Two new analytic approximations of the Chandrasekhar's H function for isotropic scattering

The Chandrasekhar's H functions play a central role in theoretical descriptions of radiative transfer in planetary and stellar atmospheres. In the present work our aim is to obtain in a simple way new analytic approximations of the Chandrasekhar's function for isotropic scattering, which would be sufficiently simple but more accurate than the existing approximations. We apply the mean value theorem for definite integrals in the nonlinear integral equation for the Chandrasekhar's H function. In this way the integral equation is formally solved but the solution depends on a new unknown parameter. We determine this parameter approximately, from the condition that the obtained H function matches the zero-order moment of the H function, which is known exactly, as accurately as possible in the whole range of the single particle albedo. The result gives our first analytic approximation for the H function. Using it as a starting approximation in the corresponding integral equation, after only one iteration which may be performed analytically, we obtain our second analytic approximation. The maximum relative error of our first analytic approximation, which is very simple in structure, is below 2.5%. The accuracy of our second approximation is within 0.07%, so that it highly surpasses the accuracy of the other analytic approximations available in the literature.