Detection of multiple phase space overdensities of GSE stars by orbit integration

In N-body simulations, nearly radial mergers can form shell-like overdensities in the sky position and phase space ($r-v_r$) due to the combination of dynamical friction and tidal stripping. The merger event of Gaia-Sausage-Enceladus has provided a unique opportunity to study the shells in the phase space. To search for them, we integrate the orbits of 5949 GSE-related halo K giants from the LAMOST survey and record their positions at all time intervals in $r-v_r$ diagram. After the subtraction of a smoothed background, we find six significant and complete thin chevron-like overdensities. The apocenters $r_\mathrm{apo}$ of stars in the six chevrons are around 6.75, 12.75, 18.75, 25.25, 27.25, and 30.25 kpc. These chevrons reveal the multiple pile-ups of GSE stars at different apocenters. The application of a different Milky Way mass $M_\mathrm{vir}$ will change the opening angles of these chevrons, while leave their apocenters almost unchanged. By comparing with a recent study of the phase space overdensities of local halo stars from Gaia RVS survey, our results are more inclined to a medium $M_\mathrm{vir}$ of $10^{12}\,M_\odot$. The application of a non-axisymmetric Galactic potential with a steadily rotating bar has a blurring effect on the appearance of these chevron-like overdensities, especially for the chevrons with $r_\mathrm{apo} > 20$ kpc.