An accreting stellar binary model for active periodic fast radio bursts
aa r X i v : . [ a s t r o - ph . H E ] F e b Draft version February 16, 2021
Typeset using L A TEX twocolumn style in AASTeX63
An accreting stellar binary model for active periodic fast radio bursts
Can-Min Deng,
1, 2
Shu-Qing Zhong, and Zi-Gao Dai
1, 2, 3,4 Department of Astronomy, School of Physical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China;[email protected] CAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology ofChina, Hefei 230026, Anhui, China School of Astronomy and Space Science, Nanjing University, Nanjing 210093, China Key laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210093, China
ABSTRACTIn this letter, we propose an accreting stellar binary model for understanding the active periodicFast 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. TheCO will accrete the material of the DS and then drive magnetic blobs. FRBs would be produced bythe shock process between the magnetic blobs and the stellar wind of the DS through the synchrotronmaser mechanism. We show that this model can in principle sufficiently produce highly active FRBswith a long lifetime, and also can naturally explain the periodicity and the duty cycle of the activityas appeared in FRBs 180916.J0158+65 and 121102. The radio nebula excited by the long-term in-jection of magnetic blobs into the surrounding environment also can account for the persistent radiosource associated with FRB 121102. In addiction, we discuss in detail the possible multi-wavelengthcounterparts of FRB 180916.J0158+65 in the context of this model. Multi-wavelength observations inthe future will verify or falsify this model.
Keywords: fast radio bursts —stars: black hole—stars: neutron star—radiation mechanisms: non-thermal INTRODUCTIONFast radio bursts (FRBs) are intense radio transientswith extremely short duration, and their physical originis a mystery (Cordes & Chatterjee 2019; Petroff et al.2019). Observationally, some FRBs showed repeatbursts, but most bursts did not (Spitler et al. 2016;Fonseca et al. 2020; James et al. 2020). It is also amystery whether all of FRBs have the same origin(Palaniswamy et al. 2018; Caleb et al. 2018, 2019).Chime/Frb Collaboration et al. (2020) found a 16.35-day periodicity in FRB 180916.J0158+65 with a 4.0-dayphase window, implying a duty cycle D ≃ .
24 of ac-tivity. In addiction, Rajwade et al. (2020) reported apossible 157-day periodicity in the FRB 121102 with aduty cycle D ≃ .
56. These are very important cluesfor studying the physical origin of the FRBs. For such aperiodicity, it may be explained, either by the orbital pe-riod of the binary sysytem (Zhang 2020; Lyutikov et al.2020; Ioka & Zhang 2020; Gu et al. 2020; Mottez et al.2020; Dai & Zhong 2020), or precession of the emit-ter (Yang & Zou 2020; Levin et al. 2020; Zanazzi & Lai2020; Tong et al. 2020; Katz 2020). Recently, CHIME/FRB Collaboration et al. (2020)and Bochenek et al. (2020) reported a single FRB (FRB200428) with two pulses in association with an activeGalactic magnetar SGR 1935+2154, which is locatedat a distance ∼ ∼ Myr, which inline with the hypothesis that the high-mass X-ray bi-naries or gamma-ray binaries are the progenitors ratherthan the magnetars (Tendulkar et al. 2020). Moreover,Pastor-Marazuela et al. (2020) rule out the scenario inwhich companion winds cause FRB periodicity by usingsimultaneous Apertif and LOFAR data. Motivated bythose observation results mentioned above, we proposean alternative model for understanding the highly ac-tive repeating FRBs with periodicity in this letter. Themodel consists of a stellar compact object (CO) and adonor star (DS) with its filled Roche lobe, in which theCO can be a neutron star (NS) or a black hole (BH).We will show that this model can explain the activelyrepeating FRBs themselves and their periodicity behav-iors, as the cases FRBs 180916.J0158+65 and 121102. THE MODELIn this section, we propose a binary model for activerepeating FRBs. The binary system consists of a stel-lar CO and a DS. The CO may be a BH or a NS. Asillustrated in Figure 1, when the DS fills its Roche lobe,significant mass transfer will occur from the DS to theCO. Then an accretion disc will form around the CO.As we know, jets are widely present in the accretionsystems. However, in addition to the usual continuousjets, the accretion process may also produce episodicjets. The energetic, collimating episodic jets had beenobserved in active galactic nuclei, stellar binaries, andprotostars. For instance, in the X-ray binaries, episodicjets are usually observed during their X-ray outburstsand intense radio fares (Fender et al. 2004; Zhang & Yu2015). Practical models and numerical simulation sug-gested that the episodic jets may be driven by mag-netic instability in the accretion disc (Yuan et al. 2009;Yuan & Zhang 2012; Zhao et al. 2020). Because of shearand turbulent motion of the accretion flow, a flux ropesystem is expected to form near the disc. The energy isaccumulated and stored in the system until a thresholdis reached, then the system loses its equilibrium and theenergy will be released in a catastrophic way, i.e. eject-ing episodic magnetic blobs. By assuming the accretionflow is advection dominated, the available isotropic freemagnetic energy of one blob is (Yuan & Zhang 2012) E f ≃ f − , − α − − β − ˙ M a − M ⊙ / yr ! (cid:18) M M ⊙ (cid:19) ˆ r / erg . (1) Here we consider the magnetic blobs is ejected in a colli-mated angle, and f b = 10 − is the beaming factor, whichis equivalent to a opening angle of several degrees. α isthe viscous parameter, β is the ratio of the magneticpressure over the total pressure in the accretion disc.We adopt α = 0 .
01 and β = 0 . M a is the mass accretion rate, M is the mass of the CO,ˆ r is the radius, where the magnetic blobs is formed, inunits of 2 GM/c . Here and hereafter, we employ theshort-hand notation q x = q/ x in cgs units.On the other hand, the binary system is sur-rounded by the stellar wind from the DS. There-fore, we suggest that interaction between the mag-netic blobs and the stellar wind could lead to shockformation, and then this shock process would pro-duced powerful coherent radiation (FRBs) through thesynchrotron maser mechanism analogy to the flaringmagnetar models (Lyubarsky 2014; Beloborodov 2017;Metzger et al. 2019). Taking FRB 180916.J0158+65 asa template, the average energy of the bursts is ∼ erg (CHIME/FRB Collaboration et al. 2020). Accord-ing to equation (1), E FRB = 10 ǫ − E f , erg, a massaccretion rate ∼ − M ⊙ / yr is required to account forFRB 180916.J0158+65. ǫ ∼ − is the radiation effi-ciency of the radio bursts, which is suggested by the nu-merical modeling (Plotnikov & Sironi 2019). Note thatthis high accretion rate ˙ M a ∼ − M ⊙ / yr can occurin high mass Roche lobe filling binaries, such as SS433which is accreting at a rate of ∼ − M ⊙ / yr (Fabrika2004). Moreover, Wiktorowicz et al. (2015) showed thatthe Roche lobe overflow rate can be up to 10 − M ⊙ / yrin if the DS is evolving in the Hertzsprung gap.The wind emerged from the DS is filled around theCO, and its density distribution n w can be estimated as n w = ˙ M w πm p R v w , R ≃ (cid:0) r + a (cid:1) / (2)where ˙ M w is the wind mass loss rate of the DS, v w isthe speed of the wind. a a is the semimajor axis of thebinary, r is the distance from the CO. For r ≪ a , wehave n w ≃ . × ˙ m w , − β − , − a − cm − (3)where ˙ m w = ˙ M w / ( M ⊙ / yr), β w = v w /c , c is the speedof light. We adopt β w = 0 .
01 as typical values for mas-sive stars, and adopt ˙ m w = 10 − for self-consistent ofthe working model. And it might be true, noting thatKaspi et al. (1996) found a strong constraint on the stel-lar wind of a B-star < − M ⊙ / yr. a = 10 cm ismotivated by the analysis for FRB 180916.J0158+65 insection 3. α pericenter yx O FRBs persistent radiomediumstellar windshocknebulamagnetic blobs (c)(a) stellar wind
FRBs magnetic blobs interaction (b)
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. Magneticblobs are ejected from the accretion disc due to magnetic instability. The magnetic blobs may accelerate to the Lorentz factorΓ ∼ σ (Yuan & Zhang 2012), where σ is the initial magnetization parameter of the blobs. Then the blobs interact with thestellar wind, which is driven by the DS and immerses itself around the binary system, to induce shocks powering FRBs bythe synchrotron maser mechanism; (c) The interactions of the magnetic blobs and the stellar wind produce FRBs. Long termenergy injection into the surrounding medium procduce a nebulae which powers the persistent raido source associated with theFRB. Therefore, according to Sari & Piran (1995), theLorentz factor of the shocked ejecta during the early re-verse shock crossing phase is Γ = ( n ej Γ / n w ) / , where n ej ≃ E f / (4 πr m p c ) δt Γ is the co-moving density inthe unshocked ejecta a radius r , Γ ej is the initial Lorentzfactor of the unshocked ejecta, δt is the duration of thecentral energy activity. Then we haveΓ ( r ≪ r dec ) = (cid:18) E f β w M w c δt (cid:19) / (cid:16) ra (cid:17) − / , (4)where r dec ≃ . × E / , ˙ m − / , − β / , − δt / − a / cm (5)is the deceleration radius. One sees that it satisfies r dec ≪ a . Therefore, the Lorentz factor at the decel-eration radius isΓ ( r dec ) ≃ E / , ˙ m − / , − β / , − δt − / − a / (6)Then we can estimate the emission frequency of the syn-chrotron maser, peaking at (Plotnikov & Sironi 2019) ν pk ≈ ν p ≈ . E / , ˙ m / , − β − / , − δt − / − a − / GHz(7)where ν p = ( e n e /πm e ) / is the plasma frequency ofthe medium ahead of the forward shock i.e. n e = n w , e is the charge of the electron, m e is the mass of theelectron. The optical depth due to free–free absorption is esti-mated as τ ff ∼ e k B m e c ( 2 π k B m e ) / ¯ g ff r dec n ( r dec ) ν − T − / ∼ − ¯ g ff ˙ m w , − β − , − δt − a − T − / . (8)where k B is the Boltzmann’s constant, ¯ g ff is themean Gunter factor, T = 10 K is the tempera-ture of the wind of the DS. For hν/k B T ≪ g ff ≃ ( √ /π ) ln(2 . k B T /hν ), one has ¯ g ff ( ν / Γ j, , T ) ≃ . τ T ∼ σ T r dec n e ( r dec ) ∼ − , is very small. However, due tothe extremely high brightness temperature of FRBs, in-duced scattering process becomes important (Lyubarsky2008; Lyubarsky & Ostrovska 2016). The effective op-tical depth due to the induced Compton scattering isestimated by (Lyubarsky & Ostrovska 2016) τ IC ( v pk ) ∼
110 3 σ T c πm e n e ( r dec ) E FRB r ν ≃ ǫ − E / , ˙ m / , − β − / , − δt / − a − / , (9)which is too opaque for emission at ν pk . However,at frequencies ≫ ν pk the radiation could still es-cape. According to equation (7), we have τ IC ( ν ) = τ IC ( ν pk )( ν/ν pk ) − if we assume the spectrum as E FRB ∝ ν − (Metzger et al. 2019). As a result, the peak fre-quency of the radiated spectrum moves to ν m where τ IC ( ν m ) = 3, that is ν m ≃ . ǫ / − E / , ˙ m / , − β − / , − δt − / − a − / GHz(10)One sees that, for FRB 180916.J0158+65, the sub-GHz emission can travel through the surrounding en-vironment. In addiction, this peak frequency willevolve with the shock decelerates as ν m ∝ t − / ,and this temporally decreasing peak frequency mayexplain the observed downward drifting frequencystructure in the sub-pulses of some repeating FRBs(CHIME/FRB Collaboration et al. 2019; Hessels et al.2019).For FRB 121102, the typical energy ∼ erg(Wang & Zhang 2019), and a ∼ . cm (see discus-sions in section 3), then we have ν m ≃ . m / , − GHz.This is consistent with the fact that FRB 121102 hasfew detection at sub-GHz band (Houben et al. 2019;Josephy et al. 2019), which may be because the DS inthe case of FRB 121102 has a stronger stellar wind com-pared to FRB 180916.J0158+65. CONSTRAINTS FROM THE PERIODICITYAND DUTY CYCLEPeriodicities has been observed in some X-raysources, and mechanisms have been proposed to ex-plain this behavior, mainly including orbital periodmodulation(Strohmayer 2009) and disk/jet precession(Begelman et al. 2006; Foster et al. 2010), also seen in areview Kaaret et al. (2017).For periodic FRBs, under the framework of our model,the periodicity might be explained alternatively by theprecession of the jets (Katz 2020). However, FRB180916.J0158+65 shows that the signals is concentratedin a narrow active window with a duty cycle D ≃ . q = M/M x , M x is the massof the CO, M is the mass of the DS. Then the orbital period of the binary is T = 2 πq / (1 + q ) − / a / ( GM ) − / (11)where G is the gravitational constant, a is the semima-jor axis. The effective radius R L , of the Roche-lobe ofthe DS at the periastron can be estimated as (Eggleton1983) R L , a (1 − e ) = 0 . q / . q / + ln (cid:0) q / (cid:1) ≡ χ, (12)where e is the orbital eccentricity. With the assumptionthat the DS is filled with its Roche lobe, then its averagedensity is ¯ ρ = 3 M πf R , , (13)where f RL = R/R L , & R is the radiu of the DS. We expect that f RL is justslightly greater than 1, .e., f RL − ≪
1. Combiningequations (11)-(13), one gets T ≃ (3 π ) / f q (1 − e ) − / ( G ¯ ρ ) − / , (14)where f q = q / (1 + q ) − / χ − / . Note that f q ∈ (1 . , .
8) for q & .
1. For simplicity, we take f q = 1 . q & .
1. Thus one can simply deduce the average densityof the DS as¯ ρ ≃ (1 − e ) − (cid:18) T . (cid:19) − ¯ ρ ⊙ , (15)where ¯ ρ ⊙ ≃ . / cm is the current average densityof the Sun. It is worth noting that equation (15) is ageneralization of equation (4.10) in Frank et al. (2002),¯ ρ ∝ T − , namely generalized from the case of circularorbit to the case of eccentric orbit. Next, we discuss theconstraint on e from the observed duty cycle D.Without loss of generality, taking the CO as the ref-erence, DS moves on an elliptic orbit with respect tothe CO. The distant from the CO to the DS is r ( θ ) = a (1 − e ) / (1 + e cos θ ). The DS is at periastron when θ = 0, where θ is the angle between the vector diameterfrom the CO to the DS and the polar axis in the polarcoordinate system. For the DS at different positions, itsRoche lobe radius is determined by R L ,θ = χr ( θ ). As-sume that when θ = ± α ( α < π ), the DS can just fill itsRoche lobe, i.e., R L ,α = R . Then we have1 + e e cos α = f RL (16)Therefore, according to the Kepler’s second law, theduty cycle of activity can be calculated as D = ∆ S / S, −3 −2 −1 λ10 −3 −2 −1 e D=0.24D=0.56
Figure 2.
The orbital eccentricity e as a function of theRoche lobe overfilling parameter of the DS λ for the dutycycle D=0.24 (FRB 180916.J0158+65) and D=0.56 (FRB121102), respectively. 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 ellipticorbit. That is D = 12 π (cid:0) − e (cid:1) / Z α − α dθ (1 + e cos θ ) , (17)where, by letting λ = 1 − f − , α = π, λ (1 + e ) / (2 e ) >
12 arcsin p λ (1 + e ) / (2 e ) , λ (1 + e ) / (2 e ) < λ (1+ e ) / (2 e ) >
1, it means that the DS fills its Rochelobe throughout the cycle, thus D=1. Therefore, basedon the observed duty cycle, we can discuss the constraintto e . Figure 2 shows e as a function of λ for D = 0 . .
56 (FRB 121102),respectively. Note that λ = 1 − f − = ( R − R L , ) /R de-scribes the degree of the Roche lobe overfilling of the DSat periastron. One naturally expects λ ≪
1, otherwisethe Roche lobe overflow would be violent. By adopting λ .
1, we have e .
22 for FRB 180916.J0158+65and e .
08 for FRB 121102, which are also showed infigure 2. It can be seen that the required orbital eccen-tricity is not large.Now we can discuss what kind of DS is needed, in or-der to explain the periodicity of FRBs 180916.J0158+65and 121102. For FRB 180916.J0158+65 T = 16 .
35 dayand e .
22, one gets ¯ ρ ≃ (1 . − . × − ¯ ρ ⊙ accord-ing to equation (15). Similarly, for FRB 121102 T = 157day and e .
08, one gets ¯ ρ ≃ (1 . − . × − ¯ ρ ⊙ . Itindicates that the DSs must be either giants or super-giants. For red supergiants, their average density can be as low as 10 − ¯ ρ ⊙ . Therefore, we expect that this modelcan explain a period up to T ∼ λ is un-clear. However, neither the masses of the DS and theCO are well constrained. Nevertheless, we can take M x = 1 . M ⊙ and M = 30 M ⊙ as an example un-der the assumption of λ = 0 . e ≃ .
22 and ¯ ρ ≃ . × − ¯ ρ ⊙ ( λ =0.1). Then the radius of the DS R ≃ R ⊙ . Ac-cording to equation (11), the orbital semi-major axis a ≃ R ⊙ ( T / . / ≈ . cm, thus the apas-tron and periastron distances are r a = a (1+ e ) ≃ R ⊙ and r p = a (1 − e ) ≃ R ⊙ , respectively. The Roche loberadius of the DS at periastron R L , = χr p ≃ R ⊙ . Sim-ilarly, for FRB 121102 we get e ≃ .