The HST large programme on $\omega$ Centauri - II. internal kinematics
  In this second installment of the series, we look at the internal kinematics
of the multiple stellar populations of the globular cluster $\omega$ Centauri
in one of the parallel Hubble Space Telescope (HST) fields, located at about
3.5 half-light radii from the center of the cluster. Thanks to the over
15-year-long baseline and the exquisite astrometric precision of the HST
cameras, well-measured stars in our proper-motion catalog have errors as low as
$\sim 10\ \mu$as yr$^{-1}$, and the catalog itself extends to near the
hydrogen-burning limit of the cluster. We show that second-generation (2G)
stars are significantly more radially anisotropic than first-generation (1G)
stars. The latter are instead consistent with an isotropic velocity
distribution. In addition, 1G have excess systemic rotation in the plane of the
sky with respect to 2G stars. We show that the six populations below the
main-sequence (MS) knee identified in our first paper are associated to the
five main population groups recently isolated on the upper MS in the core of
cluster. Furthermore, we find both 1G and 2G stars in the field to be far from
being in energy equipartition, with $\eta_{\rm 1G}=-0.007\pm0.026$ for the
former, and $\eta_{\rm 2G}=0.074\pm0.029$ for the latter, where $\eta$ is
defined so that the velocity dispersion $\sigma_\mu$ scales with stellar mass
as $\sigma_\mu \propto m^{-\eta}$. The kinematical differences reported here
can help constrain the formation mechanisms for the multiple stellar
populations in $\omega$ Centauri and other globular clusters. We make our
astro-photometric catalog publicly available.
