Action-angle Space Research Articles

Boxy/peanut shaping of a mature galactic bar in action-angle space

We study vertical resonant trapping and resonant heating of orbits. These two processes both lead to the growth of a boxy/peanut-shaped bulge in a typical $N$-body model. For the first time, we study this by means of the action variables and resonant angles of the actual orbits that compose the model itself. We used the resonant angle instead of the frequency ratio, which allowed us to clearly distinguish between these two processes in numerical simulations. We show that trapping and heating occur simultaneously, at least at the stage of a mature bar, that is, some orbits quickly pass through vertical resonance while at the same time, a substantial number of orbits remains trapped into this stage for a long time. Half of all bar orbits spend more than 2.5 Gyr in vertical resonance over an interval of 4 Gyr. Half of the orbits trapped into the bar over the last 3 Gyr of simulation remain captured in vertical resonance for more than 2 Gyr. We conclude that in the later stages of the bar evolution, the process of vertical trapping dominates in the ongoing process that causes the boxy/peanut shape of a bar in a typical $N$-body model. This contradicts the results of several recent works.

The Age Distribution of Stellar Orbit Space Clumps

The orbit distribution of young stars in the Galactic disk is highly structured, from well-defined clusters to streams of stars that may be widely dispersed across the sky, but are compact in orbital action-angle space. The age distribution of such groups can constrain the timescales over which conatal groups of stars disperse into the “field.” Gaia data have proven powerful in identifying such groups in action-angle space, but the resulting member samples are often too small and have too narrow a color–magnitude diagram (CMD) coverage to allow robust age determinations. Here, we develop and illustrate a new approach that can estimate robust stellar population ages for such groups of stars. This first entails projecting the predetermined action-angle distribution into the 5D space of positions, parallaxes, and proper motions, where much larger samples of likely members can be identified over a much wider range of the CMD. It then entails isochrone fitting that accounts for: (a) widely varying distances and reddenings; (b) outliers and binaries; (c) sparsely populated main-sequence turnoffs, by incorporating the age information of the low-mass main sequence; and (d) the possible presence of an intrinsic age spread in the stellar population. When we apply this approach to 92 nearby stellar groups identified in 6D orbit space, we find that they are predominantly young (≲1 Gyr), mono-age populations. Many groups are established (known) localized clusters with possible tidal tails, while others tend to be widely dispersed and manifestly unbound. This new age-dating tool offers a stringent approach to understanding on which orbits stars form in the solar neighborhood and how quickly they disperse into the field.

Robust clustering of the local Milky Way stellar kinematic substructures with Gaia eDR3

ABSTRACT Understanding local stellar kinematic substructures in the solar neighbourhood helps build a complete picture of the formation of the Milky Way, as well as an empirical phase space distribution of dark matter that would inform detection experiments. We apply the clustering algorithm hdbscan on the Gaia early third data release to identify a list of stable clusters in velocity space and action-angle space by taking into account the measurement uncertainties and studying the stability of the clustering results. We find 1405 (497) stars in 23 (6) robust clusters in velocity space (action-angle space) that are consistently not associated with noise. We discuss the kinematic properties of these structures and study whether many of the small clusters belong to a similar larger cluster based on their chemical abundances. They are attributed to the known structures: the Gaia Sausage-Enceladus, the Helmi Stream, and globular cluster NGC 3201 are found in both spaces, while NGC 104 and the thick disc (Sequoia) are identified in velocity space (action-angle space). Although we do not identify any new structures, we find that the hdbscan member selection of already known structures is unstable to input kinematics of the stars when resampled within their uncertainties. We therefore present the stable subset of local kinematic structures, which are consistently identified by the clustering algorithm, and emphasize the need to take into account error propagation during both the manual and automated identification of stellar structures, both for existing ones as well as future discoveries.

Characterizing the 3D Kinematics of Young Stars in the Radcliffe Wave

We present an analysis of the kinematics of the Radcliffe Wave, a 2.7 kpc long sinusoidal band of molecular clouds in the solar neighborhood recently detected via 3D dust mapping. With Gaia DR2 astrometry and spectroscopy, we analyze the 3D space velocities of ∼1500 young stars along the Radcliffe Wave in action-angle space, using the motion of the wave’s newly born stars as a proxy for its gas motion. We find that the vertical angle of young stars—corresponding to their orbital phase perpendicular to the Galactic plane—varies significantly as a function of position along the structure, in a pattern potentially consistent with a wavelike oscillation. This kind of oscillation is not seen in a control sample of older stars from Gaia occupying the same volume, disfavoring formation channels caused by long-lived physical processes. We use a “wavy midplane” model to try to account for the trend in vertical angles seen in young stars, and find that while the best-fit parameters for the wave’s spatial period and amplitude are qualitatively consistent with the existing morphology defined by 3D dust, there is no evidence for additional velocity structure. These results support more recent and/or transitory processes in the formation of the Radcliffe Wave, which would primarily affect the motion of the wave’s gaseous material. Comparisons of our results with new and upcoming simulations, in conjunction with new stellar radial velocity measurements in Gaia DR3, should allow us to further discriminate between various competing hypotheses.

Conservation of radial actions in time-dependent spherical potentials

ABSTRACT In slowly evolving spherical potentials, Φ(r, t), radial actions are typically assumed to remain constant. Here, we construct dynamical invariants that allow us to derive the evolution of radial actions in spherical central potentials with an arbitrary time dependence. We show that to linear order, radial actions oscillate around a constant value with an amplitude $\propto \dot{\Phi }/\Phi \, P(E,L)$. Using this result, we develop a diffusion theory that describes the evolution of the radial action distributions of ensembles of tracer particles orbiting in generic time-dependent spherical potentials. Tests against restricted N-body simulations in a varying Kepler potential indicate that our linear theory is accurate in regions of phase-space in which the diffusion coefficient $\tilde {D}(J_r) \lt 0.01\, J_r^2$. For illustration, we apply our theory to two astrophysical processes. We show that the median mass accretion rate of a Milky Way (MW) dark matter (DM) halo leads to slow global time-variation of the gravitational potential, in which the evolution of radial actions is linear (i.e. either adiabatic or diffusive) for ∼84 per cent of the DM halo at redshift z = 0. This fraction grows considerably with look-back time, suggesting that diffusion may be relevant to the modelling of several Gyr old tidal streams in action-angle space. As a second application, we show that dynamical tracers in a dwarf-size self-interacting DM halo (with $\sigma /m_\chi = 1\, {\rm cm^2g^{-1}}$) have invariant radial actions during the formation of a cored density profile.

From birth associations to field stars: mapping the small-scale orbit distribution in the Galactic disc

ABSTRACT Stars born at the same time in the same place should have formed from gas of the same element composition. But most stars subsequently disperse from their birth siblings, in orbit and orbital phase, becoming ‘field stars’. Here, we explore and provide direct observational evidence for this process in the Milky Way disc, by quantifying the probability that orbit-similarity among stars implies indistinguishable metallicity. We define the orbit similarity among stars through their distance in action-angle space, Δ(J, θ), and their abundance similarity simply by Δ[Fe/H]. Analysing a sample of main-sequence stars from Gaia DR2 and LAMOST, we find an excess of pairs with the same metallicity (Δ[Fe/H] < 0.1) that extends to remarkably large separations in Δ(J, θ) that correspond to nearly 1 kpc distances. We assess the significance of this effect through a mock sample, drawn from a smooth and phase-mixed orbit distribution. Through grouping such star pairs into associations with a friend-of-friends algorithm linked by Δ(J,θ), we find 100s of mono-abundance groups with ≥3 (to ≳20) members; these groups – some clusters, some spread across the sky – are over an order-of-magnitude more abundant than expected for a smooth phase-space distribution, suggesting that we are witnessing the ‘dissolution’ of stellar birth associations into the field.

Signatures of resonance and phase mixing in the Galactic disc

ABSTRACT Gaia DR2 has provided an unprecedented wealth of information about the kinematics of stars in the Solar neighbourhood, and has highlighted the degree of features in the Galactic disc. We confront the data with a range of bar and spiral models in both action-angle space, and the RG–vϕ plane. We find that the phase mixing induced by transient spiral structure creates ridges and arches in the local kinematics which are consistent with the Gaia data. We are able to produce a qualitatively good match to the data when combined with a bar with a variety of pattern speeds, and show that it is non-trivial to decouple the effects of the bar and the spiral structure.

Staying away from the bar: the local dynamical signature of slow and fast bars in the Milky Way

Both the three-dimensional density of red clump giants and the gas kinematics in the inner Galaxy indicate that the pattern speed of the Galactic bar could be much lower than previously estimated. Here, we show that such slow bar models are unable to reproduce the bimodality observed in local stellar velocity space. We do so by computing the response of stars in the Solar neighbourhood to the gravitational potential of slow and fast bars, in terms of their perturbed distribution function in action-angle space up to second order, as well as by identifying resonantly trapped orbits. We also check that the bimodality is unlikely to be produced through perturbations from spiral arms, and conclude that, contrary to gas kinematics, local stellar kinematics still favour a fast bar in the Milky Way, with a pattern speed of the order of almost twice (and no less than 1.8 times) the circular frequency at the Sun's position. This leaves open the question of the nature of the long flat extension of the bar in the Milky Way.

THE TIME EVOLUTION OF GAPS IN TIDAL STREAMS

ABSTRACT We model the time evolution of gaps in tidal streams that are caused by the impact of a dark matter subhalo, while these orbit a spherical gravitational potential. To this end, we make use of the simple behavior of orbits in action-angle space. A gap effectively results from the divergence of two nearby orbits whose initial phase-space separation is, for very cold thin streams, largely given by the impulse induced by the subhalo. We find that in a spherical potential, the size of a gap increases linearly with time for sufficiently long timescales. We have derived an analytic expression that shows how the growth rate depends on the mass of the perturbing subhalo, its scale, and its relative velocity with respect to the stream. We have verified these scalings using N-body simulations and find excellent agreement. For example, a subhalo of mass 10 8 M ⊙ directly impacting a very cold thin stream on an inclined orbit can induce a gap that may reach a size of several tens of kiloparsecs after a few gigayears. The gap size fluctuates importantly with phase on the orbit, and it is largest close to pericenter. This indicates that it may not be fully straightforward to invert the spectrum of gaps present in a stream to recover the mass spectrum of the subhalos.

Anomalous beam diffusion near beam-beam synchrobetatron resonances

The diffusion process near low order synchro-betatron resonances driven by beam-beam interactions at a crossing angle is investigated. Macroscopic observables such as beam emittance, lifetime and beam profiles are calculated. These are followed with detailed studies of microscopic quantities such as the evolution of the variance at several initial transverse amplitudes and single particle probability distribution functions. We present evidence to show that the observed diffusion is anomalous and the dynamics follows a non-Markovian continuous time random walk process. We derive a modified master equation to replace the Chapman-Kolmogorov equation in action-angle space and a fractional diffusion equation to describe the density evolution for this class of processes.

A Balescu-Lenard-type kinetic equation for the collisional evolution of stable self-gravitating systems

A kinetic equation for the collisional evolution of stable, bound, self-gravitating and slowly relaxing systems is established, which is valid when the number of constituents is very large. It accounts for the detailed dynamics and self-consistent dressing by collective gravitational interaction of the colliding particles, for the system's inhomogeneity and for different constituents' masses. It describes the coupled evolution of collisionally interacting populations, such as stars in a thick disc and the molecular clouds off which they scatter. The kinetic equation derives from the BBGKY hierarchy in the limit of weak, but non-vanishing, binary correlations, an approximation which is well justified for large stellar systems. The evolution of the 1-body distribution function is described in action–angle space. The collective response is calculated using a biorthogonal basis of pairs of density–potential functions. The collision operators are expressed in terms of the collective response function allowed by the existing distribution functions at any given time and involve particles in resonant motion. These equations are shown to satisfy an H theorem. Because of the inhomogeneous character of the system, the relaxation causes the potential as well as the orbits of the particles to secularly evolve. The changing orbits also cause the angle Fourier coefficients of the basis potentials to change with time. We derive the set of equations which describes this coupled evolution of distribution functions, potential and basis Fourier coefficients for spherically symmetric systems. In the homogeneous limit, which sacrifices the description of the evolution of the spatial structure of the system but retains the effect of collective gravitational dressing, the kinetic equation reduces to a form similar to the Balescu–Lenard equation of plasma physics.

Hierarchical phase space structure of dark matter haloes: Tidal debris, caustics, and dark matter annihilation

Most of the mass content of dark matter haloes is expected to be in the form of tidal debris. The density of debris is not constant, but rather can grow due to formation of caustics at the apocenters and pericenters of the orbit, or decay as a result of phase mixing. In the phase space, the debris assemble in a hierarchy which is truncated by the primordial temperature of dark matter. Understanding this phase structure can be of significant importance for the interpretation of many astrophysical observations and in particular dark matter detection experiments. With this purpose in mind, we develop a general theoretical framework to describe the hierarchical structure of the phase space of cold dark matter haloes. We do not make any assumption of spherical symmetry and/or smooth and continuous accretion. Instead, working with correlation functions in the action-angle space, we can fully account for the hierarchical structure (predicting a two-point correlation function ~ (\Delta J)^{-1.6} in the action space), as well as the primordial discreteness of the phase space. As an application, we estimate the boost to the dark matter annihilation signal due to the structure of the phase space within virial radius: the boost due to the hierarchical tidal debris is of order unity, whereas the primordial discreteness of the phase structure can boost the total annihilation signal by up to an order of magnitude. The latter is dominated by the regions beyond 20% of the virial radius, and is largest for the recently formed haloes with the least degree of phase mixing.

Subharmonic resonance in the non-linear Mathieu equation

In this paper, we present an O(ε) perturbation method that utilizes Lie transform perturbation theory and elliptic functions to investigate subharmonic resonances in the non-linear Mathieu equation x ̈ +(δ+ε cos ωt) x+αx 3=0. It is assumed that the parametric perturbation, ε cos ωt , is small and that the coefficient of the non-linear term, α, is positive but not necessarily small. We derive analytic expressions for features (width and location of equilibria) of resonance bands in a Poincaré section of action-angle space that are associated with 2m : 1 subharmonic periodic solutions. In contrast to previous perturbation treatments of this problem, the unperturbed system is non-linear and the transformation to action-angle variables involves elliptic functions. We are, therefore, not restricted to a neighborhood of the origin in our investigation. The Hamiltonian structure of the unperturbed vector field, an integrable vector field, provides us with a framework for developing an analysis of the perturbed orbit structure. The methodolgy revolves around employing Lie transform perturbation theory for constructing the so-called “resonance Kamiltonian”, K r, whose level curves correspond to invariant curves of a Poincaré map for the non-linear Mathieu equation. Explicit knowledge of K r enables us to derive analytic expressions for the resonance bands in a Poincaré section of action-angle space that are associated with 2m : 1 subharmonic periodic solutions. Predictions of the perturbation method are compared to results obtained by direct numerical integration of the non-linear Mathieu equation. The integrable nature of the unperturbed ( ε=0) non-linear Mathieu equation is preserved under the perturbation method. Consequently, the method is unable to predict the appearance of chaos.

The Stability of “Perfect Elliptic Disks.” II. A Minimal Streaming Case

Two-dimensional realizations of self-consistent models for the elliptic disks were tested for global stability by gravitational N-body integration. The family of perfect elliptic disk potentials have two isolating integrals; time-independent distribution functions f(E, I2), which self-consistently reproduce the density distribution, can be found numerically, using a modified marching scheme to compute the relative contributions of each member in a library of orbits. The possible solutions are not unique; for a given ellipticity, the models can have a range of angular momenta. Here results are presented for cases with minimal angular momentum, hence maximal random motion. As in previous work, N-body realizations were constructed using a modified quiet start technique to place particles on these orbits uniformly in action-angle space, making the initial conditions as smooth as possible. The most elliptical models initially showed bending instabilities; by the end of the run they had become slightly rounder. The most nearly axisymmetric models tended to become more elongated, reminiscent of the radial orbit instability in spherical systems. Between these extremes, there is a range of axial ratios 0.305 b/a 0.570 for which the minimum streaming models appear to be stable.

The stability of perfect elliptic disks. 1: The maximum streaming case

view Abstract Citations (4) References (54) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Stability of Perfect Elliptic Disks. I. The Maximum Streaming Case Levine, Stephen E. ; Sparke, Linda S. Abstract Self-consistent distribution functions are constructed for two- dimensional perfect elliptic disks (for which the potential is exactly integrable) in the limit of maximum streaming; these are tested for stability by N-body integration. To obtain a discrete representation for each model, simulated annealing is used to choose a set of orbits which sample the distribution function and reproduce the required density profile while carrying the greatest possible amount of angular momentum. A quiet start technique is developed to place particles on these orbits uniformly in action-angle space, making the initial conditions as smooth as possible. The roundest models exhibit spiral instabilities similar to those of cold axisymmetric disks; the most elongated models show bending instabilities like those seen in prolate systems. Between these extremes, there is a range of axial ratios 0.25 <~ b/a <~ 0.6 within which these models appear to be stable. All the methods developed in this investigation can easily be extended to integrable potentials in three dimensions. Publication: The Astrophysical Journal Pub Date: June 1994 DOI: 10.1086/174262 Bibcode: 1994ApJ...428..493L Keywords: Distribution Functions; Elliptic Functions; Elliptical Orbits; Galactic Structure; Lenticular Bodies; Many Body Problem; Stability; Two Dimensional Bodies; Annealing; Computerized Simulation; Elongation; Numerical Analysis; Prolateness; Astronomy; GALAXIES: ELLIPTICAL AND LENTICULAR; CD; GALAXIES: KINEMATICS AND DYNAMICS; GALAXIES: STRUCTURE; METHODS: ANALYTICAL; METHODS: NUMERICAL full text sources ADS | data products SIMBAD (5) Related Materials (1) Part 2: 1998ApJ...503..125L

Excited atoms in strong microwaves: Classical resonances and localization in experimental final-state distributions.

Highly excited hydrogen atoms have been exposed to an intense pulse of microwaves with frequency near the initial classical electron orbit frequency. Measurements were made of the probability distribution for different bound states after the pulse. Structure in such distributions is found to be near classical, and arises from the presence of a classical resonance island in action-angle space. The measured distributions also yield values of electron localization length in action space that are in agreement with quantum localization theory and with numerical values based on quantum mechanics.

Non-linear diffusion in Hamiltonian systems exhibiting chaotic motion

The exact evolution equation for the angle averaged phase space density in action-angle space is derived from the Liouville equation using projection operator techniques. This equation involves a correlation function of the initial value of the phase space density with the angle dependent part of the Hamiltonian and a correlation function of the angle dependent part of the Hamiltonian and a correlation function of the angle dependent part of the Hamiltonian with itself. Each of these correlation functions develops in time with angle projected dynamics. We show their relation to the correlation functions which develop in time with usual Hamiltonian dynamics. These correlation functions are then studied in the standard model of Chirikov, and we conclude that they behave as e -σt cos(Ωt + φ) in regions of irregular motion. We conjecture that angle averaged correlation functions behave this way in general, and we give an argument based on the mixing property of the Hamiltonian system. Our argument goes beyond the usual mixing, so we regard it as a quasi-mixing hypothesis. Under this hypothesis the equation for the angle averaged phase space density becomes a diffusion equation which incorporates much of the non-linear dynamics of Hamiltonian systems exhibiting chaotic motion.