The age-velocity dispersion relation of the Galactic discs from
  LAMOST-Gaia data
  We present the age-velocity dispersion relation (AVR) in three dimensions in
the solar neighbourhood using 3,564 commonly observed sub-giant/red-giant
branch stars selected from LAMOST, which gives the age and radial velocity, and
\emph{Gaia}, which measures the distance and proper motion. The stars are
separated into metal-poor (${\rm [Fe/H]<-0.2}$\,dex and metal-rich (${\rm
[Fe/H]>-0.2}$\,dex) groups, so that the metal-rich stars are mostly
$\alpha$-poor, while the metal-poor group are mostly contributed by
$\alpha$-enhanced stars. Thus, the old and metal-poor stars likely belong to
the chemically defined thick disc population, while the metal-rich sample is
dominated by the thin disc. The AVR for the metal-poor sample shows an abrupt
increase at $\gtrsim7$\,Gyr, which is contributed by the thick disc component.
On the other hand, most of the thin disc stars with ${\rm [Fe/H]>-0.2}$\,dex
display a power-law like AVR with indices of about 0.3--0.4 and 0.5 for the
in-plane and vertical dispersions, respectively. This is consistent with the
scenario that the disc is gradually heated by the spiral arms and/or the giant
molecular clouds. Moreover, the older thin disc stars ($>7$\,Gyr) have a
rounder velocity ellipsoid, i.e. $\sigma_\phi/\sigma_{\rm z}$ is close to 1.0,
probably due to the more efficient heating in vertical direction. Particularly
for the old metal-poor sample located with $|z|>270$\,pc, the vertical
dispersion is even larger than its azimuthal counterpart. Finally, the vertex
deviations and the tilt angles are plausibly around zero with large
uncertainties.
