Vertical Phase Mixing across the Galactic Disk

By combining the {\it LAMOST} and {\it Gaia} data, we investigate the vertical phase mixing across the Galactic disk. Our results confirm the existence of the phase space snail shells (or phase spirals) from 6 to 12 kpc. We find that grouping stars by the guiding radius ($R_{g}$), instead of the present radius ($R$) further enhances the snail shell signal in the following aspects: (1) clarity of the snail shell shape is increased; (2) more wraps of the snail shell can be seen; (3) the phase space is less affected by the observational bias, i.e., the lack of stars close to the disk mid-plane due to extinction; (4) the phase space snail shell is amplified in greater radial ranges. The quantitatively measured snail shell shapes are similar, except that the $R_{g}$-based snail shells show more wraps with better contrast. These lines of evidence lead to the conclusion that the guiding radius (angular momentum) is a fundamental parameter tracing the phase space snail shell across the Galactic disk. Results of our test particle simulations with impulse approximation verify that particles grouped according to the guiding radius reveal more prominent and well-defined, sharper snail shell features in the number density phase space. As a guideline for future vertical phase mixing study, it is recommended to use the guiding radius with additional constraints on the orbital hotness (ellipticity) to improve the clarity of the phase space snail shell in the Galactic disk.