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The stellar luminosity is $L_{\star }=4\pi R_{\star }^2{\sigma }_{\mathrm{SB}}{T}_{\star }^4$. At separation $a$, the planet intercepts a disk of area $\pi R_p^2$. For Bond albedo $A_B$ and uniform emission over its full surface, energy balance givesThis planetary equilibrium temperature neglects intrinsic heat. Dayside-only or nonuniform emission changes the redistribution factor.

For the single-layer greenhouse model, assume a black infrared surface, an atmosphere transparent to incoming stellar radiation, and infrared absorptivity/emissivity $\alpha$. Let $T_s$ be surface temperature and $T_a$ layer temperature. Kirchhoff's law makes the layer's emission in each direction $\alpha {\sigma }_{\mathrm{SB}}{T}_a^4$. Its balance isso $T_a^4=T_s^4/2$ for $\alpha >0$. Surface balance includes the downward layer emission:ThereforeThe no-atmosphere limit is $T_s=T_e$, and a completely IR-absorbing single layer gives $T_s=2^{1/4}{T}_e$. The outgoing flux $(1-\alpha )\sigma T_s^4+\alpha \sigma T_a^4$ remains $\sigma T_e^4$. The layer delays escape by absorption and re-emission; it does not permanently retain a fraction $\alpha$ of every emitted photon, which would incorrectly predict a divergent temperature at $\alpha =1$.

Use the single-layer greenhouse model with a blackbody surface and one isothermal atmospheric layer. The layer is transparent to incoming stellar radiation, absorbs a fraction $\alpha$ of the surface's thermal radiation, and has the same thermal emissivity $\alpha$ by Kirchhoff's law of thermal radiation. Here $0\le \alpha \le 1$, and $T_e$ is the planetary equilibrium temperature defined by the absorbed global stellar flux ${\sigma }_{\mathrm{SB}}{T}_e^4$. Neglect convection, latent heat transport and intrinsic heat.

The layer absorbs $\alpha {\sigma }_{\mathrm{SB}}{T}_s^4$ and emits $\alpha {\sigma }_{\mathrm{SB}}{T}_a^4$ upward and downward. For $\alpha >0$, its energy balance givesThe surface receives the absorbed stellar flux plus downward atmospheric emission, soEliminating $T_a$ yieldsThe outgoing top-of-atmosphere flux is $(1-\alpha ){\sigma }_{\mathrm{SB}}{T}_s^4+\alpha {\sigma }_{\mathrm{SB}}{T}_a^4={\sigma }_{\mathrm{SB}}{T}_e^4$, confirming overall energy conservation. At $\alpha =0$, $T_s=T_e$ and the decoupled layer's temperature is not determined; at $\alpha =1$, $T_s=2^{1/4}{T}_e$. This warming is the restriction of thermal escape by the absorbing and emitting layer.

Let $F_{\mathrm{abs}}={\sigma }_{\mathrm{SB}}{T}_{\mathrm{eq}}^4$ be the globally averaged absorbed stellar flux. The single-layer greenhouse model has an atmosphere transparent to incoming visible light and with infrared absorptivity and emissivity $\alpha$. Treat the surface as a blackbody and neglect internal heating. The layer emits $\alpha {\sigma }_{\mathrm{SB}}{T}_a^4$ both upward and downward.

Atmospheric radiative balance givesSurface radiative balance givesEliminating $T_a$ yieldsThe transparent limit has $T_s=T_{\mathrm{eq}}$, and a fully infrared-opaque layer has $T_s=2^{1/4}{T}_{\mathrm{eq}}$. The model omits convection, wavelength-dependent opacity, and atmospheric absorption of starlight, so it captures the simplest greenhouse energy balance rather than a full surface-temperature prediction.

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.