Footprints of the Sagittarius dwarf galaxy in the $Gaia$ data set

We analyse an N-body simulation of the interaction of the Milky Way (MW) with a Sagittarius-like dSph (Sgr), looking for signatures which may be attributed to its orbital history in the phase space volume around the Sun in light of $Gaia$ DR2 discoveries. The repeated impacts of Sgr excite coupled vertical and radial oscillations in the disc which qualitatively, and to a large degree quantitatively are able to reproduce many features in the 6D $Gaia$ DR2 samples, from the median $V_{R},V_{\phi}, V_{z} $ velocity maps to the local $\delta\rho(v_{z},z)$ phase-space spiral which is a manifestation of the global disc response to coupled oscillations within a given volume. The patterns in the large-scale velocity field are well described by tightly wound spirals and vertical corrugations excited from Sgr's impacts. We show that the last pericentric passage of Sgr resets the formation of the local present-day $\delta\rho(v_z,z)$ spiral and situate its formation around 500-800 Myr. As expected $\delta\rho(vz,z)$ grows in size and decreases in woundedness as a function of radius in both the $Gaia$ DR2 data and simulations. This is the first N-body model able to explain so many of the features in the data on different scales. We demonstrate how to use the full extent of the Galactic disc to date perturbations dating from Myr to Gyr, probe the underlying potential and constrain the mass-loss history of Sgr. $\delta\rho(vz,z)$ looks the same in all stellar populations age bins down to the youngest ages which rules out a bar buckling origin.