= Solution
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 gives
$$
r\simeq\left(\frac{R_p}{R_\star}\right)^2\frac{B_\lambda(T_1)}{B_\lambda(T_\star)},\qquad
T_1=\frac{hc/(\lambda k_B)}{\ln\left[1+\frac{(R_p/R_\star)^2}{r}\left(e^{hc/(\lambda k_BT_\star)}-1\right)\right]}.
$$
Using the <Planck law> at $\lambda=20\,\mu\mathrm m$ gives \b[$T_1\simeq3.1\times10^3\,\mathrm K$], or about $3000\,\mathrm K$ at the precision justified by the radius assumption. The <Rayleigh-Jeans law> instead gives $T_1\simeq0.5T_\star\simeq2900\,\mathrm K$, 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>,
$$
\tau_\lambda(P)=\int_0^P\frac{\kappa_\lambda(P',T)}{g}\,dP'.
$$
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 $g=20\,\mathrm{m\,s^{-2}}$ and an effective continuum <mass absorption coefficient> $\kappa_{20}=2\times10^{-4}\,\mathrm{m^2\,kg^{-1}}$. Then $P_{\tau\sim1}\sim g/\kappa_{20}=10^5\,\mathrm{Pa}$, or \b[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 \b[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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-315-window-temperatures.png]
{title=Illustrative pressure-temperature profile with different infrared brightness-temperature contribution depths}
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, \b[$T_3\gtrsim T_1>T_2$]. 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.
Back to article page
