Observational Constrains on the Sgr A$^*$ Black Hole Immersed in a Dark Matter Halo: Shadow and S2 Star Orbit

TL;DR

This work develops a halo-modified black hole metric by embedding Sgr A* in Zhao's generalized double power-law DM halo, linking halo density to spacetime geometry via M_D(r) and the metric function f(r). By analyzing null and timelike geodesics, it studies how halo parameters alter the black hole shadow and the S2 star orbit, and then uses EHT shadow data and long-term S2 observations within a Bayesian framework to constrain the halo parameters. The results show the shadow expands slightly with DM, and S2 dynamics yield meaningful constraints on halo density parameters, indicating a low DM density near the Galactic Center and a shadow radius consistent with Schwarzschild within uncertainties. This approach provides a joint theoretical and phenomenological framework for probing both black hole physics and the Milky Way's dark matter distribution, with future data expected to tighten the constraints further.

Abstract

It is widely believed that Sgr A$^*$, located at the center of our Galaxy, is a supermassive black hole. Recent observations of its shadow and long-term monitoring of the S2 star have provided compelling evidence supporting this hypothesis. These observational advancements also offer valuable opportunities to explore the physical properties of the black hole and its surrounding environment. Since a dark matter halo is expected to exist in the Milky Way and around Sgr A$^*$, investigating the behavior of the Galactic Center black hole embedded in such a halo provides a crucial means to simultaneously probe both black hole physics and dark matter properties. In this work, We develop a black hole metric that incorporates a generalized double power law dark matter halo, and analyze the corresponding null and timelike geodesics to investigate how the halo parameters affect the black hole shadow and the motion of the S2 star. Furthermore, by comparing our theoretical predictions with observational data of the shadow and the S2 orbit, we constrained the dark matter halo parameters. The results of this study provide both theoretical and phenomenological insights into the nature of Sgr A$^*$ and the distribution of dark matter in our Galaxy.

Figures (10)

• Figure 1: Dark matter density profile as a function of the halo radius, for different parameter combinations $(\alpha, \beta, \gamma) = (1, 3, 1), (1, 4, 1), (1, 3, 0.5), (2, 4, 1)$.
• Figure 2: The metric function $f(r)$ is shown as a function of $r/M$ for different combinations of the profile parameters $(\alpha, \beta, \gamma) = (1,3,1), (1,4,1), (1,3,0.5), (2,4,1)$, with $\rho_s = 0.0001M^{-2}$ and $r_s = 10 M$ for simplicity. These parameter sets correspond to different dark matter–modified black hole metrics. For comparison, the Schwarzschild black hole ($\rho_s = 0$) is also shown as a reference.
• Figure 3: The metric function $f(r)$ as a function of $r/M$ for different parameter combinations $(\rho_s*M^{2}, r_s/M) = (0.0001, 10), (0.001, 10), (0.0001, 20)$, with $\alpha, \beta, \gamma = 1, 3, 1$ fixed for simplicity. The Schwarzschild black hole ($\rho_s = 0$) is shown for reference.
• Figure 4: Total number of orbits $n = \phi / 2\pi$ as a function of the impact parameter $b$, obtained for different combinations of the profile parameters $(\alpha, \beta, \gamma) = (1, 3, 1)$, $(1, 4, 1)$, $(1, 3, 0.5)$, and $(2, 4, 1)$. For simplicity, the parameters are fixed as $\rho_s = 0.0001M^{-2}$ and $r_s = 10M$. The corresponding result for the Schwarzschild black hole ($\rho_s = 0$) is also shown for comparison.
• Figure 5: Photon trajectories for impact parameters $b = 0, 2, 4, 6, 8, 10$ in Euclidean coordinates $x = r \cos\phi$ and $y = r \sin\phi$, obtained for different combinations of the profile parameters $(\alpha, \beta, \gamma) = (1, 3, 1)$, $(1, 4, 1)$, $(1, 3, 0.5)$, and $(2, 4, 1)$. For simplicity, we set $\rho_s = 0.0001M^{-2}$ and $r_s = 10M$. The corresponding result for the Schwarzschild black hole ($\rho_s = 0$) is also shown for comparison. Solid lines represent photon trajectories, while dashed lines indicate the horizons.