The Galactic disc phase spirals at different Galactic positions revealed by Gaia and LAMOST data

We have investigated the distributions of stellar azimuthal and radial velocity components $V_{Phi}$ and $V_{R}$ in the vertical position-velocity plane $Z$-$V_{Z}$ across the Galactic disc of $6.34 lesssim R lesssim 12.34$,kpc and $|Phi| lesssim 7.5^{circ}$ using a Gaia and Gaia-LAMOST sample of stars. As found in previous works, the distributions exhibit significant spiral patterns. The $V_{R}$ distributions also show clear quadrupole patterns, which are the consequence of the well-known tilt of the velocity ellipsoid. The observed spiral and quadrupole patterns in the phase space plane vary strongly with radial and azimuthal positions. The phase spirals of $V_{Phi}$ become more and more relaxed as $R$ increases. The spiral patterns of $V_{Phi}$ and $V_{R}$ and the quadrupole patterns of $V_{R}$ are strongest at $-2^{circ} < Phi < 2^{circ}$ but negligible at $4^{circ} < Phi < 6^{circ}$ and $-6^{circ} < Phi < -4^{circ}$. Our results suggest an external origin of the phase spirals. In this scenario, the intruder, most likely the previously well-known Sagittarius dwarf galaxy, passed through the Galactic plane in the direction towards either Galactic center or anti-center. The azimuthal variations of the phase spirals also help us constrain the passage duration of the intruder. A detailed model is required to reproduce the observed radial and azimuthal variations of the phase spirals of $V_{Phi}$ and $V_{R}$.