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I've been researching the question for quite some time, as I understand it the phase function is actually an approximation due to the particle-wave duality inherent in participating media such as the atmosphere or anything else.

As they are reemitted from particles, some EM waves mingle with their neighbours and amplify or kill off each other, therefore general approximate lobes are constructed from empirical data.

The Rayleigh scattering phase function is symmetrical as defined:

$$\Phi_R(\theta) = \frac{1}{4\pi}\frac{3}{4}(1 + \cos^2\theta)$$ However, I have no idea where this came from. I tried researching different phase functions, even those more modern from Henyey and Greenstein in 1941. but they just stated it is an approximation and gave the final form with no details.

Could someone show me, please?

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It looks like there is a derivation at (http://irina.eas.gatech.edu/ATOC5235_2003/Lec9.pdf ) (the coefficient differs, but that may be due to a somewhat different definition of the phase function).

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Very interesting, thank you! The coefficients are $3/4$, the $1/4\pi$ is the normalization constant ($4\pi$ - the number of steradians on a sphere). I will leave the question open for a little bit more, perhaps someone would like to weight in with more references. – ScatteredFrom Feb 2 at 15:41
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I've studied the reference, unfortunately it isn't a derivation, it just drops it in and then hacks it with the previous to express the same in terms of the magic phase function. Thank you for trying, though! – ScatteredFrom Feb 2 at 16:20

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