Dynamical traceback age of the $\beta$ Pictoris moving group
  Context: The $\beta$ Pictoris moving group is one of the most well-known
young associations in the solar neighbourhood and several members are known to
host circumstellar discs, planets, and comets. Measuring its age with precision
is basic to study several astrophysical processes such as planet formation and
disc evolution which are strongly age dependent.
  Aims: We aim to determine a precise and accurate dynamical traceback age for
the $\beta$ Pictoris moving group.
  Methods: Our sample combines the extremely precise Gaia DR2 astrometry with
ground-based radial velocities measured in an homogeneous manner. We use an
updated version of our algorithm to determine dynamical ages. The new approach
takes into account a robust estimate of the spatial and kinematic covariance
matrices of the association to improve the sample selection process and to
perform the traceback analysis.
  Results: We estimate a dynamical age of $18.5_{-2.4}^{+2.0}$ Myr for the
$\beta$ Pictoris moving group. We investigated the spatial substructure of the
association at birth time and we propose the existence of a core of stars more
concentrated. We also provide precise radial velocity measurements for 81
members of $\beta$ Pic, including ten stars with the first determination of
their radial velocities.
  Conclusions: Our dynamical traceback age is three times more precise than
previous traceback age estimates and, more important, for the first time,
reconciles the traceback age with the most recent estimates of other dynamical,
lithium depletion boundary, and isochronal ages. This has been possible thanks
to the excellent astrometric and spectroscopic precisions, the homogeneity of
our sample, and the detailed analysis of binaries and membership.
