Exact solution of a key equation in a finite planetary atmosphere by the method of laplace transform and linear singular operators

Considering the ground reflection according to Lambert's law, we establish a fundamental equation in finite planetary atmospheres. An exact form of the solution of this equation is obtained for the emergent quantities from the bounding faces in terms ofX-Y equations by the method of Laplace transform, in combination with the theory of linear singular operators.     

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.     

Exact solution of a basic equation in finite atmosphere by the method of Laplace transform and linear singular operators

A finite atmosphere having distribution of intensity at both surfaces with definite form of scattering function and source function is considered here. The basic integro-differential equation for the intensity distribution at any optical depth is subjected to the finite Laplace transform to have linear integral equations for the surface quantities under interest. These linear integral equations are transformed into linear singular integral equations by use of the Plemelj's formulae. The solution of these linear singular integral equations are obtained in terms of theX-Y equations of Chandrasekhar by use of the theory of linear singular operators which is applied in Das (1978a).     

Diffuse reflection and transmission in accordance with the phase function 3/4(1+cos2Θ)

We have considered the transport equation for the problem of diffuse reflection and transmission on Rayleigh's phase function and obtained the exact solution of this equation for angular distributions of the intensities diffusely reflected from the surfacet=0 and diffusely transmitted below the surfacet=t ₀ of a finite atmosphere of optical deptht=t ₀ using the Laplace transform and the theory of singular operators. This is an exact method.     

Exact solution of a simple time-dependent integro-differential equation by the method of Laplace transform and the theory of linear singular operators

The simplest form of the equation of transfer for a time dependent radiation field in finite atmosphere is considered. This equation of transfer is an integro-differential equation, the solution of this equation is based on the theory of separation of variables, the Laplace transform and the theory of linear singular operators. The emergent intensities from the bounding faces of the finite atmosphere are determined in terms ofX-Y equations of Chandrasekhar.     

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.     

CESAM: a free code for stellar evolution calculations

The cesam code is a consistent set of programs and routines which perform calculations of 1D quasi-hydrostatic stellar evolution including microscopic diffusion of chemical species and diffusion of angular momentum. The solution of the quasi-static equilibrium is performed by a collocation method based on piecewise polynomials approximations projected on a B-spline basis; that allows stable and robust calculations, and the exact restitution of the solution, not only at grid points, even for the discontinuous variables. Other advantages are the monitoring by only one parameter of the accuracy and its improvement by super-convergence. An automatic mesh refinement has been designed for adjusting the localisations of grid points according to the changes of unknowns. For standard models, the evolution of the chemical composition is solved by stiffly stable schemes of orders up to four; in the convection zones mixing and evolution of chemical are simultaneous. The solution of the diffusion equation employs the Galerkin finite elements scheme; the mixing of chemicals is then performed by a strong turbulent diffusion. A precise restoration of the atmosphere is allowed for.     

Long nonlinear waves in a compressible magnetically structured atmosphere

The Benjamin-Ono equation is derived for long slow sausage waves propagating in a vertical magnetic slab embedded into a stratified atmosphere, provided that the slab thickness is much smaller than the scale height of the atmosphere. The soliton propagation in a nonstratified atmosphere is discussed. The approximate formulas describing the slow evolution of the amplitude and the length of a soliton propagating in a very weakly stratified atmosphere are obtained. The exact soliton-like solution for an atmosphere with a linearly growing temperature is found.     

Solution of a time-dependent equation of transfer by theF n -method

The time-dependent equation of transfer for a finite, plane-parallel, non-radiating, and isotropically-scattering atmosphere of arbitrary stratification is solved by using theF _n -method developed by Siewert.     

Wave disturbances near the temperature jump in the transition region of the solar atmosphere

We theoretically analyzed the properties of surface waves near the temperature jump in the solar atmosphere whose dispersion relation is identical in form to the equation for waves on deep water. We found an exact solution to the model equation for the vertical velocity of the medium in such a wave. Based on the derived space-time dependence of the vertical velocity of the medium, we quantitatively explained one of the events recorded in the SOI/MDI experiment onboard the SOHO spacecraft that accompanied coronal mass ejection.     

Hydromagnetic free convection flow in the stokes problem for a porous vertical limiting surface with constant suction

This paper provides a comprehensive analysis of the effects of a uniform transverse magnetic field on the free-convection flow of a viscous incompressible and electrically conductive fluid (e.g., of a stellar atmosphere) past an impulsively started, infinite, porous, vertical limiting surface (e.g., of a star) with a constant suction. The magnetic Reynolds number is assumed small so that the induced magnetic field is considered negligible. Exact solution of the equations governing the flow is obtained in closed form with the help of the Laplace transform technique when the Prandtl numberP=1. Expressions are given for the velocity field, for the temperature field and for their related quantities. The results thus obtained are discussed quantitatively in the last section of this paper.     

Natural superrotation of a rarefied planetary atmosphere

We consider the kinetics of a rarefied rotating planetary atmosphere. The spatial distributions of the atmospheric-gas density and mean angular velocity were determined by analyzing the exact solution of the two-dimensional kinetic equation. We show that the angular velocity of the gas at some distance from the planet could be higher than that in the initial layer starting from which the atmosphere is rarefied. Our model calculations elucidate the superrotation mechanism under consideration.     

Exact solution of the transport equation for imperfect rayleigh scattering in a semi-infinite atmosphere

By performing the one-sided Laplace transform on the matrix integro-differential equation for a semi-infinite plane parallel imperfect Rayleigh scattering atmosphere we derive an integral equation for the emergent intensity matrix. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity matrix in terms of singularH-matrix and an unknown matrix. The unknown matrix has been determined considering the boundary condition at infinity to be identical with the asymptotic solution for the intensity matrix.     

Exact solution of the transport equation in finite media with a plane and uniform point source and flux normally incident at the faces from outside

In this paper we apply the Wiener-Hopf technique combined with the method of the Laplace transform, to derive an exact solution of the transport equation for neutron diffusion in an isotropically scattering plane-parallel medium of finite thickness in which are situated a plane source at the middle and a uniformly distributed point source, there being flux of beams normally incident from outside on the two extreme parallel surfaces of the medium.     

Exact solution of the problem of diffuse reflection for a combination of Rayleigh and isotropic scattering

We consider the basic vector equation of transfer for radiation in a semi-infinite atmosphere for diffuse reflection which scatters radiation in accordance with the phase matrix obtained from a combination of Rayleight and isotropic scattering. This equation will give an integral equation for emergent intensity while subjected to the Laplace transform. The integral equation will give rise to the emergent intensity matrix on application of the Wiener-Hopf technique. This is an exact method.     

Self-similar space-times

An exact solution of the transport equation in radiative transfer for an axially symmetric Rayleigh scattering problem in semi-infinite planetary atmosphere both for emergent intensity and intensity at any optical depth has been derived with the help of the Laplace transform and the Wiener-Hopf technique, and by use of the constancy of net flux. Chandrasekhar's results for emergent intensity have been verified. New expressions for theH _l andH _r functions have been obtained.     

Albedo-shifting method: Source function in a plane atmosphere

A classical problem in the theory of radiative transfer is considered: calculating the radiation field within a plane scattering atmosphere. The recently proposed albedo-shifting method is used to calculate the source function both in a semi-infinite atmosphere and in an atmophere of finite optical depth, illuminated by parallel rays. The method enables one to “suppress” scattering and obtain iterative solutions of the integral equation for the source function in only a few direct lambda iterations, even when the average number of photon scatterings in the atmosphere is very large. Translated from Astrofizika, Vol. 42, No. 4, pp. 485–500, October–December, 1999.     

Energy-dependent transport problem with generalized boundary conditions

The slowing-down Boltzmann equation for generalized boundary conditions is considered and transformed to one-speed equation in Laplace space. Exact relations between energy reflection and transmission coefficients for a problem with diffuse reflecting boundary conditions and the albedos for the problem with isotropic boundary conditions are obtained. The Galerkin method is used to calculate the energy reflection coefficient for a finite slab for different thicknesses at different mass ratiosA, target to projectile mass, at different synthetic-scattering kernels. The results for partial heat fluxes for isotropic and anisotropic-scattering dispersive medium are given. The results obtained for isotropic boundary conditions are compared well with the exact results.     

Laminar hydromagnetic thermal boundary layer in fluctuating flow past a limiting surface with variable suction

An analysis of the temperature field in the case of the two-dimensional hydromagnetic flow of a viscous incompressible and electrically conducting fluid, (e.g., of a stellar atmosphere), past a porous, infinite, limiting surface in the presence of a transverse magnetic field, is considered when (i) the free stream velocity oscillates in time about a constant mean; (ii) the suction velocity normal to the limiting surface oscillates in magnitude but not in direction about a non-zero mean; and (iii) there is no heat transfer between the fluid and the wall. Approximate solution is obtained of the energy equation and are given expressions for the temperature field and for the temperature at the limiting surface, when the magnetic Prandtl numberP _m =1 and the magnetic parameterM<1. They are shown graphically followed by a discussion.Research supported by the Alexander S. Onassis Foundation.     

Compressible Fluid Model for the Seismic Waves Generated by A Sunquake

The solar convection zone is modeled as a horizontally stratified atmosphere with a constant gravitational field and an adiabatic temperature gradient (a neutrally stratified polytrope). At equilibrium, the gas pressure and density decreases to zero at the solar surface so that the solar surface is treated as a free surface which is bounded by vacuum. The evolution of small amplitude perturbations about the equilibrium state is described by the linearized Euler equations for an inviscid compressible fluid with an adiabatic equation of state. A sunquake is initiated at time zero by means of an initial perturbation with a Gaussian velocity profile and the exact solution of the initial value problem is obtained in terms of a Fourier integral. Comparisons between theory and observations indicate that this highly simplified model is able to predict the propagation of sunquake waves across the solar surface with an error of approximately 10% or 20%.