The Integrated Pulse Profiles of Fast Radio Bursts

AO-06 (Hessels et al., 2019) is a triple-peaked burst from the repeating FRB 121102 (Spitler et al., 2016). Its dynamic spectrum shows a downward drifting time-frequency structure (Fig. 1d). Other FRB 121102 bursts like AO-02 and GB-01 also show such a feature. Several authors have investigated the cause of the formation of the feature(Metzger et al., 2019; Wang et al., 2019). Their studies are based on the dynamic spectrum dedispersed traditionally, like the one presented in Figure 1d. However, an alternative dynamic spectrum could also be considered, in which the emission time is continuous and frequency-dependent.

We suggest that the real implication of the structure showing a downward time-frequency drifting is that it is ill dedispersed. The original dynamic spectrum should be the one illustrated in Figure 1c, which is an estimation spectrum of the burst AO-06 if it is dedispersed to 563.8 pc cm^-3, instead of 560.5 pc cm^-3. In this further dedispersed spectrum, the burst occurs in the lower frequency band first, reaches its maximum in the middle time of the burst on the middle-frequency, fades out from the higher frequency band at last. An essential feature of the newly conceived dynamic spectrum is that the instantaneous spectrum of it keeps being the same triple-peaked one throughout the radiation process. The instantaneous spectrum (Fig. 1a) is similar to the integrated pulse profile obtained in the usual way; both have three peaks. In this sense, we may say that the integrated pulse profiles of FRBs are the instantaneous spectra near the sources of FRBs.

An eligible physical interpretation of the dynamic spectrum of AO-06 (Fig. 1c) needs to address two observational facts: [1] the triple-peaked instantaneous spectrum; [2] the drifting of central frequency from the low frequency to the high. Undulator radiation is a known mechanism that is capable of generating both effects at the same time.

An undulator (Fig. 2a) is a periodic array of dipole magnets with alternating polarity. It is commonly used as an insertion device in the storage ring to produce monochromatic synchrotron radiation with relativistic particles. The frequency of radiation in the direction of angle $\theta$ is determined by equation (Hofmann, 2007):

Where $\gamma$ is Lorenz factor, $c$ is the speed of light, $\lambda_{u}$ is the period of the magnetic field structure. $K$ is the vector potential of the magnetic field normalized to the rest energy of the radiation particle. The radiation produced in the weak magnetic field ($K\leq 1$) is the so-called undulator radiation. This equation is essentially describing a Doppler effect. The Doppler shift is angle-dependent; if you look on-axis, you get the highest frequency; if you go off-axis, the frequency gets lower and lower.

A frequency-swept dynamic spectrum can be produced by the process illustrated in Figure 2b, which is conceived mostly according to the circumstances between the Earth and the Sun. A beam of relativistic electrons is produced in an active source, might be a neutron star or double stars. The field in the acceleration site is strong. The radiation of AO-06 is not produced here; It is produced in a relatively weak field. After traveling about $10^{7}km$ or more, when the ambient magnetic field get weak enough($K\leq 1$), electrons meet with a quasi-periodic magnetic structure. The radiation of the burst AO-06 is produced here.

To describe the burst AO-06, an on-axis condition (where $\theta=0$) suffices. In this case (Fig. 2b), the variation of $f$, the central frequency, is only determined by the variation of $\gamma$. $f$ is proportional to $\gamma^{2}$. For the burst AO-06, a raw estimation gives that the low energy electrons arrived 3 ms earlier (Fig. 1c). With a smaller $\gamma$, they produce radiation in the lower-frequency band. The central frequency increases as the electrons arriving later have larger $\gamma$. To offer an appreciation with magnitude, if a beam of electrons with $\gamma$ increasing from $2.07\times 10^{3}$ to $2.37\times 10^{3}$, sequentially passed through a periodic structure with a period of 2000 km and a weak magnetic field ($K<1$). Then the central frequency received at a distance will drift from 1.28 to 1.68 GHz similar to the one shown in Figure 1c.

In addition to the central frequency drifting, the observed radiation intensity is also varying with frequency. A gaussian envelope has been used in our simulation to account for the energy distribution of radiation particles.

The triple-peaked spectrum illustrated in Figure 1a is the squared Fourier transform of a sinusoidal function with 28 periods over which the wave vector ($k=2\pi/\lambda$) is increased by 46.5% (hereafter the taper parameter $\eta$). This type of spectrum is known to the accelerator scientists as the radiation spectrum from a tapered undulator (Bosco & Colson, 1983; Shih & Caponi, 1982). The spectrum can be produced by passing mono-energetic electrons through a periodic magnetic structure with 28 periods over which the wave vector is increased by 46.5%.

The forming mechanism of the spectrum can be easily apprehended through the relationship between a driven motion and driven force. For the moment, we call the radiation particles by the name of electrons, although we will show later that they are probably nuclei. The undulated motion of electrons in periodic magnetic fields is a driven motion and a harmonic oscillation in the frame of reference moving with the electrons. It can be approximated by a wave function $e^{-i\omega t}$. As the magnetic field is weak, the observer is kept within the radiation cone. In such a condition, the Fourier transform of the electron’s apparent trajectory is the first-order approximation of the radiation spectrum. The on-axis undulator radiation spectrum of a single electron is the Fourier transform of $e^{-i\omega t}$ with a Doppler shift due to the transformation between different frames. When the spatial frequency (driven frequency) of the magnetic structure is linearly increasing, the frequency of oscillation (driven motion) also increases linearly. The motion of particles can be approximated by $e^{-i(\omega_{0}t+bt^{2})}$, which is merely the function $e^{-i\omega t}$ with $\omega$ substituted by a linear varying term $\omega_{0}+bt$. The key point here is the $t^{2}$ term in the phasor. Because of this, the radiation spectrum, which can be approximated by the Fourier transform of this wave train, is a Fresnel integral. This process shares a similar mathematic formulism with near field (Fresnel) diffraction of visible light from narrow slits (Bosco & Dattoli, 1983). In optics, the triple-peaked pulse profile of AO-06 corresponds to the diffraction pattern from a slit encompassing 1.5 Fresnel zones (Hecht, 2017).

From the energy perspective of view, the radiation spectrum of electrons reflects how electrons lose energy in the magnetic vector potential field through radiation. The spectrum is determined by the magnetic vector potential’s spatial distribution along the traveling path of electrons. Assuming the magnetic structure to have plane polarization and the magnetic field is weak, then the spontaneous radiation spectrum is proportional to the squared Fourier transform of the transverse magnetic vector potential (the Appendix). The vector potential $A_{\bot}$ of a tapered magnetic field and the spectrum $S(\omega)$ measured at the radio telescope can be written as:

Where $A(z)$ is the envelope of the magnetic field’s amplitude along the electrons’ path: $z$; it is approximated by a Gaussian function in our simulation and affects the symmetry of the spectrum; $k_{0}$ is the wave vector at $z=0$; $b$ is a constant and describes the linear variation of the wave vector along $z$; $\omega$ is the radiation frequency. Assuming a constant velocity along the $z$-direction, then the variable $z$ in Equation 2 can be substituted with the time variable $t$, that is the time representation discussed in the previous paragraph.

The multi-peaked feature of the spectrum comes from the Fourier component of Equation 3. A parameter $W$ can be defined as $\frac{\ln(1+\eta)\times N}{2}$ (the Appendix), where $N$ is the number of undulator periods, $\eta$ is the taper parameter. The number of peaks appearing on a spectrum is about $W/2$ (illustrated by Figure 3). For example, the double-peaked FRB 121002 (Champion et al., 2016) should have a $W$ value around four. Figure 3 shows the spectrum for a general case expressed in terms of the $W$ parameters, calculated through Equation 3 with a symmetric envelope $A(z)$. This plot is the equivalence of Figure 4 of the paper (Walker, 1988).

We will explain the simulation procedure through the simulation of the burst AO-06, which is a case with $W\sim 5.44$. The objective of the simulation is twofold. We need to generate a dynamic spectrum whose shape correlates with the observation approximately. Meanwhile, its integrated pulse profile should be similar to the observed one. We can only strike a balance between them by adjusting eleven parameters manually iteratively. In the following, we will skip the iteration process and directly use the known value of the relevant parameters.

The main steps of the simulation are:

(1) A wave train (in Figure 4) is firstly generated to represent the magnetic field potential $A_{\bot}$ in Equation 2. Two parameters determine the sine component of the Equation 2: $W=5.44$ and $\eta=0.46$ from which the b parameter can be calculated. It is equivalent to know how many periods the magnetic structure has and how it is varying. The unit length of the undulator is defined by the periodic length at $z=0$, hence $k_{0}=2\pi$. The envelope $A(z)$ is approximated by a Gaussian function defined by the variable $MuWave$ and $SigmaWave$ in the code. The real form of $A(z)$ maybe $\propto 1/r$ where r is the distance from the FRB source. $A(z)$ is related to the toroidal magnetic field. We’ll discuss this in Section 4. The values of $MuWave$ and $SigmaWave$ affect the relative heights of the peaks of the final integrated pulse profile. The length of $A(z)$ is normalized to the length of the undulator ($L\_eta$ in the code).

(2) The instantaneous spectrum in Figure 1a is computed according to the Equation 3 by doing a Fourier transform to the wave train, multiplying circular frequency, and squaring itself. The red dashed line in Figure 1a represents the background noise level, which is a free parameter, defined by the variable $bg$ in the code.

(3) The dynamic spectrum in Figure 1c is created by subtracting the background noise $bg$ from the spectrum in Figure 1a, then moving it along the white line in Figure 1c, and applying a Gaussian function to modulate the radiation intensity. The white line is a dispersion curve defined by $t=4149\frac{DM}{Frequency^{2}}$ with the dispersion measure (hereafter $DM$) $\sim-3.1$ pc cm^-3. The value of the DM is defined by the variable $dm$ in the code. The Gaussian function is defined by the variables $sigma$ and $mu$. The unit length of the Gaussian function is defined by the frequency range of the white line. The high and low-frequency boundaries of the line determine the frequency range. They are defined by the variables $fh$ and $fl$ in the code. The output is not sensitive to $fh$ and $fl$ because we can adjust the width of the Gaussian envelope through the variable $sigma$. However, sufficient values should be assigned to them to guarantee reasonable time-frequency coverage. The parameters set in this step are corresponding to the effects related to the distribution of the radiation particles. Since the distribution is unknown, we have to guess the radiation spectrum in the time-frequency domain. The reasonability of the conjecture made here will be discussed in Section 4.

(4) The dynamic spectrum in Figure 1d is obtained by dispersing the dynamic spectrum in Figure 1c by 3.3 pc cm^-3 further. The variable $dm1$ in the code is account for this DM lag.

A physical interpretation of the simulation above is that the original dynamic spectrum is oblique. It is produced by a beam of charged particles passing through a quasi-periodic magnetic structure. The Fourier transform of the potential of the magnetic field has three peaks. The space-time distribution of the particles results in a dispersive-like frequency drift. However, the current data processing method is incomplete; that the dynamic spectrum is presented vertically like the one in Figure 1d. The dynamic spectrum in Figure 1d is supposed to be observed in a position far away from the source region of FRB, to which the DM from the source of the burst is about 3.3 pc cm^-3.

The above approach can be easily generalized to simulate other triple-peaked bursts like the burst GB-01, AO-02, and AO-11 of FRB 121102 (Hessels et al., 2019), the September 17 burst of FRB 180814.J0422+73 (Amiri et al., 2019) and FRB 181017 (Farah et al., 2019). They are just the burst AO-06 with slightly different magnetic field configurations and the energy distributions of radiation particles. Naturally, the pulse of triple-peaked radio pulsars, like PSR J1757-2421 (Yuan et al., 2017), also can be reproduced with the same approach. The simulation codes of the burst AO-06, GB-01, AO-02, AO-11, and AO-13 written in Matlab language and the Matlab code generating all the figures in this paper can be downloaded online¹¹1The Matlab codes can be downloaded from: https://github.com/huangyu7991/frb.git. The output pictures of the codes are also available online for none Matlab user’s references.

The simulation shows that two important parameters can be estimated from the observation for sure. One is N, the number of period of the quasi-period structure; the other one is $\eta$, the variation speed of the period length. Besides, two parameters of the envelope of the magnetic field can also be estimated, supposing the envelope is of Gaussian shape. The relative bandwidth of the burst and the shape of the spectrum constrain these parameters. From Equation 1, $\lambda_{u}=2\gamma^{2}\lambda$ (where $\lambda$ is the observed wavelength, and assuming $\theta=0,K\approx 0$), with the precise knowledge of $\gamma$ or $\lambda_{u}$, the other one can be calculated.

The instantaneous spectra of many 4$\sim$8 GHz bursts of FRB 121102 (Michilli et al., 2018, Gajjar et al., 2018) are almost already multi-peaked. The dynamic spectrum of these high-frequency bursts can be reproduced with the approach introduced above. The difference is that the central frequency of the high-frequency instantaneous spectrum is not drifting. So we only need a function to mimic the variation of radiation intensity and need not to fit the slope of the dispersive curve.

The burst 11O of FRB 121102 (Gajjar et al., 2018) has six apparent peaks on its instantaneous spectrum. Its simulation result is presented in Figure 5d. To produce this pattern, we first create the instantaneous spectrum in Figure 5b by doing a Fourier transform to a sinusoidal wave train with N=90 and $\eta=32\%$. Then let the spectral strength vary with time as the Gaussian function shown in Figure 5a, we obtain the dynamic spectrum shown in Figure 5c. Dispersing this spectrum with DM=9.3 pc cm^-3, we get Figure 5d.

The interpretation of the simulation is: the burst 11O is produced by mono-energetic particles moving through a periodic magnetic structure with 90 periods over which the wave vector increased by 32$\%$. The number of the radiation particles or the coherence of the beam varies with time as the curve in Figure 5a. The original energy distribution in the time-frequency domain looks like the pattern in Figure 5c. However, the burst in the publication is similar to the one shown in Figure 5d, for using a DM which is 9.3 pc cm^-3 smaller than the real one.

The Gaussian function used in Figure 5a is not ideal because it is symmetric while the observed time variation of the radiation intensity is not. Nevertheless, for simplicity, we pertain to the symmetric case. It is enough to constrain the energy in the certain time-frequency domain and illustrate the basic idea. The other issue at odds with the observation is that the spectrum in Figure 5b has seven peaks, while the observation shows six apparent peaks. There must be a peak in the observation at the noise level and it is not resolved. To keep the different patches in Figure 5c separating from each other as the observation, we have no freedom to use parameters otherwise.

The burst AO-02 is another triple-peaked burst of FRB 121102 (Hessels et al., 2019). A comparison between the simulation and the observation of the burst AO-02 is in Figure 6. Its simulation can be obtained by dispersing a DM$\sim-4.5$ pc cm^-3 dynamic spectrum by 5.5 pc cm^-3. The blue and yellow image in Figure 6a is the simulated time-frequency structure. The gray-colored part of the image is created from a 3D printable data²²2The 3D printable data is downloaded from: https://www.thingiverse.com/thing:2723399/files of the burst. It can be seen that the two parts correlate to each other very well, except that the simulated separation between the two strips on the right-side does not match that of the observation. By overlapping the two dynamic spectra each other, we are not doing a strict mathematical fitting; we are showing that the process proposed by us can generate an energy distribution in time-frequency domain similar to that of an FRB. The process can give a constrained energy distribution only, just better than a pure theory full of formulas.

In the simulation of AO-06, we cannot keep the strip in the top-left corner of Figure 1d strictly vertical as the observation. Otherwise, the envelope of the other two strips cannot match the observation very well.

The discrepancies between the simulation and observation may stem from two main factors: (1) Our simulation can only reflect the radiation process but not the whole process, including propagation and data collection. (2) The slope of the original dynamic spectrum can not be described by a dispersion relationship strictly as what we have assumed.

A comparison between the simulation and the observation of the burst AO-13 (Hessels et al., 2019) brings an inconspicuous feature into the attention. Although the pulse of the burst has two peaks, it is probably a triple-peaked burst with the high-frequency part buried under the background noise level. The fact is revealed by the residual radiation in the high-frequency range, marked by the black arrow in Figure 7. A small amount of observed radiation(gray colored) locates in the position where the third strip is supposed to appear. We intentionally lower the background noise level of the image to let the leftmost strip appeared.