Effects of Gravitational Wave Radiation of Eccentric Neutron Star-White Dwarf Binaries on the Periodic Activity of Fast Radio Burst Sources

We revisit the eccentric neutron star (NS)-white dwarf (WD) binary model for the periodic activity of fast radio burst (FRB) sources, by including the effects of gravitational wave (GW) radiation. In this model, the WD fills its Roche lobe at the periastron and mass transfer occurs from the WD to the NS. The accreted materials can be fragmented and arrive at the NS episodically, resulting in multiple bursts through curvature radiation. Consequently, the WD may be kicked away owing to the conservation of angular momentum. To initiate the next mass transfer, the WD has to refill its Roche lobe through GW radiation. In this scenario, whether the periodic activity can show up relies on three timescales, i.e., the orbital period $P_{\rm orb}$, the timescale $T_{\rm GW}$ for the Roche lobe to be refilled, and the time span $T_{\rm frag}$ for all the episodic events corresponding to each mass transfer process. Only when the two conditions $T_{\rm GW} \gtrsim P_{\rm orb}$ and $T_{\rm frag} < P_{\rm orb}$ are both satisfied, the periodic activity will manifest itself and the period should be equal to $P_{\rm orb}$. In this spirit, the periodic activity is more likely to show up for relatively long periods ($P_{\rm orb} \gtrsim$ several days). Thus, it is reasonable that FRBs 180916 and 121102, the only two sources having been claimed to manifest periodic activity, both correspond to relatively long periods.