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3.4: Comparison with blackbody radiation – Planck's law*

This page is optional. It gives you a deeper understanding of the shape of the AM0 spectrum: it can be understood
in terms of a rather universal law, the Planck's law [7][8]. It quantifies the light intensity
radiated by a blackbody as a function of wavelength. A blackbody is a body that absorbs all light that falls on it.

You need to know some basics of quantum physics to understand this law. If you don't, you may skip the rest of this
page and look at the following
video.
If you are familiar with quantum physics, you may know that it is common
to calculate the amount of particles *N* within an energy range by multiplying their density of states *D* by
their occupation probability *f* in these states, i.e., *N*(*E*) = *D*(*E*)⋅*f*(*E*)⋅*dE*.
In case of photons in vacuum, this is:

For a derivation, follow this link.
To compare it with the AM0 spectrum, we need an energy flux *I*, which is
the number of particles *N* times their energy *E* times their velocity *c*. We are not interested in the
sun's total energy flux, but only into a solid angle Ω (hence, we divide by the area of the unit sphere,
4π). Also, *I* scales with the area *dA* of the radiating body as seen by the observer.

This results in **Planck's law** (the energy flux from a blackbody into a solid angle *d*Ω from an area *dA*):

For the conversion from *E* to *λ*, the equation from the previous page and the chain rule
(*dE* = *dE*/*dλ* ⋅ *dλ*) is used.

The figure below [1] indicates that Planck's law matches the solar spectrum pretty well, apart from emission lines
(like the H line) and absorption lines (for example Ca II). This means that **we can consider the sun as a black body radiating
at a temperature near 6000 K**.

For more details about the comparison of the black body with the AM0 spectrum, see this
video.

For calculations of Ω and *A*, see this
video.

**Figure:** The AM0 spectrum (black line) vs blackbody radiation at 4550 K, 5775 K and 6500 K.