X-ray Quasi-periodic Eruptions driven by Star-Disc Collisions : Application to GSN069 and Probing the Spin of Massive Black Holes

X-ray quasi-periodic eruptions (QPEs) are discovered recently in active galaxies with unknown driven mechanism. Under the assumption that QPEs are caused by star-disc 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 semimajor axis of $\sim 365^{+54}_{-49}r_{\rm g}$ around a MBH with $M_\bullet=3.0^{+0.9}_{-0.6} \times10^5M_\odot$. 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 disc after 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 $\lesssim 100-150$s, the spin of massive black hole 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 MBH and tests of no-hair theorem.