Turbulence influenced line formation in stellar atmospheres

The influence of stochastic velocity fields with finite correlation lengths on the formation of spectral lines is taken into account without restrictions to specific velocity models. To construct a perturbation theory treating the influence of stochastic motion on the radiative transfer we start with the stochastic transfer equation for plane-parallel atmospheres and expand it to an infinite hierarchical system. An appropriate cut-off of the infinite system admits to achieve exactly the micro- and macroturbulent limit at every level of approach. The formalism is derived for n-point correlations of the absorption coefficient and specialised for 2-point correlations.     

Zur Dynamotheorie kosmischer Magnetfelder. I. Gleichungen für sphärische Dynamomodelle

This paper deals with the mathematical treatment of special models of hydromagnetic dynamos. For the models considered here the conducting medium occupies a spherical region surrounded by vacuum. Both laminar and turbulent dynamos are included. The partial vector differential equations governing the models are transformed into an infinite set of differential equations for scalar functions by applying a method previously used by BULLARD and GELLMAN and on the basis of a representation of vector fields as a sum of a poloidal and a toroidal part. The scalar functions depend only on a radial coordinate and possibly on the time. In both the stationary and the periodic case, an infinite set of ordinary differential equations results which may be treated numerically. A series of relations for computing various dynamo models is prepared.     

A Certain Class of Laplace Transforms with Applications to Reaction and Reaction-Diffusion Equations

A class of Laplace transforms is examined to show that particular cases of this class are associated with production-destruction and reaction-diffusion problems in physics, study of differences of independently distributed random variables and the concept of Laplacianness in statistics, α-Laplace and Mittag-Leffler stochastic processes, the concepts of infinite divisibility and geometric infinite divisibility problems in probability theory and certain fractional integrals and fractional derivatives. A number of applications are pointed out with special reference to solutions of fractional reaction and reaction-diffusion equations and their generalizations.     

Stochastic gravitational fluctuations of stellar or galactic systems with a fractal distribution of field sources

The stochastic gravitational fluctuations for a fractal mass distribution are analyzed by means of a functional integral approach. A general method is developed for evaluating the stochastic properties of vectorial additive random fields generated by a variable number of point sources obeying inhomogeneous Poisson statistics. A closed expression for the generating functional of the field is given in terms of the generating functional of the sources. The moments of the resulting vectorial field are finite if the correlation functions of the sources have short tails. In this case all cumulants of the field can be computed exactly: they are averages of the central moments of sources computed in terms of the probability density of the position of a source. The method is applied for analyzing the stochastic gravitational fluctuations generated by a fractal distribution of field sources (stars or galaxies). For a Newtonian force law the correlation functions of the sources are slowly decaying, the cumulants of the stochastic gravitational field are infinite and the probability density of the field intensityF is given by a Lévy fractal stable law with a scaling exponentH depending on the fractal dimensiond _f of the distribution of stars or galaxies:H =d _f /2.     

研究Chandler摆动的一个随机激发模型

基于同运动激发了Chandler摆动的假设，提出一个研究Chandler摆动的随机非线性模型，它是由一个随机性阶梯函数和一个具有阻尼的线性谐振子的Euler中心差分格式混合而成，研究此模型中各个参数与Chandler摆动振幅的关系，通过对基于实测资料获得的有效大气角动量时间序列的统计分析，初步发现其中含有随机噪声成份，它可以激发目前观测到的Chandler摆动振幅的28－40％，最后，对Chandler摆胡机激发的假设作了一些讨论。     

An exact solution of two-dimensional steady MGD flows

Steady-plane flow of an inviscid, electrically-conducting, compressible fluid with infinite electrical conductivity is considered and a single partial differential equation is obtained which involves two functions. Appropriate specialization of these functions generate new exact solutions of the orginal equations.     

The relative motion of two spheroidal rigid bodies

We consider two spheroidal rigid bodies of comparable size constituting the components of an isolated binary system. We assume that (1) the bodies are homogeneous oblate ellipsoids of revolution, and (2) the meridional eccentricities of both components are small parameters.We obtain seven nonlinear differential equations governing simultaneously the relative motion of the two centroids and the rotational motion of each set of body axes. We seek solutions to these equations in the form of infinite series in the two meridional eccentricities.In the zero-order approximation (i. e., when the meridional eccentricities are neglected), the equations of motion separate into two independent subsystems. In this instance, the relative motion of the centroids is taken as a Kepler elliptic orbit of small eccentricity, whereas for each set of body axes we choose a composite motion consisting of a regular precession about an inertial axis and a uniform rotation about a body axis.The first part of the paper deals with the representation of the total potential energy of the binary system as an infinite series of the meridional eccentricities. For this purpose, we had to (1) derive a formula for representing the directional derivative of a solid harmonic as a combination of lower order harmonics, and (2) obtain the general term of a biaxial harmonic as a polynomial in the angular variables.In the second part, we expound a recurrent procedure whereby the approximations of various orders can be determined in terms of lower-order approximations. The rotational motion gives rise to linear differential equations with constant coefficients. In dealing with the translational motion, differential equations of the Hill type are encountered and are solved by means of power series in the orbital eccentricity. We give explicit solutions for the first-order approximation alone and identify critical values of the parameters which cause the motion to become unstable.The generality of the approach is tantamount to studying the evolution and asymptotic stability of the motion.Research performed under NASA Contract NAS5-11123.     

Stochasticity in dynamical systems and the curvature of the associated riemannian manifold

It is shown that no conflicts need to arise between results on the stochastic properties of a dynamical system obtained through the method of the surface of section mapping and results obtained through the Hedlund-Hopf-Lobachevsky-Hadamard theorem.     

A SOLAR α - ω DYNAMO STUDY AND ITS TRANSITION TO CHAOS

In this contribution we present a nonlinear dynamo model, described by an infinite dimensional system of differential equations, whose solutions depend on the essential parameter D, the dynamo number. The solutions and the bifurcation points of the system are determined with the help of a new developed computer code. We show that, depending on D, stationary, oscillatory and chaotic solutions, which are characterized by Lyapunov exponents, result. We find that the solar dynamo may operate either in the chaotic or in the stable limit cycle domain, depending on the characteristic value of the dynamo number or the motion of the convection zone.     

Modelling the IDV Emissions of the BL Lac Objects with a Langevin Type Stochastic Differential Equation

In this paper, we introduce a simplified model for explaining the observations of optical intra-day variability (IDV) of the BL Lac Objects. We assume that the source of the IDV are the stochastic oscillations of an accretion disk around a supermassive black hole. The stochastic fluctuations on the vertical direction of the accretion disk are described by using a Langevin type equation with a damping term and a random, white noise type force. Furthermore, preliminary numerical simulation results are presented, which are based on the numerical analysis of the Langevin stochastic differential equation.     

On the solving of the Hill's differential equation through the method of operational calculus

Using a method previously applied to the treatment of the Mathieu differential equation, we solve the Hill's differential equation of lunar theory through the way of operational calculus, which avoids the cumbersome infinite determinants of the classical procedure. The one-sided Laplace transformation changes it into a difference equation with an infinite number of terms and variable coefficients. When its first member is divided by a suitable factor, this difference equation is the image of an integral equation of the Volterra type which is equivalent to the initial Hill's differential equation. Solution of this Volterra integral equation is unique and it is the general solution of the Hill's differential equation. This solution is a series in the powers of a small dimensionless parameter ² which appears as a factor in the second member of the Hill's differential equation. We reduce it to the sum of its terms of degree 12 with respect to which is the precision usually required in a lunar theory and we write down effectively the coefficients of the terms in ², ( ²)² and the coefficient of the term in ( ²)³ which depends upon the initial valuey(0) of the Hill's differential equation.     

Gravitational perturbations of equatorial orbits

Asymptotic solutions are developed for the motion of a geocentric satellite in the equatorial plane due to gravitational perturbations such as nonsphericity (especially oblateness) of the primary body. Axisymmetric potentials are considered. A class of transformations is developed and the equations of motion are solved by the method of generalized multiple scales. Further it is shown that the equations of motion can be transformed into the required form to within any specified degree of accuracy. The transformations form an Abelian group of infinite order which leaves the differential equations of motion invariant. Solutions are developed in terms of elementary functions instead of elliptic or other higher transcendental functions and are shown to agree with known results.This investigation was carried out under NASA Grant NGR-31-001-152 with the author as a consultant to Princeton University.     

Numerical simulations of granular dynamics: I. Hard-sphere discrete element method and tests

We present a new particle-based (discrete element) numerical method for the simulation of granular dynamics, with application to motions of particles on small solar system body and planetary surfaces. The method employs the parallel N-body tree code pkdgrav to search for collisions and compute particle trajectories. Collisions are treated as instantaneous point-contact events between rigid spheres. Particle confinement is achieved by combining arbitrary combinations of four provided wall primitives, namely infinite plane, finite disk, infinite cylinder, and finite cylinder, and degenerate cases of these. Various wall movements, including translation, oscillation, and rotation, are supported. We provide full derivations of collision prediction and resolution equations for all geometries and motions. Several tests of the method are described, including a model granular “atmosphere” that achieves correct energy equipartition, and a series of tumbler simulations that show the expected transition from tumbling to centrifuging as a function of rotation rate.     

Conditionally periodic solutions of the canonical systems of differential equations in non-autonomous resonant case

A formal method of constructing of conditionally periodic solutions of canonical systems of differential equations in the vicinity of a commemsurability of frequencies is proposed. The method is a union of the rapid convergence method and (well-known in celestial mechanics) Delaunay-Zeipel's method of canonical transformations. For a successful application of the method an existence of stationary resonant solutions of an averaged system of the differential equations is necessary.     

Unsteady hydromagnetic free-convection flow with radiative heat transfer in a rotating fluid

We study the unsteady free-convection flow near a moving infinite flat plate in a totating medium by imposing a time-dependent perturbation on a constant plate temperature. The temperatures involved are assumed to be very large so that radiative heat transfer is significant, which renders the problem very nonlinear even on the assumption of a differential approximation for the radiative flux. When the perturbation is small, the transient flow is tackled by the Laplace transform technique. Complete first-order solutions are deduced for an impulsive motion.     

The analysis of irregularly observed stochastic astronomical time-series – I. Basics of linear stochastic differential equations

The theory of low-order linear stochastic differential equations is reviewed. Solutions to these equations give the continuous time analogues of discrete time autoregressive time-series. Explicit forms for the power spectra and covariance functions of first- and second-order forms are given. A conceptually simple method is described for fitting continuous time autoregressive models to data. Formulae giving the standard errors of the parameter estimates are derived. Simulated data are used to verify the performance of the methods. Irregularly spaced observations of the two hydrogen-deficient stars FQ Aqr and NO Ser are analysed. In the case of FQ Aqr the best-fitting model is of second order, and describes a quasi-periodicity of about 20 d with an e-folding time of 3.7 d. The NO Ser data are best fitted by a first-order model with an e-folding time of 7.2 d.     

Existence of the solution of Szebehely's equation in three dimensions using a two-parametric family of orbits

The three-dimensional inverse problem is investigated. A quasi-linear system of partial differential equations is derived for the determination of the potential. The solution of this system is studied by a method of differential geometry. A necessary condition for the solution is derived and the determination of the potential is reduced to algebraic equations written in vectorial form. A few examples are also given.     

A multi-spiral set of solutions of a 3-D Hamiltonian system

Various sets of periodic solutions of a 3-D Hamiltonian system crossing perpendicularly thez=0 plane are presented. These sets form a main multi-spiral pattern and two secondary ones which have three focal points. The main pattern is inside a stochastic region that surrounds a simple complex unstable periodic orbit, while the two secondary patterns are parts of a stochastic sea. Through these regions the stochastic region communicates with the stochastic sea.     

Unsteady magnetic boundary-layer flow of power-law non-Newtonian conducting fluid through a porous medium past an infinite porous flat plate

In this paper, the nonsteady flow of non-Newtonian power-law conducting fluid through a porous medium past an infinite porous plate is investigated. The system is stressed by a constant transverse magnetic field. The velocity outside the boundary layer depends exponentially on time. The rheological effects are shown and discussed on the shear stress in terms of rheological parameter of power-law fluid. The approximate solution in a closed form were obtained by using the Galerkin method. Also the effect of the magnetic field and permeability parameter are discussed.