Repeated partial disruptions in a WD-NS or WD-BH merger modulate the prompt emission of long-duration merger-type GRBs

The progenitors of gamma-ray bursts (GRBs) have long been an unresolved issue. GRB 230307A stands out as an exceptionally bright event, belonging to the long-duration GRBs but also exhibiting a late emission component reminiscent of a kilonova. Together with the similar events GRBs 060614 and 211211A, they make up a new sub-group of GRBs with intriguing progenitors. If such long-duration merger-type GRBs originated from the coalescence of a white dwarf (WD) with a neutron star (NS) or a black hole (BH), as proposed in the recent literature, then the larger tidal disruption radius of the WD, together with a non-negligible residual orbital eccentricity, would make repeated partial tidal disruptions inevitable. This may modulate the mass accretion and jet launching process at the NS or BH, resulting in a quasi-periodic modulation (QPM) in the light curve of the GRB, on the orbital period. The detection of potential QPMs during the early episode of prompt emission of these three GRBs supports this scenario, and the relatively slow QPM ($>$ 1 s) suggests that the lighter object can not be a NS. We propose that the progenitor system of GRBs 230307A, 060614, and 211211A consist of a WD of mass 1.3 $M_\odot$, 0.9 $M_\odot$ and 1.4 $M_\odot$, respectively, and a NS (or BH). After several cycles of modulations, the WD is completely destructed, and the accretion of the remaining debris dominates the extended emission episode.