Mean number of photon scatterings in a spherical shell

The expression for the mean number of photon scattering in absorbing and scattering media, when the media is a spherical shell of finite thickness, has been derived. Also the expression of probability of such emergence of photons absorbed is given here.     

Polarization in spectral lines

The general formalism, presented in a previous paper of this series (Landi Degl'Innocenti, 1983a), is particularized to deduce the radiative transfer equations for polarized radiation and the statistical equilibrium equations for a multi-level atom in the zero-magnetic field, collisionless regime. The formulae are developed both in the standard representation and in the representation of the statistical tensors. For resonance scattering in a two-level atom, in the limiting case of complete depolarization of the ground level, we recover the classical results for Rayleigh scattering and we derive the expression of the phase matrix in terms of ordinary rotation matrices. The law of scattering is then generalized to take properly into account the influence of the anisotropy of the radiation field on the atomic polarization of the ground level (depopulation pumping).     

Compton attenuation coefficient in scattering by maxwellian electrons

Equations for the coefficient of attenuation of radiation due to Compton scattering by a thermal electron gas with an arbitrary temperature are summarized. A new representation is also obtained for the integral through which the attenuation coefficient, averaged over a relativistic, Maxwellian energy distribution of electrons, is expressed. This representation enables one to efficiently calculate the coefficient for a hightemperature electron gas. The accuracy of an approximate expression for the attenuation coefficient, corresponding to the assumption that scattering is isotropic in the rest frame of an electron, is also estimated. Translated from Astrofizika, Vol. 43, No. 3, pp. 473-482, July–September, 2000.     

Comments on ‘the angle-dependent redistribution functionsR III andR IV’ by S. J. McKenna

It is shown that the complicated form of the line absorption probability function for scattering in subordinate lines, derived by McKenna (1984), is an artifact of adopting a mathematically inconvenient expression for the corresponding redistribution function. It is demonstrated that the absorption probability must be given by the Voigt function.     

Scattering physics

The theory of polarized scattering in a stellar atmosphere is formulated, first within the framework of classical physics, then in terms of quantum mechanics. The expression for the redistribution matrix that describes partial redistribution in polarization and frequency is derived for the general case when the magnetic field is of arbitrary strength. The special cases of weak fields (the Hanle limit) and zero fields (non-magnetic scattering) are discussed. Observational examples of spectral signatures in linear polarization are presented, which show effects of hyperfine structure, interference between fine structure components, and molecular scattering.     

Radiative transfer in an infinite cylinder

The problem of radiation transfer in a cylinder with diffuse reflectivity and containing an energy source is connected with the source-free radiation transfer problem with isotropic boundary condition. Equation for the radiation heat flux is obtained for a polynomial source. In the special case of isotropic scattering, the radiation heat flux is given in terms of the albedo of the second problem. An expression is also given for the net radiation heat flux.     

Particle flux associated with stochastic processes

The Lagrange expansion, which may be used to derive the Fokker-Planck equation, is here used to derive the corresponding expression for the flux of particles subject to a stochastic scattering process. The coefficients which occur in this expression are, in general, not the same as the coefficients which occur in the Fokker-Planck equation itself. In the special case that the particle distribution involves only one independent variable, the particle flux is determined by the familiar Fokker-Planck coefficients. Evaluation of particle flux is of special interest in the study of stochastic acceleration.     

Electron precipitation in a non-uniform glow aurora

Electron intensities at 5 keV >18 keV and >45 keV, were measured on a Petrel rocket flown from Kiruna, Sweden, into a non-uniform glow aurora during the recovery phase of a magnetic bay. The intensities depend on pitch-angle in a way that is consistent with the precipitation being caused by pitch-angle diffusion from reservoirs of geomagnetically-trapped electrons. The scattering process that causes pitch-angle diffusion, and leads to three regions of relatively high intensity, appears to have properties different from the scattering process that leads to two intervening regions of low intensity. A spatial structure in electron reservoir intensity, is attributed mainly to variations in the rate of erosion by pitch-angle diffusion of an initially nearly-uniform reservoir intensity. An expression is derived for the minimum lifetime of trapped electrons undergoing strong pitch-angle diffusion.     

The radiation force on small particles

A more complete expression for the radiation force on a small particle in the solar system is given which includes the effect of asymmetry of the thermal reradiation and also of inelastic scattering such as fluorescence. Both the Poynting-Robertson drag and the Yarkovsky effect are affected by such asymmetries and are incorporated into the formalism. For non-spherical particles the direction of the radiation force will no longer coincide with the solar irradiation.     

A new method for exact solution of transfer equations in finite media

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.     

Radiative bulk viscosity

Viscous resistance to changes in the volume of a gas arises when different degrees of freedom have different relaxation times. Collisions tend to oppose the resulting departures from equilibrium and, in so doing, generate entropy. Even for a classical gas of hard spheres, when the mean free paths or mean flight times of constituent particles are long we find a non-vanishing bulk viscosity. Here we apply a method recently used to uncover this result for a classical rarefied gas to radiative transfer theory, and derive an expression for the radiative stress tensor for a grey medium with absorption and Thomson scattering. We determine the transport coefficients through the calculation of the comoving entropy generation. When scattering dominates absorption, the bulk viscosity may be much larger than either the shear viscosity or the thermal conductivity.     

SCR Distribution Function in the Radial IMF Approximation

The transport of cosmic rays in the interplanetary medium is considered in terms of the kinetic equation describing the energetic particle scattering by magnetic irregularities and their focusing by the regular interplanetary magnetic field. The analytical expression for solar cosmic ray distribution function in the approximation of radial regular magnetic field is obtained and the evolution of energetic particle angular distribution is analyzed. The obtained results can be used for the analysis of ground-level enhancements of cosmic ray intensity.     

A simple approximation forJ v (σ v ,κ v ,τ v,B v ), and its application for temperature-correction procedures

For the case of isotropic coherent scattering plus absorption a simple expression is given (Equation 12) to compute the mean intensity of the radiationJ _v (as a function of optical depth _v ) if the scattering coefficient _v , the absorption coefficientK _v and the Planck functionB _v are given as a function of depth. In general the accuracy of this approximation is of the order of a few percent.A fairly simple temperature-correction procedure for the case when scattering is important is described.     

International conference on planetary satellites

Further analysis of visual (V) wavelength photometric function data for Saturn's ring is presented. Evidence indicating both that primary scattering dominates, and that mutual shadowing is an irrelevant concept, is reviewed. Simple anisotropic scattering radiative transfer models are used to define the probable ranges in the single scattering albedo, and in the general shape of the scattering phase function of the individual particles. Limitations on the mean perpendicular optical thickness of the ring are also obtained. Results indicate that the ring particles are highly efficient back-scatterers of visual radiation. Macroscopic particles account for the basic shape of the scattering phase function. Based on an infinite optical thickness for the ring, a minimum single scattering albedo ≈0.75 is found. Use of conservative scattering leads to a minimum optical thickness ≈0.7. The analysis is consistent with the ring particles being centimeter-size pieces of ice.     

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.     

Polarization Models for Rayleigh Scattering Planetary Atmospheres

We present Monte Carlo simulations for the polarization of light reflected from planetary atmospheres. We investigate dependencies of intensity and polarization on three main parameters: single scattering albedo, optical depth of a scattering layer, and albedo of a Lambert surface underneath. The main scattering process considered is Rayleigh scattering, but isotropic scattering and enhanced forward scattering on haze particles are also investigated. We discuss disk integrated results for all phase angles and radial profiles of the limb polarization at opposition. These results are useful to interpret available limb polarization measurements of solar system planets and to predict the polarization of extra-solar planets as a preparation for VLT/SPHERE. Most favorable for a detection are planets with an optically thick Rayleigh-scattering layer. The limb polarization of Uranus and Neptune is especially sensitive to the vertically stratified methane mixing ratio. From limb polarization measurements constraints on the polarization at large phase angles can be set.     

Effects of scattering and dust grain size on the temperature structure of protoplanetary discs: a three-layer approach

The temperature in the optically thick interior of protoplanetary discs is essential for the interpretation of millimetre observations of the discs, for the vertical structure of the discs, for models of the disc evolution and the planet formation, and for the chemistry in the discs. Since large icy grains have a large albedo even in the infrared, the effect of scattering of the diffuse radiation in the discs on the interior temperature should be examined. We have performed a series of numerical radiation transfer simulations, including isotropic scattering by grains with various typical sizes for the diffuse radiation as well as for the incident stellar radiation. We also have developed an analytic model including isotropic scattering to understand the physics concealed in the numerical results. With the analytic model, we have shown that the standard two-layer approach is valid only for grey opacity (i.e. grain size ≳10 μm) even without scattering. A three-layer interpretation is required for grain size ≲10 μm. When the grain size is 0.1–10 μm, the numerical simulations show that the isotropic scattering reduces the temperature of the disc interior. This reduction is nicely explained by the analytic three-layer model as a result of the energy loss by scatterings of the incident stellar radiation and of the warm diffuse radiation in the disc atmosphere. For grain size ≳10 μm (i.e. grey scattering), the numerical simulations show that the isotropic scattering does not affect the interior temperature. This is nicely explained by the analytic two-layer model; the energy loss by scattering in the disc atmosphere is exactly offset by the 'green-house effect' due to the scattering of the cold diffuse radiation in the interior.     

Scattering matrices and expansion coefficients of martian analogue palagonite particles

We present measurements of ratios of elements of the scattering matrix of martian analogue palagonite particles for scattering angles ranging from 3° to 174° and a wavelength of 632.8 nm. To facilitate the use of these measurements in radiative transfer calculations we have devised a method that enables us to obtain, from these measurements, a normalized synthetic scattering matrix covering the complete scattering angle range from 0° to 180°. Our method is based on employing the coefficients of the expansions of scattering matrix elements into generalized spherical functions. The synthetic scattering matrix elements and/or the expansion coefficients obtained in this way, can be used to include multiple scattering by these irregularly shaped particles in (polarized) radiative transfer calculations, such as calculations of sunlight that is scattered in the dusty martian atmosphere.     

The effects of scattering on quiet Sun emission at frequencies less than 200 MHz

The problem of predicting the radio emission from the quiet Sun for meter and decameter waves may be formulated in terms of a standard radiative transfer problem with conservative scattering. One may therefore avoid the numerical complications involved in using a ray-tracing approach which incorporates a Monte-Carlo routine for representing scattering. The brightness and extent of the radio Sun are calculated for several values of the electron temperature and scattering parameter of Steinberg et al. (1971). It is concluded that the temperature and density model of Newkirk (1967) for solar maximum reasonably represents the observations. However, some observations appear to be inconsistent with scattering models and further observations are needed. It is shown (in Appendix B) that the standard ray-tracing technique incorporating a Monte-Carlo routine for the scattering may be replaced by a diffusion approximation.     

Solar system observations by the Wisconsin Ultraviolet Photopolarimeter Experiment– III. The first ultraviolet linear spectropolarimetry of the Moon

Portions of the Moon were observed by the Wisconsin Ultraviolet Photopolarimeter Experiment ( WUPPE ) on 1995 March 12, 14 and 17, and represent the first ultraviolet (UV) spectropolarimetric observations of the Moon. The polarimetric observations confirm that a change in the dominant scattering process occurs in the UV, changing from volume scattering in the near-UV to surface scattering in the far-UV. The data are investigated empirically. It is found that Umov's relationship holds when the polarization is perpendicular to the scattering plane. It is also found that the degree of polarization can be modelled by a phase-angle-dependent polarization modified by a wavelength-dependent depolarization factor. The scattering function for each observation is determined.