Exploring the nature of orbits in a galactic model with a massive nucleus

In the present article, we use an axially symmetric galactic gravitational model with a disk–halo and a spherical nucleus, in order to investigate the transition from regular to chaotic motion for stars moving in the meridian (r,z) plane. We study in detail the transition from regular to chaotic motion, in two different cases: the time independent model and the time evolving model. In both cases, we explored all the available range regarding the values of the main involved parameters of the dynamical system. In the time dependent model, we follow the evolution of orbits as the galaxy develops a dense and massive nucleus in its core, as mass is transported exponentially from the disk to the galactic center. We apply the classical method of the Poincaré (r,p_r) phase plane, in order to distinguish between ordered and chaotic motion. The Lyapunov Characteristic Exponent is used, to make an estimation of the degree of chaos in our galactic model and also to help us to study the time dependent model. In addition, we construct some numerical diagrams in which we present the correlations between the main parameters of our galactic model. Our numerical calculations indicate, that stars with values of angular momentum L_z less than or equal to a critical value L_zc, moving near to the galactic plane, are scattered to the halo upon encountering the nuclear region and subsequently display chaotic motion. A linear relationship exists between the critical value of the angular momentum L_zc and the mass of the nucleus M_n. Furthermore, the extent of the chaotic region increases as the value of the mass of the nucleus increases. Moreover, our simulations indicate that the degree of chaos increases linearly, as the mass of the nucleus increases. A comparison is made between the critical value L_zc and the circular angular momentum L_z0 at different distances from the galactic center. In the time dependent model, there are orbits that change their orbital character from regular to chaotic and vise versa and also orbits that maintain their character during the galactic evolution. These results strongly indicate that the ordered or chaotic nature of orbits, depends on the presence of massive objects in the galactic cores of the galaxies. Our results suggest, that for disk galaxies with massive and prominent nuclei, the low angular momentum stars in the associated central regions of the galaxy, must be in predominantly chaotic orbits. Some theoretical arguments to support the numerically derived outcomes are presented. Comparison with similar previous works is also made.     

The evolution of chaos in active galaxy models with an oblate or a prolate dark halo component

The evolution of chaotic motion in a galactic dynamical model with a disk, a dense nucleus and a flat biaxial dark halo component is investigated. Two cases are studied: (i) the case where the halo component is oblate and (ii) the case where a prolate halo is present. In both cases, numerical calculations show that the extent of the chaotic regions decreases exponentially as the scale‐length of the dark halo increases. On the other hand, a linear relationship exists between the extent of the chaotic regions and the flatness parameter of the halo component. A linear relationship between the critical value of the angular momentum and the flatness parameter is also found. Some theoretical arguments to support the numerical outcomes are presented. An estimation of the degree of chaos is made by computing the Lyapunov Characteristic Exponents. Comparison with earlier work is also made (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determining the character of motion in quiet and active galaxies with a satellite companion

A galaxy model with a satellite companion is used to study the character of motion for stars moving in the x ‐y plane. It is observed that a large part of the phase plane is covered by chaotic orbits. The percentage of chaotic orbits increases when the galaxy has a dense nucleus of massM_n. The presence of the dense nucleus also increases the stellar velocities near the center of the galaxy. For small values of the distance R between the two bodies, low energy stars display a chaotic region near the centre of the galaxy, when the dense nucleus is present, while for larger values of R the motion in active galaxies is regular for low energy stars. Our results suggest that in galaxies with a satellite companion, the chaotic character of motion is not only a result of galactic interaction but also a result caused by the dense nucleus. Theoretical arguments are used to support the numerical outcomes. We follow the evolution of the galaxy, as mass is transported adiabatically from the disk to the nucleus. Our numerical results are in satisfactory agreement with observational data from M51‐type binary galaxies (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Do active galaxies have a massive halo component?

We present a dynamical model for an active galaxy. Our model is a mass model with a disk, nucleus, and halo components. Numerical calculations and theoretical evidence show, that for a fixed value of mass of the galaxy the stellar velocities in the central region decrease as the mass of halo increases. Furthermore, the motion tends to be regular while, when the halo component is absent, the majority of orbits are chaotic. The dynamical evolution of the system is also studied when mass is transported from the halo to the disk and the nucleus. Our results are compared to the recently obtained observation data for active galaxies.     

Disk galaxies and dynamical systems with non-negative curvature

The stochastic and regular properties of disk stellar systems being dynamical systems with non-negative curvature are investigated. It is shown that the existence of regions of regular (ordered) and stochastic motion is their typical property (RS-systems). The stochastic regions of two-dimensional systems consisting of ergodic components with positive KS-entropy is shown not to be AnosovU-systems and to have no transversal fibers. As an example the Hénon-Heiles system is studied: the existence ofstrongly stable solutions is proved. The results indicate the crucial role of chaos and order in the dynamics of spiral galaxies of different classes.     

The nature of orbits in a prolate elliptical galaxy model with a bulge and a dense nucleus

We study the transition from regular to chaotic motion in a prolate elliptical galaxy dynamical model with a bulge and a dense nucleus.Our numerical investigation shows that stars with angular momentum Lz less than or equal to a critical value Lzc,moving near the galactic plane,are scattered to higher z,when reaching the central region of the galaxy,thus displaying chaotic motion.An inverse square law relationship was found to exist between the radius of the bulge and the critical value Lzc of the angular m...     

Orbit classification in the meridional plane of a disk galaxy model with a spherical nucleus

We investigate the regular or chaotic nature of star orbits moving in the meridional plane of an axially symmetric galactic model with a disk and a spherical nucleus. We study the influence of some important parameters of the dynamical system, such as the mass and the scale length of the nucleus, the angular momentum or the energy, by computing in each case the percentage of chaotic orbits, as well as the percentages of orbits of the main regular resonant families. Some heuristic arguments to explain and justify the numerically derived outcomes are also given. Furthermore, we present a new method to find the threshold between chaos and regularity for both Lyapunov Characteristic Numbers and SALI, by using them simultaneously.     

Chaos in N-body systems

Here is a selection of applications of what is now called theory of dynamical systems in galactic dynamics and N-body systems. The study of chaotic motions in potentials used as a model for elliptical galaxies is a first example of these applications. The interest in this problem stems from the fact that there are now many theoretical and observational evidences that the overall potentials of galaxies are indeed non-integrable. There are classes of objects, for example small and intermediate luminosity elliptical galaxies, for which the presence of the famous third integral is not necessary or others in which we observe peculiarities in their photometry or kinematics. We address here some of these issues and their implications in modifying our current understanding of the structure and evolution of galaxies.More in general, there is the natural question of how the systems we see have settled to their present status and what would happen if some external cause perturbs it. This issue is related to the question of the stochasticity involved in the general N-body dynamics, especially when N is very large. An N-body dynamical system is definitely chaotic, as shown by several numerical investigations, at least for N not very large. However, this statement must be reconciled with the picture of non-collisional equilibrium of big systems. The second part of this review presents a survey of numerical experiments and an interpretation of the results obtained using standard chaoticity indicators.     

The S(c) spectrum machine to visualize the motion in galaxies

The behavior of the orbits in a galaxy model composed of an harmonic core and a strong bar potential is studied. Numerical calculations show that a large number of orbits display chaotic motion. These orbits are low angular momentun orbits. The percentage of chaotic orbits increases as the angular velocity of the system increases or the strength of the harmonic term decreases. A new dynamical parameter, the S(c) spectrum, is introduced and used to detect the island motion and the evolution of the sticky regions. Comparison to previously obtained results reveals the leading role of the new spectrum. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Relationship between the thickness of stellar disks and the relative mass of a dark galactic halo

We analyze the R-and K _s-band photometric profiles for two independent samples of edge-on galaxies. The thickness of old stellar disks is shown to be related to the relative masses of the spherical and disk components of galaxies. The radial-to-vertical scale length ratio for galactic disks increases (the disks become thinner) with increasing total mass-to-light ratio of the galaxies, which reflects the relative contribution of the dark halo to the total mass, and with decreasing central deprojected disk brightness (density). Our results are in good agreement with numerical models of collisionless disks that evolved to a marginally stable equilibrium state. This suggests that, in most galaxies, the vertical stellar-velocity dispersion, on which the equilibrium-disk thickness depends, is close to a minimum value that ensures disk stability. The thinnest edge-on disks appear to be low-brightness galaxies in which the dark-halo mass far exceeds the stellar-disk mass.     

Spiral-Vortex Structure in the Gaseous Disks of Galaxies

General ideas, as well as experimental and theoretical efforts concerning the prediction and discovery of new structures in the disks of spiral galaxies – giant anticyclones - are reviewed. A crucial point is the development of a new method to restore the full vector velocity field of the galactic gas from the line-of-sight velocity field. This method can be used to get self-consistent solutions for the following problems: 1) determination of non-circular velocities associated with spiral-vortex structure; 2) determination of fundamental parameters of this structure: pattern speed, corotation radius, location of giant anticyclones; 3) refinement of galactic rotation curves taking into account regular non-circular motion in the spiral density wave, which makes it possible to build more accurate models of the mass distribution in the galaxy; 4) refinement of parameters of the rotating gaseous disk: inclination angle, center of rotation and position angle of the major dynamical axis, systematic velocity. The method is demonstrated using the restoration of the velocity field of the galaxy NGC 157 as an example. Results for this and some other spiral galaxies suggest that giant anticyclones are a universal property of galaxies with grand design structure.     

Stellar velocity dispersion and mass estimation for galactic disks

Available velocity dispersion estimates for the old stellar population of galactic disks at galactocentric distances r?2L (where L is the photometric radial scale length of the disk) are used to determine the threshold local surface density of disks that are stable against gravitational perturbations. The mass of the disk M_d calculated under the assumption of its marginal stability is compared with the total mass M_t and luminosity L _B of the galaxy within r=4L. We corroborate the conclusion that a substantial fraction of the mass in galaxies is probably located in their dark halos. The ratio of the radial velocity dispersion to the circular velocity increases along the sequence of galactic color indices and decreases from the early to late morphological types. For most of the galaxies with large color indices (B–V)₀>0.75, which mainly belong to the S0 type, the velocity dispersion exceeds significantly the threshold value required for the disk to be stable. The reverse situation is true for spiral galaxies: the ratios M_d/L_B for these agree well with those expected for evolving stellar systems with the observed color indices. This suggests that the disks of spiral galaxies underwent no significant dynamical heating after they reached a quasi-equilibrium stable state.     

盘星系的内禀扁度研究

盘星系的内禀扁度对计算星系的空间倾角非常重要．对从LEDA数据库中选取的14988个盘星系进行内禀扁度的统计分析．研究表明，盘星系的内禀扁度与星系的形态密切相关．整体上说，透镜星系和不规则星系的内禀扁度qo大于旋涡星系，早型旋涡星系的内禀扁度qo大于晚型旋涡星系，其中Scd星系的内禀扁度qo最小．利用所得的qo-T关系，还对16个已知倾角的亮星系进行了倾角计算，发现与其他方法估计的空间倾角符合得较好．     

The effect of projection on the observed gas velocity fields in barred galaxies

The problem of determining the pattern of gas motions in the central regions of disk spiral galaxies is considered. Two fundamentally different cases—noncircular motions in the triaxial bar potential and motions in circular orbits but with orientation parameters different from those of the main disk—are shown to have similar observational manifestations in the line-of-sight velocity field of the gas. A reliable criterion is needed for the observational data to be properly interpreted. To find such a criterion, we analyze two-dimensional nonlinear hydrodynamic models of gas motions in barred disk galaxies. The gas line-of-sight velocity and surface brightness distributions in the plane of the sky are constructed for various inclinations of the galactic plane to the line of sight and bar orientation angles. We show that using models of circular motions for inclinations i>60° to analyze the velocity field can lead to the erroneous conclusions of a “tilted (polar) disk” at the galaxy center. However, it is possible to distinguish bars from tilted disks by comparing the mutual orientations of the photometric and dynamical axes. As an example, we consider the velocity field of the ionized gas in the galaxy NGC 972.     

星系盘厚度效应的研究

在三维引力Poisson方程严格解基础上，探讨了有限厚星系盘基盘的动力学性质，并进一步讨论了盘的厚度效应对银河系所需晕质量的影响。研究了扰动盘的动力学性质，通过将扰动引力势Poisson方程的严格解与林家翘、徐遐生提出的自维持密度波理论相结合，建立了三维旋涡星系有限厚盘上密度波的色散关系。在此色散关系的基础上讨论了盘的局域稳定性，研究了旋涡星系旋臂的形态、三维盘状星系密度波的群速度。研究表明厚度是星系盘研究中不容忽略的重要参量。另外在有限厚盘星系密度波色散关系的基础上还探讨了一种确定星系厚度的新方法。     

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.     

A galactic formation model based on post-big bang fragmentation

A new model of galactic formation is presented. A primeval distribution of angular momentum is derived, which rests on the postulated presence of mass flows at the early stage of the post-big bang universal expansion, when fragmentation occurs. The shape of any particular fragment and its orientation with respect to the mass flow predestine the morphology of the galaxy that will be produced in the subsequent collapse.A fragment-to-disk mapping transform allows examination of the galactic disk mass distributions that result from the postulates of the model. One class of distribution adequately describes the observed luminosity profiles of elliptical galaxies. Another class indicates the allocation of a large mass fraction to an extended disk, and the intrinsic two-sided symmetry of the model further indicates that the disk formation process will begin with the establishment of two major spiral arms.     

Chaos in cuspy triaxial galaxies with a supermassive black hole: a simple toy model

This paper summarises an investigation of chaos in a toy potential which mimics much of the behaviour observed for the more realistic triaxial generalisations of the Dehnen potentials, which have been used to model cuspy triaxial galaxies both with and without a supermassive black hole. The potential is the sum of an anisotropic harmonic oscillator potential, ${\text{V}}_{\text{0}} = \frac{1}{2}\left( {a^2 x^2 + b^2 y^2 + c^2 z^2 } \right)$ , and aspherical Plummer potential, ${\text{V}}_{\text{P}} = M_{BH} /\sqrt {r^2 + \varepsilon ^2 } $ , with $r^2 = x^2 + y^2 + z^2$ . Attention focuses on three issues related tothe properties of ensembles of chaotic orbits which impact on chaotic mixing and the possibility of constructing self-consistent equilibria:(1) What fraction of the orbits are chaotic? (2) How sensitive are the chaotic orbits, that is, how large are their largest (short time) Lyapunov exponents? (3) To what extent is the motion of chaotic orbits impeded by Arnold webs, that is, how 'sticky' are the chaotic orbits? These questions are explored as functions of the axis ratio a: b: c, black hole mass M _BH, softening length ε, and energy E with the aims of understanding how the manifestations of chaos depend onthe shape of the system and why the black hole generates chaos. The simplicity of the model makes it amenable to a perturbative analysis. That it mimics the behaviour of more complicated potentials suggests that much of this behaviour should be generic.