Accretion Disk Luminosity and Topological Characteristics for a Schwarzschild Black Hole Surrounded by King Dark Matter Halo

TL;DR

This work investigates a Schwarzschild black hole embedded in an isotropic King dark matter halo with density $\rho(r)=\rho_0\big(1+(r/R)^2\big)^{-3/2}$, examining horizon structure, photon trajectories, accretion-disk properties, and a topological thermodynamics framework. The authors derive how the King DM parameters $R$ and $\rho_0$ modify the horizon radius, Hawking temperature, and allow potential remnant radii, while also altering the photon sphere, shadow, ISCO, and disk radiation profiles. By comparing the predicted shadow to EHT constraints for Sgr A$^*$ and M87$^*$, they obtain upper bounds on the DM parameters and show that DM broadens the disk emission region and reddens the spectrum, with the ISCO and efficiency enhanced relative to Schwarzschild. A topological analysis yields a conventional critical point for remnants and a RN-like classification for the generalized free energy, linking the King DM scenario to broader black-hole thermodynamic universality classes. Overall, the study provides a cohesive set of observationally relevant predictions for black holes in dense dark-matter environments and outlines avenues for extending to rotating spacetimes and alternative halo models.

Abstract

This study delves into the intricate properties of a Schwarzschild black hole enveloped by King dark matter in an isotropic configuration. The thermodynamic characteristics of this black hole are meticulously analyzed, and the dynamics of massive and massless particles in its vicinity are investigated. In examining the trajectories of massless particles, the shadow cast in the presence of King dark matter is explored, revealing virtual ranges for the corresponding parameters. For the dynamics of massive particles, the radius of the innermost stable circular orbit, angular momentum, energy, and angular velocity of a test particle within the King dark matter framework surrounding the black hole are calculated. The effect of King dark matter on the accretion disk energy flux, effective radiation temperature, differential luminosity, and spectral luminosity are then investigated. The stability of the photon sphere in the presence of King dark matter is also studied, and finally, the thermodynamic potentials of this black hole are examined from a topological perspective.

Figures (17)

• Figure 1: Density profile \ref{['den']} for $\rho_0=1/(37.96 ) M_\odot/pc^3$ and several values of $R$.
• Figure 2: Variation of the lapse function in terms of $r/M$ for $\rho_0 M^2=1$.
• Figure 3: Variation of horizon radius $r_h$ in terms of $R/M$ and $\rho_0 M^2$. Increasing either $R/M$ or $\rho_0 M^2$ enhances the magnitude of the horizon radius.
• Figure 4: Hawking temperature versus $r_h$ for the case $\rho_0=1$. In the Hawking temperature curve of a Schwarzschild black hole, no phase transition is observed. Increasing the effect of the parameter $R$ beyond a certain threshold leads to the existence of two phase transitions in the Hawking temperature. Increasing the size of this parameter causes the Hawking temperature of the black hole to become zero at some point, indicating a remnant radius, and in this case, only one phase transition is observed in the Hawking temperature of the black hole.
• Figure 5: Left panel: Remnant radius curves for varying $R$. Right panel: The colored area indicates the range of variables for which a remnant radius exist.