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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 a circular orbit of radius $a$, a spherical planet, bolometric Bond albedo $A_B$, negligible internal heat, unit thermal emissivity and complete redistribution of absorbed heat over the sphere. The stellar luminosity is $L_s=4\pi R_s^2\sigma T_s^4$, and the planet absorbs the incident radiative flux through its projected cross-section:Thermal reradiation is $P_{\mathrm{emit}}=4\pi R_p^2\sigma T_{\mathrm{eq}}^4$. Equating these gives the planetary equilibrium temperatureThe planet's radius cancels. A non-black thermal emissivity $ϵ$ divides the absorbed flux by $ϵ$ in the fourth-power balance. For uniform dayside-only reradiation, the emitting area is $2\pi R_p^2$ and $T_{\mathrm{day}}=2^{1/4}{T}_{\mathrm{eq}}$. With no local redistribution, the substellar point has $T_{\mathrm{sub}}=\sqrt{2}{T}_{\mathrm{eq}}$, while other points depend on incidence angle. These are different temperature conventions, not contradictory formulas.

The planetary equilibrium temperature is not the surface greenhouse temperature or the internal effective temperature of a planet. If intrinsic cooling matters and both powers escape through the same emitting area, the total effective temperature obeys $T_{\mathrm{eff}}^4=T_{\mathrm{int}}^4+T_{\mathrm{eq}}^4$.

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.

For a plane-parallel atmosphere with optical depth increasing downward, the radiative transfer equation isIntegrating over solid angle gives the first radiation-field momentIn radiative equilibrium, matter has no net local radiative heating, so the opacity-weighted frequency integral of $J_{\nu }-S_{\nu }$ vanishes. After converting each optical-depth derivative to physical depth and integrating over frequency,Thus the bolometric internal flux is constant with depth, as stated by constant flux in a plane-parallel radiative-equilibrium atmosphere.

In local thermal equilibrium $S_{\nu }=B_{\nu }(T)$. Define the planet's internal effective temperature of a planet by the constant outward flux. The standard Planck law integral givesThe corresponding intrinsic luminosity is $L_{\mathrm{int}}=4\pi R_p^2\sigma T_{\mathrm{eff}}^4$.

The internal effective temperature of a planet $T_{\mathrm{int}}$ parametrizes intrinsic cooling, while the irradiation temperature $T_{\mathrm{irr}}$ parametrizes incident stellar flux before the redistribution factor $f$. The quantityis the visible-to-thermal mean-opacity ratio, and $\tau$ is downward thermal optical depth. Under the Eddington two-stream boundary condition,while the common semi-grey choice is $c=\sqrt{3}$.

For a young giant at $40\mathrm{au}$, stellar heating is weak and a still-large $T_{\mathrm{int}}$ dominates. Its pressure-temperature profile rises steadily inward, approximately as $T^4\propto b+\tau$, and joins a deep convective adiabat.

For a hot Jupiter close to a Sun-like star, $T_{\mathrm{irr}}\gg T_{\mathrm{int}}$. It has a broad, nearly isothermal irradiated radiative layer, followed by a deep rise where intrinsic flux and increasing opacity matter. If $\gamma >1$, absorption of starlight above the thermal photosphere can create an atmospheric thermal inversion.

For a temperate sub-Neptune around an M dwarf, irradiation and internal cooling can be more comparable. Its profile generally has a moderate radiative layer above a convective interior; near-infrared stellar radiation, molecular opacity, clouds, and hazes determine whether the upper profile is weakly inverted or decreases outward.