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. in Gaia Data Release 2. These spirals are found by projecting 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 φ . 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 a 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 components. 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 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 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 distribution 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 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 intractable. Here we assume a finite series representation of the DF, constructed from the spectrum of the system’s 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.

The stellar disc of the Milky Way exhibits clear departures from planarity, the most conspicuous manifestation being the Galactic Warp but also includes an apparent corrugation pattern in number counts around 15kpc from the Galactic centre, a wave like pattern in the vertical velocities of stars as a function of guiding radius, asymmetries about the midplane in both number counts and bulk motions, and phase spirals in the z–vz projection of the local stellar distribution function. We discuss the physics of these phenomena and, in particular, suggest a possible avenue for inferring the vertical force in the Solar Neighbourhood from phase spirals. We apply Dynamic Mode Decomposition, a technique widely used in the realm of fluid mechanics, to simulations of disc galaxy simulations. This method appears to be particularly well-suited to the study of nonlinear processes such as the coupling of warps and spirals, first discussed by Masset and Tagger.