Polarization line radiative transfer in the atmospheres of magnetic white dwarfs

Some physical mechanisms which can affect the Zeeman line profiles of magnetic white dwarfs are studied. The pure absorption polarization transfer equation is solved including these mechanisms. The broadening of lines in magnetic white dwarfs is briefly discussed.     

Rassvet: Backward Monte Carlo radiative transfer in spherical-shell planetary atmospheres

Validation of global numerical models of planetary atmospheres requires simulating images and spectra from the IR to UV spectral regions in order to compare them with remote observations. This paper describes Rassvet, a 3-D spherical-shell backward Monte Carlo radiative transfer model developed for such simulations. It utilizes a new methodology for calculating atmospheric brightness in scattered sunlight by introducing the concept of an “effective emission source”. This allows for the accumulation of the scattered contribution along the entire path of a ray and the calculation of the atmospheric radiation when both scattered sunlight and thermal emission contribute to the remote measurement - which was not possible in previous models. A “polychromatic” algorithm is extended for applications with the backward Monte Carlo method and implemented in the model. It allows for the calculation of radiative intensity for several wavelengths simultaneously, resulting in improved efficiency. The capabilities of the model are demonstrated by simulating remote measurements from the atmosphere of Io.     

A note on the variational method of Stokes and DeMarcus for radiative transfer in planetary atmospheres

A generalized functional which yields the Milne integral equation on variation and whose extremum value is proportional to the reflectivity at arbitrary emergent angle is proposed. A similar functional exists for computing the transmissivity at arbitrary emergent angle. This work is a generalization of the variational method of Stokes and DeMarcus (1971, Icarus14, 307) based on the principle of reciprocity. In the special case of trial functions that are linear in the undetermined parameters, the calculation is greatly simplified. The computational value of our variational principle is demonstrated.     

The effect of radiative transfer on shear-flow instability in the atmospheres on mars and venus

The results of two theoretical investigations concerning the destabilizing effects of radiative transfer on stably stratified shear flows are applied to the CO₂ atmospheres Mars and Venus. It is found that radiatively modified critical Richardson numbers remain below plausible atmospheric values throughout the stratospheres of both planets. Above certain altitudes, however, in the upper stratospheres of these planets (≈50 km on Mars and ≈100 km on Venus), critical Richardson numbers begin to increase significantly above the nonradiating critical value. This trend continues until, in the lower thermosphere, critical Richardson numbers eventually surpass atmospheric values. This effect could lead to observably greater turbulent mixing in the upper atmospheres of Mars and Venus than might be expected from terrestrial observation and from nonradiating theoretical calculations.     

Multidimensional radiative transfer: Applications to planetary coronae

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.     

A novel methodology for radiative transfer in a planetary atmosphere

A novel methodology for evaluating the field of anisotropically scattered radiation within a homogeneous slab atmosphere of arbitrary optical thickness is provided. It departs from the traditional radiative transfer approach in first considering that the atmosphere is illuminated by an isotropic light source. From the solution of this problem, it subsequently proceeds to that for the more conventional case of monodirectional illumination. The azimuthal dependence of the field is separated in the usual manner by an harmonic expansion, leaving a problem in four dimensions (=optical depth, ₀=thickness, , =directions of incidence and scattering) which, as is well known, is numerically extremely inconvenient. Two auxiliary radiative transfer formulations of increasing dimensionality are considered: (i) a transfer equation for the newly introduced functionb ^m(,,₀) with Sobolev's function ^m(,₀) playing the role of a source-function. Because the incident direction does not intervene, ^m is simply expressed as a single integral term involvingb ^m. For bottom illumination, an analogous equation holds for the other new functionh ^m(,,₀). However, simple reciprocity relations link the two functions so that it is only necessary to considerb ^m; (ii) a transfer equation for the other new functiona ^m(,,,₀) with a source-function provided by Sobolev's functionD ^m(,,₀). For bottom illumination, another functionf ^m(,,,₀) is introduced; by a similar argument using reciprocity relations,f ^m is reduced toa ^m rendering necessary only the consideration ofa ^m. However, a fundamental decomposition formula is obtained which shows thata ^m is expressible algebraically in terms of functions of a single angular variable. The functions ^m andD ^m are shown to be the values in the horizontal plane ofb ^m anda ^m, respectively. The other auxiliary functionsX ^m andY ^m are also expressed algebraically in terms ofb ^m. These results enable one to proceed to the final step of evaluating the radiation field for monodirectional illumination. The above reductions toalgebraic relations involving only the functionb ^m appear to be more advantageous than Sobolev's (1972) recent approach; they also circumvent some basic numerical difficulties in it. We believe the present approach may likewise prove to be superior to most (if not all) other methods of solution known heretofore.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory under Contract No. NAS-7-100 sponsored by the National Aeronautics and Space Administration.     

On nonstationary radiative transfer in a spectral line in stellar atmospheres

We consider nonstationary radiative transfer in a line in stellar atmospheres simulated as a stationary semi-infinite plane-parallel medium. We assume complete frequency redistribution in the elementary act of scattering. We assume that the time a photon spends in the medium is determined only by the mean time spent in the absorbed state. We obtain an explicit expression for the resolvent of the nonstationary integral equation of transfer, which is a bilinear expansion with respect to the eigenfunctions found in [12] for the corresponding stationary transfer equation.Translated fromAstrofizika, Vol. 37, No. 3, 1994.I am grateful to the American Astronomical Society for financial support of this work.     

Invariant imbedding and Chandrasekhar's planetary problem of radiative transfer

In connection with Chandrasekhar's planetary problem of radiative transfer the total scattering and the diffuse transmission functions have been discussed by several authors (cf. Chandrasekhar, 1950; van de Hulst, 1948; Sobolev, 1948; Bellman,et al., 1967; Kagiwada and Kalaba, 1971). With the aid of the Bellman-Krein formula for the resolvent kernel of the auxiliary equation governing the source function, we show how the invariant imbedding equations governing the diffuse scattering and transmission functions can readily be obtained. So far as we know, the Cauchy system of the functional equations for the scattering and transmission functions is new and is well-suited for the numerical computation.Supported by the National Science Foundation under Grant No. GP 29049, and by the Atomic Energy Commission under Grant No. AT (40-3)-113 Project 19.     

Rayleigh scattering in planetary atmospheres

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.     

Multiple scattering in planetary atmospheres

Methods for solving radiative transfer problems within the extended visible spectrum in planetary atmospheres are reviewed for use by the nonspecialist. Emphasis is placed on rapid, approximate procedures for the determination of such quantities as the plane and spherical (Bond) albedo, surface illumination, absorbed energy, limb darkening, phase curve, and spectra. Precise numerical methods and analytical results are also discussed. Recent approaches to such complications as atmospheric inhomogeneity and reflection from a porous regolith are described briefly.     

Lightning generation in planetary atmospheres

The possibilities of lightning generation on other planets are considered, and the basic conditions that exist in terrestrial clouds during lightning discharges and the various theories of charge separation are reviewed. Recent measurements of cloud structure and whistlers, as well as optical observation of lightning on Jupiter, suggest that charge separation and lightning discharges occur on other planets in ways similar to those in which they occur on Earth. Using these terrestrial ideas, it is concluded that lightning on Venus will probably be found in clouds that are located in regions of convection such as those observed downwind of the subsolar point. It is also possible that if volcanoes on Venus are erupting, they too can produce lightning discharges in their plumes although it seems unlikely that this process can account for the observed rate of discharge. Jovian lightning is most probably generated in the lower water-ice clouds. These clouds are of moderate temperatures and have strong convection and large mass loading, all important ingredients for electrical buildup. Lightning is all but ruled out for Mars, even though some electrification is possible owing to the large dust storms on that planet.     

Abundances of isotopes in planetary atmospheres

Carbon and oxygen isotopes show no large anomalies on Venus (10–15%) or Mars (<5%); the high value of¹⁵N/¹⁴N found on Mars is explained by non-thermal escape of nitrogen. The isotopes of non-radiogenic noble gases in the atmosphere of Mars exhibit abundance patterns similar to those in the primordial component of meteoritic gases and in the Earth's atmosphere. This implies that gas fractionation took place in the inner solar nebula prior to planet formation. The relatively high value of¹²⁹Xe on Mars emphasizes its deficiency on Earth, implying a difference in accretion histories of volatiles for the two planets. In the outer solar system, we find normal isotope ratios for nitrogen and carbon on Jupiter, and for carbon on Saturn, but precision is low (±15% at best). Controversy exists about the correct value of D/H, with current estimates ranging from 2.3±1.1 to 5.1±0.7×10^–5. Planetary missions planned for the next few years should add considerably to the quantity and quality of these data.Paper presented at the Conference on Protostars and Planets, held at the Planetary Science Institute, University of Arizona, Tucson, Arizona, between January 3 and 7, 1978.     

Diffusion of radiation in planetary atmospheres

The problem of interaction of the solar radiation with the turbid Earth atmosphere, containing complicated polydispersive aerosol systems, is discussed in this paper. Equations for computing the angular functions ofn-th order scattering are derived. On the basis of these functions the spectral radiance, radiation flows and radiation balance of the atmosphere in the short-wave spectral range are calculated. The relations obtained can be used to calculate the complex index of refraction, distribution function and other characteristics of the submicron aerosol fraction, by solving the inverse problems.     

Formulation of molecular diffusion in planetary atmospheres

We report on a formulation of molecular diffusion for ionized multi-component atmospheres that is valid in the diffusion and small electron mass limits. The formulation is based on the construction of successive approximations of the diffusion matrix by means of the projective iterative algorithm of Ern and Giovangigli [Projected iterative algorithms with application to multicomponent transport. Lin. Alg. and its Appl. 250, 289–315], and allows naturally for different temperatures for the neutral, ion and electron constituents of the gas. The reported expressions incorporate the effect of electric forces preventing charge separation, are explicit in the driving forces and mass conservative. Yet approximate, their accuracy can be easily tested and improved upon by going to a higher approximation of the diffusion matrix. We have illustrated the formulation with a model that solves the composition of Mars’ atmosphere. The continuity equations of the model are linearized and marched in time with an implicit numerical scheme, allowing thus for large time steps. It is found that the first and second approximations of the diffusion matrix are probably optimal trade-offs between computational cost and accuracy. Finally, the formulation is tested against more conventional approximations of the molecular diffusion velocities of neutral and ion species, showing the importance of the various assumptions that may restrict their applicability.     

Spectroscopy of planetary atmospheres in our Galaxy

About 20 years after the discovery of the first extrasolar planet, the number of planets known has grown by three orders of magnitude, and continues to increase at neck breaking pace. For most of these planets we have little information, except for the fact that they exist and possess an address in our Galaxy. For about one third of them, we know how much they weigh, their size and their orbital parameters. For less than 20, we start to have some clues about their atmospheric temperature and composition. How do we make progress from here?We are still far from the completion of a hypothetical Hertzsprung–Russell diagram for planets comparable to what we have for stars, and today we do not even know whether such classification will ever be possible or even meaningful for planetary objects. But one thing is clear: planetary parameters such as mass, radius and temperature alone do not explain the diversity revealed by current observations. The chemical composition of these planets is needed to trace back their formation history and evolution, as happened for the planets in our Solar System. As in situ measurements are and will remain off-limits for exoplanets, to study their chemical composition we will have to rely on remote sensing spectroscopic observations of their gaseous envelopes.In this paper, we critically review the key achievements accomplished in the study of exoplanet atmospheres in the past ten years. We discuss possible hurdles and the way to overcome those. Finally, we review the prospects for the future. The knowledge and the experience gained with the planets in our solar system will guide our journey among those faraway worlds.     

Rayleigh-Cabannes scattering in planetary atmospheres III. The Milne problem in conservative atmospheres

The discrete ordinale method by Chandrasekhar is used to solve the conservative Milne problem in a homogeneous plane-parallel atmosphere which scatters the radiation according to the Rayleigh-Cabannes law.The approximate solution which is supposed to converge uniformly to an exact one when increasing the order of approximation is obtained explicitly. In addition to a tabulation of the Hopf vector for different factors of depolarization, the extrapolation distance, the values of c, q and the Rubenson degrees of polarization at the limb are given.     

Rayleigh-Cabannes scattering in planetary atmospheres II. Constant internal sources in nonconservative atmospheres

The vector equation of radiative transfer is solved for non-conservative homogeneous plane-parallel atmosphere using the method of discrete ordinates. The scattering processes in the atmosphere bounded by a Lambert bottom are described by the Rayleigh-Cabannes phase matrix. The primary radiation field is generated by constant internal sources. A package of FORTRAN subroutines is compiled to find the axial radiation field for such an atmosphere at arbitrary optical depth.