Constructive Theory of Scalar Characteristic Equations of the Theory of Radiation Transport: II. Algorithms for Finding Solutions and Their Analytic Representations

We present methods for finding discrete spectra and derive analytic expressions for the eigenfunctions of scalar characteristic equations of the theory of radiation transport. We obtain new two-term recursion formulas and analytic representations for solutions of infinite tridiagonal systems of linear algebraic equations. We obtain analytic forms of the resolvents of scalar characteristic equations for phase functions square integrable on the closed interval [−1, 1]. In addition, we derive a general analytic expression for the Green function of a two-dimensional (with respect to the angular variables) integro-differential equation of the radiation transport for the case in which the phase functions satisfy the Holder condition on the closed interval

Analytical representations for the truncated spectral characteristics of the four-point coherence function of a laser beam in a turbulent medium

New analytical representations for the truncated spectral characteristics of the four-point coherence function of a laser beam propagating in a turbulent medium are obtained. These representations are valid for any level of fl uctuation of the refractive index in air. They turn into exact analytical representations previously derived for two particular cases by using an integral-functional equation for truncated spectral characteristics of the four-point coherence function. A constructive procedure for obtaining approximate analytical expressions for the four-point coherence function of a laser beam propagating in a turbulent medium is proposed

The effective technique of solving of reduced characteristics equations of radiation transfer theory

An integral equation of radiation transfer theory is considered. The method to solution of this equation based on its reduction to an infinity system of linear algebraic equations is presented. The solvability conditions in L2-space are given and the explicit series solution is constructed

Exact solution of the standard transfer problem in a stellar atmosphere

We come back to the analytical solution of the standard transfer problem in a stellar atmosphere. It consists in solving the radiative transfer equation in a homogeneous and isothermal plane-parallel atmosphere, with light scattering taken as isotropic and monochromatic. The literature on the subject is reviewed and the existing solution in a finite slab is improved thanks to the introduction of non classical auxiliary functions. Eleven-figure tables of the solution are given for typical values of the input parameters currently met in stellar atmospheres.Comment: 23 pages, 1 figur

Analytical representations for the truncated spectral characteristics of the four-point coherence function of a laser beam in a turbulent medium

Получены новые аналитические представления для усеченных спектральных характеристик четырехточечной функции когерентности лазерного пучка излучения, распространяющегося в турбулентной среде. Эти представления верны для любого уровня флуктуаций показателя преломления воздуха. Для двух частных случаев они переходят в точные соотношения, выведенные ранее c использованием интегро-функционального уравнения для усеченной спектральной характеристики четырехточечной функции когерентности. Предложена конструктивная процедура получения приближенных аналитических выражений для самой четырехточечной функции когерентности лазерного пучка излучения, распространяющего в турбулентной среде. New analytical representations for the truncated spectral characteristics of the four-point coherence function of a laser beam propagating in a turbulent medium are obtained. These representations are valid for any level of fluctuations of the refractive index in air. For two particular cases they turn into exact analytical representations previously derived by the authors with using of an integro-functional equation for truncated spectral characteristic of the four-point coherence function. A constructive procedure for obtaining approximate analytical expressions of the four-point coherence function of a laser beam propagating in a turbulent medium is proposed

Integral equations for the H- X- and Y-functions

We come back to a non linear integral equation satisfied by the function H, which is distinct from the classical H-equation. Established for the first time by Busbridge (1955), it appeared occasionally in the literature since then. First of all, this equation is generalized over the whole complex plane using the method of residues. Then its counterpart in a finite slab is derived; it consists in two series of integral equations for the X- and Y-functions. These integral equations are finally applied to the solution of the albedo problem in a slab.Comment: 20 pages, JQSRT, accepted 9 July 200

Constructive Theory of Scalar Characteristic Equations of the Theory of Radiation Transport: II. Algorithms for Finding Solutions and Their Analytic Representations

We present methods for finding discrete spectra and derive analytic expressions for the eigenfunctions of scalar characteristic equations of the theory of radiation transport. We obtain new two-term recursion formulas and analytic representations for solutions of infinite tridiagonal systems of linear algebraic equations. We obtain analytic forms of the resolvents of scalar characteristic equations for phase functions square integrable on the closed interval [−1, 1]. In addition, we derive a general analytic expression for the Green function of a two-dimensional (with respect to the angular variables) integro-differential equation of the radiation transport for the case in which the phase functions satisfy the Holder condition on the closed interva