Implications of King Dark Matter Distribution on Oscillatory Dynamics around Schwarzschild Black Holes

This study examines the motion of test particles in the spacetime of a non-rotating black hole (BH) model srounded by King dark matter halos. In addition, the geometry is described by a modified static and spherically symmetric metric, where the lapse function depends on the mass of the source, the radius of the scale R, and the central density ρ of the system. In this case, the motion of the particles is studied using the effective potential method, which helps to find the conditions for stable and unstable circular orbits, as well as the location of the innermost stable circular orbits (ISCOs). Also, the effective force is obtained to show how dark matter effects change the usual gravitational attraction near the BH. In addition, small changes around circular orbits are analyzed to explain harmonic oscillations and the related epicyclic frequencies. In this case, the frequencies measured by both local and distant observers are discussed, and the periastron precession rate is calculated to show how it differs from the result of GR. In this context, these results show that the King dark matter halos changes the orbital stability, oscillation behavior, and precession motion, showing possible observable effects of King dark matter halos gravitational fields.