Dynamics of tidal spiral arms: Machine learning-assisted identification of equations and application to the Milky Way

TL;DR

This work investigates tidally induced spiral arms in the Milky Way, prompted by the Sagittarius encounter, by combining idealized test-particle simulations with a data-driven PDE-discovery method (SINDy). It identifies a linear PDE for small perturbations that yields two waves with pattern speeds $ ext{Ω}_p= ext{Ω}\nobreak ext{±} obreakrac{ ext{κ}}{m}$, and a novel nonlinear PDE for large perturbations, both validated against analytic derivations and an MW–Sgr-like N-body analogue. The authors then analytically interpret the linear and nonlinear PDEs, derive explicit solutions for wave amplitudes and phase-space evolution, and test the models against Gaia data by fitting the $L_Z- ext{V}_R$ wave, obtaining plausible Sgr-passage parameters but noting limitations in amplitude balance that hint at more complex perturbation histories. Overall, the study demonstrates a data-driven–theory hybrid path to simple, predictive models of tidal spirals, with potential extensions to include self-gravity, dynamical friction, and observational data from Gaia and external galaxies.

Abstract

Understanding the spiral arms of the Milky Way (MW) remains a key open question in galactic dynamics. Tidal perturbations, such as the recent passage of the Sagittarius dwarf galaxy (Sgr), could play a significant role in exciting them. We aim to analytically characterize the dynamics of tidally induced spiral arms, including their phase-space signatures. We ran idealized test-particle simulations resembling impulsive satellite impacts, and used the Sparse Identification of Non-linear Dynamics (SINDy) method to infer their governing Partial Differential Equations (PDEs). We validated the method with analytical derivations and a realistic $N$-body simulation of a MW-Sgr encounter analogue. For small perturbations, a linear system of equations was recovered with SINDy, consistent with predictions from linearised collisionless dynamics. In this case, two distinct waves wrapping at pattern speeds $Ω\pm κ/m$ emerge. For large impacts, we empirically discovered a non-linear system of equations, representing a novel formulation for the dynamics of tidally induced spiral arms. For both cases, these equations describe wave properties like amplitude and pattern speed, and their shape and temporal evolution in different phase-space projections. We fit the Gaia $L_Z-V_R$ waves with the linear model, providing a reasonable fit and plausible parameters for the Sgr passage. However, the predicted amplitude ratio of the two waves is inconsistent with observations, supporting a more complex origin for this feature (e.g. multiple passages, bar, spiral arms). We merge data-driven discovery with theory to create simple, accurate models of tidal spiral arms that match simulations and provide a simple tool to fit Gaia and external galaxy data. This methodology could be extended to model complex phenomena like self-gravity and dynamical friction. (ABR)

Figures (11)

• Figure 1: Kinematic signature evolution of the $m=2$ (upper panels) and $m=1$ (lower panels) kinematic kicks. We show snapshots of the velocity perturbation fields at five different times ($t = 0, 0.1, 0.2, 0.3, \text{ and } 0.4\,$Gyr). For each kick mode, the top row displays the radial velocity perturbation, $\Delta V_R$, while the bottom row shows the azimuthal velocity perturbation, $\Delta V_\phi$. The solid and dash-dotted curves overlaid on the $\Delta V_R$ panels indicate the theoretical spiral loci wrapping at pattern speeds $\Omega - \kappa/m$ and $\Omega + \kappa/m$, respectively. The panels demonstrate the clear appearance of two spiral wave patterns for both $m=2$ and $m=1$ modes, with the theoretical loci accurately matching the observed evolution of the waves in the velocity fields.
• Figure 2: Evolution of the test-particle system after a large, $m=2$ impact, designed with a $\gamma$ correction (Eq. \ref{['eq:m_kick_large']}) to excite a single spiral pattern. These panels show snapshots of the velocity field at four different times ($t=0$, $0.4$, $0.8$, and $1.2\,$Gyr). The top row shows the radial velocity $V_R$, and the bottom row shows the residual azimuthal velocity, $\Delta V_\phi = V_\phi - V_c$. The spiral pattern winds up over time at the expected rate $\Omega - \kappa/m$. The large amplitude of the perturbation introduces non-linear effects, causing the regions of negative $V_R$ (blue) to become more concentrated into arcs, and producing sharp sign changes in $\Delta V_\phi$. One-dimensional radial profiles extracted from these maps along $\phi=0^\circ$ (dash-dot regions) are presented in Figure \ref{['fig:velocity_maps_large_r']}.
• Figure 3: One-dimensional radial profiles of the velocity perturbations resulting from the large, $m=2$ impact simulation shown in Figure \ref{['fig:velocity_maps_large_xy']}. These profiles show $V_R$ (top row) and $\Delta V_\phi$ (bottom row) as a function of radius $R$, at the same four times ($t=0$, $0.4$, $0.8$, and $1.2\,$Gyr). The profiles are averaged on small radial bins at fixed azimuth ($\phi=0^\circ$, corresponding to the dashed-dot regions in Figure \ref{['fig:velocity_maps_large_xy']}). The solid black lines represent the results from the large impact simulation. The dashed blue lines show the velocity profiles resulting from the dominant $\Omega - \kappa/m$ wave (specifically the $\Omega - \kappa/2$ wave for $m=2$) in a small impact simulation (extracted from the blue curves in Fig. \ref{['fig:small_1d']}), scaled up for comparison. These black 1D profiles reveal the characteristic triangular wave shape in $V_R$ and the sawtooth pattern in $\Delta V_\phi$, which are key signatures of the non-linear velocity structures generated by the large initial perturbation.
• Figure 4: Schematic example of the usage of SINDy to infer the Navier-Stokes equation in a simulation. A. Collect snapshots representing the solution of the PDE. B. Compute numerical derivatives and organize the data into a comprehensive matrix $\Theta$ that includes the candidate terms for the PDE. C. Employ sparse regression techniques to isolate the active terms in the PDE. D. Active terms in the library ($\mathbf{x}(t)$, in the text) are informative of the underlying PDE. Reproduced from brunton2023mlpde, who adapted it from rudy2017sindypde.
• Figure 5: Radial profiles of the coefficients $A$ (blue), $B$ (coral), and $C$ (purple) from the linear PDE system (Eq. \ref{['eq:lin_sys_sindy']}) found by SINDy, governing the evolution of velocity perturbations in small impact simulations (Eq. \ref{['eq:m_kick']}). The black lines show the corresponding analytical values derived in Sect. \ref{['sect:analytic_derivation']} (Eq. \ref{['eqn:lin_syst']}). The excellent agreement between the SINDy-discovered coefficients and the analytical predictions demonstrates the success in recovering the underlying linear dynamics.