An accreting stellar binary model for active periodic fast radio bursts

In this work, we propose an accreting stellar binary model for understanding the active periodic fast radio bursts (FRBs). The system consists of a stellar compact object (CO) and a donor star (DS) companion in an eccentric orbit, where the DS fills its own Roche lobe near the periastron. The CO accretes the material from the DS and then drive relativistic magnetic blobs. The interaction between the magnetic blobs and the stellar wind of the DS produces a pair of shocks. We find that both of the reverse shock and the forward shock are likely to produce FRBs via synchrotron maser mechanism. We show that this system can in principle sufficiently produce highly active FRBs with a long lifetime, and also can naturally explain the periodicity and the duty cycle of the activity as appeared in FRBs 180916 and 121102. The radio nebula excited by the long-term injection of magnetic blobs into the surrounding environment may account for the associated persistent radio source. In addiction, we discuss the possible multi-wavelength counterparts of FRB 180916 in the context of this model. Finally, we encourage the search for FRBs in the ultraluminous X-ray sources.


INTRODUCTION
Fast radio bursts (FRBs) are intense radio transients with extremely short duration, and their physical origin is a mystery (Cordes & Chatterjee 2019;Petroff et al. 2019;Xiao et al. 2021). Observationally, some FRBs showed repeat bursts, but most bursts did not (Spitler et al. 2016;Fonseca et al. 2020;James et al. 2020). It is also a mystery whether all of FRBs have the same origin (Palaniswamy et al. 2018;Caleb et al. 2018Caleb et al. , 2019. Chime/Frb Collaboration et al. (2020) found a 16.35day periodicity in FRB 180916 with a 4.0-day phase window, implying a duty cycle D ≃ 0.24 of activity. In addiction, Rajwade et al. (2020) reported a possible 157-day periodicity in the FRB 121102 with a duty cycle D ≃ 0.56. These are very important clues for studying the physical origin of the FRBs. For such a periodicity, it may be explained, either by the orbital period of the binary sysytem Lyutikov et al. 2020;Ioka & Zhang 2020;Gu et al. 2020;Mottez et al. 2020;Dai & Zhong 2020), or precession of the emitter (Yang & Zou 2020;Levin et al. 2020;Zanazzi & Lai 2020;Tong et al. 2020;Katz 2020).
Recently, CHIME/FRB Collaboration et al. (2020) and Bochenek et al. (2020) reported a single FRB (FRB 200428) with two pulses in association with an active Galactic magnetar SGR 1935+2154, which is located at a distance ∼ 9 kpc (Zhong et al. 2020). Moreover, it is found that an X-ray burst is almost simultaneous with the FRB Mereghetti et al. 2020;Ridnaia et al. 2020;Tavani et al. 2020). But strangely, no more FRBs were observed during the active phase of this magnetar . In any case, this discovery shows that at least a part of the FRBs are produced by the magnetars (Lyubarsky 2014;Beloborodov 2017;Metzger et al. 2019). However, an important lesson learned from FRB 200428 is that extragalactic analogues of Galactic magnetars could explain some of the FRB population, but much more active sources are still required to be invoked to explain the highly active periodic repeaters like FRBs 121102 and 180916 (CHIME/FRB Collaboration et al. 2020). If the active repeaters are also powered by magnetars, they must be produced by a type of rare and active magnetars not seen in the Milky Way (Mereghetti et al. 2020;Lu et al. 2020).
In addition, the observations on the environment of FRB 180916 found that it is 250 pc away from the nearest young stellar clump and suggested that its age may be ∼ Myr, which in line with the hypothesis that the high-mass X-ray binaries or gammaray binaries are the progenitors rather than the young magnetars (Tendulkar et al. 2020). Moreover, Pastor-Marazuela et al. (2020) rule out the scenario in which companion winds cause FRB periodicity by using simultaneous Apertif and LOFAR data. Motivated by those observation results mentioned above, we propose an alternative model for understanding the highly active repeating FRBs with periodicity in this work. The model consists of a stellar compact object (CO) and a donor star (DS) with its filled Roche lobe, in which the CO can be a neutron star (NS) or a black hole (BH). We will show that this model can explain the actively repeating FRBs themselves and their periodicity behaviors, as the cases FRBs 180916 and 121102. Sridhar et al. (2021) also link the periodic FRBs to the accreting binaries. In Sridhar et al. (2021), the FRBs are produced by the interaction of the intermittent jets (high luminosity)with the quiescent jet (low luminosity). However, in this work, we discuss the process by which magnetic blobs interact with the donor star's wind to produce FRBs. This is the key difference between this work and theirs.

THE MODEL
In this section, we propose a binary model for active repeating FRBs. The binary system consists of a stellar CO and a DS. The CO may be a BH or a NS. As illustrated in Figure 1, when the DS fills its Roche lobe, significant mass transfer will occur from the DS to the CO. Then an accretion disc will form around the CO. As we know, jets are widely present in the accretion systems. However, in addition to the usual continuous jets, the accretion process may also produce episodic jets. The energetic, collimating episodic jets had been observed in active galactic nuclei, stellar binaries, and protostars. For instance, in the X-ray binaries, episodic jets are usually observed during their X-ray outbursts and intense radio fares (Fender et al. 2004;Zhang & Yu 2015). Practical models and numerical simulation suggested that the episodic jets may be driven by magnetic instability in the accretion disc (Yuan et al. 2009;Yuan & Zhang 2012;Zhao et al. 2020). Because of shear and turbulent motion of the accretion flow, a flux rope system is expected to form near the disc. The energy is accumulated and stored in the system until a threshold is reached, then the system loses its equilibrium and the energy will be released in a catastrophic way, i.e. eject-ing episodic magnetic blobs. By assuming the accretion flow is advection dominated, the available isotropic free magnetic energy of one blob is (Yuan & Zhang 2012) (1) Here we consider the magnetic blobs is ejected in a collimated angle, and f b = (1 − cos θ)/2 is the beaming factor, θ is the jet opening angle 1 . Here, we consider a beaming factor f b ∼ 10 −3 . α is the viscous parameter, β is the ratio of the magnetic pressure over the total pressure in the accretion disc. We adopt α = 0.01 and β = 0.1 as typical values.Ṁ a is the mass accretion rate, M is the mass of the CO,r is the radius, where the magnetic blobs is formed, in units of 2GM/c 2 . Here and hereafter, we employ the short-hand notation q x = q/10 x in cgs units.
On the other hand, the binary system is surrounded by the stellar wind from the DS. Therefore, we suggest that interaction between the magnetic blobs and the stellar wind could lead to shock formation, and then this shock process would produced powerful coherent radiation (FRBs) through the synchrotron maser mechanism analogy to the flaring magnetar models (Lyubarsky 2014;Beloborodov 2017;Metzger et al. 2019). Taking FRB 180916 as a template, the typical energy of the bursts is ∼ 10 37 − 10 38 erg (CHIME/FRB Collaboration et al. 2020). In fact, it turns out that the typical energy of FRB 121102 and of FRB 180916 are similar, found by recent observations (Pastor-Marazuela et al. 2020;. Also one will see from Eq.(17), E f = 10 42 erg is reasonable to power FRB 180916 and 121102. According to equation (1), a mass accretion rate ∼ 10 −5 M ⊙ /yr is required. Note that this high accretion rateṀ a ∼ 10 −5 M ⊙ /yr can occur in high mass Roche lobe filling binaries, such as SS433 which is accreting at a rate of ∼ 10 −4 M ⊙ /yr (Fabrika 2004). Moreover, Wiktorowicz et al. (2015) showed that the Roche lobe overflow rate can be up to 10 −3 M ⊙ /yr in if the DS is evolving in the Hertzsprung gap.
The wind emerged from the DS is filled around the CO, and its density distribution n w can be estimated as  Figure 1. Schematic illustration of the model in this work: (a) the DS fills its Roche lobe when orbiting near the periastron, and mass transfer to the CO occurs; (b) The transfer of mass leads to the formation of accretion disc around the CO. Magnetic blobs are ejected from the accretion disc due to magnetic instability. The magnetic blobs may accelerate to the Lorentz factor Γ ∼ σ0 (Yuan & Zhang 2012), where σ0 is the initial magnetization parameter of the blobs. Then the blobs interact with the stellar wind, which is driven by the DS and immerses itself around the binary system, to induce shocks powering FRBs by the synchrotron maser mechanism; (c) The interactions of the magnetic blobs and the stellar wind produce FRBs. Long term energy injection into the surrounding medium procduce a nebulae which powers the persistent raido source associated with the FRB.
whereṀ w is the wind mass loss rate of the DS, v w is the speed of the wind. a a is the semimajor axis of the binary, r is the distance from the CO. For r ≪ a, we have whereṁ w =Ṁ w /(M ⊙ /yr), β w = v w /c, c is the speed of light. We adopt β w = 0.01 as typical values for massive stars, and adoptṁ w = 10 −11 for self-consistent of the working model. a = 10 13 cm is motivated by the analysis for FRB 180916 in section 3. Therefore, according to Sari & Piran (1995), the Lorentz factor of the blastwave during the early reverse shock crossing phase is Γ = (n ej Γ 2 ej /4n w ) 1/4 , where n ej ≃ E f /(4πr 2 m p c 3 δtΓ 2 ej ) is the co-moving density in the ejecta at a radius r, Γ ej is the initial Lorentz factor of the ejecta, δt is the duration of the central energy activity. Then we have  (5) is the deceleration radius. One sees that it satisfies r dec ≪ a. Therefore, the Lorentz factor at the deceleration radius is On the other hand, we get the co-moving density in the ejecta at r dec (7) Combining Eq.(3) with Eq. (7), one gets that the reverse shock will become relativistic, which is the case considered in this work, if according to the condition Γ 2 ej > f ≡ n ej /n w (Sari & Piran 1995). In the context of this model, the reverse shock is ultrarelativistic only for Γ ej ≫ 10 2 , and is mildly relativistic for Γ ej ∼ 200 which is the easier case to achieve. Can accreting stellar-mass compact objects produce ejecta with Γ ej > 100? As argued in Sridhar et al. (2021), it cannot be excluded that highly super-Eddington systems are capable for brief periods of generating outflows with such a large Γ ej . Especially for ejecta with a large initial magnetization σ 0 ≫ 1, which is the case in this work, it can be accelerated to a high Lorentz factor Γ ej ∼ σ 3/2 0 (Drenkhahn 2002). Therefore, assuming the reverse shock is mildly relativistic, then Eq.(7) is reduced to (9) How, if the reverse shock is ultrarelativistic, this is the upper limit for n ej .
In previous studies that discussed the production of FRBs by magnetized shocks, both forward and reverse shocks have been involved (Lyubarsky 2014;Waxman 2017;Beloborodov 2017;Metzger et al. 2019;Beloborodov 2020). In this work, we consider the synchrotron maser emission produced by both reverse shock and forward shock. And the equipartition of energy in the reverse and forward shock is assumed in this work for simplicity. The theory of the synchrotron maser instability has been developed both for isotropic (Sagiv & Waxman 2002;Gruzinov & Waxman 2019) and ring-like (Lyubarsky 2006) particle distributions in momentum space, also see a review Lyubarsky (2021). In our scenario, we consider a weakly magnetized ejecta at the deceleration radius and a stellar wind which is itself also weakly magnetized (Weber & Davis 1967;Sakurai 1985;Ignace et al. 1998;ud-Doula & Owocki 2002;Puls et al. 2008;Harvey-Smith et al. 2010) 2 . For the weakly magnetized plasma i.e. σ ≪ 1, the characteristic frequency measured in the rest frame of the synchrotron maser may be given by (Sagiv & Waxman 2002;Waxman 2017;Gruzinov & Waxman 2019; Lyubarsky 2021) where σ is the magnetization parameter of the plasma with magnetic field of B and number density of n, The magnetization σ is a very uncertain and difficult parameter to know, so it can be regarded as a free parameter in principle in this work. As an example, in the case of gamma-ray bursts, after the highly magnetized jets accelerated through magnetic dissipation process, the remaining magnetization σ ej may be small ∼ 0.1 which can naturally account for the observed high radiative efficiency of most gamma-ray bursts (Zhang & Yan 2011). The magnetization σ w of the DS's wind is more difficult to determine, but perhaps we can make a very rough estimation. Suppose the DS have a surface magnetic field strength B b ∼ 10 2 G, a radius R b ∼ 10R ⊙ , and a mass loss rateṁ w ∼ 10 −10 (the range ofṁ w covered in this work is 10 −9 − 10 −11 ). One gets the magnetization of the wind, at r dec , σ w ∼ 10 −3 (Lamers & Cassinelli 1999;Harvey-Smith et al. 2010). In addition, it can be seen from Eq.(10) that the result is only slightly dependent on σ, so it is not unreasonable to directly use a rough values of σ for orders of estimation within the range of parameters considered in this model. Therefore, we adopt σ ej = 0.1 for the ejector at r dec and σ w = 10 −3 for the DS's wind, respectively. In the context of this work, the material shocked by the reverse shock and forward shock are both baryon-dominated plasma. Therefore, the plasma frequency ν p is determined by (Sagiv & Waxman 2002) where γ e and γ p are the Lorentz factor of the electrons and the protons, respectively. Assuming that the electrons are near equilibrium with the protons with γ e m e ∼ γ p m p in the downstream, we get the characteristic frequency of the synchrotron maser produced by the reverse shock and the forward shock, respectively, measured in the observed frame, GHz, for reverse shock GHz, for forward shock .
One sees that the maser emission is determined by proton, in contrast to the case for the pair plasma.
2 In this regard, the discussions in detail of the magnetization and magnetic field pattern of the ejecta and the stellar winds from the DS and their contribution to the rotation measure will be appear in a forthcoming work.
The optical depth due to free-free absorption is estimated as where k B is the Boltzmann's constant,ḡ ff is the mean Gaunt factor, T w = 10 4 K is the tempera-ture of the wind of the DS. For hν/k The optical depth to Thomson scattering, τ T ∼ σ T r dec n w (r dec ) ∼ 10 −9 , is very small. However, due to the extremely high brightness temperature of FRBs, induced scattering process becomes important (Lyubarsky 2008;Lyubarsky & Ostrovska 2016). Because the electrons in downstream of the shocks are ultrarelativistic, the induce Compton scattering caused by them can be negligible (Wilson & Rees 1978). Therefore, the optical depth due to induced Compton scattering is mainly contributed by the DS' wind, which is given by It can be seen that the the emission at ν pk cannot be transmitted freely due to the induced Compton scattering. Here, E FRB ∼ 10 39 ǫ −3 E f,42 erg is applied, ǫ is the efficiency of the synchrotron maser around ν = ν pk . The radiation efficiency ǫ of synchron maser is highly uncertain. The PIC simulations show that the efficiency ǫ depends on the magnetization and temperature for pair plasma. Plotnikov & Sironi (2019) found that ǫ ∼ 10 −2 for an upstream magnetization σ ∼ 0.1, and it decreases for σ > 1. When the upstream plasma is non-relativistic, the ǫ would be independent of the temperature of the plasma for σ > 1 (Babul & Sironi 2020), but it remains unclear for σ ≪ 1. However, for proton plasma which is the case considered in this work, the efficiency ǫ is even less known and has yet to be investigated in detail (Lyubarsky 2021). From the point of view of the model's consistency with observations, the efficiency should not be too low to account for the FRBs' energetics, so we assume the efficiency to be ǫ ∼ 10 −3 in this work. According to equation (15), we have τ IC (ν) = τ IC (ν pk )(ν/ν pk ) −(3+s) if we assume the spectrum as E FRB ∝ ν −s . As a result, the observed peak frequency of the radiated spectrum moves to ν m where τ IC (ν m ) = 3, GHz, for forward shock (16) One sees that ν m can account for the emission of FRBs around GHz . In addiction, during the reverse shock crossing phase t < δt, this peak frequency will evolve with the shock decelerates as ν m ∝ t −(2+3s)/4(3+s) for the reverse shock, and decelerates as ν m ∝ t −(4+3s)/8(3+s) for the forward shock. This temporally decreasing peak frequency may explain the observed downward drifting frequency structure in the sub-pulses of some repeating GHz for the forward shock, by adopting s = 1.5 (Macquart et al. 2019). This is consistent with the fact that FRB 121102 has few detection at sub-GHz band (Houben et al. 2019; Josephy et al. 2019), which may be because the DS in the case of FRB 121102 has a stronger stellar wind oḟ m w ∼ 10 −9 than the case of FRB 180916 ofṁ w ∼ 10 −11 .
By definition, the radiation energy around ν pk is E FRB (ν pk ) ≡ ǫE f , then the radiation energy around ν m is given by E FRB (ν m ) = (ν m /ν pk ) −s E FRB (ν pk ), namely, Since the maser energy is mainly concentrated around ν pk in fact, the efficiency near the observed frequency may be much lower than 10 −3 , which can be clearly seen from the above equation. That is, the efficiency near the observed frequency is equal to the ǫ multiplied by a factor of less than unity, such as 5 −s/(3+s) and 10 −7s/(3+s) for reverse shock and forward shock, respectively. Then a low radiation efficiency ∼ 10 −5 (≪ 10 −3 ) of the radio bursts at the observed frequency band found in the observations of the galactic burst FRB 200428 (Margalit et al. 2020) can be understood. Moreover, one sees that the observed energy of the synchrotron maser from the reverse shock is significantly greater than that from forward shock. It indicates that the observed energy may exhibit a bimodal distribution in a single repeating FRB. The energetic bursts come from reverse shock, while the less energetic bursts come from forward shock. This may provide an explanation for the bimodal burst energy distribution in FRB 121102 found by FAST recently . A more detailed analysis will be presented elsewhere.
In this model, according to and Eq. (17), the bursts energy may be adjusted mainly by the model parameters such as E f ,ṁ w and a. For different FRB sources, there may be different model parameters, which results in different observed energies. Moreover, FRBs may be generated by both reverse and forward shock and observed by telescopes with different frequencies and thresholds, so it is very easy to generate diverse observed distributions of energy. And in order to compare directly with observations one needs to do detailed modeling, but it is beyond the scope of this work. One thing needs special attention, Eq. (16) and Eq. (17) are only valid when τ IC > 3. And if τ IC ≤ 3, Eq. (16) and Eq. (17) will degenerate to ν m = ν pk and E FRB,39 = 1.
Periodicities has been observed in some X-ray sources, and mechanisms have been proposed to explain this behavior, mainly including orbital period modulation (Strohmayer 2009) and disk/jet precession (Begelman et al. 2006;Foster et al. 2010), also seen in a review Kaaret et al. (2017).
For periodic FRBs, under the framework of our model, the periodicity might be explained by the precession of the jets (Katz 2020(Katz , 2021Sridhar et al. 2021). However, FRB 180916 shows that the signals is concentrated in a narrow active window with a duty cycle D ≃ 0.24, which is very different from the cases in the X-ray binaries (Strohmayer 2009;Foster et al. 2010). As discussed in the previous section, FRB emission is produced when the DS fills its own Roche lobe. Therefore, we introduce an eccentric orbit modulation mechanism to explain the narrow duty cycle. The periodicity may be explained by periodic orbital motion of the binary and the duty cycle may be explained by the eccentricity of the orbit. It can be naturally understood by the picture: the DS fills its Roche lobe when it is near the periastron where the flaring jets from and the FRB emission is then produced, if the orbit of the binary is eccentric. After the DS is away from the periastron, the process of producing FRB emission stops due to significant decrease in accretion rate.
Define the mass ratio q = M/M x , M x is the mass of the CO, M is the mass of the DS. Then the orbital period of the binary is T = 2πq 1/2 (1 + q) −1/2 a 3/2 (GM ) −1/2 where G is the gravitational constant, a is the semimajor axis. The effective radius R L,0 of the Roche-lobe of the DS at the periastron can be estimated as (Eggleton 1983) where e is the orbital eccentricity. With the assumption that the DS is filled with its Roche lobe, then its average density isρ where f RL = R/R L,0 1 is the Roche lobe filling factor, R is the radiu of the DS. We expect that f RL is just slightly greater than 1, .e., f RL − 1 ≪ 1. Combining equations (18)-(20), one gets where f q = q 1/2 (1 + q) −1/2 χ −3/2 . Note that f q ∈ (1.4, 1.8) for q 0.1. For simplicity, we take f q = 1.5 because the parameter range of interest in this work is q 0.1. Thus one can simply deduce the average density of the DS asρ whereρ ⊙ ≃ 1.4 g/cm 3 is the current average density of the Sun. It is worth noting that equation (22) is a generalization of equation (4.10) in Frank et al. (2002), ρ ∝ T −2 , namely generalized from the case of circular orbit to the case of eccentric orbit. Next, we discuss the constraint on e from the observed duty cycle D. Without loss of generality, taking the CO as the reference, DS moves on an elliptic orbit with respect to the CO. The distant from the CO to the DS is r(θ) = a(1 − e 2 )/(1 + e cos θ). The DS is at periastron when θ = 0, where θ is the angle between the vector diameter from the CO to the DS and the polar axis in the polar coordinate system. For the DS at different positions, its Roche lobe radius is determined by R L,θ = χr(θ). Assume that when θ = ±α (α < π), the DS can just fill its Roche lobe, i.e., R L,α = R. Then we have Therefore, according to the Kepler's second law, the duty cycle of activity can be calculated as D = ∆S/S, where ∆S is the area swept out by r(θ) from −α to α, and S is the total area enclosed by the DS's elliptic orbit. That is where, by letting π, λ(1 + e)/(2e) 1 2 arcsin λ(1 + e)/(2e), λ(1 + e)/(2e) < 1 (25) For λ(1 + e)/(2e) 1, it means that the DS fills its Roche lobe throughout the cycle, thus D=1. Therefore, based on the observed duty cycle, we can discuss the constraint to e. Figure 2 shows e as a function of λ for D = 0.24 (FRB 180916) and D = 0.56 (FRB 121102), respectively. Note that λ = 1 − f −1 RL = (R − R L,0 )/R describes the degree of the Roche lobe overfilling of the DS at periastron. One naturally expects λ ≪ 1, otherwise the Roche lobe overflow would be violent. By adopting λ 0.1, we have e 0.22 for FRB 180916 and e 0.08 for FRB 121102, which are also showed in figure 2. It can be seen that the required orbital eccentricity is not large. Now we can discuss what kind of DS is needed, in order to explain the periodicity of FRBs 180916 and 121102. For FRB 180916 T = 16.35 day and e 0.22, one getsρ ∼ ×10 −3.5ρ ⊙ according to equation (22). Similarly, for FRB 121102 T = 157 day and e 0.08, one getsρ ∼ 10 −6ρ ⊙ . It indicates that the DSs might be supergiants. For red supergiants, their average density can be as low as 10 −8ρ ⊙ . Therefore, we expect that this model can explain a period up to T ∼ 10 3 day when the companion is a red supergiant. Note that, however, there is some uncertainty on the phase window of FRB 180916. It is pointed out in Chime/Frb Collaboration et al. (2020) that 50% of the CHIME bursts are detected in a 0.6-day phase window, with the event rate dropping rapidly towards the edges of the active phase, and the duty cycle would be D=0.04 if 0.6 day is the width of the active phase. Moreover, Pastor-Marazuela et al. (2020) found that its activity window is narrower at higher frequencies, namely, the full-width at half-maximum of Apertif bursts is 1.1 day compared to CHIME bursts' 2.7 day. If one adopts D=0.04, e 0.65 and the density of the DS for FRB 180916 would beρ ∼ 10 −2.4ρ ⊙ , which is in line with the massive OB stars.
Moreover, it should be noted that the density obtained according to Eq.(22) is the average density which means that it is not necessarily the true density of DS. For example, Be stars, although they themselves are only ∼ 10R ⊙ , but their accretion disk radius can be as large as ∼ 100R ⊙ (Rivinius et al. 2013). The CO can ac-crete material from the disk of the star, although it does not accrete material from the star itself directly (Karino 2021). In this case, for example, for a Be star with a mass 10M ⊙ and an accretion disk radius ∼ 10 − 200R ⊙ , the effective average density would be 10 −2ρ ⊙ to 10 −6ρ ⊙ . It can be seen that the Be companion is also consistent with the density requirements in this model.
On the other hand, according to the results in section 2, the model requires a relatively weak stellar wind environment, specificallyṀ w ∼ 10 −11 M ⊙ /yr for FRB 180916 andṀ w ∼ 10 −9 M ⊙ /yr for FRB 121102 may be appropriate. It is also strongly constrained by the small DM variations observed in FRB 180916 at the low frequency bands (Pleunis et al. 2021). Therefore, it is unlikely that the DSs are supergiants, as they tend to have much stronger wind, unless they happen to be in periods of cooling and weak wind or if the wind is highly inhomogeneous/clumpy (Puls et al. 2008;Stairs et al. 2001). As mentioned above, a cold OB star companion withṀ w ∼ 10 −11 M ⊙ /yr may also reasonable for FRB 180916. However, the DSs are more likely to be Be stars because their polar wind may be relatively weak (Kervella & Domiciano de Souza 2006;Kanaan et al. 2008), which could provide a unified picture for the cases of FRB 180916 and FRB 121102. But then again, current observations are not enough to tell us exactly what kind of stars the DSs are, and future observations are needed to provide more clues. In any case, this model's requirement for massive stars as DSs is consistent with the fact that FRB 121102 and FRB 180916 are associated with the starforming regions (Chatterjee et al. 2017;Marcote et al. 2020;Tendulkar et al. 2020).
Given the presence of DS's wind, one needs to consider its contribution to the dispersion measure (DM). According to Eq.(3) and Eq.(5), one can roughly estimate the DM associated with the local wind, DM loc ∼ n w (r dec )r dec ∼ 0.03 E 1/4 f,42ṁ 3/4 w,−9 β −3/4 w,−2 δt 1/4 −3 a −3/2 13 pc cm −3 which is small enough to be negligible compared to the total DM of FRBs, even if a relatively large wind rateṁ w ∼ 10 −9 is adopted. Therefore, we do not expect that there is a obvious periodic modulation in the observed DM for FRB 180916B (Pastor-Marazuela et al. 2020), and of course the same is true for FRB 121102.
Interestingly, the DM of FRB 121102 seems to have a slow growth trend with a rare of ∼ 0.85 pc cm −3 yr −1 . Based on the above analysis, it is clear that the evolution of the DS's wind is not sufficient to lead to such an outcome. The DM variation may depend on the environment in which FRB sources are lo-cated. For example, an FRB source in an expanding SNR around a nearly neutral ambient medium during the deceleration phases or in a growing H II region can increase DM (Yang & Zhang 2017). We're going to discuss this issue, in the context of this model, in detail elsewhere.

SUMMARY AND DISCUSSIONS
In this work, we propose a model for understanding the highly active periodic FRBs. The system consists of a stellar CO and a DS, in which the DS fills with its own Roche lobe. The CO accretes material from the DS and ejects relativistic magnetic blobs. The interaction between the magnetic blobs and the stellar wind of the DS produces a pair of shocks, the reverse shock traveling through the ejecta and the forward shock traveling to the wind of the DS. We find that both of these shocks are likely to produce FRBs. The energy of the FRBs from the reverse shock is greater than that from forward shock. It indicates that the observed energy in a single repeating FRB may exhibit a bimodal distribution. This may provide an explanation for the bimodal burst energy distribution in FRB 121102 found by FAST recently .
Moreover, such a Roche lobe filling accretion system can in principle sufficiently powers the highly active periodic FRBs with a long lifetime. The orbital motion of the binary can explain the periodicity of the FRBs such as FRBs 180916 and 121102, if the DSs are giants/supergiants or Be stars. To produce a narrow duty cycle of the activity, such as FRB 180916, the orbit of the binary needs to be moderately eccentric, with the DS fills the Roche lobe only near the periastron. It should be noted that for our model to work, it requires (1) a sufficiently large accretion rate and (2) weak stellar wind from the DS. If not, if the accretion is too weak, the FRB energy will be too low to be observed. If the stellar wind is too strong, GHz radiation cannot pass through freely. And we realize that this is reasonable because FRBs would have been observed in a large number of binary systems in the Milky Way if it weren't for binary systems with the right conditions to produce FRBs. It is these particular low wind binary systems that produce such rare sources like FRB 180916 and FRB 121102. Therefore, if this model is correct, it gives us a great opportunity to study these special binary systems.
Recent observations of FRB 180916 revealed that the bursts activity is frequency dependent, namely its activity window is both narrower and earlier at higher frequencies (Pastor-Marazuela et al. 2020;Pleunis et al. 2021). The causes of this observed phenomenon may be complicated. It may be due to a combination of the absorption of FRBs by the surrounding invironment and the instrument selection effects, as show in . It may also arise from the structured and beaming effects of the jet (Sridhar et al. 2021). But we won't discuss this issue in this work because it requires detailed modeling in the contex of this model, and we plan to study it in detail elsewhere.
The more recent observation found subsecond periodicity in FRB 20191221A, which may indicate that the central engine of this FRB is NS, and the period corresponds to the rotation period of NS (The CHIME/FRB Collaboration et al. 2021). However, the duration of this FRB is actually ∼ 3 seconds, which is different from any known FRBs, and it has not yet been discovered whether it repeats, so it may be an entirely new class of FRBs (The CHIME/FRB Collaboration et al. 2021). Nevertheless, a similar subsecond periodicity is known to exist in some X-ray binaries, which may result from the rotation of the accreting NSs (Kaaret et al. 2017;Patruno & Watts 2021). A similar periodicity is also expected in gamma-ray bursts (GRBs), although no unambiguous periodicity has been found in the GRB pulses (Tarnopolski & Marchenko 2021, and references therein). In this model, the rotation of the accreting NS (or the fluctuations in the accretion disc) may also modulate the accretion process and thus the generation of the jets, and whether it ultimately results in the production of periodicity in FRB pulses deserves further study.
It expects that, in the context of this model, the energy injection by the long-term blobs ejection from the system into the surrounding environment may excite a radio nebula which can explain the persistent radio source associated with FRB 121102, inspired by the fact that the Galactic X-ray binaries SS433 does power such a similar radio nebula (Fabrika 2004). If this model is correct, the luminosity of the persistent radio source may be estimated as L R ∝Ė γ FRB , wherė E FRB ≡ E FRB ℜ, E FRB is the typical energy of the FRB, ℜ is the repetition rate. Here we adopt γ = 1 for a rough estimation although the exact valu is expected to be slightly greater than 1 in the context of synchrotron radiation (Dai et al. 2017). The typical isotropic energy ∼ 10 37 erg with a repetition rate ℜ ∼ 10 −1 h −1 for FRB 180916 (Pastor-Marazuela et al. 2020), and a similar isotropic energy but with a much higher repetition rate ℜ ∼ 10 2 h −1  for FRB 121102, we haveĖ FRB 180916 /Ė FRB 121102 ∼ 10 −3 . Therefore, we predict that the persistent radio emission associated with FRB 180916 would be 10 35 erg/s, which is in line with observational limit (Marcote et al. 2020).
In addiction, it is expected that the accreting CO will also have persistent X-ray emission. Assuming solar abundances, the X-ray luminosity of the accreting accretor is (Shakura & Sunyaev 1973;Poutanen et al. 2007) L X ≈ 1.3×10 38 erg /s   ṁ Mx M⊙ ,ṁ 1 (1 + lnṁ) Mx M⊙ , 1 ṁ 100 (26) whereṁ =Ṁ /Ṁ Edd , the super-Eddington accretion rateṀ Edd ≃ 2.3 × 10 −8 (M x /M ⊙ )M ⊙ /yr. Based on the discussions in section 2, we have L X ∼ 10 38 erg /s for a accretion rateṀ a = 10 −5 M ⊙ /yr, which is below the detection limit both for FRB 180916 and FRB 121102 Scholz et al. 2020). However, we expects to detect the X-ray emission from the accretor for the sources at close distance i.e. a few tens of Mpc. In addiction, we have confirmed that the prompt gammaray emissions radiated by the thermalized electrons behind the (reverse and forward) shocks associated with FRB 121102 and FRB 180916 are also expected too dim to be detected by the current gamma-ray detectors.
Finally, we anticipate that if those active periodic FRBs are observed at close distances in the future, multiband observations will verify or falsify our model. Also, we encourage the search for FRBs in the ultraluminous X-ray sources.