Stochastic transfer of polarized radiation in finite cloudy atmospheric media with reflective boundaries

The problem of monoenergetic radiative transfer in a finite planar stochastic atmospheric medium with polarized (vector) Rayleigh scattering is proposed. The solution is presented for an arbitrary absorption and scattering cross sections. The extinction function of the medium is assumed to be a continuous random function of position, with fluctuations about the mean taken as Gaussian distributed. The joint probability distribution function of these Gaussian random variables is used to calculate the ensemble-averaged quantities, such as reflectivity and transmissivity, for an arbitrary correlation function. A modified Gaussian probability distribution function is also used to average the solution in order to exclude the probable negative values of the optical variable. Pomraning-Eddington approximation is used, at first, to obtain the deterministic analytical solution for both the total intensity and the difference function used to describe the polarized radiation. The problem is treated with specular reflecting boundaries and angular-dependent externally incident flux upon the medium from one side and with no flux from the other side. For the sake of comparison, two different forms of the weight function, which introduced to force the boundary conditions to be fulfilled, are used. Numerical results of the average reflectivity and average transmissivity are obtained for both Gaussian and modified Gaussian probability density functions at the different degrees of polarization.     

Partial frequency redistribution of polarized radiation

Resonance polarization, which is created by the scattering of an anisotropic radiation field in regions of zero or weak magnetic fields, is strongly dependent on the frequency redistribution taking place during the scatterings. Here we discuss the frequency redistribution matrix relevant to resonance lines, concentrating on linear polarization. First we analyze in detail the redistribution matrix in a zero magnetic field given by the theory of Omont, Smith and Cooper (1972), revisited by Domke and Hubeny (1988). We explain that the linear polarization maxima which may appear in the wings of the Stokes Q profiles of strong resonances lines such as the Ca I 4227 Å line are coherent frequency redistribution effects. Various approximate forms of the frequency redistribution matrix are also examined. For resonance polarization in a weak magnetic field, we suggest a new expression for the redistribution matrix which can be used at all line frequencies, and is consistent with the condition that the Hanle effect acts only in the line core.     

On the solution of the radiative transfer equations for polarized radiation

A general formalism is presented to obtain an analytical solution of the radiative transfer equations for polarized radiation when the absorption Mueller matrix for the Stokes parameters is constant along the ray-path. The formalism is then applied to find an analytical approximation of the Stokes parameters profiles for a typical chromospheric line having an optical-depth dependence of the source function of the form S() = ( + )^1/2(1 + )^1/2.     

The transfer of polarized radiation in spectral lines: Solar-type stellar atmospheres

The polarization structure in several spectral lines in solar type stars is computed using the method described by McKenna (1981, 1984a). The frequency redistribution function used for these calculations is a linear combination ofR _II andR _III. The line profiles and polarization structures have been computed for several weak solar resonance lines includingKi 7664 Å, Sri 4607 Å, Baii 4554 Å, for various polar angles along the stellar disk. Both the line profiles and polarization structures as well as the center to limb behavior of the line center polarization agree well with observations.The somewhat stronger resonance line Cai 4227 Å shows a different polarization structure when compared to the weaker solar resonance lines. It is found that for strong resonance lines the proper redistribution function to be used is a linear combination ofR _III andR _v (see McKenna, 1981, 1984b; Heinzel, 1981). The major reason for this is that for strong resonance lines both the upper and lower levels are broadened by collisions. This violates the assumptions upon which the redistribution functionsR _II andR _III are based.     

Interactions of plasmas with magnetic field boundaries

The magnetopause, the boundary layer, or current sheath, which separates the magnetosphere from the solar wind, is the particular interaction considered in this paper.The collision free electron skin depth, $\text{ξ}{}_{\mathrm{e}}\text{=}\text{c}\text{ω}{}_{\mathrm{pe}}$, where c is the velocity of light and ω_pe, is the plasma frequency, gives a classical measure of the penetration depth of a collisionless plasma by an electromagnetic field. This penetration depth is small compared with the dimensions of the magnetosphere and hence the boundary layer may be conveniently considered in one dimension.In General all one dimensional solutions lie within an order of magnitude of the value of ξ_e, the only exception being the important one, in which the electric field perpendicular to the current sheath plane is not present, either due to a particular trapped particle distribution or due to a short circuiting end effect. For this exception the thickness is increased by the factor ($\text{m}{}_{\mathrm{i}}\text{i/m}{}_{\mathrm{e}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.The current sheath solutions discussed are equilibrium solutions but not necessarily stable equilibrium solutions.The extension of the models to three dimensions has a larger effect than might at first be expected. The effect may be intuitively understood as a consequence of flux conservation in the sheath. The one dimensional solutions then correspond to the current sheath profiles at the thinnest point of the three dimensional sheath.     

Self-excitation effect in the scattering of electromagnetic radiation in a strong magnetic field

It is pointed out that the electron scattering of incident electromagnetic radiation of frequency ω_i includes also a component with the cyclotron frequency ω_L.Except in the limit ω_i ⪢ ω_L, this “self-excited effect” modifies appreciably the cross section. The modified cross section for a classical non-relativistic electron is given in this paper for both a single electron and a cold plasma. The spectral distribution of electron scattering is discussed and the Thomson absorption coefficient of a magnetized thermal equilibrium plasma is obtained for some typical cases. The results obtained here may be relevant to astrophysical processes, such as the spectrum formation of x-ray pulsars.     

Scattering and transmission functions of radiation by finite atmospheres with reflecting surfaces

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.     

The effect of the radiation pressure on the orbital evolution of geosynchronous objects

The effect of the radiation pressure and Poynting-Robertson effect on the evolution of the orbits of geosynchronous satellites is studied, depending on their area to mass ratio. The qualitative changes of the orbital evolution caused by these disturbances are considered. The reflection coefficient of the satellite’s surface was assumed to be 1.44. In the vicinity of the stable point with the longitude of 75° the exit from the libration resonance mode was registered when the area to mass ratio value changed from 5.9 to 6.0 m²/kg; in the vicinity of the unstable point at 345° with the area to mass ratio of 1.4 it occurred at 1.5 m²/kg. Re-entry to Earth occurs at values of the area to mass ratio above 32.2 m²/kg, and hyperbolic exit from the low-Earth orbit occurs at values of the area to mass ratio over 5267 m²/kg. At high values of the area to mass ratio, slopes of initially equatorial orbits can reach 49°. It is shown that due to the Poynting-Robertson effect the secular decrease in the semimajor axis of orbit in libration resonance region is 3–4 orders of magnitude less than outside of it.     

The effect of non-uniform magnetic field on the slow mode oscillations

In this paper, the slow MHD mode oscillations of the coronal plasma are studied. The aim is to identify the effect of structuring (such as magnetic field, temperature, density, and pressure) on the frequencies of oscillations. We modelled the coronal medium as a low-β plasma with longitudinally density and pressure stratifications and a weakly inhomogeneous magnetic field varied slowly with height and radial directions. The linearized ideal MHD equations reduced to a single Klein–Gordon differential equation for square of oscillatory frequencies. The eigenfunctions and analytical dispersion relations are derived. The dispersion relations were solved numerically. In the case of uniform magnetic field, the previous studies verified. Our numerical results show that, the frequencies and their ratios are very sensitive functions of pressure scale height, and slightly varying functions of inhomogeneity parameter of magnetic field. By changing the magnetic field strength between the apex and footpoints of the loop about 50%, the frequencies ratio are changed about 5%. We concluded that, the pressure scale height and temperature gradient are first order effects and inhomogeneity of magnetic field is a second order effect on the slow mode oscillations.     

The effect of Joule dissipation on field line resonance

Damped hydromagnetic eigenmodes are calculated numerically for a simple inhomogeneous plasma that is assumed to be contained within rigid perfectly conducting walls, and subject only to Joule dissipation. It is found that E_∥ must be included in order to obtain well-behaved damped eigenmodes that include resonant field lines, even though E_∥ is relatively very small. The thickness of the resonant region in the equatiorial plane, estimated from the model, is of order 10^?3 of an L number, which seems about two orders of magnitude too small to match observed long period micropulsations. The fact that mid-latitude micropulsations sometimes lead in phase at the lower latitudes is shown probably to be an effect of the local increase in resonant period with latitude.     

The effect of photon capture and field ionisation on high magnetic field pulsars

We have investigated the formation of polar gaps in pulsars as envisaged by Ruderman and Sutherland (1975) in the presence of photon conversion into positronia (Shabad and Usov, 1985; Heroldet al., 1985) and dissociation of positronia by strong electric fields. For the emission of curvature radiation near the pulsar's surface, we have used formulae which are valid in the extreme relativistic regime. We find that the gap formation is not affected, only the gap height is generally slightly increased. In the sample of pulsars that we have studied, in most cases there is likely to be enough plasma in the gap to initiate processes producing radio emission in accordance with the theory. In some cases we found the possibility of emission of high-energy photons outside the pulsar magnetosphere, as was suggested earlier by us (Bhatiaet al., 1987) in conformity with some of the 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.     

The ionizing radiation field in the galactic centre and the nature of LINERs

The Galactic centre contains a low-ionization nebula that has been previously interpreted as plasma photoionized by relatively cool O stars. We consider the possibility that this material is instead ionized by more energetic continua and described by a relatively low ionization parameter (ratio of ionizing photon to nucleon density). We find that the predicted spectrum is more sensitive to ionization parameter and cloud density than to details of the ionizing continuum shape. The Galactic centre spectrum can be generated by stars with a range of temperatures irradiating clouds with multiple components of differing densities. The present calculations also show that optical emission from many LINERs (low-ionization nuclear emission-line regions) can be generated by relatively hot, yet normal O stars embedded in gas clouds similarly characterized by a range of densities.     

The effect of solar radiation pressure on the orbit of a cylindrical satellite

The effect of solar radiation pressure on the orbits of cylindrical satellites is considered. The cylinder is assumed to reflect radiation both specularly and diffusely. The resultant forces on a stationary cylindrical satellite are given. The evolution of the satellite's orbit is described for two particular modes of rotation. In both cases the satellites are assumed to be in circular Sun-synchronous orbits.     

Order-of-scattering of partially polarized radiation in inhomogeneous,anisotropically scattering atmospheres

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.     

Order-of-scattering of partially polarized radiation in inhomogeneous,anisotropically scattering atmospheres

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.