The solution of a time-dependent equation of transfer in nonconservative finite media

A time-dependent nonconservative radiative transfer equation in a finite media has been solved. Firstly, the time-dependent function is converted into a time-independent function by the use of a Laplace transform and the transformed equation is then solved by a method using linear operator theory.     

Exact solution of a key equation in a finite planetary atmosphere by the method of laplace transform and linear singular operators

Considering the ground reflection according to Lambert's law, we establish a fundamental equation in finite planetary atmospheres. An exact form of the solution of this equation is obtained for the emergent quantities from the bounding faces in terms ofX-Y equations by the method of Laplace transform, in combination with the theory of linear singular operators.     

Exact solution of a key equation in a finite stellar atmosphere by the method of Laplace transform and linear singular operators

The key equation which commonly appears for radiative transfer in a finite stellar atmosphere having ground reflection according to Lambert's law is considered in this paper. The exact solution of this equation is obtained for surface quantities in terms of theX-Y equations of Chandrasekhar by the method of Laplace transform and linear singular operators. This exact method is widely applicable for obtaining the solution for surface quantities in a finite atmosphere.     

Exact solution of the problem of diffuse reflection for a combination of Rayleigh and isotropic scattering

We consider the basic vector equation of transfer for radiation in a semi-infinite atmosphere for diffuse reflection which scatters radiation in accordance with the phase matrix obtained from a combination of Rayleight and isotropic scattering. This equation will give an integral equation for emergent intensity while subjected to the Laplace transform. The integral equation will give rise to the emergent intensity matrix on application of the Wiener-Hopf technique. This is an exact method.     

Exact solution of a simple time-dependent integro-differential equation by the method of Laplace transform and the theory of linear singular operators

The simplest form of the equation of transfer for a time dependent radiation field in finite atmosphere is considered. This equation of transfer is an integro-differential equation, the solution of this equation is based on the theory of separation of variables, the Laplace transform and the theory of linear singular operators. The emergent intensities from the bounding faces of the finite atmosphere are determined in terms ofX-Y equations of Chandrasekhar.     

A new method for exact solution of transfer equations in finite media

In this paper we develop a new exact method combined with finite Laplace transform and theory of linear singular operators to obtain a solution of transport equation in finite plane-parallel steady-state scattering atmosphere both for angular distribution of radiation from the bounding faces of the atmosphere and for intensity of radiation at any depth of the atmosphere. The emergent intensity of radiation from the bounding faces are determined from simultaneous linear integral equations of the emergent intensity of radiation in terms ofX andY equations of Chandrasekhar. The intensity of radiation at any optical depth for a positive and negative direction parameter is derived by inversion of the Laplace transform in terms of intergrals of the emergent intensity of radiation. A new expression of theX andY equation is also derived for easy numerical computation. This is a new and exact method applicable to all problems in finite plane parallel steady scattering atmosphere.     

Closed solution of the Hill differential equation for short arcs and a local mass anomaly in the central body

The Hill differential equation describes the relative motion of a satellite w.r.t. a circular reference orbit. The deviations in the orbit are caused by a residual acceleration, which is small compared to the effect of the central gravitational field. In this paper, the acceleration of a local mass anomaly in the central body is considered, which rotates w.r.t. the inertial frame with a constant angular velocity. The mass anomaly is modeled by a superposition of radial base functions. The potential and the gradient of each base function are represented in the orbit by the Keplerian elements, the rotation rate of the central body and the parameters of the base function, i.e. the position of its center, the shape and a scaling factor. The inhomogeneous solution of the Hill differential equation for short arcs is found by means of the Laplace transform. A few lower orders of the solution require an additional Laplace transform, to consider the so-called resonance cases. The final deviations are described in a closed and differentiable formula of the Keplerian elements and the parameters of the base function.     

Exact solution of the transport equation for imperfect rayleigh scattering in a semi-infinite atmosphere

By performing the one-sided Laplace transform on the matrix integro-differential equation for a semi-infinite plane parallel imperfect Rayleigh scattering atmosphere we derive an integral equation for the emergent intensity matrix. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity matrix in terms of singularH-matrix and an unknown matrix. The unknown matrix has been determined considering the boundary condition at infinity to be identical with the asymptotic solution for the intensity matrix.     

Exact solution of the problem of diffuse reflection and transmission by the method of Laplace transform and linear singular operators

The basic integro-differential equation is subjected to a one-sided finite Laplace transform to obtain linear integral equations of angular distribution of bounding faces. These linear integral equations have been transformed into linear singular integral equations which have been solved exactly to get the emergent distributions from the bounding faces by the theory of linear singular operators. Some solutions of linear singular integral equations have also been derived for future use in radiative transfer problems.     

Exact solution of a problem in a stellar atmosphere using the Laplace transform and the Wiener-Hopf technique

The general equation for radiative transfer of line scattering intensity — including the effects of scattering, absorption and thermal emission — in the Milne-Eddington model is considered here. The scattering function is assumed to be quadratically anisotropic in the cosine of the scattering angle, and Planck's intensity function is assumed for thermal emission. The exact solutions for emergent intensity from the bounding face and the intensity at any optical depth are obtained by the method of the Laplace transform in combination with the Wiener-Hopf technique.     

An exact solution of the equation of transfer for coherent scattering in an exponential atmosphere

An exact solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, of the form $$B_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method of the Laplace transform and Wiener-Hopf technique.     

Rayleigh scattering: Volterra equation for the matrix source function

Multiple Rayleigh scattering is examined in a semi-infinite atmosphere with uniformly distributed primary sources of partially polarized radiation. The resulting linear polarization is described by a 2×2 matrix transfer equation. A matrix generalization of Rybicki's two point Q-integral is obtained for this case. It is shown that the Volterra equation for the matrix source function for this problem is a particular case of our Q integral. Applying the Laplace transform to it yields the matrix form of the Ambartsumyan-Chandrasekhar H-equation. The Volterra equation for Sobolev's matrix resolvent function is another simple consequence of this equation. Translated from Astrofizika, Vol. 52, No. 2, pp. 301–310 (May 2009).     

Time-dependentX- andY-functions by integral equation method

The time-dependent equation of radiative transfer for isotropic scattering has been solved by integral equation technique in terms ofX- andY-functions appropriate for the problem. It is seen thatX- andY-functions are reducible to the corresponding function for steady-state problems by simply changing the Laplace transform parameters-i.e., byS0.     

Exact solution of the equation of radiative transfer for semi-infinite plane-parallel isotropic scattering atmosphere with scattering albedo ω0 ≤ 1 by a method based on Laplace transform and Wiener-Hopf technique

By performing the one-sided Laplace transform on the scalar integro-differential equation for a semi-infinite plane-parallel isotropic scattering atmosphere with a scattering albedo ₀ 1, an integral equation for the emergent intensity has been derived. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity. The intensity at any optical depth for a positive scattering angle is also derived by inversion. The intensity at any optical depth for a negative scattering angle is also derived in terms of Cauchy's principal value using Plemelj's formulae.     

Exact solution of a basic equation in finite atmosphere by the method of Laplace transform and linear singular operators

A finite atmosphere having distribution of intensity at both surfaces with definite form of scattering function and source function is considered here. The basic integro-differential equation for the intensity distribution at any optical depth is subjected to the finite Laplace transform to have linear integral equations for the surface quantities under interest. These linear integral equations are transformed into linear singular integral equations by use of the Plemelj's formulae. The solution of these linear singular integral equations are obtained in terms of theX-Y equations of Chandrasekhar by use of the theory of linear singular operators which is applied in Das (1978a).     

Exact solution of the problem of diffuse reflection and transmission in Rayleigh scattering by use of the Laplace transform and the theory of linear singular operators

We have considered six scalar equations which are obtained from the vector transport equation for radiative transfer to the problem of diffuse reflection and transmission in finite plane-parallel Rayleigh scattering atmosphere. By use of the Laplace transform and the theory of linear singular operators these equations have been solved exactly to get the angular distribution of the intensity diffusely reflected from the surface and diffusely transmitted below the surface.     

Exact solution of the transport equation in finite media with a plane and uniform point source and flux normally incident at the faces from outside

In this paper we apply the Wiener-Hopf technique combined with the method of the Laplace transform, to derive an exact solution of the transport equation for neutron diffusion in an isotropically scattering plane-parallel medium of finite thickness in which are situated a plane source at the middle and a uniformly distributed point source, there being flux of beams normally incident from outside on the two extreme parallel surfaces of the medium.     

Energy albedo for multiregion slabs

An asymptotic solution for the equation of radiative transfer in an inhomogeneous medium was obtained on the basis of the corresponding solutions for homogeneous sub-layers in the slowing down region. Function relations between the reflection and transmission coefficients for the whole slab and those of the sublayers are given. The invariant embedding concepts are used to get the reflection and transmission coefficients for the sub-layers. We assumed different models for the slowing-down kernels. Laplace transform was used to transform the Boltzmann equation to one velocity approximation with re-scaled mean-free path and single-scattering albedo. Numerical results are given for energy albedo as a function of the mass number of the host medium.     

Monoenergetic-source solutions of the steady-state cosmic-ray equation of transport

A non spherically-symmetric monoenergetic-point-source solution of the steady-state equation of transport for cosmic-rays in the interplanetary region, in which monoenergetic particles are released isotropically and continuously from a fixed heliocentric position is derived by a Laplace transform method. The solution is for a spherically-symmetric model of the propagating region incorporating anisotropic diffusion, with a diffusion tensor symmetric about the radial direction, and the solar wind velocity is radial and of constant speedV. The spherically-symmetric monoenergeticsource solution of Webb and Gleeson (1973) and of Toptygin (1973) is regained from the spherically-symmetric component of the point-source solution.     

Diffuse reflection and transmission in accordance with the phase function 3/4(1+cos2Θ)

We have considered the transport equation for the problem of diffuse reflection and transmission on Rayleigh's phase function and obtained the exact solution of this equation for angular distributions of the intensities diffusely reflected from the surfacet=0 and diffusely transmitted below the surfacet=t ₀ of a finite atmosphere of optical deptht=t ₀ using the Laplace transform and the theory of singular operators. This is an exact method.