Narrow spectra of repeating fast radio bursts: A magnetospheric origin

The waves are coherently enhanced significantly when the emitting electrons form charged bunches, which is the fundamental unit of coherent emission. Different photon arrival delays are attributed to charges moving in different trajectories, so that charges in the horizontal plane (a light solid blue line denotes a bunch shown in Figure 1) have roughly the same phases. The bunch requires a longitude size smaller than the halfwavelength, which is $\sim 10$ cm for GHz wave.

Inside a bunch, the total emission intensity is the summation of that of single charges with coherently adding. Consider a single charge that moves along a trajectory $\boldsymbol{r}_{s}(t)$. The energy radiated per unit solid angle per unit frequency interval is interested in considering spectrum, which is given by (Rybicki & Lightman, 1979; Jackson, 1998)

where $\boldsymbol{\beta}_{s\perp}$ is the component of $\beta_{s}$ in the plane that is perpendicular to the LOS:

$\boldsymbol{\epsilon}_{1}$ is the unit vector pointing to the center of the instantaneous circle, $\boldsymbol{\epsilon}_{2}=\boldsymbol{n}\times\boldsymbol{\epsilon}_{1}$ is defined. If there is more than one charged particle, one can use subscripts $(i,\,j,\,k)$ to describe any charge in the three-dimensional bunch. The subscripts contain all the information about the location. We assume that bunches are uniformly distributed in all directions of ($\hat{\boldsymbol{r}},\,\hat{\boldsymbol{\theta}},\hat{\boldsymbol{\varphi}}$). Equation (3) can be written as

where $N_{\mathrm{e}}=N_{i}N_{j}N_{k}$. The emission power from charges with the same phase is $N_{\mathrm{e}}^{2}$ enhanced. Detailed calculations are referred to Appendix A.

The bunches thus act like single macro charges in some respects. Define a critical frequency of curvature radiation $\omega_{\mathrm{c}}=3c\gamma^{3}/(2\rho)$, where $\rho$ is the curvature radius. Spectra of curvature radiation from a single charge can be described as a power law for $\omega\ll\omega_{\mathrm{c}}$ and exponentially drop when $\omega\gg\omega_{\mathrm{c}}$. The spectrum and polarization properties of a bunch with a longitude size smaller than the half-wavelength and half-opening angle $\varphi_{t}\lesssim 1/\gamma$, are similar to the case of a single charge. In general, the spectra can be characterized as multisegmented broken power laws by considering a variety of geometric conditions of bunch (Yang & Zhang, 2018; Wang et al., 2022c).

We consider two general cases that the perturbations exist both parallel ($E_{1,\|}$) and perpendicular ($E_{1,\perp}$) to the local $\boldsymbol{B}$-field independently. The perturbations can produce extra parallel $c\dot{\beta}_{\|}$ and perpendicular accelerations $c\dot{\beta}_{\perp}$ to the local $\boldsymbol{B}$-field, as shown in Figure 1. The perturbation electric fields are much smaller than the $E_{\|}$, which is $\sim 10^{7}$ esu, sustaining a constant Lorentz factor (Wang et al., 2019). Affected by these accelerations, electrons not only emit curvature radiation, but also emissions caused by the perturbations on the curved trajectory. As a general discussion, we do not care about what is the perturbation in this Section but require that the velocity change slightly, i.e., $\delta\beta\ll\beta$.

Four Stokes parameters can be used to describe the polarization properties of a quasi-monochromatic electromagnetic wave (see Appendix B). Based on Equation (42), the electric vectors of the emitting wave reflect the motion of charge in a plane normal to the LOS. For instance, if the LOS is located within the trajectory plane of the single charge, the observer will see electrons moving in a straight line so that the emission is $100\%$ linearly polarized. Elliptical polarization would be seen if the LOS is not confined to the trajectory plane. We define that $\varphi<\theta_{c}$ as on-beam and $\varphi>\theta_{c}$ as off-beam. For the bunch case, similar on-beam and off-beam cases can also be defined in which $\theta_{c}$ is replaced by $\varphi_{t}+\theta_{c}$ (Wang et al., 2022a). The general conclusion of the bunching curvature radiation is that high linear polarization appears for the on-beam geometry, whereas high circular polarization is present for the off-beam geometry.

We consider the polarization properties of the bunch structure case (Section 2.1.2), and the perturbation scenarios including the parallel and the perpendicular cases (Section 2.2). For the quasi-periodically distributed bulk of bunches scenario, both $A_{\|}$ and $A_{\perp}$ multiply the same factor, so that the polarization properties are similar with a bulk of bunches, which bunches are uniformly distributed (e.g., Wang et al. 2022c). For the perturbation scenario, the perturbation on the curved trajectory can bring new polarization components, which generally obey the on(off)-beam geometry. However, the intensity of the perturbation-induced components is too small then the polarization profiles are mainly determined by the bunching curvature radiation. Consequently, the polarization properties for both the bunching mechanism and the perturbation scenarios can be described by the uniformly distributed bulk of bunches.

We take $\gamma=10^{2}$, $\varphi_{t}=10^{-2}$ and $\rho=10^{7}$ cm. The beam angle at $\omega=\omega_{\rm c}$ is 0.02. According to the calculations in Appendix B, we simulate the $I$, $L$, $V$, and PA profiles, shown in Figure 5. These four parameters can completely describe all the Stokes parameters. The total polarization fraction is $100\%$ because the electric vectors are coherently added. The polarization position angle (PA) for the linear polarization exhibits some variations but within $30^{\circ}$. The differential of $V$ is the largest at $\varphi=0$ and becomes smaller when $|\varphi|$ gets larger. High circular polarization with flat fractions envelope would appear at off-beam geometry. In this case, the flux is smaller but may still be up to the same magnitude as the peak of the profile (e.g., Liu et al. 2023b). The spectrum at the side of the beam becomes flatter due to the Doppler effect (Zhang, 2021). The sign change of $V$ would be seen when $\varphi=0$ is confined into the observation window (Wang et al., 2022a).

An Alfvén wave packet can be launched from the stellar surface during a sudden crustal motion and starquake. The wave vector is not exactly parallel to the field line so the Alfvén waves have a non-zero electric current along the magnetic field lines. Starquakes as triggers of FRBs are always accompanied by Alfvén wave launching. We investigate whether Alfvén waves as the perturbations discussed in Section 2.2 can affect the spectrum.

The angular frequencies of the torsional modes $\Omega$ caused by a quake are calculated as follows (also see Appendix C). We chose the equation of state (EOS A) (Pandharipande, 1971) and EOS WFF3 (Wiringa et al., 1988), which describe the core of neutron stars, as well as consider two different proposed EOSs for the crusts, including EOSs for NV (Negele & Vautherin, 1973) and DH (Douchin & Haensel, 2001) models. We match the various core EOS to two different EOSs for the crust. The crust-core boundary for the NV and DH EOSs is defined at $\rho_{m}\approx 2.4\times 10^{14}\,\rm g\,cm^{-3}$, and $\rho_{m}\approx 1.28\times 10^{14}\,\rm g\,cm^{-3}$, respectively.

The frequencies of torsional modes are various with a strong magnetic field (Messios et al., 2001). As already emphasized by Sotani et al. (2007), the shift in the frequencies would be significant when the magnetic field exceeds $\sim 10^{15}$ G. In the presence of magnetic fields, frequencies are shifted as,

where ${}_{\ell}{\alpha}_{n}$ is a coefficient depending on the structure of the star, ${}_{\ell}f_{n}^{(0)}$ is the frequency of the non-magnetized star, and $B_{\mu}\equiv(4\pi\mu)^{1/2}$ is the typical magnetic field strength.

In Fig. 6, we show the effects of the magnetic field on the frequencies of the torsional modes. The magnetic field strength is normalized by $B_{\mu}=4\times 10^{15}$ G. Different dashed lines in Fig. 6 are our fits to the calculated numerical data with a high accuracy. For $B>B_{\mu}$, we find that the frequencies follow a quadratic increase against the magnetic field, and tend to become less sensitive to the NS parameters. The oscillation frequencies exceed $10^{3}$ Hz for $n=2$ and $\Omega\sim 10^{4}$ Hz for typical magnetars.

Alfvén waves are launched from the surface at a frequency of $\Omega$, which is much faster than the angular frequency of the spin, and propagate along the magnetic field lines. Modifying the Goldreich-Julian density can produce electric fields to accelerate magnetospheric charged particles. The charged particles oscillate at the same frequency as the Alfvén waves in the comoving frame. In the lab frame, the perturbation has $\omega_{a}=\Omega\gamma^{2}/\gamma^{2}_{\rm A}$ due to the Doppler effect, where $\gamma_{\rm A}=\sqrt{1+B^{2}/(4\pi\rho_{m}c^{2})}$. $\omega_{a}$ is much smaller than $10^{8}\Omega_{4}\gamma^{2}_{2}\,\rm{Hz}$ because of the strong magnetic field and low mass density in the magnetosphere.

Therefore, by considering that Alfvén waves play a role in the electric perturbations discussed in Section 2.2, the spectra have no spike from 100 MHz to 10 GHz, which is the observation bands for FRBs so far. The spectral spike would appear at the observation bands when the perturbations are global oscillations, meaning that all positions in the magnetosphere oscillate simultaneously with $\Omega$ rather than a local Alfvén wave packet.

We consider the propagation of electromagnetic waves in magnetospheric relativistic plasma. The magnetosphere mainly consists of relativistic electron-positron pair plasma, which can affect the radiation created in the inner region. We assume that the wavelength is much smaller than the scale lengths of the plasma and of the magnetic field in which the plasma is immersed. The scale lengths of the magnetic field can be estimated as $B/(\partial B/\partial r)\sim r$. The plasma is consecutive and its number density is thought to be multiples of the Goldreich-Julian density (Goldreich & Julian, 1969), leading to the same scale lengths as that of the magnetic field. Both the scale lengths of magnetic field and plasma are much larger than the wavelength $\lambda=30\nu^{-1}_{9}$ cm, even for outer magnetosphere $r\sim 4.8\times 10^{9}P_{0}$ cm.

Whatever a neutron star or a magnetar, the magnetospheric plasma is highly magnetized. It is useful to define the (non-relativistic) cyclotron frequency and plasma frequency:

where $n_{\mathrm{p}}=\mathcal{M}n_{\rm GJ}=\mathcal{M}B/(ceP)$, in which $n_{\rm GJ}$ is the Goldreich-Julian density, and $\mathcal{M}$ is a multiplicity factor. In the polar-cap region of a pulsar, a field-aligned electric field may accelerate primary particles to ultra-relativistic speed. The pair cascading may generate a lot of plasma with lower energy and a multiplicity factor of $\mathcal{M}\sim 10^{2}-10^{5}$ (Daugherty & Harding, 1982). The relationships that $\omega\ll\omega_{B}$ and $\omega_{\mathrm{p}}\ll\omega_{B}$ are satisfied at most regions inside the magnetosphere.

The vacuum polarization may occur at the magnetosphere for high energy photons and radio waves by the plasma effect in a strong magnetic field (Adler, 1971). At the region of $r\gtrsim 10R$ where FRBs can be generated, in which $R$ is the stellar radius, the magnetic field strength $B<B_{Q}=m_{\mathrm{e}}^{2}c^{3}/(e\hbar)$ with assumption of $B_{\rm s}=10^{15}$ at stellar surface. In the low-frequency limit $\hbar\omega\ll m_{\mathrm{e}}c^{2}$, the vacuum polarization coefficient can be calculated as $\delta_{V}=1.6\times 10^{-4}(B/B_{Q})^{2}=2.65\times 10^{-8}B_{12}^{2}$ (Heyl & Hernquist, 1997; Wang & Lai, 2007). Therefore, the vacuum polarization can be neglected in the following discussion about the dispersion relationship in the magnetosphere.

Waves have two propagation modes: O-mode and X-mode. In the highly magnetized relativistic case, the dispersion of X-mode photon is $n\approx 1$. O-mode photons have two branches in the dispersion relationship, which are called subluminous O-mode ($n>1$) and superluminous O-mode ($n<1$), shown as Figure 11. The subluminous O-mode cuts off at $1/\cos\theta_{B}$, where $\theta_{B}$ is the angle between angle between $\boldsymbol{\hat{k}}$ and $\boldsymbol{\hat{B}}$. Following the calculation shown in Appendix D, the dispersion relation of magnetospheric plasma for subluminous O-mode photons can be found as (Arons & Barnard, 1986; Beskin et al., 1988)

with

where $\beta_{n}=n_{\|}^{-1}=\omega/(k_{\|}c)$, $\gamma_{n}=(1-\beta_{n}^{2})^{-1/2}$ and $f(u)$ is the distribution function of plasma.

Wave forces can exist many collective waves in a beam and exchange energy between waves and particles. If the distribution function is ${\mathrm{d}}f/{\mathrm{d}}v<0$ close to the phase speed of the wave, the number of particles with speeds $v<\omega/k$ will be larger than that of the particles having $v>\omega/k$. Then the transmission of energy to the wave is then smaller than that absorbed from the wave, leading to a Landau damping of the wave. According to the dispersion relationship given in Section 4.1, there is no linear damping of the X-mode since wave-particle resonance is absent. The superluminous branch of the O-mode also has no Landau damping because the phase speed exceeds the speed of light. Landau damping only occurs at the subluminous O-mode photons.

The Landau damping decrement can be calculated by the imaginary part of the dispersion relationship (Boyd & Sanderson, 2003). We let $k=k_{r}+ik_{i}$, where $k_{i}\ll k_{r}\approx k$. If a wave has a form of $E\propto\exp(-i\omega t+ikr)$, the imaginary $k_{i}$ can make the amplitude drop exponentially. The imaginary part of Equation (24) by considering a complex waveform of $k$ is given by

where $g_{r}$ and $g_{i}$ denote the real and imaginary parts of Equation (25). Consider emission at radius $r_{e}$ along a field line whose footprint intersects the crust at magnetic colatitude $\Theta_{e}(R/r_{e})^{1/2}$ for $\Theta_{e}\ll 1$, one can obtain the angle between $\boldsymbol{k}$ and $\boldsymbol{B}$ at each point:

Consequently, the optical depth for waves emitted at $r_{e}$ to reach $r$ is

Even the distribution function for the magnetospheric electrons and positrons is not known, a relativistic thermal distribution is one choice. Since the vertical momentum drops to zero rapidly, for instance, a 1-D Jüttner distribution is proposed here (e.g., Melrose et al. 1999; Rafat et al. 2019):

where $\rho_{T}=m_{\mathrm{e}}c^{2}/(k_{\mathrm{B}}T)$, in which $T$ is the temperature of plasma and $k_{\mathrm{B}}$ is the Boltzmann constant. Alternatively, another widely used distribution is a Gaussian distribution (e.g., Egorenkov et al. 1983; Asseo & Melikidze 1998)

where $u_{m}$ can be interpreted as the average $u$ over the distribution function. Besides a thermal distribution, power law distributions $f(\gamma_{\rm p})=(p-1)\gamma_{\rm p}^{-p}$, where $p$ is the power law index ($p>2$), are representative choice to describe the plasma in the magnetosphere (e.g., Kaplan & Tsytovich 1973).

The optical depths as a function of wave frequency with four different distributions are shown in Figure 7. Here, we assume that $r_{e}=10^{7}$ cm, $\mathcal{M}=10^{3}$, $\gamma_{\rm p}=10$ and $\Theta_{e}=0.05$. For all considered distributions, optical depths decrease with frequencies which means that the Landau damping is more significant for lower-frequency subluminous O-mode photons. The optical depths are much larger than 1 for the FRB emission band so that the subluminous O-mode photon can not escape from the magnetosphere. Superluminous O-mode photons which frequency higher than $\omega_{\rm cut}=\omega_{\mathrm{p}}\gamma_{\rm p}^{-3/2}$ can propagate. The bunching curvature radiation is exponentially dropped at $\omega>\omega_{\mathrm{c}}$. If the superluminous O-mode cut-off frequency is close to $\omega_{\mathrm{c}}$, the spectrum of the emission would be narrow. A simple condition could be given when a dipole field is assumed:

where the emission region is considered at closed field lines in which $\rho\sim r$. The cut-off frequency $\omega_{\rm cut}\propto r^{-3/2}$ which evolves steeper than $\omega_{\mathrm{c}}\propto r^{-1}$. The bandwidth is suggested to become wider as the burst frequency gets lower.

Coherent curvature radiation can propagate out of the magnetosphere via superluminous O-mode with a frequency cut-off, leading to narrow spectra likely. However, the condition that the frequency cut-off is close to the characteristic frequency of curvature radiation could not be satisfied at all heights, leading to narrow spectra sometimes. The waves can also propagate via X-mode photon which has no frequency cut-off. The spectrum of X-mode photons ought to be broad. The waves propagate via whether X-mode or O-mode is hard to know because the real magnetic configuration is complex. Narrow spectra have been found in some FRBs, while some FRBs seem to have broad spectra, and there are also some not sure due to the limitation of bandwidth of the receiver (Zhang et al., 2023).

In principle, the spectrum of a radio magnetar would be narrow, if the emission is generated from $r>10R$ propagating via O-mode. However, radio pulses are rarely detected among magnetars. For normal pulsars, the lower magnetic fields may induce lower number density so that the Landau damping is not as strong as in magnetars.

The self-absorption is also associated with the emission. We assume that the momentum of the particle is still along the magnetic field line even after having absorbed the photon. The cross-section can be obtained through the Einstein coefficients when $h\nu\ll\gamma_{s}m_{\mathrm{e}}c^{2}$ (e.g., Locatelli & Ghisellini 2018):

where $\sigma_{\mathrm{c}}$ is the curvature cross section and $j_{s}$ is the emissivity.

There are special cases which the stimulated emission becomes larger than the true absorption. As a result, the total cross section becomes negative, and there is a possible maser mechanism. However, such maser action is ineffective when the magnetic energy density is much higher than that for the kinetic energy of charges (Zhelezniakov & Shaposhnikov, 1979). The isotropic equivalent luminosity for the curvature radiation is $\sim 10^{42}\,\rm erg\,s^{-1}$ if $N_{\mathrm{e}}=10^{22}$ at $r=10^{7}$ cm for $\gamma=10^{2}$ (Kumar et al., 2017). The number density in a bunch is estimated as $n_{\mathrm{e}}\sim 10^{11}\,\rm cm^{-3}$ by Wang et al. (2022c), so that $n_{\mathrm{e}}\gamma m_{\mathrm{e}}c^{2}\ll B^{2}/(8\pi)$. Thus, the maser process is ignored in the following discussion.

By inserting the curvature power per unit frequency (see e.g., Jackson 1998), the total cross section integrated over angles is given by (Ghisellini & Locatelli, 2018)

where $\Gamma(\nu)$ is the Gamma function. As discussed in Section 2, the charges are approximately monoenergetic due to the rapid balance between $E_{\|}$ and the curvature damping. The absorption optical depth is

The absorption is more significant for lower frequencies.

Figure 8 exhibits the comparison between purely bunching curvature radiation and it after the absorption. We adopt that $\gamma_{s}=10^{2}$, $n_{\mathrm{e}}=10^{11}\,\rm cm^{-3}$ and $\rho=10^{7}$ cm. The plotted spectra are normalized to the $F_{\nu}$ of curvature radiation with $\varphi_{t}=10^{-3}$ at $\omega=\omega_{\mathrm{c}}$. The spectral index is $5/3$ at lower frequencies due to the self-absorption, which is steeper than pure curvature radiation, but it is still wide-band emission. Alternatively, the $E_{\|}$ can let electrons and positrons decouple, which leads to a spectral index of $8/3$ at lower frequencies (Yang et al., 2020). The total results of these independent processes can make a steeper spectrum with an index of $13/3$.

Curvature radiation is formed since the perpendicular acceleration on the curved trajectory and its spectrum is naturally wide-band. Spectra of bunching curvature radiation are more complex due to the geometric conditions and energy distribution (e.g., Yang & Zhang 2018; Wang et al. 2022c). Despite considering the complex electron energy distributions, the spectrum is still hard to be narrow-band.

A quasi-periodically distributed bunches can create such narrow-band emission. The radiation from the bulk of bunches is quasi-periodically enhanced and the bandwidth depends on $\omega_{M}$ and the threshold of the telescope. As shown in Figure 2, narrow-band emissions with roughly the same bandwidth are expected to be seen from multi-band observations simultaneously. The energy is radiated into several multiples of $2\pi\omega_{M}$ with narrow bandwidths and the required number of particles in the bunch is smaller than that of pure curvature radiation.

Inverse Compton scattering (ICS) occurs when low-frequency electromagnetic waves enter bunches of relativistic particles (Zhang, 2022). The bunched particles are induced to oscillate at the same frequency in the comoving frame. The outgoing ICS frequency in the lab-frame is calculated as

where $\omega_{\rm in}$ is the angular frequency of the incoming wave and $\theta_{\rm in}$ is the angle between the incident photon momentum and the electron momentum. This ICS process could produce a narrow spectrum for a single electron by giving a low-frequency wave of narrow spectrum (Zhang, 2023). In order to generate GHz waves, dozens of kHz incoming wave is required for bunches with $\gamma=10^{2}$ and $\theta_{\rm in}$ should be larger than $0.5$. This requires the low-frequency waves to be triggered from a location very far from the emission region. The X-mode of these waves could propagate freely in the magnetosphere, but O-mode could not unless the waves’ propagation path approaches a vacuum-like environment.

As discussed in Section 2, the Lorentz factor is constant due to the balance between $E_{\|}$ and radiation damping. $\theta_{\rm in}$ varies slightly as the LOS sweeps across different field lines. If the incoming wave is monochromatic, the outgoing ICS wave would have a narrow value range matching the observed narrow-band FRB emissions. The polarization of ICS from a single charge is strongly linearly polarized, however, the electric complex vectors of the scattered waves in the same direction are added for all the particles in a bunch. Therefore, circular polarization would be seen for off-beam geometry (Qu & Zhang, 2023), similar to the case of bunching curvature radiation but they are essentially different mechanisms(Wang et al., 2022b). Doppler effects may be significant at off-beam geometry so that the burst duration and spectral bandwidth would get larger. The model predicts that a burst with higher circular polarization components tends to have wider spectral bandwidth. This may differ from the quasi-periodically distributed bunch case, in which the mechanism to create narrow spectra is independent of polarization.

The perturbation discussed in Section 2.2 can let charges oscillate at the same frequency in the comoving frame. This process is essentially the same as the ICS emission. The acceleration caused by the perturbation is proposed to be proportional to $\exp(-i\omega_{a}t)$, which leads to the wave frequency of $\omega_{a}(1-\beta\cos\theta_{\rm in})$ due to the Doppler effect in the lab frame. Taking the wave frequency into Equation (13), the result is the same as Equation (36). We let $F(t)=\int E(t)em^{-1}_{\rm e}{\rm d}t$ and ignore the radiation damping as a simple case. The dimensionless velocity is then given by $\beta=F(t)/\sqrt{1+F(t)^{2}}$. By assuming $F(t)$ has a monochrome oscillating form, according to Equation (42) if $F(t)\ll 1$, the result is reminiscent of the perturbation cases and ICS. If $F(t)\gg 1$, we have $\beta\sim 1$ which implies a wide-band emission.

Some large-scale interference processes, e.g., scintillation, gravitational lensing, and plasma lensing, could change the radiation spectra via wave interference (Yang, 2023). The observed spectra could be coherently enhanced at some frequencies, leading to the modulated narrow spectra. We consider that the multipath propagation effect of a given interference process (scintillation, gravitational lensing, and plasma lensing) has a delay time of $\delta t$, then the phase difference between the rays from the multiple images is $\delta\phi\sim 2\pi\nu\delta t$, where $\nu$ is the wave frequency. Thus, for GHz waves, a significant spectral modulation requires a delay time of $\delta t\sim 10^{-10}\nu_{9}^{-1}\leavevmode\nobreak\ {\rm s}$, which could give some constraints on scintillation, gravitational lensing, or plasma lensing, if the observed narrow-band is due to these processes.

For scintillation, according to Yang (2023), the plasma screen should be at a distance of $\sim 10^{15}$ cm from the FRB source, and the medium in the screen should be intermediately dense and turbulent. For plasma lensing, the lensing object needs to have an average electron number density of $\sim 10\leavevmode\nobreak\ {\rm cm^{-3}}$ and a typical scale of $\sim 10^{-3}$ AU at a distance of $\sim 1$ pc from the FRB source. The possibility of gravitational lensing could be ruled out because the gravitational lensing event is most likely a one-time occurrence, meanwhile, it cannot explain the remarkable burst-to-burst variation of the FRB spectra.