Identifying resonances of the Galactic bar in Gaia DR2: II. Clues from angle space

The Milky Way disk exhibits intricate orbit substructure of still-debated dynamical origin. The angle variables (θφ, θR)—which are conjugates to the actions (Lz, JR), and describe a star’s location along its orbit—are a powerful diagnostic to identify l:m resonances via the orbit shape relation ΔθR/Δθφ = −m/l. In the past, angle signatures have been hidden by survey selection effects (SEs). Using test particle simulations of a barred galaxy, we demonstrate that Gaia should allow us to identify the Galactic bar’s Outer Lindblad Resonance (l = +1, m = 2, OLR) in angle space. We investigate strategies to overcome SEs. In the angle data of the Gaia DR2 RVS sample, we independently identify four candidates for the OLR and therefore for the pattern speed Ωbar. The strongest candidate, Ωbar ∼ 1.4Ω0, positions the OLR above the ‘Sirius’ moving group, agrees with measurements from the Galactic center, and might be supported by higher-order resonances around the ‘Hercules/Horn’. But it misses the classic orbit orientation flip, as discussed in the companion study on actions. The candidate Ωbar ∼ 1.2Ω0 was also suggested by the action-based study, has the OLR at the ‘Hat’, is consistent with slow bar models, but still affected by SEs. Weaker candidates are Ωbar = 1.6 and 1.74Ω0. In addition, we show that the stellar angles do not support the ‘Hercules/Horn’ being created by the OLR of a fast bar. We conclude that—to resolve if ‘Sirius’ or ‘Hat’ are related to the bar’s OLR—more complex dynamical explanations and more extended data with well-behaved SEs are required.

Identifying resonances of the Galactic bar in Gaia DR2: I. Clues from action space

ABSTRACT Action space synthesizes the orbital information of stars and is well suited to analyse the rich kinematic substructure of the disc in the second Gaia data release's radial velocity sample. We revisit the strong perturbation induced in the Milky Way disc by an m = 2 bar, using test particle simulations and the actions (JR, Lz, Jz) estimated in an axisymmetric potential. These make three useful diagnostics cleanly visible. (1) We use the well-known characteristic flip from outward to inward motion at the outer Lindblad resonance (OLR; l = +1, m = 2), which occurs along the axisymmetric resonance line (ARL) in (Lz, JR), to identify in the Gaia action data three candidates for the bar’s OLR and pattern speed Ωbar: 1.85Ω0, 1.20Ω0, and 1.63Ω0 (with ∼0.1Ω0 systematic uncertainty). The Gaia data is therefore consistent with both slow and fast bar models in the literature, but disagrees with recent measurements of ∼1.45Ω0. (2) For the first time, we demonstrate that bar resonances – especially the OLR – cause a gradient in vertical action 〈Jz〉 with Lz around the ARL via ‘Jz-sorting’ of stars. This could contribute to the observed coupling of 〈vR〉 and 〈|vz|〉 in the Galactic disc. (3) We confirm prior results that the behaviour of resonant orbits is well approximated by scattering and oscillation in (Lz, JR) along a slope ΔJR/ΔLz = l/m centred on the l:m ARL. Overall, we demonstrate that axisymmetrically estimated actions are a powerful diagnostic tool even in non-axisymmetric systems.

Diffusion of radial action in a galactic disc

Context. The stellar migration of the galactic disc stars has been invoked to explain the dispersion of stellar metallicity observed in the solar neighbourhood. Aims. We seek to identify the dynamical mechanisms underlying stellar migration in an isolated galaxy disc under the influence of a bar. Our approach is to analyse the diffusion of dynamical quantities. Methods. We extend our previous work by exploring Chirikov’s diffusion rate (and derived timescale) of the radial action JR in an idealised N-body simulation of an isolated disc galaxy. We limit our study to the evolution of the disc region well after the formation of the bar, in a regime of adiabatic evolution. Results. The JR diffusion timescale TD(JR) is less than 3 Gyr for roughly half the galaxy mass. It is always much shorter than the angular momentum diffusion timescale TD(Lz) outside the stellar bar. In the disc, ⟨TD(JR)⟩ ∼ 1 Gyr. All non-axisymmetric morphological structures that are characteristic of resonances and waves in the disc are associated to particles with TD(JR) < 3 Gyr and TD(Lz) > 10 Gyr. Short TD(JR) can be explained by the gradual de-circularisation of initially circular orbits (JR = 0) under the effect of intermittent. Inner Linblad resonance scattering by wave trains propagating in the disc, well beyond the outer Lindblad resonance of the bar (OLR). This leads to a moderate secular heating of the disc beyond the bar’s OLR for 7 Gyr, which is comparable to solar neighbourhood observations. The complex multi-wave structure, mixing permanent and intermittent modes, allows for multiple resonance overlaps.

The shearing sheet and swing amplification revisited

ABSTRACT The principal results of the classic analysis of the shearing sheet and swing amplification by Julian and Toomre (JT) are re-derived in a more accessible way and used to gain a better quantitative understanding of the dynamics of stellar discs. The axisymmetric limit of the shearing sheet is derived and used to re-derive Kalnajs’ 1965 dispersion relation and Toomre’s 1964 stability criterion for axisymmetric disturbances. Using the shearing sheet to revisit Toomre’s important 1969 paper on the group velocity implied by the Lin–Shu–Kalnajs (LSK) dispersion relation, we discover that two wavepackets emerge inside corotation: one each side of the inner Lindblad resonance. An extended form of the JT equation is used to investigate the impact of there being a deficit or surplus of stars in a narrow range of angular momenta. Swing amplification of leading waves introduced by such a groove gives rise to transient trailing spirals that extend further in radius and live longer at smaller azimuthal wavenumbers. Although the LSK dispersion relation provides useful interpretations of wavepackets, the shearing sheet highlights the limitations of the LSK approach to disc dynamics. Disturbances do not avoid an annulus around corotation, as the LSK dispersion relation implies. While disturbances of the shearing sheet have a limited life in real space, they live on much longer in velocity space, which Gaia allows us to probe extensively. c++ code is provided to facilitate applications of winding spiral waves.

Trapped orbits and solar-neighbourhood kinematics

ABSTRACT Torus mapping yields constants of motion for stars trapped at a resonance. Each such constant of motion yields a system of contours in velocity space at the Sun and neighbouring points. If Jeans’ theorem applied to resonantly trapped orbits, the density of stars in velocity space would be equal at all intersections of any two contours. A quantitative measure of the violation of this principle is defined and used to assess various pattern speeds for a model of the bar recently fitted to observations of interstellar gas. Trapping at corotation of a bar with pattern speed near $36\, \mathrm{Gyr}^{-1}$ is favoured and trapping at the outer Lindblad resonance is disfavoured. As one moves around the Sun the structure of velocity space varies quite rapidly, both as regards the observed star density and the zones of trapped orbits. The data seem consistent with trapping at corotation.

Angle-action variables for orbits trapped at a Lindblad resonance

ABSTRACT The conventional approach to orbit trapping at Lindblad resonances via a pendulum equation fails when the parent of the trapped orbits is too circular. The problem is explained and resolved in the context of the Torus Mapper and a realistic Galaxy model. Tori are computed for orbits trapped at both the inner and outer Lindblad resonances of our Galaxy. At the outer Lindblad resonance, orbits are quasi-periodic and can be accurately fitted by torus mapping. At the inner Lindblad resonance, orbits are significantly chaotic although far from ergodic, and each orbit explores a small range of tori obtained by torus mapping.

Tracing Hercules in Galactic azimuth with Gaia DR2

The second data release of the Gaia mission has revealed, in stellar velocity and action space, multiple ridges, the exact origin of which is still debated. Recently, we demonstrated that a large Galactic bar with pattern speed 39 km s−1 kpc−1 creates most of the observed ridges. Among these ridges, the Hercules moving group would then be associated with orbits trapped at the co-rotation resonance of the bar. Here we show that a distinctive prediction of such a model is that the angular momentum of Hercules at the Sun’s radius must significantly decrease with increasing Galactocentric azimuth (i.e. when getting closer to the major axis of the bar). We show that this dependence of the angular momentum of trapped orbits on the azimuth on the other hand does not happen close to the outer Lindblad resonance of a faster bar, unless the orbital distribution is still far from phase-mixed, namely for a bar perturbation younger than ∼2 Gyr. Using Gaia DR2 and Bayesian distances from the StarHorse code, and tracing the average Galactocentric radial velocity as a function of angular momentum and azimuth, we show that the Hercules angular momentum changes significantly with azimuth as expected for the co-rotation resonance of a dynamically old large bar.

Stellar bars in counter-rotating dark matter haloes: the role of halo orbit reversals

Disc galaxies can exchange angular momentum and baryons with their host dark matter (DM) haloes. These haloes possess internal spin, λ, which is insignificant rotationally but does affect interactions between the baryonic and DM components. While statistics of prograde and retrograde spinning haloes in galaxies is not available at present, the existence of such haloes is important for galaxy evolution. In the previous works, we analysed dynamical and secular evolution of stellar bars in prograde spinning haloes and the DM response to the bar perturbation, and found that it is modified by the resonant interactions between the bar and the DM halo orbits. In this work, we follow the evolution of stellar bars in retrograde haloes. We find that this evolution differs substantially from evolution in rigid unresponsive haloes, discussed in the literature. First, we confirm that the bar instability is delayed progressively along the retrograde λ sequence. Secondly, the bar evolution in the retrograde haloes differs also from that in the prograde haloes, in that the bars continue to grow substantially over the simulation time of 10 Gyr. The DM response is also substantially weaker compared to this response in the prograde haloes. Thirdly, using orbital spectral analysis of the DM orbital structure, we find a phenomenon we call the orbit reversal – when retrograde DM orbits interact with the stellar bar, reverse their streaming and precession, and become prograde. This process dominates the inner halo region adjacent to the bar and allows these orbits to be trapped by the bar, thus increasing efficiency of angular momentum transfer by the inner Lindblad resonance. We demonstrate this reversal process explicitly in a number of examples.

On the ridges, undulations & streams in Gaia DR2: Linking the topography of phase-space to the orbital structure of an N-body bar

We explore the origin of phase-space substructures revealed by the second Gaia data release in the disc of the Milky Way, such as the ridges in the Vφ-r plane, the undulations in the Vφ-r-Vr space and the streams in the Vφ-Vr plane. We use a collisionless N-body simulation with co-spatial thin and thick discs, along with orbit integration, to study the orbital structure close to the Outer Lindblad Resonance (OLR) of the bar. We find that a prominent, long-lived ridge is formed in the Vφ-r plane due to the OLR which translates to streams in the Vφ-Vr plane and examine which closed periodic and trapped librating orbits are responsible for these features. We find that orbits which carry out small librations around the x1(1) family are preferentially found at negative Vr, giving rise to a ‘horn’-like feature, while orbits with larger libration amplitudes, trapped around the x1(2) and x1(1) families, constitute the positive Vr substructure, i.e. the Hercules-like feature. This changing libration amplitude of orbits will translate to a changing ratio of thin/thick disc stars, which could have implications on the metallicity distribution in this plane. We find that a scenario in which the Sun is placed close to the OLR gives rise to a strong asymmetry in Vr in the Vφ-Vr plane (i.e. Hercules vs. ‘the horn’) and subsequently to undulations in the Vφ-r-Vr space. We also explore a scenario in which the Sun is placed closer to the bar corotation and find that the bar perturbation alone cannot give rise to the these features.

Density waves and the viscous overstability in Saturn’s rings

This paper considers resonantly forced spiral density waves in a dense planetary ring that is close to the threshold for viscous overstability. We solved numerically the hydrodynamical equations for a dense thin disk in the vicinity of an inner Lindblad resonance with a perturbing satellite. Our numerical scheme is one-dimensional so that the spiral shape of a density wave is taken into account through a suitable approximation of the advective terms arising from the fluid orbital motion. This paper is a first attempt to model the co-existence of resonantly forced density waves and short-scale free overstable wavetrains as observed in Saturn’s rings, by conducting large-scale hydrodynamical integrations. These integrations reveal that the two wave types undergo complex interactions, not taken into account in existing models for the damping of density waves. In particular we found that, depending on the relative magnitude of both wave types, the presence of viscous overstability can lead to the damping of an unstable density wave and vice versa. The damping of the short-scale viscous overstability by a density wave was investigated further by employing a simplified model of an axisymmetric ring perturbed by a nearby Lindblad resonance. A linear hydrodynamic stability analysis as well as local N-body simulations of this model system were performed and support the results of our large-scale hydrodynamical integrations.