A discrete ordinates method of radiative transfer in one-dimensional   spherical geometry
  A new set of discrete ordinates is proposed for one-dimensional radiative transfer in spheres with central symmetry. The set is structured with un-normalized circular functions. This resulted in a conservative and closed set of discrete ordinates equations that need not starter intensity: N equations with N unknowns. The angular derivative of the transfer equation is represented by a set of angular parameters that are calculated directly and without recursion from the abscissas and weights of the quadrature and without the constraint of asymptotic conditions. An analytic solution is obtained in infinite homogeneous and heterogeneous cold media for monochromatic radiation. Graphical and tabulated data validate the proposed set of discrete ordinates at all radial positions ranging from 10^(-7) to 10^7 within round-off errors 
Charles H. Aboughantous