Radiation equilibrium

The radiation balance is the state that a closed system of radiant reaches bodies when all bodies the same end temperature have adopted: every body emits as much radiation as it absorbs .

In the case of the participation of gray bodies in the radiation equilibrium, their heat radiation performance follows the extended Stefan-Boltzmann law :

${\displaystyle P=\epsilon \cdot \sigma \cdot A\cdot T^{4}}$,

being for black bodies . ${\displaystyle \epsilon =1}$

Example earth

The radiant power of the sun to earth is (approximately):

${\displaystyle P=E_{0}\cdot \pi \cdot r_{\mathrm {E} }^{2}=1{,}74\cdot 10^{17}\,\mathrm {W} }$

where the power represents the solar constant and the earth's radius . The area used here is the cross-section of the earth. For the sake of simplicity, it is also assumed that no radiation is reflected, i.e. the earth has an albedo of zero. ${\displaystyle P}$ ${\displaystyle E_{0}}$ ${\displaystyle r_{\mathrm {E} }}$

For a temperature equilibrium, the same power must be emitted via black body radiation. According to the Stefan-Boltzmann law, the following applies to the radiated power of an ideal black body :

${\displaystyle {\begin{aligned}T^{4}&={\frac {P}{\sigma \cdot A}}={\frac {1{,}74\cdot 10^{17}\,\mathrm {W} }{\sigma \cdot 4\cdot \pi \cdot r_{\mathrm {E} }^{2}}}\\T&=278{,}5\,\mathrm {K} \end{aligned}}}$

with the Stefan-Boltzmann constant and the surface of the earth as an area. ${\displaystyle \sigma }$

This corresponds to an average earth temperature of (only through black body radiation). Taking into account the albedo, the result is a radiation temperature with a greenhouse effect on the earth's surface. ${\displaystyle 5{,}4\mathrm {^{\circ }\mathrm {C} } }$ ${\displaystyle -19\mathrm {^{\circ }\mathrm {C} } }$ ${\displaystyle +15\mathrm {^{\circ }\mathrm {C} } }$

See also

• Prévost's theorem