Spherical Orbital Dynamics and Relativistic Precession in Kerr-MOG Spacetime

We study the dynamics and relativistic precessions of massive particles on spherical orbits around Kerr-MOG black holes in scalar-tensor-vector gravity (STVG). By employing the Hamilton-Jacobi formalism, we derive conserved quantities and analyze how the MOG parameter $α$ and orbital tilt angle $ζ$ influence the innermost stable spherical orbits (ISSOs) and orbital stability. We compute the nodal and periastron precession frequencies, finding that nodal precession increases monotonically with both black hole spin and MOG parameter, while periastron precession exhibits a more complex behavior: MOG amplifies curvature-induced effects, which can be partially counteracted by spin. Furthermore, to complement the orbital analysis, we examine the Lense-Thirring spin precession of a gyroscope and demonstrate its sensitivity to the MOG parameter, spin, and orbital tilt angle. These results reveal distinctive signatures of modified gravity in orbital dynamics and provide a potential observational probe to test deviations from general relativity near rotating black holes.