Motion of spinning particles around black hole in a dark matter halo

TL;DR

The paper addresses how a Hernquist-type dark matter halo around a Schwarzschild black hole alters the motion of spinning test particles. It employs the Mathisson-Papapetrou-Dixon (MPD) equations with the Tulczyjew spin-supplementary condition to derive the effective potential, four-momentum, and four-velocity for equatorial spin-aligned/anti-aligned orbits, and analyzes MBO, ISCO, and periodic orbits. The results show that the halo lowers the effective potential and shifts MBO and ISCO inward, with the magnitude of these shifts increasing with halo compactness $\mathcal{C}$ and particle spin $s$; periodic orbits also exhibit halo-induced modifications, suggesting potential observational signatures in accretion disks and gravitational-wave contexts. These findings enhance understanding of black holes in dark matter halos and offer a pathway to constrain halo properties through strong-field dynamics and future observations.

Abstract

The motion of a rapidly rotating object in curved spacetime is affected by the spin-curvature force, an effect captured in the motion of spinning test particles. Recently, Cardoso et al.~[Phys. Rev. D 105, L061501 (2022)] found an exact solution describing a black hole immersed in a Hernquist distribution of dark matter. In this work, we investigate the motion of spinning particles around this black hole. We use the Mathison-Papapetrou-Dixon equation and the Tulczyjew spin-supplementary condition to calculate the effective potential, four-momentum, and four-velocity of the spinning particle. The equatorial motion of spinning test particles and the properties of the marginally bound orbits, innermost stable circular orbits, and periodic orbits are further studied. We find that the existence of dark matter halos can significantly change the orbital eccentricity, energy, and the marginally bound orbits, innermost stable circular orbits, and periodic orbits parameters of spinning test particles. Compared to the Schwarzschild black hole, dark matter halos bring the marginally bound orbit and innermost stable circular orbit of a spinning test particle closer to the event horizon. These results could help us understand the properties of black holes in dark matter halos.

Figures (11)

• Figure 1: The shapes of the effective potential for different parameters.
• Figure 2: Plots of the effective potential $V_{\text{eff}}$ and the corresponding orbits for a spinning test particle. The red dashed lines in the effective potentials represent that the energy of particles are $\bar{e}=0.947$. The red dots in the orbits indicate the initial position of the particles.
• Figure 3: Plots of the radial velocity $\dot{r}$ and angular velocity $\dot{\phi}$ for different compactness $\mathcal{C}$.
• Figure 4: Properties of the circular orbits for the spinning test particle in the $(s-l)$ parameter space. The range of $\bar{s}$ and $\bar{l}$ is $(-8, 8)$. In the gray region, the test particle can have a stable timelike circular orbit. In the yellow region, the test particle has only the unphysical spacelike circular orbits. In the blue region, the particle does not have circular orbits.
• Figure 5: Plots of the radius $r_{\text{MBO}}$ and orbital angular momentum $l_{\text{MBO}}$ of the spinning test particle for the different parameters $\mathcal{C}$ and $s$. The light yellow areas indicate that the particle orbits are spacelike.