TRANSFER PROBLEMS IN SPHERICAL GEOMETRY AND MODIFIED SPHERICAL HARMONIC METHOD

Abstract

The radiation field in an atmosphere scattering according to well known physical laws was studied by Lord Rayleigh as early as 1871. But the credit of formulating the relation between the intensity and the physical properties of the stellar atmosphere in the form of the equation of transfer goes to A. Schuster [3] and K. Schwarzchild [3]. Within the course of the past sixty years this equation had been extensively used to solve various problems connected with the transfer of radiation in the atmosphere of the planets, sun, stars, nebulae and galaxies. In the past twenty years, interest in transfer equation received a fresh stimulus in the hands of the neutron physicists. They found that the mathematical equations governing neutron diffusion show formal resemblance to those in the theory of radiative transfer. Mathematically speaking, the problem is one of solving the integro-differential equations of transfer, of varying degrees of complexity, subject is given boundary conditions. By standard methods one can reduce these integro-differential equations to integral equations. Only for a restricted class of problems (particularly in plane geometry) was it possible to obtain the exact solution by (i) Using discrete ordinate method and making n -> oo [7] (ii) Introducing Laplace transforms [7] (iii) Using the probabilistic method of Sobolev and Uenno [9]

So it as found necessary to devise approximate methods of solution of sufficient power and flexibility to solve large class of problems which defy exact solution. Most problems with spherical symmetry belong to this class, the problem of radiation through a homogeneous sphere being the only exception. Among the various approximate methods used to solve the transfer problems, the spherical harmonic method ranks as one of the earliest. Wick and Chandrasekhar developed the method of discrete ordinates and used it for solving various complex problems of radiative transfer and neutron transport. On examination, these two methods were found to be equivalent. In the single-interval spherical harmonic method, the specific intensity was expanded into a series of Legendre polynomials. The major difficulty that arose in using this method for a plane-parallel atmosphere or a finite spherical (extended) atmosphere was the inability to use the exact boundary condition at the free surface. For neutron transport problems Mark and Marshak used equivalent boundary conditions at the free surface to circumvent this difficulty. But a unique scheme for the choice of such boundary conditions was not found to suit all physical situations. Kourganoff on the other hand indicated that the double-interval representation of intensity, as suggested by Yvon, could prove a useful tool to attack problems of this type. Mertens demonstrated, with considerable success, Yvon’s method for certain problems of plane-parallel atmosphere. But this representation was discontinuous at one particular direction (? = o) for all depths. This feature is contrary to the physical logic of the problem. Moreover, this defect restricts the extension of the method to problems with spherical symmetry. In an attempt to overcome this limitation the author, in his M.Sc thesis, proposed a modified double-interval spherical harmonic method which retained the major advantages of Yvon’s method but made it more flexible and adaptable to a wider class of problems in different geometries. The method was applied to solve transfer problems in plane-parallel atmospheres scattering isotropically and anisotropically (with Rayleigh’s phase function). In the present thesis it is shown that the transfer and transport problems in spherical geometry can be brought within the scope of competence of this method without much difficulty. The material in the thesis is divided into two parts. In Part I, the first chapter is devoted to the general notions and definitions leading to equations of transfer and neutron transport necessary for the subsequent chapters. In Chapter II (Part I) we have tried to give a critical appreciation of the existing works on single and double interval spherical harmonic methods used in plane and spherical geometry. We have scanned the method of Yvon, specially about the feasibility of extending the method to spherical geometry and have stated the advantage of the modified double-interval method in this respect. In Part II of the thesis we have applied the modified double-interval spherical armonic method to solve some problems with spherical symmetry. Chapter III is devoted to the solution of the problem of neutron transport with a finite spherical core, which is the spherical analogue of the Milne problem for the half-plane. In chapter IV we have considered the classical problem of diffusion of radiation due to a point source through a homogenous sphere of finite radius. In chapter V we have solved the equation of transfer for a finite, spherically symmetric, conservative scattering stellar atmosphere. In the concluding chapter (chapter VI) we have shown that the transfer equation for conservative scattering planetary nebular shell can also be solved by the same method.