Captured are circularized: A relativistic treatment of extreme mass ratio inspirals crossing accretion disks

A small body orbiting around an accreting massive object and periodically crossing its accretion disk is a common configuration in astrophysics. In this work, we investigate the secular evolution of extreme mass-ratio inspirals (EMRIs), in which a stellar-mass object (SMO), e.g., a star or a stellar-mass black hole (sBH), collides with the accretion disk of a central supermassive black hole (SMBH), within a fully relativistic framework. We find (1) the disk always tends to align the SMO no matter what the initial orbital inclination $\iota$ relative to the disk is, (2) the final orbital eccentricity of the SMO captured by the disk is always low though the orbital eccentricity may temporarily grow when the orbital inclination $\iota$ is large and the SMO is an sBH, and (3) via collisions with the accretion disk only, only a small fraction of sBHs that are initially close to the SMBH and close to the disk can be captured by the disk within typical disk lifetime of active galactic nuclei. Two-body scatterings between SMOs in the nuclear stellar cluster play an essential role in randomly kicking sBHs towards the disk and boosting the capture rate.