Compositions of the Hercules-Aquila Cloud and Virgo Over-density

Based on a sample of K giant from Large sky Area Multi-Object fiber Spectroscopic Telescope (LAMOST) Data Release 8 and a sample of RR Lyrae (RRL) from \textit{Gaia} Data Release 3, we investigate the compositions of the Hercules-Aquila Cloud (HAC) and Virgo Over-density (VOD) and their collective contribution to the tilt and triaxiality of the stellar halo ($r\,\textless\,40\,{\rm kpc}$) as well as two breaks at $\approx15\,{\rm kpc}$ and 30\,kpc. We apply the Gaussian mixture model (GMM) to divide the stellar halo into the isotropic component and the radially biased anisotropic component, namely Gaia-Sausage-Enceladus (GSE), and find that both HAC and VOD are dominated by the GSE debris stars with weights of $0.67^{+0.09}_{-0.07}$ and $0.57^{+0.07}_{-0.06}$, respectively. In addition, using the K giants with orbital parameters, we identify the member stars of known substructures, including GSE, Sagittarius (Sgr), Helmi Streams, Sequoia, Thamnos, Pontus, Wukong, and Metal-weak Thick Disk (MWTD), to probe the compositions of low-eccentricity stars in the HAC and VOD regions. In density fittings of the RRL sample, we note that the absence of HAC and VOD has a weak effect on the shape of halo. Finally, we find that the radially biased anisotropic halo contributes majorly to the stellar halo that can be modelled with a tilted triaxial ellipsoid and a doubly broken power law with breaking radii at $18.08^{+2.04}_{-3.22}\,{\rm kpc}$ and $33.03^{+1.30}_{-1.21}\,{\rm kpc}$. This has important significance for understanding the status of large diffuse over-densities in the Milky Way.