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Use a single-layer greenhouse model whose atmosphere is transparent to incident stellar light and has infrared emissivity $ϵ$. Let $T_s$ and $T_a$ be the surface and atmospheric temperatures. Atmospheric balance givesAt the top of the atmosphere, the escaping flux is the directly transmitted surface radiation plus upward atmospheric emission:ThereforeFor a perfectly infrared-opaque one-layer atmosphere, $T_s=2^{1/4}{T}_{\mathrm{eq}}$; for $ϵ=0$, $T_s=T_{\mathrm{eq}}$.

In an infrared-opaque single-layer greenhouse model, the atmospheric layer obeys $2\sigma T_a^4=\sigma T_s^4$. The outgoing planetary flux is $\sigma T_a^4$, while the globally averaged absorbed stellar flux is $(1-A_B)L/(16\pi a^2)$. Radiative equilibrium therefore givesHence the orbit at which the prescribed surface temperature can be maintained isThis assumes uniform redistribution, constant Bond albedo, unit longwave emissivity, a transparent atmosphere to starlight, and no internal heat. Without the greenhouse layer, replace $8\pi$ by $16\pi$.

Assume uniform global temperatures, all visible light not scattered back to space is absorbed by the surface, and use Kirchhoff's law of thermal radiation so that the atmospheric infrared emissivity is $\alpha$. If $F_{\ast }=\sigma T_{\ast }^4(R_{\ast }/a{)}^2$, the globally averaged absorbed stellar flux isThe atmosphere absorbs $\alpha \sigma T_{\mathrm{surf}}^4$ from the surface and emits from both faces, soSurface balance isTherefore the single-layer greenhouse model givesVisible scattering cools the surface, whereas infrared absorption and downward re-emission warm it.