DRAFT VERSION JULY 26, 2021Typeset using LATEX twocolumn style in AASTeX631

Kinematics of Crab Giant Pulses

AKANKSHA BIJ ,1, 2 HSIU-HSIEN LIN ,1 DONGZI LI ,1, 3 MARTEN H. VAN KERKWIJK ,4 UE-LI PEN ,1, 5, 6, 7, 8

WENBIN LU ,3 ROBERT MAIN ,4, 8 JEFFREY B. PETERSON ,9 BRENDAN QUINE ,10, 11 AND KEITH VANDERLINDE 4, 12

1Canadian Institute of Theoretical Astrophysics, 60 Saint George Street, Toronto, ON M5S 3H8, Canada2Department for Physics, Engineering Physics and Astrophysics, Queen’s University, Kingston, ON, K7L 3N6, Canada

3Cahill Center for Astronomy and Astrophysics, California Institute of Technology, 1216 E California Boulevard, Pasadena, CA 91125, USA4David A. Dunlap Department of Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, ON M5S 3H4, Canada

5Department of Physics, University of Toronto, 60 Saint George Street, Toronto, ON M5S 1A7, Canada6Canadian Institute for Advanced Research, CIFAR Program in Cosmology and Gravity, Toronto, ON, M5G 1Z8, Canada

7Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada8Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121, Bonn, Germany

9Department of Physics, Carnegie Mellon University, 5000 Forbes Ave. Pittsburgh PA 15213 USA10Thoth Technology Inc., 33387 Highway 17, Deep River, Ontario K0J 1P0, Canada

11Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada12Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, ON M5S 3H4, Canada

ABSTRACTThe Crab Pulsar’s radio emission is unusual, consisting predominantly of giant pulses, with durations of

about a micro-second but structure down to the nano-second level, and brightness temperatures of up to 1037 K.It is unclear how giant pulses are produced, but they likely originate near the pulsar’s light cylinder, wherecorotating plasma approaches the speed of light. We report observations in the 400–800 MHz frequency band,where the pulses are broadened by scattering in the surrounding Crab nebula. We find that some pulse frequencyspectra show strong bands, which vary during the scattering tail, in one case showing a smooth upward drift.While the banding may simply reflect interference between nano-second scale pulse components, the variationis surprising, as in the scattering tail the only difference is that the source is observed via slightly longer paths,bent by about an arcsecond in the nebula. The corresponding small change in viewing angle could neverthelessreproduce the observed drift by a change in Doppler shift, if the plasma that emitted the giant pulses movedhighly relativistically, with a Lorentz factor γ ∼ 104 (and without much spread in γ). If so, this would supportmodels that appeal to highly relativistic plasma to transform ambient magnetic structures to coherent GHz radioemission, be it for giant pulses or for potentially related sources, such as fast radio bursts.

1. CRAB GIANT PULSES

The Crab pulsar (PSR B0531+21) is a young pulsar,formed in supernova SN 1054, and powering a pulsar windnebula in the Crab nebula supernova remnant. It was the firstout of eleven radio pulsars found to emit giant pulses (Staelin& Reifenstein 1968). Giant Pulses (GPs) are much shorterand brighter than regular pulses and occur in narrow phaseranges, typically limited to the edges of the main profile (fora description, e.g., Lyutikov 2007 and references therein). Inthe case of the Crab, their intrinsic duration ranges from theorder of nanoseconds to microseconds (Hankins et al. 2003),while their flux densities can reach brightness temperaturesof 1037 K (Hankins et al. 2003; Hankins & Eilek 2007).

The energetic GPs from Crab share numerous similaritieswith another type of energetic bursts, the Fast Radio Bursts(FRBs). Both emit coherently with high polarization frac-tions; both have short durations, with sub-structure down toeven shorter timescales (Hankins et al. 2003; Nimmo et al.2021; Majid et al. 2021); and while FRBs are brighter, theluminosities of the most energetic GPs approach those of

the faintest FRBs (Hankins et al. 2003; Nimmo et al. 2021;Majid et al. 2021), and the gap is likely to be reduced asmore nearby FRBs are found. Given these similarities, ithas been suggested that FRBs are GPs from (very) young,(very) rapidly rotating pulsars (Cordes & Wasserman 2016;Lyu et al. 2021). Thus, constraints on the emission mecha-nism of GPs may also help elucidate the emission mechanismof FRBs.

Giant pulses likely originate near the pulsar’s light cylin-der, where corotating plasma approaches the speed of light(Eilek & Hankins 2016), but it is not yet clear what physicalmechanism is responsible. Indeed, as a result, theoretical es-timates of the Lorentz factor γ range from ∼1 to ∼107 (e.g.,Eilek & Hankins 2016; Petrova 2004; Istomin 2004; Lyu-tikov 2007; Lyubarsky 2019; Philippov et al. 2019; Mach-abeli et al. 2019; Lyutikov 2021).

For the Crab, unusually, the emission from giant pulsesleads to the two largest components in the mean pulse profile:the main pulse and interpulse. From detailed studies, Hank-ins et al. (2003); Hankins & Eilek (2007) found that there aretwo types of emission patterns. The first type, in the main

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pulse and the interpulses at frequencies of up to 4 GHz, hasmicro-second duration bursts composed of nano-second du-ration shots. The second, observed exclusively at higher fre-quency, between 5 and 30 GHz, for a phase window near theinterpulse (and hence called the high-frequency interpulse),has similar overall durations of a few microseconds, but hasa spectrum showing proportionaly spaced emission bands.

In this work, we characterize Crab giant pulses at lowerfrequencies (400–800 MHz), where frequency-dependentscattering from the Crab Nebula becomes more pronounced,causing substantial smearing due to time delays from theemission travelling varying path lengths (Rankin & Coun-selman 1973). Although we are unable to resolve the giantpulse ‘nanoshot’ substructure at lower frequencies, the mul-tiple lines of sight offer a unique opportunity to resolve theemission region (Main et al. 2021), and, as we find here, toconstrain properties of the emitting sources.

We describe our data in Section 2, and the variety of giantpulse features we find in Section 3, comparing with what wasseen at higher frequency. In Section 4, we highlight spectralbands seen in some pulses, and how those drift in frequencywithin the scattering tail. We interpret them in terms of avarying Doppler shift in Section 5, and explore alternativeinterpretations in Section 6, before discussing implicationsin Section 7.

2. OBSERVATIONS AND DATA REDUCTION

We observed the Crab pulsar with the 46m antenna at theAlgonquin Radio Observatory (ARO) on 2015 July 23 and24 (12.5 hours spread over 6 scans), and 2018 April 25 (1.5hours over 2 scans). Data were taken at 400-800 MHz withtwo linear polarizations. The raw voltage streams (baseband)are processed using a CHIME acquisition board (Banduraet al. 2016), which digitizes the signals, passes them througha polyphase filter, channelizes them to 1024 frequency chan-nels, and then records these in standard VDIF format (Whit-ney et al. 2009). Different receivers were used for the obser-vations in 2015 and 2018, which had significantly differentfrequency responses and gains. We gain and flux-calibratedthese as described in Appendix A.

The observation performed on 2015 July 24 partially over-laps with an observations of Crab pulsar at the John A. GaltTelescope at the Dominion Radio Astrophysical Observatory(DRAO). These observations were originally taken for verylong baseline interferometry with ARO, but here we will usethem only to rule out instrumental effects at ARO. They cov-ered the same frequency band and were taken using a similarreceiver and backend.

To search for giant pulses, we first coherently dedispersedthe raw data to a reference frequency of 800 MHz, using adispersion measure (DM) of 56.7708 pc/cm3 for 2015 and56.7562 pc/cm3 for 2018, obtained by using the closest DMvalue from the Jodrell Bank ephemeris (Lyne et al. 1993) asan initial guess and then optimized by adjusting the 4th sig-nificant figure such that the rising edge of the pulse is aligned,by-eye, in frequency. Next, we sum the power from all fre-quency channels and both polarizations, apply a rolling box-

car window of size 1.3 ms, and flag any peaks more than5σ above the noise. We calculate the rotational phase withthe TEMPO2 package (Hobbs et al. 2006), using the Jo-drell Bank ephemeris (Lyne et al. 1993). However, whileour clocks were tied to a maser, they had small random off-sets at the start of each integration. Hence, while we can stilldistinguish main pulses from interpulses, otherwise only therelative phases are useful within each continuous integration.

For each pulse, we then created flux-calibrated dynamicspectra.

3. A VARIETY OF DYNAMIC SPECTRA

We show the resulting dynamic spectra for a selection ofbright giant pulses in Figure 1. This comparison readily re-veals that there are large variations. Some of the types we seecan be related to what was seen previously, but others cannot.

In particular, from the figure, as well as from a catego-rization of the 148 brightest pulses by eye (see App. B), wefind that most pulses are ‘regular’ (e.g., panels a–e and l ofFigure 1), showing only a smooth variation with frequency,and a scattering tail that becomes increasingly long towardslower frequency (with the scattering time longer in 2018 thanin 2015). In these, the scattering has likely blended togetherthe short time-scale variations seen at higher frequencies (1–43 GHz; Hankins et al. 2003; Hankins & Eilek 2007; Eilek &Hankins 2016), where one often sees multiple micro-burstsover a few to a few tens of micro-second, with sub-structuredown to the nano-second level.

Some other triggers (e.g., g and k) are ‘multi-peak’ pulses,with more than one obvious peak. These likely reflect multi-ple indepedent giant pulses occurring in a single rotation, asalso seen at higher frequencies (Eilek & Hankins 2016), buthere partially blended together due to the scattering.

More unusal are the pulses that differ in their frequencyspectrum. Specifically, some show ‘partial’ emission (e.g.,i), where the pulse is bright in only a sub-band of frequen-cies rather than throughout the band, while three others show‘banded’ spectra (f , h, and j), with the bands spaced by∼20to ∼ 50 MHz. Possibly similar spectral structure at frequen-cies comparable to ours has been reported for giant pulsesfrom the millisecond pulsars M28A (Bilous et al. 2015) andPSR B1937+21 (McKee et al. 2019), but what is differentfor the three banded pulses we observe is that the bands ap-pear to vary over the scattering tail, shifting in frequency orchanging in relative strength.

Among the three banded pulses, j is the clearest example,with the bands consistently moving up in frequency as thepulse proceeds. Pulse f also appear to have some driftingbands, but the direction is less clear: above 500 MHz, thereseems to be several bands that drift downward, while below500 MHz, some bands may be drifting up.

4. ORIGIN OF THE SPECTRAL BANDS

At first glance, the drifting bands are reminiscent of a sur-prising earlier observation of the Crab, by Hankins & Eilek(2007), who found that one particular pulse component, thehigh-frequency interpulse, showed bands with a typical spac-

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Figure 1. Dynamic spectra of 12 bright giant pulses. Each image shows flux-calibrated intensity over a 0.84 ms time span and the full 400 MHzfrequency range, with the dedispersed data binned by 4 in both time and frequency (giving 10.24µs time resolution and 1.56MHz frequencyresolution). The red marks to the left of the images indicate noisy frequency channels (determined from the channel standard deviation beforegain correction). The top panels show frequency-averaged pulse profiles (in units of kJy), while the side panels show the pulse fluence spectraintegrated over the full 0.84 ms (in units of kJyms). Panels a–f and j–l are the brightest pulses in 2015 and 2018, respectively (ordered bydescending brightness), while panels g–i are three further 2015 pulses chosen for their particular profiles. Note that the faint curved featurevisible in panels a–f , starting at ∼ 0.4ms at 800 MHz and leaving the panels at ∼ 650MHz, is a reduction artefact (it represents incorrectlydedispersed leakage from neighbouring channels).

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ing proportional to frequency, ∆f ' 0.06f , which oftendrift upward over the few-µs duration of the pulse (see alsoEilek & Hankins 2016). It would be tempting to associate ourbanded pulses with these, especially as our observed spacingbetween bands, of ∆f ' 40 MHz at f ' 500 MHz, roughlyfollows the same scaling relation. Nevertheless, the associ-ation seems unlikely, because the banded spectra we see areall from giant pulses associated with the Crab’s main pulsecomponent, for which banded spectra have never been seenat high frequencies. In contrast the high-frequency interpulseis, as its name suggests, not visible at our relatively low fre-quencies.

A physical clue comes from the fact that the drifts andother variations in the banded spectra occur on the sametimescale as the scattering, of a few ms, which is much longerthan the duration of giant pulses. Thus, the variation must berelated to the fact that as time proceeds, one sees radiationfrom the pulse that has followed increasingly longer pathstowards us, having been bent slightly by matter in the Crabnebula.

However, because the scattering timescale is much longerthan the inverse of the band spacing, ∼ 25 ns, it is unlikelythat the scattering itself causes the bands. Given this, as wellas the fact that the banding patterns are not shared betweenneighbouring pulses while at least for pulse h they are seen intwo telescopes simultaneously, we conclude that the bandingis not due to instrumental or scattering effects, but intrinsic tothe pulsar emission (for details, see App. C and D). Indeed,banding itself seems not unexpected given the behaviour seenat high frequency, where giant pulses in the main pulse com-ponent often show several bright nano-shots spaced closelytogether. If sometimes the broadband nano-shots are spacedby a few to a few tens of ns, this would at lower frequenciesnaturally lead to ‘partial’ and ‘banded’ spectra with structureon the scale of 10–100 MHz.

5. A POSSIBLE PHYSICAL MODEL

If the banding in the spectra is indeed intrinsic to the emis-sion, what could cause the changes during the scattering tail?Inspired by the fact that in pulse j the whole pattern movesin concert, we suggest that as the viewpoint shifts slightlyduring the scattering tail, one sees a difference in Dopplershift. Specifically, we envision a situation like that depictedin Figure 2, where a group of charged particles moves rela-tivistically, with a high Lorentz factor γ. Any emission fromthese particles will be beamed forward, within a beam ofsize ∼ 1/γ. For that beam to be observable, it must includethe line of sight, i.e., the particles should move at an angleθr . 1/γ relative to the line of sight. The scattering screenwill then bend light towards us which was originally emit-ted in a slightly different angle, either closer or further awayfrom the direction the particles are moving in and thus witha different projected velocity.

Quantitatively, for the line of sight we expect emission tobe Doppler shifted by a factor D given by,

D =1

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Figure 2. The model geometry: ‘S’ is a blob of radiating plasmamoving at an angle θr with respect to the observer ‘R’, with a ve-locity ~v sufficiently relativistic that its emission is strongly beamedforward in the observer’s frame (as indicated by the shading). Atthe start of the burst, the observer sees emission that travelled di-rectly to the observer, but in the scattering tail the emission took aless direct path, initially leaving the source at an angle δ relative tothe direct line of sight, before being bent towards the observer in theCrab Nebula (at about 1 pc from the pulsar). If, as drawn, the anglesare in the same direction, then as one proceeds along the scatteringtail to larger δ, one will see a steadily increasing blueshift.

where β = v/c and γ = 1/√

1− β2 as per convention. Theapproximate equality is for highly relativistic motion, withγ � 1, for which β ' 1− 1/2γ2 and cos θr ' 1− 1

2θ2r (the

latter because θr has to be small, . 1/γ).The above implies that for a fractional change in the view-

ing angle from θr to θr + δ, as resolved by our scatteringscreen, the resulting change in observed frequency fr willbe,

∆frfr

=∆D

D=−γ2(2θrδ + δ2)

1 + γ2(θr + δ)2. (2)

For an observer in the center of the beam, with θr � δ <1/γ, one expects ∆fr/fr ≈ −γ2δ2, indicating a downwarddrift. For the statistically more likely case depicted in Fig-ure 2, with an observer at the edge of the beam, i.e., withθr ' 1/γ , one finds ∆fr/fr ≈ γδ (where the sign wouldbe opposite if scattering was in the other direction, and caseswhere δ and θr are not aligned on the sky would be interme-diate).

The scattering angle δ can be estimated using the knowngeometry for the Crab pulsar. The distance of the screen isabout d ' 1 pc from the pulsar (Lawrence et al. 1995; Martinet al. 2021). The scattering time measured from the dynamicspectrum of the 2018 drifting pulse is approximately τ '0.5 ms at 450 MHz. Since from the geometry,

cτ =d

cos δ− d ' d

2δ2, (3)

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we infer δ2 ' 1 × 10−11 and thus δ ' 0.6 arcsec. Giventhat we observe about a 4% change in observed frequency,this implies γ ' 0.04/δ ' 104. Furthermore, given thatwe see the structure throughout the tail, the range in γ can-not be very large, ∆γ/γ . 50%, i.e., the plasma must berelatively cold. This is consistent with the presence of rel-atively narrow bands: we measure full width at half max-imum w ' 20 MHz, which would suggest a stricter con-straint, ∆γ/γ . w/f ' 4%.

If the above holds, then as a function of time t relative tothe start of the pulse, the Doppler shift should roughly varyas ∆D(t)/D ∝ δ(t) ∝

√t. This scaling with the square root

of time seems consistent from what one approximately seesfor pulse j in Fig. 1.

To explore this further, we compared the evolution of theobserved spectrum with a simple model that starts with theinitial spectrum but shifts and dims it over time, assumingthe√t scaling above for the shifts and the standard exponen-

tial decay (Lang 1971) for the dimming, in which the decaytimescale scales as τ ∝ 1/f4. Specifically, we assume thatthe intensity I(t, f) as a function of time since the start of theburst t and frequency f follows,

I(t, f) = I(0, f/Drel(t)) D3rel(t) e−t/τ(f)

Drel(t) =D(t)/D(0) = 1 + a√t/τref

τ(f) = τref(f/fref)−4, (4)

where the term D3rel accounts for the fact that the Doppler

effect not only shifts the frequencies, but also boosts the in-tensity due to beaming and time contraction. For the initialspectrum I(0, f), we use a smoothed version of the averageover the first 24 time samples (61µs) of the pulse.

In the expression for the decay time τ(f), either the ref-erence frequency fref or the associated scattering time τref

should be fixed. Since we normalize the time in the Dopplershift term to τref to make the scale factor a easier to inter-pret, we found it most convenient to fix the reference scat-tering time, to τref = 250µs. Then we roughly adjustedthe two other parameters, finding that for a ' 0.028 andfref = 465 MHz the model reproduces the observations quitewell (see Fig. 3), especially taking into account its simplicity.

In our scenario, different giant pulses would be emittedin different directions relative to the line of sight, and thusbands could drift up, down, or not at all. Furthermore, forpulse j, we implicitly assumed that the scattering is asym-metric around the line of sight, dominated by the side inwhich pulse j was emitted. From observations of other pul-sars, we know scattering geometries vary as the pulsar movesand probes different parts of the scattering screen (Hill et al.2005). For the Crab, similarly, variations in scattering ge-ometry on timescales of weeks are suggested by echoes andchanges in scattering time (McKee et al. 2018; Driessen et al.2019). At times when the screens is more symmetric, onewould expect to observe both upward and downward-driftingfeatures simultaneously. Since the structure of the scatteringscreen could be quite complex and the scattering strength isfrequency dependent, an uneven distribution of material on

different sides of the screen could result in the overall driftdirection being frequency dependent. Indeed, this may helpunderstand the observed dynamic spectra of pulse f (Fig. 1),which appears to have downward drifts at short delay, espe-cially in the top half of the band and upwards shifts at longerdelays in the bottom of the band.

An interesting aspect of our proposed Doppler-shift inter-pretation is the expectation that beyond the shift in frequencyby a factor Drel there will also be a boost in intensity by afactor D3

rel. Since pulse j is drifting up, this boost factor willincrease during the tail, and hence the scattering tail shouldbe slightly brighter, by about 10%, than would be the case fora pulse for which the Doppler factor did not vary. To see ifthis is indeed the case, we also compare in Fig. 3 the bright-ness in the scattering tail of the drifting pulse j with that ofthe brightest other pulse within 5 s as well as that of the aver-age of all pulses within 30 s (see also App. C and Fig. 7). Onesees that the scattering tail of pulse j does appear slightlyboosted. Furthermore, if one were to remove the Dopplerboosting from the model, it would fall closer to the profiles ofthe other pulses. However, the significance of the differenceis limited, with the differences between pulse j and the com-parisons not much larger than those between pulse j and themodel. A stronger test may be possible at lower frequency,where the range in scattering angle, and hence in Dopplershift, would be larger.

6. ALTERNATIVE INTERPRETATIONS

The drifts we observe in the banded structure of somepulses almost certainly reflect the change in viewing geom-etry that occurs during the scattering tail. So far, we haveassumed that the source of the burst intrinsically emitted abanded spectrum, and that the drift corresponded to a changein Doppler shift, from which we derived that the source hadto move highly relativistically.

Here, we discuss what happens if one relaxes the assump-tion that the source produces bands intrinsically, instead re-quiring that the banding reflects interference between multi-ple nanoshots, which do not necessarily originate from thesame region but are causally connected. In this scenario,the banding in frequency comes from interference betweenmultiple nanoshots, and the width of the bands thus reflectsthe inverse of the arrival time separation between differentnanoshots.

In this picture, during the scattering tail, the nanoshots areviewed at an angle δ away from the direct line of sight, result-ing in a change of path difference ∆x⊥δ between nanoshots,where ∆x⊥ is the distance between the nanoshots perpen-dicular to the line of sight (i.e., projected on the sky). As theinterference pattern of multiple nanoshots moves by abouthalf of the pattern spacing in the scattering tail, ∆x⊥δ '0.5λ. Given our observation wavelength λ ' 0.5 m andδ ' 0.6 arcsec, we infer a separation ∆x⊥ ∼ 100 km.

Given the typical observed frequency spacing between thebands of 40 MHz, in the observer’s frame the nanoshots needto be separated in time by about 25 ns. Since at the speedof light this corresponds to only 7.5 m, for the nanoshots to

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Drifting Model Drifting Pulse j Neighbourhood Brightest Neighbourhood Average

Figure 3. Comparison of pulse j with the drifting bands with a simple model in which the initial profile is shifted in frequency and attenuatedas time progresses. Top row: a The observed dynamic spectrum from Figure 1, b the approximate model, and c the residuals from the model.d: Spectra of pulse j as a function of time, with the model overdrawn. e: Light curves throughout the scattering tail in various sub bands, withflux normalized to peak flux and shown in a logarithmic scale to ensure that exponential decay with time looks linear. The curves correspondto the drifting pulse j, the model, the next-brightest pulse within 5 s and a stack of all pulses within 30 s (see Fig. 7). For the model, we startedwith the observed spectrum, a smoothed version of the average over the first 24 time samples (61µs) of the pulse, and then shifted and dimmedit as described by Eq. 4. Given that scattering screen is unlikely to be as simple as assumed, the model reproduces the observations remarkablywell. One also sees that pulse j appears to dim slightly more slowly than the comparison pulses, as is expected in our model due to increasingDoppler boosting.

be causally connected, the signal must appear to move su-perluminally on the sky, by a factor of roughly 104. Thisis possible if the shots represent bursts arising from a singlesource that emits bursts (or causes bursts to be emitted) as ittravels almost directly towards the observer at γ ∼ 104.

The above recovers the conclusions from Section 5, per-haps not surprising since neither mathematically nor phys-ically can one distinguish between a blob moving rela-tivistically that is emitting a banded spectrum or multiplenanoshots. An implication in either case then is that the phys-ical separation along the line of sight is a factor γ larger thanthat on the sky, or a factor γ2 relative to the observed timedelay, i.e., about 108×25 lt-ns ∼ 2.5 lt-s (or about 500 timesthe light cylinder radius cP/2π ' 1600 km).

Instead of requiring the emission region of differentnanoshots to be separated by ∼ 100 km, alternatively, wecould also consider nanoshots to be echos of a single im-pulse. Here, the picture would be that the light emitted atan angle δ′ away from the line of sight encounters a largeoverdensity region of pulsar wind d′ away from the emis-sion region, which scatters it. Similar to the calculationabove, to explain the frequency widths of the bands, the scat-tered images, referred to as echoes, have to be delayed byτ ′ = 1

2d′δ′2/c ∼ 25 ns. In order for the nebular screen to

resolve the different echoes in the scattering tail, they haveto be separated by d′δ′ ∼ 100 km during the first scattering.This yields δ′ ∼ 10−4 rad = 20 arcsec and d′ ∼ 2.5 lt-s, i.e.,

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also for this case something has to happen at a distance ofabout 500 times the light cylinder radius.

It is not obvious whether it is possible to produce suchechoes in the pulsar wind, with the required rather largebending angles δ′, especially given the observation that gi-ant pulses are composed of multiple nanoshots also at highfrequency, where scattering would be less effective. How-ever, there are observational differences between this sce-nario and the Doppler shift scenario. In particular, in this sce-nario, Doppler boosting does not play a role and hence onedoes not expect variations in the scattering tails of individualburst. This unlike for the Doppler shift scenario, where espe-cially at low frequencies one expects to occasionally observea strongly boosted tail or a drop in flux corresponding to thescattered light coming from outside the relativistic beam.

7. RAMIFICATIONS

We conclude that while our observations provide direct ev-idence that small changes in viewing direction during thescattering time result in different spectra, it is evidence onlyfor highly relativistic motion if the shifts are interpreted asDoppler shifts. In principle, it may be possible to get similarshifts in a model in which the banding results from interfer-ence of a few pulses that arise in different locations, as longas those locations are sufficiently far apart that their separa-tion changes by 3% as seen from the screen. For all cases,however, this implies an extent of the region from whichemission is seen by the screen of about 100 km as projectedon the sky, and of about 2.5 lt-s along the line of sight.

For the case that the plasma that emits the giant pulsemoves highly relativistically, a prediction is that togetherwith the frequency drift, there also is a Doppler boost in in-tensity, scaling as ∝ D3. Thus, independent of any banding,pulses which drift upward should be slightly brighter in thescattering tail than the average, while those that drift down-ward should be fainter. Unfortunately, in our data, the evi-dence for this effect is only marginal (see Sect. 5 and Fig. 3).

It should be possible to find more conclusive evidence atlower frequencies, where the scattering time is longer (Popovet al. 2006; Karuppusamy et al. 2012; Eftekhari et al. 2016),corresponding to larger changes in viewing geometry (onegenerically expects τ ∝ 1/f4 and thus δ ∝ 1/f2, whichyields δ ' 6 arcsec at 150 MHz and δ ' 40 arcsec at60 MHz). Indeed, once the scattering angle δ becomes largerthan the beam opening angle 1/γ, one should sometimes seea suppression of the scattering tail. For a given pulse, theexact effect would depend on geometry and the pulse’s valueof γ, but generically one predicts that individual pulses willno longer share the same scattering profile but rather show arange of behaviour around the average.

Taking our model results at face value, our inference ofhighly relativistic (but relatively cold) motion provides astrong constraint on the emission mechanism, since for dif-ferent models vastly different values and distributions of γare expected, ranging anywhere from γ ∼ 1 to γ ∼ 107 (e.g.,Eilek & Hankins 2016; Petrova 2004; Istomin 2004; Lyu-tikov 2007; Lyubarsky 2019; Philippov et al. 2019; Mach-

abeli et al. 2019; Lyutikov 2021). A model that seems tomatch particularly well is the one recently proposed by Lyu-tikov (2021), in which the giant pulses are produced by pairplasma blobs created in reconnection events outside of thelight cylinder, moving relativistically through ambient mag-netic structures that causes the electrons and positrons tobunch up (and thus have a relatively narrow distribution in γ)and emit coherent GHz radio emission via the free electronlaser mechanism (Colgate & Noerdlinger 1971). In this pic-ture, for the parameters of the Crab pulsar, the plasma has tomove with γ & 103. More generally, a large γ value makesit easier to understand the high apparent brightness tempera-tures of the nanoshots.

The Crab’s giant pulses are exceeded in brightness tem-perature only by the Fast Radio Bursts (FRBs). Since FRBsshare quite a number of properties with giant pulses, it maybe that they also arise from plasma that moves highly rela-tivistically (Lu et al. 2020). Repeating FRBs may be the clos-est analogues. While those often show multiple burst sub-structure referred to as the “downward march” (CHIME/FRBCollaboration et al. 2019) which given its persistent direc-tion is unlikely to be related, it may still be worthwhileto look carefully at FRBs with long scattering tails. Mostpromising may be those for which significant scattering hasalready been inferred, such as FRB 121102 (Josephy et al.2019), FRB 181017.J1705+68 (CHIME/FRB Collaborationet al. 2019), and FRB 190117.J2207+17 (Fonseca et al. 2020)(though the first two are likely scattered in the milky way,making them less attractive candidates). Also for the FRBs,high-time resolution observations at lower frequencies wouldbe particularly valuable.

A. B. thanks Thierry Serafin Nadeau and Rebecca Lin forhelp with the data reduction. We also thank Laura New-burgh and Andre Renard for their critical contributions tothe ARO 46m front-end and back-end systems, respectively,the latter and Tom Landecker for conducting the overlappingDRAO observations, Sasha Philippov, Maxim Lyutikov, Wa-diasingh Zorawar and Sterl Phinney for discussions about thetheoretical interpretation, all members of the scintillometrygroup for lively discussions, and the referee for commentsthat helped make our manuscript clearer. U.-L. P. receivessupport from Ontario Research Fund—Research ExcellenceProgram (ORF-RE), Natural Sciences and Engineering Re-search Council of Canada (NSERC) [funding reference num-ber RGPIN-2019-067, CRD 523638-201, 555585-20], Cana-dian Institute for Advanced Research (CIFAR), CanadianFoundation for Innovation (CFI), Simons Foundation, ThothTechnology Inc, which owns and operates ARO, and Alexan-der von Humboldt Foundation. W. L. was supported by theDavid and Ellen Lee Fellowship at Caltech. Computationswere performed on the Niagara supercomputer at the SciNetHPC Consortium. SciNet is funded by: the Canada Foun-dation for Innovation; the Government of Ontario; OntarioResearch Fund - Research Excellence; and the University ofToronto.

8 BIJ ET AL.

Facility: Algonquin Radio Observatory:46m,DRAO:26m

Software: astropy (Astropy Collaboration et al. 2018),Baseband ((van Kerkwijk et al. 2020)), tempo2 (Hobbs et al.2006), numpy (Harris et al. 2020), matplotlib (Hunter 2007)

APPENDIX

A. RELATIVE AND ABSOLUTE GAIN CALIBRATION

There were substantial differences between the frequency-dependent gains of the two polarizations, in particular in the2015 data. To help calibrate the differential gains, we calcu-lated mean net fluxes in each polarization using bright pulses,using that the intrinsic polarization of giant pulses is quiterandom (Hankins et al. 2016), and hence an average shouldserve as an unpolarized source. We choose the brightest 1000pulses for each observating scan, subtracting off the off-pulseaverage, aligning and then averaging them to get the fre-quency profile. We then use a smoothed profile as our relativegain, dividing all our intensities by them.

We show the result in Figure 4. One sees that in the firstsection in 2015, the signal in Y was very weak. Fortunately,that section contained none of the pulses of specific interest.In the other 2015 sessions, the Y polarization shows a strongdrop at the bottom of the band, while no such drop is seen inthe X polarization.

To flux-calibrate the gain-corrected data, we use the Crabnebula itself, which contributes significantly to the off-pulseemission. To get its net contribution, we subtract the aver-age off-pulse signal from pointings towards other pulsars inthe same run from the off-pulse signal from our pointings to-wards the Crab. We then use that the nebular flux spectrum ofthe nebula is Fneb(f) ' 955 Jy(f/1 GHz)−0.27 (like Cordeset al. (2004), we take the normalization from Allen (1973)and slope from Bietenholz et al. (1997)), so that the fluxof a giant pulse can be estimated from the measured (gain-corrected) intensities by,

FGP(f) = Fneb(f)IGP(f)

Ineb(f). (A1)

B. CATEGORIZATION OF THE BRIGHTER PULSES

We classified all 148 selected giant pulses by eye, usingthe categories given in Section 3. We summarize the resultsin Figure 5 and Table 1. From Figure 5, one sees that thereare no obvious trends or clustering of pulse features with timeand phase.

We note that all three banded pulses are very bright, withthe drifting pulses j and f among the brightest of the pulseswe detect. This suggests our classification may be biasedagainst faint pulses with banded structure. It would be inter-esting to verify whether this is indeed the case, but we leavethat to future work since here we are concerned with under-standing the individual banded pulses themselves.

In contrast, numerous multi-peak pulses are found amongthe relatively fainter pulses. This can be understood from thefact that in any given pair, at least one is likely to be faint, as

Table 1. Giant pulse categorization.

Fraction IPFeature N (%) (%)

All . . . . . . . . . . . . 148 100 10

Regular . . . . . . . 129 87 10

Multi-peak . . . . 9 6 0

Partial . . . . . . . . . 7 5 30

Banded . . . . . . . 3 2 0

Driftinga . . . . . . 2 1.3 0

a‘Drifting’ is a sub-category of ‘Banded’.

NOTE—N is the number of pulses that havea given feature, fraction the relative occur-rence rate, and IP is the fraction that oc-curred in the interpulse phase.

faint pulses are much more common, and a faint additionalpulse is easier to detect if the other pulse in the pair is faint aswell. Also here, we leave a more rigorous analysis for futurework.

Among other bright pulses, b and d in Figure 1 show hintsof narrow bands, but the bands were too ambiguous for thosepulses to be labelled as ‘banded’. Similarly, while all thebanded and multi-peak pulses are found among main-pulses,the low event-rate of interpulses does not allow us to rule outthe possibility that these features also exist in the interpulse.

C. EXCLUDING INSTRUMENTAL ORIGINS

In principle, it might be possible to produce apparent band-ing due to non-linearities, drop-outs, etc., in ways that mightnot be immediately obvious once data is dedispersed. Hence,we looked in some detail at our data.

We started with possible problems with the data. First, weruled out that the banding and drifting effect is due to satura-tion, by verifying that the raw VDIF data for the time aroundthe upper drifting pulse are not saturated: We find that boththe real and imaginary parts of the channel samples, of bothpolarizations and for all samples, have probability distribu-tions that are approximately the same before, during, andafter the sweep of the pulse (see Fig. 6). Other non-lineareffects also seem unlikely, as the banding is also seen wherethe pulse is weaker and possible non-linear effect thus lessstrong: e.g., at high frequency for pulses f and h (where sig-nals arrive before the brighter low-frequency signals due to

9

Figure 4. Gain determination using average spectra of giant pulses. For each of our 8 scans, the top panel shows the average background-subtracted giant-pulse spectra in X (left, blue) and Y polarization (Right, gray). Overdrawn are smoothed versions of the spectra (dotted, black),which we use as a proxy for the instrumental gain. The bottom panels show the ratio between the raw and smoothed curves.

Figure 5. Categorized giant pulses as a function of time and phase. Marker styles reflect the different pulse categories (as labelled; seeTable 1), while marker sizes scale with the peak flux. Breaks on the time axis are used to remove periods between scans, with the large breakbetween the six scans in 2015 and the two scans in 2018 indicated by the dashed vertical line, and time on each side counted from the start ofthe first scan of each year. The top and bottom panels show phase windows around the main and interpulse components, respectively. Note thata constant phase offset has been applied in each scan to account for inaccuracies in the absolute clock times.

10 BIJ ET AL.

Figure 6. Raw dynamic spectra of thegiant pulse with drifting bands binned by64 in time (163.84 µs) and 4 in frequency(1.5625 MHz). One sees that there are nodrop outs or broad-band RFI events thatcould, after dedispersion, lead to bandsin the frequency spectra. The struc-ture around 475 MHz is low-level radio-frequency interference, which makes thatpart of the spectrum somewhat less reli-able.

Figure 7. Comparison of giant pulsesshowing banding in their spectra withneighbouring pulses. Left: The three giantpulses with banding; Middle: Next bright-est single pulse within 5 s of each bandedpulse; Right: Average of giant pulses ina 30 s windows centered on the bandedpulse. All spectra were constructed in thesame way as those in Fig. 1. One seesthat the banding does not persist beyondthe banded pulses.

11

ARO

0.0 0.2 0.4 0.6time (ms)

400

500

600

700

Freq

uenc

y (M

Hz)

DRAO

0.0 0.2 0.4 0.6time (ms)

Figure 8. Dynamic spectra of banded giant pulse h in Figure 1observed at two telescopes, ARO (Left) and DRAO (Right). Thetime and frequency resolution are the same as in Figure 1, and thedata were reduced the same way, except without gain and flux cal-ibration. While the DRAO data suffer from more artefacts (mostlybecause some channels are missing, causing the striated horizontalpattern), it is clear that they show the same frequency bands seen atARO, thus confirming those are not instrumental in origin.

dispersion), and in the scattering tail for all pulses (which ar-rives after the bright peaks). We also looked for drop-outsin time, which in the dedispersed time stream might showup as bands1. We found no such drop-outs (see Fig. 6). Fi-nally, since some giant pulses are strongly polarized and theintensity spectra might be affected by polarization leakage,we confirmed that the spacing of none of the bands matchesthat expected from the known Faraday rotation.

Beyond the individual time segments, we also checkedfor possible longer term problems with the instrumental re-sponse using data recorded close in time to each bandedpulse. In Figure 7, we compare the dynamic spectra of thethree banded pulses with the next brightest pulse near each(at most 5 s away), and with the sum of all pulses near it(within a 30 s window centered on the pulse). One sees thatthe neighbouring pulses do not share the bands, indicatingthat the bands are intrinsic to each pulse.

The strongest evidence against an instrumental origin ofthe banded feature is that one banded pulse (h in Fig. 1) is si-multaneously observed at DRAO and ARO (while the AROtimestamp has some uncertainty due clocks not being syn-chronized with GPS, it is easily sufficient to identify a givenCrab cycle). In Fig. 8, we show the burst as detected at thetwo telescopes (after de-dispersion and subtraction of the off-pulse, but without gain correction). One sees that the bandingis similar in both.

1 Any “bands” induced by time variability in the observed data would driftupwards slightly in the dedispersed data. Given the dispersion mea-sure of ∼ 57pc/cm3 of the Crab, however, the expected drift rate of∼ 0.19MHz/ms at 450MHz is far smaller than the drift rates shown bysome of our banded pulses.

D. EXCLUDING INTERPLANETARY ANDINTERSTELLAR SCINTILLATION

The light from the pulsar is bent due to electron densityfluctuations in the surrounding Crab nebula, causing delaysseen in the scattering tail. On the way to us, the light alsoencounters electron density fluctuations in the interstellarmedium and in the solar wind, which cause interstellar andinterplanetary scintillation, respectively.

Interplanetary scintillation (IPS) generally causes onlyweak modulation in our frequency band, but since we ob-served an unusual event, we will nevertheless considerwhether a strong IPS event could reproduce the drifting bandsthat we observed. The typical timescale on which IPS mod-ulates flux is ∆t ∼ few s. Given a typical velocity in thesolar wind of v ∼ 500 km/s, the size of the scattering in-volved is thus L = v∆t ∼ 103 km, which corresponds toscattering angles of θ = L/λ ∼ 0.1 arcsec (for wavelengthλ = 0.5 m [600 MHz]). For a typical distance to a densityperturbation d ∼ 1 AU, the expected scattering time is thenτ = 1

2dθ2/c ∼ 0.1 ns, which corresponds to a de-correlation

bandwith ∼ 1/2πτ ∼ 500 MHz. The latter is larger thanour observation bandwidth and hence IPS would appear as amodulation in time, affecting all frequencies simultaneously.

Since the pulse is dispersed in the interstellar medium, theIPS time modulation would appear as bands in frequency af-ter de-dispersion. However, like the instrumental effects dis-cussed above, the bands would not drift noticeably. More-over, due to the proximity of the source of interplanetaryscintillation, the scintillation pattern seen by two telescopesseparated by more than the size L ∼ 103 km should not bethe same, yet we detected one banded pulse simultaneouslyat ARO and DRAO (which are separated by about 3000 km).Thus, we conclude that the banding we observe is not intro-duced by IPS.

The Crab pulsar is known to be affected by interstellarscintillation. Interpolating from the observed angular broad-ening of the Crab in Very Long Baseline Interferometry (Van-denberg 1976; Rudnitskii et al. 2016), we infer a scatteringangle θ ∼ 1.5 mas at 600 MHz for interstellar scintillation.For a screen roughly halfway to the pulsar, at dscreen '1 kpc, this implies a scattering time τ = 1

2deffθ2/c ' 5µs

(where the effective distance deff = dscreendpsr/(dpsr −dscreen) ' 2 kpc). Given this, the de-correlation bandwidthis ∼1/2πτ ' 30 kHz. Since this value is much smaller thanthe 390 kHz width of our frequency channels, it would be un-likely to be observable, let alone able to explain the observed∼40 MHz band spacing.

In addition, no frequency modulation would be expectedif the Crab pulsar, broadened by nebular scattering, is not apoint source to the interstellar screen. Since the physical sizeof the interstellar scattering region is roughy L = θdscreen '1.5 AU, it will resolve any source at the distance of the Crabthat is larger than (λ/L)(dpsr − dscreen) ' 0.5 mAU. Thiscan be compared with the size of the nebular scattering re-gion: as seen from the pulsar, we inferred from the observedscattering time that it subtends an angle δ ' 0.6 arcsec (see

12 BIJ ET AL.

Sec. 5), which, at ∼ 1 pc from the pulsar, implies a scatter-ing disk size of ∼ 0.6 AU. Thus, as was already noted byVandenberg (1976), the scattering region in the nebula is too

large for there to be any scintillation due to the interstellarmedium.

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