Context. The impact of Sagittarius and other satellite galaxies such as the Large Magellanic Cloud on our Galaxy and in particular its disc is gradually being disclosed. Simulations tailored to the interaction of the Milky Way and Sagittarius show rings and spiral arms appearing in the Galaxy disc. However, spiral arms can also be induced by the bar or by disc instabilities. Aims. We aim to study the dynamics of tidally induced spiral arms in the context of the different encounters with Sagittarius and their kinematic signatures in the shape of ridges and waves in angular momentum, similar to those detected with Gaia DR2. Methods. We build toy models of the interaction between a host and a satellite galaxy using orbital integrations after a tidal distant encounter. We derive analytically the shape of the structures seen in phase space as a function of time for simple power-law potential models. We compare these models to a more realistic N-body simulation of the Milky-Way Sagittarius-like interaction and also to real data. Results. As previously found, an impulsive distant tidal approach of a galactic satellite generates a kick in velocities that leads to a 2-armed spiral structure. The arms are made of orbits in between their apocentres and pericentres, thus, correspond to regions with average negative galactocentric radial velocity. The 2-arm pattern rotates at an angular speed of Ω − 1/2κ which depends on Galactocentric radius, thus causing a wind-up with time. This winding produces ridges in the R-V φ projection with alternating signs of V R and oscillations of V R in the L Z -φ space, similar to those observed in the Gaia data. The frequency of these kinematic features increases with time, offering a powerful means to infer the potential and the perturbation's onset time and azimuthal phase. Simple Fourier analysis allows to date the impact times of simple models and even to date perturbations induced from subsequent pericentric passages that appear as simultaneous waves. For the MW, the Fourier analysis indicates a superposition of two different frequencies, confirming previous studies. Assuming that both are due to impulsive and distant pericentre passages, we find perturbation times <0.4 Gyr and in the range of 0.7-1.8 Gyr. The latter is compatible with a previous pericentre of Sagittarius and would be associated to about 4 wraps of the spiral arms in the observed radial range. Conclusions. Further work on the self-gravitating response of galactic discs and possible degeneracies with secular processes induced by the bar is necessary. An exploration of the Milky Way's kinematics over a larger radial and azimuthal range can help to better discern and date the various perturbations. Our study is a first step towards shedding some more light into the elusive structure and dynamics of the spiral arms of the Galaxy.

Context. The impact of Sagittarius and other satellite galaxies such as the Large Magellanic Cloud on our Galaxy and in particular its disc is gradually being disclosed. Simulations tailored to the interaction of the Milky Way (MW) and Sagittarius show rings and spiral arms appearing in the Galaxy disc. However, spiral arms can also be induced by the bar or by disc instabilities. Aims. We aim to study the dynamics of tidally induced spiral arms in the context of the different encounters with Sagittarius and determine their kinematic signatures in the shape of ridges and waves in angular momentum, similar to those detected with Gaia DR2. Methods. We built toy models of the interaction between a host and a satellite galaxy using orbital integrations after a tidal distant encounter. We derived analytically the shape of the structures seen in phase space as a function of time for simple power-law potential models. We compared these models to a more realistic N-body simulation of the MW Sagittarius-like interaction and also to real data from Gaia DR3. Results. As previously found, an impulsive distant tidal approach of a galactic satellite generates a kick in velocities that leads to a two-armed spiral structure. The arms are made of orbits in between their apocentres and pericentres, thus, they correspond to regions with average negative galactocentric radial velocity. The two-arm pattern rotates at an angular speed of ω − 1/2κ which depends on Galactocentric radius, thus causing a wind-up with time. This winding produces ridges in the R-V φ projection with alternating signs of V R and oscillations of V R in the L Z -φ space, similar to those observed in the Gaia data. The frequency of these kinematic features increases with time, offering a powerful means to infer the potential and the perturbation's onset time and azimuthal phase. Fourier analysis allows us to date the impact times of simple models and even to date perturbations induced from subsequent pericentric passages that appear as simultaneous waves. For the MW, the Fourier analysis indicates a superposition of two different frequencies, confirming previous studies. Assuming that both are due to impulsive and distant pericentre passages, we find perturbation times <0.6 Gyr and in the range of 0.8-2.1 Gyr. The latter is compatible with a previous pericentre of Sagittarius and would be associated to about four wraps of the spiral arms in the observed radial range. Conclusions. Further work on the self-gravitating response of galactic discs and possible degeneracies with secular processes induced by the bar is necessary. Our study is a first step towards shedding more light on the elusive structure and dynamics of the spiral arms of the Galaxy.