Periodic orbits and observational accretion disk around a Schwarzschild-like black hole surrounded by dark matter halo

TL;DR

The study addresses how a King-type dark matter halo modifies the spacetime around a Schwarzschild-like black hole and affects timelike and null geodesics and disk observables. It derives the halo-augmented metric $f(r)$, analyzes equatorial timelike geodesics with $V_{\rm eff}=f(r)\left(1+\frac{L^2}{r^2}\right)$, and investigates $R_{\mathrm{MBO}}$, $L_{\mathrm{MBO}}$, $R_{\mathrm{ISCO}}$, $L_{\mathrm{ISCO}}$, and $E_{\mathrm{ISCO}}$ as functions of halo parameters $(r_s,\rho_s)$. It then studies periodic orbits described by $(z,w,v)$ and null geodesics, finding outward shifts of periodic orbits and increases in the photon sphere and critical impact parameter $b_c$, alongside broader lensing features; it finally computes thin-disk radiation and images via the Novikov–Thorne model, showing disk dimming and larger redshift ranges. These results offer observational diagnostics to constrain DM halos and test GR in the strong-field regime.

Abstract

In this work, we investigate the dynamics of periodic orbits and the properties of accretion disks around a Schwarzschild-like black hole (BH) immersed in a King-type dark matter (DM) halo. Our analysis focuses on how the presence of the King DM halo influences both the behavior of periodic orbits and the radiative characteristics of the accretion disk. We begin by examining time-like periodic geodesic orbits for various configurations characterized by different energy and angular momentum values, represented by the integers $(z, w, v)$. Furthermore, we explore the effects of the King DM halo on time-like periodic geodesics, marginally bound orbits, and innermost stable circular orbits, thereby providing a deeper understanding of how the DM halo environment modifies the behavior of these stable orbits and timelike particle geodesics. Finally, we analyze the null geodesics and the accretion disk properties by studying their direct and secondary images, redshift distributions, and radiation fluxes as observed at infinity for a range of inclination angles. This approach allows us to gain valuable insights into the spacetime geometry of a Schwarzschild-like BH within the King-type DM halo, its physical and radiative properties in the accretion disk, and the corresponding observational implications.

Figures (16)

• Figure 1: Left: MBO radius as a function of the DM halo density $\rho_s$ for various scale radii $r_s$. Right: Variation of the MBO angular momentum with the DM halo density $\rho_s$ for different $r_s$ values.
• Figure 2: The ISCO parameters $R_{\mathrm{ISCO}}$, $L_{\mathrm{ISCO}}$, and $E_{\mathrm{ISCO}}$ as functions of the DM halo density $\rho_s$ for different values of the scale radius $r_s$.
• Figure 3: Top: The rational number $q$ as a function of the energy $E$ for orbits around a Schwarzschild BH surrounded by a King DM halo, shown for different values of the parameters $r_s$ (top-left panel) and $\rho_s$ (top-right panel). The orbital angular momentum is fixed at $L = \tfrac{1}{2}(L_{\mathrm{MBO}} + L_{\mathrm{ISCO}})$. Bottom: The rational number $q$ as a function of the orbital angular momentum $L$ for orbits around a Schwarzschild BH in a King DM halo, plotted for different values of $r_s$ and $\rho_s$. The energy is fixed at $E = 0.96$.
• Figure 4: Comparison of periodic orbits with parameters $(1, 1, 0)$ for the Schwarzschild BH with and without a King DM halo.
• Figure 5: Periodic orbits corresponding to various $(z, w, v)$ values around a Schwarzschild BH in a King DM halo. Parameters: $\rho_s = 0.1$, $r_s = 0.3$, and $L = \tfrac{1}{2}(L_{\mathrm{MBO}} + L_{\mathrm{ISCO}})$.