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

Context. 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. Aims. We aim to analytically characterise the dynamics of tidally induced spiral arms, including their phase-space signatures. Methods. We ran idealised 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. Results. 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 ± κ/m emerge, where Ω and κ are the azimuthal and epicyclic frequencies, and m is the azimuthal mode number. 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, along with their shape and temporal evolution in different phase-space projections. In the realistic simulations, we recovered the same equation. However, the fit is sub-optimal, pointing to missing terms in our analysis, such as velocity dispersion and self-gravity. We fit the Gaia LZ−〈VR〉 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). Conclusions. We merged 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 such as self-gravity and dynamical friction.