X-Ray Quasi-periodic Eruptions Driven by Star–Disk Collisions: Application to GSN069 and Probing the Spin of Massive Black Holes

X-ray quasi-periodic eruptions (QPEs) were discovered recently in active galaxies with an unknown driving mechanism. Under the assumption that QPEs are caused by star–disk collisions, we adopt full relativistic method and show that both the orbital parameters of the star and also the mass and spinning of the massive black hole (MBH) can be revealed by using the time of arrival (TOA) of the QPEs. By applying the model to the observed QPEs of GSN069, we find that the star is in a near-circular orbit (${e}_{\bullet }={0.05}_{-0.02}^{+0.02}$) with a semimajor axis of $\sim {365}_{-49}^{+54}{r}_{{\rm{g}}}$ around an MBH with ${M}_{\bullet }={3.0}_{-0.6}^{+0.9}\times {10}^{5}\,{M}_{\odot }$. The alternative short and long recurring time of the QPEs of GSN069 can be well explained by the small eccentricity and the orbital precession of the star. We find that the QPEs of GSN069 are possibly driven by a striped stellar core colliding with accretion disk after a partial tidal disruption event around the MBH. For GSN069-like galaxies, if continuous X-ray monitoring of QPE events can be accumulated with uncertainties of TOA ≲100–150 s, the spin of the MBH can be constrained by fitting to QPEs. Our results show that the timing of QPEs can provide a unique probe for measuring the spinning of the MBH and tests of the no-hair theorem.