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Coalescing neutron star binary (NSB) systems are primary candidates for$r$-process enrichment of galaxies. The recent detection of $r$-processelements in ultra-faint dwarf (UFD) galaxies and the abundances measured inclassical dwarfs challenges the NSB merger scenario both in terms ofcoalescence time scales and merger locations. In this paper, we focus on thedynamics of NSBs in the gravitational potentials of different types of hostgalaxies and on its impact on the subsequent galactic enrichment. We find that,for a $\sim t^{-1}$ delay time distribution, even when receiving a low kick($\sim 10~{\rm km~s^{-1}}$) from the second supernova explosion, in shallowdwarf galaxy potentials NSBs tend to merge with a large off-set from the hostgalaxy. This results in a significant geometrical dilution of the amount ofproduced $r-$process elements that fall back and pollute the host galaxy gasreservoir. The combination of dilution and small number statistics produces alarge scatter in the expected $r$-process enrichment within a single UFD orclassical dwarf galaxy. Comparison between our results and observed europiumabundances reveals a tension that even a systematic choice of optimisticparameters in our models cannot release. Such a discrepancy could point to theneed of additional $r$-process production sites that suffer less severedilution or to a population of extremely fast merging binaries.

Neutron star binary orbits in their host potential: effect on early r-process enrichment