Polarization of singly scattered sunlight ($\mathcal{P}=\frac{{\sin }^2\chi }{1+{\cos }^2\chi }$)

In ideal single Rayleigh scattering of unpolarized incident light by an isotropic small scatterer, observed intensities perpendicular and parallel to the scattering plane are proportional to 1 and ${\cos }^2\chi$. Their normalized difference is $\mathcal{P}={\sin }^2\chi /(1+{\cos }^2\chi )$. At a right-angle scattering geometry the output is fully linear and perpendicular to that plane. Multiple scattering, large particles and anisotropic scatterers change this ideal sky-polarization law.