A comparison of the anelastic and subseismic approximations for low-frequency gravity modes in stars

A comparative study of the properties of the anelastic and subseismic approximations is presented. The anelastic approximation is commonly used in astrophysics in compressible convection studies, whereas the subseismic approximation comes from geophysics where it is used to describe long-period seismic oscillations propagating in the Earth's outer fluid core. Both approximations aim to filter out the acoustic waves while retaining the density variations of the equilibrium configuration. However, they disagree on the form of the equation of mass conservation. We show here that the anelastic approximation is in fact the only consistent approximation as far as stellar low-frequency oscillations are concerned. We also show that this approximation implies Cowling's approximation which neglects perturbations of the gravity field. Examples are given to demonstrate the efficiency of the anelastic approximation.     

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.     

Pulsational properties of V2109 Cyg

In this study V2109 Cyg (a pulsating δ Scuti star) has been modelled. In treating the oscillation equations perturbation in gravitational potential energy has been taken into account. Both radial and nonradial oscillations are treated with adiabatic approximation. The so called radial fundamental frequency (5.3745 c/d) and the nonradial frequency (5.8332 c/d) were obtained within a satisfactory precision. It was found that the Cowling approximation introduced more error as one went from low overtones to high overtones in radial oscillations. A similar trend was observed in nonradial case with low values of l. By keeping the effective temperatures almost the same as with V2109 Cyg two more models with different masses have also been calculated to see the effect of inclusion of perturbation in gravitational potential energy on oscillation frequencies in different masses. Conclusion arrived is that one must be careful to employ the Cowling approximation especially for high nonradial oscillation frequencies. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical derivation of the probability for the escape of stars from colliding galaxies

We use the impulse approximation to derive analytical formulae for the escape probability from a simple binary system that interacts with a third body. The binary system is made up of a mass-less body in circular orbit around a massive object, and we assume that the two massive bodies follow a Schuster (or Plummer) distribution. Within the ranges imposed by the impulsive approximation to the parameters of the encounter, we find good agreement between our results and those obtained from numerical experiments.     

Excluded-volume approximation for supernova matter

A general scheme of the excluded-volume approximation as applied to multicomponent systems with an arbitrary degree of degeneracy has been developed. This scheme also admits an allowance for additional interactions between the components of a system. A specific form of the excluded-volume approximation for investigating supernova matter at subnuclear densities has been found from comparison with the hard-sphere model. The possibility of describing the phase transition to uniform nuclear matter in terms of the formalism under consideration is discussed.     

Nonlinear Generation of Fluting Perturbations by Kink Mode

We study the excitation of fluting perturbations in a magnetic tube by an initially imposed kink mode. We use the ideal magnetohydrodynamic (MHD) equations in the cold-plasma approximation. We also use the thin-tube approximation and scale the dependent and independent variables accordingly. Then we assume that the dimensionless amplitude of the kink mode is small and use it as an expansion parameter in the regular perturbation method. We obtain the expression for the tube boundary perturbation in the second-order approximation. This perturbation is a superposition of sausage and fluting perturbations. The amplitude of the fluting perturbation takes its maximum at the middle of the tube, and it monotonically decreases with the distance from the middle of the tube.     

Sensitivity Kernels for Time-Distance Inversion

Inversion of local-area helioseismic time-distance data has so far only been done in the ray approximation (Kosovichev, 1996). Since this is a high-frequency approximation its applicability can be questioned for the solar case. Bogdan (1997) showed that for a simple solar model the localized wave packets do follow, but are not confined to, the ray path. We use an approximation based on the first Fresnel zone that has been developed in geophysics by Snieder and Lomax (1996) to go beyond the ray approximation in the inversions. We have calculated sensitivity kernels using both approximations. To test them we use a finite-difference forward modeling of the whole wave field in an acoustic medium reminiscent of the Sun. We use the finite-difference modeling to calculate sensitivity kernels for the full wave field and compare this with the other kernels. The results show that the Fresnel-zone-based kernels are in good agreement with the sensitivity obtained from the modeling. Thus these new kernels represent a significant step forward in the inversion of time-distance data.     

2-post-Newtonian approximation of a nonstationary spherically symmetric star in flat space-time theory of gravitation

Starting from a previously stated theory of gravitation in flat space-time the 2-post-Newtonian approximation of a nonstationary spherically symmetric star is derived. The theory forces retarded effects for one part of the solutions of the 2-post-Newtonian approximation. The equations of motion and the conserved total energy of a nonstationary star are given up to 2-post-Newtonian order. The solution in the exterior of the body is studied to this accuracy. Birkhoff's theorem is violated. For the special case of a static spherically symmetric star we get deviations of the exterior solution in the 2-post-Newtonian approximation relative to Einstein's theory.     

Sensitivity of the orbital content of a model stellar system to the potential approximation used to describe it

We have classified orbits in a stationary triaxial stellar system created from a cold dissipationless collapse of 100,000 particles. In order to integrate the orbits, two potential approximations with different fitting functions were used in turn. We found that the relative amount of chaotic versus regular orbits does depend on the chosen approximation of potential, even though both models resulted in very good fits of the underlying exact potential. On the other hand, the content of regular orbits, i.e., its distribution among main families, does not strongly depend of the potential approximation, being therefore a more robust signature of the gravitational system under study.     

Estimation of electron temperature by VLF waves propagating in directions near the resonance cone

A ray tracing computer program for non-ducted whistler mode waves in a warm plasma in the magnetosphere is developed, where electron temperature effects are taken into account. The refractive index is calculated from the warm-plasma approximation and is used in the ray tracing after its accuracy has been checked by comparison with the hot-plasma solution without approximation. The ray paths do not depend appreciably on electron temperature. However, there are regions where the waves are heavily damped by Landau damping. By paying attention to this damping region, the electron temperature can be estimated from a satellite observation of the Doppler shift and damping of a ground-based VLF signal.     

On the theory of type III radio bursts in the nonhomogeneous interplanetary space (quasi-linear approximation)

Within the limits of geometrical optics frequency characteristics of perturbations of one-dimensionally non-uniform system “electron beam-solar wind plasma” are investigated in linear approximation on the basis of Maxwell equations closed by the derived constitutive equation. The beam is generated by the active region during solar flares and it appears as a source of type III radio emission in the interplanetary space. The appropriate dispersion equation is solved. Resonance interaction of wave with electron beam appears to happen only in two space points. Such transient (pointwise) mechanism of resonance throws light on one of the basic problems of physics of electron beams generated by solar flares: incomparably more long-term time of their existence compared to the time of existence resulting from the former theoretical estimates of velocity of beam energy loss on radiation within the limits of homogeneous medium. The degree and time of electron beam dissipation were determined in quasi-linear approximation.     

The amplitude of the opposition effect due to weak localization of photons in discrete disordered media

Weak localization of photons in discrete disordered media is considered as a possible physical mechanism of the opposition effect of some atmosphereless bodies. The amplitude of the opposition effect is calculated by using the rigorousvector multiple-scattering theory and thescalar approximation. It is shown that the scalar approximation can significantly overestimate the amplitude of the opposition effect. Thus, this approximation should not be used in interpreting the observational data, and some previous results obtained with this approximation may require substantial revision.     

Spurious contribution to cosmic ray scattering calculations

The quasi-linear theory for cosmic ray propagation is a well-known and widely accepted theory. In this paper, we discuss the different contributions to the pitch-angle Fokker–Planck coefficient from large and small scales for slab geometry using the damping model of dynamical turbulence. These examinations will give us a hint on the limitation range where the quasi-linear approximation is a good approximation.     

Non-Linear Gravitational Clustering Using the Zel'dovich Approximation

As an alternative to computationally expensive N-body simulations for gravitional clustering, the Zel'dovich approximation (ZA) was studied in 3D, 2D and 1D. Plots of the density contrast were compared against linear theory and the exact solution. The ZA was found to perform very well in the linear regime, better than linear theory, and to give a good approximation well into the non-linear regime. This revised version was published online in July 2006 with corrections to the Cover Date.     

Angular distribution of radiation in an isotropically scattering slab

The equation of radiative transfer in an isotropically scattering slab subject to general boundary conditions is solved. The Padé approximation technique is used to calculate the reflected and transmitted angular distributions. Numerical results for angular distributions through and at the boundaries of a finite slab are calculated by the Padé approximation technique. The results for a Padé approximation of [0/1] are compared with results obtained by the Galerkin method.     

Closure tests for mean field magnetohydrodynamics using a self-consistent reduced model

The mean electromotive force and α effect are computed for a forced turbulent flow using a simple non-linear dynamical model. The results are used to check the applicability of two basic analytic ansätze of mean-field magnetohydrodynamics – the second-order correlation approximation (SOCA) and the τ approximation. In the numerical simulations the effective Reynolds number Re is 2–20, while the magnetic Prandtl number P _m varies from 0.1 to 10⁷. We present evidence that the τ approximation may be appropriate in dynamical regimes where there is a small-scale dynamo. Catastrophic quenching of the α effect is found for high P _m. Our results indicate that for high P _m SOCA gives a very large value of the α coefficient compared with the 'exact' solution. The discrepancy depends on the properties of the random force that drives the flow, with a larger difference occurring for δ-correlated force compared with that for a steady random force.     

Reflection effect in close binaries. II. Distribution of emergent radiation from the irradiated component along the line of sight

We have calculated the effects of irradiation from a point source observed at infinity. Plane-parallel approximation and spherically-symmetric approximations are employed in calculating the self-radiation field for the sake of comparison. It is found that there are considerable changes in the radiation received at infinity between the approximation of plane-parallel stratification and spherical symmetry.     

Relativistic corrections to the Kompaneets equation

We study the Sunyaev–Zeldovich effect for clusters of galaxies. We explore the relativistic corrections to the Kompaneets equation in terms of two different expansion approximation schemes, namely, the Fokker–Planck expansion approximation and delta function expansion approximation. We show that two expansion approximation formalisms are equivalent under the Thomson approximation, which is extremely good approximation for the CMB photon energies. This will clarify the situation for existing theoretical methods to analyse observation data.