N. Voglis

Chaotic motion and spiral structure in self-consistent models of rotating galaxies

Dissipationless N-body models of rotating galaxies, iso-energetic to a non-rotating model, are examined as regards the mass in regular and in chaotic motion. Iso-energetic means that they have the same mass and the same binding energy and they are near the same scalar virial equilibrium, but their total amount of angular momentum is different. The values of their spin parameters λ are near the value λ= 0.22 of our Galaxy. We distinguish particles moving in regular and in chaotic orbits and we show that the spatial distribution of these two sets of particles is much different. The rotating models are characterized by larger fractions of mass in chaotic motion (up to the level of ≈65 per cent) compared with the fraction of mass in chaotic motion in the non-rotating iso-energetic model (which is on the level of ≈32 per cent). Furthermore, the Lyapunov numbers of the chaotic orbits in the rotating models become by about one order of magnitude larger than in the non-rotating model. This impressive enhancement of chaos is produced, partly by the more complicated distribution of mass, induced by the rotation, but mainly by the resonant effects near corotation. Chaotic orbits are concentrated preferably in values of the Jacobi integral around the value of the effective potential at the corotation radius. We find that density waves form a central rotating bar embedded in a thin and a thick disc with exponential mean radial profile of the surface density. A surprising new result is that long living spiral arms are excited on the disc, composed almost completely by chaotic orbits. The bar excites an m= 2 mode of spiral waves on the surface density distribution of the disc, emanating from the corotation radius. The bar goes temporarily out of phase with respect to an excited spiral wave, but it comes in phase again in less that a period of rotation. As a consequence, spiral arms show an intermittent behaviour. They are deformed, fade or disappear temporarily, but they grow again re-forming a well-developed spiral pattern. Spiral arms are discernible up to 20 or 30 rotations of the bar (lasting for about a Hubble time). The relative power of the spiral m= 2 mode with respect to all other fluctuations on the surface density is initially about 50 per cent, but it is reduced by a factor of about 2 or 3 at the end of the Hubble time.

Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

In the presence of a strong m = 2 component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L 1 or L 2 . The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine-Gordon equation. Numerical examples are given in an N-body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar-spiral pattern.

Large-scale structure in the HI Parkes All-Sky Survey : filling the voids with HI galaxies?

We estimate the two-point correlation function in redshift space of the recently compiled HI Parkes All-Sky Survey neutral hydrogen (HI) sources catalogue, which if modelled as a power law, xi(r) = (r(0)/r)(gamma), the best-fitting parameters for the HI selected galaxies are found to be r(0) = 3.3 +/- 0.3 h(-1) Mpc with gamma = 1.38 +/- 0.24. Fixing the slope to its universal value gamma = 1.8, we obtain r(0) = 3.2 +/- 0.2 h(-1) Mpc. Comparing the measured two-point correlation function with the predictions of the concordance cosmological model (Omega(Lambda) = 0.74), we find that at the present epoch the HI selected galaxies are antibiased with respect to the underlying matter fluctuation field with their bias value being b(0) similar or equal to 0.68. Furthermore, dividing the HI galaxies into two richness subsamples we find that the low-mass HI galaxies have a very low present bias factor (b(0) similar or equal to 0.48), while the high-mass HI galaxies trace the underlying matter distribution as the optical galaxies (b(0) similar or equal to 1). Using our derived present-day HI galaxy bias we estimate their redshift-space distortion parameter, and correct accordingly the correlation function for peculiar motions. The resulting real-space correlation length is r(0)(re) = 1.8 +/- 0.2 h(-1) Mpc r(0)(re) = 3.9 +/- 0.6 h(-1) Mpc for the low- and high-mass HI galaxies, respectively. The low-mass HI galaxies appear to have the lowest correlation length among all extragalactic populations studied to date. In order to corroborate these results we have correlated the IRAS-Point Source Catalogue for Redshift (PSCz) reconstructed density field, smoothed over scales of 5 h(-1) Mpc, with the positions of the HI galaxies, to find that indeed the HI galaxies are typically found in negative overdensity regions (delta rho/rho(PSCz) less than or similar to 0), even more so the low-mass HI galaxies. Finally, we also study the redshift evolution of the HI galaxy linear bias factor and find that the HI-galaxy population is antibiased up to z similar to 1.3. While at large redshifts similar to 3, we predict that the HI galaxies are strongly biased. Our bias evolution predictions are consistent with the observational bias results of Ly alpha galaxies.

Transition spectra of dynamical systems

The spectra of ‘stretching numbers’ (or ‘local Lyapunov characteristic numbers’) are different in the ordered and in the chaotic domain. We follow the variation of the spectrum as we move from the centre of an island outwards until we reach the chaotic domain. As we move outwards the number of abrupt maxima in the spectrum increases. These maxima correspond to maxima or minima in the curve a(θ), where a is the stretching number, and θ the azimuthal angle. We explain the appearance of new maxima in the spectra of ordered orbits. The orbits just outside the last KAM curve are confined close to this curve for a long time (stickiness time) because of the existence of cantori surrounding the island, but eventually escape to the large chaotic domain further outside. The spectra of sticky orbits resemble those of the ordered orbits just inside the last KAM curve, but later these spectra tend to the invariant spectrum of the chaotic domain. The sticky spectra are invariant during the stickiness time. The stickiness time increases exponentially as we approach an island of stability, but very close to an island the increase is super exponential. The stickiness time varies substantially for nearby orbits; thus we define a probability of escape Pn(x) at time n for every point x. Only the average escape time in a not very small interval Δx around each x is reliable. Then we study the convergence of the spectra to the final, invariant spectrum. We define the number of iterations, N, needed to approach the final spectrum within a given accuracy. In the regular domain N is small, while in the chaotic domain it is large. In some ordered cases the convergence is anomalously slow. In these cases the maximum value of ak in the continued fraction expansion of the rotation number a = [a0,a1,... ak,...] is large. The ordered domain contains small higher order chaotic domains and higher order islands. These can be located by calculating orbits starting at various points along a line parallel to the q-axis. A monotonic variation of the sup {q}as a function of the initial condition q0 indicates ordered motions, a jump indicates the crossing of a localized chaotic domain, and a V-shaped structure indicates the crossing of an island. But sometimes the V-shaped structure disappears if the orbit is calculated over longer times. This is due to a near resonance of the rotation number, that is not followed by stable islands.

Stickiness and cantori

We study the phenomenon of stickiness in the standard map. The sticky regions are limited by cantori. Most important among them are the cantori with noble rotation numbers, that are approached by periodic orbits corresponding to the successive truncations of the noble numbers. The size of an island of stability depends on the last KAM torus. As the perturbation increases, the size of the KAM curves increases. But the outer KAM curves are gradually destroyed and in general the island decreases. Higher-order noble tori inside the outermost KAM torus are also destroyed and when the outermost KAM torus becomes a cantorus, the size of an island decreases abruptly. Then we study the crossing of the cantori by asymptotic curves of periodic orbits just inside the cantorus. We give an exact numerical example of this crossing (non-schematic) and we find how the asymptotic curves, after staying for a long time near the cantorus, finally extend to large distances outwards. Finally, we find the relation between the forms of the sticky region and asymptotic curves.

The coalescence of invariant manifolds and the spiral structure of barred galaxies

In a previous paper (Voglis et al. 2006a, paper I) we demonstrated that, in a rotating galaxy with a strong bar, the unstable asymptotic manifolds of the short period family of unstable periodic orbits around the Lagrangian points L1 or L2 create correlations among the apocentric positions of many chaotic orbits, thus supporting a spiral structure beyond the bar. In the present paper we present evidence that the unstable manifolds of all the families of unstable periodic orbits near and beyond corotation contribute to the same phenomenon. Our results refer to a N-Body simulation, a number of drawbacks of which, as well as the reasons why these do not significantly affect the main results, are discussed. We explain the dynamical importance of the invariant manifolds as due to the fact that they produce a phenomenon of ‘stickiness’ slowing down the rate of chaotic escape in an otherwise non-compact region of the phase space. We find a stickiness time of order 100 dynamical periods, which is sufficient to support a long-living spiral structure. Manifolds of different families become important at different ranges of values of the Jacobi constant. The projections of the manifolds of all the different families in the configuration space produce a pattern due to the ‘coalescence’ of the invariant manifolds. This follows closely the maxima of the observed m = 2 component near and beyond corotation. Thus, the manifolds support both the outer edge of the bar and the spiral arms.

“Stickiness” in mappings and dynamical systems

Abstract We present results of a study of the so-called “stickiness” regions where orbits in mappings and dynamical systems stay for very long times near an island and then escape to the surrounding chaotic region. First we investigated the standard map in the form xi+1 = xi+yi+1 and y i+1 = y i + K 2π · sin (2πx i ) with a stochasticity parameter K = 5, where only two islands of regular motion survive. We checked now many consecutive points—for special initial conditions of the mapping—stay within a certain region around the island. For an orbit on an invariant curve all the points remain forever inside this region, but outside the “last invariant curve” this number changes significantly even for very small changes in the initial conditions. In our study we found out that there exist two regions of “sticky” orbits around the invariant curves: A small region I confined by Cantori with small holes and an extended region II is outside these cantori which has an interesting fractal character. Investigating also the Sitnikov-Problem where two equally massive primary bodies move on elliptical Keplerian orbits, and a third massless body oscillates through the barycentre of the two primaries perpendicularly to the plane of the primaries—a similar behaviour of the stickiness region was found. Although no clearly defined border between the two stickiness regions was found in the latter problem the fractal character of the outer region was confirmed.

Chaos and secular evolution of triaxial N-body galactic models due to an imposed central mass

We investigate the response of triaxial non-rotating N-body models of elliptical galaxies with smooth centers, initially in equilibrium, under the presence of a central mass assumed to be due mainly to a massive central black hole. We examine the fraction of mass in chaotic motion and the resulting secular evolution of the models. Four cases of the size of the central mass are investigated, namely m = 0.0005, 0.0010, 0.0050, 0.0100 in units of the total mass of the galaxy. We find that a central mass with value m < 0.005 increases the mass fraction in chaotic motion from the level of 25-35% (that appears in the case of smooth centers) to the level of 50-80% depending on the value of m and on the initial maximum ellipticity of the system. However, most of this mass moves in chaotic orbits with Lyapunov numbers too small to develop chaotic diffusion in a Hubble time. Thus their secular evolution is so slow that it can be neglected in a Hubble time. Larger central masses (m ≥ 0.005) give initially about the same fractions of mass in chaotic motion as for smaller m, but the Lyapunov numbers are concentrated to larger values, so that a secular evolution of the self-consistent models is prominent. These systems evolve in time tending to a new equilibrium. During their evolution they become self-organized by converting chaotic orbits to ordered orbits of the Short Axis Tube type. The mechanism of such a self-organization is investigated. The rate of this evolution depends on m and on the value of the initial maximum ellipticity of the system. For m = 0.01 and a large initial maximum ellipticity E max 7, equilibrium can be achieved in one Hubble time, forming an oblate spheroidal configuration. For the same value of m and initial maximum elipticity E max 3.5, or for E max 7, but m = 0.005, oblate equilibrium configurations can also be achieved, but in much longer times. Furthermore, we find that, form = 0.005 and E max 3.5, triaxial equilibrium configurations can be formed. The fraction of mass in chaotic motion in the equilibrium configurations is in the range of 12-25%.

Special Features of Galactic Dynamics

This is an introductory article to some basic notions and currently open problems of galactic dynamics. The focus is on topics mostly relevant to the so-called ‘new methods’ of celestial mechanics or Hamiltonian dynamics, as applied to the ellipsoidal components of galaxies, i.e., to the elliptical galaxies and to the dark halos and bulges of disk galaxies. Traditional topics such as Jeans theorem, the role of a ‘third integral’ of motion, Nekhoroshev theory, violent relaxation, and the statistical mechanics of collisionless stellar systems are ?rst discussed. The emphasis is on modern extrapolations of these old topics. Recent results from orbital and global dynamical studies of galaxies are then shortly reviewed. The role of various families of orbits in supporting self-consistency, as well as the role of chaos in galaxies, are stressed. A description is then given of the main numerical techniques of integration of the N-body problem in the framework of stellar dynamics and of the results obtained via N-Body experiments. A ?nal topic is the secular evolution and selforganization of galactic systems.

Order and Chaos in Self-Consistent Galactic Models

We construct and compare two different self-consistent N-body equilibrium configurations of galactic models. The two systems have their origin in cosmological initial conditions selected so that the radial orbit instability appears in one model and gives an E5 type elliptical galaxy, but not in the other that gives an E1 type. We examine their phase spaces using uniformly distributed orbits of test particles in the resulting potential and compare with the distribution of the orbits of the real particles in the two systems. The main types of orbits in both cases are box, tube and chaotic orbits. One main conclusion is that the orbits of the test particles in the 3-dimensional potential are foliated in a way quite close to the foliation of invariant tori in a 2-dimensional potential. The real particles describe orbits having similar foliation. However, their distribution is far from being uniform. The difference between the two models of equilibrium is realized mainly by different balances of the populations of real particles in box and tube orbits.