  The hydronium cation plays a crucial role in interstellar oxygen and water
chemistry. While its spectroscopy was extensively investigated earlier, the
collisional excitation of H$_3$O$^+$ is not well studied yet. In this work we
present state-to-state collisional data for rotational de-excitation of both
ortho- and para-H$_3$O$^+$ due to para-H$_2$ impact. The cross sections are
calculated within the close-coupling formalism using our recent, highly
accurate rigid-rotor potential energy surface for this collision system. The
corresponding thermal rate coefficients are calculated up to 100 K. For
para-H$_3$O$^+$ the lowest 20 rotation-inversion states were considered in the
calculations, while for ortho-H$_3$O$^+$ the lowest 11 states are involved (up
to $j\leq5$), so all levels with rotational energy below 420 K (292 cm$^{-1}$)
are studied. In order to analyse the impact of the new collisional rates on the
excitation of H$_3$O$^+$ in astrophysical environments radiative transfer
calculations are also provided. The most relevant emission lines from an
astrophysical point of view are studied, taking into account the transitions at
307, 365, 389 and 396 GHz. We show that our new collisional data have a
non-negligible impact (from a few percents up to about a factor of 3) on the
brightness and excitation temperatures of H$_3$O$^+$, justifying the revision
of the physical conditions in the appropriate astrophysical observations. The
calculated rate coefficients allow one to recalculate the column density of
hydronium in interstellar clouds, which can lead to a better understanding of
interstellar water and oxygen chemistry.
