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

In a previous paper (Voglis et al., 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 L ₁ or L ₂ create correlations among the apocentric positions of many chaotic orbits, thus supporting a spiral structure beyond the bar. In this 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 the order of 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.     

Global Dynamics in Self-Consistent Models of Elliptical Galaxies

We compare two different N-body models simulating elliptical galaxies. Namely, the first model is a non-rotating triaxial N-body equilibrium model with smooth center, called SC model. The second model, called CM model, is derived from the SC by inserting a central mass in it, so that all possible differences between the two models are due to the effect of the central mass. The central mass is assumed to be mainly due to a massive central black hole of mass about 1% of the total mass of the galaxy. By using the fundamental frequency analysis, the two systems are thoroughly investigated as regards the types of orbits described either by test particles, or by the real particles of the systems at all the energy levels. A comparison between the orbits of test particles and the orbits of real particles at various energy levels is made on the rotation number plane. We find that extensive stable regions of phase space, detected by test particles remain empty, i.e. these regions are not occupied by real particles, while many real particles move in unstable regions of phase space describing chaotic orbits. We run self-consistently the two models for more than a Hubble time. During this run, in spite of the noise due to small variations of the potential, the SC model maintains (within a small uncertainly) the number of particles moving on orbits of each particular type. In contrast, the CM model is unstable, due to the large amount of mass in chaotic motion caused by the central mass. This system undergoes a secular evolution towards an equilibrium state. During this evolution it is gradually self-organized by converting chaotic orbits to ordered orbits mainly of the short axis tube type approaching an oblate spheroidal equilibrium. This is clearly demonstrated in terms of the fundamental frequencies of the orbits on the rotation number plane and the time evolution of the triaxiality index.     

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.     

Regular motions in double bars – I. Double-frequency orbits and loops

Bars in galaxies are mainly supported by particles trapped around stable periodic orbits. These orbits represent oscillatory motion with only one frequency, which is the bar driving frequency, and miss free oscillations. We show that a similar situation takes place in double bars: particles get trapped around parent orbits, which in this case represent oscillatory motion with two frequencies of driving by the two bars, and which also lack free oscillations. Thus the parent orbits, which constitute the backbone of an oscillating potential of two independently rotating bars, are the double-frequency orbits. These orbits do not close in any reference frame, but they map on to closed curves called loops. Trajectories trapped around the parent double-frequency orbit map on to a set of points confined within a ring surrounding the loop.     

Unravelling the mystery of the M31 bar

The inclination of M31 is too close to edge-on for a bar component to be easily recognized and is not sufficiently edge-on for a boxy/peanut bulge to protrude clearly out of the equatorial plane. Nevertheless, a sufficient number of clues allow us to argue that this galaxy is barred. We use fully self-consistent N -body simulations of barred galaxies and compare them with both photometric and kinematic observational data for M31. In particular, we rely on the near-infrared photometry presented in a companion paper. We compare isodensity contours to isophotal contours and the light profile along cuts parallel to the galaxy major axis and offset towards the north, or the south, to mass profiles along similar cuts on the model. All these comparisons, as well as position–velocity diagrams for the gaseous component, give us strong arguments that M31 is barred. We compare four fiducial N -body models to the data and thus set constraints on the parameters of the M31 bar, as its strength, length and orientation. Our 'best' models, although not meant to be exact models of M31, reproduce in a very satisfactory way the main relevant observations. We present arguments that M31 has both a classical and a boxy/peanut bulge. Its pseudo-ring-like structure at roughly 50 arcmin is near the outer Lindblad resonance of the bar and could thus be an outer ring, as often observed in barred galaxies. The shape of the isophotes also argues that the vertically thin part of the M31 bar extends considerably further out than its boxy bulge, that is, that the boxy bulge is only part of the bar, thus confirming predictions from orbital structure studies and from previous N -body simulations. It seems very likely that the backbone of M31's boxy bulge is families of periodic orbits, members of the x₁-tree and bifurcating from the x₁ family at its higher order vertical resonances, such as the x1v3 or x1v4 families.     

Numerical simulations of interacting gas-rich barred galaxies: vertical impact of small companions

We investigate the dynamical effects of an interaction between an initially barred galaxy and a small spherical companion using an N -body/smoothed-particle-hydrodynamics algorithm. In the models described here the small companion passes through the disc of the larger galaxy nearly perpendicular to its plane. The impact positions and times are varied with respect to the phase of the bar and the dynamical evolution of the disc.
The interactions produce expanding ring structures, offset bars, spokes and other asymmetries in the stars and gas. These characteristic signatures of the interaction are present in the disc for about 1 Gyr. We find that in some cases it is possible to destroy the bar while keeping the disc structure. In general, the central impacts cause larger damage to the bar and the disc than the peripheral ones. The interaction tends to accelerate the transition from a strongly-barred galaxy to a weakly- or non-barred galaxy. The final disc morphology is determined more by the impact position relative to the bar rather than the impact time.     

Bar-induced evolution of dark matter cusps

The evolution of a stellar bar transforms not only the galactic disc, but also the host dark matter halo. We present high-resolution, fully self-consistent N -body simulations that clearly demonstrate that dark matter halo central density cusps flatten as the bar torques the halo. This effect is independent of the bar formation mode and occurs even for rather short bars. The halo and bar evolution is mediated by resonant interactions between orbits in the halo and the bar pattern speed, as predicted by linear Hamiltonian perturbation theory. The bar lengthens and slows as it loses angular momentum, a process that occurs even in rather warm discs. We demonstrate that the bar and halo response can be critically underestimated for experiments that are unable to resolve the relevant resonant dynamics; this occurs when the phase space in the resonant region is undersampled or plagued by noise.     

The morphology of galactic rings exterior to evolving bars: test-particle simulations

The morphology of the outer rings of early-type spiral galaxies is compared to integrations of massless collisionless particles initially in nearly circular orbits. Particles are perturbed by a quadrupolar gravitational potential corresponding to a growing and secularly evolving bar. We find that outer rings with R1R2 morphology and pseudo-rings are exhibited by the simulations even though they lack gaseous dissipation. Simulations with stronger bars form pseudo-rings earlier and more quickly than those with weaker bars. We find that the R1 ring, perpendicular to the bar, is fragile and dissolves after a few bar rotation periods if the bar pattern speed increases by more than ∼8 per cent, bar strength increases (by ≳140 per cent) after bar growth or the bar is too strong  ( Q_T > 0.3)  . If the bar slows down after formation, pseudo-ring morphology persists and the R2 ring perpendicular to the bar is populated due to resonance capture. The R2 ring remains misaligned with the bar and increases in ellipticity as the bar slows down. The R2 ring becomes scalloped and does not resemble any ringed galaxies if the bar slows down more than 3.5 per cent suggesting that bars decrease in strength before they slow down this much. We compare the morphology of our simulations to B -band images of nine ringed galaxies from the Ohio State University Bright Spiral Galaxy Survey, and we find a reasonable match in morphologies to R1R2' pseudo-rings seen within a few bar rotation periods of bar formation. Some of the features previously interpreted in terms of dissipative models may be due to transient structure associated with recent bar growth and evolution.     

H i and CO observations of Arp 104: a spiral–elliptical interacting pair

We present data probing the spatial and kinematical distribution of both the atomic (H  i ) and molecular (CO) gas in NGC 5218, the late-type barred spiral galaxy in the spiral–elliptical interacting pair, Arp 104. We consider these data in conjunction with far-infrared and radio-continuum data, and N -body simulations, to study the galaxies interactions, and the star formation properties of NGC 5218. We use these data to assess the importance of the bar and tidal interaction on the evolution of NGC 5218, and the extent to which the tidal interaction may have been important in triggering the bar. The molecular gas distribution of NGC 5218 appears to have been strongly affected by the bar; the distribution is centrally condensed with a very large surface density in the central region. The N -body simulations indicate a time-scale since perigalacticon of  ∼3 × 10⁸ yr  , which is consistent with the interaction having triggered or enhanced the bar potential in NGC 5218, leading to inflow and the large central molecular gas density observed. Whilst NGC 5218 appears to be undergoing active star formation, its star formation efficiency is comparable to a 'normal' SBb galaxy. We propose that this system may be on the brink of a more active phase of star formation.     

A bar in the inner halo of barred galaxies – I. Structure and kinematics of a representative model

N -body simulations argue that the inner haloes of barred galaxies should not be spherical, nor even axisymmetric, but triaxial. The departure from sphericity is the strongest near the centre and decreases outwards; typical axial ratios for the innermost parts are of the order of 0.8. The halo shape is prolate-like in the inner parts up to a certain radius and then turns to oblate-like. I call this inner halo structure the 'halo bar' and analyse here in depth its structure and kinematics in a representative model. It is always considerably shorter than the disc bar. It lags the disc bar by only a few degrees at all radii and the difference between the two bar phases increases with distance from the centre. The two bars turn with roughly the same pattern speed. This means that the halo bar is a slow bar, since its corotation radius is much larger than its length. The bisymmetric component in the halo continues well outside the halo bar in the form of an open spiral, trailing behind the disc bar. The inner parts of the halo display some mean rotation in the same sense as the disc rotation. This is more important for particles nearer to the equatorial plane and decreases with increasing distance from it, but is always much smaller than the disc rotation.     

On the relation between orbital structure and observed bar morphology

We investigate the morphological relation between the orbits of the central family of periodic orbits ( x ₁ family) and the bar itself using models of test particles moving in a barred potential. We show that different bar morphologies may have as a backbone the same set of x ₁ periodic orbits. We point out that by populating initially axisymmetric stellar discs exponentially with test particles in circular, or almost circular motion, we may end up with a response bar which reveals a shape different in crucial details from that of the individual stable x ₁ orbits. For example, a bar model in which the x ₁ orbits are pure ellipses may have a much more complicated response morphology. This depends on the particular invariant curves around x ₁, which are populated in each model.     

The Monitor project: JW 380 – a 0.26-, 0.15-M⊙, pre-main-sequence eclipsing binary in the Orion nebula cluster

We present a general recipe for constructing N -body realizations of galaxies comprising near spherical and disc components. First, an exact spherical distribution function for the spheroids (halo and bulge) is determined, such that it is in equilibrium with the gravitational monopole of the disc components. Second, an N -body realization of this model is adapted to the full disc potential by growing the latter adiabatically from its monopole. Finally, the disc is sampled with particles drawn from an appropriate distribution function, avoiding local-Maxwellian approximations. We performed test simulations and find that the halo and bulge radial density profile very closely match their target model, while they become slightly oblate due to the added disc gravity. Our findings suggest that vertical thickening of the initially thin disc is caused predominantly by spiral and bar instabilities, which also result in a radial re-distribution of matter, rather than scattering off interloping massive halo particles.     

A new formula describing the scaffold structure of spiral galaxies

We describe a new formula capable of quantitatively characterizing the Hubble sequence of spiral galaxies including grand design and barred spirals. Special shapes such as ring galaxies with inward and outward arms are also described by the analytic continuation of the same formula. The formula is   r (φ) = A /log [ B tan   (φ/2 N )]  . This function intrinsically generates a bar in a continuous, fixed relationship relative to an arm of arbitrary winding sweep. A is simply a scale parameter while B , together with N , determines the spiral pitch. Roughly, greater N results in tighter winding. Greater B results in greater arm sweep and smaller bar/bulge, while smaller B fits larger bar/bulge with a sharper bar/arm junction. Thus B controls the 'bar/bulge-to-arm' size, while N controls the tightness much like the Hubble scheme. The formula can be recast in a form dependent only on a unique point of turnover angle of pitch – essentially a one-parameter fit, aside from a scalefactor. The recast formula is remarkable and unique in that a single parameter can define a spiral shape with either constant or variable pitch capable of tightly fitting Hubble types from grand design spirals to late-type large barred galaxies. We compare the correlation of our pitch parameter to Hubble type with that of the traditional logarithmic spiral for 21 well-shaped galaxies. The pitch parameter of our formula produces a very tight correlation with ideal Hubble type suggesting it is a good discriminator compared to logarithmic pitch, which shows poor correlation here similar to previous works. Representative examples of fitted galaxies are shown.     

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)     

Stochasticity in N-body simulations of disc galaxies

We demonstrate that the chaotic nature of N -body systems can lead to macroscopic variations in the evolution of collisionless simulations containing rotationally supported discs. The unavoidable stochasticity that afflicts all simulations generally causes mild differences between the evolution of similar models but, in order to illustrate that this is not always true, we present a case that shows an extreme bimodal divergence. The divergent behaviour occurs in two different types of code, and is independent of all numerical parameters. We identify and give explicit illustrations of several sources of stochasticity, and also show that macroscopic variations in the evolution can originate from differences at the round-off error level. We obtain somewhat more consistent results from simulations in which the halo is set-up with great care compared with those started from more approximate equilibria, but we have been unable to eliminate diverging behaviour entirely because the main sources of stochasticity are intrinsic to the disc. We show that the divergence is only temporary and that halo friction is merely delayed, for a substantial time in some cases. We argue that the delays are unlikely to arise in real galaxies, and that our results do not affect dynamical friction constraints on halo density. Stochastic variations in the evolution are inevitable in all simulations of disc–halo systems, irrespective of how they were created, although their effect is generally far less extreme than we find here. The possibility of divergent behaviour complicates comparison of results from different workers.     

Regular and chaotic motion in softened gravitational systems

The stability of the dynamical trajectories of softened spherical gravitational systems is examined, both in the case of the full N -body problem and that of trajectories moving in the gravitational field of non-interacting background particles. In the latter case, for   N 10 000  , some trajectories, even if unstable, had exceedingly long diffusion times, which correlated with the characteristic e-folding time-scale of the instability. For trajectories of   N ≈100 000  systems this time-scale could be arbitrarily large – and thus appear to correspond to regular orbits. For centrally concentrated systems, low angular momentum trajectories were found to be systematically more unstable. This phenomenon is analogous to the well-known case of trajectories in generic centrally concentrated non-spherical smooth systems, where eccentric trajectories are found to be chaotic. The exponentiation times also correlate with the conservation of the angular momenta along the trajectories. For N up to a few hundred, the instability time-scales of N -body systems and their variation with particle number are similar to those of the most chaotic trajectories in inhomogeneous non-interacting systems. For larger N (up to a few thousand) the values of the these time-scales were found to saturate, increasing significantly more slowly with N . We attribute this to collective effects in the fully self-gravitating problem, which are apparent in the time variations of the time-dependent Liapunov exponents. The results presented here go some way towards resolving the long-standing apparent paradoxes concerning the local instability of trajectories. This now appears to be a manifestation of mechanisms driving evolution in gravitational systems and their interactions – and may thus be a useful diagnostic of such processes.     

Chaotic spiral galaxies

We study the role of asymptotic curves in supporting the spiral structure of a N-body model simulating a barred spiral galaxy. Chaotic orbits with initial conditions on the unstable asymptotic manifolds of the main unstable periodic orbits follow the shape of the periodic orbits for an initial interval of time and then they are diffused outwards along the spiral structure of the galaxy. Chaotic orbits having small deviations from the unstable periodic orbits, stay close and along the corresponding unstable asymptotic manifolds, supporting the spiral structure for more than 10 rotations of the bar. Chaotic orbits of different Jacobi constants support different parts of the spiral structure. We also study the diffusion rate of chaotic orbits outwards and find that the orbits that support the outer parts of the galaxy are diffused outwards more slowly than the orbits supporting the inner parts of the spiral structure.     

Gas-driven evolution of stellar orbits in barred galaxies

We carry out a detailed orbit analysis of gravitational potentials selected at different times from an evolving self-consistent model galaxy consisting of a two-component disc (stars+gas) and a live halo. The results are compared with a pure stellar model, subject to nearly identical initial conditions, which are chosen so as to make the models develop a large-scale stellar bar. The bars are also subject to hose-pipe (buckling) instability which modifies the vertical structure of the disc. The diverging morphological evolution of both models is explained in terms of gas radial inflow, the resulting change in the gravitational potential at smaller radii, and the subsequent modification of the main families of orbits, both in and out of the disc plane.   We find that dynamical instabilities become milder in the presence of the gas component, and that the stability of planar and 3D stellar orbits is strongly affected by the related changes in the potential — both are destabilized, with the gas accumulation at the centre. This is reflected in the overall lower amplitude of the bar mode and in the substantial weakening of the bar, which appears to be a gradual process. The vertical buckling of the bar is much less pronounced and the characteristic peanut shape of the galactic bulge almost disappears when there is a substantial gas inflow towards the centre. Milder instability results in a smaller bulge, the basic parameters of which are in agreement with observations. We also find that the overall evolution in the model with a gas component is accelerated because of the larger central mass concentration and the resulting decrease in the characteristic dynamical time.     

The dynamical evolution of substructure

The evolution of substructure embedded in non-dissipative dark haloes is studied through N -body simulations of isolated systems, both in and out of initial equilibrium, complementing cosmological simulations of the growth of structure. We determine by both analytic calculations and direct analysis of the N -body simulations the relative importance of various dynamical processes acting on the clumps, such as the removal of material by global tides, clump–clump heating, clump–clump merging and dynamical friction. The ratio of the internal clump velocity dispersion to that of the dark halo is an important parameter; as this ratio approaches a value of unity, heating by close encounters between clumps becomes less important, while the other dynamical processes continue to increase in importance. Our comparison between merging and disruption processes implies that spiral galaxies cannot be formed in a protosystem that contains a few large clumps, but can be formed through the accretion of many small clumps; elliptical galaxies form in a more clumpy environment than do spiral galaxies. Our results support the idea that the central cusp in the density profiles of dark haloes is the consequence of self-limiting merging of small, dense haloes. This implies that the collapse of a system of clumps/substructure is not sufficient to form a cD galaxy, with an extended envelope; plausibly, subsequent accretion of large galaxies is required. The post-collapse system is in general triaxial, with rounder systems resulting from fewer, but more massive, clumps. Persistent streams of material from disrupted clumps can be found in the outer regions of the final system, and at an overdensity of around 0.75, can cover 10 to 30 per cent of the sky.