Loop Orbits Research Articles

Lagrangian fillings in $A$-type and their Kálmán loop orbits

We compare two constructions of exact Lagrangian fillings of Legendrian positive braid closures, the Legendrian weaves of Casals–Zaslow, and the decomposable Lagrangian fillings of Ekholm–Honda–Kálmán, and show that they coincide for large families of Lagrangian fillings. As a corollary, we obtain an explicit correspondence between Hamiltonian isotopy classes of decomposable Lagrangian fillings of Legendrian (2,n) torus links described by Ekholm–Honda–Kálmán and the weave fillings constructed by Treumann and Zaslow. We apply this result to describe the orbital structure of the Kálmán loop and give a combinatorial criteria to determine the orbit size of a filling. We follow our geometric discussion with a Floer-theoretic proof of the orbital structure, where an identity studied by Euler in the context of continued fractions makes a surprise appearance. We conclude by giving a purely combinatorial description of the Kálmán loop action on the fillings discussed above in terms of edge flips of triangulations.

Face-on structure of barlenses and boxy bars: an insight from spectral dynamics

ABSTRACT Based on the spectral analysis of individual orbits of stars from different N-body models, we show that the face-on morphology of the so-called ‘face-on peanut’ bars (boxy bars) and barlenses is supported by different types of orbits. For ‘face-on peanut’ bars, the so-called boxy orbits come to the fore, and they are responsible for the unusual morphology of the bar in the central regions. In the models with compact bulges, the bars show a barlens morphology in their central parts. We found that the barlens supporting orbits come in two types, one of which gives a square-like shape and the other have a rosette-like shape in the frame co-rotating with the bar. Such a shape is typical for orbits around stable loop orbits in nearly axisymmetric potentials only slightly distorted by the bar. They were already known from some of the previous studies but their role in barlens shaping was barely investigated. Although quite simple, the rosette-like orbits are found to be the main building block of a barlens in our models. The detailed analysis of all bar orbits also allowed us to distinguish the x2 orbital family and isolate the structure supported by orbits trapped around this family. The x2 family is well-known, but, apparently, for the first time in N-body models we have revealed the structure it supports by means of spectral dynamics and highlight its contribution to the barlens. We found that the x2 family population increases with an increase in central matter concentration.

Orbits in elementary, power-law galaxy bars – 1. Occurrence and role of single loops

Orbits in galaxy bars are generally complex, but simple closed loop orbits play an important role in our conceptual understanding of bars. Such orbits are found in some well-studied potentials, provide a simple model of the bar in themselves, and may generate complex orbit families. The precessing, power ellipse (p-ellipse) orbit approximation provides accurate analytic orbit orbits in symmetric galaxy potentials. It remains useful for finding and fitting simple loop orbits in the frame of a rotating bar with bar-like and symmetric power-law potentials. Second order perturbation theory yields two or fewer simple loop solutions in these potentials. Numerical integrations in the parameter space neighborhood of perturbation solutions reveal zero or one ac- tual loops in a range of such potentials with rising rotation curves. These loops are embedded in a small parameter region of similar, but librating orbits, which have a subharmonic frequency superimposed on the basic loop. These loops and their librat- ing companions support annular bars. Solid bars can be produced in more complex potentials, as shown by an example with power-law indices varying with radius. The power-law potentials can be viewed as the elementary constituents of more complex potentials. Numerical integrations also reveal interesting classes of orbits with multiple loops. In two-dimensional, self-gravitating bars, with power-law potentials, single loop orbits are very rare. This result suggests that gas bars or oval distortions are unlikely to be long-lived, and that complex orbits or three-dimensional structure must support self-gravitating stellar bars.

Torus-fitting method for obtaining action variables in two-dimensional Galactic potentials

A phase-space distribution function of the steady state in galaxy models that admits regular orbits overall in the phase-space can be represented by a function of three action variables. This type of distribution function in Galactic models is often constructed theoretically for comparison of the Galactic models with observational data as a test of the models. On the other hand, observations give Cartesian phase-space coordinates of stars. Therefore it is necessary to relate action variables and Cartesian coordinates in investigating whether the distribution function constructed in galaxy models can explain observational data. Generating functions are very useful in practice for this purpose, because calculations of relations between action variables and Cartesian coordinates by generating functions do not require a lot of computational time or computer memory in comparison with direct numerical integration calculations of stellar orbits. Here, we propose a new method called a torus-fitting method, by which a generating function is derived numerically for models of the Galactic potential in which almost all orbits are regular. We confirmed the torus-fitting method can be applied to major orbit families (box and loop orbits) in some two-dimensional potentials. Furthermore, the torus-fitting method is still applicable to resonant orbit families, besides major orbit families. Hence the torus-fitting method is useful for analyzing real Galactic systems in which a lot of resonant orbit families might exist.

Quaternion solution for the rock’n’roller: Box orbits, loop orbits and recession

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock’n’roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.

Tidal disruption of globular clusters in dwarf galaxies with triaxial dark matter haloes

We use N-body simulations to study the tidal evolution of globular clusters (GCs) in dwarf spheroidal (dSph) galaxies. Our models adopt a cosmologically motivated scenario in which the dSph is approximated by a static NFW halo with a triaxial shape. We apply our models to five GCs spanning three orders of magnitude in stellar density and two in mass, chosen to represent the properties exhibited by the five GCs of the Fornax dSph. We show that only the object representing Fornax's least dense GC (F1) can be fully disrupted by Fornax's internal tidal field--the four denser clusters survive even if their orbits decay to the centre of Fornax. For a large set of orbits and projection angles we examine the spatial and velocity distribution of stellar debris deposited during the complete disruption of an F1-like GC. Our simulations show that such debris appears as shells, isolated clumps and elongated over-densities at low surface brightness (>26 mag/arcsec^2), reminiscent of substructure observed in several MW dSphs. Such features arise from the triaxiality of the galaxy potential and do not dissolve in time. The kinematics of the debris depends strongly on the progenitor's orbit. Debris associated with box and resonant orbits does not display stream motions and may appear "colder"/"hotter" than the dSph's field population if the viewing angle is perpendicular/parallel to progenitor's orbital plane. In contrast, debris associated with loop orbits shows a rotational velocity that may be detectable out to few kpc from the galaxy centre. Chemical tagging that can distinguish GC debris from field stars may reveal whether the merger of GCs contributed to the formation of multiple stellar components observed in dSphs.

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 center out to the radius where non-rotating waves are stabilized by the model's outwardly rising Q-profile.

Dynamics of a single ion in a perturbed Penning trap. Sextupolar perturbation

In the frame work of classical mechanics, we study the nonlinear dynamics of a single ion trapped in a Penning trap perturbed by an electrostatic sextupolar perturbation. The perturbation is caused by a deformation in the configuration of the electrodes. By using a Hamiltonian formulation, we obtain that the system is governed by three parameters: the z-component of the canonical angular momentum P φ - which is a constant of the motion because the perturbation we assume is axial-symmetric -, the parameter δ that determines the ratio between the axial and the cyclotron frequencies, and the parameter a which indicates how far from the ideal design the electrodes are. We study the case P φ = 0. By means of surfaces of section, we show that the phase space structure is made of three fundamental families of orbits: arch, loop and box orbits. The coexistence of these kinds of orbits depends on the parameter δ. The escape is also explained on the basis of the shape of the potential energy surface as well as of the phase space structure.

Dynamical modeling of the stellar nucleus of M 31

We present stellar dynamical models of the lopsided, double{peaked nucleus of M 31, derived from Hubble Space Telescope (HST) photometry. A Schwarzschild{type method, in conjunction with Richardson{Lucy deconvolution, was employed to construct steadily rotating, hot, stellar disks. The stars orbit a massive dark object, on prograde and retrograde quasi{periodic loop orbits. Our results support Tremaine's eccentric disk model, extended to include a more massive disk, non zero pattern speed (), and dierent viewing angle. Most of the disk mass populated prograde orbits, with'3.4% on retrograde orbits. The best ts to photometric and kinematic maps were disks with' 16 km s 1 pc 1 . We speculate on the origins of the lopsidedness, invoking recent work on the linear overstability of nearly Keplerian disks, that possess even a small amount of a counter { rotating component. Accretion of material { no more massive than a globular cluster { onto a preexisting stellar disk, will account for the mass in our retrograde orbits, and could have stimulated the lopsidedness seen in the nucleus of M 31.

Integrals of Motion in Kuzmin‐like Potentials

We have identified an orthogonal curvilinear coordinate system (ξ1,ξ2,ξ3) for which surfaces of constant ξ1 exactly coincide with equipotential surfaces in the potential. The three principal coordinates are ξ1 ≡ [x2 + y2 + (a + |z|)2]1/2, ξ2 ≡ arctan[(|z|/z)(x2 + y2)1/2/(a + |z|)], and ξ3 ≡ arctan[(y/x)]. When the equation of motion is written in terms of this Kuzmin coordinate system, the integrals of motion for particles moving in Kuzmin-like potentials are easily recognized. Specifically, any time-independent potential Φ that is a function only of ξ1 will exhibit the same number of integrals of motion as does a spherically symmetric potential. In such Kuzmin-like potentials, the vector ≡1ξ1 × is the analog of the specific angular momentum vector in spherically symmetric potentials. Furthermore, in potentials of the form Φ ∝ 1/ξ1, all three Cartesian components of the vector ≡ [ × +1(ξ1Φ)] (an analog of the Laplace-Runge-Lenz vector) also prove to be integrals of the motion. For two specific potentials, the potential and a logarithmic Kuzmin-like potential, we have derived analytical expressions defining the surfaces of section for particles in polar orbits, and in both cases we have identified the domain of occupancy of box orbits, loop orbits, and periodic orbits.

Integrable models of galactic discs with double nuclei

We introduce a new class of 2-D mass models, whose potentials are of Stackel form in elliptic coordinates. Our model galaxies have two separate strong cusps that form double nuclei. The potential and surface density distributions are locally axisymmetric near the nuclei and become highly non-axisymmetric outside the nucleus. The surface density diverges toward the cuspy nuclei with the law �/ r 2 . Our model is sustained by four general types of regular orbits: butterfly, nucleuphilic banana, horseshoe and aligned loop orbits. Horseshoes and nucleuphilic bananas support the existence of cuspy regions. Butterflies and aligned loops control the non-axisymmetric shape of outer regions. Without any need for central black holes, our distributed mass models resemble the nuclei of M31 and NGC4486B. It is also shown that the self-gravity of the stellar disc can prevent the double nucleus to collapse.

The pendulum dilemma of fish orbits

The shape of a galaxy is constrained both by mechanisms of formation (dissipational vs. 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.

Building Galaxy Models with Schwarzschild Method and Spectral Dynamics

AbstractTremendous progress has been made recently in modelling the morphology and kinematics of centers of galaxies. Increasingly realistic models are built for central bar, bulge, nucleus and black hole of galaxies, including our own. The newly revived Schwarzschild method has played a central role in these theoretical modellings. Here I will highlight some recent work at Leiden on extending the Schwarzschild method in a few directions. After a brief discussion of (i) an analytical approach to include stochastic orbits (Zhao 1996), and (ii) the “pendulum effect” of loop and boxlet orbits (Zhao, Carollo, de Zeeuw 1999), I will concentrate on the very promising (iii) spectral dynamics method, with which not only can one obtain semi-analytically the actions of individual orbits as previously known, but also many other physical quantities, such as the density in configuration space and the line-of-sight velocity distribution of a superposition of orbits (Copin, Zhao & de Zeeuw 1999). The latter method also represents a drastic reduction of storage space for the orbit library and an increase in accuracy over the grid-based Schwarzschild method.

Stellar dynamics around black holes in galactic nuclei

We classify orbits of stars that are bound to central black holes in galactic nuclei. The stars move under the combined gravitational influences of the black hole and the central star cluster. Within the sphere of influence of the black hole, the orbital periods of the stars are much shorter than the periods of precession. We average over the orbital motion and end up with a simpler problem and an extra integral of motion: the product of the black hole mass and the semimajor axis of the orbit. Thus the black hole enforces some degree of regularity in its neighbourhood. Well within the sphere of influence, (i) planar, as well as three-dimensional, axisymmetric configurations — both of which could be lopsided — are integrable, (ii) fully three-dimensional clusters with no spatial symmetry whatsoever must have semi-regular dynamics with two integrals of motion. Similar considerations apply to stellar orbits when the black hole grows adiabatically. We introduce a family of planar, non-axisymmetric potential perturbations, and study the orbital structure for the harmonic case in some detail. In the centred potentials there are essentially two main families of orbits: the familiar loops and lenses, which were discussed by Sridhar and Touma. We study the effect of lopsidedness, and identify a family of loop orbits, the orientation of which reinforces the lopsidedness. This is an encouraging sign for the construction of self-consistent models of eccentric discs around black holes, such as in M31 and NGC 4486B.

Dynamics of galaxies with triaxial haloes

We study the stability of trajectories near the disk plane of galaxy models with a triaxial dark matter halo component. We also examine the effect of weak discreteness noise, rapidly rotating bar perturbations and weak dissipation on these trajectories. The latter effect is studied both by adding a dissipative component to the force law and by using particle simulations. If the matter distribution is triaxial and has a constant density core, dissipation leads to inflow of material inside the core radius to the centre (since no non-self-intersecting closed orbits exist in the central areas). This leads to the formation of central masses which in turn destabilise the trajectories of any stars formed in these regions. In particular, even if the gas settles by dissipation into a flat disk, stars formed in that disk will later form a bulge like distribution the extent of which would be related to the core radius of the halo and the original asymmetry in the plane. The process of gas inflow is regulated by the fact that too large and condensed a central mass leads to the creation of stable closed loop orbits in the central area around which the gas can move. This would appear to stop the accumulation of central mass before it becomes large enough for rapid loss of halo triaxiality. It was found that weak discreteness noise can increase the fraction of such trajectories significantly and can therefore have important consequences for the modelling of galaxies. The addition of rapidly rotating bar perturbations also increases the degree of instability dramatically. So if bars can form in triaxial haloes they are likely to be quickly destroyed leaving a bulge like structure behind (which may explain the absence of bars in surveys of high redshift galaxies). If they do survive, their main effect on the gas dynamics is to create attractors other than the centre, around which the gas can move. We discuss some possible consequences of the aforementioned effects. In particular, it is suggested that the halo core radius and initial asymmetry may be important in determining the relative disk-halo contribution to the rotation curve of a galaxy - and hence its Hubble type.

Cusps without chaos

We present cuspy, non-axisymmetric, scale-free mass models of discs, whose gravitational potentials are of Stäckel form in parabolic coordinates. A black hole may be added at the centre, without in any way affecting the Stäckel form ; the dynamics in these potentials is, of course, fully integrable. The surface density Σdisc ∝ 1/ry, where 0 < γ < 1 corresponds to steep cusps for which the central force diverges. Thus cusps, black holes, and non-axisymmetry are not a sure recipe for chaos, as is generally assumed. A new family of orbits, lens orbits, emerges to replace the box orbits of models of elliptical galaxies that have constant-density cores. Loop orbits are conspicuous by their absence. Both lenses and bananas (the other family of orbits) can be elongated in the direction of the density distribution, a property that is favourable for the construction of non-axisymmetric, self-consistent equilibrium models of elliptical galaxies.

Signatures of bulge triaxiality from kinematics in Baade's window

We study a sample of 62 Baade's Window (1,-4) K giants that have published proper motions, radial velocity and metallicity. We construct the velocity ellipsoids for the metal rich stars and metal poor stars. We find a vertex deviation characteristic of a non-axisymmetric system. The long axes of the velocity ellipsoids of two samples point to two nearly perpendicular directions (lv,bv)=(-65+-9,+14+-9) and (25+-14,-11+-14) respectively. The vertex deviations of the velocity ellipsoids cannot be consistently explained by ANY oblate model. We populate orbits in a realistic bar potential with a Gaussian velocity distribution, allowing us to simulate and interpret observations. We conclude that the data are consistent with a triaxial bulge pointing towards (l,b) with l<0 and b=0 as suggested by earlier work. We also predict that low latitude abs(b)<4 deg bulge fields should show the vertex deviation more strongly and would therefore be the best locations for future proper motion studies. The metal rich and metal poor stars appear to occupy the direct boxy and retrograde loop orbits. The correlations between the metallicity and the orbit families can develop if the Bulge forms dissipatively on a sufficiently long time scale. However, it is difficult to reconcile the existence of such correlations with a violent history for the potential of the inner Galaxy.

Dynamical friction in a non-rotating triaxial galaxy: the effect on box orbits

The effect of dynamical friction in the deceleration of globular clusters (GCs) on box orbits in triaxial potentials is investigated by means of numerical computations in Schwarzschild's self-consistent model. It is shown that, because of lack of conservation of angular momentum, GCs pass inside the galactic core where they are powerfully decelerated. Typical energy decay times are 10 times shorter than for roundish loop orbits of comparable energy. An analytic approximation for the energy decay time is also derived, and it is shown that it is independent of the detailed shape of the orbit, and of the detailed velocity structure of the underlying galactic model

Stellar orbits in angle variables. II - Two-dimensional orbits in a rotating potential

view Abstract Citations (2) References (10) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Stellar Orbits in Angle Variables. II. Two-dimensional Orbits in a Rotating Potential Guerra, David V. ; Ratcliff, Stephen J. Abstract An algorithm has devised to calculate two-dimensional quasi-periodic orbits in the plane of rotation of a nonaxisymmetric galactic potential which is slowly rotating. The method used is an extension of that used by Ratcliff, Chang, and Schwarzschild, in which the orbit is specified by its frequencies and the Fourier amplitudes are solved for directly. A sample of orbits is calculated for a range of rotation speeds and the results are compared to previous work by de Zeeuw and Merritt in the same rotating potential. The retrograde loop orbits, at least at moderate energies, form a continuous family with the boxes, while the prograde loops remain distinct and separate. Publication: The Astrophysical Journal Pub Date: January 1990 DOI: 10.1086/168219 Bibcode: 1990ApJ...348..127G Keywords: Dynamic Characteristics; Elliptical Galaxies; Many Body Problem; Stellar Orbits; Stellar Rotation; Algorithms; Equations Of Motion; Fourier Series; Astrophysics; GALAXIES: INTERNAL MOTIONS; STARS: STELLAR DYNAMICS full text sources ADS |

Stellar orbits in angle variables

view Abstract Citations (21) References (4) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Stellar orbits in angle variables Ratcliff, S. J. ; Chang, K. M. ; Schwarzschild, M. Abstract Following recent basic developments by Binney and Spergel, the equations of motion for a star in a galactic potential have been transformed from time to angle variables. A numerical procedure has been devised to solve the transformed equations for two-dimensional box and loop orbits in the equatorial plane of a triaxial potential. In this procedure an orbit is specified not by its starting values in phase space, but either by its frequencies or by its action values. A substantial set of orbits has been derived by this procedure for one potential. The results for their physical and geometrical characteristics are displayed in action diagrams. Publication: The Astrophysical Journal Pub Date: April 1984 DOI: 10.1086/161925 Bibcode: 1984ApJ...279..610R Keywords: Celestial Mechanics; Equations Of Motion; Orbit Calculation; Stellar Motions; Stellar Orbits; Algorithms; Iteration; Partial Differential Equations; Transformations (Mathematics); Astronomy full text sources ADS |