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1.
In the present paper, with the aid of invariance principles in connection with the scattering matrix, we get the exact solution of diffuse reflection and transmission problems by finite inhomogeneous, anisotropically scattering atmospheres bounded by reflecting sufaces. On making use of the reflection and transmission integral operators, we show how to obtain the non-linear integro-differential equations for these operators, which do not depend on the initial condition. Then, we have a system of the required integro-differential equations for the scattering and transmission functions. The obtained result is new, so far as we know. Finally, using the scattering matrix, we reduce the diffuse reflection and transmission problems for planetary atmospheres with reflecting surfaces to the standard diffuse reflection and transmission problems.Supported by the National Science Foundation under Grant No. GP29049 and the Atomic Energy Comminission, Division of Research under Contract No. AT(40-3)-133, Project 19.  相似文献   

2.
In this paper we consider the time-dependent diffuse reflection and transmission problems for a homogeneous anisotropically-scattering atmosphere of finite optical depth and solve it by the principle of invariance. Also we consider the time-dependent diffuse reflection and transmission of parallel rays by a slab consisting of two anisotropic homogeneous layers, whose scattering and transmission properties are known. It is shown how to express the time-dependent reflected and transmitted intensities in terms of their components. In a manner similar to that given by Tsujita (1968), we assumed that the upward-directed intensities of radiation at the boundary of the two layers are expressed by the sum of products of some auxiliary functions depending on only one argument. Then, after some analytical manipulations, three groups of systems of simultaneous integral equations governing the auxiliary functions are obtained.  相似文献   

3.
This series of papers is devoted to multiple scattering of light in plane parallel, inhomogeneous atmospheres. The approach proposed here is based on Ambartsumyan's method of adding layers. The main purpose is to show that one can avoid difficulties with solving various boundary value problems in the theory of radiative transfer, including some standard problems, by reducing them to initial value problems. In this paper the simplest one dimensional problem of diffuse reflection and transmission of radiation in inhomogeneous atmospheres with finite optical thicknesses is considered as an example. This approach essentially involves first determining the reflection and transmission coefficients of the atmosphere, which, as is known, are a solution of the Cauchy problem for a system of nonlinear differential equations. In particular, it is shown that this system can be replaced with a system of linear equations by introducing auxiliary functions P and S. After the reflectivity and transmissivity of the atmosphere are determined, the radiation field in it is found directly without solving any new equations. We note that this approach can be used to obtain the required intensities simultaneously for a family of atmospheres with different optical thicknesses. Two special cases of the functional dependence of the scattering coefficient on the optical thickness, for which the solutions of the corresponding equations can be expressed in terms of elementary functions, are examined in detail. Some numerical calculations are presented and interpreted physically to illustrate specific features of radiative transport in inhomogeneous atmospheres.  相似文献   

4.
In recent years Sobolev's -function of radiative transfer has been discussed in connection with the resolvent of Milne's integral equation such that it plays an important role in the determination of the radiation field in semi-infinite (or finite) atmospheres with internal sources (cf. Sobolev, 1963). In the present paper, the part of Sobolev's -function in plane-parallel and spherical, isotropically scattering, atmospheres with internal source distribution is investigated from analytical and numerical aspects. With the aid of invariant imbedding (cf. Bellmanet al., 1968), we computed Sobolev's -function of Milne's integral equation for the planar case by solving the Cauchy system for the auxiliary function and Chandrasekhar'sX- andY-functions. The corresponding -function for the spherical case is readily obtained from the for the planar case.Investigation supported by the National Science Foundation under Grant No. GP 29049, the Atomic Energy Commission, Division of Research under Contract No. AT(04-3)-113, Project 19, and the National Institutes of Health under Grant No. 16197-05.  相似文献   

5.
The determination of the average path-length of photons emerging from a finite planeparallel atmosphere with molecular scattering is discussed. We examine the effects of polarisation on the average path-length of the emergent radiation by comparing the results with those obtained for the atmosphere where the scattering obeys the scalar Rayleigh function. Only the axial radiation field is considered for both cases.To solve this problem we have used the integro-differential equations of Chandrasekhar for the diffuse scattering and transmission functions (or matrices). By differentiation of these equations with respect to the albedo of single scattering we obtain new equations the solution of which gives us the derivatives of the intensities of the emergent radiation at the boundaries.As in the case of scalar transfer the principles of invariance by Chandrasekhar may be used to find an adding scheme to obtain both the scattering and transmission matrices and their derivatives with respect to the albedo of single scattering. These derivatives are crucial in determining the average path length.The numerical experiments have shown that the impact of the polarisation on the average pathlength of the emergent radiation is the largest in the atmospheres with optical thickness less than, or equal to, three, reaching 6.9% in the reflected radiation.  相似文献   

6.
In a manner similar to that given in a preceding paper (cf. Castiet al., 1970), in this paper we show how to get a Cauchy system for the scattering and transmission matrices of the azimuth-independent term of partially polarized light in accordance with Rayleigh scattering in a homogeneous atmosphere bounded by a specular reflector. This set of integro-differential equations for the scattering and transmission matrices is suitable for numerical computation by high-speed digital computer.  相似文献   

7.
The problem of the passage of a plane electromagnetic wave through an arbitrary, inhomogeneous dielectric layer bounded on two sides by two different homogeneous, semi-infinite media is considered. Algebraic relations are obtained between the amplitudes of transmission and reflection (the scattering amplitudes) for the problem under consideration and the wave scattering amplitudes when the layer is bounded on both sides by a vacuum. It is shown that for s and p polarized fields the scattering problem (a boundary-value problem) can be formulated as a Cauchy problem directly for the s and p wave equations. It is also shown that the problem of finding the field inside the layer also reduces to a Cauchy problem in the general case.  相似文献   

8.
Differential equations are derived, following the methods ofLifshitz (1946) andLifshitz andKhalatnikov (1963), for density perturbations in isotropic, spatially homogeneous cosmological models of arbitrary space curvature. The unperturbed models contain both matter and radiation. An explicit third-order equation is obtained for the time development of the perturbations, and it is shown that one of the solutions is not covariantly defined. The two remaining solutions are compared with existing solutions for the limiting cases of radiation-filled and dust-filled models. The results ofBonnor's (1957) Newtonian analysis are shown to be a valid limiting case of our equation when the pressurep is finite, but small compared with the densityp timesc 2.A detailed analysis is given of a model containing coupled radiation (p=pc 2/3) and dust (p=0). It is shown that density perturbations with long wavelengths are unstable (with slow growth rate) for all time. The instability exists because for a long-wavelength disturbance, the time scale governing the propagation of pressure effects (which stabilize perturbations) is longer than the time scale for which pressure falls to the point of ineffectiveness. The present value of the critical wavelength is 60 Mpc in models based on flat space sections in which the present background radiation temperature is 3 °K.The research reported herein was supported in part by the Atomic Energy Commission under contract number AT(11-1)-34, Project Agreement No. 125, and by the National Science Foundation, under Grant GP-4975.  相似文献   

9.
The Combined Operations Method is utilised to solve Diffuse Reflection and Transmission Problems in inhomogeneous, isotropically and non-coherently scattering, spherical shell media. The source function is assumed to be frequency independent. TheN-solution of an auxiliary equation is sought, and tractable equations for the scattering and transmission functions are established. The solution of the problem for a scattering and emitting medium have been considered for a perfectly absorbing core.  相似文献   

10.
A standard problem of radiative transfer theory — calculating the diffuse reflection and transmission of radiation by a plane scattering atmosphere — is considered. The recently proposed albedoshift method is used to calculate the X and Y functions (and the H function) for the case of anisotropic scattering with a Henyey-Greenstein indicatrix. The method enables one to “suppress” scattering and obtain iterative solutions of high accuracy in only a few iterations, even when the mean number of photon scatterings in the atmosphere is very large. Translated from Astrofizika, Vol. 41. No. 4, pp. 623–646, October–December, 1998.  相似文献   

11.
A method of analysis is presented for solving the radiative transfer problem in an absorbing, emitting, inhomogeneous, and anisotropically scattering plane-parallel medium with specular and diffuse reflecting boundaries and internal source (problem 1). Exact relations for the radiation heat flux at the boundaries of problem 1 are obtained in terms of the radiation density and albedos of the corresponding source-free medium with specular reflecting boundaries (problem 2). Two coupled integral equations for the radiation density and the second moment of the radiation intensity for problem 2 with Rayleigh phase functions are obtained. The Galerkin method is used to solve these equations. Albedos of problem 2 are compared with theF n method. Numerical results for radiation heat fluxes at the boundaries of problem 1 are tabulated for different forms of the internal source.  相似文献   

12.
In order to choose a numerical method for solving the time dependent equations of radiative transport, we obtain an exact solution for the time dependent radiation field in a one dimensional infinite medium with monochromatic, isotropic scattering for sources with an arbitrary spatial distribution and an arbitrary time variation of their power. The Lax-Wendroff method seems to be the most suitable. Because it is assumed that radiation delay is caused by the finite speed of light, the following difficulty arises when the numerical method is used: the region of variation of the variables (position τ and time t) is triangular (the inequality τ ≤ t). This difficulty is overcome by expanding the unknown functions in series in terms of small values of the time and position. By comparing the numerical and exact solutions for a point source with a given time dependence for its power and with pure scattering, the number of steps in the variables required to obtain a desired accuracy is estimated. This numerical method can be used to calculate the intensity and polarization of the radiation from sources in the early universe during epochs close to the recombination epoch. __________ Translated from Astrofizika, Vol. 51, No. 1, pp. 109–123 (February 2008).  相似文献   

13.
In this paper an analytical solution for the differential equations governing the motion of Artificial Satellite under the oblateness of the earth will be developed. We start with the differential equations in terms of Euler parameters. To compact algebra, we introduce the Cayley-Klein type complex variables. Comparison between numerical and analytical final states withJ 2 will be given for a test case.  相似文献   

14.
A multigroup, diffusion scheme is derived from the Boltzmann equation for efficient numerical computation of neutrino transport in a collapsing stellar core. This scheme treats all neutrino scattering processes correctly with a Legendre expansion of the scattering rates. Modifications of the diffusion coefficient and the addition of a flux limiter extend the applicability of the scheme to neutrino transparent regimes and improve the agreement of the solution in both opaque and transparent regimes with those obtained analytically for some test problems. In addition, a more accurate treatment of the neutrino flowAcross shock discontinuities is developed through a shock matching condition. The solutions given by this multigroup, flux-limited diffusion scheme are compared with those of a modifiedS N code.Supported in part by the National Science Foundation (Grant AST75-05012 and AST 77-17572).  相似文献   

15.
The method of computing the radiation field in an infinite circular cylinder proposed in Part I is now applied to the case of isotropic scattering with sources on the boundary and axis of the cylinder, as well as for a uniform distribution of sources inside the cylinder. For the simplest kernel we obtain exact solutions of the basic integral equation in explicit form. For scattering in a spectral line with complete frequency redistribution and a power absorption profile we develop an asymptotic theory for the case when the optical radius of the cylinder is large. We solve the asymptotic equations for the basic characteristics of the scattering in closed form for conservative scattering and find its asymptotics. We obtain estimates of the mean number of scatterings with a layered source, and also the mean and variance of the number of scatterings with a uniform source distribution.Translated fromAstrofizika, Vol. 37, No. 4, 1994.This work was carried out with the financial support of the Russian Basic Research Fund (grant 93-02-2957).  相似文献   

16.
We show that Hovenier's Exit Function equation describing reflection and transmission by a plane-parallel layer can be obtained from the Invariant Imbedding equations. As an immediate extension we obtain a similar equation for an Exit Function defined in terms of reflection and transmission functions for successive orders of scattering. These equations allow the reflection and transmission functions of a homogeneous atmosphere of arbitrary optical thickness to be obtained from angle integrations of only one function.A technique based on successive iterations is developed to solve Hovenier's equation. The numerical behavior of this equation is then investigated employing a few representative (i.e., isotropic, Rayleigh, and Henyey-Greenstein) phase functions with the following conclusions. (i) As long as the deviation from isotropy is small (cos 0.15), the Exit Function equation can be numerically solved with an efficiency comparable to that of the standard Doubling technique, which is one of the fastest algorithms available. (ii) The reflection function generated from the Exit Function is usually more accurate than the corresponding transmission function, particularly in the case of large optical thickness. (iii) As the degree of anisotropy increases, so does the difficulty in obtaining the numerical solution for the Exit Function. The solution of the equation depends sensitively on the treatment of the numerical singularities which arise from the integrands and also on the initial approximation employed for the iteration. An improved scheme is required for numerically obtaining the Exit Function in order for this method to yield accurate reflection and transmission functions for strongly anisotropic scattering.  相似文献   

17.
New methods are proposed for finding the Ambartsumyan functions φ(η) for a half space and φ(η, τ) and ψ(η, τ) for finite layers, as well as their analogs with complete frequency redistribution, X (z, τ) and Y (z, τ). Substantial simplifications are obtained for monochromatic conservative scattering. Besides the Ambartsumyan functions, expressions for several of their angular moments are obtained directly in terms of the basis functions u ±. A system of differential equations is obtained for the basis functions. A system of equations without the characteristic pseudosingularities is obtained for φ(η, τ) and ψ(η, τ) instead of the classical system of nonlinear equations. Some aspects of the numerical realization of the proposed method are also discussed.  相似文献   

18.
Asymptotic solutions are developed for the motion of a geocentric satellite in the equatorial plane due to gravitational perturbations such as nonsphericity (especially oblateness) of the primary body. Axisymmetric potentials are considered. A class of transformations is developed and the equations of motion are solved by the method of generalized multiple scales. Further it is shown that the equations of motion can be transformed into the required form to within any specified degree of accuracy. The transformations form an Abelian group of infinite order which leaves the differential equations of motion invariant. Solutions are developed in terms of elementary functions instead of elliptic or other higher transcendental functions and are shown to agree with known results.This investigation was carried out under NASA Grant NGR-31-001-152 with the author as a consultant to Princeton University.  相似文献   

19.
In this paper we develop a new method, combined with Laplace transformation and Wiener-Hopf technique, to obtain unique solutions of transport equations in finite media. For this purpose we consider the simple transfer equation for diffuse reflection by a plane-parallel finite atmosphere scattering radiation with moderate anisotropy. It is transformed, by Laplace transformation, into two coupled linear integral equations which are then reduced to two uncoupled Fredholm integral equations admitting of unique solutions by the method of iteration for values of the breadth of the atmosphere greater than that specified, depending on the scattering process.  相似文献   

20.
An efficient method is proposed to calculate scalar Ambarzumian-Chandrasekhar's and Hopf's functions. This method is based on the approximation of Sobolev's resolvent function using exponent series, the coefficients of which are readily found from approximate characteristic equation and from a system of linear algebraic equations.The approximate expressions for the above functions are given. For checking purposes the calculations were carried out in single, double, and quadruple precision. For isotropic, Rocard, and Rayleigh scattering we present a sample of results in 14 significant figures.The Hopf function for isotropic and Rayleigh scattering is presented in 18 significant figures and the well-known Hopf constantq() is found in 59 significant figures.  相似文献   

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