Estimating outflow masses and velocities in merger simulations: Impact of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>r</mml:mi></mml:math>-process heating and neutrino cooling

The determination of the mass, composition, and geometry of matter outflows in black hole--neutron star and neutron star--neutron star binaries is crucial to current efforts to model kilonovae and to understand the role of neutron star merger in $r$-process nucleosynthesis. In this manuscript, we review the simple criteria currently used in merger simulations to determine whether matter is unbound and what the asymptotic velocity of ejected material will be. We then show that properly accounting for both heating and cooling during $r$-process nucleosynthesis is important to accurately predict the mass and kinetic energy of the outflows. These processes are also likely to be crucial to predict the fall-back timescale of any bound ejecta. We derive a model for the asymptotic velocity of unbound matter and binding energy of bound matter that accounts for both of these effects and that can easily be implemented in merger simulations. We show, however, that the detailed velocity distribution and geometry of the outflows can currently only be captured by full three-dimensional fluid simulations of the outflows, as nonlocal effect ignored by the simple criteria used in merger simulations cannot be safely neglected when modeling these effects. Finally, we propose the introduction of simple source terms in the fluid equations to approximately account for heating/cooling from $r$-process nucleosynthesis in future seconds-long three-dimensional simulations of merger remnants, without the explicit inclusion of out-of-nuclear statistical equilibrium reactions in the simulations.