Constructing the Milky Way Stellar Halo in the Galactic Center by Direct Orbit Integration

The halo stars on highly radial orbits should inevitably pass the center regions of the Milky Way. Under the assumption that the stellar halo is in dynamical equilibrium and axisymmetric, we integrate the orbits of $\sim 10,000$ halo K-giants at $5\leq r \leq 50$ kpc cross-matched from LAMOST DR5 and $Gaia$ DR3. By carefully considering the selection function, we construct the stellar halo distribution at the entire regions of $r \leq 50$ kpc. We find that a double-broken power-law function well describes the stellar halo density distribution with shallower slopes in the inner regions and the two breaks at $r=10$ kpc and $r=25$ kpc, respectively. The stellar halo becomes flatter from outer to inner regions but has $q\sim 0.5$ at $r \lesssim 5$ kpc. The stellar halo becomes isotropic with a slight prograde rotation in the inner 5 kpc, and reaches velocity dispersions of $\sim 250\rm \ km\ s^{-1}$. We get a weak negative metallicity gradient of $-0.005$ dex kpc$^{-1}$ at $5\leq r \leq 50$ kpc, while there is an excess of relative metal-rich stars with [Fe/H]$>-1$ in the inner 10 kpc. The halo interlopers at $r \leq 5$ kpc from integration of our sample has a mass of $\sim1.2 \times 10^8\ M_{\odot}$ ($\sim 4.7 \times 10^7\ M_{\odot}$ at [Fe/H]$<-1.5$), which can explain 50-100% of the metal-poor stars with [Fe/H]$<-1.5$ directly observed in the Galactic central regions.