# 6.8: What is the total irradiance of any object?

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If the Planck distribution function spectral irradiance is integrated over all wavelengths, then the total irradiance emitted into a hemisphere is given by the Stefan–Boltzmann Law:

$F_{s}=\sigma T^{4}$

where σ is called the Stefan-Boltzmann constant (5.67 x 10–8 W m–2 K–4). Fs has SI units of W m–2, where the m2 refers to the surface area of the object that is radiating.

The Stefan–Boltzmann law (total) irradiance applies to an object that radiates according to the Planck distribution function spectral irradiance. If we look at the figure below, we see that the solar spectrum at the top of the atmosphere is similar to the Planck distribution function but does not follow it perfectly. However, the Planck distribution function with the same total irradiance as the sun has a temperature of 5777 K, as in the second figure. Solar spectrum and atmospheric absorbing gases from 240 nm to 2.5 µm wavelengths. Credit: Nick84 [ CC BY-SA 3.0], via Wikimedia Commons The Planck distribution function spectral irradiance, Pe (equation 6.4), emitted into a hemisphere for the sun, Tsun = 5777 K. Credit: W. Brune

Exercise

Clouds radiate. Assume two spherical clouds, one with a radius of 100 m and a temperature of 275 K and a second with a radius of 100 m and a temperature of 230 K. Assuming that they both radiate according to the Planck distribution function, calculate the emission for each cloud in W m–2 and in W. Which cloud is radiating more total energy and by how much?