  Hydronium (H$_3$O$^+$) was first detected in 1986 in interstellar molecular
clouds. It was reported in many galactic diffuse and dense regions, as well as
in extragalactic sources. H$_3$O$^+$ plays a major role both in interstellar
oxygen and water chemistry. However, despite the large number of H$_3$O$^+$
observations, its collisional excitation was investigated only partially. In
the present work we study the state-to-state rotational de-excitation of
$ortho$- and $para$-H$_3$O$^+$ in collisions both with $ortho$- and
$para$-H$_2$. The cross sections are calculated within the close-coupling
formalism using a highly accurate potential energy surface developed for this
system. The rate coefficients are computed up to $300$ K kinetic temperature.
Transitions between the lowest 21 rotation-inversion states were studied for
$para$-H$_3$O$^+$, and the lowest 11 states for $ortho$-H$_3$O$^+$, i.e. all
levels with rotational energies below 430 K ($\sim 300$ cm$^{-1}$) are
considered (up to $j\leq5$). In order to estimate the impact of the new rate
coefficients on the astrophysical models for H$_3$O$^+$, radiative transfer
calculations were also carried out. We have examined how the new collisional
data affect the line intensities with respect to older data previously used for
the interpretation of observations. By analysing all detected transitions we
find that our new, accurate rate coefficients have a significant impact
(typically within a factor of 2) on radiation temperatures, allowing more
accurate estimation of column densities and relative abundances of hydronium,
especially in warm molecular clouds, paving the path towards better
interpretation of interstellar water and oxygen chemistry.
