Galactic seismology: the evolving "phase spiral" after the Sagittarius dwarf impact

In 2018, the ESA \Gaia\ satellite discovered a remarkable spiral pattern ("phase spiral") in the $z-V_z$ phase plane throughout the solar neighbourhood, where $z$ and $V_z$ are the displacement and velocity of a star perpendicular to the Galactic disc. In response to Binney \& Sch\"onrich's analytic model of a disc-crossing satellite to explain the \Gaia\ data, we carry out a high-resolution, N-body simulation (N$\:\approx 10^8$ particles) of an impulsive mass ($2\times 10^{10}$ \Msun) that interacts with a cold stellar disc at a single transit point. The disc response is complex since the impulse triggers a superposition of two distinct bisymmetric ($m=2$) modes $-$ a density wave and a corrugated bending wave $-$ that wrap up at different rates. Stars in the {\it faster} density wave wrap up with time $T$ according to $\phi_D(R,T)=(\Omega_D(R) + \Omega_{\rm o})\:T$ where $\phi_D$ describes the spiral pattern and $\Omega_D =\Omega(R) -\kappa(R)/2$, where $\kappa$ is the epicyclic frequency. While the pattern speed $\Omega_{\rm o}$ is small, it is non-zero. The {\it slower} bending wave wraps up according to $\Omega_B\approx\Omega_D/2$ producing a corrugated wave. The bunching effect of the density wave triggers the phase spiral as it rolls up and down on the bending wave ("rollercoaster" model). The phase spiral emerges slowly about $\Delta T \approx 400$ Myr after impact. It appears to be a long-lived, disc-wide phenomenon that continues to evolve over most of the 2~Gyr simulation. Thus, given Sagittarius' (Sgr) low total mass today ($M_{\rm tot}\sim 3\times 10^8$ \Msun\ within 10 kpc diameter), we believe the phase spiral was excited by the disc-crossing dwarf some $1-2$ Gyr {\it before} the recent transit. For this to be true, Sgr must be losing mass at 0.5-1 dex per orbit loop.