Finding $r$-II sibling stars in the Milky Way with the Greedy Optimistic Clustering algorithm

$R$-process enhanced stars with [Eu/Fe]$\ge +0.7$ (so-called $r$-II stars) are believed to have formed in an extremely neutron-rich environment in which a rare astrophysical event (e.g., a neutron star merger) occurred. This scenario is supported by the existence of an ultra-faint dwarf galaxy, Reticulum~II, where most of the stars are highly enhanced in $r$-process elements. In this scenario, some small fraction of dwarf galaxies around the Milky Way were $r$ enhanced. When each $r$-enhanced dwarf galaxy accreted to the Milky Way, it deposited many $r$-II stars in the Galactic halo with similar orbital actions. To search for the remnants of the $r$-enhanced systems, we analyzed the distribution of the orbital actions of $N=161$ $r$-II stars in the Solar neighborhood by using the Gaia EDR3 data. Since the observational uncertainty is not negligible, we applied a newly-developed {\it greedy optimistic clustering method} to the orbital actions of our sample stars. We found six clusters of $r$-II stars that have similar orbits and chemistry, one of which is a new discovery. Given the apparent phase-mixed orbits of the member stars, we interpret that these clusters are good candidates for remnants of completely disrupted $r$-enhanced dwarf galaxies that merged with the ancient Milky Way.