Spectra of stretching numbers and helicity angles in dynamical systems

We define a stretching number (or Lyapunov characteristic number for one period) (or stretching number) a = In % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada% Wcaaqaaiabe67a4jaadshacqGHRaWkcaaIXaaabaGaeqOVdGNaamiD% aaaaaiaawEa7caGLiWoaaaa!3F1E!\[\left| {\frac{{\xi t + 1}}{{\xi t}}} \right|\]as the logarithm of the ratio of deviations from a given orbit at times t and t + 1. Similarly we define a helicity angle as the angle between the deviation % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaam% iDaaaa!3793!\[\xi t\]and a fixed direction. The distributions of the stretching numbers and helicity angles (spectra) are invariant with respect to initial conditions in a connected chaotic domain. We study such spectra in conservative and dissipative mappings of 2 degrees of freedom and in conservative mappings of 3-degrees of freedom. In 2-D conservative systems we found that the lines of constant stretching number have a fractal form.     

The use of symplectic algorithm in the computation of Lyapunov characteristic numbers

It is found that, in the computation of Lyapunov characteristic numbers, symplectic algorithms are clearly more advantageous than non-symplectic ones.     

Investigation of dynamical chaotization in stellar systems with double massive centers using the Ricci curvature criterion

The Ricci curvature criterion is used to investigate the relative instability of various configurations of N-body gravitational systems. Systems with double massive centers are shown to be less stable than homogeneous systems or systems with single massive centers. In general, this is a confirmation that the Ricci curvature criterion is efficient in studying N-body systems by means of relatively simple computations.     

Generalized Lyapunov characteristic indicators and corresponding Kolmogorov like entropy of the standard mapping

Lyapunov characteristic indicators are currently defined as the mean, i.e. the first moment, of the distribution of the local variations of the tangent vector to the flow. Higher moments of the distribution give further informations about the fluctuations around the average.     

Television spectra of meteors

The first results of the television observations of meteors at the Ondejov Observatory are presented. It is shown that three spectral components may be distinguished in meteors: cool meteoric, hot meteoric and hot atmospheric. The intensity ratio of these components varies strongly even in meteors of the same velocity and within the records of single meteors. This is evidence for variations in the ablation process and in the formation of the shock wave. The so called calcium anomaly is in fact only a demonstration of these variations.     

Covariant formulation of the dynamical equations of quantum vortices in type II superconductors

We have derived the closed system of covariant equations which describe the motion of quantum vortices regarded as a two-2dimensional polarized liquid. We have obtained the covariant expressions of the forces acting on the vortices; from the equilibrium condition of these forces we have deduced the equation satisfied by the velocity field of the fluid. It is shown that this velocity field depends on the friction coefficient, the density of vortices and the superconducting current. From this closed system of equations we derived the relaxation equation when a variable magnetic field is applied. Published in Astrofizika, Vol. 50, No. 3, pp. 381–391 (August 2007).     

A numerical solution of the dynamical instability of a thin accretion torus

We consider the non-axisymmetric, dynamical instability of a thin accretion torus with a non-zero shift of corotation radius. By numerical method we evaluated the wave number dependence of the linear rate of growth of instability and the co-rotation shift. The rate of growth is only slightly affected by the non-zero co-rotation shift, while the dispersion relation in the case of a shift is the same as that of the linear KdV equation. This shows that the “planet-like” solution found in numerical simulations of thin tori is very probably analogous to the soliton solution of the KdV equation.     

Chemical and dynamical evolution of spiral galaxies

We present the very first results of a new 3D numerical model for the formation and evolution of spiral galaxies along the Hubble sequence. We take into account the hydrodynamical properties of the gas with an SPH method while we use a tree code for the gravitational forces of the dark matter and stars. The chemical evolution is also fully included, with both SNe Ia and SNe II explosions being followed, and this will allows us to predict abundances of various chemical species, abundance ratios and their radial distributions. This revised version was published online in September 2006 with corrections to the Cover Date.     

High resolution spectra of galaxies

We present inversion techniques which aim at recovering the composite nature and the kinematics of a stellar population from its high resolution absorption line spectrum. The originality of the combined inversion is its potential to recover both the stellar content and the kinematics simultaneously. These techniques use new synthetic high resolution spectra produced by PéGASE and minimization algorithms. We apply them to mock data representing the bulge and disk population of the inner region of spiral galaxies. This revised version was published online in August 2006 with corrections to the Cover Date.     

Approximate general solution of 2-degrees of freedom dynamical systems using `les solutions precieuses'

This paper presents the procedure of a computational scheme leading to approximate general solution of the axi-symmetric,2-degrees of freedom dynamical systems. Also the results of application of this scheme in two such systems of the non-linear double oscillator with third and fifth order potentials in position variables. Their approximate general solution is constructed by computing a dense set of families of periodic solutions and their presentation is made through plots of initial conditions. The accuracy of the approximate general solution is defined by two error parameters, one giving a measure of the accuracy of the integration and calculation of periodic solutions procedure, and the second the density in the initial conditions space of the periodic solutions calculated. Due to the need to compute families of periodic solutions of large periods the numerical integrations were carried out using the eighth order, variable step, R-K algorithm, which secured for almost all results presented here conservation of the energy constant between 10^-9 and 10^-12 for single runs of any and all solutions. The accuracy of the approximate general solution is controlled by increasing the number of family curves and also by `zooming' into parts of the space of initial conditions. All families of periodic solutions were checked for their stability. The computation of such families within areas of `deterministic chaos' did not encounter any difficulty other than poorer precision. Furthermore, on the basis of the stability study of the computed families, the boundaries of areas of `order' and `chaos' were approximately defined. On the basis of these results it is concluded that investigations in thePoincaré sections have to disclose 3 distinct types of areas of `order' and 2 distinct types of areas of `chaos'. Verification of the `order'/`chaos' boundary calculation was made by working out several Poincaré surfaces of sections. This revised version was published online in July 2006 with corrections to the Cover Date.     

Determination of the dynamical structure of galaxies using optical spectra

Galaxy spectra are a rich source of kinematical information since the shapes of the absorption lines reflect the movement of stars along the line-of-sight. We present a technique with which to build directly a dynamical model for a galaxy by fitting model spectra, calculated from a dynamical model, to the observed galaxy spectra. Using synthetic spectra from a known galaxy model we demonstrate that this technique indeed recovers the essential dynamical characteristics of the galaxy model. Moreover, the method allows a statistically meaningful error analysis on the resulting dynamical quantities.     

Optical spectra of four objects identified with variable radio sources

We obtained optical spectra of four objects identified with variable radio sources. Three objects (0029+0554, 0400+0550, 2245+0500) were found to be quasars with redshifts of 1.314, 0.761, and 1.091. One object (2349+0534) has a continuum spectrum characteristic of BL Lac objects. We analyze spectra of the radio sources in the range 0.97–21.7 GHz for the epoch 1997 and in the range 3.9–11.1 GHz for the epoch 1990, as well as the pattern of variability of their flux densities on time scales of 1.5 and 7 years.     

Hard spectra of X-ray pulsars from INTEGRAL data

We present spectra for 34 accretion-powers X-ray pulsars and one millisecond pulsar that were within the field of view of the INTEGRAL observatory over two years (December 2002–January 2005) of its in-orbit operation and that were detected by its instruments at a statistically significant level (> 8σ in the energy range 18–60 keV). There are seven recently discovered objects of this class among the pulsars studied: 2RXP J130159.6-635806, IGR/AX J16320-4751, IGR J16358-4726, AX J163904-4642, IGR J16465-4507, SAX/IGR J18027-2017, and AX J1841.0-0535. We have also obtained hard X-ray (>20 keV) spectra for the accretion-powered pulsars RX J0146.9+6121, AX J1820.5-1434, and AX J1841.0-0535 for the first time. We analyze the evolution of spectral parameters as a function of the intensity of the sources and compare these with the results of previous studies.     

Analysis of COMPTEL gamma-ray burst locations and spectra

In its first three years of operation, the COMPTEL instrument on theCompton Gamma-Ray Observatory has measured the locations (mean accuracy 1°) and spectra (0.75-30 MeV) of 18 gamma-ray bursts and continues to observe new events at a rate of 1/month. With good angular resolution and sensitivity at MeV energies, the growing COMPTEL burst catalog is an important new piece of evidence in the on-going GRB mystery. The COMPTEL burst locations are consistent with an isotropic distribution of sources, yet the spatial coincidence of two of the bursts indicates the possibility of repetition. The COMPTEL burst spectra are in most cases consistent with a single power law model with spectral index in the range 2–3. However, two bursts show evidence of a spectral break in the MeV range. Measurement of rapid variability at MeV energies in the stronger bursts provides evidence that either the sources are nearby (within the Galaxy) or the gamma-ray emission is relativistically beamed. We present an overview of analysis results obtained from the COMPTEL burst catalog concentrating on the search for burst repetition and the implications of highly variable MeV emission.     

The dynamical stability of W Ursae Majoris-type systems

Theoretical study indicates that a contact binary system would merge into a rapidly rotating single star due to tidal instability when the spin angular momentum of the system is more than a third of its orbital angular momentum. Assuming that W Ursae Majoris (W UMa) contact binary systems rigorously comply with the Roche geometry and the dynamical stability limit is at a contact degree of about 70 per cent, we obtain that W UMa systems might suffer Darwin's instability when their mass ratios are in a region of about 0.076–0.078 and merge into the fast-rotating stars. This suggests that the W UMa systems with mass ratio   q ≤ 0.076  cannot be observed. Meanwhile, we find that the observed W UMa systems with a mass ratio of about 0.077, corresponding to a contact degree of about 86 per cent would suffer tidal instability and merge into the single fast-rotating stars. This suggests that the dynamical stability limit for the observed W UMa systems is higher than the theoretical value, implying that the observed systems have probably suffered the loss of angular momentum due to gravitational wave radiation (GR) or magnetic stellar wind (MSW).     

Dynamics of rotating triple systems

We investigate the dynamical evolution of 100 000 rotating triple systems with equal-mass components. The system rotation is specified by the parameter ω=?c²E, where c and E are the angular momentum and total energy of the triple system, respectively. We consider ω=0.1,1, 2, 4, 6 and study 20 000 triple systems with randomly specified coordinates and velocities of the bodies for each ω. We consider two methods for specifying initial conditions: with and without a hierarchical structure at the beginning of the evolution. The evolution of each system is traced until the escape of one of the bodies or until the critical time equal to 1000 mean system crossing times. For each set of initial conditions, we computed parameters of the final motions: orbital parameters for the final binary and the escaping body. We analyze variations in the statistical characteristics of the distributions of these parameters with ω. The mean disruption time of triple systems and the fraction of the systems that have not been disrupted in 1000 mean crossing times increase with ω. The final binaries become, on average, wider at larger angular momenta. The distribution of their eccentricities does not depend on ω and generally agrees with the theoretical law f(e)=2e. The velocities of the escaping bodies, on average, decrease with increasing angular momentum of the triple system. The fraction of the angles between the escaping-body velocity vector and the triple-system angular momentum close to 90° increases with ω. Escapes in the directions opposite to rotation and prograde motions dominate at small and large angular momenta, respectively. For slowly rotating systems, the angular momentum during their disruption is, on average, evenly divided between the escaping body and the final binary, whereas in rapidly rotating systems, about 80% of the angular momentum is carried away by the escaping component. We compare our numerical simulations with the statistical theory of triple-system disruption.     

The occulting effect of the transition region on emergent spectra of accretion discs

There may exist a transition region from the geometrically thick outer region to the geometrically thin inner region in accretion discs. In this paper we calculate the spectra emitted by the portions of the inner region which are not occulted by the transition region, and deduce the emergent spectra of accretion discs to be observed, taking into account the reflection effect of radiation by the transition region. Our results show that the difference between these emergent spectra of discs and the standard ones of steady accretion disc models is significant when the central portion of discs is occulted by the transition region. It might play a role in the explanation of continuous spectra for some dwarf novae.     

Main features of dynamical escape from three-dimensional triple systems

The dynamical evolution of triple systems with equal and unequal-mass components and different initial velocities is studied. It is shown that, in general, the statistical results for the planar and three-dimensional triple systems do not differ significantly. Most (about 85%) of the systems disrupt; the escape of one component occurs after a triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (about 15%) of stable or long-lived systems is formed if the angular momentum is non-zero. Averages, distributions and coefficients of correlations of evolutionary characteristics are presented: the life-time, angular momentum, numbers of wide and close triple approaches of bodies, relative energy of escapers, minimum perimeter during the last triple approach resulting in escape, elements of orbits of the final binary and escaper.     

Fourier analysis of local Lyapunov characteristic exponents for satellite-type motions

For a binary star system with equal masses the transition from regular to stochastic motion has been determined by means of the Lyapunov Characteristic Numbers (LCN) in a paper of Gonczi and Froeschlé (1981), where the initial distance of the massless body to the second primary was used as parameter of integration. This time we analyse the Fourier spectra of the local LCNs (ln _k ⁱ ) of this system for different intervals of time.     

Estimate of the Transition Value of Librational Invariant Curves

We investigate the break-down threshold of librational invariant curves. As a model problem, we consider a variant of a mapping introduced by M. Hénon, which well describes the dynamics of librational motions surrounding a stable invariant point. We verify in concrete examples the applicability of Greene's method, by computing the instability transition values of a sequence of periodic orbits approaching an invariant curve with fixed noble frequency. However, this method requires the knowledge of the location of the periodic orbits within a very good approximation. This task appears to be difficult to realize for a libration regime, due to the different topology of the phase space. To compute the break-down threshold, we tried an alternative method very easy to implement, based on the computation of the fast Lyapunov indicators and frequency analysis. Such technique does not require the knowledge of the periodic orbits, but again, it appears very difficult to have a precision better than Greene's method for the computation of the critical parameter.