Precessing and periodic orbits around hairy black holes in Horndeski's Theory

We investigate the dynamics of neutral timelike particles around a hairy black hole in Horndeski's theory, which is characterized by a coupling parameter with the dimension of length. With deriving the particles' relativistic periastron precessions, a preliminary bound on the hairy black hole is obtained by using the result of the S2 star's precession with GRAVITY. It is tighter than the previous result constrained by the shadow size from EHT observations of M87* by about 3-4 orders of magnitude. We also analyse the particles' periodic motions around the hole in the strong gravitational field. It clearly shows that small variations in the coupling parameter can make the neutral particles' motions back and forth from the quasi-periodic orbits to the periodic orbits or no bound orbit. Our present work might provide hints for distinguishing the hairy black hole in Horndeski's theory from the classical hole by using the particles' dynamics in the strong gravitational field.