Matrix method evaluating an internal radiation field in a plane-parallel atmosphere

Suppose that the atmosphere consists of homogeneous sublayers whose scattering and transmission matrices are known, then it is shown in this paper how to determine the intensity and polarization parameters of internal radiation field by the matrix method.     

Polarization effect on radiative transfer in planetary composite atmospheres with interacting interface

A procedure of computing the radiance and the polarization parameters of radiation diffusely reflected and transmitted by an inhomogeneous, plane-parallel terrestrial atmosphere bounded by a ruffled ocean surface is discussed with the aid of the adding method. If the atmosphere and the ocean are simulated by a number of homogeneous sublayers, the matrices of radiation reflected and transmitted diffusely by the atmosphere-ocean system can be expressed in terms of these matrices of sublayers by using only a couple of iterative equations in which the polarity effect of radiation is included. Furthermore, the upwelling radiance and the polarization degree of radiation at the top of the atmosphere can be calculated by using a single iterative equation without requiring the equation for the diffuse transmission matrix of radiation. The ruffled ocean surface can be treated as an interacting interface, where the transmitted radiation from beneath the ocean surface into the atmosphere is also taken into account into the derivation of equations. Finally, sample computations of the upwelling radiance and the polarization degree of radiation from the top of the atmosphere are carried out at the wavelength of 0.60 micron.     

A new approach of the adding method for the computations of emergent radiation of an inhomogeneous plane-parallel planetary atmosphere

The procedure of computing the intensity and the polarization parameters of radiation diffusely reflected and transmitted by an inhomogeneous, plane-parallel planetary atmosphere is discussed with the aid of the adding method. If the atmosphere is simulated by a number of homogeneous sublayers (aerosols and ozone may be included), the matrices of radiation diffusely reflected and transmitted by the atmosphere can be expressed in terms of these matrices of sublayers by using only a couple of iterative equations with the polarity effect of radiation. This procedure is to be extended to the model atmosphere bounded by the surface reflector with a quite arbitrary phase matrix.     

A hybrid mode of diffuse and specular reflector for computation of the emergent radiation by the adding method

A method of computing the diffuse reflection and transmission radiation from an inhomogenous, plane-parallel planetary atmosphere bounded by the hybrid surface of a diffuse and specular reflector is discussed by using the addition method. If the atmosphere is simulated by a number of homogeneous sublayers, the radiation diffusely reflected and transmitted by the atmosphere can be expressed in terms of the diffuse reflection and transmission matrices of radiation of sublayers (Laciset al., 1974; Takashima, 1973, 1975). With this method (Takashima, 1975), the troublesome treatment due to the effect of polarity of radiation is overcome. Moreover, if the surface reflects radiation in accordance with the Lambert law as well as a quite arbitrary phase matrix (Takashima, 1974), the addition method can be easily extended. It is shown in this paper that the addition method is suitable for numerical computation even if the surface reflects light according to the hybrid mode of the diffuse and specular law (Uenoet al., 1974; Mukai, 1976).On leave from the Meterological Research Institute, Tokyo, Japan.     

A method of computing the emergent radiation by the atmosphere in the region ranging from ultraviolet to infrared

A method of computing the diffuse reflection and transmission radiation by an inhomogeneous, plane-parallel planetary atmosphere with internal emission source is discussed by use of the adding method. If the atmosphere is simulated by a number of homogeneous sub-layers, the radiation diffusely reflected or transmitted by the atmosphere can be expressed in terms of the reflection and transmission matrices of the radiation of sub-layers. The diffusely transmitted radiation due to the internal emission source can be also easily computed in the same manner. These equations for the emergent radiation are in a quite general form and are applicable to radiative transfer in the atmosphere in the region from ultraviolet to infrared radiation. With this method, the tiresome treatment due to the polarity effect of radiation is overcome.     

Radiative Transfer in Inhomogeneous Atmospheres. I

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.     

Elemental abundance analyses with DAO spectrograms — XVIII. The double-lined spectroscopic binary 46 Draconis

Equipartition magnetic fields can dramatically affect the polarization of radiation emerging from accretion disc atmospheres in active galactic nuclei. We extend our previous work on this subject by exploring the interaction between Faraday rotation and absorption opacity in local, plane-parallel atmospheres with parameters appropriate for accretion discs. Faraday rotation in pure scattering atmospheres acts to depolarize the radiation field by rotating the polarization planes of photons after last scattering. Absorption opacity in an unmagnetized atmosphere can increase or decrease the polarization compared to the pure scattering case, depending on the thermal source function gradient. Combining both Faraday rotation and absorption opacity, we find the following results. If absorption opacity is much larger than scattering opacity throughout the atmosphere, then Faraday rotation generally has only a small effect on the emerging polarization because of the small electron column density along a photon mean free path. However, if the absorption opacity is not too large and it acts alone to increase the polarization, then the effects of Faraday rotation can be enhanced over those in a pure scattering atmosphere. Finally, while Faraday rotation often depolarizes the radiation field, it can in some cases increase the polarization when the thermal source function does not rise too steeply with optical depth. We confirm the correctness of the analytic calculation by Silant'ev of the high magnetic field limit of the pure scattering atmosphere, which we incorrectly disputed in our previous paper.     

On the propagation of magneto-acoustic-gravity waves

An analysis of magneto-acoustic-gravity waves in the case of an isothermal atmosphere permeated by a uniform magnetic field is presented. The general solution is expressed in terms of generalized hypergeometric functions. It can be used in numerical simulation of oscillations in a magnetic atmosphere.

It is shown that the elliptically polarized magneto-acoustic-gravity waves consist of a pair of surface waves and a pair of body waves above the cut-off frequency. The body waves along the magnetic field are similar to acoustic waves in an atmosphere and their cut-off frequency is unaffected by magnetic field. The transverse oscillation decreases with height. For the usual boundary condition, the longitudinal oscillation decreases with height; however, in some cases, it may contain terms that increase with height. The solution is singular on a family of ellipses in the frequency - horizontal wave number plane. Near these ellipses, the wave components grow indefinitely.     

Time-dependent scattering and transmission function in an anisotropic two-layered atmosphere

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.     

Average photon path-length of radiation emerging from finite atmospheres

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.     

关于磁声重力波传播性质的探讨

本文对充满垂直均匀磁场的等温大气内的磁声重力波做了严格的解析分析,并将其通解表述成广义超几何函数的形式。该解可用于对磁大气内振荡现象的进一步数值模拟研究。对解的分析澄清了若干磁声重力波的传播性质。     

The effect of surface characteristics on diffuse reflection radiation at λ=0.40 μm

The diffuse radiation in the upward direction at the top and at an internal level of an inhomogeneous atmosphere is computed at =0.40 m. The surface is assumed to reflect light in accordance with a hybrid mode of a diffuse and specular reflector. The objective is to estimate the effect of underlying surface characteristics in terms of the diffuse radiation field. By making use of these results, accuracy in monitoring the atmospheric aerosols would be increased for the use of remote sensing satellite techniques. Junge power lawv ^*=3 is adopted for the size distribution of aerosols (1963), while the data given by McClatchyet al. (1971) is used for the number density of aerosols with height distribution. It is noted from the computations that the diffuse reflection radiation is affected by the surface characteristics, even if the albedo of the surface is a fixed constant and very small.On leave from the Meterological Research Institute, Tokyo, Japan.bl]References     

Effect of anisotropic collisions on solar scattering polarization

Scattering of anisotropic radiation by atoms,ions or molecules is sufficient to generate linear polarization observable in stars and planets' atmospheres,circumstellar environments,and in particular in the Sun's atmosphere.This kind of polarization is called scattering polarization(SP) or second solar spectrum(SSS) if it is formed near the limb of the solar photosphere.Generation of linear SP can typically be reached more easily than circular SP.Interestingly,the latter is often absent in observations and theories.Intrigued by this,we propose to demonstrate how circular SP can be created by anisotropic collisions if a magnetic field is present.We also demonstrate how anisotropic collisions can result in the creation of circular SP if the radiation field is anisotropic.We show that under certain conditions,linear SP creation is accompanied by the emergence of circular SP which can be useful for diagnostics of solar and astrophysical plasmas.We treat an example and calculate the density matrix elements of tensorial order k=1 which are directly associated with the presence of circular SP.This work should encourage theoretical and observational research to be increasingly oriented towards circular SP profiles in addition to linear SP in order to improve our analysis tools of astrophysical and solar observations.     

The transfer of polarized radiation in spectral lines: Formalism and solutions in simple cases

A general treatment of the transfer of polarized radiation in spectral lines assuming a Rayleigh phase function and a general law of frequency redistribution is derived. It is shown how nine families of coupled integral equations for the moments of the radiation field arise which are necessary to fully describe the state of polarization of the emergent radiation from a plane-parallel, semi-infinite atmosphere. The special case of angle independent redistribution functions is derived from the general formalism, and it is shown how the nine families of integral equations reduce to the six linearly independent integral equations derived by Collins (1972). To serve as a test of the formulation, solutions for isothermal atmospheres are given.     

Source vector in Rayleigh-Cabannes scattering atmosphere: the nonconservative milne problem

We consider the radiative transfer in a nonconservative homogeneous plane-parallel semi-infinite planetary atmosphere where the scattering processes are described by the Rayleigh-Cabannes phase matrix and where the primary sources are in infinitely deep layers. If we use the superposition principle we derive the Cauchy problem for the source vector.As a by-product the external field of radiation for the problem described is obtained using the principle of invariance by Chandrasekhar. The respective formulae for the radiation field in the deep layers and for the extrapolation distance are given. It is shown that the Rubenson degree of polarization even in the case of near-conservative atmospheres reaches the asymptotic regime at rather small values of the optical depth. The-plane reliefs of the characteristic equation, extrapolation distance and the normalized components of the source vector at the boundary are given along with a sample of zeros of the characteristic equation.     

Study of interplanetary magnetic field with Ground State Alignment

We demonstrate a new way of studying interplanetary magnetic field—Ground State Alignment (GSA). Instead of sending thousands of space probes, GSA allows magnetic mapping with any ground telescope facilities equipped with spectropolarimeter. The polarization of spectral lines that are pumped by the anisotropic radiation from the Sun is influenced by the magnetic realignment, which happens for magnetic field (<1 G). As a result, the linear polarization becomes an excellent tracer of the embedded magnetic field. The method is illustrated by our synthetic observations of the Jupiter’s Io and comet Halley. Polarization at each point was constructed according to the local magnetic field detected by spacecrafts. Both spatial and temporal variations of turbulent magnetic field can be traced with this technique as well. The influence of magnetic field on the polarization of scattered light is discussed in detail. For remote regions like the IBEX ribbons discovered at the boundary of interstellar medium, GSA provides a unique diagnostics of magnetic field.     

The effect of reflecting boundaries on the polarized radiation field

The in fluence of the different reflective boundaries on the polarized radiation field in a conservative homogeneous plane-parallel atmosphere is studied. The equation of transfer for molecular scattering is solved using the method of discrete ordinates and the results are checked by using the invariant imbedding approach. The results of a long series of numerical experiments have shown that in general a reflecting boundary at the bottom depolarizes the radiation field while the depolarizing effect of an isotropic scatterer is much larger than that of a specular reflector or a perfect absorber.     

Spectral line formation in a mesoturbulent atmosphere

The formation of spectral lines in a turbulent atmosphere with a spatially correlated velocity field is examined. A new approach for solving this problem is proposed which is not based on the Fokker-Planck formalism. The invariant imbedding method makes it possible to reduce the problem of finding the mean radiant intensity observed in a line to solving a system of differential equations. This possibility is based on determining the mean intensity of the radiation emerging from the medium for a fixed value of the turbulent velocity at its boundary. A separate integral equation is derived for this quantity. The dependence of the line profile, integrated intensity, and width on the mean correlation length and the average value of the hydrodynamic velocity is studied. It is shown that the transition from a microturbulent regime to a macroturbulent regime occurs within a comparatively narrow range of variation in the correlation length. The proposed method yields a solution to the problem for a family of inhomogeneous atmospheres with different optical thicknesses, which makes it easy to determine the radiation field inside the turbulent medium. This approach can be generalized in various ways, in particular, it can be applied without significant changes to the case where the correlation length depends on position within the atmosphere. __________ Translated from Astrofizika, Vol. 50, No. 2, pp. 219–232 (May 2007).     

The diurnal heat budget of the thermosphere

Detailed numerical calculations of thermospheric heat sources and sinks are presented and their relative importance is discussed in reference to the energy balance phenomena of the neutral atmosphere. It is shown that the thermal energy available from the absorption in the Schumann-Runge continuum leading to photo-dissociation of O₂ is by far the largest energy source in the lower thermosphere. Other sources of varying importance in different altitude ranges are: (1) energy from photoelectrons; (2) energy exchange from thermal plasma; (3) chemical reaction (ion-electron dissociative recombination) energy gain; (4) kinetic and dissipative energy associated with the neutral wind. The energy sinks of importance are (1) thermal conduction at the lower boundary (120km); and (2) radiative cooling of atomic oxygen.It is shown that the combined energy from processes 2–4 constitutes only a small fraction of the total energy available from photoelectrons and is in phase with the latter. These secondary sources (processes 2–4), therefore, do not constitute a significant energy source and their contribution can be simply incorporated into photoelectron energy (process 1) by defining an effective photo-ionization heating efficiency. The heating efficiencies for photo-ionization (including processes 2–4) and photo-dissociation are estimated to be 0.5 and 0.3, respectively.As the important heat input (photo-dissociation) and loss (conduction and radiation) rates are basically governed by the O₂ and O densities, any diurnal or seasonal variation in these constituents at the lower boundary would have profound effects on the thermal structure of the overlying atmosphere. For this and other reasons, it is suggested that a choice of lower boundary much below 120km, e.g. near the mesopause level (90 km), should be more appropriate for general thermospheric studies.