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 spaces are less affected by the lack
of stars closer to the disk mid-plane due to extinction; (4) the phase space
snail shell is amplified in greater radial ranges. Compared to the $R$-based
snail shell, the quantitatively measured 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 $R_{g}$ reveal well-defined and
sharper snail shell features. By comparing the radial profiles of the pitch
angle between observation and simulation, the external perturbation can be
constrained to $\sim$500$-$700 Myr ago. For future vertical phase mixing study,
it is recommended to use the guiding radius with additional constraints on
orbital hotness (ellipticity) to improve the clarity of the phase snail.
