Altermatt Lecture:   The Solar Spectrum


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:

DOS of photons in vacuum

Bose-Einstein distribution

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 dΩ from an area dA):

Planck's law with dE

Planck's law with dlambda

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

AM0 vs blackbody radiation

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


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