= Solution
Let $F_{\rm abs}=\sigma_{\rm SB}T_{\rm eq}^4$ 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 $\alpha\sigma_{\rm SB}T_a^4$ both upward and downward.
Atmospheric radiative balance gives
$$
\alpha\sigma_{\rm SB}T_s^4=2\alpha\sigma_{\rm SB}T_a^4,
\qquad T_a^4=T_s^4/2\quad(\alpha>0).
$$
Surface radiative balance gives
$$
\sigma_{\rm SB}T_s^4=\sigma_{\rm SB}T_{\rm eq}^4+\alpha\sigma_{\rm SB}T_a^4.
$$
Eliminating $T_a$ yields
$$
\boxed{T_s=T_{\rm eq}\left(1-\frac\alpha2\right)^{-1/4}.}
$$
The transparent limit has $T_s=T_{\rm eq}$, and a fully infrared-opaque layer has $T_s=2^{1/4}T_{\rm eq}$. 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.
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