共查询到20条相似文献，搜索用时 578 毫秒
1.
We have considered a homogeneous atmosphere scattering anisotropically with Dirac -function type time-dependent incidence. We used the method of integral operator developed by Ambartsumian and the theory ofN-solutions developed by Busbridge to find the correspondingH-function (in semi-infinite atmosphere) andX- andY-functions (in finite atmosphere). 相似文献
2.
The equation for radiative transfer in the case of resonance radiation for isotropic scattering has been solved by the method of the Laplace transformation and linear singular operators. The solution for emergent intensities have come out in terms ofX- andY-functions. 相似文献
3.
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
4.
The equation of transfer for the case of non-coherent scattering (Hummer, 1968; Ivanov, 1973; McCormick and Siewert, 1970) has been considered. The correspondingX- andY-functions have been derived by a combination of eigenfunction method developed by Case, and from the principle of invariance as developed by Chandrasekhar (1960). 相似文献
5.
Takao Fukui 《Astrophysics and Space Science》1988,149(1):13-18
The bi-variational technique is used to calculated Chandrasekhar'sX- andY-functions and their first two moments for an isotropic homogeneous finite slab. Numerical results obtained are compared with published results. 相似文献
6.
The nonlinear integral equations forX- andY-functions have been developed for an inhomogeneous atmosphere scattering anisotropically using the principle of invariance. The anisotropy is represented by means of a phase function expressed in terms of finite-order Legendre polynomials. 相似文献
7.
We discuss a simple method of linearization and decoupling of the integral equations satisfied by time-dependentX - andY -functions which play an important rôle in the study of non-stationary radiative transfer problems. 相似文献
8.
Tõnu Viik 《Astrophysics and Space Science》1995,225(2):205-219
The determination of the average path-length of photons in a finite isotropically scattering plane-parallel homogeneous atmosphere is discussed. To solve this problem we have used the kernel approximation method which easily allows us to find the derivatives of the intensity with respect to optical depth, optical thickness and albedo of single scattering.In order to check the results we have used another approach by exploiting the set of integrodifferential equations of Chandrasekhar for theX- andY-functions. This approach allows us to find the average path length only at the boundaries of the atmosphere but on the other hand it gives also the dispersion of the path-length distribution function, thus generating the input parameters for determining the approximate path-length distribution function. It occurred that the set so obtained is stable and the results are highly accurate.As a by-product we obtain the first two derivatives of theX- andY-functions with respect to the albedo of single scattering and optical thickness, and the mixed derivative. 相似文献
9.
Expressions for time-dependentX- andY-functions for a one-speed neutron transport problem in a finite slab have been derived using a technique combining invariant imbedding method and eigenfunction expansion method. The atmosphere has been considered to scatter isotropically. 相似文献
10.
A study of pulsational properties with evolution has been done for a 15.6M
star withX
e
=0.90 andY
e
=0.08. Pulsational properties in the hydrogen-burning stages have been compared with those in helium-burning stages. A comparison with observed characteristics of Cepheids, classical Cepheids and supergiant variables has been made during the course of its evolution. In addition, models of 5,9 and 15M
withX
e
=0.708,Y
e
=0.272 have also been studied for pulsational properties during the helium burning stage. It is also seen that pulsational instability is sensitive to changes in initial chemical composition and opacity parameters,n ands. A low helium abundance could be a reason for the stability of the models, even when lying in the instability strip of the H-R diagram. 相似文献
11.
Rabindra Nath Das 《Astrophysics and Space Science》1979,61(1):169-173
The application of the Wiener-Hopf technique to the coupled linear integral equation ofX- andY-equations gives rise to the Fredholm equations with simpler kernels.X-equation is expressed in terms ofY-equation and vice-versa. These are unique in representation with respect to coupled linear constraints. 相似文献
12.
Rabindra Nath Das 《Astrophysics and Space Science》2010,326(1):91-103
In radiative transfer, the intensities of radiation from the bounding faces of a scattering atmosphere of finite optical thickness
can be expressed in terms of Chandrasekhar’s X- and Y-functions. The nonlinear nonhomogeneous coupled integral equations which the X- and Y-functions satisfy in the real plane are meromorphically extended to the complex plane to frame linear nonhomogeneous coupled
singular integral equations. These singular integral equations are then transformed into nonhomogeneous Riemann–Hilbert problems
using Plemelj’s formulae. Solutions of those Riemann–Hilbert problems are obtained using the theory of linear singular integral
equations. New forms of linear nonhomogeneous decoupled expressions are derived for X- and Y-functions in the complex plane and real plane. Solutions of these two expressions are obtained in terms of one known N-function and two new unknown functions N
1- and N
2- in the complex plane for both nonconservative and conservative cases. The N
1- and N
2-functions are expressed in terms of the known N-function using the theory of contour integration. The unknown constants are derived from the solutions of Fredholm integral
equations of the second kind uniquely using the new linear decoupled constraints. The expressions for the H-function for a semi-infinite atmosphere are obtained as a limiting case. 相似文献
13.
M. T. Edalati 《Astrophysics and Space Science》1978,59(2):443-462
The aim of the present paper will be to evaluate numerically Jacobian and other functions which have been discussed in more detail in a previous paper of this series (Edalati, 1978b, Paper XXII), and also choose the most convenient moments to obtain a good determination for the unknown eclipse parametersa andc
0. More than 12 different pairs ofg-functions for real values ofm have been investigated numerically and diagrammatically. The behaviour ofg-functions depends but very little on different combination of the moments, and related diagrams are approximately the same asg
2 andg
4 (Kopal and Demircan, 1978, Paper XIV).The behaviour of the vanishing Jacobian, arising from different pairs ofg-functions for real values ofm1 has been shown diagrammatically in terms ofa andc
0. Accordingly, we obtain the optimum combination of the moments (i.e.,A
6,A
7,A
8 andA
9) ing-functionsg
7 andg
8. It has been noted that the behaviour of theg-functions which depend on the combinations of the higher order moments (i.e.,m5) have been ruled out, because the proportional error of the momentsA
2m
increases with increasing values of realm. The automated method has been tested successfully on the light curve of RT Per (Mancusoet al., 1977; Edalati, 1978a). Finally, a comparison is given of the elements of RT Per arising from two different pairs ofg-functions, i.e.,g
2,g
4 (Edalati, 1978a) andg
7,g
8 for the light curves analysis. 相似文献
14.
Das Gupta represented theH-functions of transport problems for the albedo [0, 1] in the formH(z)=R(z)–S(z) (see Das Gupta, 1977) whereR(z) is a rational function ofz andS(z) is regular on [–1, 0]
c
. In this paper we have representedS(z) through a Fredholm integral equation of the second kind with a symmetric real kernelL(y, z) as
. The problem is then solved as an eigenvalue problem. The kernel is converted into a degenerate kernel through finite Taylor's expansion and the integral equation forS(z) takes the form:
(which is solved by the usual procedure) where
r
's are the discrete eigenvalues andF
r
's the corresponding eigenfunctions of the real symmetric kernelL(y, z). 相似文献
15.
M. T. Edalati 《Astrophysics and Space Science》1978,59(2):333-345
The theoretical values of the momentsA
2m
for any type of eclipses, expressed in terms of the elementsL
1,a andc
0, have been derived in the simple forms of rapidly convergent expansions to the series of Chebyshev polynomials, Jacobi polynomials and KopalJ-integrals (Kopal, 1977c) and hold good for any real (not necessarily integral) value ofm0.The aim of the present paper has been to establish explicit expressions for the Jacobian and its fast enough computation in the light changes of close eclipsing systems, arising from the partial derivative of different pairs ofg-functions (Kopal and Demircan, 1978, Paper XIV) with respect toa andc
0
2
, for any type of eclipses (be these occultations or transit, partial, total or annular) and for any arbitrary degreel of the adopted law of limb-darkening. The functional behaviour of this Jacobian would determine the reasonable light curve in connection with geometrical determinacy of the parametersa andc
0. In the expansion of Jacobian, the terms consist of two polynomials which satisfy certain three-term recursion relations having the eclipse parametersa andc
0, as their arguments.Closed form expressions forf-functions, as well as of the Jacobian (e.g.,m=1, 2, 3), obtaining in the case of total eclipses, are given for a comparative discussion with the theoretical values of Jacobian derived from partial derivative of different pairs ofg-functions.The numerical magnitude of Jacobian would determine the best combination of the momentsA
2m
in the different pairs ofg-functions and definite results would follow in the subsequent paper of this series (Edalati, 1978c, Paper XXIV). 相似文献
16.
Models for the motions of flare loops and ribbons 总被引：1，自引：0，他引：1
We have found a conformal mapping which is valid for any magnetic boundary condition at the photosphere and which can be used to determine the evolution of an open, two-dimensional magnetic field configuration as it relaxes to a closed one. Solutions obtained with this mapping are in quasi-static equilibrium, and they contain a vertical current sheet and have line-tied boundary conditions. As a specific example, we determine the solution for a boundary condition corresponding to a submerged, two-dimensional dipole below the photosphere. We assume that the outer edges of the hottest X-ray loops correspond to field lines mapping from the outer edges of the H ribbon to the lower tip of the current sheet where field lines reconnect at aY-type neutral line which rises with time. The cooler H loops are assumed to lie along the field lines mapping to the inner edges of the flare ribbons. With this correspondence between the plasma structures and the magnetic field we determine the shrinkage that field lines are observed to undergo as they are disconnected from the neutral line. During the early phase of the flare, we predict that shrinkage inferred from the height of the H and X-ray loops is close to 100% of the loop height. However, the shrinkage should rapidly decrease with time to values on the order of 20% by the late phase. We also predict that the shrinkage in very large loops obeys a universal scaling law which is independent of the boundary condition, provided that the field becomes self-similar (i.e., all field lines have the same shape) at large distances. Specifically, for any self-similar field containing aY-type neutral line, the observed shrinkage at large distances should decrease as (X/X
R)–2/3, where X is the ribbon width andX
Ris the ribbon separation. Finally, we discuss the relation between the electric field at the neutral line and the motions of the flare loops and ribbons. 相似文献
17.
Rabindra Nath Das 《Astrophysics and Space Science》1981,76(2):441-463
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. 相似文献
18.
19.
Ther-centroids and Franck-Condon factors for the bands of theA
2 –X
2+ of CP,C
3 –X
3 of SiC, andB
2+ –X
2+ of CO+ molecules have been determined. The Franck-Condon factors are evaluated by the approximate analytical method of Jarmain and Fraser. The absence of the bands in these systems is explained. 相似文献
20.
Vibrational transition probabilities, namely Franck—Condon factors and
-centroids have been evaluated by an approximate analytical method for the (A–X), (A–X), and (A–X) system of YO molecule. Morse potential energy curves forX
2+,A
22,A22, andA22, states of YO have been constructed using the latest spectroscopic data. The value of
-centroids for the band have been found to decrease linearly with the corresponding wavelengths. We show results for two new transitions of (A–X) and (A–X) and five new bands of (A–X) of YO in the umbral spectrum of the Sun. 相似文献