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For $a\gg a_{AB}$, both stars illuminate the circumbinary planet at approximately distance $a$. Treat them as blackbodies, neglect internal heating, and assume Bond albedo $A_B$ and uniform global reradiation. The absorbed and emitted powers areThus the equilibrium temperature of a circumbinary planet isFor an example, take $R_A=R_{\odot }$, $R_B=0.3R_{\odot }$, $T_A=6000\mathrm{K}$, $T_B=3000\mathrm{K}$, $a=1\mathrm{au}$, $a_{AB}=0.05\mathrm{au}$, circular coplanar orbits, and $A_B=0$. ThenThe cooler star contributes only $0.3^2(3000/6000{)}^4\simeq 0.0056$ of the hotter star's luminosity in this example.

An opaque isothermal atmosphere emits $F_{\lambda }=\pi B_{\lambda }(T_p)$, with no absorption or emission bands. The Wien displacement law puts the maximum of this wavelength spectrum near ${\lambda }_{\max }\simeq 10\mu \mathrm{m}$. At observer distance $d$, the planetary spectral flux is $\pi B_{\lambda }(T_p)(R_p/d{)}^2$.