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.
