= Zero-dimensional planetary energy balance
{title2=$C\dot T=S(1-\alpha)/4-\sigma T^4$}
Consider an idealized <planet> that is a <sphere> of <radius> $R$, illuminated by parallel sunlight of <irradiance> $S$, measured on a plane perpendicular to the rays. Its intercepting <projected area> is $\pi R^2$, whereas its total surface <area> is $4\pi R^2$. Therefore the incoming <power> averaged per unit total surface <area> is
$$
\frac{S\pi R^2}{4\pi R^2}=\frac S4,
\qquad P_{\rm abs}=(1-\alpha)S\pi R^2,
$$
where $\alpha$ is the broadband <Bond albedo>. The factor four is a <sphere> geometry result, not an atmospheric transmission factor. Local sunlight still varies over the illuminated hemisphere and vanishes on the dark side.
Now stipulate a single spatially uniform <temperature> $T$, no atmosphere, no internal heating, and <emissivity> one for outgoing <thermal radiation>. This is a thermal <blackbody> model using the <Stefan–Boltzmann law>; the prescribed solar-band reflection is treated separately, so the body is not a perfect absorber at all <wavelengths>. Let $C>0$ be the effective <heat capacity> per unit surface <area>, in $\mathrm{J\,m^{-2}\,K^{-1}}$. <Conservation of energy> gives the <ordinary differential equation>
$$
C\frac{dT}{dt}=F(T),\qquad
F(T)=\frac S4[1-\alpha(T)]-\sigma T^4.
$$
Here $F$ is net heating per unit surface <area>, in $\mathrm{W\,m^{-2}}$, and $\sigma$ is the <Stefan-Boltzmann constant>. Use $\sigma=5.670374419\times10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}$, rounded from the exact constant reported by NIST. https://physics.nist.gov/cgi-bin/cuu/Value?sigma%7Csearch_for=stefan+boltzmann[NIST, CODATA value: Stefan-Boltzmann constant]{ref} The uniform <temperature> is a model assumption: generally the surface average of $T^4$ differs from the fourth power of the surface-average <temperature>.
For fixed <Bond albedo> $\alpha$, <radiative equilibrium> requires $F(T_*)=0$, giving
$$
T_* = \left[\frac{S(1-\alpha)}{4\sigma}\right]^{1/4}.
$$
With the illustrative <solar constant> $S=1360\,\mathrm{W\,m^{-2}}$, the global incoming average is $340\,\mathrm{W\,m^{-2}}$:
|| Fixed <Bond albedo>
|| Reflected average ($\mathrm{W\,m^{-2}}$)
|| Absorbed average ($\mathrm{W\,m^{-2}}$)
|| Equilibrium <temperature> ($\mathrm K$)
| $0.30$
| $102$
| $238$
| $254.531$
| $0.60$
| $204$
| $136$
| $221.300$
Doubling <albedo> here lowers the equilibrium <temperature> by $33.231\,\mathrm K$, rather than halving it: the absorbed fraction changes from $0.7$ to $0.4$, and <thermal radiation> scales as $T^4$. These stipulated <Bond albedos> are whole-planet examples, not measurements of bare water and ice. The model predicts neither <Earth>'s modern surface <temperature> nor its response to actual <climate change>; it omits the <greenhouse effect>, spatial <heat transfer> and seasons.
\Image[/albedo-planet-energy-balance.png]
{title=From a sunlit disk to a whole <sphere>: geometry and fixed-<albedo> energy budgets}
{description=The disk intercepts sunlight over area pi R squared; a uniform emitting sphere has four times that area. Both fixed-albedo examples use the same 1360 watts per square metre illumination and thermal emissivity one.}
Each fixed-<albedo> <equilibrium> is a <stable equilibrium>: for $T_*>0$, the <derivative> is $F'(T_*)=-4\sigma T_*^3<0$. A small <temperature> increase produces net cooling, and a small decrease produces net heating. This is the simplest expression of the stabilizing <Planck feedback>. Allowing <albedo> to fall with <temperature> adds a competing term, as in <ice-ocean albedo feedback>.
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