We present a new, computationally efficient, energy-integrated approximation for neutrino effects in hot and dense astrophysical environments such as supernova cores and compact binary mergers and their remnants. Our new method, termed ILEAS for Improved Leakage-Equilibration-Absorption Scheme, improves the lepton-number and energy losses of traditional leakage descriptions by a novel prescription of the diffusion time-scale based on a detailed energy integral of the flux-limited diffusion equation. The leakage module is supplemented by a neutrino-equilibration treatment that ensures the proper evolution of the total lepton number and medium plus neutrino energies as well as neutrino-pressure effects in the neutrino-trapping domain. Moreover, we employ a simple and straightforwardly applicable ray-tracing algorithm for including re-absorption of escaping neutrinos especially in the decoupling layer and during the transition to semi-transparent conditions. ILEAS is implemented on a three-dimensional (3D) Cartesian grid with a minimum of free and potentially casedependent parameters and exploits the basic physics constraints that should be fulfilled in the neutrino-opaque and free-streaming limits. We discuss a suite of tests for stationary and time-dependent proto-neutron star models and post-merger blackhole-torus configurations, for which 3D ILEAS results are demonstrated to agree with energy-dependent 1D and 2D two-moment (M1) neutrino transport on the level of 10-15 percent in basic neutrino properties. This also holds for the radial profiles of the neutrino luminosities and of the electron fraction. Even neutrino absorption maps around torus-like neutrino sources are qualitatively similar without any fine-tuning, confirming that ILEAS can satisfactorily reproduce local losses and re-absorption of neutrinos as found in sophisticated transport calculations.

We present a coherent study of the impact of neutrino interactions on the r-process element nucleosynthesis and the heating rate produced by the radioactive elements synthesised in the dynamical ejecta of neutron star-neutron star (NS-NS) mergers. We have studied the material ejected from four NS-NS merger systems based on hydrodynamical simulations which handle neutrino effects in an elaborate way by including neutrino equilibration with matter in optically thick regions and re-absorption in optically thin regions. We find that the neutron richness of the dynamical ejecta is significantly affected by the neutrinos emitted by the post-merger remnant, in particular when compared to a case neglecting all neutrino interactions. Our nucleosynthesis results show that a solar-like distribution of r-process elements with mass numbers $A \gtrsim 90$ is produced, including a significant enrichment in Sr and a reduced production of actinides compared to simulations without inclusion of the nucleonic weak processes. The composition of the dynamically ejected matter as well as the corresponding rate of radioactive decay heating are found to be rather independent of the system mass asymmetry and the adopted equation of state. This approximate degeneracy in abundance pattern and heating rates can be favourable for extracting the ejecta properties from kilonova observations, at least if the dynamical component dominates the overall ejecta. Part II of this work will study the light curve produced by the dynamical ejecta of our four NS merger models.

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