2026AJ....171..160L

To unravel the formation history of the Milky Way, we estimate the accretion times of six phase-space substructures in the stellar halo, using the orbital frequencies toward two spatial directions (r, ϕ) in spherical coordinates. These substructures, identified in our previous studies, are located in the solar neighbourhood and therefore have high-precision astrometry from Gaia. The uncertainties of the results are determined using the Monte Carlo method, and the significance is established through comparison with random halo samples. The results for the substructure GL-1 in both directions show good consistency and high significance (4.3σ and 3.9σ), yielding a combined accretion time of 5.6 ± 0.1 Gyr ago, where the uncertainties quoted are statistical only. The substructures GL-4 and GR-1, with smaller pericenters, exhibit higher significance in the less massive potential of the Milky Way, implying that the more massive potential may overestimate the central mass, especially the bulge. The accretion times of GL-4 and GR-1 are 6.9 ± 0.3 Gyr with a confidence of 3.7σ, and 2.0 ± 0.1 Gyr with a confidence of 4.4σ, respectively. Further constraints on the accretion times of phase-space substructures require more precise astrometric data, e.g., by Gaia DR4, China Space Station Survey Telescope and Roman space telescope.