Rotation and anisotropy of galaxies revisited

The use of the tensor virial theorem (TVT) as a diagnostic of anisotropic velocity distributions in galaxies is revisited. The TVT provides a rigorous global link between velocity anisotropy, rotation and shape, but the quantities appearing in it are not easily estimated observationally. Traditionally, use has been made of a centrally averaged velocity dispersion and the peak rotation velocity. Although this procedure cannot be rigorously justified, tests on model galaxies show that it works surprisingly well. With the advent of integral-field spectroscopy it is now possible to establish a rigorous connection between the TVT and observations. The TVT is reformulated in terms of sky-averages, and the new formulation is tested on model galaxies.     

Stellar and gas dynamics of late-type barred-spiral galaxies: NGC 3359, a test case

We study the dynamics of a model for the late-type barred-spiral galaxy NGC 3359 by using both observational and numerical techniques. The results of our modelling are compared with photometric and kinematical data. The potential used is estimated directly from observations of the galaxy. It describes with a single potential function, a barred-spiral system with an extended spiral structure. Thus, the study of the dynamics in this potential has an interest by itself. We apply orbital theory and response models for the study of the stellar component, and smoothed particle hydrodynamics for modelling the gas. In particular, we examine the pattern speed of the system and the orbital character (chaotic or ordered) of the spiral arms. We conclude that the spiral pattern rotates slowly, in the sense that its corotation is close to or even beyond the end of the arms. Although a single, slow pattern speed could, under certain assumptions, characterize the whole disc, the comparison with the observational data indicates that probably the bar and the spirals have different angular velocities. In our two pattern speeds model, the best fit is obtained with a bar ending close to its 4:1 resonance and a more slowly rotating spiral. Assuming an 11 Mpc distance to the galaxy, a match of our models with the observed data indicates a pattern speed of about  39 km s⁻¹ kpc⁻¹  for the bar and about  15 km s⁻¹ kpc⁻¹  for the spiral. We do not find any indication for a chaotic character of the arms in this barred-spiral system. The flow in the region of the spirals can best be described as a regular 'precessing-ellipses flow'.     

Fast detection of chaotic behavior in galactic potentials

In the present paper we apply a new method of distinction between ordered and chaotic motion in galactic potentials. The method uses the Fourier Transform of a series of time intervals each one representing the time that elapsed between two successive points on the Poincaré surface of section. Examples of the methods ability to achieve an early and clear detection of an orbit's behavior are provided using two galactic potentials. The new method can also be applied in order to have an early distinction between ordered and sticky orbits. The method is generalized in order to be used in models with more than two dimensions. Finally we have tried to find an one‐number index to give us the nature of the orbit instead of checking by eye the whole spectrum. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling the dynamical profile of a BL Lacertae object

A simple dynamical model for a BL Lacertae active galaxy is presented. The model consists of a logarithmic potential with an additional term representing internal perturbations. The time independent and the evolving model are investigated. In both cases we search for regular and chaotic motion and study the velocity distribution near the centre of the system. Numerical calculations suggest that responsible for the chaotic phenomena is the internal perturbation, the flattening parameter and the dense nucleus. The radius of the nucleus also affects the maximum velocity in the central regions of the galaxy. Our numerical outcomes are supported by theoretical arguments and analytical calculations. A linking of our numerical outcomes to observational data is also presented. (© 2008 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)     

Density cusps: restrictions on non-axisymmetric models

Galactic nuclei are now generally thought to have density cusps in their centres, and the evidence is mounting that as a consequence they are unlikely to be triaxial. Self-consistent stellar dynamical models of non-axisymmetric cusps would be an interesting counter-argument to this conclusion. We consider 2D analogues of triaxial cusps: a sequence of non-axisymmetric, cuspy discs first described by Sridhar & Touma. Scale-free models with potential Φ ∝  r ^α are examined in detail. It is shown analytically for 0 < α ≲ 0.43 that self-consistent models with positive phase-space density do not exist. Numerical solutions of the combined Vlasov and Poisson equations suggest that the whole sequence of models with 0 < α < 1 are also unphysical. Together with existing work on cusps, we conclude on purely theoretical grounds that galactic nuclei are not expected to be triaxial.     

Chaos in a quasar model with a disk and a massive nucleus

We present a quasar model with a rotating disk and a massive nucleus. We use this model in order to characterize the motion in the model (regular or chaotic) and to connect the extent of the chaotic regions to the physical parameters of the model. Numerical experiments suggest that, there are connections between the extent of the chaotic areas and the parameters of the system. Furthermore, it is shown that the form of numerically found relationships can be expressed analytically. Comparison to previous work is also made. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical potential–density pairs from complex-shifted Kuzmin–Toomre discs

The complex-shift method is applied to the Kuzmin–Toomre family of discs to generate a family of non-axisymmetric flat distributions of matter. These are then superposed to construct non-axisymmetric flat rings. We also consider triaxial potential–density pairs obtained from these non-axisymmetric flat systems by means of suitable transformations. The use of the imaginary part of complex-shifted potential–density pairs is also discussed.     

Do high-velocity clouds form by thermal instability?

We examine the proposal that the H  i 'high-velocity' clouds (HVCs) surrounding the Milky Way and other disc galaxies form by condensation of the hot galactic corona via thermal instability. Under the assumption that the galactic corona is well represented by a non-rotating, stratified atmosphere, we find that for this formation mechanism to work the corona must have an almost perfectly flat entropy profile. In all other cases, the growth of thermal perturbations is suppressed by a combination of buoyancy and thermal conduction. Even if the entropy profile were nearly flat, cold clouds with sizes smaller than  10 kpc  could form in the corona of the Milky Way only at radii larger than  100 kpc  , in contradiction with the determined distances of the largest HVC complexes. Clouds with sizes of a few kpc can form in the inner halo only in low-mass systems. We conclude that unless even slow rotation qualitatively changes the dynamics of a corona, thermal instability is unlikely to be a viable mechanism for formation of cold clouds around disc galaxies.     

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)     

A new dynamical spectrum for galactic potentials

We investigate the properties of motion, using the distribution of the values of a new dynamical parameter, in two different galactic potentials. The first is a potential made up of harmonic oscillators while the second is a logarithmic potential. We call the distribution functions of the new parameter the S(r) dynamical spectrum. A comparison between the spectrum of stretching numbers S(α) and the new spectrum is made. The results of our numerical calculations suggest that the S(r) spectrum is a better discriminant between different types of orbits while requiring considerable less computation time. An application of the new dynamical spectrum to barred galaxy models is also presented.     

Dynamical evolution of two associated galactic bars

We study the dynamical interactions of mass systems in equilibrium under their own gravity that mutually exert and ex‐perience gravitational forces. The method we employ is to model the dynamical evolution of two isolated bars, hosted within the same galactic system, under their mutual gravitational interaction. In this study, we present an analytical treatment of the secular evolution of two bars that oscillate with respect to one another. Two cases of interaction, with and without geometrical deformation, are discussed. In the latter case, the bars are described as modified Jacobi ellipsoids. These triaxial systems are formed by a rotating fluid mass in gravitational equilibrium with its own rotational velocity and the gravitational field of the other bar. The governing equation for the variation of their relative angular separation is then numerically integrated, which also provides the time evolution of the geometrical parameters of the bodies. The case of rigid, non‐deformable, bars produces in some cases an oscillatory motion in the bodies similar to that of a harmonic oscillator. For the other case, a deformable rotating body that can be represented by a modified Jacobi ellipsoid under the influence of an exterior massive body will change its rotational velocity to escape from the attracting body, just as if the gravitational torque exerted by the exterior body were of opposite sign. Instead, the exchange of angular momentum will cause the Jacobian body to modify its geometry by enlarging its long axis, located in the plane of rotation, thus decreasing its axial ratios. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

From kinematics to dynamics in thin galactic discs

We present a method for recovering the distribution functions of edge-on thin axisymmetric discs directly from their observable kinematic properties. The most generally observable properties of such a stellar system are the line-of-sight velocity distributions of the stars at different projected radii along the galaxy. If the gravitational potential is known, then the general two-integral distribution function can be reconstructed using the shapes of the high-velocity tails of these line-of-sight distributions. If the wrong gravitational potential is adopted, then a distribution function can still be constructed using this technique, but the low-velocity parts of the observed velocity distributions will not be reproduced by the derived dynamical model. Thus, the gravitational potential is also tightly constrained by the observed kinematics.     

Energy relaxation in galaxies induced by an external environment and/or incoherent internal pulsations

This paper explores the phenomenon of energy relaxation for stars in a galaxy embedded in a high-density environment that is subjected continually to perturbations reflecting the presence of other nearby galaxies and/or incoherent internal pulsations. The analysis is similar to earlier analyses of energy relaxation induced by binary encounters between nearby stars and between stars and giant molecular clouds in that the perturbations are idealized as a sum of near-random events which can be modelled as diffusion and dynamical friction. However, the analysis differs in one important respect: because the time-scale associated with these perturbations need not be short compared with the characteristic dynamical time t _D for stars in the original galaxy, the diffusion process cannot be modelled as resulting from a sequence of instantaneous kicks, i.e. white noise. Instead, the diffusion is modelled as resulting from random kicks of finite duration, i.e. coloured noise, characterized by a non-zero autocorrelation time t _c. A detailed analysis of coloured noise generated by sampling an Ornstein–Uhlenbeck process leads to a simple scaling in terms of t _c and an effective diffusion constant D . Interpreting D and t _c following early work by Chandrasekhar (the 'nearest neighbour approximation') implies that, for realistic choices of parameter values, energy relaxation associated with an external environment and/or internal pulsations could be important on times short compared with the age of the Universe.