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1.
A. G. Nikoghossian 《Astrophysics》2004,47(2):248-259
This paper is a continuation of a study of radiative transfer in one-dimensional inhomogeneous atmospheres. Two of the most important characteristics of multiple scattering in these media are calculated: the photon escape probability and the average number of scattering events. The latter is determined separately for photons leaving the medium and for photons that have undergone thermalization in the medium. The problem of finding the radiation field in an inhomogeneous atmosphere containing energy sources is also examined. It is assumed that the power of these sources, as well as the scattering coefficient, can vary arbitrarily with depth. It is shown that knowledge of the reflection and transmission coefficients of the atmosphere makes it possible to reduce all these problems to solving some first order linear differential equations with specified initial conditions. A series of new analytic results are obtained. Numerical calculations are done for two types of atmosphere with different depth dependences for the scattering coefficient. These are interpreted physically. 相似文献
2.
A. G. Nikoghossian 《Astrophysics》2004,47(1):104-116
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
3.
A complete set of transfer equations required for the order-of-scattering analysis of partially polarized radiation in inhomogeneous, anisotropically scattering atmospheres is provided. The equations have been derived for both a local study using the radiative transfer equation and its associated auxiliary equation for the source-matrix, and a global study in terms of the scattering and transmission matrices; they account for the polarity of the scattering medium. Their derivations for the finite order scattering and the finitely cumulative scattering, in particular, have yielded important new equations expressing the invariance principles and the integro-differential recurrences for the scattering and transmission matrices. These novel expressions contain as a special case Bellmanet al's (1972) equations for the simpler case of isotropic scattering of unpolarized light in homogeneous atmospheres. 相似文献
4.
A. G. Nikoghossian 《Astrophysics》2009,52(1):1-23
The purpose of this article is to provide some insight into Ambartsumian’s methods in the theory of radiative transfer, their
applications, and further development. Two of these methods are emphasized--the invariance principle and the method of addition
of layers, proposed by Ambartsumian in the 1940’s. The difference between these methods and the classical approach for solving
radiative transfer problems is discussed. We discuss only a small portion of the subsequent work by others that we believe
reveals, in a more intuitive way, the essence and significance of Ambartsumian’s methods and their efficiency for applications.
Thus, for example, a separate section is devoted to applications of the Lagrangian formalism to radiative transfer and it
is shown that the invariance principle is a special case of a more general variational principle that reflects an invariance
with respect to translational transformation of the optical depth. Our discussion of the method of addition of layers points
out its generality and the major role it has played in the later creation of such methods as Bellman’s invariant imbedding
method and the method for solving radiative transfer problems in inhomogeneous media. The latter method has yielded a number
of new analytic results. The concluding section is a brief summary of Ambartsumian’s results in the nonlinear theory of radiative
transfer, where he was a pioneer in the study of the class of multilevel problems. This article also sets out to demonstrate
the place and role of Ambartsumian’s methods in the theory of radiative transfer, which, to a great extent, set the path along
which this theory developed for many years to come.
Translated from Astrofizika, Vol. 52, No. 1, pp. 5–27 (February 2009). 相似文献
5.
A. G. Nikoghossian 《Astrophysics》2004,47(3):412-421
The approach proposed in the previous parts of this series of papers is used to solve the radiative transfer problem in scattering and absorbing multicomponent atmospheres. Linear recurrence relations are obtained for both the reflectance and transmittance of these kinds of atmospheres, as well as for the emerging intensities when the atmosphere contains energy sources. Spectral line formation in a one-dimensional inhomogeneous atmosphere is examined as an illustration of the possibility of generalizing our approach to the matrix case. It is shown that, in this case as well, the question reduces to solving an initial value problem for linear differential equations. Some numerical calculations are presented. 相似文献
6.
In this paper, the Combined Operational Method developed by Busbridge (1961) in connection with the radiative transfer problems in plane-parallel atmospheres has been extended to similar problems in isotropic scattering, homogeneous spherical media. The relevant auxiliary equation has been formulated, the scattering function defined and the integro-differential equation for such function deduced. For a medium having radial distribution of source in addition to the incident flux at the outer surface, the integro-differential equations for source function and emergent intensity have been established. 相似文献
7.
The computation of synthetic spectra for planetary atmospheres in which multiple scattering is important usually requires lengthy numerical work. We obtain solutions to the appropriate equation of radiative transfer by use of a variational principle whose extremum is essentially the reflectivity. When considering light diffusely reflected from the atmospheres of the outer planets, the technique reduces the computational work enormously and is applicable to models in which the albedo varies strongly with optical depth as well. 相似文献
8.
Viik Tõnu 《Earth, Moon, and Planets》1989,46(3):261-284
The vector equation of radiative transfer is solved both for conservative and non-conservative planetary atmospheres using the method of discrete ordinates. The atmosphere, bounded by a Lambert bottom, is considered plane-parallel and homogeneous. The scattering in the atmosphere obeys the Rayleigh or Rayleigh-Cabannes law. The compiled package of FORTRAN codes allows us to find the Stokes parameters for such an atmosphere at arbitrary optical depth. 相似文献
9.
A collocation method is used to obtain numerical values for the radiative transfer in absorbing, emitting, linearly anisotropically-scattering-inhomogeneous finite slab media. The results for the isotropic scattering are compared with those obtained by Thynell and Özisik (1986) for the homogeneous slab and with those obtained by Garcia and Siewert (1981) for the inhomogeneous slab. 相似文献
10.
Reciprocity and symmetry relationships, representing local invariants for the scattering phase-matrix, are derived for twelve cases of particle assemblies studied by van de Hulst (1957) including situations of scattering in an arbitrary direction, in the near forward and near backward directions. These relations are used to generate corresponding relations representing global invariants for the scattering and transmission matrices of atmospheres consisting of such assemblies. The latter relations are obtained from the matrix integro-differential equations for scattering and transmission; they apply to single scattering, any finite order of scattering, and after an arbitrary cumulation of scattering orders (finite or infinite). Our results are summarized in Tables I and II for general inhomogeneous atmospheres and for particular inhomogeneous atmospheres that are symmetrical with respect to their central level. The latter case includes homogeneous atmospheres as a special case. The largest set of local relations obtained contains three independent relations (called universal, reversal, exchange) which can further be combined to yield four additional dependent relations. This circumstance happens in three out of the above twelve cases. In the remaining cases fewer relations (both independent and dependent) remain valid. Likewise, a maximal set of three independent global relations is obtained for general inhomogeneous atmospheres; they too can be linearly combined to yield seven other dependent relations. For the symmetrically inhomogeneous atmospheres, three independent and seven dependent additional relations are obtained. On the basis of these tables, it becomes a trivial matter to provide the local and global invariants (both the independent and the dependent relations) for any assembly of particles and atmospheric inhomogeneity. A mixture of Rayleigh-Cabannes scattering by anisotropic molecules or extremely small particles and Mie scattering by large isotropic particles is considered for illustration. Lastly, the group properties of these invariants are studied.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the U.S. National Aeronautics and Space Administration. 相似文献
11.
TheF
N
method is used to solve radiative transfer problems, based on the general anisotropically scattering model, in multi-layer atmospheres. 相似文献
12.
H. F. Machali 《Astrophysics and Space Science》1993,208(1):33-39
The time-dependent equation of radiative transfer for a finite, plane-parallel, non-radiating, and isotropically scattering atmosphere of arbitrary stratification is solved by using the integral equation method. The medium is taken to be inhomogeneous. The Laplace transform is used in the time domain. It is seen that the obtained solutions are reducible to the corresponding ones for steady-state problems by simply changing the Laplace transform parameter to zero. 相似文献
13.
J. Stahlberg 《Astronomische Nachrichten》1990,311(1):49-53
The influence of stochastic velocity fields with finite correlation lengths on the formation of spectral lines is taken into account without restrictions to specific velocity models. To construct a perturbation theory treating the influence of stochastic motion on the radiative transfer we start with the stochastic transfer equation for plane-parallel atmospheres and expand it to an infinite hierarchical system. An appropriate cut-off of the infinite system admits to achieve exactly the micro- and macroturbulent limit at every level of approach. The formalism is derived for n-point correlations of the absorption coefficient and specialised for 2-point correlations. 相似文献
14.
A new method for determining various quantities describing the radiation field in an inhomogeneous, plane-parallel atmosphere
is proposed in this two-part paper. The essence of this method is the reduction of the boundary value problems which arise
during the customary statement of various astrophysical problems associated with solving the radiative transfer equations
to initial value problems. Compared to previous attempts in this area, the proposed method is universal and simple. The first
part of this paper deals with one-dimensional media. Scalar, as well as vector–matrix problems relating to the diffusion of
radiation in spectral lines with frequency redistribution are examined. 相似文献
15.
A. R. Degheidy M. Sallah A. Elgarayhi S. M. Shaaban 《Astrophysics and Space Science》2014,349(1):329-337
Radiation transfer in atmospheric aerosol media with general boundary conditions has been studied for anisotropic scattering. The considered aerosol medium assumed to have specular and diffused reflecting boundary surfaces and in the presence of internal source. The radiation transfer scattering parameters as single scattering albedo, asymmetry factor, scattering, absorption, extinction efficiencies and anisotropic scattering coefficient have been calculated using the Mie theory. The problem with general boundary conditions is solved in terms of the solution of source-free problem with simply boundary conditions. Pomraning-Eddington approximation is used to solve the source-free problem. For the sake of comparison, a weight function is introduced and used in two special forms. The calculated partial heat fluxes with the two methods are compared and showed good agreement. Some of our results are found in a good agreement with published data. 相似文献
16.
A numerical solution to the integral equation for radiative transfer by resonance reradiation in an isothermal spherical atmosphere is described. The method presented is 100 times more efficient than earlier spherical radiative transfer models. The new model can accommodate density variations in the full three dimensional space and includes effects due to the presence of pure absorbers. Complete frequency redistribution is assumed for photon scattering. Applications of this model to the problem of solar photons scattered by atomic hydrogen in the atmospheres of Venus, Earth and Mars are described, and limb and disk profiles, as well as equivalent mean disk intensities for Venus, Earth and Mars, are presented. 相似文献
17.
The infinite medium Green’s function is used to solve the half-space albedo, slab albedo and Milne problems for the unpolarized Rayleigh scattering case; these problems are the most classical problems of radiative transfer theory. The numerical results are obtained and are compared with previous ones. 相似文献
18.
The bivariational method is used to solve the radiative transfer problems for an anisotropic inhomogeneous finite slab. Numerical results are given for the albedo. 相似文献
19.
John K. Hillier 《Icarus》1997,130(2):328-335
It has been proposed that composite particles containing internal scatterers may provide the explanation for the fact that most photometric studies of planetary surfaces based on Hapke's model of bidirectional reflectance have found the planetary particles to exhibit moderately backscattering phase functions. However, an implicit assumption made in this explanation is that the scattering by composite particles containing multiple internal inclusions in a planetary surface can still be adequately computed using standard radiative transfer theory assuming the composite particles to be the fundamental individual scatterers even though such particles are necessarily in close proximity to each other. In this paper, this assumption is explored by examining the effects of close packing on the light scattering by spherical particles containing isotropic internal scatterers using a Monte Carlo routine. As expected, classical radiative transfer (assuming a random distribution of scattering particles) coupled with the assumption that the composite particle is the fundamental scatterer provides a good approximation in the high porosity limit. However, even for porosities as high as 90% the effects of close packing are clearly seen with the radiative transfer calculation underestimating the scattering by ∼10% at high incidence, emission, and phase angles. As the porosity is lowered further, the discrepancy becomes more severe and can reach 50% or more. In contrast, assuming the individual scatterer properties in the radiative transfer calculation leads to a substantial overestimate of the scattering even for porosities as low as 27.5%. This suggests that parameters derived using the classical radiative transfer theory will yield results intermediate between those of the composite as a whole and those of the internal scatterers. Thus, one should exercise caution in interpreting the results of models based on classical radiative transfer theory in terms of the physical properties of the surface particles and, where possible, the bidirectional reflectance of densely packed composite particles should be computed using more accurate methods such as the stochastic radiative transfer theory. 相似文献
20.
For radiative transfer in plane-parallel emitting, absorbing, and scattering media, the two-stream approximation, and its various modifications or related methods, is probably mathematically the most simple to use. Unfortunately this physical approximation produces errors that are neither analytically known nor controllable. For externally (Sun) driven problems, many error studies exist for reflectivity, transmissivity, and certain defined albedos. A two-stream accuracy study for internally (thermal) driven problems is presented in this paper by comparison with a recently developed “exact” adding/doubling method. The resulting errors in external (or boundary) radiative intensity and flux are usually larger than those for the externally driven problems and vary substantially with the radiative parameters. Error predictions for a specific problem are difficult. An unexpected result was that the exact method is computationally as fast as the two-stream approximation for nonisothermal media. Although the adding/doubling method is mathematically and conceptually more complex, it may be used as a developed code with no essential sacrifice in computing time. 相似文献