2/5$, so $w_b^3=136/5$. Since $T_b^{12}/T_e^{12}=w_b^3/64$, the grey-atmosphere convection onset with seventeenth-power opacity isThe purely radiative atmosphere is used only up to this boundary.

The convective relation $P=K{T}^{5/2}$ is spatially uniform in its entropy coefficient $K$ at each instant; $K$ can change as the star radiates and contracts. At the convection boundary $w_b$ is a fixed number, so $P_b\propto g^{1/2}{T}_e^{-8}$ at fixed composition and opacity normalization. ThusOn the other hand, the central scalings for the $n=3/2$ stellar polytrope give $P_c\propto G{M}^2/R^4$ and $T_c=\mu P_c/(\mathcal{R}{\rho }_c)\propto \mu GM/(\mathcal{R}R)$. Hence, at fixed composition,Equating the interior and atmosphere expressions yields $T_e^{21/2}\propto M{R}^{1/2}$. Applying the Stefan–Boltzmann law then proves the Hayashi contraction with seventeenth-power opacity relation

At fixed mass write $L=A{M}^{8/21}{R}^{\alpha }$, with constant $A$ and $\alpha =46/21$. The contraction equation is $\dot{R}=-C{R}^{\alpha +2}$, where $C=7A{M}^{8/21}/(3G{M}^2)$. Direct integration givesMultiplying through by $R^{\alpha +1}$ and substituting the current luminosity givesFor the stated very large initial radius, or after that initial-radius term becomes negligible,This is the luminosity-age relation within the contraction model, rather than an assertion that a physical star begins at literally infinite radius.

The quasar proximity effect is weaker Lyman-alpha forest absorption near a quasar's emission redshift. Extra photons from the quasar increase the photoionization rate and reduce the neutral fraction without necessarily reducing the total gas density.

For isotropic specific intensity $J_{\nu }$ and a steady, isotropically radiating quasar with spectral luminosity $L_{\nu }$, the background and local rates areHere $r$ is a proper separation small enough that attenuation and expansion across it can initially be neglected, and ${\sigma }_{\nu }$ is the hydrogen photoionization cross-section. At fixed density and temperature, photoionization equilibrium makes the neutral column density, and hence a given line's optical depth, scale asThe observed quasar luminosity supplies ${\Gamma }_Q(r)$; the distance at which the absorption changes significantly estimates where ${\Gamma }_Q\sim {\Gamma }_{\mathrm{bg}}$. Adopting a background spectral shape then converts the inferred rate to its specific intensity, often quoted near the hydrogen ionization energy threshold.

Important uncertainties include an enhanced gas density contrast around the host, the unobserved ionizing spectrum, anisotropic emission, source lifetime and variability, the systemic redshift and continuum placement, attenuation between source and absorber, and fluctuations between lines of sight. Higher gas density enhances recombination and can partially hide the effect, biasing a simple background estimate upward. Dall'Aglio, Wisotzki and Worseck's 2008 measurements illustrate the observed sightline scatter and the roles of variability and clustering.

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.

Let $z$ increase outward and define the inward optical depth by $d\tau /dz=-\bar{\kappa }\rho$. With constant outward internal radiative flux $F=\sigma T_{\mathrm{int}}^4$, radiative diffusion givesThe constant $q$ is a boundary condition; diffusion alone does not determine it. For an unirradiated grey atmosphere with the Eddington closure approximation and Eddington surface boundary condition, $q=2/3$, soThe internal effective temperature of a planet is defined by its intrinsic cooling flux, not by its incident stellar heating. Since $dT/d\tau >0$ and $d\tau /dz<0$, temperature decreases outward. Towards the thin upper layers, $T\to 2^{-1/4}{T}_{\mathrm{int}}\simeq 0.84T_{\mathrm{int}}$; if the density and optical-depth gradient vanish there, $dT/dz\to 0$. This is the upper nearly isothermal atmosphere. The diffusion approximation itself fails at small optical depth: the grey boundary closure supplies the approximate continuation. Small irradiation perturbs this intrinsic-flux solution.

Young, self-luminous gas giants at wide orbital separations can have intrinsic cooling dominate their photospheric budget. They are favorable for exoplanet direct imaging, especially in the infrared, where their own thermal radiation is easier to separate from the host light. A strongly irradiated hot Jupiter instead has a large stable outer radiative region set mainly by stellar heating; it can be nearly isothermal over a broad pressure range or develop an atmospheric thermal inversion if stellar light is absorbed sufficiently high. Inversion is not inevitable for every strongly irradiated planet. Deep convection begins below its radiative-convective boundary.

The intrinsic curve follows the grey formula. The irradiated curves illustrate possible shapes only; they are not solutions for a specified opacity model. Smaller optical depth corresponds to greater altitude.

Assume $R_p/R_{\star }=0.10$, $T_{\star }=5800\mathrm{K}$, uniform disk-averaged thermal emission, a nearly blackbody stellar continuum, negligible reflected light at $20\mu \mathrm{m}$, and a flat response over the narrow band. If the eclipse depth is normalized to the out-of-eclipse star-plus-planet flux, the planet/star flux ratio is $r=0.005/(1-0.005)=0.005025$; using $r=0.005$ changes the estimate by less than one percent. The exoplanet secondary eclipse then givesUsing the Planck law at $\lambda =20\mu \mathrm{m}$ gives $T_1\simeq 3.1\times 10^3\mathrm{K}$, or about $3000\mathrm{K}$ at the precision justified by the radius assumption. The Rayleigh-Jeans law instead gives $T_1\simeq 0.5T_{\star }\simeq 2900\mathrm{K}$, a useful rough check. This is a brightness temperature, not the bolometric planetary equilibrium temperature. With zero Bond albedo and global redistribution, the latter would be only about $1250\mathrm{K}$ at $0.05\mathrm{AU}$; a thermal window can sample hotter, deeper gas.

A continuum photon normally emerges from an optical depth of order unity. In hydrostatic equilibrium,The emission-pressure degeneracy of a brightness temperature means that pressure cannot be uniquely recovered from the measured brightness temperature without $g$ and an opacity model. For an explicit nominal estimate, take $g=20\mathrm{m}{\mathrm{s}}^{-2}$ and an effective continuum mass absorption coefficient ${\kappa }_{20}=2\times 10^{-4}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-1}$. Then $P_{\tau \sim 1}\sim g/{\kappa }_{20}=10^5\mathrm{Pa}$, or approximately one bar. In a hydrogen-rich atmosphere, collision-induced absorption by transient ${\mathrm{H}}_2$ pairs supplies continuum opacity even without ordinary molecular lines; its density dependence makes the constant-opacity calculation an illustrative estimate. With a normal deep atmospheric pressure-temperature profile, temperature increases at larger pressures, eventually following an interior adiabatic temperature gradient.

Two mechanisms for $T_1\ne T_2$ are wavelength-dependent sampling of a vertically varying temperature, and wavelength-dependent weighting of a horizontally nonuniform dayside. In the first, carbon monoxide-rich opacity near $4.5\mu \mathrm{m}$ can sample cooler gas around $0.1$ bar while the $20\mu \mathrm{m}$ continuum samples hotter gas around one bar. In the second, hot and cool areas contribute different proportions of the total flux in different bands because the Planck function depends nonlinearly on temperature. The vertical mechanism alone produces the illustrated pair of temperatures; a single isothermal, horizontally uniform blackbody would have the same brightness temperature in both bands.

If the J-band observation is thermal emission, its low-opacity window normally samples at least as deeply as the continuum at $20\mu \mathrm{m}$. On the illustrated non-inverted profile, $T_3\gtrsim T_1>T_2$. The ordering is conditional: molecular-line absence alone does not specify the relative continuum opacities, and an inversion or reflected-light contamination can change the inference. The exoplanet emission spectrum measures weighted contribution regions rather than three exact thermometers at known pressures.

Take $R_p$ to be the radius of an opaque planetary disc and $H$ the geometric thickness of the model atmosphere, not necessarily one atmospheric scale height. Assume a uniform stellar specific intensity, a fully projected non-grazing transit, negligible planetary emission in the measured band, and no scattering or refraction returning light to the beam. The cylindrical approximation assigns the same slant optical depth ${\tau }_{\lambda }$ to every ray through the annulus.

The opaque disc blocks area $\pi R_p^2$. The annulus has projected area $\pi [(R_p+H{)}^2-R_p^2]$, and the radiative transfer equation transmits fraction $e^{-{\tau }_{\lambda }}$ through it. Its blocked fraction is therefore $1-e^{-{\tau }_{\lambda }}$. Dividing the missing light by the unobscured stellar-disc light gives the exoplanet transmission spectrumFor $H\ll R_p$, the annulus model for transmission spectroscopy becomesThe optically thin excess is $2R_{p}H{\tau }_{\lambda }/R_s^2$; the optically thick limit is the area ratio $(R_p+H{)}^2/R_s^2$ before the thin-annulus approximation. For a realistic atmosphere, the slant optical depth varies with the ray's impact parameter $b$, giving insteadLimb darkening replaces the simple area weighting by the local stellar specific intensity. The constant-depth model is therefore an explicit geometric approximation, not the slant-depth law of a spherical hydrostatic atmosphere.

The two measurements sample different atmospheric regions and different radiative transfer geometries. An exoplanet emission spectrum probes the dayside; an exoplanet transmission spectrum probes the morning and evening limbs of the day-night terminator. Several alternatives to an exoplanet cloud deck can therefore suppress the transmission feature.

Temperature, mean molecular weight, limb chemistry, continuum opacity and the dayside temperature gradient are alternatives or degeneracies to test. In particular, the three-scale-height amplitude alone cannot establish different CO abundances between the hemispheres.

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.

At the stellar surface boundary condition, match the interior to a stellar atmosphere, rather than setting the temperature and pressure to zero at an arbitrary radius. At a photosphere of radius $R$, the enclosed mass is $M$, the luminosity is $L$, and the outward radiative flux defines the effective temperature through $F=L/(4\pi R^2)={\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4$. With no external illumination the incoming specific intensity satisfies $I_{\nu }(0,\mu )=0$ for $\mu <0$. A deeper boundary is matched to the nearly isotropic radiative diffusion field. A grey atmosphere commonly matches near $\tau =2/3$; under the Eddington surface boundary condition this gives $T=T_{\mathrm{eff}}$. For roughly constant gravity and opacity, the total pressure rises by $g\tau /\kappa$ relative to its outer boundary value; if radiation supplies appreciable support, the gas-pressure gradient uses the effective gravity $g-\kappa F/c$ instead.

Take $z$ increasing outward and $\mu$ the outward direction cosine. If $j_{\nu }$ is emission per unit path length and solid angle, the radiative transfer equation isDefine inward-increasing optical depth by $d{\tau }_{\nu }=-\rho {\kappa }_{\nu }dz$ and the source function by $S_{\nu }=j_{\nu }/(\rho {\kappa }_{\nu })$. Then $\mu {\partial }_{{\tau }_{\nu }}{I}_{\nu }=I_{\nu }-S_{\nu }$. This states the emission-coefficient convention; if $j_{\nu }$ is defined per unit mass instead, the emission term is $\rho j_{\nu }$.