The pendulum dilemma of fish orbits

The shape of a galaxy is constrained both by mechanisms of formation (dissipational versus dissipationless) and by the available orbit families (the shape and amount of regular and stochastic orbits). It is shown that, despite the often very flattened shapes of banana and fish orbits, these boxlet orbits generally do not fit a triaxial galaxy in detail because, similar to loop orbits, they spend too little time at the major axis of the model density distribution. This constraint from the shape of fish orbits is relaxed at (large) radii where the density profile of a galaxy is steep.     

The tidal stripping of satellites

We present an improved analytic calculation for the tidal radius of satellites and test our results against N -body simulations.
The tidal radius in general depends upon four factors: the potential of the host galaxy, the potential of the satellite, the orbit of the satellite and the orbit of the star within the satellite . We demonstrate that this last point is critical and suggest using three tidal radii to cover the range of orbits of stars within the satellite. In this way we show explicitly that prograde star orbits will be more easily stripped than radial orbits; while radial orbits are more easily stripped than retrograde ones. This result has previously been established by several authors numerically, but can now be understood analytically. For point mass, power-law (which includes the isothermal sphere), and a restricted class of split power-law potentials our solution is fully analytic. For more general potentials, we provide an equation which may be rapidly solved numerically.
Over short times (≲1–2 Gyr ∼1 satellite orbit), we find excellent agreement between our analytic and numerical models. Over longer times, star orbits within the satellite are transformed by the tidal field of the host galaxy. In a Hubble time, this causes a convergence of the three limiting tidal radii towards the prograde stripping radius. Beyond the prograde stripping radius, the velocity dispersion will be tangentially anisotropic.     

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.     

Orbital dynamics of three-dimensional bars – I. The backbone of three-dimensional bars. A fiducial case

In this series of papers we investigate the orbital structure of three-dimensional (3D) models representing barred galaxies. In the present introductory paper we use a fiducial case to describe all families of periodic orbits that may play a role in the morphology of three-dimensional bars. We show that, in a 3D bar, the backbone of the orbital structure is not just the x1 family, as in two-dimensional (2D) models, but a tree of 2D and 3D families bifurcating from x1. Besides the main tree we have also found another group of families of lesser importance around the radial 3:1 resonance. The families of this group bifurcate from x1 and influence the dynamics of the system only locally. We also find that 3D orbits elongated along the bar minor axis can be formed by bifurcations of the planar x2 family. They can support 3D bar-like structures along the minor axis of the main bar. Banana-like orbits around the stable Lagrangian points build a forest of 2D and 3D families as well. The importance of the 3D x1-tree families at the outer parts of the bar depends critically on whether they are introduced in the system as bifurcations in z or in   z˙   .     

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.     

Orbital dynamics of three-dimensional bars – II. Investigation of the parameter space

We investigate the orbital structure in a class of three-dimensional (3D) models of barred galaxies. We consider different values of the pattern speed, of the strength of the bar and of the parameters of the central bulge of the galactic model. The morphology of the stable orbits in the bar region is associated with the degree of folding of the x1 characteristic. This folding is larger for lower values of the pattern speed. The elongation of rectangular-like orbits belonging to x1 and to x1-originated families depends mainly on the pattern speed. A detailed investigation of the trees of bifurcating families in the various models shows that major building blocks of 3D bars can be supplied by families initially introduced as unstable in the system, but becoming stable at another energy interval. In some models without radial and vertical 2:1 resonances we find, except for the x1 and x1-originated families, also families related to the z -axis orbits, which support the bar. Bifurcations of the x2 family can build a secondary 3D bar along the minor axis of the main bar. This is favoured in the slowly rotating bar case.     

Dark matter halo response to the disc growth

We consider the sensitivity of the circular-orbit adiabatic contraction approximation to the baryon condensation rate and the orbital structure of dark matter haloes in the Λ cold dark matter (ΛCDM) paradigm. Using one-dimensional hydrodynamic simulations including the dark matter halo mass accretion history and gas cooling, we demonstrate that the adiabatic approximation is approximately valid even though haloes and discs may assemble simultaneously. We further demonstrate the validity of the simple approximation for ΛCDM haloes with isotropic velocity distributions using three-dimensional N -body simulations. This result is easily understood: an isotropic velocity distribution in a cuspy halo requires more circular orbits than radial orbits. Conversely, the approximation is poor in the extreme case of a radial orbit halo. It overestimates the response of a core dark matter halo, where radial orbit fraction is larger. Because no astronomically relevant models are dominated by low angular momentum orbits in the vicinity of the disc and the growth time-scale is never shorter than a dynamical time, we conclude that the adiabatic contraction approximation is useful in modelling the response of dark matter haloes to the growth of a disc.     

Stellar dynamics in razor-thin discs with massive nuclear black holes

The bifurcations of orbit-averaged dynamics are studied in a class of razor-thin discs with central black holes. The model used here consists of a perturbed harmonic oscillator Hamiltonian augmented with a GM r potential. Through a sequence of conformal and canonical transformations, we reduce the phase-space flows of the system to a set of non-linear differential equations on a sphere. Based on the critical points of the averaged system, we classify orbit families and reveal the existence of six types of periodic motions: circular , long - and short-axis elliptical , long - and short-axis radial and inclined radial orbits. Long-axis elliptical orbits and their surrounding tubes have significant features: whilst they keep stars away from the centre, they elongate in the same direction as the density profile. These properties are helpful in the construction of self-consistent equilibria.     

Tidal tails of star clusters

Based on recent findings of a formation mechanism of substructure in tidal tails by Küpper et al., we investigate a more comprehensive set of N -body models of star clusters on orbits about a Milky Way like potential. We find that the predicted epicyclic overdensities arise in any tidal tail no matter which orbit the cluster follows as long as the cluster lives long enough for the overdensities to build up.
The distance of the overdensities along the tidal tail from the cluster centre depends for circular orbits only on the mass of the cluster and the strength of the tidal field, and therefore decreases monotonically with time, while for eccentric orbits the orbital motion influences the distance, causing a periodic compression and stretching of the tails and making the distance oscillate with time. We provide an approximation for estimating the distance of the overdensities in this case.
We describe an additional type of overdensity which arises in extended tidal tails of clusters on eccentric orbits, when the acceleration of the tidal field on the stellar stream is no longer homogeneous. Moreover, we conclude that a pericentre passage or a disc shock is not the direct origin of an overdensity within a tidal tail. Escape due to such tidal perturbations does not take place immediately after the perturbation but is rather delayed and spread over the orbit of the cluster. All observable overdensities are therefore of the mentioned two types. In particular, we note that substructured tidal tails do not imply the existence of dark matter substructures in the haloes of galaxies.     

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 m= 1 instabilities and lopsidedness in disc galaxies

Lopsidedness is common in spiral galaxies. Often, there is no obvious external cause, such as an interaction with a nearby galaxy, for such features. Alternatively, the lopsidedness may have an internal cause, such as a dynamical instability. In order to explore this idea, we have developed a computer code that searches for self-consistent perturbations in razor-thin disc galaxies and performed a thorough mode-analysis of a suite of dynamical models for disc galaxies embedded in an inert dark matter halo with varying amounts of rotation and radial anisotropy.
Models with two equal-mass counter-rotating discs and fully rotating models both show growing lopsided modes. For the counter-rotating models, this is the well-known counter-rotating instability, becoming weaker as the net rotation increases. The m = 1 mode of the maximally rotating models, on the other hand, becomes stronger with increasing net rotation. This rotating m = 1 mode is reminiscent of the eccentricity instability in near-Keplerian discs.
To unravel the physical origin of these two different m = 1 instabilities, we studied the individual stellar orbits in the perturbed potential and found that the presence of the perturbation gives rise to a very rich orbital behaviour. In the linear regime, both instabilities are supported by aligned loop orbits. In the non-linear regime, other orbit families exist that can help support the modes. In terms of density waves, the counter-rotating m = 1 mode is due to a purely growing Jeans-type instability. The rotating m = 1 mode, on the other hand, grows as a result of the swing amplifier working inside the resonance cavity that extends from the disc centre out to the radius where non-rotating waves are stabilized by the model's outwardly rising Q profile.     

Finding how many isolating integrals of motion an orbit obeys

The correlation dimension, that is the dimension obtained by computing the correlation function of pairs of points of a trajectory in phase space, is a numerical technique introduced in the field of non-linear dynamics in order to compute the dimension of the manifold in which an orbit moves, without the need of knowing the actual equations of motion that give rise to the trajectory. This technique has been proposed in the past as a method to measure the dimension of stellar orbits in astronomical potentials, that is the number of isolating integrals of motion the orbits obey. Although the algorithm can in principle yield that number, some care has to be taken in order to obtain good results. We studied the relevant parameters of the technique, found their optimal values, and tested the validity of the method on a number of potentials previously studied in the literature, using the Smaller Alignment Index (SALI), Lyapunov exponents and spectral dynamics as gauges.     

Active galactic nuclei and the minor merger hypothesis

We have investigated the dynamics of the merging process in the minor merger hypothesis for active galactic nuclei. We find that for a satellite galaxy to be able to merge directly with the nucleus of the host galaxy (for example, to give rise to the compact dust discs which are seen in early-type active galaxies), the initial orbit of the satellite is required to be well aimed. For the case of the host galaxy being a disc galaxy, if the initial orbits of the satellites are randomly oriented with respect to the host galaxy, then the orbits of those which reach the host nuclear regions in a reasonable time are also fairly randomly oriented once they reach the nucleus. We note that this result might be able to provide an explanation of why the jet directions in the nuclei of Seyfert galaxies are apparently unrelated to the plane of the galaxy discs.     

The kinematics of lopsided galaxies

Lopsidedness is a common feature in galaxies, both in the distribution of light and in the kinematics. We investigate the kinematics of a model for lopsided galaxies that consists of a disc lying off-centre in a dark halo, and circling around the halo centre. We search for families of stable, closed, non-crossing orbits, and assume that gas in our galaxies moves on these orbits. Several of our models show strong lopsided gas kinematics, especially those in which the disc spins around its axis in a retrograde sense compared with its motion around the halo centre. We are able to reproduce the H  i velocity map of the kinematically lopsided galaxy NGC 4395.
The lopsidedness in our models is most pronounced in the models where the halo provides a relatively large fraction of the total mass at small radii. This may explain why the gas shows lopsidedness more frequently in late-type galaxies, which are dominated by dark matter. Surfaces of section show large regions of irregular orbits in the models where the halo density is low. This may indicate that these models are unstable.     

Regular black hole motion and stellar orbital resonances

The motion of a black hole about the centre of gravity of its host galaxy induces a strong response from the surrounding stellar population. We treat the case of a harmonic potential analytically and show that half of the stars on circular orbits in that potential shift to an orbit of lower energy, while the other half receive a positive boost and recede to a larger radius. The black hole itself remains on an orbit of fixed amplitude and merely acts as a catalyst for the evolution of the stellar energy distribution function f ( E ). We show that this effect is operative out to a radius of approximately three to four times the hole's influence radius, R _bh. We use numerical integration to explore more fully the response of a stellar distribution to black hole motion. We consider orbits in a logarithmic potential and compare the response of stars on circular orbits, to the situation of a 'warm' and 'hot' (isotropic) stellar velocity field. While features seen in density maps are now wiped out, the kinematic signature of black hole motion still imprints the stellar line-of-sight mean velocity to a magnitude ≃13 per cent the local rms velocity dispersion σ. A study in three dimensions suggests a reduced effect for polar orbits.     

Resonances and bifurcations in axisymmetric scale-free potentials

We investigate an analytical treatment of bifurcations of families of resonant 'thin' tubes in axisymmetric galactic potentials. We verify that the most relevant bifurcations are due to the (1:1) resonance producing the 'inclined' orbits through two different mechanisms: from the disc orbit and from the 'thin' tube associated with the vertical oscillation. The closest resonances occurring after these are the (4:3) resonance in the oblate case and the (2:1) resonance in the prolate case. The (1:1) resonances are treated in a straightforward way using a second-order truncated normal form. The higher order resonances are instead cumbersome to investigate, because the normal form has to be truncated to a high degree and the number of terms grows very rapidly. We therefore adopt a further simplification giving analytical formulae for the values of the parameters at which bifurcations ensue and compare them with selected numerical results. Thanks to the asymptotic nature of the series involved, the predictions are reliable well beyond the convergence radius of the original series.     

A novel algorithm for generating libration point orbits about the collinear points

This paper presents a numerical algorithm that can generate long-term libration points orbits (LPOs) and the transfer orbits from the parking orbits to the LPOs in the circular-restricted three-body problem (CR3BP) and the full solar system model without initial guesses. The families of the quasi-periodic LPOs in the CR3BP can also be constructed with this algorithm. By using the dynamical behavior of LPO, the transfer orbit from the parking orbit to the LPO is generated using a bisection method. At the same time, a short segment of the target LPO connected with the transfer orbit is obtained, then the short segment of LPO is extended by correcting the state towards its adjacent point on the stable manifold of the target LPO with differential evolution algorithm. By implementing the correction strategy repeatedly, the LPOs can be extended to any length as needed. Moreover, combining with the continuation procedure, this algorithm can be used to generate the families of the quasi-periodic LPOs in the CR3BP.     

Ram pressure stripping the hot gaseous haloes of galaxies in groups and clusters

We use a large suite of carefully controlled full hydrodynamic simulations to study the ram pressure stripping of the hot gaseous haloes of galaxies as they fall into massive groups and clusters. The sensitivity of the results to the orbit, total galaxy mass, and galaxy structural properties is explored. For typical structural and orbital parameters, we find that ∼30 per cent of the initial hot galactic halo gas can remain in place after 10 Gyr. We propose a physically simple analytic model that describes the stripping seen in the simulations remarkably well. The model is analogous to the original formulation of Gunn & Gott, except that it is appropriate for the case of a spherical (hot) gas distribution (as opposed to a face-on cold disc) and takes into account that stripping is not instantaneous but occurs on a characteristic time-scale. The model reproduces the results of the simulations to within ≈10 per cent at almost all times for all the orbits, mass ratios, and galaxy structural properties we have explored. The one exception involves unlikely systems where the orbit of the galaxy is highly non-radial and its mass exceeds about 10 per cent of the group or cluster into which it is falling (in which case the model underpredicts the stripping following pericentric passage). The proposed model has several interesting applications, including modelling the ram pressure stripping of both observed and cosmologically simulated galaxies and as a way to improve present semi-analytic models of galaxy formation. One immediate consequence is that the colours and morphologies of satellite galaxies in groups and clusters will differ significantly from those predicted with the standard assumption of complete stripping of the hot coronae.     

Boxy/peanut 'bulges': comparing the structure of galaxies with the underlying families of periodic orbits

The vertical profiles of disc galaxies are built by the material trapped around stable periodic orbits, which form their 'skeletons'. Therefore, knowledge of the stability of the main families of periodic orbits in appropriate 3D models enables one to predict possible morphologies for edge-on disc galaxies. In a pilot survey we compare the orbital structures that lead to the appearance of 'peanut'- and 'X'-like features with the edge-on profiles of three disc galaxies (IC 2531, NGC 4013 and UGC 2048). The subtraction from the images of a model representing the axisymmetric component of the galaxies reveals the contribution of the non-axisymmetric terms. We find a direct correspondence between the orbital profiles of 3D bars in models and the observed main morphological features of the residuals. We also apply a simple unsharp masking technique in order to study the sharpest features of the images. Our basic conclusion is that the morphology of the boxy 'bulges' of these galaxies can be explained by considering disc material trapped around stable 3D periodic orbits. In most models, these building-block periodic orbits are bifurcated from the planar central family of a non-axisymmetric component, usually a bar, at low-order vertical resonances. In such a case, the boxy 'bulges' are parts of bars seen edge-on. For the three galaxies we study, the families associated with the 'peanut' or 'X'-shape morphology are probably bifurcations at the vertical 2/1 or 4/1 resonance.     

The haloes of merger remnants

We perform collisionless N -body simulations of 1:1 galaxy mergers, using models which include a galaxy halo, disc and bulge, focusing on the behaviour of the halo component. The galaxy models are constructed without recourse to a Maxwellian approximation. We investigate the effect of varying the galaxies' orientation, their mutual orbit and the initial velocity anisotropy or cusp strength of the haloes upon the remnant halo density profiles and shape, as well as on the kinematics. We observe that the halo density profile (determined as a spherical average, an approximation we find appropriate) is exceptionally robust in mergers, and that the velocity anisotropy of our remnant haloes is nearly independent of the orbits or initial anisotropy of the haloes. The remnants follow the halo anisotropy – local density slope (β–γ) relation suggested by Hansen & Moore in the inner parts of the halo, but β is systematically lower than this relation predicts in the outer parts. Remnant halo axis ratios are strongly dependent on the initial parameters of the haloes and on their orbits. We also find that the remnant haloes are significantly less spherical than those described in studies of simulations which include gas cooling.