Secular evolution of quasi-periodic eruptions

Quasi-periodic eruptions (QPEs) are intense repeating soft X-ray bursts with recurrence times about a few hours to a few weeks from galactic nuclei. More and more analyses show that QPEs are the result of collisions between a stellar mass object (SMO, a stellar mass black hole or a main sequence star) and an accretion disk around a supermassive black hole (SMBH) in galactic nuclei. In this work, we propose an osculating trajectory method for efficiently calculating secular evolution of extreme mass ratio inspirals (EMRIs) that are perturbed by an accretion disk. This method accelerates the calculation of EMRI orbital evolution by orders of magnitude and lays the foundation for analyzing long-term QPE observations. Applying this method to orbital analyses of GSN 069 and eRO-QPE2, the two most stable QPE sources, we find informative constraints on the SMBH mass, the radiation efficiency of QPEs, the SMO nature, the accretion disk surface density and the accretion disk viscosity. Combining all the QPE sources available, we find the QPE EMRIs can be divided into two populations according to their orbital eccentricities, where the orbital periods and the SMBH masses in the low-eccentricity population follow a scaling relation $T_{\rm obt}\propto M_{\bullet}^n$ with $n\approx 0.8$.