RADIATIVE TRANSFER PROBLEMS IN SCATTERING MEDIA WITH CURVATURE

Abstract

In the past, much of the theory of radiative transfer had been devoted to the study of transport processes in plan- parallel media. However, there exists a number of problems in stellar atmospheres, planetary atmospheres, meteorology and neutron transport where the curvature of the medium plays a significant role. Out of the search for solving such problems have emerged in recent years a number of new methodologies for tackling in particular the transfer problems in spherical and cylindrical geometries. Broadly speaking these methods can be classified in to three groups- ( a) those depending on the moments of intensity; ( b) schemes based on Ambartzumian’s physical technique and ( c) those connected with the solution of the relevant integral equations of transfer. The solution of transfer equation for homogenous, isotropically scattering atmospheres is rather in an advanced state, Those in inhomogeneous atmospheres generally load to the numerical solutions of in integro-differential equations under specialised type of boundary conditions. Attempts at obtainin analytical solutions of integral equations of transfer in the above case are still in the rudimentary stage. It is our intention in this thesis to explore the possibility of extending the scope of some of the existing methods and developing some new techniques for obtaining the analytic solutions of transfer problems in spherical and cylindrical geometries (if possible). Part II and Part III of the thesis record the results of such attempts. Part I of the thesis may be considered to be an introduction to such studies. It consists of two chapters. Chapter I contains the basic notion of radiative transfer and schematic derivations of the relevant equations of transfer in spherical and cylindrical geometries. Chapter II gives a critical survey of existing literature dealing with transfer problems in curved media. The results given in Part II and III of the thesis are new. In Part II, a probabilistic model of representing photon diffusion in terms of four probability functions as proposed by Uesugi [56] for plane-parallel atmosphere is extended to the case of curved geometries. Careful attention is paid to the problem of “geometric convergence" and "oblique incidence" , the main distinguishing criterion between plane-parallel and spherical and cylindrical media. In Part III of the thesis, it has boon shown that the Fredholm type of integral equation for transfer problems in inhomogeneous, isotropically scattering spherical ( Chapter V) and infinite to cylindrical shell media ( Chapter VI) can be solved by the method of Fredholm theorem for general L2-kernel. It is shown that the process of breaking up of a L2-kernel occurring in such integral equations into a sum of Pincherle- Goursat kernel and another L2-kernel proves equally effective in the case of spherical and infinite cylindrical shell media. In both case the attenuation coefficicnt is supposed to vary as some inverse power of radial distance.