A Binary Comb Model for Periodic Fast Radio Bursts

With the observed period (Equation (1)) identified as the binary orbital period, the semimajor axis of the binary is

$$\begin{eqnarray}&&a={\left({GM}\right)}^{1/3}{\left({P}_{\mathrm{orb}}/2\pi \right)}^{2/3}\sim 4\times {10}^{12}\,\mathrm{cm}\,{M}_{1}^{1/3}{P}_{\mathrm{orb},16}^{2/3},\end{eqnarray} \tag{ 2 }$$

where $M={m}_{A}+{m}_{B}=10{M}_{\odot }{M}_{1}$ is the total mass of the binary and ${P}_{\mathrm{orb},16}={P}_{\mathrm{orb}}/16$ days. The separation between the stars ranges from ${a}_{\min }=a(1-e)$ to ${a}_{\max }=a(1+e)$ for an eccentricity e. For a massive star companion, the total mass is M ∼ (10–50) M_⊙. For main-sequence stars, the stellar radius is about ∼3 × 10¹¹ cm ${M}_{1}^{0.57}$ for M₁ ≳ 0.1 (Kippenhahn & Weigert 1990), which is smaller than the binary separation. For a neutron star companion, the total mass is M ∼ 2.8M_⊙ and hence a ∼ 2.6 × 10¹² cm.

2.2.1. Massive Star Companion

For a massive star case, the wind density around the FRB source is

$$\begin{eqnarray}&&{n}_{w}(0)\sim \displaystyle \frac{\dot{M}}{4\pi {a}^{2}{m}_{p}V}\sim 9\times {10}^{5}\,{\mathrm{cm}}^{-3}\,{\dot{M}}_{-9}{a}_{12.6}^{-2}{V}_{3.3}^{-1},\end{eqnarray} \tag{ 3 }$$

where a_12.6 = a/4 × 10¹² cm, V = 2 × 10³ km s⁻¹ V_3.3 is the wind velocity, and m_p is the proton mass. We adopt a mass-loss rate $\dot{M}={10}^{-9}{M}_{\odot }\,{\mathrm{yr}}^{-1}{M}_{-9}$ of main-sequence B stars as the fiducial value because they are popular (see also Section 4.1). Note that the mass-loss rate of O7 and later stars is a factor of 10–10² lower than theoretically expected (Puls et al. 2008; Smith 2014). The B star becomes a rapidly rotating Be star through a mass-exchange episode before the FRB pulsar is born (e.g., Postnov & Yungelson 2014). The equatorial mass-loss rates of Be stars may be larger by a factor of ∼10² (Nieuwenhuijzen & de Jager 1988).

The optical depth to Thomson scattering is small τ_T ∼ σ_Tn_w(0) a ∼ 2 × 10⁻⁶ around the FRB source where σ_T is the Thomson cross section. The optical depth to free–free absorption is ${\tau }_{\mathrm{ff}}\sim a{\alpha }_{\nu }^{\mathrm{ff}}\sim 0.06\,{\bar{g}}_{\mathrm{ff}}{T}_{4}^{-3/2}{\nu }_{9}^{-2}$ for fiducial parameters where ${\alpha }_{\nu }^{\mathrm{ff}}=(4{q}^{6}/3{m}_{e}{kc}){\left(2\pi /3{{km}}_{e}\right)}^{1/2}{T}^{-3/2}{n}_{w}{\left(0\right)}^{2}{\nu }^{-2}{\bar{g}}_{\mathrm{ff}}$ is the free–free absorption coefficient at frequency ν = 1 GHz ν₉ and temperature T = 10⁴ K T₄ (Lyutikov et al. 2020).

More important are the induced scattering processes (Wilson & Rees 1978; Thompson et al. 1994; Lyubarsky 2008) because the brightness temperature of the FRB is extremely high (e.g., T_FRB ∼ 10³² K) and the scattering probability of bosons is enhanced by the occupation number of the final state kT_FRB/hν ∼ 2 × 10³³ T_FRB,32ν₉⁻¹. The optical depth to the induced Compton scattering at a radius $r={10}^{13}\,\mathrm{cm}\,{r}_{13}$ is estimated by

$$\begin{eqnarray}&&{\tau }_{C}\sim \displaystyle \frac{3{\sigma }_{T}}{32{\pi }^{2}}\displaystyle \frac{{n}_{w}(r){Lc}{\rm{\Delta }}t}{{r}^{2}{m}_{e}{\nu }^{3}}\sim 30\,{\dot{M}}_{-9}{V}_{3.3}^{-1}{r}_{13}^{-4}{\left(L{\rm{\Delta }}t\right)}_{38}{\nu }_{9}^{-3},\end{eqnarray} \tag{ 4 }$$

where $L{\rm{\Delta }}t={10}^{38}\,\mathrm{erg}{\left(L{\rm{\Delta }}t\right)}_{38}$ is the FRB isotropic luminosity times duration, and the wind density decreases as ${n}_{w}(r)\,\sim {n}_{w}(0){[a/(a+r)]}^{2}$ (∝r⁻² at r ≫ a). Here r is measured from the FRB source, not from the companion. Using a simple criterion for observability τ_C < 10 (Lyubarsky 2008), the photospheric radius for the induced Compton scatterings is

$$\begin{eqnarray}&&{r}_{\mathrm{ph}}^{C}\sim 1\times {10}^{13}\,\mathrm{cm}\,{\left(L{\rm{\Delta }}t\right)}_{38}^{1/4}{\dot{M}}_{-9}^{1/4}{V}_{3.3}^{-1/4}{\nu }_{9}^{-3/4}.\end{eqnarray} \tag{ 5 }$$

The above expression is easy to understand as follows. The photon occupation number is given by

$$\begin{eqnarray}&&{\mathscr{N}}=\displaystyle \frac{{c}^{2}{L}_{\nu }}{8{\pi }^{2}{\theta }_{b}^{2}{r}^{2}h{\nu }^{3}}\sim \displaystyle \frac{{c}^{2}L}{8{\pi }^{2}{\theta }_{b}^{2}{r}^{2}h{\nu }^{4}},\end{eqnarray} \tag{ 6 }$$

where ${L}_{\nu }$ is the isotropic specific luminosity. For induced Compton scattering, the scattered photon lies within the half-opening angle of the photon beam θ_b, so that the cross section is ${\sigma }_{C}\sim {\sigma }_{{\rm{T}}}{\mathscr{N}}{\theta }_{b}^{2}/4$. In each scattering, a photon loses a fraction ${\varepsilon }_{C}\sim h\nu {\theta }_{b}^{2}/2{m}_{e}{c}^{2}$ of its energy. Then the effective optical depth is estimated by ${\tau }_{C}\sim {\varepsilon }_{C}{\sigma }_{C}{n}_{w}r$, which reproduces Equation (4) within a factor of π/3 if we replace $r{\theta }_{b}^{2}/2$ by cΔt because the induced scattering occurs only if the scattered ray remains within the zone illuminated by the scattering radiation (Lyubarsky 2008). If ${\theta }_{b}\lt {\left(2c{\rm{\Delta }}t/r\right)}^{1/2}\sim 8\,\times {10}^{-3}{\left({\rm{\Delta }}t/10\mathrm{ms}\right)}^{1/2}{r}_{13}^{-1/2}$, this replacement is not necessary. Note that the Planck constant h is canceled in the product of ${\mathscr{N}}$ and ${\varepsilon }_{C}$.

The induced Raman scattering by emitting Langmuir waves could be even more significant. The optical depth at $r={10}^{14}\,\mathrm{cm}\,{r}_{14}$ is estimated by

$$\begin{eqnarray}&&{\tau }_{R}\sim {\tau }_{C}\nu /{\nu }_{p}\sim 9\,{\dot{M}}_{-9}^{1/2}{V}_{3.3}^{-1/2}{r}_{14}^{-3}{\left(L{\rm{\Delta }}t\right)}_{38}{\nu }_{9}^{-2},\end{eqnarray} \tag{ 7 }$$

where ${\nu }_{p}={[{q}^{2}{n}_{w}(r)/(\pi {m}_{e})]}^{1/2}$ is the plasma frequency, if the scattering angle is not too small and the decay of plasmons is weak (Thompson et al. 1994; Lyubarsky 2008). The photospheric radius for induced Raman scattering is

$$\begin{eqnarray}&&{r}_{\mathrm{ph}}^{R}\sim 1\times {10}^{14}\,\mathrm{cm}\,{\left(L{\rm{\Delta }}t\right)}_{38}^{1/3}{\dot{M}}_{-9}^{1/6}{V}_{3.3}^{-1/6}{\nu }_{9}^{-2/3}.\end{eqnarray} \tag{ 8 }$$

Note that the Raman scattering effect just widens the beam to ${\theta }_{b}\sim 6\times {10}^{-2}{\left({n}_{w}(r)/{10}^{3}{\mathrm{cm}}^{-3}\right)}^{1/2}{T}_{4}^{-1/2}$ but temporally smears a pulse to $\sim r{\theta }_{b}^{2}/2c\gt {\rm{\Delta }}t$.

The photosphere ${r}_{\mathrm{ph}}\sim {10}^{13}$–10¹⁴ cm is larger than the separation a in Equation (2) for fiducial parameters. It is also remarkable that the photosphere is larger than the separation even for a Sun-like star with $\dot{M}\sim 2\times {10}^{-14}{M}_{\odot }$ yr⁻¹, M ∼ 2.4M_⊙, and V ∼ 800 km s⁻¹. Therefore, the stellar wind basically makes the system optically thick.

2.2.2. Neutron Star Companion

For the neutron star companion case, the wind density around the FRB source at $a\sim 2.6\times {10}^{12}\,\mathrm{cm}\,{a}_{12.4}$ is

$$\begin{eqnarray}&&{n}_{w}(0)\sim \displaystyle \frac{{L}_{w}}{4\pi {a}^{2}{m}_{e}{c}^{2}V{\rm{\Gamma }}(1+\sigma )}\sim \displaystyle \frac{5\times {10}^{3}\,{{\rm{cm}}}^{-3}}{{\rm{\Gamma }}(1+\sigma )}{L}_{w,34}{a}_{12.4}^{-2},\end{eqnarray} \tag{ 9 }$$

where we take $V\sim c$, ${\rm{\Gamma }}={[1-{\left(V/c\right)}^{2}]}^{-1/2}$ is the Lorentz factor of the wind, and σ is the ratio of Poynting flux to particle energy flux. For the fiducial value of the wind luminosity, we take that of a typical millisecond pulsar ${L}_{w}={10}^{34}{L}_{w,34}$, because a millisecond pulsar is usually formed in a neutron star binary system and is the one with the higher spin-down rate as observed in our Galaxy (e.g., Tauris et al. 2017).

The optical depth to the induced Compton scattering is easy to estimate in the comoving frame of the wind to take relativistic effects into account,

$$\begin{eqnarray}\begin{array}{rcl}{\tau }_{C} & \sim & \displaystyle \frac{3{\sigma }_{T}}{32{\pi }^{2}}\displaystyle \frac{{n}_{w}^{{\prime} }(r)L^{\prime} c{\rm{\Delta }}t^{\prime} }{{r}^{{\prime} 2}{m}_{e}{\nu }^{{\prime} 3}}\\ & \sim & \displaystyle \frac{8{\delta }^{2}}{{{\rm{\Gamma }}}^{2}(1+\sigma )}{L}_{w,34}{r}_{12.5}^{-4}{\left(L{\rm{\Delta }}t\right)}_{38}{\nu }_{9}^{-3},\end{array}\end{eqnarray} \tag{ 10 }$$

where the relations with the lab-frame quantities are $L^{\prime} =L/{\delta }^{4}$, $\nu ^{\prime} =\nu /\delta$, ${\rm{\Delta }}t^{\prime} =\delta {\rm{\Delta }}t$, $r^{\prime} =r/\delta$, and ${n}_{w}^{{\prime} }={n}_{w}/{\rm{\Gamma }}$, and $\delta \,={[{\rm{\Gamma }}(1-(V/c)\cos {\theta }_{w})]}^{-1}$ is the Doppler factor for an angle ${\theta }_{w}$ between the photon and the wind direction. Note that L/r² transforms as a flux. The photosphere is located at

$$\begin{eqnarray}&&{r}_{\mathrm{ph}}^{C}\sim 2\times {10}^{12}\,\mathrm{cm}\,{\delta }^{1/2}{{\rm{\Gamma }}}^{-1/2}{\left(1+\sigma \right)}^{-1/4}{L}_{w,34}^{1/4}{\left(L{\rm{\Delta }}t\right)}_{38}^{1/4}{\nu }_{9}^{-3/4}.\end{eqnarray} \tag{ 11 }$$

The photospheric radius for the induced Raman scattering is estimated from ${\tau }_{R}\sim {\tau }_{C}\nu ^{\prime} /{\nu }_{p}^{{\prime} }\sim 10$ as

$$\begin{eqnarray}&&{r}_{\mathrm{ph}}^{R}\sim 3\times {10}^{13}\,\mathrm{cm}\,{\delta }^{1/3}{{\rm{\Gamma }}}^{-1/3}{\left(1+\sigma \right)}^{-1/6}{L}_{w,34}^{1/6}{\left(L{\rm{\Delta }}t\right)}_{38}^{1/3}{\nu }_{9}^{-2/3}.\end{eqnarray} \tag{ 12 }$$

Note that δ ∼ Γ for ${\theta }_{w}\ll {{\rm{\Gamma }}}^{-1}$, and $\delta \sim 2/({\rm{\Gamma }}{\theta }_{w}^{2})$ for ${\theta }_{w}\gg {{\rm{\Gamma }}}^{-1}$. Although the Lorentz factor Γ and magnetization parameter σ are quite uncertain, the dependence is weak, so that the pulsar wind makes the system optically thick and temporally smears a pulse to $\sim {r}_{\mathrm{ph}}{\theta }_{w}^{2}/2c\gt {\rm{\Delta }}t$ for a large parameter space.

For the wind from the FRB pulsar, the Doppler factor is δ ∼ Γ since θ_w ∼ 0. An FRB pulse is likely generated below the photosphere and is temporally smeared to $\sim {r}_{\mathrm{ph}}/2c{{\rm{\Gamma }}}^{2}$ via scatterings. However, this is shorter than the pulse width Δt for large Lorentz factors, and the FRB pulsar itself is observable.

The wind from the companion basically makes the system optically thick. In order to make an FRB observable by an Earth observer, the wind from the FRB pulsar should open a way to the observer. Namely, a cosmic comb retains a clear funnel for the FRB to propagate (Figure 1). Since all the bursts arrive in a 4 day phase window of the 16 day period, the half-opening angle of the comb should be⁶

$$\begin{eqnarray}&&{\theta }_{c}\gtrsim \pi /4.\end{eqnarray} \tag{ 13 }$$

In principle, the opening angle can be arbitrarily small for a highly eccentric orbit because the polar angle swept by the FRB pulsar during the 4 day phase becomes smaller for higher eccentricity around the apocenter. In this case the observable viewing angle is also small.

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Near the FRB pulsar with a distance much smaller than the binary separation, the comb structure is obtained by a problem that a wind-blowing star moves with a constant velocity in a uniform density. The shape of the contact discontinuity is obtained analytically as

$$\begin{eqnarray}&&r(\theta )={r}_{0}\csc \theta \sqrt{3[1+(\pi -\theta )\cot \theta ]},\end{eqnarray} \tag{ 14 }$$

in a thin shock limit (Wilkin 1996), where r(θ) is the distance from the FRB pulsar, θ is the polar angle from the axis of symmetry, and r₀ is the minimum size at θ = π (toward the companion). This solution is consistent with numerical simulations of pulsar bow shocks (Bucciantini 2002; Vigelius et al. 2007).

The above solution is applicable only up to the binary separation r(θ) ∼ a because the radial dependence of the companion's wind becomes similar to that of the wind from the FRB pulsar (i.e., the approximation of a uniform density breaks down). Because of the same radial dependence (e.g., fluxes ∝ r⁻²), the polar angle of the contact discontinuity asymptotically becomes constant, which determines the opening angle of the comb (Figure 1). Requiring r(π/4) ≳ a based on Equation (13), we find that the comb size on the side of the companion should be larger than

$$\begin{eqnarray}&&{r}_{0}\gtrsim 0.22a\sim 9\times {10}^{11}\,\mathrm{cm}\,{M}_{1}^{1/3}{P}_{\mathrm{orb},16}^{2/3}.\end{eqnarray} \tag{ 15 }$$

We stress that the above condition is a necessary condition. The duty cycle is also related to the solid angle ${\rm{\Delta }}{\rm{\Omega }}$ in which the bulk of FRBs are concentrated. (Note that this is different from the beaming angle of each FRB $\delta {\rm{\Omega }}\sim \pi {\theta }_{b}^{2}$.) This is even implied by the observations because the European Very-long-baseline-interferometry Network (EVN) at 1.7 GHz detected bursts at the leading edge of the activity cycle observed at 400–800 MHz, while the Effelsberg radio telescope at 1.4 GHz detected no bursts during the middle of the cycle (The CHIME/FRB Collaboration et al. 2020). Only a plasma eclipse cannot explain the high-frequency deficit in the middle phase because high-frequency photons are generally transmittable. However, we should await more observations to confirm the periodicity at high frequencies.

On the other hand, in order for the FRB pulsar to be combed, the companion wind pressure must win at half of the separation, so that the comb size should satisfy

$$\begin{eqnarray}&&{r}_{0}\lesssim 0.5a\sim 2\times {10}^{12}\,\mathrm{cm}\,{M}_{1}^{1/3}{P}_{\mathrm{orb},16}^{2/3}.\end{eqnarray} \tag{ 16 }$$

Therefore, if the comb triggers FRBs, the comb size is constrained to a relatively narrow range 0.22a ≲ r₀ ≲ 0.5a in Equations (15) and (16). Remember that the eccentricity relaxes the lower limit while the inclination tightens it.

One more necessary condition arises because the comb tail is spiraled by the orbital motion at a radius

$$\begin{eqnarray}&&{r}_{s}\sim \displaystyle \frac{{{VP}}_{\mathrm{orb}}}{2\pi }\displaystyle \frac{{\theta }_{c}}{\pi /4}\sim 4\times {10}^{13}\,\mathrm{cm}\,{P}_{\mathrm{orb},16}{V}_{3.3}(4{\theta }_{c}/\pi ),\end{eqnarray} \tag{ 17 }$$

for the massive star case, and ${r}_{s}\sim 7\times {10}^{15}$ cm ${P}_{\mathrm{orb},16}(4{\theta }_{c}/\pi )$ for the neutron star case with V ∼ c. This spiral radius should be larger than the photosphere; otherwise, the wind eventually shields the line of sight as shown in Figure 1. This condition is marginally satisfied for the massive star case as the photospheric radius is r_ph ∼ (10¹³–10¹⁴) cm in Equations (5) and (8), while it is satisfied for the neutron star case. We can also predict a lack of sources with P_orb ≲ 10 days for the massive star case and P_orb ≲ 0.1 days (with some dependence on Γ and σ) for the neutron star case.

The comb size necessary for the duty cycle in Equation (15) is usually larger than that of the light cylinder of the FRB pulsar with a spin period P_FRB = 1 s P_FRB,0,

$$\begin{eqnarray}&&{r}_{{\ell }}\sim {{cP}}_{\mathrm{FRB}}/(2\pi )\sim 5\times {10}^{9}\,\mathrm{cm}\,{P}_{\mathrm{FRB},0}.\end{eqnarray} \tag{ 18 }$$

Then the ram pressure balance between the winds from the companion and the FRB pulsar is expressed by

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{w}}{4\pi {a}^{2}V}\sim \displaystyle \frac{{B}_{p}^{2}}{8\pi }{\left(\displaystyle \frac{R}{{r}_{{\ell }}}\right)}^{6}{\left(\displaystyle \frac{{r}_{{\ell }}}{{r}_{0}}\right)}^{2},\end{eqnarray} \tag{ 19 }$$

where R ∼ 10 km is the neutron star radius, ${L}_{w}\sim \dot{M}{V}^{2}$ for the massive star case, and V ≈ c for the neutron star case. For the massive star case, the opacity condition in Equation (15) gives

$$\begin{eqnarray}&&{B}_{p}\gtrsim 3\times {10}^{13}\,{\rm{G}}\,{P}_{\mathrm{FRB},0}^{2}{\dot{M}}_{-9}^{1/2}{V}_{3.3}^{1/2},\end{eqnarray} \tag{ 20 }$$

where the equality holds if the comb also stimulates an FRB in Equation (16). For the neutron star case, Equation (15) gives

$$\begin{eqnarray}&&{B}_{p}\gtrsim 4\times {10}^{12}\,{\rm{G}}\,{P}_{\mathrm{FRB},0}^{2}{L}_{w,34}^{1/2},\end{eqnarray} \tag{ 21 }$$

where the equality holds if Equation (16) is also true. These conditions are presented in Figure 2.

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The total energy of FRBs during the whole lifetime ${t}_{\mathrm{life}}={10}^{4}\,\mathrm{yr}\,{t}_{\mathrm{life},4}$ is

$$\begin{eqnarray}\begin{array}{rcl}E & = & \dot{N}{t}_{\mathrm{life}}L{\rm{\Delta }}t({\rm{\Delta }}{\rm{\Omega }}/4\pi )\\ & \sim & 4\times {10}^{42}\,\mathrm{erg}\,{\dot{N}}_{25}{t}_{\mathrm{life},4}{\left(L{\rm{\Delta }}t\right)}_{38}{\rm{\Delta }}{{\rm{\Omega }}}_{0.6\pi },\end{array}\end{eqnarray} \tag{ 22 }$$

where the observed burst rate is $\dot{N}\sim 25$ yr${}^{-1}{\dot{N}}_{25}$, the true energy of each burst is smaller by a factor of δΩ/4π, and the total number of bursts is increased by a factor of ΔΩ/δΩ (Zhang 2020). We take ${\rm{\Delta }}{\rm{\Omega }}\sim 2\pi (1-\cos (\pi /4))\sim 0.6\pi$ as a fiducial value as implied by the duty cycle in Equation (13). This energy can be supplied by the magnetic energy E_B > E if

$$\begin{eqnarray}&&{B}_{p}\gtrsim 5\times {10}^{12}\,{\rm{G}}\,{\dot{N}}_{25}^{1/2}{t}_{\mathrm{life},4}^{1/2}{\left(L{\rm{\Delta }}t\right)}_{38}^{1/2}{\rm{\Delta }}{{\rm{\Omega }}}_{0.6\pi }^{1/2}.\end{eqnarray} \tag{ 23 }$$

This is marked as the yellow shaded region in Figure 2.⁷

Combining Equations (20), (21), and (23), we find the FRB pulsar parameters in the range that includes magnetars and young high-B pulsars with B_p ∼ 10¹³–10¹⁵ G and P_FRB ∼ 1–10 s for a lifetime t_life ∼ 10⁴ yr (Figure 2). With these parameters, the FRB pulsar has enough energy and enough luminosity for pertaining a funnel in the wind from the companion while it is still combed by the wind to trigger FRBs.

Notice that Galactic binary neutron star systems do not satisfy the energy constraint in Equation (23), because their ages are much older, i.e., t_life ≫ 10⁴ yr (e.g., Tauris et al. 2017). Gamma-ray binary systems such as PSR B1259-63 (Aharonian et al. 2005) and PSR J2032+4127 (Lyne et al. 2015) also do not satisfy the energy constraint. Only a relatively new born pulsar could have crust or magnetic configurations that can frequently trigger FRBs via crust cracking or magnetic reconnection. These intrinsic triggering factors could also explain why weaker Galactic analogs are not observed. The pulsar B of the double pulsar system J0737-3039 has ${P}_{\mathrm{FRB}}\sim 2.7735\,{\rm{s}}$ and B_p = 4.9 × 10¹¹ G. It also does not satisfy the opacity constraint in Equation (21). The Galactic high-mass X-ray binaries do not satisfy the opacity constraint because the neutron stars are accreting matter. Although many Galactic neutron stars satisfy both the opacity and energy constraints, they are not in binaries. In any case, we do not exclude the possibility that Galactic binary neutron star systems may emit FRB-like signals with much lower luminosities at a much lower rate. Detections or nondetections of such signals in long-term monitoring of these systems may lend support or pose constraints on the scenario proposed in this Letter.

From the opacity constraint, the interaction radius in Equation (15) is larger than the light cylinder in Equation (18). The magnetic field at the interaction radius is weaker than that of the inner magnetosphere near the neutron star surface where the most energy E_B is stored to produce an FRB. Then binary interactions cannot change the inner magnetic structure to trigger an FRB unless a much tighter binary system is invoked. In Earth's magnetosphere, for example, magnetic reconnection takes place in the dayside magnetopause and the near-Earth plasma sheet. No reconnection happens near Earth's surface.

On the other hand, binary interactions may change the current structure (electron density) in the inner magnetosphere, which may be related to the coherent condition for FRB emission. In intermittent pulsars (Kramer et al. 2006) and mode-switching pulsars (Lyne et al. 2010), the change of the spin-down rate (possibly related to the change of the magnetospheric structure) is connected with the change of the radio emission condition.

Based on the energy budget argument, the FRB energy should be dissipated from the inner magnetosphere of the FRB pulsar itself, likely due to crust cracking or magnetic reconnection. This intrinsic trigger would be similar to that of ordinary magnetars or young, high-magnetic-field pulsars. However, we speculate that binary interaction may provide the condition to facilitate the FRB coherent radiation mechanism, so that only a small fraction of the intrinsic trigger events can lead to FRBs (see Section 4.2 for more detailed arguments). The interface of the interaction region and the light cylinder of the FRB pulsar makes a connection between the external wind and inner magnetosphere (see Figure 1), powering an aurora (Perreault & Akasofu 1978; Ebihara & Tanaka 2020) or lightening (Scott et al. 2014) similar to that in Earth's magnetosphere. Possibly the massive star companion provides substantial seed photons for creating high-energy particles. The sudden release of energy in the inner magnetosphere launches a strong particle outflow. When this outflow interacts with the aurora plasma, two-stream instability may drive particle bunches that meet the coherent condition for FRBs (Kashiyama et al. 2013; Kumar et al. 2017; Yang & Zhang 2018). Alternatively, coherent radiation may be generated by the conversion from reconnection-driven fast magnetosonic waves to electromagnetic waves (Philippov et al. 2019; Lyubarsky 2020). The aurora particles might change the physical parameters such as the conversion radius and Lorentz factor, and hence the coherent condition. These mechanisms can reproduce the FRB properties such as polarization and downward-drifting subpulses (Wang et al. 2019).

For FRB 180916.J0158+6, the change of dispersion measure (DM) is constrained as ΔDM < 0.1 pc cm⁻³ (The CHIME/FRB Collaboration et al. 2020). This is easily satisfied for the neutron star companion case. For the massive star companion case, the contribution to ΔDM only comes from the radius beyond the photosphere. Considering the difference in the path length during the 4 day active phase, we find that the DM variation is limited by

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}\mathrm{DM} & \lesssim & {n}_{w}({r}_{\mathrm{ph}}){r}_{\mathrm{ph}}[1-\cos (\pi /4)]\\ & \sim & 0.05\ \mathrm{pc}\ {\mathrm{cm}}^{-3}\,{M}_{-9}{V}_{3.3}^{-1}{r}_{\mathrm{ph},14}^{-1},\end{array}\end{eqnarray} \tag{ 24 }$$

which is consistent with the observation. More precise observations could detect the DM variation. Note that at the photosphere, the electric field of the FRB radiation is too weak to accelerate electrons to relativistic energies on the timescale of ${\left(2\pi \nu \right)}^{-1}$ to reduce the DM (Lu & Phinney 2019; Yang & Zhang 2020). The outflow associated with FRBs could also bring additional ΔDM (Yamasaki et al. 2019).

We expect an even larger mass-loss rate for a more massive star and, hence, a larger ΔDM. Thus, a main-sequence B star companion is preferred for the periodic FRB 180916.J0158+65. About 20 CHIME bursts do not show a large ΔDM, possibly except for source 5 in Fonseca et al. (2020). Since more massive stars are less abundant, this observation is consistent with our scenario, even though more samples are needed to verify the massive star companion case.

The rotation measure (RM) of the source is measured as RM $\sim -114.6\pm 0.6\,\mathrm{rad}$ m⁻² (CHIME/FRB Collaboration et al. 2019). For the neutron star companion case, the expected RM is small. For the massive star companion case, there is no evidence of magnetic field for Be stars, and the dipole field component of 50% of the stars is probably weaker than 50 G (Wade et al. 2014). At the photosphere, this corresponds to $\lesssim 50\,{\rm{G}}\,{\left(3\times {10}^{11}\mathrm{cm}/{r}_{\mathrm{ph}}\right)}^{2}\sim 5\,\times \,{10}^{-4}$ G. The expected absolute value of RM is $\sim {q}^{3}{\left(2\pi {m}^{2}{c}^{4}\right)}^{-1}{\int }_{{r}_{\mathrm{ph}}}{n}_{w}{B}_{\parallel }{dr}\lesssim 20\,\mathrm{rad}$ m⁻², less than the observations. The nondetection of RM variation implies that the RM comes from a further distance such as the persistent emission region (Yang et al. 2020).

The persistent radio counterpart is constrained to have a luminosity νL_ν < 1.3 × 10³⁶ erg s⁻¹ at 1.7 GHz by the continuum EVN data and νL_ν < 7.6 × 10³⁵ erg s⁻¹ at 1.6 GHz by the VLA data (Marcote et al. 2020). In our model the wind luminosities are less than these constraints. In the massive star case, the companion mainly shines in the optical band, also consistent with the observations.

Given the burst fluxes ∼1 Jy and the distance to the periodic FRB ∼149 Mpc (The CHIME/FRB Collaboration et al. 2020), a typical luminosity of each burst is L ∼ 10⁴⁰ erg s⁻¹. At this luminosity, the event rate density from all the observed FRBs is about

$$\begin{eqnarray}&&{{ \mathcal R }}_{\mathrm{FRB}}(\gt L)\lesssim {\int }_{L}\phi (L){dL}\sim {10}^{6}\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}{L}_{40}^{-0.8}\end{eqnarray} \tag{ 25 }$$

by extrapolating and integrating the luminosity function derived from FRBs above 10⁴² erg s⁻¹ (Luo et al. 2020): $\phi (L){dL}={\phi }^{* }{\left(L/{L}^{* }\right)}^{\alpha }\exp \left(-L/{L}^{* }\right){dL}/{L}^{* },$ where α = −1.8, ϕ* ∼ 339 Gpc⁻³ yr⁻¹ and L* ∼ 2.9 × 10⁴⁴ erg s⁻¹. Here the "≲" sign in Equation (25) denotes the possibility of a luminosity function cutoff below ${10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. We note that if this is the case, the argument below is even tighter, so that the current estimate is rather conservative.

If the majority of observed FRBs are dominantly produced by similar sources discussed in this Letter, the true volumetric birth rate of such sources is

$$\begin{eqnarray}&&{\mathscr{R}}\sim \displaystyle \frac{{{ \mathcal R }}_{\mathrm{FRB}}}{\dot{N}{t}_{\mathrm{life}}}\displaystyle \frac{4\pi }{{\rm{\Delta }}{\rm{\Omega }}}\lesssim 30\,{\mathrm{Gpc}}^{-3}\,{\mathrm{yr}}^{-1}\,{\dot{N}}_{25}^{-1}{t}_{\mathrm{life},4}^{-1}{\rm{\Delta }}{{\rm{\Omega }}}_{0.6\pi }^{-1}.\end{eqnarray} \tag{ 26 }$$

This is much smaller than the supernova rate density ∼10⁵ Gpc⁻³ yr⁻¹ and the magnetar birth rate density ∼10⁴ Gpc⁻³ yr⁻¹ (Kaspi & Beloborodov 2017) for a reasonable lifetime ${t}_{\mathrm{life}}\gg 1\,\mathrm{yr}$. The expected number of FRB sources is only $\lesssim 0.03{\dot{N}}_{25}^{-1}{\rm{\Delta }}{{\rm{\Omega }}}_{0.6\pi }^{-1}$ in our Galaxy. This suggests that a special condition is necessary to limit the amount of sources to produce FRBs and binary interaction is an attractive solution. Thus, our model is not like a single magnetar model. Binary interaction is an important ingredient of the model.

For the massive star companion case, the reduction factors from the supernova rate density include the fraction of pulsars that satisfies the FRB condition (may be comparable to the magnetar fraction) by ∼0.1 (Kaspi & Beloborodov 2017), the fraction of the right binary separation by ∼0.1 (Moe & Di Stefano 2017), the survival fraction of the kick at the first supernova by ∼0.1 (Postnov & Yungelson 2014; Tauris et al. 2017), the mass ratio and so on by ∼0.3 (Moe & Di Stefano 2017). Thus, it is natural to have a small birth rate similar to Equation (26).

For the neutron star companion case, the birth rate of binary neutron stars with a separation of a ∼ 10¹² cm is estimated as ∼10² Gpc⁻³ yr⁻¹ by the population synthesis (Belczynski et al. 2002), which is smaller by a factor of ∼10 than the merger rate derived from gravitational-wave observations (Abbott et al. 2017, 2020). This is also consistent with the number of Galactic binary neutron star systems (Tauris et al. 2017). By multiplying the magnetar fraction ∼0.1, it also gives a small birth rate similar to Equation (26).