Exploring effective force, trajectories, QPOs and center-of-mass energy around a black hole surrounded by pseudo-isothermal dark matter halo

Abstract

In this work, we analyze the motion of test particles around a spherically symmetric non-rotating black hole embedded in a pseudo-isothermal dark matter halo, investigating how the black hole mass $M$, central halo density ${\rho }_0$ and core radius $R_0$ influence particle dynamics. We have used the effective potential approach to evaluate the stability of circular orbits, derive analytical formulas for the specific energy and angular momentum of the test particles, and calculate the innermost stable circular orbit as well as the effective force functioning on the particles. To gain a deeper understanding of orbital behavior, we employ numerical integration to simulate and visualize particle trajectories in the vicinity of the black hole. In addition, we explore epicyclic oscillations near the equatorial plane, derive expressions for radial, vertical, and orbital frequencies, which give some information about motion as perceived by both local and distant observers. Moreover, we have tested the precession of the periastron and evaluated the influence of model parameters on this relativistic phenomenon. We have analyzed high-energy particle collisions within this type of black hole, from which we can easily formulate an analytical equation for the center-of-mass energy. In this context, our results demonstrate that the combined effects of the black hole and the surrounding dark matter halo significantly alter the motion of test particles, influencing their stability, orbital evolution, and collision dynamics.