# How did Lord Rayleigh derive/determine the phase function for his scattering model?

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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People of the future: I have just confirmed by hand that the given phase function is correctly normalized (i.e., the coefficient should indeed be $\frac{3}{16\pi}$). Specifically, this causes it to integrate to one over the unit sphere, thus conserving probability. – imallett Nov 29 '14 at 5:43

If the incident radiation is unpolarized, it can be seen as the sum of two mutually incoherent terms: the first one due to a polarization perpendicular to the scattering plane and the second one contained in this plane. For each term, the scattered radiation preserves the polarization.

• The first term does not vary with $\theta$, since the induced dipole is also perpendicular to the scattering plane, and its emission is constant along the equator

• The amplitude of the second one goes as $\cos (\theta)$ (and the intensity as $\cos ^2(\theta)$), because the dipole does not emit along its direction. See the last slide of this for an illustration.

Notes: a) the formula also holds for circularly polarized light. b) for other polarization states, the two terms must be weighted according to their contribution to the incident field.

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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 '13 at 15:41
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 '13 at 16:20
The only thing that that derivation lacks is the asymptotic form of the electric field due to a dipole radiator, here, which they write as $\vec{E} = \frac{1}{c^2 r}\frac{\partial \vec{p}}{\partial t} \hat p \cdot \hat r$. Basically, the incident radiation induces an oscillating dipole moment which then re-radiates with the well-known formula.I would consider akhmeteli's answer sufficient to get the ball rolling. +1 from me – lionelbrits Nov 30 '13 at 21:40