Radius of the innermost stable circular orbit of black-holes with localized Astrophysical Dark Matter halo

We investigate the geodesic structure of a static, spherically symmetric black-hole in the background of a localized Astrophysical Dark Matter (ADM)-halo. The halo is modeled by a core-radius ($r_0$), regularized density profile that vanishes at the event horizon and exponential factor ensures the finite total halo mass. We solved the Einstein field equations numerically with equation of state with $p=w\rho$ for stiff matter $w=1$ scenario. By analyzing the effective potential and its first and second derivatives, we evaluated the characteristic radius of the time-like innermost stable circular orbit ($r_{\text{ISCO}}$) of massive particle in the vicinity of hairy black-hole. Our numerical analysis is consistent with the recently proposed theoretical upper bound $r_{\text{ISCO}}\le 6$ normalized to unit mass black-hole. We showed that the ADM-halo shifts $r_{\text{ISCO}}$ inward with respect to the Schwarzschild (bald) scenario and the shifts are larger for larger halo core-radius. In particular our numerical finding enumerate $r_{\text{ISCO}}$ are approximately $5.97$, $5.61$, $5.53$ for $r_0=2.0,10.0,20.0$ respectively. We believe that our analysis with ADM-halo model shed light on probing black-hole hair through high-energy astrophysical observables.