Comparing maps to symplectic integrators in a galactic type Hamiltonian

We obtain thex - p _xPoincare phase plane for a two dimensional, resonant, galactic type Hamiltonian using conventional numerical integration, a second order symplectic integrator and a map based on the averaged Hamiltonian. It is found that all three methods give good results, for small values of the perturbation parameter, while the symplectic integrator does a better job than the mapping, for large perturbations. The dynamical spectra are used to distinguish between regular and chaotic motion.     

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.     

Report on the dynamical evolution of an axially symmetric quasar model

The role of the angular momentum in the regular or chaotic character of motion in an axially symmetric quasar model is examined. It is found that, for a given value of the critical angular momentumL _zc , there are two values of the mass of the nucleusM _n for which transition from regular to chaotic motion occurs. The [L _zc – M _n ] relationship shows a linear dependence for the time independent model and an exponential dependence for the evolving model. Both cases are explained using theoretical arguments together with some numerical evidence. The evolution of the orbits is studied, as mass is transported from the disk to the nucleus. The results are compared with the outcomes derived for galactic models with massive nuclei.     

Statistics of the Big Blue Bump Feature in a Low Redshift Sample of Active Galactic Nuclei

We present a statistical analysis of the big blue bump (BBB) feature for a large heterogeneous sample of 95 optically selected and soft X-ray bright, low redshift active galactic nuclei (AGNs). This sample covers a sufficiently broad luminosity range, allowing us to test the luminosity dependence of the spectral energy distribution in the BBB region. Following the works of Zheng et al., Laor et al. and Kriss et al., we introduce the broad band spectral index from 1050 Å to0.5 keV (α _UV-SX ), compare its distribution with that of the soft X-ray spectral index (α _SX ) obtained by ROSAT PSPC, and find that the two indices have equal average-values within 1 ～ 2σuncertainties, whether in the whole sample, in luminosity divisions or in subsamples. These equalities also have no obvious luminosity dependence. This indicates that a single power law can describe the overall UV toX-ray spectrum in a statistical sense, or the broad band UV to soft X-ray spectrum is the soft X-ray spectral extension on an average. Thus, our results support Laor et al.'s conjecture about the BBB peak aroundFUV 1050 Å from a statistical viewpoint. As we further test whether the equality holds for individual objects within measure errors, χ₂ test refuse to accept it. In addition, our statistical results, from the luminosity divisions and on the correlation of spectral indices with luminosity (M _B), imply that the luminosity dependence of α _UV and α _UV-SX is mainly due to absorption in low luminosity AGNs.     

A mapping for the study of the 1/1 resonance in a galactic type Hamiltonian

We derive an algebraic mapping for an autonomous, two-dimensional galactic type Hamiltonian in the 1/1 resonance case. We use the mapping to study the stability of the periodic orbits. Using the x — p _x Poincaré surface section, we compare the results of the mapping with those found by the numerical integration of the full equations of motion. For small values of the perturbation the results of the two methods are in very good agreement while satisfactory agreement is obtained for larger perturbations.     

Towards the Unified Theory of Slow Modes in Stellar Galactic Discs

This paper considers the formation of stellar galactic structures, which are assumed to be slow modes in a disc of orbits precessing at different speeds. The mode pattern speeds, Ω_p, are eigen-values of a Fredholm integral operator. Its general analysis shows the existence of two types of eigen-functions, bar-like and spiral. The bars grow through the immediate action of mode gravitational fields on the stars near the corotation and the outer Lindblad resonance. The excitation of spirals is due to the inner Lindblad resonance. Apparently, the commonly used swing amplification mechanism does not play any role in the formation of both bar-modes and grand design spiral modes. However, it can be essential in the formation of transient excitations when the normal global mode cannot be organized. This revised version was published online in July 2006 with corrections to the Cover Date.     

Frequency analysis of a dynamical system

Frequency analysis is a new method for analyzing the stability of orbits in a conservative dynamical system. It was first devised in order to study the stability of the solar system (Laskar, Icarus, 88, 1990). It is a powerful method for analyzing weakly chaotic motion in hamiltonian systems or symplectic maps. For regular motions, it yields an analytical representation of the solutions. In cases of 2 degrees of freedom system with monotonous torsion, precise numerical criterions for the destruction of KAM tori can be found. For a 4D symplectic map, plotting the frequency map in the frequency plane provides a clear representation of the global dynamics and describes the actual Arnold web of the system.     

A Symplectic Mapping Model for the Study of 2:3 Resonant Trans-Neptunian Motion

A symplectic mapping is constructed for the study of the dynamical evolution of Edgeworth-Kuiper belt objects near the 2:3 mean motion resonance with Neptune. The mapping is six-dimensional and is a good model for the Poincaré map of the real system, that is, the spatial elliptic restricted three-body problem at the 2:3 resonance, with the Sun and Neptune as primaries. The mapping model is based on the averaged Hamiltonian, corrected by a semianalytic method so that it has the basic topological properties of the phase space of the real system both qualitatively and quantitatively. We start with two dimensional motion and then we extend it to three dimensions. Both chaotic and regular motion is observed, depending on the objects' initial inclination and phase. For zero inclination, objects that are phase-protected from close encounters with Neptune show ordered motion even at eccentricities as large as 0.4 and despite being Neptune-crossers. On the other hand, not-phase-protected objects with eccentricities greater than 0.15 follow chaotic motion that leads to sudden jumps in their eccentricity and are removed from the 2:3 resonance, thus becoming short period comets. As inclination increases, chaotic motion becomes more widespread, but phase-protection still exists and, as a result, stable motion appears for eccentricities up to e = 0.3 and inclinations as high as i = 15°, a region where plutinos exist.     

The Spin-Orbit Resonant Rotation of Mercury: A Two Degree of Freedom Hamiltonian Model

The paper develops a hamiltonian formulation describing the coupled orbital and spin motions of a rigid Mercury rotation about its axis of maximum moment of inertia in the frame of a 3:2 spin orbit resonance; the (ecliptic) obliquity is not constant, the gravitational potential of mercury is developed up to the second degree terms (the only ones for which an approximate numerical value can be given) and is reduced to a two degree of freedom model in the absence of planetary perturbations. Four equilibria can be calculated, corresponding to four different values of the (ecliptic) obliquity. The present situation of Mercury corresponds to one of them, which is proved to be stable. We introduce action-angle variables in the neighborhood of this stable equilibrium, by several successive canonical transformations, so to get two constant frequencies, the first one for the free spin-orbit libration, the other one for the 1:1 resonant precession of both nodes (orbital and rotational) on the ecliptic plane. The numerical values obtained by this simplified model are in perfect agreement with those obtained by Rambaux and Bois [Astron. Astrophys. 413, 381–393]. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Galactic Center: a laboratory for AGN?

Summary. This paper reviews the physical state of stars and Interstellar Matter in the Galactic Bulge (radius kpc from the dynamical center of the Galaxy), in the Nuclear Bulge (kpc) and in the Sgr A Radio and GMC Complex, i.e. the central \,pc of our Galaxy. The Galactic Bulge is devoid of cold Interstellar Matter and consists mainly of old stars, while the Nuclear Bulge accounts for of the mass of all of the Interstellar Matter in the Galaxy. A similar ratio holds for the formation rate of medium and high mass stars in Bulge and Disk. The metal abundance of the Interstellar Matter in the Galactic Bulge is found to be . The H-to-CO conversion factors to be applied to molecular gas in the Central Region are by factors 3 (Arimoto et al. 1996) to 10 (Sodroski et al. 1995) lower than in the solar vicinity. Hence, most H masses derived for the Central Region appear to be considerably overestimated. The Nuclear Bulge is pervaded by a thermal plasma (K) which is responsible for the diffuse free-free emission. Lyman continuum photon and dust IR luminosity of the Nuclear Bulge again account for of the respective total luminosities of the Galaxy. Magnetic fields in the Nuclear Bulge are strong (up to mG) as compared with the Galactic Disk (a few tens of G). The field lines are oriented parallel to the galactic plane inside giant molecular clouds and perpendicular to the plane in the intercloud medium. The compact source Sgr A* is close to or at the dynamical center of the Galaxy. Its radio spectrum with a high frequency cut-off at GHz, a low frequency turnover at GHz and a flux density dependence in between can be explained by synchrotron emission from quasi-monoenergetic relativistic electrons. Due to an extinction between Sun and Galactic Center corresponding to , an intrinsic weakness of this source in the near infrared, and a strong background emission from warm dust there are only upper limits available for the flux density of Sgr A* in the far, mid and near infrared and X-ray regime. The size of Sgr A* in the radio regime is cm, its dereddened K-band flux density is mJy, its luminosity has upper limits of (if radiation comes from an Accretion Disk) and (if black-body radiation from an object with a single temperature of K is assumed). If anyone of the soft X-ray sources detected by ROSAT actually coincides with Sgr A*, its X-ray luminosity would be less than a few . With a dark mass of Sgr A* is the best candidate for a starving black hole, although there are no observational indications for the presence of a (Standard) Accretion Disk. While the radio/IR spectrum of Sgr A* is purely nonthermal, the spectrum integrated over the central parsec resembles that of a Seyfert galaxy. Sgr A* is embedded in the Hii region Sgr A West with part of the ionized gas forming a minispiral. Sgr A West is surrounded by the Circum Nuclear Disk, an irregular shaped assembly of molecular gas which extends from pc and rotates around the Galactic Center with an estimated dynamical time scale of \,yr. The total luminosity of of the central parsec is due to the radiation of early-type stars of which have now been directly identified as luminous blue supergiants. It is still debated, however, if these stars can also account for all of the ionization of Sgr A West. In addition, the central parsec contains red giants, AGB stars, and a few super giants of which the brightest are now identified by direct imaging. These stars – together with a few million low mass main sequence stars – account for the bulk of the 2.2\,m emission. The spatial distributions of the three stellar populations in the central pc are remarkably different. Sgr A* is – along the line-of-sight – presumably located close to the center of the Hii region Sgr A West, which in turn is located in front of the extended (pc) synchrotron source Sgr A East, which appears to be the remnant of a gigantic explosion (of the order of the energy of a single supernova explosion) which took place yr ago inside the GMC Sgr A East Core. X-ray observations show within pc a pervasive hot (keV) plasma of expansion age of yr. Both phenomena – as well as the formation of the Circum Nuclear Disk – may have the same origin. Influx of material is observed within the Nuclear Bulge on all distance scales. In the Nuclear Bulge (pc) as well as in the Circum Nuclear Disk (pc) inflow towards the Galactic Center occurs primarily in the galactic plane and amounts to a few . The accretion rate into the central Black Hole, deduced from the luminosity of Sgr A*, however, appears to be lower by at least five orders of magnitude (assuming standard disk accretion). But in an equilibrium state only part of the infalling mass which is not accreted by the Black Hole can be consumed by star formation. A mass inflow rate varying with time is a more natural explanation. Comparing the physical state of the Center of our Galaxy with that of Active Galactic Nuclei derived from observations and modelling, we find that most of the basic characteristics of an AGN are also present in the Galactic Center. Lacking are, however, both the evidence for a standard Accretion Disk and a hard UV spectrum with accompanying high excitation emission lines in the Galactic Center which are characteristic for AGN. The luminosity of the central parsec, , amounts to only of the total luminosity of the Galaxy of . Seen from a distance of M31 (kpc) with an angular resolution of (corresponding to a linear size of pc) the Center of our Galaxy would appear as a mildly active nucleus with some starburst activity and would probably be classified as a weak Seyfert galaxy. The synchrotron spectrum of Sgr A*, however, would be completely masked by reprocessed stellar light (i.e. free-free and dust emission). Received: October 21, 1996     

Spatial Structure of Regular and Chaotic Orbits in Self-Consistent Models of Galactic Satellites

In several previous papers we had investigated the orbits of the stars that make up galactic satellites, finding that many of them were chaotic. Most of the models studied in those works were not self-consistent, the single exception being the Heggie and Ramamani (1995) models; nevertheless, these ones are built from a distribution function that depends on the energy (actually, the Jacobi integral) only, what makes them rather special. Here we built up two self-consistent models of galactic satellites, freezed theirs potential in order to have smooth and stationary fields, and investigated the spatial structure of orbits whose initial positions and velocities were those of the bodies in the self-consistent models. We distinguished between partially chaotic (only one non-zero Lyapunov exponent) and fully chaotic (two non-zero Lyapunov exponents) orbits and showed that, as could be expected from the fact that the former obey an additional local isolating integral, besides the global Jacobi integral, they have different spatial distributions. Moreover, since Lyapunov exponents are computed over finite time intervals, their values reflect the properties of the part of the chaotic sea they are navigating during those intervals and, as a result, when the chaotic orbits are separated in groups of low- and high-valued exponents, significant differences can also be recognized between their spatial distributions. The structure of the satellites can, therefore, be understood as a superposition of several separate subsystems, with different degrees of concentration and trixiality, that can be recognized from the analysis of the Lyapunov exponents of their orbits.     

A Symplectic Mapping Model as a Tool to Understand the Dynamics of 2/1 Resonant Asteroid Motion

We present a 3-D symplectic mapping model that is valid at the 2:1 mean motion resonance in the asteroid motion, in the Sun-Jupiter-asteroid model. This model is used to study the dynamics inside this resonance and several features of the system have been made clear. The introduction of the third dimension, through the inclination of the asteroid orbit, plays an important role in the evolution of the asteroid and the appearance of chaotic motion. Also, the existence of the secondary resonances is clearly shown and their role in the appearance of chaotic motion and the slow diffusion of the elements of the orbit is demonstrated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Generalized Lyapunov exponents indicators in Hamiltonian dynamics: An application to a double star system

The Lyapunov characteristic numbers (LCNs) which are defined as the mean value of the distribution of the local variations of the tangent vectors to the flow (=ln _k ⁱ ) (see Froeschlé, 1984) have been found to be sensitive indicators of stochasticity. So we computed the distribution of these local variations and determined the moments of higher order for the integrable and stochastic regions in a binary star system with =0.5.     

Effective Hamiltonian for the D'Alembert Planetary Model Near a Spin/Orbit Resonance

The D'Alembert model for the spin/orbit problem in celestial mechanics is considered. Using a Hamiltonian formalism, it is shown that in a small neighborhood of a p:q spin/orbit resonance with (p,q) different from (1,1) and (2,1) the 'effective' D'Alembert Hamiltonian is a completely integrable system with phase space foliated by maximal invariant curves; instead, in a small neighborhood of a p:q spin/orbit resonance with (p,q) equal to (1,1) or (2,1) the 'effective' D'Alembert Hamiltonian has a phase portrait similar to that of the standard pendulum (elliptic and hyperbolic equilibria, separatrices, invariant curves of different homotopy). A fast averaging with respect to the 'mean anomaly' is also performed (by means of Nekhoroshev techniques) showing that, up to exponentially small terms, the resonant D'Alembert Hamiltonian is described by a two-degrees-of-freedom, properly degenerate Hamiltonian having the lowest order terms corresponding to the 'effective' Hamiltonian mentioned above.     

The mass distribution in the innermost regions of spiral galaxies

We use high-spatial resolution (100 pc) rotation curves of 83 spiral galaxies to investigate the mass distribution of their innermost kpc. We show that, in this region, the luminous matter completely accounts for the gravitational potential and no dark component is required. The derived I-band disk mass-to-light ratios agree well with those obtained from population synthesis models and correlate with color in a similar way. We find strict upper limits of 10⁷ M for the masses of compact bodies at the center of spirals, ruling out that these systems host the remnants of the quasar activity.     

Morphology and kinematics of the ionised gas in early-type galaxies

We present results of our ongoing study of the morphology and kinematics of the ionised gas in 48 representative nearby elliptical and lenticular galaxies using the SAURON integral-field spectrograph on the 4.2m William Herschel Telescope. Making use of a recently developed technique, emission is detected in 75% of the galaxies. The ionised-gas distributions display varied morphologies, ranging from regular gas disks to filamentary structures. Additionally, the emission-line kinematic maps show, in general, regular motions with smooth variations in kinematic position angle. In most of the galaxies, the ionised-gas kinematics is decoupled from the stellar counterpart, but only some of them present signatures of recent accretion of gaseous material. The presence of dust is very common in our sample and is usually accompanied by gas emission. Our analysis of the [Oiii]/Hβ emission-line ratios, both across the whole sample as well as within the individual galaxies, suggests that there is no unique mechanism triggering the ionisation of the gas.