Young stellar cluster dilution near supermassive black holes: The impact of vector resonant relaxation on neighbour separation

We investigate the rate of orbital orientation dilution of young stellar clusters in the vicinity of supermassive black holes. Within the framework of vector resonant relaxation, we predict the time evolution of the two-point correlation function of the stellar orbital plane orientations as a function of their initial angular separation and diversity in orbital parameters (semimajor axis, eccentricity). As expected, the larger the spread in initial orientations and orbital parameters, the more efficient the dilution of a given set of co-eval stars, with a characteristic time-scale set up by the coherence time of the background potential fluctuations. A Markovian prescription that matches numerical simulations allows us to efficiently probe the underlying kinematic properties of the unresolved nucleus when requesting consistency with a given dilution efficiency, imposed by the observed stellar disc within the 1 arcsec of Sgr A∗. As a proof of concept, we compute maps of constant dilution times as a function of the semimajor axis cusp index and fraction of intermediate-mass black holes in the old background stellar cluster. This computation suggests that vector resonant relaxation should prove useful in this context since it impacts orientations on time-scales comparable to the stars' age.