Constraints on Schwarzschild Black Hole in a Generalized Dehnen-Type $(1,4,\gamma)$ Dark Matter Halo via the S2 Star Orbit around Sgr A$^\star$

TL;DR

This study models the orbit of the S2 star around Sgr A* using a generalized Schwarzschild black hole embedded in a Dehnen-type dark matter halo, constraining dark matter parameters through observational data.

Contribution

It introduces a fully generalized Dehnen-type dark matter halo model for black hole environments and constrains its parameters using stellar orbit data with MCMC analysis.

Findings

Best-fit dark matter halo parameters with uncertainties.

95% confidence upper bounds on halo parameters.

Demonstrates stellar orbits can constrain dark matter distribution.

Abstract

The distribution of dark matter (DM) halo around supermassive black holes (BHs) may leave observable imprints on stellar dynamics near galactic centers. Motivated by this, we investigate the orbital motion of the S2 star in the spacetime of a recently derived generalized Schwarzschild BH solution embedded in a Dehnen-type $(1,4,\gamma)$ DM halo, considering it as a possible model for Sgr A$^{\star}$ at the center of the Milky Way. Unlike previous studies restricted to specific values of the halo parameter $\gamma$, the present solution describes the fully generalized case with arbitrary $\gamma$. We derive the corresponding equations of motion and obtain the associated perihelion shift over one orbital period. Using observational data of the S2 star, we constrain the parameters of the Schwarzschild--Dehnen BH-DM system through a Markov Chain Monte Carlo (MCMC) analysis. Our results yield the best-fit values $\gamma = 1.18^{+1.03}_{-0.81}$ $(1.23^{+1.01}_{-0.85})$, $\rho_s = 0.37^{+0.42}_{-0.29}$ $(0.31^{+0.44}_{-0.26})$, and $r_s = 0.05^{+0.05}_{-0.03}$ $(0.14^{+0.18}_{-0.10})$ for observational data of Do et al.~\cite{Do19} and Gillessen et al.~\cite{Gillessen17ApJ}, respectively. We further obtain the corresponding 95\% confidence upper bounds: $\gamma < 2.66$ $(2.67)$, $\rho_s < 0.93$ $(0.92)$, and $r_s < 0.16$ $(0.52)$. These results demonstrate that precise stellar orbit measurements can provide meaningful constraints on the DM halo distributions surrounding supermassive BHs and may offer insights into the DM environment of Sgr A$^{\star}$ at the center of the Milky Way.