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 study the secular evolution of extreme mass-ratio inspirals (EMRIs), where a stellar-mass object (SMO)—such as a star or a stellar-mass black hole (sBH)—collides with the accretion disk surrounding a central supermassive black hole (SMBH). Using a perturbation method applied to EMRI geodesics, we find the following: (1) the disk tends to align the SMO regardless of its initial inclination $\iota$ relative to the disk, (2) the final orbital eccentricity of an SMO captured by the disk is low, although the eccentricity can temporarily increase when the initial inclination $\iota$ is large and the SMO is an sBH, (3) through collisions with the accretion disk alone, only a small fraction of sBHs that initially lie close to both the SMBH and the disk can be captured within the typical disk lifetime of active galactic nuclei. Two-body scatterings among SMOs in the nuclear stellar cluster play an essential role, randomly kicking sBHs toward the disk and significantly boosting the capture rate.

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• General relativity
• Astronomical black holes