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 $V_{R}$.