Numerical search for periodic solutions in the vicinity of the figure-eight orbit: slaloming around singularities on the shape sphere

We present the results of a numerical search for periodic orbits with zero angular momentum in the Newtonian planar three-body problem with equal masses focused on a (narrow) search window bracketing the figure-eight initial conditions. We found eleven solutions that can be described as some power of the “figure-eight” solution in the sense of the topological classification method. One of these solutions, with the seventh power of the “figure-eight”, is a choreography. We show numerical evidence of its stability.     

Probing the Nekhoroshev Stability of Asteroids

We apply the spectral formulation of the Nekhoroshev theorem to investigate the long-term stability of real main belt asteroids. We find numerical indication that some asteroids are in the so-called Nekhoroshev stability regime, that is they are on chaotic orbits but their motion is stable over very long times. We have analyzed the motion of bodies in different regions of the belt, to assess the sensitivity of our method. We found that it allows us to clearly discriminate between different dynamical regimes, such as the one described by the Nekhoroshev stability, the one well described by the KAM theory, and the unstable chaotic regime in which diffusion in phase space can be detected over time spans much shorter than the age of the solar system.     

Expansion of the planetary disturbing function

Some methods are described for the expansion of the disturbing function in planetary theory. One method uses the classical binomial expansion theorem or a successive approximation process derived from it. Another method is a direct application of the Laplace series expansions. For both methods it is proposed to first prepare the series to be manipulated by a scaling operation. These methods can be applied either in a literal or in a numerical form, or any combination of both, but they are especially designed for use on a large scale digital computer with standard Poisson series programs. No usage is made of Newcomb operators or derivatives of Laplace coefficients.     

Free-free opacity in strong magnetic fields

A way of computing the absorption cross-section for photons on electrons undergoing free-free transitions in magnetic plasma is described. Theoretical expressions for the free-free cross sections in magnetic plasma are given in a representation in which they can be easily compared with the classical results in the absence of the magnetic field. The results of numerical computations of these cross-sections are also presented and discussed. Finally the free-free cross-sections are averaged over the electron states in magnetic plasma in thermal equilibrium, yielding the opacity coefficient as a function of photons frequency. The results of numerical computations are given in graphical form.     

Recent developments of integrating the gravitational problem ofN-bodies

This paper discusses the formulation and the numerical integration of large systems of differential equations occurring in the gravitational problem ofn-bodies.Different forms of the pertinent differential equations of motion are presented, and various regularizing and smoothing transformations are compared. Details regarding the effectiveness and the efficiency of the Kustaanheimo-Stiefel and of other methods are discussed. In particular, a method is described in which some of the phase variables are treated in the regularized system and others in the ordinary system. This mixed method of numerical regularization offers some advantages.Several numerical integration techniques are compared. A high order Runge-Kutta method, Steffensen's method, and a finite difference method are investigated, especially with regard to their adaptability to regularization.The role of integrals and integral invariants is displayed in controlling the accuracy of the numerical integration.Numerical results are described with 5, 25 and 500 bodies participating. These examples compare the various integration techniques, several regularization methods and different logics in treating binaries.     

On the physics of resonant disk-satellite interaction

Within the framework of a single derivation, we study the transfer of angular momentum in a disk subjected to a linear perturbation at Lindblad resonance, whenever the physics include friction, nonstationarity, or self-gravitation, pressure, and viscosity. Each of the above physical processes can be described by one parameter which indicates the main physics at work and the resonance width. We show that dissipation or waves are not formally necessary for a torque to appear, but only for the problem to remain stationary and/or linear. In this framework, the torque exerted at an isolated resonance is independent of the particular physics at work. We consider applications to numerical simulations, the impulse approximation, planetesimal accretion, and edges and gaps in planetary rings.     

Numerical simulations of the Hyades dynamics and the nature of the moving Hyades cluster

We present our numerical simulations of the dynamical evolution of the Hyades open cluster. The simulations were performed usinga modified NBODY6 algorithm that included tidal forces and a realistic orbit of the cluster in a gravitational field described by the Miyamoto-Nagai potential. Our goal was to study the nature of movingclu sters. We show that the stars that were earlier cluster members could be later identified within a sphere of 50 pc in diameter around the Sun. The number of such stars for the chosen initial mass and virial radius of the cluster does not exceed ten. The maximum space velocity of these stars relative to the core of the current cluster does not exceed 3 km s^?1. Our numerical simulations confirm the assumption that some of the moving clusters near the Sun could consist of stars that have escaped from open clusters in the course of their dynamical evolution.     

Computing secular motion under slowly rotating quadratic perturbation

We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potential that can be approximated to sufficient accuracy by expanding it to second order in the coordinates. After averaging over time to obtain the secular Hamiltonian, we use angular momentum and eccentricity vectors as elements. The method of variation of constants then leads to a set of equations of motion that are simple and regular, thus allowing efficient numerical integration. Some possible applications are briefly described.     

Numerical model of the influence function of deformable mirrors based on Bessel Fourier orthogonal functions

A numerical model is presented to simulate the influence function of deformable mirror actuators. The numerical model is formed by Bessel Fourier orthogonal functions, which are constituted of Bessel orthogonal functions and a Fourier basis. A detailed comparison is presented between the new Bessel Fourier model, the Zernike model, the Gaussian influence function and the modified Gaussian influence function. Numerical experiments indicate that the new numerical model is easy to use and more accurate compared with other numerical models. The new numerical model can be used for describing deformable mirror performances and numerical simulations of adaptive optics systems.     

Computation of Partial Derivatives of Perturbed Motion Using the Arcs of Osculating and Superosculating Orbits

The methods for analytical determination of partial derivatives of the current parameters of motion with respect to their initial values are described. The methods take into account principal perturbations and are based on the use of the osculating and superosculating intermediate orbits constructed earlier by the author. These orbits ensure the first-, second-, and third-order contact to the real trajectory at the initial time. The solution for parameters of the intermediate motion and partial derivatives of these parameters is given in a universal closed form. The partial derivatives on long time intervals are computed using a step-by-step procedure combined with the Encke method of special perturbations, in which the intermediate orbits are used as the reference. The numerical results show that the new approach can be efficiently used for solving the problem of differential correction of orbits of asteroids and comets on the basis of observational data.     

Accurate integration of geostationary orbits with Burdet's focal elements

The theory of Burdet's focal elements is outlined. The differential equations are presented, and the initial value problem is described together with the transformation to rectangular coordinates and classical elements. The focal elements are well defined for zero eccentricity and inclination. They can be adopted for the computation of elliptic, parabolic and hyperbolic motion. For the numerical integration of near-geostationary orbits a comparison of the efficiency is made between focal elements, KS theory and rectangular coordinates. For this class of orbits, a higher accuracy has been obtained by integrating elements than integrating rectangular coordinates.     

Notes on obtaining the eigenvalues of Laplace''s tidal equation

A review is undertaken of the various forms that have been obtained for the recurrence relation from which the eigenvalues of Laplace's tidal equation may be obtained. Such forms are shown to be analytically consistent and are discussed in relation to their subsequent numerical evaluation. By determining eigenvalues of equivalent depth for a given frequency of oscillation instead of the other way around the problem becomes a straightforward one of matrix diagonalization. If solutions are based on normalized Legendre functions the matrix is symmetric. A method of evaluating the related wind functions is described.     

Remarks on the numerical integration of near-parabolic orbits

For near-parabolic orbits the distinction between coordinates and elements disappears provided the KS-technique is used. In KS-variables a pure parabolic motion is described by linear functions. Advantage is taken of that fact for establishing numerical procedures in perturbed near-parabolic cases.     

Opacity errors in model atmospheres

Different issues involved in the numerical evaluation of continuous absorption by hydrogenlike atoms and ions have been analyzed. Usually, the calculation of the opacity coefficient is explicitly performed for a predetermined number of levels and the higher ones are treated as an integral; it is shown that there are some cases where as many as 17 discrete levels need to be added before switching to the integral approximation if an error smaller than 0.1% is wanted in the results (a condition not always fulfilled with model atmosphere codes being used). Also, approximations to Gaunt factors are analyzed. Though in some cases approximations can be used which are very good, there are other occasions when it should be advisable to switch to interpolating in original tables, or to use more elaborated expressions (such as Hummer's for the Gaunt free-free factor). A numerical procedure for solving the non linear Pg-Pe-T relation is described in an Appendix.     

Some aspects of the solution of vector transfer equation in a magnetized medium

A simplification of the numerical method of solving the vector transfer equation, given earlier by Nagendra and Peraiah (1985a), is described for problems which involve only absorption. This allows us to attempt to solve under realistic conditions and with reduced computing efforts, the important problems of polarization of light emerging from magnetized stars. For the purpose of illustration, the equations described are used for solving the continuum and Zeeman line transfer problems.     

Power-series solutions of the Lane-Emden equation

In this paper the problem of representing the numerical solutions of the Lane-Emden equation analytically by means of a convergent power series has been considered. Our results show that it is possible to represent the numerical solutions of the Lane-Emden equation by means of a power series which can be convergent in the whole interior of a polytropic model.     

Magnetic Flux Transport in the ISM Through Turbulent Ambipolar Diffusion

Under ideal MHD conditions the magnetic field strength should be correlated with density in the interstellar medium (ISM). However, observations indicate that this correlation is weaker than expected. Ambipolar diffusion can decrease the flux-to-mass ratio in weakly ionized media; however, it is generally thought to be too slow to play a significant role in the ISM except in the densest molecular clouds. Turbulence is often invoked in other astrophysical problems to increase transport rates above the (very slow) diffusive values. Building on analytical studies, we test with numerical models whether turbulence can enhance the ambipolar diffusion rate sufficiently to explain the observed weak correlations. The numerical method is based on a gas-kinetic scheme with very low numerical diffusivity, thus allowing us to separate numerical and physical diffusion effects.     

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 perturbations of the five outer planets by the four inner ones

The expressions given by Clemence were checked by the comparison of numerical values of their second differentials with the numerical values of the perturbing forces. Agreement was good in most cases, save that the use of the second differentials unduly magnified some terms rightly neglected by Clemence. In the appendix the work of Clemence is compared with that of Carpenter and the differences are found to be all smaller than the contributions from the secular variation of the eccentricity, which are so small that they were neglected by Clemence. The differences can be removed by considering the secular variation of the eccentricity and by making a small adjustment in the integration constants.     

Integration rules from the equivalent differential equation method applied to equations of motion

The numerical integration of equations of motion necessarily implies the presence of errors that depend on initial conditions as well as the different physical parameters under consideration. More particularly, dumping or dissipative terms can appear and it is especially interesting to determine its causes. The equivalent differential equation method may allow the errors from a certain numerical scheme to be analyzed and, together with other considerations, can help us to eliminate or reduce them.