  The evolution of Europa's water-product exosphere over its 85-hour day, based
on current models, has not been shown to exhibit any diurnal asymmetries. Here
we simulate Europa's exosphere using a 3-D Monte Carlo routine including, for
the first time, the role of Europa's rotation on the evolution of exospheric
molecules throughout the orbit. We focus on O$_2$, sputtered by a trailing
hemisphere source with a temperature-dependence under isotropic plasma
conditions. We find that the O$_2$ component, while global, is not homogenous
in Europa local time. Rather, the O$_2$ accumulates at the dusk hemisphere.
When rotation is explicitly excluded, no diurnal asymmetries exist. We find
that the assumed thermal-dependence on the O$_2$ source is critical for a
diurnal asymmetry: the diurnal surface temperature profile is imprinted on to
the near-surface O$_2$ atmosphere, due to the small hop times of the
non-adsorbing O$_2$ effectively rotating with Europa. Altogether, our
simulations conclude that the dusk-over-dawn asymmetry is driven by Europa's
day-night O$_2$ cycle synchronized with Europa's orbital period based on our
model assumptions on O$_2$ production and loss. This conclusion is in agreement
with the recent understanding that a non-adsorbing, rotating O$_2$ source
peaking at noon will naturally accumulate from dawn-to-dusk, should the O$_2$
lifetime be sufficiently long compared to the orbital period. Lastly we compare
hemispherically-averaged dusk-over-dawn ratios to the recently observed oxygen
emission data by the Hubble Space Telescope. We find that while the simulations
are globally consistent with brighter oxygen emission at dusk than at dawn, the
orbital evolution of the asymmetries in our simulations can be improved by
ameliorating the O$_2$ source & loss rates, and possibly adsorption onto the
regolith.
