The Secular Periodic Evolution of X-ray Quasi-periodic Eruptions Driven by Star-disc Collisions

We study the secular periodic evolution of quasi-periodic eruptions (QPEs) for GSN069 and eRO-QPE2 assuming that they are driven by star-disc collisions. We set up numerical simulations and compared them with the observed periodic decay of $\sim -3160\pm720$ s yr$^{-1}$ in GSN069 and $\sim -370\pm40$ s yr$^{-1}$ in eRO-QPE2. We find that: (1) Stellar mass black holes are unlikely the orbiters in these two sources, as their periodic decay are on the order of $<10$ s yr$^{-1}$; (2) A naked degenerate core (including white dwarf) is unlikely the orbiter in GSN069, as the decay is on the order of $<200$ s yr$^{-1}$. However, it is possible in eRO-QPE2, although the required surface density of the accretion disc is relatively high (e.g., $\Sigma\gtrsim10^7\sim 10^8$ g cm$^{-2}$); (3) Both the orbiters in GSN069 and eRO-QPE2 can be solar-like main-sequence stars (MSs). However, each collision can lead to gradual ablation of the stellar envelope in the order of $10^{-5}\sim 10^{-3}M_\odot$. To reproduce the observed decay while surviving for $\gtrsim 3$ yr, the surface density of the disc needs to be within a certain range. For example, given a $1M_\odot$ MS orbiter the surface density of the disc gas should be in the range of $3\times10^5\sim 2\times10^6$g cm$^{-2}$ for GSN069 or $5\times10^4\sim 10^6$ g cm$^{-2}$ for eRO-QPE2. In both of these two sources, the MS can not survive for more than $\sim 12$ yr. We expect that future observations of these two sources can help to distinguish whether the orbiters are degenerated compact objects or gaseous stars.