We propose a method for constructing the time-dependent phase space distribution function (DF) of a collisionless system from an isolated kinematic snapshot. In general, the problem of mapping a single snapshot to a time-dependent function is intract
able. Here we assume a finite series representation of the DF, constructed from the spectrum of the systems Koopman operator. This reduces the original problem to one of mapping a kinematic snapshot to a discrete spectrum rather than to a time-dependent function. We implement this mapping with a convolutional neural network (CNN). The method is demonstrated on two example models: the quantum simple harmonic oscillator, and a self-gravitating isothermal plane. The latter system exhibits phase space spiral structure similar to that observed in Gaia Data Release 2.
We investigate the spatiotemporal structure of simulations of the homogeneous slab and isothermal plane models for the vertical motion in the Galactic disc. We use Dynamic Mode Decomposition (DMD) to compute eigenfunctions of the simulated distributi
on functions for both models, referred to as DMD modes. In the case of the homogeneous slab, we compare the DMD modes to the analytic normal modes of the system to evaluate the feasibility of DMD in collisionless self gravitating systems. This is followed by the isothermal plane model, where we focus on the effect of self gravity on phase mixing. We compute DMD modes of the system for varying relative dominance of mutual interaction and external potential, so as to study the corresponding variance in mode structure and lifetime. We find that there is a regime of relative dominance, at approximately $ 4:1 $ external potential to mutual interaction where the DMD modes are spirals in the $ (z,v_z) $ plane, and are nearly un-damped. This leads to the proposition that a system undergoing phase mixing in the presence of weak to moderate self gravity can have persisting spiral structure in the form of such modes. We then conclude with the conjecture that such a mechanism may be at work in the phase space spirals observed in Gaia Data Release 2, and that studying more complex simulations with DMD may aid in understanding both the timing and form of the perturbation that lead to the observed spirals.
We investigate the connection between the vertical structure of stellar discs and the formation of bars using high-resolution simulations of galaxies in isolation and in the cosmological context. In particular, we simulate a suite of isolated galaxy
models that have the same Toomre Q parameter and swing amplification parameter but that differ in the vertical scale height and velocity dispersion. We find that the onset of bar formation occurs more slowly in models with thicker discs. Moreover, thicker discs and also discs evolved in simulations with larger force softening also appear to be more resilient to buckling, which acts to regulate the length and strength of bars. We also simulate disc-halo systems in the cosmological environment using a disc-insertion technique developed in a previous paper. In this case, bar formation is driven by the stochastic effects of a triaxial halo and subhalo-disc interactions and the initial growth of bars appears to be relatively insensitive to the thickness of the disc. On the other hand, thin discs in cosmological halos do appear to be more susceptible to buckling than thick ones and therefore bar strength correlates with disc thickness as in the isolated case. More to the point, one can form discs in cosmological simulations with relatively weak bars or no bars at all provided the discs as thin as the discs we observe and the softening length is smaller than the disc scale height.
We discuss the physical mechanism by which pure vertical bending waves in a stellar disc evolve to form phase space spirals similar to those discovered by Antoja et al. ( arXiv:1804.10196) in Gaia Data Release 2. These spirals were found by projectin
g Solar Neighbourhood stars onto the $z-v_z$ plane. Faint spirals appear in the number density of stars projected onto the $z-v_z$ plane, which can be explained by a simple model for phase wrapping. More prominent spirals are seen when bins across the $z-v_z$ plane are coloured by median $v_R$ or $v_phi$. We use both toy model and fully self-consistent simulations to show that the spirals develop naturally from vertical bending oscillations of a stellar disc. The underlying physics follows from the observation that the vertical energy of a star (essentially, its radius in the $z-v_z$ plane) correlates with its angular momentum or, alternatively, guiding radius. Moreover, at fixed physical radius, the guiding radius determines the azimuthal velocity. Together, these properties imply the link between in-plane and vertical motion that lead directly to the Gaia spirals. We show that the cubic $R-z$ coupling term in the effective potential is crucial for understanding the morphology of the spirals. This suggests that phase space spirals might be a powerful probe of the Galactic potential. In addition, we argue that self-gravity is necessary to properly model the evolution of the bending waves and their attendant phase space spirals.
We use $N$-body simulations to investigate the excitation of bending waves in a Milky Way-like disc-bulge-halo system. The dark matter halo consists of a smooth component and a population of subhaloes while the disc is composed of thin and thick comp
onents. Also considered is a control simulation where all of the halo mass is smoothly distributed. We find that bending waves are more vigorously excited in the thin disc than the thick one and that they are strongest in the outer regions of the disc, especially at late times. By way of a Fourier decomposition, we find that the complicated pattern of bending across the disc can be described as a superposition of waves, which concentrate along two branches in the radius-rotational frequency plane. These branches correspond to vertical resonance curves as predicted by a WKB analysis. Bending waves in the simulation with substructure have a higher amplitude than those in the smooth-halo simulation, though the frequency-radius characteristics of the waves in the two simulations are very similar. A cross correlation analysis of vertical displacement and bulk vertical velocity suggests that the waves oscillate largely as simple plane waves. We suggest that the wave-like features in astrometric surveys such as the Second Data Release from textit{Gaia} may be due to long-lived waves of a dynamically active disc rather than, or in addition to, perturbations from a recent satellite-disc encounter.
We present a new method for embedding a stellar disc in a cosmological dark matter halo and provide a worked example from a {Lambda}CDM zoom-in simulation. The disc is inserted into the halo at a redshift z = 3 as a zero-mass rigid body. Its mass and
size are then increased adiabatically while its position, velocity, and orientation are determined from rigid-body dynamics. At z = 1, the rigid disc is replaced by an N-body disc whose particles sample a three-integral distribution function (DF). The simulation then proceeds to z = 0 with live disc and halo particles. By comparison, other methods assume one or more of the following: the centre of the rigid disc during the growth phase is pinned to the minimum of the halo potential, the orientation of the rigid disc is fixed, or the live N-body disc is constructed from a two rather than three-integral DF. In general, the presence of a disc makes the halo rounder, more centrally concentrated, and smoother, especially in the innermost regions. We find that methods in which the disc is pinned to the minimum of the halo potential tend to overestimate the amount of adiabatic contraction. Additionally, the effect of the disc on the subhalo distribution appears to be rather insensitive to the disc insertion method. The live disc in our simulation develops a bar that is consistent with the bars seen in late-type spiral galaxies. In addition, particles from the disc are launched or kicked up to high galactic latitudes.
We study the spontaneous generation and evolution of bending waves in $N$-body simulations of two isolated Milky Way-like galaxy models. The models differ by their disc-to-halo mass ratios, and hence by their susceptibility to the formation of a bar
and spiral structure. Seeded from shot noise in the particle distribution, bending waves rapidly form in both models and persist for many billions of years. Waves at intermediate radii manifest as corrugated structures in vertical position and velocity that are tightly wound, morphologically leading, and dominated by the $m=1$ azimuthal Fourier component. A spectral analysis of the waves suggests they are a superposition of modes from two continuous branches in the Galactocentric radius-rotational frequency plane. The lower-frequency branch is dominant and is responsible for the corrugated, leading, and warped structure. Over time, power in this branch migrates outward, lending credence to an inside-out formation scenario for the warp. Our power spectra qualitatively agree with results from linear perturbation theory and a WKB analysis, both of which include self-gravity. Thus, we conclude that the waves in our simulations are self-gravitating and not purely kinematic. These waves are reminiscent of the wave-like pattern recently found in Galactic star counts from the Sloan Digital Sky Survey and smoothly transition to a warp near the discs edge. Velocity measurements from textit{Gaia} data will be instrumental in testing the true wave nature of the corrugations. We also compile a list of minimum requirements needed to observe bending waves in external galaxies.
We present a disc-halo N-body model of the low surface brightness galaxy UGC 628, one of the few systems that harbours a slow bar with a ratio of corotation radius to bar length of $mathcal{R} equiv R_c/a_b sim 2$. We select our initial conditions us
ing SDSS DR10 photometry, a physically motivated radially variable mass-to-light ratio profile, and rotation curve data from the literature. A global bar instability grows in our submaximal disc model, and the disc morphology and dynamics agree broadly with the photometry and kinematics of UGC 628 at times between peak bar strength and the onset of buckling. Prior to bar formation, the disc and halo contribute roughly equally to the potential in the galaxys inner region, giving the disc enough self gravity for bar modes to grow. After bar formation there is significant mass redistribution, creating a baryon dominated inner and dark matter dominated outer disc. This implies that, unlike most other low surface brightness galaxies, UGC 628 is not dark matter dominated everywhere. Our model nonetheless implies that UGC 628 falls on same the relationship between dark matter fraction and rotation velocity found for high surface brightness galaxies, and lends credence to the argument that the disc mass fraction measured at the location where its contribution to the potential peaks is not a reliable indicator of its dynamical importance at all radii.
A satellite galaxy or dark matter subhalo that passes through a stellar disk may excite coherent oscillations in the disk perpendicular to its plane. We determine the properties of these modes for various self-gravitating plane symmetric systems (Spi
tzer sheets) using the matrix method of Kalnajs. In particular, we find an infinite series of modes for the case of a barotropic fluid. In general, for a collisionless system, there is a double series of modes, which include normal modes and/or Landau-damped oscillations depending on the phase space distribution function of the stars. Even Landau-damped oscillations may decay slowly enough to persist for several hundred Myr. We discuss the implications of these results for the recently discovered vertical perturbations in the kinematics of solar neighborhood stars and for broader questions surrounding secular phenomena such as spiral structure in disk galaxies.
We explore the hypothesis that a passing satellite or dark matter subhalo has excited coherent oscillations of the Milky Ways stellar disk in the direction perpendicular to the Galactic midplane. This work is motivated by recent observations of spati
ally dependent bulk vertical motions within ~ kpc of the Sun. A satellite can transfer a fraction of its orbital energy to the disk stars as it plunges through the Galactic midplane thereby heating and thickening the disk. Bulk motions arise during the early stages of such an event when the disk is still in an unrelaxed state. We present simple toy-model calculations and simulations of disk-satellite interactions, which show that the response of the disk depends on the relative velocity of the satellite. When the component of the satellites velocity perpendicular to the disk is small compared with that of the stars, the perturbation is predominantly a bending mode. Conversely, breathing and higher order modes are excited when the vertical velocity of the satellite is larger than that of the stars. We argue that the compression and rarefaction motions seen in three different surveys are in fact breathing mode perturbations of the Galactic disk.
