Scattering Cross Sections of Magnetized Particles within Intense Electromagnetic Waves: Application to Fast Radio Bursts
Yu-Chen Huang Shu-Qing Zhong Zi-Gao Dai








0009-0009-7749-8998]Yu-Chen Huang
Department of Astronomy, University of Science and Technology of China, Hefei 230026, China; daizg@ustc.edu.cn
School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China



0000-0002-1766-6947]Shu-Qing Zhong
Department of Astronomy, University of Science and Technology of China, Hefei 230026, China; daizg@ustc.edu.cn
School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China




0000-0002-7835-8585]Zi-Gao Dai
Department of Astronomy, University of Science and Technology of China, Hefei 230026, China; daizg@ustc.edu.cn
School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, China






	
	
	Recently, Beloborodov suggested that there exists a resonance phenomenon between an extremely intense electromagnetic wave and internal magnetized particles. The particles exchange energy with the wave at frequent resonance events and then reach the radiation reaction limit immediately. This process greatly enhances the scattering cross section of the particles. Note that these results only involve an extraordinary (X) mode wave. In this paper, we focus on an intense ordinary (O) mode wave propagating through magnetized particles and compare it with the case of the X-mode wave. Our result shows that the scattering cross section of the particles in the O-mode wave is significantly smaller than that in the X-mode wave. This has important implications for the transparency of a fast radio burst (FRB) inside the magnetosphere of a magnetar. We argue that there is a strong scattering region in the stellar magnetosphere, within which an O-mode wave is more transparent than an X-mode wave for an FRB. 
	
	















§ INTRODUCTION






Some highly magnetized neutron stars, known as magnetars, can generate extremely intense transient radio emissions called fast radio bursts (FRBs) <cit.>. Currently, the radiation mechanism of FRBs is not well understood. Even in the magnetar models, there are  still different viewpoints on the location where FRBs are generated. Some models propose that FRBs could be produced inside the magnetosphere <cit.>, while some other models suggest that FRBs could take place outside the magnetosphere <cit.>. It was also proposed that FRBs could be generated near the light cylinder <cit.>. More details about the models can be seen in some recent reviews <cit.>.


The typical luminosity of cosmological FRBs is approximately 10 orders of magnitude larger than that of regular radio pulses from pulsars. Therefore, a strong wave effect in FRBs likely introduces new characteristics distinct from those radio pulses and can constrain theoretical models <cit.>. To describe this effect, a nonlinear parameter can be introduced, which is defined as

    a_0=eE_0/mcω≈2.3×10^4 ν_9^-1 L_iso,42^1/2r_9^-1,

where c is the speed of light, and e and m are electron charge and mass. E_0, L_iso=(10^42 erg s^-1)L_iso,42, and ν=ω/2π=(10^9 Hz)ν_9  represent the electric field strength, isotropic luminosity, and frequency of an FRB, respectively. r=(10^9 cm)r_9 is the magnitude of the light cylinder radius,

    R_LC=cP/2π≈(4.8× 10^9 cm)(P/1 s),

where P is the period of the magnetar. The nonlinear parameter in Eq. (<ref>) can also be written as a_0=v_ec/c, where v_ec=eE_0/mω can be understood as the nonrelativistic characteristic oscillation velocity of an electron in the wave. For a strong wave with a_0≫ 1, the classical nonrelativistic theorem describing the Thomson scattering fails, and the strong wave effect is supposed to be taken into account. The strong wave effect of FRBs has been extensively discussed in the near-source environment <cit.>, while there have been few works focusing on the effect within the magnetosphere.










Recently, <cit.> proposed a novel description of the scattering of magnetized particles in such a strong electromagnetic wave. For instance, an FRB traveling outward through the magnetosphere of a magnetar likely encounters a region with

    1<ω_B/ω<a_0,

where ω_B=eB_bg/mc is the electron cyclotron frequency in a background magnetic field. This condition can be satisfied near the outer magnetosphere where

    ω_B/ω≈2.8×10^3 B_s,15ν_9^-1r_9^-3.

We adopt a dipole background field with surface strength B_s=(10^15 G)B_s,15. Within the region where Eq. (<ref>) holds, the wave exchanges energy with internal charged particles at resonance events, and pushes the particles to the radiation reaction limit. In this case, the energy loss of the particles due to radiation equals the energy gained from resonance. The scattering cross section of particles can then be semianalytically derived and is found to be significantly enhanced compared with the classical Thomson scattering.







We note that the above consideration only involves an extraordinary (X) mode wave, whose electric field is perpendicular to the propagation direction and the background magnetic field. In fact, both ordinary (O) mode and X-mode waves can be generated within the framework of magnetospheric coherent emission models, such as curvature radiation <cit.>. It is therefore necessary to investigate the scattering of magnetized particles in an intense O-mode wave, whose electric field lies in the plane defined by the propagation direction and the background magnetic field. 






In this paper, we first solve the equation of motion of a charged particle in an O-mode wave and an X-mode wave, respectively. We then study the resonance phenomenon between the wave and the particle in both modes in Section <ref>. The derivation of the cross sections in different modes is conducted in Section <ref>. Based on these results, we further investigate the implications for FRBs in Section <ref>. The final summary and discussion are presented in Section <ref>. In this paper, the notation Q_n=Q/10^n and cgs units are adopted.













§ PARTICLE MOTION IN AN INTENSE WAVE WITHIN A BACKGROUND MAGNETIC FIELD



The equation of motion of a charged particle within an intense electromagnetic wave and a background magnetic field is

    du/dt=e/mc(E+β×B+β×B_bg),

where u=γβ is the spatial component of the normalized four-velocity u^α=dx^α/cdτ=(γ,γβ). We assume that the incident wave is linearly polarized. As shown in Figure <ref>, its electric field, magnetic field, and propagation direction are along the x-, y- and z-axes, respectively. For simplicity, we also assume that the incident wave is a monochromatic plane wave, i.e., eE/mcω=e_xa_0sin(ωξ) and eB/mcω=e_ya_0 sin(ωξ), where ξ≡ t-z/c. The background magnetic field can point in any direction, which is given by B_bg=B_bg(sinθcosϕ, sinθsinϕ,cosθ). 







Eq.  (<ref>) can be written as three scalar equations:

    dU_x/dt   =ω_B/γ(u_y  cosθ-u_z sinθsinϕ),
    du_y/dt   = ω_B/γ(u_z  sinθcosϕ-u_x cosθ),
    du_ξ/dt   =ω_B/γ(u_y sinθcosϕ-u_x sinθsinϕ),

where U_x=u_x+a_0cos(ωξ) and u_ξ=dξ/dτ=γ-u_z. The solutions of motion of an initial rest particle are exhibited in Figure <ref>, in which the wave is O-mode (upper panels) or X-mode (lower panels). The motion of the particle can be regarded as the superposition of a large-amplitude oscillation and a small-amplitude oscillation, as shown in the right two panels in Figure <ref>. According to Eq. (<ref>), we can infer and estimate that the frequency of the large-amplitude oscillation is the Larmor frequency ω_L=ω_B/γ. The small-amplitude oscillation is driven directly by the electromagnetic field of the wave. The wave undergoes one cycle with ωδξ=ω(1-β_z)δ t∼2π, providing an approximate angular frequency ω(1-β_z) for the small-amplitude oscillation. In the O-mode wave, the motion of the particle in the x-axis direction can be solved analytically, i.e., u_x=a_0[1-cos(ωξ)]. In the X-mode wave, the motion of the particle in the y-axis direction vanishes, i.e., u_y=0.










Figure <ref> also shows that the energy of the particle can be pumped to a different value at every resonance event. This pumping effect was found by <cit.>. We find that the value of the pumped Lorentz factor of the particle in an O-mode wave is apparently different from that in an X-mode wave. As we demonstrate below, the gained Lorentz factors of the particle at resonance in these two modes are totally different.




The pumped energy of a particle at a resonance event is provided by the electric field of the wave, that is, mc dγ/dt=eE β_x.
This equation can be equivalently written as

    dγ=a_0/γu_x sin(ωξ)/1-β_zω dξ.

In the vicinity of resonance, β_z^res∼cosδψ∼1-δψ^2/2, where δψ=ω_Lδ t. The resonance event occurs within one small-amplitude oscillation, i.e., ω(1-β_z)δ t∼ 2π. Therefore, one arrives at

    δψ∼(ω_L/ω)^1/3.

For the particle in the O-mode wave, one has u_x=a_0[1-cos(ωξ)]. Substituting these relations into Eq. (<ref>) gives the pumped Lorentz factor at a resonance event

    δγ_O≈ H_O a_0^2 γ^-1/3(ω_B/ω)^-2/3sin(ωξ_0)[1-cos(ωξ_0)],

where ωξ_0 is the phase of the wave at resonance. The coefficient H_O≈1.39, which can be obtained by fitting the data of a_0^-2γ^1/3(ω_B/ω)^2/3δγ and sin(ωξ_0)[1-cos(ωξ_0)] at resonance events in the numerical solution, as shown in the left panel of Figure <ref>. For the particle in the X-mode wave, u_x near a resonance event can be estimated as 

    u_x^res   =[γ^2 (1-β_z^2)-1]^1/2∼(γ^2 δψ^2-1)^1/2
       ∼[γ^2(ω_L/ω)^2/3-1]^1/2∼γ^2/3(ω_B/ω)^1/3,

where we have used the relation γ^2-u_x^2-u_z^2=1. A further calculation of Eq. (<ref>) gives

    δγ_X ≈ H_X a_0 γ^1/3(ω_B/ω)^-1/3sin(ωξ_0).

The fitting result is shown in the right panel of Figure <ref>, in which H_X≈ 1.05 is obtained. The derivation of Eq. (<ref>) is independent of whether the wave is O mode or X mode. Therefore, one can observe that the difference in the pumped Lorentz factor between both modes is solely determined by u_x, i.e., the motion of the charged particle along the electric field direction. This motion behaves differently in O-mode and X-mode waves, resulting in a difference in the pumped Lorentz factor during resonance.



In a sufficiently long time, the pumping effect of the Lorentz factor behaves like a random walk process, and hence can theoretically boost the energy of the particle to high values. Nevertheless, the maximal Lorentz factor can be limited by radiation losses. A radiation reaction limit can be reached when the energy loss due to radiation equals to the energy gained  from resonance. In the following section, we demonstrate that after reaching such an equilibrium state, a significant difference in the Lorentz factors of the particle emerges between the O-mode and X-mode waves. The difference results in distinct scattering cross sections of the particle.













§ SCATTERING CROSS SECTIONS IN DIFFERENT POLARIZED MODES





The total emission power of a relativistic electron is <cit.>

    P_e=2e^2/3c^3γ^4(a_⊥^2+γ^2 a_∥^2),

where a_⊥ and a_∥ represent the acceleration perpendicular and parallel to the velocity, respectively. The emission power can be equivalently expressed in terms of the electromagnetic field as

    γ̇_emmc^2=c/4πσ_T[ (γE+u×B+u×B_bg)^2-(u·E)^2],

where γ̇_em is the loss rate of Lorentz factor due to radiation. σ_T=8π r_e^2/3 is the Thomson scattering cross section, and r_e=e^2/mc^2 is the classical electron radius. When B≫ B_bg, one has γ̇_em∼σ_T E^2 u_ξ^2  /4π m c^2. Averaging this loss rate over the period of a small-amplitude oscillation gives

    γ̇_em∼1/3r_e a_0^2 ω^2 /cu_ξ^2,

where u_ξ^2≈u_ξ^2 has been used. 


The radiation reaction limit requires that, within each large-amplitude oscillation, the energy loss due to radiation equals to the energy gained from resonance, i.e., ⟨γ̇_em⟩Δ t=δγ, where the angle bracket denotes the time average during one large-amplitude oscillation, with a timescale Δ t∼ 2π/ω_L. The average radiative loss rate ⟨γ̇_em⟩=r_e a_0^2 ω^2 ⟨ u_ξ^2⟩ /3c∼ r_e a_0^2 ω^2 γ^2 /3c. Therefore, in the O-mode wave, the Lorentz factor at the radiation reaction limit is

    γ_rrl,O≈(3H_O/2πc/r_e  ω)^3/10(ω_B/ω)^1/10.

For the X-mode wave, the Lorentz factor at radiation reaction limit

    γ_rrl,X≈(3H_X/2πc/r_e a_0 ω)^3/8(ω_B/ω)^1/4.

Notice that the condition ⟨γ̇_em⟩Δ t<γ must be satisfied to ensure that the energy loss during each large-amplitude oscillation is smaller than the total energy. This condition implies that there exists an upper limit for the particle Lorentz factor in both modes, i.e.,

    γ<γ_ul=(3/2πc/r_e a_0^2 ω)^1/2(ω_B/ω)^1/2.

Substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>) gives a_0<H_O^-3/10a_c≈ a_c for the O mode, and a_0< H_X^-3/5a_c≈ a_c for the X mode, where a characteristic nonlinear parameter is defined as

    a_c=(3cω_B^2/2π r_e ω^3)^1/5.


The pumping effect of the Lorentz factor resembles a random walk process, and therefore the time to reach the radiation reaction limit can be simply obtained through an analogy with the random walk process, i.e., t_rrl∼(γ_rrl/δγ)^2 2π/ω_L, which gives

    ω t_rrl,O/2π∼ H_O^-9/10(a_0/ω_B/ω)^3/2(a_c/a_0)^11/2

for the O-mode wave, and 

    ω t_rrl,X/2π∼ H_X^-9/8(a_0/ω_B/ω)^3/2(a_c/a_0)^35/8

for the X-mode wave. Since the timescale is short, once particles are within the wave, they can reach the radiation reaction limit immediately.


The scattering cross section is defined as the time-averaged radiation power over the incident flux of the wave,

    σ=⟨γ̇_em⟩ mc^2/⟨ S⟩∼γ^2 σ_T,

where we have used ⟨ S⟩=m c a_0^2 ω^2/8π r_e. Due to the short time to reach the radiation reaction limit, the scattering cross section of a particle in the O-mode wave can be estimated as

    σ_O∼γ_rrl,O^2 σ_T ≈(3H_O/2πc/r_e  ω)^3/5(ω_B/ω)^1/5σ_T.

The scattering cross section in an X-mode wave can be given by

    σ_X∼γ_rrl,X^2 σ_T≈(3H_X/2πc/r_e a_0 ω)^3/4(ω_B/ω)^1/2σ_T.

Note that the scattering cross section in Eq. (<ref>) does not depend on the wave intensity, which is a unique property for the intense O-mode wave. One can naturally define the ratio of these two cross sections,

    f_σ=σ_O/σ_X≈ H_O^3/5H_X^-3/4(a_c/a_0)^-3/4≈(a_c/a_0)^-3/4.

When a_0<a_c, one has f_σ< 1. This implies that the scattering cross section of a charged particle in an O-mode wave is smaller than that in an X-mode wave.



The aforementioned calculations assume that the propagation direction is perpendicular to the background magnetic field. For a wave propagating obliquely, the scattering process can be considered in a frame that moves relativistically along the background magnetic field. In order to keep the propagation direction perpendicular to the background magnetic field in the moving frame, the Lorentz factor of this frame should be

    γ_f=1/sinθ,

where θ is the angle between the propagation direction and the background magnetic field in the lab frame. The wave frequency and cyclotron frequency in the moving frame are (a prime symbol denotes the quantity in the moving frame)

    ω^'=ω/γ_f,ω_B^'=ω_B.

The condition a_0^'<a_c^' therefore can be rewritten as

    a_0<a_c γ_f^3/5,

where a_0=a_0^' is Lorentz invariant. The ratio between O-mode and X-mode scattering cross sections is then transformed into

    f_σ^'≈ (sinθ)^9/20(a_c/a_0)^-3/4.

One can see that in the moving frame, the ratio is further reduced, but does not show a significant difference with Eq. (<ref>).







§ IMPLICATIONS FOR FRBS





If an FRB originates inside the magnetosphere, it is likely to be generated in the open magnetic field line region at a distance of several tens of neutron star radii. Therefore, the angle between the propagation direction of the FRB and the background magnetic field near the emission site is small <cit.>. However, this angle is not extremely small near the outer magnetosphere. To simplify the calculation, we consider the case of perpendicular propagation. According to Eqs. (<ref>) and (<ref>), the region where scattering is most severe, i.e., 1<ω_B/ω<a_0 can be written as

    0.4 L_iso,42^-1/4B_s,15^1/2< r_9<14.1 ν_9^-1/3B_s,15^1/3.

The condition a_0<a_c further constrains the region where strong scattering occurs. For a typical FRB traveling through the magnetosphere of a magnetar, this can be written as

    a_c/a_0≈0.3ν_9^2/5L_iso,42^-1/2B_s,15^2/5r_9^-1/5>1.

Observations show that the luminosity of FRBs falls within a broad range of 10^38–10^46 erg s^-1 <cit.>. Within such a wide range, the existence of a strong scattering region determined by both Eqs. (<ref>) and (<ref>) is reasonable. We plot the region in the luminosity–radius parameter space in Figure <ref>. Within this region, the strong scattering in different modes mentioned previously is valid. Different values of a_c/a_0 or f_σ are marked by dashed lines in this figure. One can see that the scattering cross section of a particle in an O-mode wave is significantly smaller than that in an X-mode wave. In the gray region where a_0>a_c, the upper limit of the Lorentz factor from Eq. (<ref>) is smaller than the Lorentz factor at the radiation reaction limit from Eqs. (<ref>) and (<ref>). Radiation then suppresses further growth of the Lorentz factor. The characteristic scattering cross sections of particles in O-mode and X-mode waves can be considered approximately equal since the maximal achievable Lorentz factors are equal in both modes.





The optical depth of an FRB in the strong scattering region is given by

    τ_O,X=∫ξ n_GJσ_O,X dr,

where n_GJ≈7× 10^4  cm^-3B_s,15 P^-1 r_9^-3 is the Goldreich-Julian density, and ξ is the pair multiplicity parameter. In the following discussion, we fix the surface magnetic field strength B_s=10^15 G, the period P=3 s, and the wave frequency ν=10^9  Hz. The scattering cross section of a particle can then be written as σ_O≈3.3×10^8 r_9^-3/5σ_T in an O-mode wave, and σ_X≈1.4×10^8 L_iso,42^-3/8r_9^-3/4σ_T in an X-mode wave. As shown in Figure <ref>, since the upper bound of the strong scattering region is a piecewise function, we divide the region into part I and part II and calculate the value of optical depth for a given value of luminosity. In part I, the integral in Eq. (<ref>) is calculated over 0.4L_iso,42^-1/4<r_9<14.1. In part II, the integral is calculated over 0.4L_iso,42^-1/4<r_9<0.3^5 L_iso,42^-5/2. An FRB passing through part II will inevitably traverse the gray region, where the scattering cross section in both modes can be estimated as σ_O^gray≈σ_X^gray≈γ_ul^2 σ_T≈4.3×10^7L_iso,42^-1r_9^-1σ_T. The integral in this region is calculated over 0.3^5 L_iso,42^-5/2<r_9<14.1. The optical depth as a function of luminosity is exhibited in Figure <ref>. One can clearly see that due to different scattering cross sections, there is a significant difference between the values of optical depth of the O-mode and X-mode waves.














§ SUMMARY AND DISCUSSION





A novel semianalytical approach was developed to describe the behavior of a magnetized particle in a strong X-mode electromagnetic wave <cit.>. We have used a similar method and solved the motion of a magnetized particle in a strong O-mode wave. We found that the particle behaves similarly to that in the X-mode wave: it exchanges energy with the wave at frequent resonance events, and reaches the radiation reaction limit in a short time, leading to a significant enhancement in the scattering cross section. However, the pumped Lorentz factor of the particle at resonance in an O-mode wave exhibits a remarkable difference from that in an X-mode wave, as indicated by Eqs. (<ref>) and (<ref>). Consequently, the maximal Lorentz factors as well as scattering cross sections of particles in both modes are different. We further obtained the ratio of these two cross sections and found that this value is usually less than 1, implying that a charged particle in the X-mode wave is more likely to have a larger scattering cross section. We also briefly discussed the case of oblique propagation and found that it does not show a significant difference unless the angle between the magnetic field and the propagation direction is particularly small. 




The above results have important implications if FRBs are generated inside the magnetosphere. When an FRB propagates outward, it likely encounters a strong scattering region determined by Eqs. (<ref>) and (<ref>), as plotted in Figure <ref>. The values of the optical depth of an O-mode wave and an X-mode wave are shown in Figure <ref>. Our results indicate that an FRB is more transparent if it is an O-mode wave. We emphasize that this conclusion holds only in the strong scattering region, as it is possible for an FRB not to encounter such a region during its propagation.




The topic of whether an FRB can escape from the magnetosphere of a magnetar or not has been pointed out as an argument against the magnetospheric origin of FRBs. <cit.> argued that the photons emitted by charged particles in an FRB are capable of inducing the creation of electron–positron pairs, which increase the multiplicity parameter in the magnetosphere. This process then blocks the propagation of an FRB in the magnetosphere. However, this argument was challenged by several other perspectives. First, an FRB traveling in open field line regions probably reduces this effect <cit.>. Second, the large radiation pressure possibly helps an FRB break out from the magnetosphere <cit.>. Third, a self-cleaning effect may sweep away the plasma on the path of an FRB <cit.>. These effects can alleviate the criticism that an FRB can not escape the magnetosphere of a magnetar. In conclusion, we argue that if an FRB is partially scattered by magnetized particles, rather than being completely blocked,  an O-mode wave remains more transparent than an X-mode wave.







§ ACKNOWLEDGMENTS

We are very grateful to the referee for the careful and thoughtful suggestions that have helped us improve this manuscript substantially. We thank Biao Zhang, Zi-Bin Zhang, Yuanhong Qu, Yue Wu, and Ze-Nan Liu for their useful discussions. This work is supported by  the National SKA Program of China (grant No. 2020SKA0120300) and National Natural Science Foundation of China (grant No. 12393812). S.Q.Z. acknowledges support from the National Natural Science Foundation of China (grant No. 12247144) and China Postdoctoral Science Foundation (grant Nos. 2021TQ0325 and 2022M723060). 






