= Solution
Let $z$ increase outward and define the inward <optical depth> by $d\tau/dz=-\bar\kappa\rho$. With constant outward internal <radiative flux> $F=\sigma T_{\rm int}^4$, <radiative diffusion> gives
$$
\frac{dT}{d\tau}=\frac{3F}{16\sigma T^3},\qquad \frac{dT^4}{d\tau}=\frac34T_{\rm int}^4,\qquad T^4=\frac34T_{\rm int}^4(\tau+q).
$$
The 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>, $q=2/3$, so
$$
\boxed{T(\tau)=T_{\rm int}\left[\frac34\left(\tau+\frac23\right)\right]^{1/4}.}
$$
The <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, $T\to2^{-1/4}T_{\rm int}\simeq0.84T_{\rm int}$; if the density and optical-depth gradient vanish there, $dT/dz\to0$. 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>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-59-temperature.png]
{title=Intrinsic grey-atmosphere cooling profile compared with illustrative irradiated hot-Jupiter profiles}
{height=440}
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.
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