Origin of the two-armed vertical phase-spiral in the inner Galactic disk

TL;DR

Gaia revealed a two-armed vertical phase spiral in the inner Galactic disk, challenging the view that only short-lived, non-adiabatic perturbations can produce such structures. The authors develop a minimal 1D vertical model combining an isothermal slab, a softened periodic spiral perturbation, and diffusion from small-scale kicks; through linear and nonlinear analyses and test-particle simulations, they show that vertical resonances near the spiral arms, together with diffusion, generate a steadily rotating, open two-armed phase spiral without requiring impermanently short perturbations. The results reproduce the observed amplitude and rotation characteristics and extend to a multi-resonance scenario where several local spirals cohere into a global pattern, suggesting that inner-disk two-armed phase spirals are a natural outcome of common spiral structures and stochastic heating. This framework provides a robust mechanism for interpreting Gaia DR3 data and highlights the important, sometimes constructive, role of diffusion in preserving phase-space structures in galaxies. The work also clarifies that the Galactic bar is unlikely to be the dominant driver for the observed inner-disk two-armed spiral, focusing attention on spiral-arm resonances and diffusion as the primary agents.

Abstract

Gaia recently revealed a two-armed spiral pattern in the vertical phase-space distribution of the inner Galactic disk (guiding radius $R_\textrm{g} \sim 6.2$ kpc), indicating that some non-adiabatic perturbation symmetric about the mid-plane is driving the inner disk out of equilibrium. The non-axisymmetric structures in the disk (e.g., the bar or spiral arms) have been suspected to be the major source for such a perturbation. However, both the lifetime and the period of these internal perturbations are typically longer than the period at which stars oscillate vertically, implying that the perturbation is generally adiabatic. This issue is particularly pronounced in the inner Galaxy, where the vertical oscillation period is shorter and therefore adiabatically shielded more than the outer disk. We show that two-armed phase spirals can naturally form in the inner disk if there is a vertical resonance that breaks the adiabaticity; otherwise, their formation requires a perturber with an unrealistically short lifetime. We predict analytically and confirm with simulations that a steadily rotating (non-winding) two-armed phase spiral forms near the resonance when stars are subject to both periodic perturbations (e.g., by spiral arms) and stochastic perturbations (e.g., by giant molecular clouds). Due to the presence of multiple resonances, the vertical phase-space exhibits several local phase spirals that rotate steadily at distinct frequencies, together forming a global phase spiral that evolves over time. Our results demonstrate that, contrary to earlier predictions, the formation of the two-armed phase spiral does not require transient perturbations with lifetimes shorter than the vertical oscillation period.

Figures (15)

• Figure 1: The two-armed phase-spiral discovered in Gaia DR3. We select stars with parallax error $p/\sigma_p > 3$, magnitude $G<15$, cylindrical distance from the sun $d < 1\,{\rm kpc}$, angular momentum $J_\varphi \in [1400,1600] \,{\rm kpc}^2 \,{\rm Gyr}^{-1}$, and azimuthal angle variable $|\theta_\varphi - {\varphi_\odot}| < 5 {^{\circ}}$. The angle variables are computed using the Milky Way potential from McMillan (2017) and the Stäckel fudge Binney2012Stackel as implemented in AGAMA Vasiliev2019AGAMA. Top panel shows the full distribution, while the bottom panel shows the fractional density contrast relative to the smooth distribution obtained using a Gaussian filter with scale $80 \,{\rm pc}$ in $z$ and $6.4 \,{\rm kpc}\,{\rm Gyr}^{-1}$ in $v_z$.
• Figure 2: Top: Potential of the isothermal slab fitted to the Milky Way potential from McMillan (2017) at $R=6.2 \,{\rm kpc}$. Bottom: Vertical orbital frequency as a function of the vertical action for stars with $(J_R,J_\varphi)=(0,Rv_{\rm c})$.
• Figure 3: Spectra of the spiral arms' potential along an in-plane unperturbed orbit with guiding radius $R_{\rm g}=6.2\,{\rm kpc}$ for three different values of the radial action $J_R$. The brackets in the figures denote the set of integers $(N_R,N_\varphi)$, corresponding to the frequency $\omega=N_R\Omega_R+N_\varphi(\Omega_\varphi-\Omega_{\rm p})$. The two-armed phase spiral was observed from stars with large radial actions ($J_R\simeq80\,{\rm kpc}^2\,{\rm Gyr}^{-1}$, right panel). These stars are subject to a variety of resonances at high frequencies.
• Figure 4: A steadily rotating (non-winding) two-armed phase-spiral predicted by the linearized kinetic equation (\ref{['eq:kinetic_eq_lin_sol']}) in the time-asymptotic limit. The disk is subject to a periodic vertical perturbation by galactic spiral arms in the presence of small-scale random kicks. The black curve marks the resonance.
• Figure 5: Steady-state solutions of the non-linearized kinetic equation (\ref{['eq:kinetic_equation_slowAA']}), describing the evolution of the disk subject to both persistent spiral perturbations and stochastic perturbations. We plot the perturbation $f_1$ as well as the full distribution $f=f_0+f_1$ in the slow angle-action space and in the $(z,v_z)$ space. We also overlay the contours of the Hamiltonian (\ref{['eq:newHamiltonian']}), which are stationary in the slow angle-action space but rotate with time in the $(z,v_z)$ space. As the strength of diffusion $\Delta$ increases from left to right, the perturbed distribution near the resonance becomes asymmetric about $\theta_{\rm s} = 0$, resulting in a spiral pattern in the $(z,v_z)$ space. The strength of diffusion in our standard model is $\Delta \simeq 0.2$ (middle column).