Coherent Inverse Compton Scattering by Bunches in Fast Radio Bursts

The general picture of the hereby discussed mechanism is the following (Figure 1): The sudden cracking of the crust of a magnetar excites crustal quakes and plasma oscillations near the surface of the magnetar. Low-frequency electromagnetic waves with angular frequency ω₀ would be generated by the coherently oscillating charged particles and would propagate across the magnetosphere. The waves encounter bunches of relativistic particles with a typical Lorentz factor γ and a net charge N_e,b e, which moves along magnetic field lines in the outer magnetosphere or in current sheets outside of the magnetosphere. In the rest frame of the particle bunch, the incident wave frequency is boosted to $\sim \gamma (1-\beta \cos {\theta }_{i}){\nu }_{0}$. The bunched particles are induced to oscillate at the same frequency in the comoving frame, which is transferred to a lab-frame outgoing ICS frequency,

$$\begin{eqnarray}&&\nu \simeq {\gamma }^{2}{\nu }_{0}(1-\beta \cos {\theta }_{i})=(1\ \mathrm{GHz})\ {\gamma }_{2.5}^{2}{\nu }_{\mathrm{0,4}}(1-\beta \cos {\theta }_{i}),\end{eqnarray} \tag{ 3 }$$

where ν₀ = ω₀/2π and θ_i is the angle between the incident photon momentum and the electron momentum. Similar to CR, such an ICS mechanism does not depend intrinsically on the dispersive properties of the plasma and may be treated by assuming vacuum wave properties. Different from the CR mechanism, which appeals to the curved trajectory of charged particles for acceleration, the ICS mechanism invokes an oscillating electromagnetic field from the low-frequency waves to accelerate bunched particles.

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Such a mechanism is not new in the pulsar literature. The linear acceleration emission (LAE) invokes an oscillating E_∥ along the direction of particle motion (e.g., Melrose 1978; Rowe 1995). A free-electron laser (FEL) invokes the interaction between an electromagnetic disturbance (wiggler) with a relativistically moving bunch to power pulsar radio emission (e.g., Fung & Kuijpers 2004) or FRB emission (Lyutikov 2021a). Lyubarskii (1996) introduced the induced upscattering of longitudinal oscillations to interpret pulsar radio emission. Qiao & Lin (1998) considered low-frequency electromagnetic waves generated from regular discharges of an inner vacuum gap in the pulsar polar cap region and showed that ICS off these waves may reproduce many observational features of pulsar radio emission (Qiao et al. 2001; Xu et al. 2000).

Because FRBs are rarely produced from their sources (e.g., the Galactic magnetar SGR 1935+2154 only produced one detected FRB since its discovery, CHIME/FRB Collaboration et al. 2020, Bochenek et al. 2020; while many X-ray bursts have been produced without association with any FRBs, Lin et al. 2020), it is reasonable to assume that something extraordinary must have happened to trigger an FRB. We envisage the low-frequency electromagnetic waves generated near the surface of the neutron star as one condition to produce FRBs. This may be related to violent oscillations of charged particles near the surface triggered by, e.g., strong oscillations of the neutron star crust during a starquake (e.g., Thompson & Duncan 2001; Beloborodov & Thompson 2007; Wang et al. 2018; Dehman et al. 2020; Yang & Zhang 2021). The bulk of the global oscillation energy would be carried by Alfvén waves that propagate along the magnetic field lines in the form of a magnetic pulse, which would deposit energy in the form of particle energy and radiation at a large distance, as envisaged in most FRB models (e.g., Kumar & Bošnjak 2020; Lyubarsky 2020; Yuan et al. 2020). However, the same shearing oscillations of the crust would induce oscillations of charges in the near-surface magnetosphere in the direction perpendicular to the magnetic field lines, so that a small fraction of the oscillation energy may be radiated by these charges coherently and nearly isotropically as X-mode electromagnetic waves with angular frequency ω₀, similar to an antenna in a radio station on Earth. Strictly speaking, the electromagnetic waves discussed here are electromagnetic modes propagating in a plasma, which may be equivalent to some forms of fast magnetosonic waves. However, because the magnetization factor σ is extremely high in the inner magnetosphere, the magnetosonic waves all propagate at essentially the speed of light and would behave like electromagnetic waves in a vacuum with a proper dispersion relation. Because the electromagnetic waves do not carry the bulk of the FRB energy but rather provide seed photons for particles to upscatter at a large radius, the brightness temperature of the low-frequency coherent emission does not need to be very high. In this model, the low-frequency waves should at least last several milliseconds (the duration of the FRB) but could last longer (in which case the FRB duration would be defined by the duration of the magnetic pulse dissipation leading to coherent ICS emission by relativistic particles).

For a cold, nonrelativistic plasma, the X-mode electromagnetic waves can propagate in the frequency regimes ω > ω_R, ω_L < ω < ω_uh, and ω < ω_lh, where ω_R and ω_L are the cutoff frequencies for the R-mode and L-mode and ω_uh and ω_lh are the upper and lower hybrid principal resonant frequencies, respectively (Boyd & Sanderson 2003). For an electron–positron pair plasma relevant for a pulsar/magnetar magnetosphere, one has ${\omega }_{{\rm{R}}}={\omega }_{\mathrm{uh}}={\omega }_{{\rm{L}}}=\sqrt{{\omega }_{p}^{2}+{\omega }_{{\rm{B}}}^{2}}$ and ω_lh = ω_B, so that the X-mode electromagnetic waves are free to propagate at $\omega \gt \sqrt{{\omega }_{p}^{2}+{\omega }_{{\rm{B}}}^{2}}$ or ω < ω_B (see the Appendix). For a magnetar with surface magnetic field B_s = 10¹⁵GB_s,15 and rotation period P, the Goldreich–Julian charge number density (Goldreich & Julian 1969; Ruderman & Sutherland 1975) is

$$\begin{eqnarray}&&{n}_{\mathrm{GJ}}\simeq \displaystyle \frac{{\rm{\Omega }}B}{2\pi {ce}}\simeq (6.9\times {10}^{7}\ {\mathrm{cm}}^{-3}){B}_{s,15}{P}^{-1}{\hat{r}}_{2}^{-3},\end{eqnarray} \tag{ 4 }$$

where Ω = 2π/P is the angular frequency, and

$$\begin{eqnarray}&&\hat{r}\equiv \displaystyle \frac{r}{R}\end{eqnarray} \tag{ 5 }$$

is the radius r normalized to the neutron star radius R = 10⁶ cmR₆. The plasma number density can be estimated as

$$\begin{eqnarray}&&n=\xi {n}_{\mathrm{GJ}},\end{eqnarray} \tag{ 6 }$$

where ξ is the multiplicity parameter due to electron–positron pair production. As a result, the plasma frequency and the Larmor frequency can be estimated as

$$\begin{eqnarray}\begin{array}{rcl}{\omega }_{p} & = & {\left(\displaystyle \frac{4\pi {{ne}}^{2}}{{m}_{e}}\right)}^{1/2}\\ & \simeq & (4.7\times {10}^{8}\ \mathrm{rad}\ {{\rm{s}}}^{-1})\ {\xi }^{1/2}{B}_{s,15}^{1/2}{P}^{-1/2}{\hat{r}}_{2}^{-3/2}\end{array}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\omega }_{{\rm{B}}}=\displaystyle \frac{{eB}}{{m}_{e}c}\simeq (1.8\times {10}^{16}\ \mathrm{rad}\ {{\rm{s}}}^{-1})\ {B}_{s,15}{\hat{r}}_{2}^{-3}.\end{eqnarray} \tag{ 8 }$$

It is also informative to evaluate these frequencies at the light cylinder of the magnetar, which has r = R_lc = c/Ω and

$$\begin{eqnarray}&&{\hat{r}}_{\mathrm{lc}}=(4.8\times {10}^{3}){{PR}}_{6}^{-1}.\end{eqnarray} \tag{ 9 }$$

This gives

$$\begin{eqnarray}&&{\omega }_{p}^{\mathrm{lc}}\simeq (1.4\times {10}^{6}\ \mathrm{rad}\ {{\rm{s}}}^{-1})\ {\xi }^{1/2}{B}_{s,15}^{1/2}{P}^{-2}{R}_{6}^{3/2}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\omega }_{{\rm{B}}}^{\mathrm{lc}}\simeq (1.6\times {10}^{11}\ \mathrm{rad}\ {{\rm{s}}}^{-1})\ {B}_{s,15}{P}^{-3}{R}_{6}^{3}.\end{eqnarray} \tag{ 11 }$$

One can see that for the low-frequency electromagnetic waves we are interested in, the condition ω₀ ≪ ω_p ≪ ω_B is satisfied throughout the magnetosphere. Therefore, the X-mode low-frequency waves are transparent to the magnetosphere in all directions for essentially all frequencies (see also Lu et al. 2019).

The above treatment assumes a cold plasma. Near-surface electromagnetic waves need to penetrate through the closed field line region to reach the emission sites in our two scenarios (Figure 1). Because the closed field line region of a magnetar is likely populated by inactive nonrelativistic particles, the cold plasma dispersion relation treatment can serve the purpose for our discussion. We note that the dispersion relation is more complicated when relativistic plasmas are considered (e.g., Arons & Barnard 1986; Lyubarskii & Petrova 1998; Melrose 2017). Nonetheless, the conclusion that X-mode low-frequency electromagnetic waves are transparent in a magnetar magnetosphere remains valid for such more complicated situations.

In the following, we perform a vacuum treatment of the ICS process for simplicity. A more rigorous treatment should consider various plasma effects (e.g., Melrose 1978; Rowe 1995; Lyubarskii 1996; Fung & Kuijpers 2004; Lyutikov 2021a). However, a vacuum treatment can catch the essential features of this family of models, as discussed below.

The Compton scattering (in the rest frame of an electron) cross section in a strong magnetic field is significantly modified from the Thomson cross section σ_T. The cross section for the two modes of photons reads (Herold 1979; Xia et al. 1985)

$$\begin{eqnarray}\begin{array}{rcl}\sigma ^{\prime} (1) & = & {\sigma }_{{\rm{T}}}\left\{{\sin }^{2}{\theta }_{i}^{{\prime} }+\displaystyle \frac{1}{2}{\cos }^{2}{\theta }_{i}^{{\prime} }\right.\\ & & \left.\times \,\left[\displaystyle \frac{\omega {{\prime} }^{2}}{{\left(\omega ^{\prime} +{\omega }_{{\rm{B}}}\right)}^{2}}+\displaystyle \frac{\omega {{\prime} }^{2}}{{\left(\omega ^{\prime} -{\omega }_{{\rm{B}}}\right)}^{2}}\right]\right\},\end{array}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\sigma ^{\prime} (2)=\displaystyle \frac{{\sigma }_{{\rm{T}}}}{2}\left[\displaystyle \frac{\omega {{\prime} }^{2}}{{\left(\omega ^{\prime} +{\omega }_{{\rm{B}}}\right)}^{2}}+\displaystyle \frac{\omega {{\prime} }^{2}}{{\left(\omega ^{\prime} -{\omega }_{{\rm{B}}}\right)}^{2}}\right].\end{eqnarray} \tag{ 13 }$$

where 1 denotes the mode that the electric vector ${\boldsymbol{E}}^{\prime}$ of the incident wave is parallel to the ( k , B ) plane, and 2 denotes the mode that ${\boldsymbol{E}}^{\prime}$ is perpendicular to the plane. Here, the primed symbols denote the rest frame of the electron (because electrons move along magnetic field lines, ω_B does not change when the frame changes). In this frame, the photon incident angle ${\theta }_{i}^{{\prime} }$ is connected to the lab-frame incident angle θ_i through

$$\begin{eqnarray}&&\sin {\theta }_{i}^{{\prime} }=\displaystyle \frac{\sin {\theta }_{i}}{\gamma (1-\beta \cos {\theta }_{i})},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\cos {\theta }_{i}^{{\prime} }=\displaystyle \frac{\cos {\theta }_{i}-\beta }{1-\beta \cos {\theta }_{i}},\end{eqnarray} \tag{ 15 }$$

where γ and β are the Lorentz factor and dimensionless speed of the electron, respectively. In the lab frame, the ICS cross section is related to the rest-frame Compton scattering cross section through $\sigma =(1-\beta \cos {\theta }_{i})\sigma ^{\prime}$ (Pacholczyk 1970). This finally gives

$$\begin{eqnarray}\begin{array}{rcl}\sigma (1) & = & {\sigma }_{{\rm{T}}}\left\{\displaystyle \frac{{\sin }^{2}{\theta }_{i}}{{\gamma }^{2}(1-\beta \cos {\theta }_{i})}+\displaystyle \frac{{\left(\cos {\theta }_{i}-{\beta }_{i}\right)}^{2}}{2(1-\beta \cos {\theta }_{i})}\right.\\ & & \times \,\left.\left[\displaystyle \frac{\omega {{\prime} }^{2}}{{\left(\omega ^{\prime} +{\omega }_{{\rm{B}}}\right)}^{2}}+\displaystyle \frac{\omega {{\prime} }^{2}}{{\left(\omega ^{\prime} -{\omega }_{{\rm{B}}}\right)}^{2}}\right]\right\},\end{array}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\sigma (2)=\displaystyle \frac{{\sigma }_{{\rm{T}}}}{2}\left[\displaystyle \frac{\omega {{\prime} }^{2}}{{\left(\omega ^{\prime} +{\omega }_{{\rm{B}}}\right)}^{2}}+\displaystyle \frac{\omega {{\prime} }^{2}}{{\left(\omega ^{\prime} -{\omega }_{{\rm{B}}}\right)}^{2}}\right].\end{eqnarray} \tag{ 17 }$$

At this point, it is informative to compare the order of magnitude of the two terms in the curly brackets. In the lab frame, θ_i can be an arbitrary angle, so that all the factors involving θ_i could be of the order of unity. For our nominal parameters γ ∼ 10^2.5 and ω₀ ∼ (2π)10⁴ and noting $\omega ^{\prime} \simeq \gamma {\omega }_{0}(1-\beta \cos {\theta }_{i})$, the first term in σ(1) is therefore ∝ γ⁻² ∼ 10⁻⁵ and the second term (also σ(2)) is $\propto {(\omega ^{\prime} /{\omega }_{{\rm{B}}})}^{2}\sim {({10}^{6}/{10}^{11})}^{2}\sim {10}^{-10}$ even at the light cylinder where ω_B is the lowest in the magnetosphere. One can therefore ignore σ(2) and the second term of σ(1), so that only (see also Qiao & Lin 1998)

$$\begin{eqnarray}&&\sigma (1)\simeq {\sigma }_{{\rm{T}}}\displaystyle \frac{{\sin }^{2}{\theta }_{i}}{{\gamma }^{2}(1-\beta \cos {\theta }_{i})}\simeq (6.65\times {10}^{-30}\ {\mathrm{cm}}^{2}){\gamma }_{2.5}^{-2}f({\theta }_{i})\end{eqnarray} \tag{ 18 }$$

is relevant, where $f({\theta }_{i})={\sin }^{2}{\theta }_{i}/(1-\beta \cos {\theta }_{i})$ is defined.

The ICS emission power of a single relativistic electron may be estimated as

$$\begin{eqnarray}\begin{array}{rcl}{P}_{e}^{\mathrm{ICS}} & \simeq & \displaystyle \frac{4}{3}{\gamma }^{2}\sigma (1){{cU}}_{\mathrm{ph}}\\ & \simeq & (2.1\times {10}^{-7}\ \mathrm{erg}\ {{\rm{s}}}^{-1})f({\theta }_{i}){\left(\delta {B}_{\mathrm{0,6}}\right)}^{2}{\hat{r}}_{2}^{-2},\end{array}\end{eqnarray} \tag{ 19 }$$

where

$$\begin{eqnarray}\begin{array}{rcl}{U}_{\mathrm{ph}} & \simeq & \displaystyle \frac{{\left(| {\boldsymbol{\delta }}{{\boldsymbol{E}}}_{{\bf{0}}}\times {\boldsymbol{\delta }}{{\boldsymbol{B}}}_{{\bf{0}}}| \right)}^{2}}{4\pi }{\hat{r}}^{-2}\\ & \simeq & (8.0\times {10}^{6}\,\mathrm{erg}\,{\mathrm{cm}}^{-3}){\left(\delta {B}_{\mathrm{0,6}}\right)}^{2}{\hat{r}}_{2}^{-2}\end{array}\end{eqnarray} \tag{ 20 }$$

is the photon energy density in the emission region, and δ E ₀ and δ B ₀ are the electric and magnetic vectors of the low-frequency waves near the surface region, which we normalize to a relatively small value δ B₀ ∼ 10⁶ G. Comparing with the CR emission power of a single electron

$$\begin{eqnarray}&&{P}_{e}^{\mathrm{CR}}={\gamma }^{4}\displaystyle \frac{2{e}^{2}c}{3{\rho }^{2}}\simeq (4.6\times {10}^{-15}\ \mathrm{erg}\ {{\rm{s}}}^{-1}){\gamma }_{2.5}^{4}{\rho }_{8}^{-2},\end{eqnarray} \tag{ 21 }$$

one can see that the ICS power is much larger, i.e., ${P}_{e}^{\mathrm{ICS}}\gg {P}_{e}^{\mathrm{CR}}$, a known fact in pulsar studies (Qiao & Lin 1998; Zhang et al. 1999). This suggests that when coherent ICS operates, the effect of coherent CR is negligibly small.

A bunch of leptons with a net charge value of N_e,b e can radiate coherently in response to the low-frequency waves, with an emitted power of $\sim {N}_{e,b}^{2}{P}_{e}^{\mathrm{ICS}}$. The observed power is boosted by a factor of γ² because the observer time is shorter by a factor of $(1-\beta \cos \theta )\sim 1/{\gamma }^{2}$ (when the angle between the charge motion direction and the line of sight is θ < 1/γ) with respect to the emission time. For a total number of bunch number N_b , the total luminosity of the emitter may be written as

$$\begin{eqnarray}&&L\simeq {N}_{b}{N}_{e,b}^{2}{P}_{e}^{\mathrm{ICS}}{\gamma }^{2}.\end{eqnarray} \tag{ 22 }$$

One may estimate

$$\begin{eqnarray}&&{N}_{e,b}=\zeta {n}_{\mathrm{GJ}}{A}_{b}\lambda ,\end{eqnarray} \tag{ 23 }$$

where A_b is the cross section of the bunch, λ = c/ν is the wavelength, and ζ is the factor denoting the net charge density with respect to n_GJ. The most conservative estimate is (Kumar et al. 2017)

$$\begin{eqnarray}&&{A}_{b}^{\min }=\pi {\left(\gamma \lambda \right)}^{2}\end{eqnarray} \tag{ 24 }$$

because it describes the causally connected area of a relativistic bunch. ¹ Because the ICS mechanism is very efficient, in the following we use this conservative limit. Plugging Equations (4), (19), (23), and (24) into Equation (22), one finally gets

$$\begin{eqnarray}\begin{array}{rcl}L & \simeq & (7.3\times {10}^{38}\ \mathrm{erg}\ {{\rm{s}}}^{-1}){\zeta }^{2}{N}_{b,5}{\gamma }_{2.5}^{6}{\nu }_{9}^{-6}\\ & & \times \,{B}_{s,15}^{2}{P}^{-2}f({\theta }_{i})\delta {B}_{0,6}^{2}{\hat{r}}_{2}^{-8}.\end{array}\end{eqnarray} \tag{ 25 }$$

One may compare Equation (25) with the true FRB luminosity from the observations, i.e., ${L}^{\mathrm{obs}}={L}_{\mathrm{iso}}^{\mathrm{obs}}\max (\pi /{\gamma }^{2},\pi {\theta }_{j}^{2})$, where θ_j is the half-opening angle of the FRB jet. For the narrow jet scenario with a solid angle defined by the γ⁻¹ cone, one has ${L}^{\mathrm{obs}}\simeq (3\times {10}^{38}\ \mathrm{erg}\ {{\rm{s}}}^{-1}){L}_{\mathrm{iso},43}^{\mathrm{obs}}{\gamma }_{2.5}^{-2}$. One can see that the coherent ICS mechanism can easily account for the typical FRB luminosity by only requiring a moderate number of N_b ∼ 10⁵ even with the most conservative estimate of the coherent bunch cross section ${A}_{b}^{\min }$. From Equation (22), one can immediately see that the required ${N}_{b}{N}_{e,b}^{2}$ for the coherent ICS model is smaller than the CR model by a factor ${P}_{e}^{\mathrm{CR}}/{P}_{e}^{\mathrm{ICS}}\sim 2.2\times {10}^{-8}$ to interpret the same FRB luminosity for the same electron Lorentz factor γ.

A bunch with N_e,b particles cools more efficiently than single particles because the emission power is $\sim {N}_{e,b}^{2}$ times the individual particle emission power. The bunch quickly loses the total energy, which itself is not large enough to power FRB emission. In order to power FRB emission, a sustained E_∥ is needed to continuously pump energy to the bunch. Such a scenario is relevant for coherent CR by bunches (Kumar et al. 2017). Below we investigate the case for coherent ICS emission. The cooling timescale of the coherent ICS bunches may be estimated as

$$\begin{eqnarray}\begin{array}{rcl}{t}_{c,b} & = & \displaystyle \frac{{N}_{e,b}\gamma {m}_{e}{c}^{2}}{{N}_{e,b}^{2}{P}_{e}^{\mathrm{ICS}}}\simeq (2.1\times {10}^{-15}\ {\rm{s}})\\ & & \times \,{\zeta }^{-1}{\nu }_{9}^{3}{\gamma }_{2.5}^{-1}{B}_{s,15}^{-1}P{\left[f({\theta }_{i})\right]}^{-1}\delta {B}_{0,6}^{-2}{\hat{r}}_{2}^{5}.\end{array}\end{eqnarray} \tag{ 26 }$$

Because this is much shorter than the typical FRB duration, similar to the bunched coherent CR mechanism, a parallel electric field is also needed to continuously pump energy to the bunch. One may estimate the strength of this E_∥ by requiring N_e,b e(ct_c,b ) ≃ N_e,b γ m_e c², which gives

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\parallel } & \simeq & \displaystyle \frac{{N}_{e,b}{P}_{e}^{\mathrm{ICS}}}{{ec}}\simeq (8.6\times {10}^{9}\ \mathrm{esu})\\ & & \times \,\zeta {\nu }_{9}^{-3}{\gamma }_{2.5}^{2}{B}_{s,15}{P}^{-1}f({\theta }_{i})\delta {B}_{0,6}^{2}{\hat{r}}_{2}^{-5}.\end{array}\end{eqnarray} \tag{ 27 }$$

This is larger than that required for the bunched coherent CR mechanism because of a larger P_e . However, in view of the steep dependence on $\hat{r}$, the required E_∥ comfortably falls into the range available for magnetars if $\hat{r}$ is somewhat greater than 100. Several mechanisms may provide this E_∥ at $\hat{r}\gg 1$. First, a slot gap near the last open field line region may extend from the neutron surface all the way to the light cylinder because the region cannot be filled with electron–positron pairs produced from a pair production cascade—a consequence of the straight line propagation of γ-rays and curved magnetic field lines (e.g., Arons & Scharlemann 1979; Muslimov & Harding 2004). Charge starvation may become more prominent in old, low-twist magnetars (Wadiasingh et al. 2020). Second, Alfvén waves propagating along magnetic field lines would enter a charge-starved region at a large-enough radius because the charge density may become insufficient to provide the required current (e.g., Kumar & Bošnjak 2020; Lu et al. 2020, but see Chen et al. 2020). Third, in the current sheet outside of the magnetosphere, an E_∥ naturally exists in the magnetic reconnection layer (e.g., Lyubarsky 2020; Kalapotharakos et al. 2018; Philippov & Spitkovsky 2018).

After discussing some general properties of this model, we discuss two specific scenarios within the framework of the magnetar repeating FRB models. The geometric configurations of the two scenarios are presented in Figure 1.

The first scenario is similar to that of Kumar & Bošnjak (2020) and Lu et al. (2020), but with coherent CR by bunches replaced by the more efficient coherent ICS by bunches. Within this scenario, neutron star crust cracking excites crustal seismic waves, which in turn excite Alfvén waves in the magnetosphere. Low-frequency electromagnetic waves (effectively fast magnetosonic waves in a high-σ medium) are generated, and the X-mode waves propagate at a speed close to the speed of light. The waves are upscattered at a large-enough altitude where E_∥ is developed, either due to charge starvation in the Alfvén waves (Kumar & Bošnjak 2020) or due to the traditional pulsar gap mechanism (Arons & Scharlemann 1979; Cheng et al. 1986; Muslimov & Harding 2004; Wadiasingh et al. 2020). Assuming that this emission site is at $\hat{r}\sim 100$, based on the scalings presented in Section 2.3, the FRB luminosities are easily interpreted even if the low-frequency waves have a relatively low amplitude with δ B₀ ∼ 10⁶ G.

In the second scenario, the emission site is in the reconnection current sheet region just outside of the light cylinder, similar to Lyubarsky (2020). However, the radiation mechanism is via coherent ICS by bunches rather than oscillations of colliding magnetic islands within the current sheet as proposed by Lyubarsky (2020). Here the bunches could be density fluctuations in the turbulent relativistic particle outflows within the reconnection layer. At this radius, the amplitude of low-frequency waves is degraded significantly with respect to that in the near-surface region. Because magnetic reconnection is enhanced by a strong magnetic pulse that is eventually responsible for the FRB, the magnetic energy density is enhanced by a factor of b ≡ B_pulse/B_wind in the current sheet region (Lyubarsky 2020). Based on magnetic flux conservation, the cross section of the magnetic tube is also compressed by the same factor b. Because the length scale in the B direction is not changed, this leads to a local plasma density increased by a factor of ∼b. According to Equations (1) and (4) of Lyubarsky (2020), one estimates b ∼ 4 × 10⁴. As a result, we can estimate the local charge density using Equation (6) with $\hat{r}={\hat{r}}_{\mathrm{lc}}$ (Equation (9)) and ζ ≃ b = 4 × 10⁴ b_4.6. Rewriting the equations presented in Section 2.3 with $\hat{r}={\hat{r}}_{\mathrm{lc}}$ (Equation (9)), one obtains:

$$\begin{eqnarray}&&{U}_{\mathrm{ph}}^{\mathrm{lc}}\simeq (3.5\times {10}^{3}\,\mathrm{erg}\,{\mathrm{cm}}^{-3})\delta {B}_{0,6}^{2}{P}^{-2}{R}_{6}^{2}\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{P}_{e}^{\mathrm{ICS},\mathrm{lc}}\simeq (9.3\times {10}^{-11}\ \mathrm{erg}\ {{\rm{s}}}^{-1})\delta {B}_{0,6}^{2}{P}^{-2}f({\theta }_{i}){R}_{6}^{2}\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}\begin{array}{rcl}{L}^{\mathrm{lc}} & \simeq & (4.4\times {10}^{38}\ \mathrm{erg}\ {{\rm{s}}}^{-1}){b}_{4.6}^{2}{N}_{b,9}{\gamma }_{2.5}^{6}{\nu }_{9}^{-6}\\ & & \times \,{B}_{s,15}^{2}{P}^{-10}f({\theta }_{i})\delta {B}_{0,6}^{2}{R}_{6}^{8}\end{array}\end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray}\begin{array}{rcl}{t}_{c,b}^{\mathrm{lc}} & \simeq & (1.3\times {10}^{-11}\ {\rm{s}}){b}_{4.6}^{-1}{\nu }_{9}^{3}{\gamma }_{2.5}^{-1}{B}_{s,15}^{-1}\\ & & \times \,{P}^{6}{\left[f({\theta }_{i})\right]}^{-1}\delta {B}_{0,6}^{-2}{R}_{6}^{-5}\end{array}\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}\begin{array}{rcl}{E}_{\parallel }^{\mathrm{lc}} & \simeq & (1.4\times {10}^{6}\ \mathrm{esu}){b}_{4.6}{\nu }_{9}^{-3}{\gamma }_{2.5}^{2}{B}_{s,15}{P}^{-6}\\ & & f({\theta }_{i})\delta {B}_{0,6}^{2}{R}_{6}^{5}.\end{array}\end{eqnarray} \tag{ 32 }$$

One can see that the FRB luminosities can still be explained with δ B₀ ∼ 10⁶ G but with a somewhat larger bunch number N_b ∼ 10⁹.

Note that in the magnetic reconnection site, the reconnection-driven electric field is perpendicular to B . This is different from the first scenario where E_∥ is along the B field direction. However,we do not know the geometric configuration of the FRB central engine, the same FRB may be interpreted by the two scenarios with different viewing geometries: The inner magnetospheric scenario has the viewing angle sweep the open field line region, whereas the reconnection scenario has the viewing angle sweep the equatorial plane (see Figure 1).

In our model, the energy budget to power FRBs is carried by particles rather than by low-frequency waves. The latter only carries a small amount of energy. Its role is to provide seed photons for coherent ICS emission to operate. Because δ B ∝ r⁻¹ and B ∝ r⁻³, with the nominal parameters δ B₀ ∼ 10⁶ G and B_s ∼ 10¹⁵ G, one estimates $\delta {B}^{\mathrm{lc}}/{B}^{\mathrm{lc}}=4.8\,\times {10}^{-6}{{PR}}_{6}^{-1}\ll 1$ at the light cylinder. As a result, the enhancement of the magnetospheric Thomson cross section (which is relevant when δ B > B; Beloborodov 2021) ² does not occur for the low-frequency waves so that they can propagate freely within the magnetar magnetosphere. ³ The energy of the emitting particles comes from the main magnetic pulse driven by the violent event itself. The energy that powers the FRB is first carried by the Alfvén waves, which advect particles to a large altitude where an E_∥ is developed in a charge-starved region in the magnetar magnetosphere (Kumar & Bošnjak 2020; Muslimov & Harding 2004; Wadiasingh et al. 2020) or in magnetic reconnection sites in the current sheet just outside of the magnetosphere (Lyubarsky 2020; Kalapotharakos et al. 2018; Philippov & Spitkovsky 2018). Particles are accelerated in these E_∥ fields so that their ultimate energy comes from the magnetic pulse itself.

Because only the X-mode low-frequency waves can propagate in the magnetosphere, the incident photon is linearly polarized and has an electric field vector perpendicular to the local magnetic field direction. The outgoing photon in an ICS process is also linearly polarized. As a result, our model predicts linearly polarized X-mode emission with a ∼100% degree of linear polarization in the X-mode. Circular polarization may develop under special conditions (Xu et al. 2000). This is in general consistent with the FRB observations (e.g., Gajjar et al. 2018; Michilli et al. 2018; Cho et al. 2020; Day et al. 2020; Luo et al. 2020).

The duration of an FRB is defined by the longer of the emission duration and the timescale for the line of sight to sweep the bundle of the magnetic field lines where emitting particles flow out (Wang et al. 2019, 2021). For a slow rotator and an emission site far from the neutron star surface, the polarization angle would be nearly constant across an individual burst, as observed in some repeating FRBs (e.g., Michilli et al. 2018). For a rapid rotator and an emission site closer to the neutron star surface, the polarization angle may display diverse swing features, as observed in some other repeating FRBs (Luo et al. 2020). Our model can account for both patterns.

The characteristic frequency of coherent ICS emission Equation (3) depends on ω₀, γ, and θ_i . For the coherent mechanism to work, particles need to continuously tap energy from E_∥ so that they are likely emitting in the radiation-reaction-limited regime. As a result, γ may remain roughly constant during the emission phase. At the large emission radius as envisaged in the two scenarios, θ_i varies little as the line of sight sweeps across different field lines. For a seismic oscillation originating from the neutron star crust, the oscillation frequency ω₀ depends on the resonance frequency of the crust, which may have a characteristic value. As a result, the characteristic frequency ν has a narrow value range, unlike CR, which is an intrinsically broadband mechanism (Yang & Zhang 2018). ⁴ This is consistent with the narrowband emission as observed from repeating FRBs (Spitler et al. 2016; CHIME/FRB Collaboration et al. 2021).

The characteristic Lorentz factor of electrons may slightly decrease with radius. If so, the outgoing ICS frequency would decrease with radius and give a radius-to-frequency mapping within the ICS model (Qiao & Lin 1998). Alternatively, as the seismic waves damp, ω₀ would gradually decrease with decreasing amplitude. This would also result in a decreasing ν of FRB emission with reduced amplitude. Both factors may offer an explanation for the observed subpulse frequency downdrifting pattern (also called the "sad trombone" effect) (Hessels et al. 2019; CHIME/FRB Collaboration et al. 2019a, 2019b). ⁵

Because the ICS emission power of individual electrons is much higher than that of CR, the required degree of coherence in the bunch is much lower. In the scalings presented in Section 2.3, the net charge parameter ζ is normalized to unity. In other words, a bunch with the Goldreich–Julian net charge density can produce the high brightness temperature of FRB emission. In fact, the model even allows the bunch charge density to be below the Goldreich–Julian density, with the luminosity compensated for by a somewhat larger total number of bunches, N_b , which is currently normalized to a small value. The only requirement to make a bunch is that there are fluctuations of net charge density in space with respect to the background number density (Yang & Zhang 2018). This alleviates traditional criticisms on the bunching mechanisms within the context of CR regarding the formation and maintenance of bunches (Melrose 1978). Since the mechanism can apply to magnetospheres with a low plasma density (see also Wadiasingh et al. 2020), another major criticism on the bunching mechanism, i.e., the suppression of coherent emission flux due to the dense plasma effect (Gil et al. 2004; Lyubarsky 2021), is also alleviated.