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Register a new userEmergence of the Gaia Phase Space Spirals from Bending Waves
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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 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 $v_phi$. 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.
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Using a single N-body simulation ($N=0.14times 10^9$) we explore the formation, evolution and spatial variation of the phase-space spirals similar to those recently discovered by Antoja et al. in the Milky Way disk, with Gaia DR2. For the first time in the literature, we use a self-consistent N-body simulation of an isolated Milky Way-type galaxy to show that the phase-space spirals develop naturally from vertical oscillations driven by the buckling of the stellar bar. We claim that the physical mechanism standing behind the observed incomplete phase-space mixing process can be internal and not necessarily due to the perturbation induced by a massive satellite. In our model, the bending oscillations propagate outwards and produce axisymmetric variations of the mean vertical coordinate and of the vertical velocity component. As a consequence, the phase-space wrapping results in the formation of patterns with various morphology across the disk, depending on the bar orientation, distance to the galactic center and time elapsed since the bar buckling. Once bending waves appear, they are supported for a long time via disk self-gravity. The underlying physical mechanism implies the link between in-plane and vertical motion that leads directly to phase-space structures whose amplitude and shape are in remarkable agreement with those of the phase-space spirals observed in the Milky Way disk. In our isolated galaxy simulation, phase-space spirals are still distinguishable, at the solar neighbourhood, 3 Gyr after the buckling phase. The long-lived character of the phase-space spirals generated by the bar buckling instability cast doubts on the timing argument used so far to get back at the time of the onset of the perturbation: phase-space spirals may have been caused by perturbations originated several Gyrs ago, and not as recent as suggested so far.
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.
Using the method that was developed in the first paper of this series, we measure the vertical gravitational potential of the Galactic disk from the time-varying structure of the phase-space spiral, using data from Gaia as well as supplementary radial velocity information from legacy spectroscopic surveys. For eleven independent data samples, we inferred gravitational potentials that were in good agreement, despite the data samples varied and substantial selection effects. Using a model for the baryonic matter densities, we inferred a local halo dark matter density of $0.0085 pm 0.0039$ M$_odot$/pc$^3 = 0.32 pm 0.15$ GeV/cm$^3$. We were also able to place the most stringent constraint to the surface density of a thin dark disk with a scale height $leq 50$ pc: an upper 95 % confidence limit of roughly 5 M$_odot$/pc$^2$ (compared to previous limit of roughly 10 M$_odot$/pc$^2$, given the same scale height). For the inferred halo dark matter density and thin dark disk surface density, the uncertainties are dominated by the baryonic model. With this level of precision, our method is highly competitive with traditional methods that rely on the assumption of a steady state. In a general sense, this illustrates that time-varying dynamical structures are not solely obstacles to dynamical mass measurements, but can also be regarded as assets containing useful information.
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 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 have investigated the distributions of stellar azimuthal and radial velocity components $V_{Phi}$ and $V_{R}$ in the vertical position-velocity plane $Z$-$V_{Z}$ across the Galactic disc of $6.34 lesssim R lesssim 12.34$,kpc and $|Phi| lesssim 7.5^{circ}$ using a Gaia and Gaia-LAMOST sample of stars. As found in previous works, the distributions exhibit significant spiral patterns. The $V_{R}$ distributions also show clear quadrupole patterns, which are the consequence of the well-known tilt of the velocity ellipsoid. The observed spiral and quadrupole patterns in the phase space plane vary strongly with radial and azimuthal positions. The phase spirals of $V_{Phi}$ become more and more relaxed as $R$ increases. The spiral patterns of $V_{Phi}$ and $V_{R}$ and the quadrupole patterns of $V_{R}$ are strongest at $-2^{circ} < Phi < 2^{circ}$ but negligible at $4^{circ} < Phi < 6^{circ}$ and $-6^{circ} < Phi < -4^{circ}$. Our results suggest an external origin of the phase spirals. In this scenario, the intruder, most likely the previously well-known Sagittarius dwarf galaxy, passed through the Galactic plane in the direction towards either Galactic center or anti-center. The azimuthal variations of the phase spirals also help us constrain the passage duration of the intruder. A detailed model is required to reproduce the observed radial and azimuthal variations of the phase spirals of $V_{Phi}$ and $V_{R}$.
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