Binary Comb Models for FRB 121102

Fast radio bursts (FRBs) are bright radio transients with durations of a few milliseconds (Lorimer et al. 2007; Thornton et al. 2013; Petroff et al. 2016). Most FRBs have an extragalactic origin, and their high brightness temperature requires that they are produced in a coherent radiation process (e.g., Katz 2018; Petroff et al. 2019; Platts et al. 2019; Zhang 2020a, for review). Some FRBs are observed to emit repeated bursts (Spitler et al. 2014, 2016; CHIME/FRB Collaboration et al. 2019a, 2019b; Bhardwaj et al. 2021), suggesting that at least some FRBs have noncatastrophic origins. Recently, an FRB-like radio burst, FRB 200428, from a galactic magnetar SGR1935+2154 was discovered to be associated with a luminous, hard X-ray burst (Bochenek et al. 2020; CHIME/FRB Collaboration et al. 2020; Mereghetti et al. 2020; Li et al. 2021a; Ridnaia et al. 2021; Tavani et al. 2021). This event suggests that at least some FRBs originate from magnetars (Ioka 2020; Lu et al. 2020; Yuan et al. 2020; Zhang 2021; Yang & Zhang 2021). Regardless of their origin, these bursts can be useful probes for studying cosmology (e.g., Ioka 2003; Inoue 2004; Takahashi et al. 2021).

Recently, one repeating FRB source detected by the Canadian Hydrogen Intensity Mapping Experiment (CHIME; CHIME/FRB Collaboration et al. 2020), i.e., FRB 180916, was found to have a 16.35 day periodicity in burst emission, with a 5 day active window in each period. Several models have been proposed to explain this periodicity. Some models explain the period by invoking the orbital period of a binary (Lyutikov et al. 2020; Ioka & Zhang 2020; Kuerban et al. 2021; Du et al. 2021; Deng et al. 2021). For the binary comb model (Ioka & Zhang 2020), a population synthesis of binary stars has also been carried out (Zhang & Gao 2020). Other models explain the period by invoking the precession of a pulsar (Levin et al. 2020; Zanazzi & Lai 2020; Yang & Zou 2020; Li & Zanazzi 2021; Katz 2021a; Sridhar et al. 2021). Current data cannot tell which of these models is the correct solution (Zhang 2020b). In addition, recent observations by the CHIME, LOw Frequency ARrray (LOFAR), the upgraded Giant Metre Wavelength Radio Telescope (uGMT), and the Apertif Radio Transient System at the Westerbork Synthesis Radio Telescope (Apertif) have revealed that the active window varies with the observed frequency of the bursts (Pastor-Marazuela et al. 2020; Pleunis et al. 2020): High frequency bursts arrive earlier than low-frequency ones, and the active window is narrower in a higher frequency.

FRB 121102, the first repeater (Spitler et al. 2014, 2016), was also found to have a periodicity (Rajwade et al. 2020; Cruces et al. 2021; Li et al. 2021b). Its period is ${159}_{-8}^{+3}$ days and its active window is ∼47–60% of the period. Its host galaxy is a dwarf galaxy at redshift z = 0.193 and its stellar mass is estimated to be ∼(4–7) × 10⁷ M_⊙ (Tendulkar et al. 2017). FRB 121102 is associated with a persistent radio counterpart with a luminosity of ∼10³⁹ erg s⁻¹ in 1–10 GHz (Chatterjee et al. 2017; Marcote et al. 2017), as proposed before the discovery (Murase et al. 2016). Its size is less than 0.7 pc and its location is offset from the optical center of the galaxy by 0.5–1 kpc. The bursts show a rotation measure that is at least two orders of magnitude higher than that of other FRBs (Michilli et al. 2018). This high rotation measure may be caused by a magnetized synchrotron nebula around the source (Beloborodov 2017; Yang & Dai 2019; Hilmarsson et al. 2021) or by the strong magnetic fields around a central black hole in the host galaxy (Zhang 2018). In addition, long-term observations over more than eight years revealed secular evolution of the dispersion measure and the rotation measure (Tabor & Loeb 2020; Cruces et al. 2021; Hilmarsson et al. 2021; Katz 2021b; Li et al. 2021a; Platts et al. 2021). Although many models have been proposed to explain these features of FRB 121102 and the persistent radio counterpart, the origin is still unknown (Yang et al. 2016; Katz 2017; Kashiyama & Murase 2017; Beloborodov 2017; Dai et al. 2017; Zhang 2018; Yang & Dai 2019; Li et al. 2020).

The cosmic comb model for FRBs (Zhang 2017, 2018; Ioka & Zhang 2020) invokes the interaction between an astrophysical gas flow (stream) and the magnetosphere of a neutron star to power FRBs. In the original paper (Zhang 2017), the energy source of the FRB was envisaged to be from the kinetic energy of the stream. Ioka & Zhang (2020) proposed a binary comb model to interpret FRB 180916 and found that with a 16 day orbital period, the kinetic energy of the companion wind is not large enough to power FRBs. Rather, FRBs are intrinsically produced from the neutron star (the FRB pulsar) and the interaction (combing) serves as defining an optically-thin funnel for FRB propagation.

In this paper, we further develop the binary comb model and apply it to FRB 121102. In Section 2, we show that the binary with the observed period may be optically thick to the FRBs by free–free absorption or induced Compton scattering. In Section 3, we identify three scenarios (i.e., three modes of the binary comb model) in which the FRBs become observable periodically from an optically thick binary. In Section 4, we show that the upper limit on the change in the dispersion measure and the luminosity of the persistent radio counterpart constrain the parameters of the host binary. In Section 5, we use the binary comb model to constrain the binary system of FRB 121102. In Section 6, we propose two scenarios to realize the frequency dependent active window observed in FRB 180916. Section 7 is devoted to a summary and discussion. In this paper, we employ the notation Q_x = Q/10^x in cgs units.

We consider a binary system with an orbital period P as the origin of periodic FRBs. The binary contains the source of FRBs (which we call the FRB pulsar). We set the mass of the FRB pulsar as M_NS, the mass of the companion as M_c, the semimajor axis of the binary as a, and the eccentricity of the binary as e. For a given total mass, M_tot = M_NS + M_c, the observed period determines the semimajor axis of the binary as

$$\begin{eqnarray}\begin{array}{rcl}a & = & {\left[\displaystyle \frac{{GM}_{{\rm{t}}{\rm{o}}{\rm{t}}}}{{\left(2\pi /P\right)}^{2}}\right]}^{1/3}\\ & = & 8.6\times {10}^{13}\,{\rm{c}}{\rm{m}}\,{\left(\displaystyle \frac{{M}_{{\rm{t}}{\rm{o}}{\rm{t}}}}{{10}^{3}{M}_{\odot }}\right)}^{1/3}{\left(\displaystyle \frac{P}{159{\rm{d}}{\rm{a}}{\rm{y}}}\right)}^{-2/3},\end{array}\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant. In this paper, we consider three types of companions, a massive star (MS), a supermassive black hole (SMBH), and an intermediate-mass black hole (IMBH). ³ We adopt the mass of the MS as 10 M_⊙, the mass of the SMBH as 10⁵ M_⊙, and the mass of the IMBH as 10³ M_⊙ (see Table 1). The MS is the lightest case, the SMBH is the heaviest, and the IMBH is in between. In any case, the companion of the FRB pulsar would be accompanied by a wind (which is called the companion wind in the paper), which is a stellar wind for the MS case and a disk wind for the SMBH and IMBH cases.

Table 1. The Parameters for the Companion Star and the Optical Depth for Each Scattering Process

model	Mc (M⊙)	a (cm)	V (c)	$\dot{M}$ (M⊙ yr−1)	τTh	τff	τIdC
MS	10	1.9 × 1013	1 × 10−2, 3 × 10−2	10−9–10−3	10−7–10−1	10−4–108	10−2–106
IMBH	103	8.6 × 1013	3 × 10−2, 1 × 10−1	10−9–10−6	10−8–10−5	10−8–10−1	10−4–1
SMBH	105	4.0 × 1014	3 × 10−2, 1 × 10−1	10−9–10−4	10−9–10−4	10−10–10	10−6–0.1

Note. M_c is the mass of the companion, a is the semimajor axis determined by the observed period, V is the velocity of the wind, $\dot{M}$ is the mass-loss rate, and τ_i is the optical depth where i = Th, ff, and IdC. These optical depths are evaluated at r = a here. In Section 3 and later sections, we evaluate the optical depth at r = R_f where R_f is the funnel size (see Section 3.1). Thus, the optical depth can be higher than the values in this table.

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For the MS, we set the mass-loss rate to be 10⁻⁹–10⁻³ M_⊙ yr⁻¹. The lower limit corresponds to a weak wind for a late O-type or an early B-type star (Puls et al. 2008; Smith 2014). The upper limit corresponds to a Wolf–Rayet-like star a few years before its supernova explosion (Smith 2014; Gal-Yam et al. 2014). We consider two cases for the wind velocity as 1 × 10⁻² c and 3 × 10⁻² c, respectively. The former is the observed maximum velocity of the wind of a hot massive star (Kudritzki & Puls 2000). The latter is an extremely high velocity case, and we will show that the persistent radio counterpart of FRB 121102 is explained only in such an extreme condition for the MS companion case (see Section 5).

For the SMBH and the IMBH cases, we assume that the mass-loss rate, $\dot{M}$, is less than the Eddington accretion rate, ${\dot{M}}_{\mathrm{Edd}}={L}_{\mathrm{Edd}}/{c}^{2}$, where L_Edd is the Eddington luminosity. If radiation efficiency is taken into account, the accretion rate can be higher than ${\dot{M}}_{\mathrm{Edd}}$. See Table 1 for the adopted parameters.

The companion wind can be opaque for the repeating FRB emission, making FRBs unobservable. We evaluate the optical depth, τ, to Thomson scattering, free–free absorption, and induced Compton scattering. In this paper, we do not consider the induced Raman scattering because it depends on unknown plasma properties. For simplicity, we use the criterion τ < 10 for the observability of FRBs in all scattering processes (see Lyubarsky 2008). ⁴

To calculate optical depth, we take polar coordinates (r, θ) in the orbital plane. The FRB pulsar is at the origin and the companion star is at r = R, θ = π, where R is the separation of the binary at a given time. R and a are related by the following equation,

$$\begin{eqnarray}&&R=\displaystyle \frac{a}{1+e\cos (f)},\end{eqnarray} \tag{ 2 }$$

where f is the true anomaly at the given time. The particle number density of the companion wind at (r, θ) is approximately

$$\begin{eqnarray}&&n(r,\theta )\simeq \displaystyle \frac{\dot{M}}{4\pi {m}_{p}V\left({r}^{2}+{R}^{2}+2{Rr}\cos \theta \right)},\end{eqnarray} \tag{ 3 }$$

where m_p is the proton mass.

The optical depth to the Thomson scattering at (r, θ) is written as

$$\begin{eqnarray}&&\begin{array}{l}{\tau }_{\mathrm{Th}}(r,\theta )={\displaystyle \int }_{r}^{\infty }{\sigma }_{T}n\left({r}^{{\prime} },\theta \right){{dr}}^{{\prime} }\\ \quad =\,\displaystyle \frac{{\sigma }_{T}\dot{M}}{4\pi {{VRm}}_{p}}{\displaystyle \int }_{r/R}^{\infty }\displaystyle \frac{{dx}}{1+{x}^{2}+2x\cos \theta }\\ \quad \sim 6.7\times {10}^{-7}\left(\displaystyle \frac{\dot{M}}{{10}^{-9}{M}_{\odot }\,{\mathrm{yr}}^{-1}}\right){\left(\displaystyle \frac{V}{{10}^{-2}c}\right)}^{-1}{R}_{13}^{-1}\\ \times \quad \left\{\begin{array}{ll}\displaystyle \frac{\pi }{2}-\varphi & (\theta \ne 0,\pi )\\ {\left(1+r/R\right)}^{-1} & (\theta =0,\pi )\end{array}\right.,\end{array}\end{eqnarray} \tag{ 4 }$$

where σ_T is the Thomson cross section, c is the speed of light, $\varphi =\arctan \tfrac{(r/R)+\cos \theta }{\sin \theta }$, and $x=r^{\prime} /R$. For FRB 121102, the system is transparent to Thomson scattering (see Table 1). This is obvious because we assume that $\dot{M}$ for the SMBH and the IMBH is less than the Eddington accretion rate (see Table 1).

The optical depth to the free–free absorption is (Rybicki & Lightman 1979)

$$\begin{eqnarray}&&\begin{array}{l}{\tau }_{{\rm{ff}}}(r,\theta )={\displaystyle \int }_{r}^{\infty }{\alpha }_{{ff}}{{dr}}^{{\rm{{\prime} }}}\\ \,\simeq \displaystyle \frac{4{q}^{6}}{3{m}_{e}{ck}}{\left(\displaystyle \frac{2\pi }{3{{km}}_{e}}\right)}^{1/2}{T}^{-3/2}{\nu }^{-2}{\bar{g}}_{{ff}}{\left(\displaystyle \frac{\dot{M}}{4\pi {{Vm}}_{p}{R}^{2}}\right)}^{2}\\ \,\times \,R{\displaystyle \int }_{r/R}^{\infty }\displaystyle \frac{{dx}}{{\left(1+{x}^{2}+2x\cos \theta \right)}^{2}}\\ \,\sim 1.8\times {10}^{-3}{T}_{4}^{-3/2}{\nu }_{9}^{-2}{\bar{g}}_{{\rm{ff}}}{\left(\displaystyle \frac{\dot{M}}{{10}^{-9}{M}_{\odot }{{\rm{yr}}}^{-1}}\right)}^{2}{\left(\displaystyle \frac{V}{{10}^{-2}c}\right)}^{-2}\\ \times \,{R}_{13}^{-3}\\ \times \,\left\{\begin{array}{ll}\displaystyle \frac{1}{2{\sin }^{3}\theta }\left(\displaystyle \frac{\pi }{2}-\varphi -\displaystyle \frac{\sin 2\varphi }{2}\right) & (\theta \ne 0,\pi )\\ \displaystyle \frac{1}{3{\left(1+r/R\right)}^{3}} & (\theta =0,\pi )\end{array}\right.,\end{array}\end{eqnarray} \tag{ 5 }$$

where q is the elementary charge, m_e is the electron mass, k is the Boltzmann constant, ν is the frequency of the FRB, T is the temperature of the wind, and ${\bar{g}}_{\mathrm{ff}}$ is the mean Gaunt factor for bremsstrahlung. The frequency and the temperature in this paper are in the small-angle classical regime (Novikov & Thorne 1973). For FRB 121102, the optical depth to the free–free absorption can be higher than 10 (Table 1).

For an FRB with luminosity L_FRB and duration Δt, the optical depth to induced Compton scattering is (Lyubarsky 2008)

$$\begin{eqnarray}&&\begin{array}{r}\begin{array}{l}{\tau }_{{\rm{I}}{\rm{d}}{\rm{C}}}(r,\theta )=\displaystyle \frac{3{\sigma }_{T}}{8\pi }\displaystyle \frac{c}{{m}_{e}{\nu }^{3}}n(r,\theta )\displaystyle \frac{{L}_{{\rm{F}}{\rm{R}}{\rm{B}}}}{4\pi {r}^{2}}{\rm{\Delta }}t\\ \,\sim 2.1\times 10\,{\nu }_{9}^{-3}\left(\displaystyle \frac{\dot{M}}{{10}^{-9}{M}_{\odot }\,{{\rm{y}}{\rm{r}}}^{-1}}\right)\\ \times \,{\left(\displaystyle \frac{V}{{10}^{-2}c}\right)}^{-1}{R}_{13}^{-4}{\left({L}_{{\rm{F}}{\rm{R}}{\rm{B}}}{\rm{\Delta }}t\right)}_{38}\\ \times \,{\left[\displaystyle \frac{1}{{x}^{2}\left(1+{x}^{2}+2x\cos \theta \right)}\right]}_{x=r/R}.\end{array}\end{array}\end{eqnarray} \tag{ 6 }$$

The above expression is valid if ${\theta }_{{\rm{b}}}\gt {(2c{\rm{\Delta }}t/r)}^{1/2}\sim 8\,\times {10}^{-3}{({\rm{\Delta }}t/10\mathrm{ms})}^{1/2}{r}_{13}^{-1/2}$, where θ_b is the half-opening angle of the FRB beam. The binary can be optically thick to induced Compton scattering (Table 1).

Strictly speaking, for a given θ, a photospheric radius for each process is obtained by solving τ_i(r_{ph, i}, θ) = 10 for r_ph,i where i = Th, ff, or IdC. The photospheric radius of the binary, r_ph, equals the largest of these radii. If the photospheric radius is larger than the typical length scale of the funnel created by the wind of the FRB pulsar (FRB pulsar wind), the binary is optically thick. The typical length scale of the funnel is calculated in Section 3.1.

The binary comb model explains the periodic activity of the FRBs by the periodic variation of the optical depth along the line of sight. There are three modes of configurations for the opacity variation. The first mode is the funnel mode (Figure 1a) proposed by Ioka & Zhang (2020) and Lyutikov et al. (2020). In the optically thick companion wind, the FRB pulsar wind would create a funnel. The FRBs are observable only when the funnel points toward Earth because the opacity is low in the funnel. The second mode is the τ-crossing mode (Figure 1b), which we find in this paper for the first time. In this mode, the orbit intersects the photosphere and the FRB pulsar goes in and out the photosphere during the period. The last mode is the inverse funnel mode (Figure 1(c)), which is also identified in this paper for the first time. In this mode, the FRB pulsar wind is stronger than that of the companion. The companion wind points toward the opposite direction of the FRB pulsar and blocks the FRBs from the FRB pulsar, producing the periodicity. Throughout this paper, we assume that the FRB pulsar wind does not absorb or scatter the FRBs.

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In this section, we assume that the binary is observed from the direction of its periapsis in an edge-on configuration. We also assume that the FRBs are radiated isotropically for simplicity. The isotropic radiation does not necessarily mean that individual FRBs are radiated spherically. Each FRB emission may be beamed and we assume that it is radiated in a random direction. The general viewing angle case and the effect of beaming are discussed in Section 5 for FRB 121102 and in the Appendix.

3.1. Funnel Mode

In the funnel mode, FRBs are observable when the funnel points toward Earth because the funnel created by the FRB pulsar wind makes the optical depth smaller. Thus, the observed duty cycle determines the half-opening angle of the funnel. Also, the funnel must extend in the radial direction to the outside of the photosphere, otherwise the FRBs are eventually blocked by the companion wind. To evaluate these conditions, we calculate three quantities: the range of the true anomaly where the FRBs are observable, the half-opening angle of the funnel, and the length of the spiral arm where the tail of the funnel is spiraled by the orbital motion.

For a given duty cycle, the range of the true anomaly where the FRBs are observable is determined as follows. As we can generally take the coordinates so that f increases with time, the FRB becomes observable when the true anomaly, f, equals π − θ_d and becomes unobservable when f = π + θ_d. The time taken to orbit from f = π − θ_d to f = π + θ_d is expressed as (Poisson & Will 2014)

$$\begin{eqnarray}&&T=2{\int }_{\pi -{\theta }_{{\rm{d}}}}^{\pi }\sqrt{\displaystyle \frac{{\left(1-{e}^{2}\right)}^{3}{a}^{3}}{{{GM}}_{\mathrm{tot}}}}\displaystyle \frac{{df}}{{\left(1+e\cos f\right)}^{2}}.\end{eqnarray} \tag{ 7 }$$

With the eccentric anomaly u, this integral is expressed only by elementary functions as

$$\begin{eqnarray}&&T=2\sqrt{\displaystyle \frac{{a}^{3}}{{{GM}}_{\mathrm{tot}}}}\left[\pi -\left({u}_{d}-e\sin {u}_{d}\right)\right],\end{eqnarray} \tag{ 8 }$$

where ${u}_{d}=2{\tan }^{-1}\left(\sqrt{\tfrac{1-e}{1+e}}\tan \tfrac{\pi -{\theta }_{{\rm{d}}}}{2}\right)$. To realize the observed duty cycle, ∼47%, T/P must be equal to 0.47 where $P=2\pi \sqrt{\tfrac{{a}^{3}}{{{GM}}_{\mathrm{tot}}}}$ is the orbital period. This condition is given as

$$\begin{eqnarray}&&\displaystyle \frac{T}{P}=\displaystyle \frac{1}{\pi }\left(\pi -{u}_{d}+e\sin {u}_{d}\right)=0.47,\end{eqnarray} \tag{ 9 }$$

which gives a relation between e and θ_d. Figure 2 shows this relation. The higher the eccentricity, the smaller the angle θ_d. This is because the FRB pulsar spends more time near the apoapsis for a higher eccentricity.

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The half-opening angle of the funnel, θ_c, is determined by the pressure balance between the FRB pulsar wind and the companion wind. Using the thin-shell approximation, in which the shocked wind creates a dense shell and its thickness is negligible compared to the distance to the star, θ_c is determined by

$$\begin{eqnarray}\begin{array}{rcl}-{\theta }_{c}+\tan {\theta }_{c} & = & \displaystyle \frac{\beta \pi }{1-\beta },\\ \beta & = & \displaystyle \frac{{L}_{{\rm{w}}}/c}{\dot{M}V},\end{array}\end{eqnarray} \tag{ 10 }$$

where L_w is the luminosity of the FRB pulsar wind (Canto et al. 1996). By definition, in the funnel mode the companion wind dominates over the FRB pulsar wind so that β is smaller than unity. The case of β ≥ 1 is considered in Section 3.3. We consider the case that β is smaller than unity in the funnel mode. ⁵ The term θ_c does not depend on the true anomaly or the semimajor axis of the binary and thus does not change with time. Here, we neglect the effect of the orbital motion on the half-opening angle because the orbital motion is much slower than the stellar wind. We assume that the shape of the funnel is a cone whose vertex is the FRB pulsar and whose half-opening angle equals θ_c. The FRBs are observable while the line of sight is inside the cone. The opening angle of the funnel becomes almost constant above a radius R_f (see Figure 1) where the pressure of the companion wind balances the FRB pulsar wind. This radius, R_f, which we call the funnel size, is determined by

$$\begin{eqnarray}&&\displaystyle \frac{\dot{M}}{4\pi V({R}^{2}+{R}_{{\rm{f}}}^{2})}{V}^{2}\sim \displaystyle \frac{{L}_{{\rm{w}}}}{4\pi {{cR}}_{{\rm{f}}}^{2}},\end{eqnarray} \tag{ 11 }$$

where we have used the approximation that ${R}^{2}+{R}_{{\rm{f}}}^{2}\,+2{{RR}}_{{\rm{f}}}\cos \theta \sim {R}^{2}+{R}_{{\rm{f}}}^{2}$. Solving Equation (11), we obtain

$$\begin{eqnarray}&&{R}_{{\rm{f}}}\sim \sqrt{\displaystyle \frac{\beta }{{\bf{1}}-\beta }}R.\end{eqnarray} \tag{ 12 }$$

In the limit of β → 1, one has ${R}_{{\rm{f}}}\sim \sqrt{1/(1-\beta )}R$, and for β ≪ 1, one has ${R}_{{\rm{f}}}\sim \sqrt{\beta }R$. We adopt R_f to evaluate the optical depths (see Section 2). Although the exact funnel size depends on the direction of the observer, we adopt R_f as the typical value of funnel size and do not consider its angular structure.

In the funnel mode, the length of the spiral arm, r_s, where the tail of the funnel is spiraled by the orbital motion (Parkin & Pittard 2008; Bosch-Ramon et al. 2015), should be larger than the photospheric radius, r_ph. We evaluate r_s by splitting the contact discontinuity of the funnel into two sections. One is a shocked cap where the orbital motion does not affect the structure of the bow shock (the red curve in Figure 3). The other is the ballistic contact discontinuity where the orbital motion changes the shape of the funnel (the green curve in Figure 3). In the ballistic contact discontinuity, the fluid elements flow ballistically with the velocity V because the densities of both winds have almost the same dependence on distance. For simplicity, we ignore the skew of the shock cap by the orbital motion discussed in Parkin & Pittard (2008) and assume that the fluid element in the ballistic contact discontinuity has a half-opening angle θ_c. To calculate r_s, we introduce two true anomalies f₁, f₂, and a time duration Δt. Here f₂ is the true anomaly when the FRB pulsar is on the line of sight from the observer to the companion, f₁ is the true anomaly such that the fluid element ejected at f₁ along the contact discontinuity reaches the line of sight at f₂ (see Figure 3), and Δt is the time duration taken for the FRB pulsar to orbit from f₁ to f₂. Using the sine theorem, we obtain the relation between r_s and the other parameters as

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{a{\left(1+e\cos {f}_{2}\right)}^{-1}+{r}_{{\rm{s}}}}{\sin {\theta }_{{\rm{c}}}} & = & \displaystyle \frac{V{\rm{\Delta }}t}{\sin \left({f}_{2}-{f}_{1}\right)}\\ & = & \,\displaystyle \frac{a{\left(1+e\cos {f}_{1}\right)}^{-1}}{\sin \left({\theta }_{{\rm{c}}}-({f}_{2}-{f}_{1})\right)}.\end{array}\end{eqnarray} \tag{ 13 }$$

For a given f₂, we obtain f₁ using the second equation and by substituting f₁ into the first equation, we obtain r_s. Assuming that ∣f₁ − f₂∣ ≪ 1 and thus ${f}_{2}-{f}_{1}\simeq \dot{\phi }{\rm{\Delta }}t$, we obtain the length of the spiral arm as

$$\begin{eqnarray}&&\begin{array}{l}{r}_{s}=\sin {\theta }_{c}\displaystyle \frac{V}{\dot{\phi }}-\displaystyle \frac{a}{1+e\cos {f}_{2}}\\ \quad =\,\sin {\theta }_{{\rm{c}}}\displaystyle \frac{V}{{\rm{\Omega }}}{\left(1-{e}^{2}\right)}^{3/2}{\left(1+e\cos {f}_{2}\right)}^{-2}\\ \qquad -\displaystyle \frac{a}{1+e\cos {f}_{2}},\end{array}\end{eqnarray} \tag{ 14 }$$

where ${\rm{\Omega }}=\sqrt{\tfrac{{{GM}}_{\mathrm{tot}}}{{a}^{3}}}$ and $\dot{\phi }={\rm{\Omega }}{\left(1-{e}^{2}\right)}^{-3/2}{\left(1+e\cos {f}_{2}\right)}^{2}$.

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There are three conditions for the funnel mode. The first condition is that the companion wind must be optically thick to FRBs so that the FRBs in the inactive phase are absorbed or scattered by the companion wind (r_ph > R_f). The second condition is that the half-opening angle of the funnel must be such that the duty cycle can be explained (θ_c = θ_d). The third condition is that the funnel must extend radially enough for FRBs not to be blocked by the companion wind. Thus, the length of the spiral arm must be larger than the photospheric radius (r_s > r_ph). These conditions for the funnel mode are represented as

$$\begin{eqnarray}&&{\theta }_{{\rm{c}}}={\theta }_{{\rm{d}}},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{r}_{{\rm{s}}}\gt {r}_{\mathrm{ph}}\gt {R}_{{\rm{f}}}.\end{eqnarray} \tag{ 16 }$$

The latter condition is evaluated at f = π ± θ_d.

3.2.  τ-crossing Mode

In the τ-crossing mode, the FRBs in the active phase become observable because the optical depth on the line of sight gets lower than 10 (Figure 1(b)). This condition requires that the photospheric radius equals the funnel size at f = π ± θ_d where θ_d is obtained by solving Equation (9). Also, the half-opening angle of the funnel must be small enough that the FRBs in the inactive phase do not become observable through the funnel. This condition requires θ_c ≤ θ_d. Thus, the conditions for the τ-crossing mode are

$$\begin{eqnarray}&&{\theta }_{{\rm{c}}}\leqslant {\theta }_{{\rm{d}}},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{r}_{\mathrm{ph}}={R}_{{\rm{f}}}.\end{eqnarray} \tag{ 18 }$$

The latter condition is evaluated at f = π ± θ_d.

3.3. Inverse Funnel Mode

4.1. Change in Dispersion Measure

The observed change in the dispersion measure, ΔDM, also gives a constraint on the host binary. The change of DM due to orbital motion must be smaller than the observed upper limit of variability, which is of the order 6 cm⁻³ pc for FRB 121102 (Li et al. 2021b). We calculate the dispersion measure, DM, from the edge of the spiral arm, r_s, to infinity. ΔDM is obtained by comparing the values at f = π (apoapsis) and at f = π + θ_d. In the calculation, we ignore the θ-dependence of the wind number density and set θ = 0 for simplicity. The dispersion measure is given by

$$\begin{eqnarray}&&\begin{array}{r}\begin{array}{l}{\rm{D}}{\rm{M}}\simeq {\displaystyle \int }_{{r}_{{\rm{s}}}}^{{\rm{\infty }}}ndr\\ \,=\,{\displaystyle \int }_{{r}_{{\rm{s}}}}^{{\rm{\infty }}}\displaystyle \frac{\dot{M}}{4\pi {m}_{p}V\left({r}^{2}+{R}^{2}+2Rr\right)}dr\\ \,=\,\displaystyle \frac{\dot{M}}{4\pi {m}_{p}V}\displaystyle \frac{1}{{r}_{{\rm{s}}}+R}\\ \,=\,3.24\times {10}^{7}\,{{\rm{c}}{\rm{m}}}^{-3}{\rm{p}}{\rm{c}}\,\displaystyle \frac{1}{{r}_{{\rm{s}}}/R+1}\left(\displaystyle \frac{\dot{M}}{{M}_{\odot }{{\rm{y}}{\rm{r}}}^{-1}}\right)\\ \times \,{\left(\displaystyle \frac{R}{{10}^{14}{\rm{c}}{\rm{m}}}\right)}^{-1}{\left(\displaystyle \frac{V}{0.01c}\right)}^{-1}.\end{array}\end{array}\end{eqnarray} \tag{ 19 }$$

Thus, the difference of DM in the active phase is given by

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Delta }}\mathrm{DM}=\displaystyle \frac{\dot{M}}{4\pi {m}_{p}V}\left({\left.\displaystyle \frac{1}{{r}_{{\rm{s}}}+R}\right|}_{f=\pi }-{\left.\displaystyle \frac{1}{{r}_{{\rm{s}}}+R}\right|}_{f=\pi +{\theta }_{{\rm{d}}}}\right)\\ \quad =\,\displaystyle \frac{\dot{M}}{4\pi {m}_{p}{VaX}(\pi )}\left[1-\displaystyle \frac{X(\pi )}{X(\pi +{\theta }_{{\rm{d}}})}\right],\end{array}\end{eqnarray} \tag{ 20 }$$

where we have defined

$$\begin{eqnarray}&&\begin{array}{l}X(f)\,=\,{\left.\displaystyle \frac{{r}_{{\rm{s}}}+R}{a}\right|}_{f}\\ \,=\,\sin {\theta }_{{\rm{c}}}\displaystyle \frac{V}{a{\rm{\Omega }}}{\left(1-{e}^{2}\right)}^{3/2}\\ \,\times \,{\left(1+e\cos (f)\right)}^{-2}+\displaystyle \frac{{M}_{{\rm{N}}{\rm{S}}}}{{M}_{c}}\displaystyle \frac{1}{1+e\cos f},\end{array}\end{eqnarray} \tag{ 21 }$$

and used Equation (14). This value must be smaller than the observed upper limit on the change of DM.

We note that Equation (20) is also used for the τ-crossing mode. In the τ-crossing mode, this is a conservative constraint on the ΔDM. If θ_c < θ_d, FRBs enter the companion wind at ∼R_f instead of ∼r_s and ΔDM becomes higher than Equation (20).

4.2. Persistent Radio Counterpart

We consider the consistency with the persistent radio counterpart of the FRB 121102 (Chatterjee et al. 2017; Marcote et al. 2017). There are two possibilities for the source of this persistent radio counterpart.

One is the synchrotron nebula emission powered by the pulsar wind of the FRB pulsar or by the magnetar flare. The nebula is formed by the interaction between the supernova ejecta and the FRB pulsar wind (Kashiyama & Murase 2017; Yang et al. 2016), by the interaction between the interstellar matter and the FRB pulsar wind ⁶ (Dai et al. 2017), or by the interaction between the supernova ejecta and the ejecta of the magnetar flare (Beloborodov 2017). In these models, the wind luminosity of the FRB pulsar, L_w, should be high enough to energize the persistent radio counterpart. If the persistent radio counterpart is due to the nebula emission,

$$\begin{eqnarray}&&{L}_{{\rm{w}}}\geqslant {L}_{{\rm{w}},\mathrm{lower}}\end{eqnarray} \tag{ 22 }$$

is required, where L_{w, lower} is the lower limit on the wind luminosity of the FRB pulsar to energize the persistent radio counterpart. We note that the value depends on the parameters of the nebula, such as the age of the nebula, the ejecta mass, the energy of the supernova, and so on. For FRB 121102, we adopt L_{w, lower} ∼ 10⁴⁰ erg s⁻¹ (Yang et al. 2016; Beloborodov 2017; Kashiyama & Murase 2017; Dai et al. 2017).

The other possibility is that the persistent radio counterpart is due to the activity of the companion. For example, for the SMBH companion case, the persistent radio counterpart might be due to the active galactic nucleus (AGN) activity of the SMBH (Zhang 2018). The kinetic luminosity of the outflow equals $\tfrac{1}{2}\dot{M}{V}^{2}$ and some fraction of this luminosity would be converted into the radio luminosity, L_radio. Thus, in this case, the kinetic luminosity must be higher than the luminosity of the persistent radio counterpart,

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\dot{M}{V}^{2}\gt {L}_{\mathrm{radio}}.\end{eqnarray} \tag{ 23 }$$

This is the least necessary condition for this case because the emission efficiency in the radio band would be less than unity.

The radio efficiency would change the constraint of Equation (23). If an _e fraction of the kinetic energy is converted to electron energy and an η_r fraction of the electron energy is radiated in the radio band, we obtain

$$\begin{eqnarray}&&{\epsilon }_{{\rm{e}}}{\eta }_{{\rm{r}}}\,\displaystyle \frac{1}{2}\dot{M}{V}^{2}\geqslant {L}_{\mathrm{radio}}\end{eqnarray} \tag{ 24 }$$

instead of Equation (23). We adopt _e η_r ∼ 10⁻³ in later discussions.

In the SMBH or IMBH case, the radio emission may originate from synchrotron radiation in the jet. In that case, some fraction of the accretion luminosity, $\sim 0.1{\dot{M}}_{\mathrm{acc}}{c}^{2}$ where ${\dot{M}}_{\mathrm{acc}}$ is the mass accretion rate of the black hole, is converted to the total jet luminosity, and some of the jet luminosity is radiated in the radio band. Besides this, some fraction of the accretion luminosity, $\sim 0.1{\dot{M}}_{\mathrm{acc}}{c}^{2}$, is converted to the outflow kinetic luminosity, $\dot{M}{V}^{2}/2$. Thus, the $\dot{M}{V}^{2}$ factor may not be directly related to L_radio so that conditions looser than Equation (23) are possible.

Using the binary comb model discussed so far, we constrain the parameters of the host binary of FRB 121102. The parameters used here are as follows. The period of FRB 121102 is 159 days and its duty cycle is 47% (Rajwade et al. 2020; Cruces et al. 2021). (The frequency dependence of the duty cycle is discussed in Section 6.) The change in the dispersion measure, ΔDM, is less than 6 cm⁻³ pc (Li et al. 2021b). Also, FRB 121102 is associated with the persistent radio counterpart whose luminosity, L_radio, is 10³⁹ erg s⁻¹ (Chatterjee et al. 2017; Marcote et al. 2017). If this persistent radio counterpart is the emission of a synchrotron nebula, the lower limit on the pulsar wind luminosity of the FRB pulsar is adopted as L_{w, lower} = 10⁴⁰ erg s⁻¹ (see Section 4.2). We fix the frequency of FRBs, ν, as 1 GHz, the energy of FRBs, L_FRBΔt, as 1 × 10³⁸ erg, and the temperature of the wind, T, as 10⁴ K. The FRB energy is adopted as L_FRBΔt = 1 × 10³⁸ erg, at which the event rate is highest in Cruces et al. (2021).

There are seven parameters for the binary, M_c, V, e, $\dot{M}$, L_w, the viewing angle, and the inclination angle. First, we consider the case that the binary is observed from the direction of its periapsis with an edge-on geometry. The effects of the viewing angle and the inclination angle are discussed later in this section and in the Appendix. In this section, for given M_c and V values, we put constraints on $\dot{M}$, L_w, and e. As in the previous section, we assume that each FRB is radiated spherically or the FRBs are radiated in random directions. The beaming effect is considered later.

Figures 4–6 show the constraints on the binary system of FRB 121102 in the cases that the companion is an SMBH (10⁵ M_⊙), an MS (10 M_⊙), and an IMBH (10³ M_⊙). The colored region is allowed in the funnel mode or the τ-crossing mode and the color contour shows the lower limit on the eccentricity. The blue line in the color contour represents the boundary between the funnel mode (above the line; Section 3.1) and the τ-crossing mode (below the line; Section 3.2). For a given eccentricity, the region allowed by the funnel mode corresponds to the line where θ_c is constant. This is because θ_d is uniquely determined for a given eccentricity (Equation (9)) and θ_d = θ_c is one of the conditions for the funnel mode (Equation (15)). The left ends of these lines (the blue line) are determined by the condition that the binary system is optically thick, r_ph > R_f, at f = π ± θ_d (Equation (16)). The right ends (part of the black line) are determined by the condition that the length of the spiral arm is longer than the photospheric radius, r_s > r_ph, at f = π ± θ_d (Equation (16)). The region allowed by the τ-crossing mode corresponds to the line where r_ph = R_f at f = π ± θ_d (Equation(18)). The right ends of these lines (the blue line) are determined by the condition that the funnel is small enough that the FRBs emitted in the inactive phase are not observable (Equation (17)). This line coincides with the line where the binary system becomes optically thick in the funnel mode. The area allowed in the τ-crossing mode does not have a left edge. This is because no matter how low $\dot{M}$ is chosen to be, by choosing a low L_w, the funnel size becomes smaller. Then, the FRBs are scattered by induced Compton scattering and the periodicity can be realized. The leftmost black line shows the parameters that realize the periodicity of FRB 121102 for the case of e = 0. The black line on the right of the yellow-colored region corresponds to the parameters for the case of e = 0.95. The light pink line in the color contour is the line where ΔDM = 6 cm⁻³ pc. The right side of this pink line is excluded because ΔDM becomes higher than the observed upper limit. The green dashed line represents L_w = L_w,lower, and the green dashed–dotted line represents ${L}_{\mathrm{radio}}=\dot{M}{V}^{2}/2$. To energize the persistent radio counterpart by wind, L_w must be higher than the green dashed line (energized by the FRB pulsar wind) or $\dot{M}$ must be higher than the green dashed–dotted line (energized by the companion wind). The magenta solid line is the line where $\beta ={L}_{{\rm{w}}}/\dot{M}{Vc}=1$ is satisfied. Above this line, the inverse funnel mode is realized. In summary, if the persistent radio counterpart is energized by wind, the allowed region is as follows: the colored region to the left of the light pink line and above the green dashed line for the effect of the radio efficiency, the colored region between the light pink line and the green dashed–dotted line ⁷ , and the region above the magenta solid line.

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For the SMBH case (Figure 4), there are allowed regions. Two different wind velocity cases, V = 3 × 10⁻² c and V = 1 × 10⁻¹ c are shown. In both cases, there are allowed regions in the inverse funnel mode where the persistent radio counterpart is energized by the disk wind or the FRB pulsar wind. In the case of V = 3 × 10⁻² c (left panel of Figure 4), the maximum value of $\tfrac{1}{2}\dot{M}{V}^{2}$ for the funnel mode or τ-crossing mode is about one hundred times higher than L_radio. Thus, if the AGN activity of the SMBH is responsible for the persistent radio counterpart, the emission efficiency in the radio band must be higher than ∼0.01. This value is higher than the adopted radio efficiency introduced in the last paragraph in Section 4.2, and so the disk wind cannot energize the persistent radio counterpart in these modes. Thus, only the FRB pulsar wind can be responsible for the persistent radio counterpart. In the case of V = 1 × 10⁻¹ c (right panel of Figure 4), the allowed region becomes larger than that for V = 3 × 10⁻² c. The AGN activity can explain the persistent radio counterpart as long as the emission efficiency in the radio band is higher than ∼4 × 10⁻⁴; that is, the disk wind can energize the persistent radio counterpart in these modes. The FRB pulsar wind can also energize the persistent radio counterpart. For the SMBH case, if the radio emission originates from the jet, the green dashed–dotted line can move to the left and the allowed region becomes larger. The stellar mass of the host galaxy of FRB 121102 is about (4–7) × 10⁷ M_⊙. The spectra of dwarf galaxies with such small masses are not well understood. Thus, the SMBH at a slight offset from the galaxy center could be responsible for the persistent radio counterpart and its observed spectrum.

For the MS case (Figure 5), two different wind velocity cases, V = 1 × 10⁻² c (the observed maximum velocity) and V = 3 × 10⁻² c (extremely high velocity) are shown. In both cases, there are allowed regions in the inverse funnel mode where the persistent radio counterpart would be energized by the FRB pulsar wind. An extremely high mass-loss rate (≳10^−1.5 M_⊙ yr⁻¹) is required to energize the persistent radio counterpart by the companion wind with a radio efficiency of 10⁻³. If V = 3 × 10⁻² c, there are allowed regions in the funnel mode where the persistent radio counterpart is energized by the FRB pulsar wind (right panel of Figure 5). If V = 1 × 10⁻² c, in the allowed region for the funnel mode or τ-crossing mode, the FRB pulsar wind luminosity is too weak to reproduce the persistent radio counterpart (left panel of Figure 5). Furthermore, the kinetic luminosity of the companion wind is lower than the radio luminosity of the persistent radio counterpart. Thus, the MS case with typical velocities V ≲ 1 × 10⁻² c is excluded in the funnel mode and τ-crossing mode.

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For the IMBH case (Figure 6), there are allowed regions in the cases of V = 1 × 10⁻¹ c and V = 3 × 10⁻² c. In both of the velocity cases, the inverse funnel mode is possible and the persistent radio counterpart is energized by the activity of the IMBH or the FRB pulsar wind. In the case of V = 1 × 10⁻¹ c, the persistent radio counterpart is energized by the synchrotron nebula or by the activity of the IMBH in the funnel mode or τ-crossing mode. In the case of V = 3 × 10⁻² c, the disk wind cannot be responsible for the persistent radio counterpart in the funnel mode and τ-crossing mode if the radio efficiency is taken into account, while the FRB pulsar is able to energize the persistent radio counterpart. As these figures show, the allowed region is highly dependent on the wind velocity of the companion wind.

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The beaming effect of each FRB loosens the constraints on the parameter space. In the funnel mode, if FRBs are beamed, the half-opening angle of the funnel, θ_c, must be larger than θ_d. This is because the beaming effect makes the active window shorter than that without the beaming effect. For a given eccentricity, on the line of θ_d = θ_c (constant eccentricity line for the funnel mode; see Figures 4–6), the observed duty cycle is realized without the beaming effect. For that eccentricity, between this line and the magenta solid line, we have θ_d < θ_c and the active window is larger than that on the θ_d = θ_c line. Therefore, for a given eccentricity, the region between the line of θ_d = θ_c and the magenta solid line is also allowed if the beaming effect is taken into account. In the τ-crossing mode, if FRBs are beamed, θ_d must be larger than the value for the unbeamed case. Thus, for a given eccentricity, the allowed region is to the left of the line allowed in the unbeamed case. The region to the right of the colored region is not allowed even if the beaming effect is taken into account.

If we consider the general viewing angle, there are two effects on the allowed region for the funnel mode. One effect is that the allowed region gets smaller. First, we consider the effect of changing the argument of periapsis. For given θ_c (i.e., L_w and $\dot{M}$) and e, if the argument of periapsis is changed, the duty cycle gets shorter (see Figures 11 and 12 in the Appendix). To make the duty cycle longer, the eccentricity must get higher. If there is an eccentricity that realizes the observed duty cycle, these parameters are also allowed. However, if there is no such eccentricity, these parameters are not allowed in the given argument of periapsis. Next, changing the inclination angle also makes the required eccentricity higher because the half-opening angle of the funnel effectively becomes smaller (see the Appendix). The other effect is that the disallowed region between the magenta solid line and the colored region in Figures 4–6 might become an allowed region. This region is not allowed because the half-opening angle is too large and the duty cycle is larger than the observed duty cycle. For given θ_c and e, if the argument of periapsis is changed, the duty cycle gets shorter (see Figures 11 and 12 in the Appendix). Thus, the observed duty cycle might be realized and these parameters becomes allowed. Changing the inclination angle also has the same effect (see Figure 12 in the Appendix). Even in this case, the region to the right of the colored region is not allowed.

For the τ-crossing mode, the allowed region would get smaller if we consider the general viewing angle. For a given e, if the argument of periapsis is changed, the region near the apoapsis gets optically thicker. Because the FRB pulsar spends more time near the apoapsis, the active window gets shorter. As a result, the allowed region would get smaller from the same discussion as in the funnel mode. Changing the inclination angle has the same effect. This is because there are more wind particles along the line of sight to the observers in I ≠ π/2.

Recent observations suggest that the beginning and the width of the active window of FRB 180916 depend on the observed frequency. Pleunis et al. (2020) compared the observations by the LOFAR high-band antenna (110–190 MHz), the uGMT (200–450 MHz), and the CHIME (400–800 MHz). Pastor-Marazuela et al. (2020) compared the observations by the LOFAR, the CHIME, and the Apertif (1220–1520 MHz) and reported that the peak activity and the width of the active phase vary with the frequency (their Figure 4). In their observations, the active window in the Apertif band begins at the same time as that in the CHIME band but ends earlier than in the CHIME band. The peak activity in the Apertif band is ∼0.7 days before that in the CHIME band. The full-width at half-maximum is 1.1 days in the Apertif band, while it is 2.7 days in the CHIME band. The active phase in the LOFAR band begins later than that in the CHIME band and ends later than in the CHIME band. The peak activity in the LOFAR band is ∼2 days after that of the CHIME, and the activity window is not estimated because of its lower number of samples.

The observed earlier active phase in a higher frequency band is actually predicted in the binary comb model. This is because higher frequency FRBs are less obscured by the plasma absorption or scattering in the companion wind. However, the observed earlier ending of the active window in a higher frequency band does not seem to support the binary comb model and Pleunis et al. (2020) and Pastor-Marazuela et al. (2020) have used this effect to disfavor the binary comb model and support a magnetar precession model. In this section, we revisit this problem and argue that it is premature to disfavor the binary comb model using the observational results of Pleunis et al. (2020) and Pastor-Marazuela et al. (2020). We suggest two scenarios in which a shorter active window is realized in the higher frequency band.

The first possibility is the sensitivity-absorption scenario. The earlier ending of the active window in the higher frequency band is due to the different sensitivities of different telescopes, with the threshold sensitivity in the higher frequency band shallower than in the lower frequency band. Let the intrinsic flux distribution profile have some structure in phase, as shown in the blue line in the right part of Figure 7. The sensitivities of the telescopes normalized by the flux profile are different between the telescopes (the top right of Figure 7). Only FRBs whose fluxes are higher than the observational threshold are observable. If the sensitivity of the telescope in the low-frequency range is higher than that in the high frequency range, fainter FRBs are observable in low frequency not in high frequency (the middle right of Figure 7). In addition to this observational threshold, low-frequency FRBs radiated in the earlier orbital phases are blocked by the wind because they propagate in a denser wind (the left side of Figure 7). Because the event rate is generally higher for fainter events, the final detected event rate in each frequency becomes frequency dependent as observed, as shown in the bottom right of Figure 7. This scenario is supported by the fact that the longest active phase is in the CHIME band where the sensitivity of the telescope is high, and by the fact that the phase of the active window is not monotonically dependent on frequency. In addition, we should note that the active window for each frequency also depends on the spectrum of the burst.

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The second possibility is the aurora scenario. In this scenario, the shorter active window in the higher frequency range is due to the intrinsic emission mechanism (Figure 8). As discussed in Ioka & Zhang (2020), the binary interaction might provide the condition to facilitate the FRB coherent radiation mechanism. Although the contact discontinuity of the funnel is larger than the magnetosphere of the FRB pulsar, some of the wind particles may enter the magnetosphere through the interface of the contact discontinuity of the funnel and the light cylinder of the FRB pulsar, forming an aurora-like inflow similar to that of Earth.

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The change in the number of the aurora particles may change the FRB frequency. One of the factors that may change the FRB frequency is the eccentricity. If the binary of FRB 180916 has nonzero eccentricity, the separation of the binary changes with time. When the separation is small, the number density of the companion wind around the contact discontinuity is high. When the number density is high, the number of the aurora particles would be large. Therefore, the smaller the separation, the more aurora particles are expected to be present. When the amount of the aurora particles is large, the plasma frequency around the emission area would be high (${\nu }_{p}\propto {n}_{{\rm{a}}}^{1/2}$, where ν_p is the plasma frequency, and n_a is the aurora particle number density). If FRBs are emitted due to plasma effects, the FRBs are expected to be emitted around the plasma frequency. Therefore, when the separation is small, the FRB frequency would be high. Because the separation changes with time, the FRB frequency also changes with time.

The FRB frequency may also change as the velocity of the companion wind received by the FRB pulsar changes with time. The velocity of the companion wind can vary with latitude, as in the case of the Sun. Thus, if the spin of the companion star and the orbital angular momentum of the binary are not parallel, the velocity of the companion wind received by the FRB pulsar can vary with time. The higher the velocity of the companion wind, the higher the kinetic energy of the wind particles. This high kinetic energy allows the aurora particles to reach more into the inner regions of the magnetosphere. The reason is as follows. In the inner region of the magnetosphere, the magnetic field is stronger, and the energy at the lowest Landau level becomes higher. Because the magnetic field does not perform any work on the aurora particles, the total energy of the particles does not increase. ⁸ Therefore, the particles cannot penetrate inside the region where the total energy of the particles is equal to the energy of the lowest Landau level. Thus, the radius where the aurora particles can reach, R_in, is given by solving the equation ⁹

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{m}_{{\rm{e}}}{V}^{2}=\displaystyle \frac{1}{2}{\rm{\hslash }}\displaystyle \frac{{eB}({R}_{\mathrm{in}})}{{m}_{{\rm{e}}}c},\end{eqnarray} \tag{ 25 }$$

where B(R_in) is the magnetic field at R_in. Assuming a dipolar magnetic field, we obtain

$$\begin{eqnarray}&&{R}_{\mathrm{in}\ }\simeq 1\times {10}^{7}{\left(\displaystyle \frac{V}{0.1c}\right)}^{-2/3}{\left(\displaystyle \frac{{B}_{\mathrm{NS}}}{{10}^{15}\,{\rm{G}}}\right)}^{1/3}\left(\displaystyle \frac{{R}_{\mathrm{NS}}}{{10}^{6}\,\mathrm{cm}}\right)\mathrm{cm},\end{eqnarray} \tag{ 26 }$$

where B_NS is the magnetic field at the neutron star surface and R_NS is the neutron star radius. If the aurora particles trigger the FRB radiation via curvature radiation, the FRB frequency (∝ γ³/ρ_c, where γ is the Lorentz factor of the particle and ρ_c is the magnetic field curvature radius) becomes higher in the inner region of the magnetosphere (Li & Zanazzi 2021).

In this paper, we develop the binary comb model to constrain the host binary for FRB 121102. We found that for each type of companion (MS, IMBH, and SMBH), there are some parameters in which the observed periodicity, the change in the dispersion measure, and the persistent radio counterpart of FRB 121102 are realized. An SMBH with a disk wind is one of the best candidates for the companion of FRB 121102. An IMBH is also a possible candidate. An MS is found to be an unfavorable companion because the stellar wind is too weak to power the persistent radio counterpart of FRB 121102. However, if the persistent radio counterpart is energized by the FRB pulsar, the MS companion is still viable because the binary may be in the inverse funnel mode (see Figure 5).

We have extended the binary comb model in two ways: by taking into account the eccentricity of the binary, and by considering the other modes not considered in the original model of Ioka & Zhang (2020). The addition of eccentricity to the model plays an essential role in explaining periodic FRBs with a large active window. In addition to the funnel mode (see Figure 1(a)) that has been considered in Ioka & Zhang (2020), we show that there are two other modes. One is the τ-crossing mode in which the period is explained by the temporal variation of the optical depth (see Figure 1(b)). The other is the inverse funnel mode in which the funnel is reversed (see Figure 1(c)).

For given companion mass and velocity of the companion wind, the allowed parameter space in the $\dot{M}$–L_w plane is constrained by the observed period, the duty cycle, the change in the dispersion measure, and the ability to provide enough energy for the persistent radio counterpart. For any type of companion, there are parameter regions where the observed signatures are realized in the inverse funnel mode. In the parameters where the periodicity and the duty cycle are realized in the funnel mode or τ-crossing mode, the other constraints (the change in the dispersion measure and the persistent radio counterpart) are important. In the MS companion case with a typical velocity V = 1 × 10⁻² c, there are some parameters where the change in the dispersion measure is less than the observed value in the funnel mode and τ-crossing mode. However, the persistent radio counterpart is not as bright as observed in these parameters. Therefore, these modes are excluded in this case, although they are allowed in the extremely high velocity case with V = 3 × 10⁻² c. In the SMBH or IMBH companion case with V = 1.0 × 10⁻¹ c (see also Zhang 2018), there is a parameter region in the funnel mode and τ-crossing mode where the upper limit on the change in the dispersion measure is satisfied and the persistent radio counterpart is as bright as observed. That said, if the radio emission originates from the jet of the central black hole, the cases of V = 3.0 × 10⁻² c are not excluded. It was found that a high velocity of the wind is important for the companion of FRB 121102.

We have also proposed two scenarios to explain the frequency dependence of the active window of FRB 180916. In the sensitivity-absorption scenario, the earlier ending of the active window at a higher frequency is due to the different sensitivities of different telescopes, and the later onset of the active phase at a low frequency is due to absorption. In the aurora scenario, the frequency dependent active window is explained by the change in the amount of the aurora particles in the magnetosphere. The separation of the binary or the wind velocity of the companion can change in orbital phases, and this change invokes a variation in the amount of aurora particles. The elaboration of these models is future work.

In the SMBH or IMBH companion case, the accretion disk of the black hole can affect the periodicity of the FRBs. The size of the accretion disk is ∼10³–10⁴ R_Sch for the AGN disk where R_Sch is the Schwarzschild radius of the black hole (Netzer 2015; Laor & Netzer 1989). This could be of the same order as the semimajor axis but is smaller than the semimajor axis. Therefore, it would not affect the active window in the case of FRB 121102. If the disk is larger than the shortest separation in the active window and θ_d is larger than π/2, the accretion disk can affect the periodicity. Also, the accretion disk has the potential to shine on its own. The upper limit on the luminosity of the persistent counterpart in optical band and in the X-ray band is ∼10⁴¹ erg s⁻¹ (Chatterjee et al. 2017). This value is ∼10⁻² L_Edd for the SMBH case and ∼L_Edd for the IMBH case. These are the upper limits on the luminosity of the black hole accretion disk.

In the future, it is expected that more periodic FRBs may be detected and they can help to determine the origin of the periodicity, e.g., binary or precession. In the binary comb model, the change in the dispersion measure ΔDM may be also periodic, providing a possible smoking-gun signature for this model. Furthermore, the variation of the rotation measure may be also periodic as proposed in Zhang (2018). Because the contribution to the rotation measure would be larger near the source than in the intergalactic medium, the periodic change in the rotation measure may be more prominent and likely be observable. Long-term RM variation of FRB 121102 has been observed showing a complicated behavior (Hilmarsson et al. 2021), and the binary scenario has not been compared to the data. In the binary comb model, $\dot{M}$ and V are important in determining the active window, and if they change during the observation, the periodicity may also change accordingly. In this case, the active window can change while the period does not change. Also, in the future, we might observe a periodic FRB from a binary in three-body systems. In this case, the period would be both increasing and decreasing in the binary comb model. By contrast, in the precession model, the period is either increasing or decreasing monotonically (Katz 2021a). If future periodic FRBs exhibit both increasing and decreasing periods, the origin of the period might be related to binaries in three-body systems.

In the SMBH or IMBH companion case, the persistent radio counterpart is expected to be created by the outflow from the black hole. It is an open question whether the radio emission due to the outflow can reproduce the observed spectrum. In the sensitivity-absorption scenario, the intrinsic flux distribution profile is assumed and its origin needs to be clarified. In the aurora scenario, the relationship between the FRB frequency and the separation or between the FRB frequency and the wind velocity needs to be clarified. These open questions will be studied in future works.

We are grateful to Kazuya Takahashi, Hamidani Hamid, Wataru Ishizaki, Koutarou Kyutoku, Susumu Inoue, Kazumi Kashiyama, Norita Kawanaka, Kohta Murase, and Shuta Tanaka for fruitful discussion and valuable comments. We also thank the participants and the organizers of the workshops with the identification number YITP-T-20-04 for their generous support and helpful comments. This work is supported by Grants-in-Aid for Scientific Research Nos. 20J13806 (TW), 20H01901, 20H01904, 20H00158, 18H01213, 18H01215, 17H06357, 17H06362, and 17H06131 (KI) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

In this Appendix, we consider the duty cycle for an arbitrary viewing angle (Figure 9) for the funnel mode. Generally, the configuration of a binary is determined by an orbital inclination angle, I, and an argument of periapsis, ω. The fundamental plane of the binary is orthogonal to the line of sight and the inclination angle, I, is defined as the angle between the fundamental plane and the orbital plane. The argument of periapsis ω is the angle from the ascending node to the periapsis, and the ascending node is where the star ascends from the fundamental plane. In the main text, we have adopted I = π/2 and ω = 3π/2.

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First, we evaluate the effect of changing the orbital inclination angle, I. For I ≠ π/2, the half-opening angle of the funnel becomes effectively smaller than that for I = π/2 (Figure 10). Thus, we introduce an effective half-opening angle, $\tilde{{\theta }_{{\rm{c}}}}$, as shown in Figure 10. We assume that the shape of the funnel is a cone whose apex angle equals θ_c. Then, $\tilde{{\theta }_{{\rm{c}}}}$ is related to the orbital inclination angle, I, and the half-opening angle of the funnel, θ_c, through the equation

$$\begin{eqnarray}&&\cos \tilde{{\theta }_{{\rm{c}}}}=\displaystyle \frac{\cos {\theta }_{{\rm{c}}}}{\sqrt{1-{\cos }^{2}{\theta }_{{\rm{c}}}{\cot }^{2}I}}.\end{eqnarray} \tag{ A1 }$$

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In the case of I = π/2 − θ_c, the effective half-opening angle equals zero because the line of sight coincides with the baseline of the cone.

Next, we evaluate the effect of changing the argument of periapsis, ω. We can observe FRBs when the true anomaly is between $\tfrac{5\pi }{2}-\omega -\tilde{{\theta }_{{\rm{c}}}}$ and $\tfrac{5\pi }{2}-\omega +\tilde{{\theta }_{{\rm{c}}}}$ (see Figure 9). Thus, the duty cycle is represented as

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{T}{P}=\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{u}_{-}}^{{u}_{+}}\displaystyle \frac{{dt}}{{du}}{du}=\displaystyle \frac{1}{2\pi }\left[({u}_{+}-e\sin {u}_{+})\right.\\ \quad \left.\,-({u}_{-}-e\sin {u}_{-})+2\pi {\rm{\Theta }}(-{u}_{+}){\rm{\Theta }}({u}_{-})\right],\end{array}\end{eqnarray} \tag{ A2 }$$

$$\begin{eqnarray}&&\begin{array}{r}\begin{array}{l}{u}_{\pm }=2{\tan }^{-1}\left(\sqrt{\displaystyle \frac{1-e}{1+e}}\tan \displaystyle \frac{{\textstyle \tfrac{5\pi }{2}}-\omega \pm \tilde{{\theta }_{{\rm{c}}}}}{2}\right)\\ \,(-\pi \lt {u}_{\pm }\lt \pi ),\end{array}\end{array}\end{eqnarray} \tag{ A3 }$$

where Θ(x) is the step function. We note that the relation between the true anomaly and the eccentric anomaly is valid in − π < f < π. Thus, in the case of u₋ > 0 and u₊ < 0, we have to calculate the integration in Equation (A2) separating u₋ < u < π and − π < u < u₊ and an additional 2π factor appears.

Using Equations (A1) and (A2), we obtain the relation between the eccentricity and the duty cycle (Figures 11 and 12). In each figure, we fix θ_c and I, and change ω from π/2 (observation from the direction of the periapsis) to 3π/2 (observation from the direction of the apoapsis). Figure 11 shows the relations for (θ_c, I) = (0.25π, 0.5π), (0.25π, 0.275π), and Figure 12 shows the relations for (θ_c, I) = (0.1π, 0.5π), (0.5π, 0.5π). In the case of e = 0, the duty cycle does not depend on ω because there is no specific direction for the circular orbit. However, in the case of e ≠ 0, the duty cycle varies depending on the argument of periapsis. In the case of $\left|\tfrac{3\pi }{2}-\omega \right|\lt \tilde{{\theta }_{{\rm{c}}}}$, the duty cycle grows with e and approaches 100% as e approaches unity. In the case of $\left|\tfrac{3\pi }{2}-\omega \right|\gt \tilde{{\theta }_{{\rm{c}}}}$, the duty cycle grows or decreases as e grows, and finally approaches 0% as e approaches unity. This is because the FRB pulsar spends more time around the apoapsis as e grows, and the FRBs radiated at the apoapsis are observable in the case of $\left|\tfrac{3\pi }{2}-\omega \right|\lt \tilde{{\theta }_{{\rm{c}}}}$, and they are not observable in the case of $\left|\tfrac{3\pi }{2}-\omega \right|\gt \tilde{{\theta }_{{\rm{c}}}}$. As these figures show, any duty cycle can be realized if the binary has nonzero eccentricity and the viewing angles are free parameters.

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