Estimating some integral parameters of phase function of large particles to analytically compute light propagation through a medium

Analytical formulas are derived for integrals of phase function of large particles, namely for light fluxes scattered singly within an arbitrary angular range and mean scattering angle squared. The relations of the first kind are obtained via direct integration, by the scattering angle, of the Fresnel's reflectivities weighted with some angular function. The Fraunhofer's diffraction and geometrical optics parts are taken into account. As a result, the light flux is expressed as a sum of elementary functions. The formulas can be obviously converted to the known relations for the single-scattering albedo and mean cosine of the phase function for a particular case of the integration over full range from) to 180 degrees. The mean scattering angle squared is used, for example, by the small- angle diffusion approximation to compute light propagation. The corresponding formula is derived by comparing the solutions to the radiative transfer equation with the said approximation and with the small-angle one. The mean scattering angle squared is particularly shown to be inversely proportional to the effective size parameter squared of particles. The proportionality coefficient is found.