A Comparison Between Repeating Bursts of FRB 121102 and Giant Pulses from Crab Pulsar and Its Applications

Front. Phys. ???, ?? (2020) DOI ??? Research article

A comparison between repeating bursts of FRB 121102 and giant pulses from Crab pulsar and its applications

Fen Lyu1,2∗, Yan-Zhi Meng1,3, Zhen-Fan Tang1, Ye Li1,4, Jun-Jie Wei1, Jin-Jun Geng3,5∗∗, Lin Lin6, Can-Min Deng7∗∗∗, Xue-Feng Wu1,8∗∗∗∗

1. Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China 2. University of Chinese Academy of Sciences, Beijing 100049, China 3. School of Astronomy and Space Science, Nanjing University, Nanjing 210023, Jiangsu, China 4. Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China 5. Institute of Astronomy and Astrophysics, University of Tübingen, Auf der Morgenstelle 10, D-72076,Tübingen 6. Department of Astronomy, Beijing Normal University, Beijing 100875, China 7. Department of Astronomy, University of Science and Technology of China, Hefei 230026, Anhui, China 8. School of Astronomy and Space Sciences, University of Science and Technology of China, Hefei 230026, Anhui, China Corresponding author E-mail: ∗[email protected], ∗∗[email protected],∗∗∗[email protected],∗∗∗∗ [email protected]

There are some similarities between bursts of repeating fast radio bursts (FRBs) and giant pulses (GPs) of pulsars. To explore possible relations between them, we study the cumulative energy distributions of these two phenomena using the observations of repeating FRB 121102 and the GPs of Crab pulsar. We find that the power-law slope of GPs (with fluence ≥130 Jy ms) is 2.85±0.10. The energy distribution of FRB 121102 can be well fitted by a smooth broken power-law function. For the bursts of FRB 37 +0.55 121102 above the break energy (1.22 ×10 erg), the best-fitting slope is 2.90−0.44, similar to the index of GPs at the same observing frequency (∼1.4 GHz). We further discuss the physical origin of the repeating FRB 121102 in the framework of the super GPs model. And we find that the super GPs model involving a millisecond pulsar is workable and favored for explaining FRB 121102 despite that the magnetar burst model is more popular. Keywords Pulsars, Radio sources, general -methods: statistical

PACS numbers 97.60.Gb, 98.70.Dk, 02.70.Rr repetition of FRB 200428 is very rare, which may be 1 Introduction different from other extragalactic repeating FRBs [13]. The progenitors of FRBs are still under heated de- Fast Radio Bursts (FRBs) are mysterious, bright as- bates, and many theoretical models have been proposed tronomical millisecond-duration radio pulses [1–3], occa- (see [19] and references therein2), despite some of them sionally discovered in pulsar searches. Up to date, more have confronted serious challenges. Repeating FRBs are than one hundred FRBs have been reported [4]1. They the best candidates to explore physical nature due to its are expected to be of cosmological origin due to high repetition. FRB 121102 is the first observed repeating dispersion measures (DM, 102 − 103 pc cm−3) in excess FRB and has been detected in the frequency range from of the DM contribution from the Milky Way. And it is 600 MHz [20] to 8 GHz [21]. For repeating FRBs, some further confirmed by the localization of the host galaxies models involving catastrophic processes have been ruled arXiv:2012.07303v1 [astro-ph.HE] 14 Dec 2020 [5–8]. out, e.g., binary neutron star (NS) mergers [22, 23], and Among the moderately observed large sample of FRBs, binary white dwarf mergers [24]. Many non-catastrophic most of them are one-off events, and only twenty of models have been put forward, such as the flaring mag- them sporadically show repeating bursts [9–11]. More- netars model [25–28], accretion process by a neutron star over, FRB 180916.J0158+65 detected by CHIME [12] in a binary system [29], the interaction of NS with an as- exhibits a ∼16 day period acitvity with unknown mech- teroid belt [30], the interaction between the NS magneto- anism. Most recently, a bright FRB 200428 has been sphere and cosmic winds [31], and the giant pulses (GPs) reported to be spatially coincident with the galactic Soft from pulsars [32, 33]. The central engine of FRB 121102 Gamma-ray Repeater (SGR) 1935+2154 [13–16], also as- is questionable, which can be constrained from the persis- sociated with a hard X-ray burst [14, 15, 17, 18]. The tent radio nebula of FRB 121102 for the magnetar-type

1https://www.frbcat.org 2https://frbtheorycat.org/index.php/Main_Page

© Higher Education Press and Springer-Verlag Berlin Heidelberg 2020 Research article model [34–38]. m telescope in Pune, India (National Centre for Radio GPs were first discovered in the Crab pulsar (PSR Astrophysics) in a 260-hours observation from February B0531+21), and then were found in several energetic to April 2019. The observation was conducted between pulsars (see Table 1 of [39] and reference therein). Dif- 1280 MHz and 1380 MHz, with 65 MHz bandwidth us- ferent from the regular pulses, the typical duration of able. Each GP was detected with a signal-to-noise ratio GPs is very short, with no clear periodicity, ranging from (S/N) of ≥ 10. nanoseconds to microseconds. The flux density could be Among 20 repeating FRBs have been reported, 2 or 4 orders of magnitude higher than the average pulse- only two of repeating FRBs have host galaxy iden- integrated flux of regular energetic pulses (e.g., [40, 41]). tified and redshift measured (FRB 121102 and FRB Among the pulsars emitting the GPs, the largest value of 180916.J0158+65; [5, 12]). FRB 121102 is the first and the magnetic field at the light cylinder Blc are the Crab currently the best representative of repeating bursts. pulsar and PSR B1937+21 [39]. The Crab pulsar is a More than one hundred bursts from FRB 121102 have remarkable source emitting numerous GPs and simulta- been detected by several different surveys, including neously has been detected across a broadband range of Green Bank Telescope (GBT) in C-band (4-8 GHz; [46]) radio frequencies from 20 MHz [42] to 46 GHz [43]. The and Arecibo telescope at 1.4 GHz [47]. The repeating distinctive feature of GPs is the energy of GPs obeys a FRB 121102 has a very large and variable Faraday ro- power-law distribution, whereas the regular ones follow tation measure (RM ∼ 105 rad m−2) which means the a Gaussian or log-normal distribution (e.g.,[40, 44]), in- burst inhabits in an extreme magneto-ionic environment dicating that the emission mechanism of GPs may be [3]. different from that of regular pulses. Meanwhile, GPs To compare with the Crab GPs sample under the same are also crucial to understand the radiation mechanism condition, we use the sample of FRB 121102 detected at of pulsars. almost the same frequency [47]. The observing frequency Repeating FRBs and GPs from pulsars share simi- of Arecibo in the source frame is 1.4 ∗ (1 + z) GHz. Due larities in many observational aspects, such as bright- to the redshift 0.193 for FRB 121102 [48], the frequency ness temperature, duration time, sub-bursts, polariza- in the source rest frame approximatively is ∼ 1.4 GHz. tion, and frequency drift. Moreover, the emission mech- In this sample, there are 41 bursts from FRB 121102 ob- anism of both of them is coherent radiation, supporting served by the 305-m Arecibo telescope. The data is from that they may have a similar physical origin. Actually, two observations taken on 2016 September 13/09:47:07 in some theoretical models, it has been suggested that and 14/09:50:12, and lasting 5967 s and 5545 s. Each FRBs can be treated as GPs from young rapidly rotat- observation detected 18 and 23 bursts with S/N ≥10, re- ing pulsars [32, 33]. spectively. Moreover, we also investigate the energy dis- The statistical properties, such as the energy distri- tribution of the repeating FRB 121102 detected by GBT butions, of repeating FRBs and GPs might help us to telescope at C-band [46]. There are 93 bursts in total in understand FRBs progenitors. In this paper, we fo- 5 hours with a convolutional neural network technique. cus on the statistical properties of the repeating FRB 121102 and GPs from the Crab pulsar. The paper is organized as follows. In Section 2, we introduce the sam- 3 The cumulative energy E distributions ple selection. The statistical analysis of the isotropic- equvivalent energy E is shown in Section 3. Discussion Energy is a crucial parameter to identify the progeni- and conclusions are given in Section 4 and Section 5. tor and to explore the radiation mechanism. We perform We assume a flat ΛCDM universe with Ωm = 0.27, and an analysis on the energy distribution of the repeating −1 −1 H0 = 70 km s Mpc throughout this work. FRB 121102 and GPs to explore the progenitor of FRB 121102, despite there are many works devoted to the statistical properties of the repeating FRBs [46, 49–52]. 2 Sample selection The isotropic-equvivalent energy E in the source rest frame within the observing bandwidth for the selected The Crab Nebula pulsar is well known for its GPs, sample can be calculated by with detected GPs numerous enough for detailed studies 2 (e.g., [40, 41, 45]). Here, GPs refer to GPs generated by E = 4πdLfν ∆ν/(1 + z) , (1) the main pulse (MP) and interpulse (IP) phase. In our analysis, we adopt a complete sample of 1153 where dL is the luminosity distance of the source, with bright GPs (with fluence ≥130 Jy ms to avoid the in- 2.0 kpc for the Crab pulsar, fν is the fluence density complete effect near the detection threshold) from the of each burst, and ∆ν is the corresponding bandwidth Crab pulsar [45]. These GPs were detected by the 15- of every burst. In this paper, the bandwidth for FRB 121102 is derived from the burst frequency edges (fhigh

Fen Lyu, Yan -Zhi Meng, Zhen-Fan Tang. et al., Front. Phys. ???, ?? (2020) ??-2 +0.46 37 and flow) from the table given in [47]. while the break energy Eb is 1.22−0.31 ×10 erg (blue Here we adopt a simple power-law model, N(> E) ∝ dashed line). In [47], the threshold energy was set at −α 37 E to fit the cumulative energy distributions of bright Eth = 2 × 10 erg considering the sensitivity limit and GPs (fluence ≥ 130 Jy ms) from the Crab pulsar. While the observed turnover. Above the Eth, the best fit is dR −2.8 for the repeating FRB 121102, we fit it by the smooth dE ∝ E , which is generally consistent with our fitting broken power-law function, i.e., result within the uncertainty. All the uncertainties quote a 68% confidence interval for the best-fitted parameters. α ω α ω −1/ω E 1 E 2 To investigate what causes the different power indices Ncum(> E) = A + , (2) Eb Eb of the energy distribution of FRB 121102 detected by the Arecibo telescope at L-band and that by GBT telescope where A is the ampltitude, α1, α2 are slopes of the two at C- band and the meaning of the break energy Eb in segments, Eb is the break energy, ω is the sharpness (or Figure 1, we also fit the energy distribution for 93 bursts smoothness) of the break of energy distribution, and it from FRB 121102 by GBT telescope with broken power- usually can be set as a constant (ω = 3). law function. As shown in Figure 2, the best fits for the We can now obtain the goodness-of-fit χd.o.f for the lower and higher energy are −1.49+0.48 and 1.33+0.18 difference between the observed cumulative distribution −0.56 −0.14 (χ = 0.63), respectively, while the break energy is N and the theoretical distribution function N d.o.f cum,obs cum 1.60+0.94 ×1037 erg. for the cumulative distribution (with npar = 4 parame- −0.44 Interestingly, with the same observed frequency (∼ 1.4 ters, E0, α1, Eb, α2 ) GHz), FRB 121102 and Crab GPs share the power-law v u Ntot 2 form of similar indices (∼ 2.8) (Figure 1). At a different u 1 X [Ncum (Ei) − Ncum,obs (Ei)] observed frequency (C-band, 4-8 GHz), the energy dis- χd.o.f = t 2 , (Ntot − npar) σ i=1 cum,i tribution of FRB 121102 detected by the GBT telescope (3) is flatter (Figure 2) than that of the Arecibo telescope with more energetic bursts. where Ntot is the number of bursts dectected by the There is a lack of bursts below near the break energy same instrument at the same frequency for the same ∼ 1.60 × 1037 erg (the black dashed line) due to the in- source. Note that the uncertainty of Ncum(> E) is taken p strument sensitivity and observational limit. So maybe as σcum,i = Ncum,obs (Ei). The best-fitting parame- the incomplete sampling below the threshold is the ma- ters and the uncertainties can be obtained by minimizing jor reason that evokes the different power-law indices of χd.o.f via the python package emcee (a Bayesian MCMC energy distribution for FRB 121102 in Figure1 and Fig- method by sampling the affine-invariant for Markov ure 2. The power-law index of energy distribution with Chain Monte Carlo chains, [53]). repeating FRB 121102 may depend on the energy band, When fitting variables via a MCMC approach, we as- the burst is more energetic at a higher observational fre- sume the various parameters in the fitting function are quency (see Figure 1 and Figure 2), which can be fur- Researchuniform distribution article in the prior probability distribution. ther tested by detecting more bursts at different energy Here are the range of parameters for every sample: 1 < bands. Note that the energy distribution of GPs might α< 5 for GPs; 50 < A < 200, -2.5 <α1 < -0.5, 36 < also depend on the energy band [40, 44], and can have the lgEb <38 and 2 < α2 < 5 for Arecibo sample; 50 < A< index ∼ 1.8 for some pulsars (such as PSR 0631+1036; 200, -3< α1 < -0.01, 36 < lgEb <38.5 and 1< α2 < 3 [54]). for GBT sample. The power-law index of energy distribution and the GPs are generally considered as the power-law tail of statistical characteristics of the other observable quan- the regular pulses for the pulsar. In the upper panel of tities of FRB121102 are claimed to be similar to those Figure 1, the yellow-green points represent the GPs from of soft gamma repeater bursts from magnetars [55]. In the Crab pulsar, which is found to be well fitted by the addition, there are several statistical analyses [50, 55– power-law model. The rollover does not appear at the 60] dedicated to the self-organizing critical (SOC) study, high-energy end, while it does show for the flux density especially some researches [50, 55, 60, 61] that indicate distribution as stated in some previous studies [40, 44]. that the repeating FRB 121102 is a SOC system [62]. The best-fit slope is 2.85 ± 0.10 (χd.o.f = 0.45), which Thus, the repeating FRB 121102, is likely to be an SOC is well consistent with that of the fluence distribution system due to some kind of instability driven above the (2.8 ± 0.3) reported in [45]. In Figure1, we present the instability threshold value. If so, the break energy in Fig- cumulative energy (E) distribution for 41 Arecibo bursts ure 1 should be the threshold energy, which determines from FRB 121102 (blue spots). Fitted by the broken the trigger mechanism of the system. It is an important power-law function given in equation (2), the best fitting quantity that affects the slope of the power-law in energy results for indices of the lower and higher energy segment distribution, which is also discussed in [60]. +0.53 +0.55 are −1.59−0.57 and 2.90−0.44 (χd.o.f = 0.36), respectively,

Fen Lyu, Yan -Zhi Meng, Zhen-Fan Tang. et al., Front. Phys. ???, ?? (2020) ??-3 Research article

FRB 121102 101 +0.55 +0.53 2 = 2.90 0.44 1 = 1.59 0.57

100

10 1

Rate (per hour) = 2.85 ± 0.10 2 10 GPs from the Crab pulsar FRB 121102 1029 1030 1037 1038 E (erg) iso 101

+13.27 A = 88.30 22.35

100

+0.53 1 = 1.59 0.57 Rate (per hour)

0.8

1.2 1 10 1 1.6 1038 1039 2.0 +0.14 E (erg) 2.4 lgEb = 37.09 0.13 iso

37.35 +7.23 A = 135.79 14.12 37.20 b E g l 37.05

36.90

+0.54 2 = 2.90 36.75 0.44

4.8

+0.48 1 = 1.49 4.2 0.56 2 3.6 0.6

3.0 1.2 1

2.4 1.8

75 100 125 150 2.4 2.0 1.6 1.2 0.8 2.4 3.0 3.6 4.2 4.8 36.75 36.90 37.05 37.20 37.35 2.4 A lgE +0.20 1 b 2 lgEb = 37.21 0.14

38.1

37.8 b E g

Fig.1 Upper panel: The cumulative energy E distribu- l iso 37.5

tions of the GPs from the Crab pulsar (yellow-green), FRB 37.2

+0.18 2 = 1.33 0.14 121102 detected by Arecibo (blue circle). The vertical blue 36.9 dotted line represents the break energy of FRB 121102 fitted, 2.4 2.1 2 and the red line represents the best fitting line; Lower panel: 1.8 1D, 2D posterior marginalized probability distributions of the 1.5 fitting parameters obtained using MCMC method for FRB 1.2 75 2.4 1.8 1.2 0.6 1.2 1.5 1.8 2.1 2.4 100 125 150 36.9 37.2 37.5 37.8 38.1

121102 detected by Arecibo telescope. The contours in the A 1 lgEb 2 2D plots are 68% and 95% confidence intervals from inside to outside. Fig.2 Upper panel: The cumulative energy distribution of FRB 121102 detected by GBT telescope at C-band, the solid red line denotes the best fit line using equation (2) with Interestingly, the power-law slope of FRB 121102 omit- the best-fit parameters labeled in the lower panel, and the ting bursts that fall bellow the break energy is generally black dashed line is the break energy; Lower panel: 1D, 2D consistent with that of the GPs from the Crab as shown posterior marginalized probability distributions of the fitting in Figure 1, which implies they may share a similar ori- parameters obtained using MCMC method. The contours in gin. If FRB 121102 is extragalactic GPs, which energies the 2D plots are 68% and 95% confidence intervals from inside can power it? We will discuss this in the next section to outside. below.

4 Discussion

As mentioned in the introduction, neutron stars are the most popular FRB central engine. Especially the super GPs model involving pulsars [32, 33, 63] and the giant ﬂare model involving magnetars [26–28, 64–66] have

Fen Lyu, Yan -Zhi Meng, Zhen-Fan Tang. et al., Front. Phys. ???, ?? (2020) ??-4 been widely discussed. Specifically for FRB 121102, it share similar complex pulses morphology and a simi- was found that its host is a dwarf galaxy with a high lar repetition rate in a single pulsar (∼ 1-100 per hour, star formation rate [5, 48], which implies that the cen- see Figure 1). Furthermore, similar to FRBs [71, 72], tral engine of FRB 121102 may locate in the remnant of GPs have strong polarization signals, which is either left- the death of a massive star. Moreover, the polarization handed or right-handed polarization [73] between linear observations give that the Faraday rotation measure of polarization and circular polarization. Besides, spectral FRB 121102 is surprisingly high ∼ 105 rad m−2, indicat- structures in repeating FRBs resemble those seen in Crab ing that FRB 121102 should be surrounded by a strongly GPs [74]. Therefore, the similarities between the GPs magnetized environment [3]. These observational evi- and FRB 121102 possibly link to the same origin and dences support that the central engine of FRB 121102 mechanism of these two phenomena. may be a magnetized NS. In the following, we explore the physical origin of the repeating FRB 121102 in the framework of the super GPs model. 5 Conclusions As shown in the upper panel of figure 1, the typical energy of the FRB 121102 and GPs are 1037 ∼ 1038 erg In this paper, we have made a comparison of the en- and 1029 ∼ 1030 erg. On the other hand, the typical ergy distributions between the repeating FRBs and the time scale of the FRB 121102 and GPs are ms and µs, GPs from the Crab pulsar. We find that FRB 121102 hence the typical luminosity are 1040 ∼ 1041 erg/s and and Crab GPs share similar power-law indices in the en- 1035 ∼ 1036 erg/s, respectively. Following [67], based on ergy distribution at the same observed frequency ∼1.4 the typical luminosity of FRB 121102 and that of GPs, GHz3. Furthermore, we explored the physical origin of we have the repeating FRB 121102 in the framework of the giant pulses model. We find that the millisecond pulsar is LFRB 5 ξ = ∼ 10 . (4) possible to produce super GPs with luminosity compa- L GP rable to FRB 121102 in a reasonable active time scale. Since the GPs are powered by the spin down of the pulsar Therefore, we argue that the super GPs model is work- Lsd, the constraint on the magnetic field strength BFRB able and favored to explain FRB 121102 despite that the of the FRB source is magnetar burst model is more popular. 2 1/2 P−3 B ≈ 0.3 ξ B , (5) Acknowledgements We thank the anonymous referee for con- 5 Crab P Crab structive comments and suggestions that improved the paper. By scaling the power of the Crab GPs to the level of This work is partially supported by the National Natural Science LFRB FRB 121102 i.e. Lsd,FRB = Lsd,GP, where BCrab ∼ Foundation of China (grant Nos. 11673068, 11725314, U1831122, LGP 12 11903019, 11533003, and 11703002), the Youth Innovation Promo- 4 × 10 G and PCrab ∼ 33 ms are the magnetic field strengthResearch and the article period of the Crab pulsar, respectively tion Association (2017366), the Key Research Program of Frontier [68]. It is seen that the millisecond pulsars may produce Sciences (grant Nos. QYZDB-SSW-SYS005 and ZDBS-LY-7014), GPs with luminosity comparable to FRB 121102. On the Strategic Priority Research Program “Multi-waveband gravi- the other hand, the active time scale is the spin-down tational wave universe” (grant No. XDB23000000) of the Chinese time scale of the pulsar, that is Academy of Sciences, the China Post- doctoral Science Founda- tion (Nos. 2018M631242 and 2020M671876), the Fundamental Re- −1 −6 −2 4 −2 tactive ∼ 50 ξ5 I45R6 BCrabPCrabP−3 yr. (6) search Funds for the Central Universities, and the National Post- doctoral Program for Innovative Talents (grant No. BX20200164). which can in principle explain FRB 121102 so far. Con- sidering the condition of tactive > 7 yrs, we constraints that the period of the initial spin P should be smaller References than 3 ms. In addition, if the radiation efficiency of FRB ηFRB is higher than that of GPs ηGP, for example an or- 1. D. R. Lorimer, M. Bailes, M. A. McLaughlin, D. J. der of magnitude higher, then we would have a longer Narkevic, and F. Crawford A Bright Millisecond Radio active time scale tactive ∼ 500 (ηFRB/10ηGP). Then the Burst of Extragalactic Origin, Sci. 318,777 (2007) constraint to the period of the initial spin is P ≤ 8 ms 2. W. Farah, C. Flynn, M. Bailes, and 24 colleagues FRB in this case. For comparison, the constraint from the persistent radio nebula gives P ≤ 7 ms [37]. 3Although the fitting results of GBT data and Arecibo data for Moreover, we note that both repeating FRBs and GPs repeating FRB 121102 are not consistent with each other, we have have high brightness temperature (≥ 1035 K), inferring discussed some possible reasons for the inconsistency in section 3 such as under different observation frequency, a lack of the bursts that their radiation mechanism should be coherent emis- below the break energy for Arecibo data, or the intrinsic physical sion [69, 70]. GPs from Crab pulsar and FRBs 121102 mechanism.

Fen Lyu, Yan -Zhi Meng, Zhen-Fan Tang. et al., Front. Phys. ???, ?? (2020) ??-5 Research article

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