  Predicting the properties of the matter ejected during and after a neutron
star merger is crucial to our ability to use electromagnetic observations of
these mergers to constrain the masses of the neutron stars, the equation of
state of dense matter, and the role of neutron star mergers in the enrichment
of the Universe in heavy elements. Our ability to reliably provide such
predictions is however limited by a broad range of factors, including the
finite resolution of numerical simulations, their treatment of magnetic fields,
neutrinos, and neutrino-matter interactions, and the approximate modeling of
the equation of state of dense matter. In this manuscript, we study
specifically the role that a small residual eccentricity and different
implementations of the same equation of state have on the matter ejected during
the merger of a $1.3M_\odot-1.4M_\odot$ binary neutron star system. We find
that a residual eccentricity $e\sim 0.01$, as measured $\sim 4-6$ orbits before
merger, causes $O(25\%-30\%)$ changes in the amount of ejected mass, mainly due
to changes in the amount of matter ejected as a result of core bounces during
merger. We note that $O(1\%)$ residual eccentricities have regularly been used
in binary neutron star merger simulations as proxy for circular binaries,
potentially creating an additional source of error in predictions for the mass
of the dynamical ejecta.
