On the Time-Frequency Downward Drifting of Repeating Fast Radio Bursts
aa r X i v : . [ a s t r o - ph . H E ] A p r Draft version April 19, 2019
Preprint typeset using L A TEX style emulateapj v. 12/16/11
ON THE TIME-FREQUENCY DOWNWARD DRIFTING OF REPEATING FAST RADIO BURSTS
Weiyang Wang , Bing Zhang , Xuelei Chen , Renxin Xu Key Laboratory for Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences, 20A DatunRoad, Beijing 100101, China University of Chinese Academy of Sciences, Beijing 100049, China School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China Department of Astronomy, School of Physics, Peking University, Beijing 100871, China and Center for High Energy Physics, Peking University, Beijing 100871, China
Draft version April 19, 2019
ABSTRACTThe newly discovered second repeating fast radio burst (FRB) source, FRB 180814.J0422+73, wasreported to exhibit a time-frequency downward drifting pattern, which is also seen in the first repeaterFRB 121102. We propose a generic geometrical model to account for the observed downward driftingof sub-pulse frequency, within the framework of coherent curvature radiation by bunches of electron-positron pairs in the magnetosphere of a neutron star. A sudden trigger event excites these coherentbunches of charged particles, which stream outwards along open field lines. As the field lines sweepacross the line of sight, the bunches seen later have traveled farther into the less curved part ofthe magnetic field lines, thus emitting at lower frequencies. We use this model to explain the time-frequency downward drifting in two FRB generation scenarios, the transient pulsar-like sparking fromthe inner gap region of a slowly rotating neutron star, and the externally-triggered magnetospherereconfiguration known as the “cosmic comb”.
Subject headings: pulsars: general - radiation mechanisms: non-thermal - radio continuum: general -stars: neutron INTRODUCTION
Fast radio bursts (FRBs) are mysterious millisecond-duration astronomical radio transients with large dis-persion measures in excess of the Galactic value (DM &
200 pc cm − , Lorimer et al. 2007; Keane et al. 2012;Thornton et al. 2013; Kulkarni et al. 2014; Petroff et al.2015, 2016; Chatterjee et al. 2017). The cosmo-logical origin of FRBs was established after FRB121102, the first repeating source (Spitler et al. 2016),was localized in a star-forming dwarf galaxy at z = 0 .
193 with an associated persistent radio source(Bassa et al. 2017; Chatterjee et al. 2017; Marcote et al.2017; Tendulkar et al. 2017) and an extreme magneto-ionic environment (Michilli et al. 2018).Recently, the CHIME/FRB Collaboration et al.(2019) reported the discovery of the second repeatingFRB source, FRB 180814.J0422+73. Very intriguingly,both FRB 121102 and FRB 180814.J0422+73 showed aninteresting sub-pulse time-frequency downward driftingpattern in at least some of their bursts. For these bursts,each burst have several sub-pulses, with the later-arrivalsub-pulses having lower frequencies (Hessels et al.2018; CHIME/FRB Collaboration et al. 2019). Thistime-frequency structure is reminiscent to the Type IIIsolar bursts and the decametric radiation from Jupiter(Bastian et al. 1998; Treumann 2006). However, it isunclear whether the same mechanism are at work, as theFRBs are at cosmological distances and have extremelyhigh brightness temperatures. Plasma lensing may causea sub-pulse drift, but both upward and downward drifts [email protected]@physics.unlv.edu are expected (Cordes et al. 2017). In contrast, only thedownward drifting is seen in the repeating FRBs. Amechanism intrinsic to the FRB source is most likelythe origin of the drift. One such mechanism has beenproposed in the framework of magnetar-wind-drivenexternal shock synchrotron maser (Metzger et al. 2019).However, in this model it is not clear why such downdrifting does not occur in consecutive individual bursts.Here we propose an alternative model by invoking co-herent curvature radiation in a neutron star (NS) mag-netosphere. Sub-pulse drifting is a well-known phe-nomenon in radio pulsars (Rankin 1990), which canbe interpreted as E × B drift in the inner gap wherethe particles are accelerated from the polar cap region(Ruderman & Sutherland 1975). Curvature radiationfrom charge bunches from pulsar magnetospheres hasbeen invoked to interpret FRB coherent radio emissionby several authors (e.g. Katz 2014; Kumar et al. 2017;Lu & Kumar 2018; Yang & Zhang 2018). In this letter,we propose a generic geometrical mechanism to accountfor the observed time-frequency downward drifting fromthe two repeating FRBs. The model is described in § § GEOMETRIC MODEL
We consider a generic model of coherent curvature ra-diation by bunches of charged particles in a NS mag-netosphere. The specific geometry does not matter, aslong as the bunches are generated abruptly and streamoutwards along open magnetic field lines. The field linessweep across the line of sight as the magnetosphere ro-tates. The observer sees emission from several bunches Wang et al.
ConeΔ r LF wavesHF wavesMagnetic axis 12
HF Bunch
LOS Δ ϕ Spin
LF Bunch Δ r Magnetic axisSpin
Fig. 1.—
A schematic diagram of the first scenario, withsparks originating from the polar gap region. The HF wavesare emitted from the lower altitudes than the LF waves. Theleft panel shows the initial configuration when the two sparksare produced around the same location. The dashed linesshow the LOS. The second spark sweeps the LOS at a higheraltitude. The right panel shows the sky map of two sparks.These two sparks sweep the LOS at different heights at dif-ferent times. from neighboring magnetic field lines. Assuming thatthe Lorentz factors of the bunches are the same fromeach other and do not evolve significantly as they streamalong the field lines, the bunches observed earlier emitcurvature radiation in more curved part of the field lines,and therefore have higher frequencies. In contrast, thebunches observed later emit in less curved part of thefield lines with lower frequencies.Figure 1 shows a schematic plot of one version of suchfield lines, where the “sparks” are produced from the in-ner magnetosphere of the open field lines of a NS. Thesparks are produced at a low height due to a sudden re-lease of energy, e.g. by magnetic reconnection or crustcracking. Several bunches are released around the sametime and continuously radiate along neighboring fieldlines. In the plot, two locations are marked for the sub-pulses of high frequency (HF) and low frequency (LF).The two locations for the two sub-pulse emission are dif-ferent in radius (∆ r ) and in azimuthal angle (∆ φ ).The emission frequency of curvature radiation reads ν = (3 / π ) γ ( c/ρ ), where ρ is the curvature radius, and c is speed of light. Assuming a constant Lorentz factor γ ofthe charges, the change in the typical curvature radiationfrequency is given by∆ ν = − cγ ∆ ρ πρ = − ν ∆ ρρ , (1)where ∆ ρ is the change in the curvature radiusbetween the two emitting points. Observationally,∆ ν/ν is of the order of 0.1 (Hessels et al. 2018;CHIME/FRB Collaboration et al. 2019), so we can in-fer ∆ ρ/ρ ∼ . γ .If the bunch scale is smaller than the half-wavelength( ∼
10 cm, for 1 GHz), the phase of emission radiated byeach particle in the bunch would be approximately the same, so coherent radio emission is produced (Melrose2017; Kumar et al. 2017; Yang & Zhang 2018). The GHzcurvature radiation time scale for such a bunch is 1 ns,which is much shorter than that of the observed pulseduration ∼ t = ∆ t φ + ∆ t r , (2)where ∆ t φ is the interval between the two emissionbeams sweeping across the LOS, and ∆ t r is the delayof emission in the radial direction for the two sparks,i.e. the retardation delay (see Fig.1). One can generallywrite ∆ t φ = ∆ r ⊥ v , (3)where ∆ r ⊥ is the projected horizontal distance betweenthe HF emitting region and the LF emitting region, and v is the projected speed of the sweeping beam. As the twosparks are generated simultaneously but the observedemissions from the two sparks are emitted at differentepochs, the delay of receiving the two signals due to theretardation delay can be estimated as∆ t r = ∆ rv e − ∆ rc ≈ ∆ r γ e c , (4)where v e ∼ c is the velocity of the electrons (or pairs) inthe bunches, and γ e is its corresponding Lorentz factor. APPLICATIONS
In this section, we apply this generic geometrical modelto two specific scenarios of FRB production. The firstscenario is a transient pulsar sparking model with theFRB originating from the pulsar inner gap region. Themagnetic field configuration in this scenario may be ap-proximated as a simple dipole. The second scenariois the cosmic comb model (Zhang 2017, 2018). Thesparks are suddenly generated upon the interaction be-tween the external plasma stream and the pulsar mag-netosphere, which flow along the open field lines in thesheath. The field line configuration is not dipolar, but ismore stretched. In both cases, the sparks propagate fromhigh-curvature regions to low-curvature regions, leadingto frequency downward drifting. We now discuss thesetwo scenarios in turn.
Polar gap sparking
For the first scenario, we consider an FRB generatedfrom the polar gap region of a pulsar. This could berelated to a young regular field pulsar (e.g. Connor et al.2016; Cordes & Wasserman 2016) or a young magnetarwith the emission coming from the inner magnetosphere(Kumar et al. 2017).We consider a scenario similar to the polar gapsparking of the regular pulsars (Ruderman & Sutherland1975). However, instead of invoking regular, continuoussparks, we envisage a sudden, violent sparking processfrom the surface, possibly triggered by an abrupt crustime-Frequency Downward Drifting of Repeating FRB 3cracking that leads to an abrupt magnetic field dissipa-tion. A significant deviation from the regular magneticfield configuration is triggered, which leads to coherentcurvature radiation by bunches of charged particles in alotus of field lines (Yang & Zhang 2018). The perturba-tion propagates along the field lines outwards, leadingto multiple sparks emitting in adjacent field line bundlestraveling with a similar Lorentz factor.Consider that the polar gap of the pulsar is enclosedwithin the last open field lines with a polar angle θ p =0 . P/
10 ms) − / , where P is the period of the pulsar.For a dipole magnetic field, a magnetic field line can bedescribed as u = R sin θr , (5)where R is the radius of the NS surface, and u is a di-mensionless constant. The curvature radius of the fieldline is (for θ . . ρ = r (1 + 3 cos θ ) / θ (1 + cos θ ) ≈ r θ . (6)For γ e = 300, the curvature radius is estimated to be ρ ≃ . × cm to produce ∼ GHz curvature radiation.For a dipolar geometry, the time for the line to sweepa phase ∆ φ is given by∆ t φ = P sin β ∆ φ π sin( α + β ) , (7)where P is the period of the pulsar, α is the magneticinclination angle and β is the impact angle of LOS withrespect to the magnetic axis. In this scenario, ∆ t φ onlydepends on the geometry of the pulsar. As an exam-ple, we assume ∆ r = 0 . ρ . From equation (4), onecan estimate the retardation time delay to be ∆ t r ≃
10 ns, which is much smaller than the observed intervaltimes between sub-pulses ∼ . −
10 ms (Hessels et al.2018; CHIME/FRB Collaboration et al. 2019). Hence,the time delay of LF waves with respect to the HF wavesis mainly given by the sweeping delay ∆ t φ .Combining equations (3), (5) and (6), one gets˙ ν = A g ν = 2 π sin( α + β )∆ uuP sin β ∆ φ ν, (8)According to equation (8), when the geometrical condi-tion of ∆ φ/ ∆ u ≃ − πu − ( P/
10 ms) − [sin( α + β ) / sin β ]is satisfied, the drifting rate is very similar towhat is observed in FRB 121102 (Hessels et al.2018). If ∆ t φ ≪ /A g , the central frequency de-creases linearly with time. This scenario matchesthe observations of FRB 180814.J0422+73 well(CHIME/FRB Collaboration et al. 2019).At the same height, electrons are in the different trajec-tories with essentially the same curvature radius. Sincedifferent field lines have slightly different curvatures, thecondition of coherence is that the bunch opening an-gle ∆ φ b should be smaller than 1 /γ e (Yang & Zhang2018). Defining ν φ = 12 c/ ( πρ ∆ φ ), the condition ν < ν φ can be translated to ∆ φ b < /γ e . Observationally, thesub-pulse interval time is of the order of milliseconds(Hessels et al. 2018; CHIME/FRB Collaboration et al.2019), ∆ t ∼ (1ms)∆ t ms . The condition ∆ φ < /γ e can be satisfied if the pulsar period satisfy P > γ e ∆ t =0 . γ e / t ms . Cosmic comb
In the cosmic comb model (Zhang 2017, 2018) a plasmastream from a nearby source interacts with a pulsar.Similar to solar wind interacting with the earth mag-netosphere, the external stream would re-structure themagnetosphere of the pulsar, forming an elongated mag-netosphere surrounded by a sheath. The FRB is seenwhen the sheath plasma sweeps the LOS. For GHz ra-dio waves, one requires γ e ∼ for the curvature ra-dius ρ ∼ cm that matches the light cylinder radius R LC = 4 . × cm ( P/ t r ∼ . × − s ∆ r γ − e, ≪ ∆ t (The convention Q n = Q/ n in cgs units is adopted). Therefore, the observed delaytime is mostly defined by the sweeping delay, which reads∆ t φ ≃ ∆ R s v s γ e ≈ (3 ms)∆ R s, v − s, − γ − e, , (9)where ∆ R s is the size of the sheath, and r ⊥ = R s /γ is theprojected distance in the sky when the emission beam isobserved, and v s ≃ . c v s, − is the velocity of the streamthat combs the magnetosphere. This is consistent withthe observed millisecond interval time of the sub-pulses.Equation (9) has properties similar to equation (7).Combining equations (1), (3) and (9), one can obtain˙ ν = A c ν = − v s γ ∆ ρρ ∆ R s ν (10)for the cosmic comb model. One can estimate that∆ ρ/ ∆ R s ≃ . ρ γ − e, . The frequency drifting rateswould decrease linearly with ν , which is consist with theobservations of FRB 121102. The drifting rate wouldbe a constant when ∆ t ≪ /A c for each multi-sub-pulsesequence. In such a situation, the result matches theobservations of FRB 180814.J0422+73. Drifting rates
Equations (8) and (10) show that both models sharethe similar feature of frequency down-drifting. In Fig-ure 3, we show the simulated sub-pulse central frequencydrift as a function of the arrival time for the parameter A g = A c = − .
01 ms − . We fix ∆ t = 1 ms but allow the Wang et al. FRBPlasma stream (a)(b)(c) O s Bunches φ Fig. 2.—
A schematic diagram of the second scenario in thecosmic comb model. The sparks are produced in the distortedsheath region which stream outwards along the field lines.For the illustrative purpose, the separations between the fieldlines are stretched. Sparks from different field lines sweepthe LOS at different times when the sparks reach differentheights. The spark observed at a later epoch emits at a lesscurved part of field line and thus has a lower frequency. Aburst with three sub-pulses are shown for illustration, withthree epochs: (a) the inner spark emission beams towards theLOS; (b) an intermediate spark emission beams towards theLOS; (c) the outer spark beams towards the LOS. fr e qu e n c y ( M H z ) -2 -1 time (ms)
10 2
Fig. 3.—
Simulated sub-burst central frequency as a functionof the arrival time. We assume A g = A c = − .
01 ms − . Thesub-burst sequences have different central frequencies withthe same interval time ∆ t = 1 ms: 6.5 GHz (red diamonds),2.2 GHz (green squares), 1.4 GHz (blue triangles), and 400MHz (black dots). central frequency to vary. From up to down, differentcurves (with different colors) stand for different centralfrequencies: 6.5 GHz (red diamonds), 2.2 GHz (greensquares), 1.4 GHz (blue triangles), and 400 MHz (blackdots). These results are generally consistent with the ob-servations of the two FRB repeaters (Hessels et al. 2018;CHIME/FRB Collaboration et al. 2019). Particle cooling and acceleration
In the above discussion, we have assumed a constant γ for both models. For typical FRB parameters, bothmodels involve rapid cooling of the emitting particles(the cooling rate increases by a factor of N e for coher-ent emission by bunches, where N e is the number of netelectrons in the bunch) and therefore require continuousacceleration of the bunched particles. Very generally, thecooling timescale of curvature radiation in the observer’srest frame can be written as (Kumar et al. 2017) t cool ∼ m e c γ π e ν N e ∼ . × − γ , ν − ( N e , ) − s . (11)Therefore, to sustain a constant Lorentz factor within alab-frame time duration of γ /ν , one requires that thereexists an electric field parallel E k to the B-field that canaccelerate electrons, which is given by E k ≃ γ e m e c ( et cool ) ∼ . × ν N e , γ − , esu . (12)For the scenario of polar gap sparking, the electronnumber may be described by (e.g. Kumar et al. 2017) N e ≃ µBc γ ν eP = 1 . × µB γ , P − − ν − , (13)where µ is the normalized fluctuation of electrons de-viated from the Goldreich–Julian density. The re-quired electric field is calculated as E k ∼ . × µB P − − γ e , ν − esu. One possible mechanism to cre-ate such an electric field is the sudden magnetic recon-nection in the magnetosphere.ime-Frequency Downward Drifting of Repeating FRB 5Within the cosmic comb model, the electron numberis given by (Yang & Zhang 2018) N e ≃ µηBR L πeR = 3 . × µηB R L ( R LC , ) − , (14)where ηR is the cross section of the bunch in nearlyparallel field lines in the combed magnetosphere, and L ∼ λ is the thickness of the bunch, which is com-parable to the wavelength λ of the emission. Therequired electric field for tthis model is then E k ∼ µηB ν R L ( R LC , ) − γ − , esu. The strong rampressure of the stream likely would trigger magnetic re-connection and provide the required electric field to ac-celerate electrons. SUMMARY AND DISCUSSION