Three body first post-Newtonian effects on the secular dynamics of a compact binary near a spinning supermassive black hole

Abstract

The binary black holes (BBHs) formed near the supermassive black holes (SMBHs) in the galactic nuclei would undergo eccentricity excitation due to the gravitational perturbations from the SMBH and therefore merger more efficiently. In this paper, we study the coupling of the three body 1st post-Newtonian (PN) effects with the spin effects from the SMBH in the hierarchical triple system. We extend previous work by including the coupling between the de Sitter precession and the Lense-Thirring precession from the SMBH spin. This coupling includes both the precessions of the inner orbit angular momentum and the Runge-Lenz vector around the outer orbit angular momentum in a general reference frame. We find the change of the (maximal) eccentricity in the neighboring Kozai-Lidov cycles due to spin effects is detectable by LISA in the future. Our general argument on the coupling of the three body 1PN effects in three body systems could be extended to any other situation as long as the outer orbital plane evolves.