Spinning particle dynamics and orbital structure around hairy black holes in Horndeski gravity

We analyze equatorial spinning-particle motion around a static hairy black hole in Horndeski gravity using the Mathisson–Papapetrou–Dixon equations with the Tulczyjew spin condition. The dynamics depends on two parameters: the scalar hair q / M and the spin of the particles s . The aligned spin ( s > 0 ) raises the effective potential barrier and shifts orbits outward. Meanwhile, antialigned spin ( s < 0 ) lowers the barrier, allowing deeper penetration. Increasing q / M tightens the timelike bound, reducing the maximum spin allowed from s max ≃ 2.38 (at q / M ≃ − 1.0 ) to s max ≃ 1.14 (at q / M ≃ 0.7 ). The ISCO radius increases in a nearly linear manner with spin, for example, for q / M = 0.0 , r ISCO grows from ∼ 6.2M to ∼ 6.9M as s goes from − 0.4 to