Finding the Ambartsumyan functions and some of their analogs

New methods are proposed for finding the Ambartsumyan functions φ(η) for a half space and φ(η, τ) and ψ(η, τ) for finite layers, as well as their analogs with complete frequency redistribution, X (z, τ) and Y (z, τ). Substantial simplifications are obtained for monochromatic conservative scattering. Besides the Ambartsumyan functions, expressions for several of their angular moments are obtained directly in terms of the basis functions u ^±. A system of differential equations is obtained for the basis functions. A system of equations without the characteristic pseudosingularities is obtained for φ(η, τ) and ψ(η, τ) instead of the classical system of nonlinear equations. Some aspects of the numerical realization of the proposed method are also discussed.     

Strong and Weak Damping of Slow MHD Standing Waves in Hot Coronal Loops

Slow magnetohydrodynamic (MHD) standing wave oscillations in hot coronal loops for both strong (i.e. τ_d/P∼ 1) and weak (i.e. τ_d/P≥ 2) damping are investigated taking account of viscosity, thermal conductivity and optically thin radiation. The individual effect of the dissipative terms is not sufficient to explain the observed damping. However, the combined effect of these dissipative terms is sufficient to explain the observed strong damping, as well as weak damping seen by SUMER. We find that, the ratio of decay time (τ_d) and period (P) of wave, i.e., τ_d/P (which defines the modes of damping, whether it is strong or weak) is density dependent. By varying density from 10⁸ to 10¹⁰ cm⁻³ at a fixed temperature in the temperature range 6 – 10 MK, observed by SUMER, we get two sets of damping: one for which τ _d/P∼ 1 corresponds to strong damping that occurs at lower density and another that occurs at higher density for which τ_d/P ≥ 2 corresponds to weak damping. Contrary to strong-damped oscillations, the effect of optically thin radiation provides some additional dissipation apart from thermal conductivity and viscosity in weak-damped oscillations. We have, therefore, derived a resultant dispersion relation including the effect of optically thin radiation. Solutions of this dispersion relation illustrate how damping time varies with physical parameters of loops in both strong and weak damping cases.     

Calculation of effective optical depth of absorption line formation in homogeneous semi-infinite planetary atmosphere during anisotropic scattering

The algorithm for determining effective optical thickness of absorption line formation in a plane-parallel homogeneous planetary atmosphere is presented. The case of anisotropic scattering is considered. The results of numerical calculations of τ _e (μ₀) at the scattering angle γ = π for some values of the single scattering albedo λ and the parameter of the Heyney-Greenstein scattering indicatrix g are given. The refined equation for the function T ^m (−μ, μ₀) is presented.     

An improved version of the implicit integral method to solving radiative transfer problems

Radiative transfer (RT) problems in which the source function includes a scattering-like integral are typical two-points boundary problems. Their solution via differential equations implies making hypotheses on the solution itself, namely the specific intensity I (τ; n) of the radiation field. On the contrary, integral methods require making hypotheses on the source function S(τ). It seems of course more reasonable to make hypotheses on the latter because one can expect that the run of S(τ) with depth is smoother than that of I (τ; n). In previous works we assumed a piecewise parabolic approximation for the source function, which warrants the continuity of S(τ) and its first derivative at each depth point. Here we impose the continuity of the second derivative S′′(τ). In other words, we adopt a cubic spline representation to the source function, which highly stabilizes the numerical processes.     

Chaos in Relativity and Cosmology

Chaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes. (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does not pass the Painlevé test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τ_m and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The "finite time LCN" is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem. (b) In the case of two fixed black holes M₁ and M₂ the orbits of photons are separated into three types: orbits falling into M₁ (type I), or M₂ (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Multiple light scattering: mean number of scatterings and related problems

This is a discussion of V. A. Ambartsumyan’s studies of the mean number of scatterings for photons in scattering media and of further work and development in this area, especially at Ambartsumyan’s St. Petersburg school. The following questions are discussed briefly: (a) the traditional method for calculating the number of scatterings from the source function and critiques of this method. (b) The equation for the number N(τ; τ₀ ) of scatterings for a photon born at optical depth τ in a plane layer of optical thickness τ₀ and its use for calculating the number of scatterings, averaged over the entire ensemble of photons for a medium with arbitrary internal sources. These questions are first considered for the case of monochromatic scattering, and then for scattering in a spectral line with complete frequency redistribution (CFR). (c) The mean path length for a resonance line photon in a scattering medium with CFR and continuum absorption: the basic equations and asymptotic behavior of an optically thick layer. (d) A review of calculations of and in media that are so thick that the CFR approximation breaks down and the effects of partial frequency redistribution (PFR) become dominant. The presentation is at a semiquantitative level in many parts of this paper, with stress on physical significance rather than the mathematics, through the use of approximate and asymptotic solutions. Translated from Astrofizika, Vol. 52, No. 1, pp. 29–45 (February 2009).     

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 transfer in a cylinder. II. Special problems. Asymptotics

The method of computing the radiation field in an infinite circular cylinder proposed in Part I is now applied to the case of isotropic scattering with sources on the boundary and axis of the cylinder, as well as for a uniform distribution of sources inside the cylinder. For the simplest kernel we obtain exact solutions of the basic integral equation in explicit form. For scattering in a spectral line with complete frequency redistribution and a power absorption profile we develop an asymptotic theory for the case when the optical radius of the cylinder is large. We solve the asymptotic equations for the basic characteristics of the scattering in closed form for conservative scattering and find its asymptotics. We obtain estimates of the mean number of scatterings with a layered source, and also the mean and variance of the number of scatterings with a uniform source distribution.Translated fromAstrofizika, Vol. 37, No. 4, 1994.This work was carried out with the financial support of the Russian Basic Research Fund (grant 93-02-2957).     

Effects of high velocities on photoionization lines

We have treated formation of spectral lines in a commoving frame where photoionization is predominant over collisional processes. We have assumed that the radiation field for causing photoionization is a function of Planck function. We have also considered the situation in which the continuum contributes to the radiation in the line. In all the models the quantityB/A (ratio of outer to inner radii) is kept equal to 10 and the total optical depth is taken to be 10³. The velocity is assumed to be varying according to the lawdV/dτ ∼ < 1/τ whereτ is the optical depth (τ > 0) in the given shell. The velocities at the innermost radius (r =A) are set equal to 0 and at the outermost radius (r =B), the maximum velocities are taken to be 0, 1, 3 and 10 Doppler units. The calculated line profiles are those seen by an observer at infinity.P Cygni-type profiles are observed in the case of a medium with no continuum absorption. For a medium with continuum absorption double peaked asymmetric profiles are noticed when the velocities are small; the two emission peaks merge into a single asymmetric peak for larger velocities.     

Unified structure of analytical solutions for radiative transfer problems in plane-parallel,homogeneous media

The basic concepts for developing a system of analytic solutions for the standard problems of radiative transfer theory are discussed. These solutions, which are found using Ambartsumyan’s layer addition method in Sobolev’s probabilistic interpretation for radiative diffusion problems, are maximally compact and easily used in numerical computations. New expressions are obtained for the resolvents and the resolvent functions, as well as a unified structure for the form of an integral representation for solving different radiative transfer problems in semi-infinite media and in finite layers. Block diagrams of the sequence of stages for solving these problems are provided, where the Ambartsumyan function φ(η) (more precisely, 1/φ(η)) plays a fundamental role in the case of semi-infinite media while the functions a(η, τ₀ ) and b(η, τ₀) play an analogous role for finite layers.     

Scattering of the Low Frequency Pulsar Radiation

We report the results of the measurements and analysis of the pulse broadening due to interstellar scattering on 43 pulsars at 102 MHz. This is the largest uniform sample of direct measurements of pulsar scatteringτ_sc, which make it feasible to analyze the dependence of this value on other pulsar parameters. The measured dependence of τ_scon dispersion measure τ_sc (DM)=40(DM/100)^2.1 is close to theoretically expected relation τ_sc (DM)∝ DM². A frequency dependence of the scattering pulse broadening is weaker than commonly accepted τ_sc ∝ ν^-4.4. This revised version was published online in July 2006 with corrections to the Cover Date.     

Early Cosmological Models with Bulk Viscosity in Brans-Dicke Theory

The effect of time dependent bulk viscosity on the evolution of Friedmann models with zero curvature in Brans-Dicke theory is studied. The solutions of the field equations with ‘gamma-law’ equation of state p = (γ-1) ρ, where γ varies continuously as the Universe expands, are obtained by using the power-law relation φ = bR ⁿ , which lead to models with constant deceleration parameter. We obtain solutions for the inflationary period and radiation dominated era of the universe. The physical properties of cosmological solutions are also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Dependence of the aerosol component of optical thickness and the relative concentration of methane on depth in atmospheres of giant planets

Based on the data on a spectral dependence of the geometric albedo of giant planet discs, we obtained depth variations in the optical thickness τ _a of the aerosol component and relative concentration γ of methane (Uranium, Neptune) lnτ _a = −0.720 + 1.507Δlnp (for −2.2085 ≤ lnp ≤ −1.0018), lnτ _a = +1.224 + 1.160Δlnp (for −1.0018 ≤ lnp ≤ −0.0595), lnτ _a = +2.318 + 0.192Δlnp (for −0.0595 ≤ lnp), γ = 0.0027 for Jupiter; lnτ _a = −0.846 + 1.598Δlnp (for −3.3619 ≤ lnp ≤ −2.0575), lnτ _a = +1.238 + 1.342Δlnp (for −2.0575 ≤ lnp ≤ −1.2074), lnτ _a = +2.379 + 0.722 (for −1.2074 ≤ lnp ≤ −0.6501), lnτ _a = +2.781 + 0.326Δlnp (for 0.6501 ≤ lnp), γ = 0.0027 for Saturn; lnτ _a = −2.694 + 0.087Δlnp (for +0.3685 ≤ lnp ≤ +1.2314), lnτ _a = −2.619 + 7.341Δlnp (for +1.2314 ≤ lnp ≤ +1.7556), lnτ _a = +1.229 + 0.956Δlnp (for +1.7556 ≤ lnp) for Uranium; lnτ _a = −1.861 + 1.248Δlnp (for +0.3204 ≤ lnp ≤ +0.9051), lnτ _a = −1.131 + 0.347Δlnp (for +0.9051 ≤ lnp) for Neptune; depth-averaged relative methane concentration lnγ = −9.982 + 2.676Δlnp(0.3584 ≤ lnp ≤ 1.5445); ln γ = −9.738 + 2.561Δlnp(0.3237 ≤ lnp ≤ 1.6156) and γ = 0.00382(lnp ≥ 1.6156); 0.00554(lnp ≥ 1.6156) for Uranium and Neptune, respectively (p is in bar).     

Albedo-shift method in the problem of anisotropic light scattering in a plane atmosphere

A standard problem of radiative transfer theory — calculating the diffuse reflection and transmission of radiation by a plane scattering atmosphere — is considered. The recently proposed albedoshift method is used to calculate the X and Y functions (and the H function) for the case of anisotropic scattering with a Henyey-Greenstein indicatrix. The method enables one to “suppress” scattering and obtain iterative solutions of high accuracy in only a few iterations, even when the mean number of photon scatterings in the atmosphere is very large. Translated from Astrofizika, Vol. 41. No. 4, pp. 623–646, October–December, 1998.     

The cyclical reversible transitions of the universe’s evolution

In this work we propose cyclical reversible transitions as the scenario in which the universe evolves, through a series consisting of reversible expansion, temporary stability, and contraction. Our model is based on the comparison between local and global time-dependent densities {ρ ₀(τ ₀),ρ(τ)} instead of the critical density ρ _c, local and global time-dependent Hubble parameters {H ₀(τ ₀),H(τ)}, and the variations {Δρ(τ),ΔH(τ)} due to cosmological chaotic fluctuations, which are generally ignored in certain oscillating models. By taking into account all these factors, a rate equation in the form of (H ₀/H)² ∝(ρ ₀/ρ) has been established, and from it we derive some others, to provide a mechanism that is responsible for the cyclical reversible transitions. Also, the problems of singularities, black hole overproduction, and the second law of thermodynamics arising in oscillating universe models are conceptually resolved.     

Nonlinear spinor field in Bianchi type-I Universe filled with viscous fluid: numerical solutions

We consider a system of nonlinear spinor and a Bianchi type I gravitational fields in presence of viscous fluid. The nonlinear term in the spinor field Lagrangian is chosen to be λ F, with λ being a self-coupling constant and F being a function of the invariants I an J constructed from bilinear spinor forms S and P. Self-consistent solutions to the spinor and BI gravitational field equations are obtained in terms of τ, where τ is the volume scale of BI universe. System of equations for τ and ε, where ε is the energy of the viscous fluid, is deduced. This system is solved numerically for some special cases.      

An exact solution of the equation of transfer with three-term scattering indicatrix in an exponential atmosphere

The general equation for radiative transfer in the Milne-Eddington model is considered here. The scattering function is assumed to be quadratically anisotropic in the cosine of the scattering angle and Planck's intensity function is assumed for thermal emission. Here we have taken Planck's function as a nonlinear function of optical depth, viz.,B _v(T)=b _o+b ₁ e ^–. The exact solution for emergent intensity from the bounding face is obtained by the method of the Laplace transform in combination with the Wiener-Hopf technique.     

Cyclical cosmo-thermodynamics

In this work, which is a supplemental to previous one, we undertake to establish some cosmological thermodynamic equations in the context of the cyclical universe as the scenario in which the universe itself is considered like an adiabatic thermodynamical system enclosed in physical volume characterized by periodic reversible transitions. Our model is based on the combination of local and global cosmological time-dependent temperatures {T ₀(τ ₀),T(τ)} and volumes {V ₀(τ ₀),V(τ)} instead of the critical temperature T _c and volume V _c; and the infinitesimal relative variations {dT/T,dV/V}, which are mainly due to the cosmological chaotic fluctuations that are generally ignored in certain oscillating models. By taking into account all these factors, certain equations in the form of d ℓ/ℓ=±η d τ/τ _H have been established and from them we derive some others to provide a mechanism that is responsible for the thermodynamic evolution of the cyclical universe.     

Time-dependent radiation transfer and a possible explanation of the interpulse in CP 0950

The time-dependent equation of radiative transfer is solved exactly and in then-th Gaussian approximation.The atmosphere is plane-parallel and semi-infinite; isotropic scattering is assumed, but the boundary condition at =0 is arbitrary.The results are used to investigate a suggested mechanism for the origin of the secondary pulses in CP 0950; it is found that a binary system of neutron stars can indeed explain formation, time delay and intensity of the observed interpulse.