Orbital dynamics and frequency spectra around a black hole with scalar hair

This study examines the behavior of test particles within the spacetime of a non-rotating Black Hole (BH) possessing fundamental scalar hair. The geometry expands the standard Schwarzschild solution within scalar-tensor frameworks, defined by a charge [Formula: see text] and coupling hair constants [Formula: see text] and [Formula: see text], which alter the gravitational potential and spacetime configuration. Through the integration of analytical techniques and numerical simulations, we obtain precise formulations for conserved quantities, including energy and angular momentum, and examine the impact of scalar hair on the effective potential, the stability of circular orbits, and the positioning of the Innermost Stable Circular Orbits (ISCOs). Our results show that increasing the scalar charge or coupling strengths deepens the effective potential, reduces the ISCOs radii, and lowers the angular momentum required for stable orbits. We analyze the effective force and demonstrate its increased attractive nature under the influence of a scalar field. Through geodesic integration, we simulate the exact particle trajectories, revealing broader and more distinct orbits compared to the Schwarzschild BH model. In this case, we also study small harmonic oscillations around stable orbits, deriving radial, vertical, and orbital frequencies for both local and distant observers. In addition, the frequency of periastron precession decreases with increasing scalar hair parameters, suggesting potential observational signatures in systems with Quasi-Periodic Oscillations (QPOs).