Massive particles and ISCO with anisotropic model in black hole spacetimes

We study neutral massive test particles in a four-dimensional, static black-hole spacetime sourced by an anisotropic fluid with p r = − ρ / ( 2 q − 1 ) and p t = − p r , where q is a positive integer. Starting from the metric, we derive the radial effective potential, the energy and angular momentum of circular timelike geodesics, and a dimensionless master equation for marginal stability. Explicit horizon and far-field expansions establish the relevant endpoint signs. We prove that the physically admissible circular-orbit branch has exactly one marginally stable radius, separating unstable from stable circular motion. The master equation is written explicitly for q = 1 , 2 , 3 , and its Schwarzschild and fixed-radius large- q limits are obtained. Numerical solutions illustrate the effective potential and characteristic radii. Radii are expressed in units of the horizon parameter φ, and the ADM mass is M = φ ( 1 + 2 p ) , where p measures the fluid contribution to the metric.