Radiative transfer in spherical shell atmospheres: I. Rayleigh scattering

A comparison is made between the plane-parallel approximation and the more realistic spherical shell approximation for the radiance reflected from a planetary atmosphere. In this paper we have considered a planet of radius 6371 km (the Earth) with a homogeneous, conservative, Rayleigh scattering atmosphere extending to a height of 100 km. We have found significant departures from the plane-parallel approximation. Radiance versus height distributions for both single and multiple scattering are presented. Results are presented for the fractional radiance from altitudes in the atmosphere which contribute to the total unidirectional reflected radiance at the top of the atmosphere. We have referred to this as the radiance versus height distribution in the sequel. These data will be very useful for both remote sensing applications and planetary spectroscopy. We have also found that gross violations of the principle of reciprocity do occur in the spherical shell approximation.     

Radiative transfer in spherical shell atmospheres: II. Asymmetric phase functions

In this paper we investigate the effects of sphericity on the radiation reflected from a planet with a homogeneous, conservative scattering atmosphere of optical thicknesses of 0.25 and 1.0. We considered a Henyey-Greenstein phase function with asymmetry factors of 0.5 and 0.7. Significant differences were found when these results were compared with the plane-parallel calculations. Also, large violations of the reciprocity theorem, which is only true for plane-parallel calculations, were noted. Results are presented for the radiance versus height distributions as a function of planetary phase angle. These results will be useful to researchers in the field of remote sensing and planetary spectroscopy.     

Radiative transfer in spherical shell atmospheres. III. Application to Venus

We have introduced a method of partitioning the radiance emerging from a planetary atmosphere in proportion to the average number of scatterings in each atmospheric layer in order to gain a more fundamental understanding of the so-called level of line formation. A realistic model of the Venus atmosphere was used to compute the radiance for a range of phase angles and two planetary colatitudes, namely, 20 and 90°. We computed the core and continuum radiances for the P(16) line of 8689-Å CO₂ band and introduced two ways of computing an effective temperature. Both definitions yielded similar results. We found that these effective temperatures varied little with phase angles up to 120°, but fell rather rapidly beyond this point. Also colder effective temperatures were found as we went from equator to pole. The results obtained are all consistent with the spectroscopic temperature determination from CO₂ band studies. We have also defined an effective optical depth, τ_eff, which we feel gives a better understanding of the level of line formation than other definitions used to date.     

Radiative transitions of the helium atom in highly magnetized neutron star atmospheres

Recent observations of thermally emitting isolated neutron stars revealed spectral features that could be interpreted as radiative transitions of He in a magnetized neutron star atmosphere. We present Hartree–Fock calculations of the polarization-dependent photoionization cross-sections of the He atom in strong magnetic fields ranging from 10¹² to 10¹⁴ G. Convenient fitting formulae for the cross-sections are given along with the related oscillator strengths for various bound–bound transitions. The effects of finite nucleus mass on the radiative absorption cross-sections are examined using perturbation theory.     

Radiative transfer in a semi-infinite plane parallel atmosphere with an inhomogeneous source function

We investigate the multiple scattering of radiation in semi-infinite homogeneous atmosphere when the sources of the radiation are distributed inhomogeneous, for example, are created by restricted beams penetrating into the medium. The case of isotropic scattering is considered. It is shown that the density of radiation and the intensity of outgoing radiation for any forms of the sources can be represented as some integrals with the real and imaginary parts of the universal H-function, which satisfies the nonlinear integral equation. We calculated the intensity of radiation emerging from the surface after multiple scattering for the case when a beam with a finite radius incident perpendicular on the medium surface. The results allowed us to estimate quantitatively when the intensity of outgoing radiation in the center of a beam coincides with that for the classical case of unbounded flux (the case considered by Chandrasekhar et al.). We compared our exact solutions with those in the diffusion approximation. For conservative medium the difference is ?20–30%, depending on the particular forms of the radiation sources. For absorbing medium the difference is much larger. Our exact semi-analytical solution can be generalized for the cases of multiple anisotropic scattering of the polarized beams. The presented simple theory can be used at the consideration of close binary systems, flare stars etc.     

Average photon path-length in isotropically scattering finite atmospheres

The determination of the average path-length of photons in a finite isotropically scattering plane-parallel homogeneous atmosphere is discussed. To solve this problem we have used the kernel approximation method which easily allows us to find the derivatives of the intensity with respect to optical depth, optical thickness and albedo of single scattering.In order to check the results we have used another approach by exploiting the set of integrodifferential equations of Chandrasekhar for theX- andY-functions. This approach allows us to find the average path length only at the boundaries of the atmosphere but on the other hand it gives also the dispersion of the path-length distribution function, thus generating the input parameters for determining the approximate path-length distribution function. It occurred that the set so obtained is stable and the results are highly accurate.As a by-product we obtain the first two derivatives of theX- andY-functions with respect to the albedo of single scattering and optical thickness, and the mixed derivative.     

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.     

Diffuse reflection and transmission of radiation by finite atmospheres with specular reflectors

It is supposed that solar rays are incident uniformly on the top of a plane-parallel, anisotropically scattering, and homogeneous atmosphere, whose bottom is bounded by a specular reflector. An initial-value solution for the scattering and transmission functions is obtained, starting with an auxiliary equation. Some results of numerical experiments for given optical parameters are presented in Figures.     

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.     

Hamilton's equations of motion for non-conservative systems

This paper deals with the Hamilton equations of motion and non conservative forces. The paper will show how the Hamilton formalism may be expanded so that the auxiliary equations for any problem may be found in any set of canonical variables, regardless of the nature of the forces involved. Although the expansion does not bring us closer to an analytical solution of the problem, it's simplicity makes it worth noticing.The starting point is a conservative system (for instance a satellite orbiting an oblate planet) with a known Hamiltonian (K) and canonical variables {Q, P}. This system is placed under influence of a non-conservative force (for instance drag-force). The idea is then to use, as far as possible, the same definitions used in the conservative problem, in the process of finding the auxiliary equations for the perturbed system.     

Radiative transfer in atmospheres of Algol-type binaries. Departures from the state of LTE in late-type components

Departures from the state of LTE in the atmospheres of the secondary components of Algol-type binaries due to anisotropy of irradiation are investigated. Temperature inversion and the abnormal law of limb darkening are the most serious consequences of anisotropy of irradiation if the ratio of the effective temperatures of the components exceeds 2–2.5, providing that collisions can be neglected.Originally presented at the IAU Colloquium No. 16, held at the University of Pennsylvania, Philadelphia, Pa., U.S.A., September 8–11, 1971.     

Noble gases in planetary atmospheres: Implications for the origin and evolution of atmospheres

The radiogenic and primordial noble gas content of the atmospheres of Venus, Earth, and Mars are compared with one another and with the noble gas content of other extraterrestial samples, especially meteorites. The fourfold depletion of ⁴⁰Ar for Venus relative to the Earth is attributed to the outgassing rates and associated tectonics and volcanic styles for the two planets diverging significantly within the first billion or so years of their history, with the outgassing rate for Venus becoming much less than that for the Earth at subsequent times. This early divergence in the tectonic style of the two planets may be due to a corresponding early onset of the runaway greenhouse on Venus. The 16-fold depletion of ⁴⁰Ar for Mars relative to the Earth may be due to a combination of a mild K depletion for Mars, a smaller fraction of its interior being outgassed, and to an early reduction in its outgassing rate. Venus has lost virtually all of its primordial He and some of its radiogenic He. The escape flux of He may have been quite substantial in Venus' early history, but much diminished at later times, with this time variation being perhaps strongly influenced by massive losses of H₂ resulting from efficient H₂O loss processes.Key trends in the primordial noble gas content of terrestial planetary atmospheres include (1) a several orders of magnitude decrease in ²⁰Ne and ³⁶Ar from Venus to Earth to Mars; (2) a nearly constant ²⁰Ne/³⁶Ar ratio which is comparable to that found in the more primitive carbonaceous chondrites and which is two orders of magnitude smaller than the solar ratio; (3) a sizable fractionation of Ar, Kr, and Xe from their solar ratios, although the degree of fractionation, especially for ³⁶Ar/¹³²Xe, seems to decrease systematically from carbonaceous chondrites to Mars to Earth to Venus; and (4) large differences in Ne and Xe isotopic ratios among Earth, meteorites, and the Sun. Explaining trends (2), (2) and (4), and (1) pose the biggest problems for the solar-wind implantation, primitive atmosphere, and late veneer hypotheses, respectively. It is suggested that the grain-accretion hypothesis can explain all four trends, although the assumptions needed to achieve this agreement are far from proven. In particular, trends (1), (2), (3), and (4) are attributed to large pressure but small temperature differences in various regions of the inner solar system at the times of noble gas incorporation by host phases; similar proportions of the host phases that incorporated most of the He and Ne on the one hand (X) and Ar, Kr, and Xe on the other hand (Q); a decrease in the degree of fractionation with increasing noble-gas partial pressure; and the presence of interstellar carriers containing isotopically anomalous noble gases.Our analysis also suggests that primordial noble gases were incorporated throughout the interior of the outer terrestial planets, i.e., homogeneous accretion is favored over inhomogeneous accretion. In accord with meteorite data, we propose that carbonaceous materials were key hosts for the primordial noble gases incorporated into planets and that they provided a major source of the planets' CO₂ and N₂.     

Stellar Occultations by turbulent planetary atmospheres: A wave-optical theory including a finite scale height

A generalized wave-optical theory of stellar occultations by a turbulent planetary atmosphere is developed. The finite scale height of the atmosphere is retained for the first time. It is found that the finite scale height of the atmosphere affects the scintillations observed during the occultation in a number of ways which are most easily understood in terms of an effective Fresnel scale. We demonstrate the validity of a phase-changing screen approximation for occultation by a turbulent atmosphere in parameter ranges of general interest. Using this approximation various statistical properties of the fluctuating intensity are calculated explicitly. We present expressions for the total scintillation power, correlation function of the intensity, the cross-correlation at two frequencies, and its application to refractivity determinations. All of these expressions are given as a function of occultation depth and of parameters of the mean atmosphere and turbulence.     

Similarity considerations and conservation laws for magneto-static atmospheres

The equations of magnetohydrostatic equilibria for a plasma in a gravitational field are investigated analytically. For equilibria with one ignorable spatial coordinate, the equations reduce to a single nonlinear elliptic equation for the magnetic potential A. Similarity solutions of the elliptic equation are obtained for the case of an isothermal atmosphere in a uniform gravitational field. The solutions are obtained from a consideration of the invariance group of the elliptic equation. The importance of symmetries of the elliptic equation also appears in the determination of conservation laws. It turns out that the elliptic equation can be written as a variational principle, and the symmetries of the variational functional lead (via Noether's theorem) to conservation laws for the equation. As an example of the application of the similarity solutions, we construct a model magnetostatic atmosphere in which the current density J is proportional to the cube of the magnetic potential, and falls off exponentially with distance vertical to the base, with an e-folding distance equal to the gravitational scale height. The solutions show the interplay between the gravitational force, the J × B force (B, magnetic field induction) and the gas pressure gradient.     

Hydrogen-model atmospheres for white dwarf stars

We present a detailed calculation of model atmospheres for DA white dwarfs. Our atmosphere code solves the atmosphere structure in local thermodynamic equilibrium with a standard partial linearization technique, which takes into account the energy transfer by radiation and convection. This code incorporates recent improved and extended data base of collision-induced absorption by molecular hydrogen. We analyse the thermodynamic structure and emergent flux of atmospheres in the range 2500 T _eff60 000 K and 6.5log  g 9.0. Bolometric correction and colour indices are provided for a subsample of the model grid. Comparison of the colours is made with published observational material and results of other recent model calculations.
Motivated by the increasing interest in helium-core white dwarfs, we analyse the photometric characteristics of these stars during their cooling, using evolutionary models recently available. Effective temperatures, surface gravities, masses and ages have been determined for some helium-core white dwarf candidates, and their possible binary nature is briefly discussed.     

Solar photo rates for planetary atmospheres and atmospheric pollutants

Unattenuated solar photo rate coefficients and excess energies for dissociation, ionization, and dissociative ionization are presented for atomic and molecular species that have been identified or are suspected to exist in the atmospheres of planets, satellites (moons), comets, or as pollutants in the Earth atmosphere. The branching ratios and cross sections with resonances have been tabulated to the greatest detail possible and the rate coefficients and excess energies have been calculated from them on a grid of small wavelength bins for the quiet and the active Sun at 1 AU heliocentric distance.