1

# Kinematics of Crab Giant Pulses

Akanksha Bij1,2 Hsiu-Hsien Lin1 Dongzi Li1,3 Marten H. van Kerkwijk4 Ue-Li
Pen1,5,6,7 Wenbin Lu3 Robert Main4,8 Jeffrey B. Peterson9 Brendan Quine10,11
Keith Vanderlinde12,13

###### Abstract

The 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 $10^{37}\,$K[1, 2]. It is
unclear how giant pulses are produced, but they likely originate near the
pulsar’s light cylinder, where corotating plasma approaches the speed of
light[3]. Here, 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 frequency spectra 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 variation is 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 nevertheless reproduce the observed drift by a change in Doppler
shift, if the plasma that emitted the giant pulses moved highly
relativistically, with a Lorentz factor $\gamma\sim 10^{4}$. If so, this would
support models[4] that appeal to highly relativistic plasma to transform
ambient magnetic structures to coherent GHz radio emission[5], be it for giant
pulses or for potentially related sources, such as fast radio bursts[6].

Canadian Institute of Theoretical Astrophysics, 60 Saint George St, Toronto,
ON M5S 3H8, Canada

Department for Physics, Engineering Physics and Astrophysics, Queen’s
University, Kingston, ON, K7L 3N6, Canada

Cahill Center for Astronomy and Astrophysics, California Institute of
Technology, 1216 E California Boulevard, Pasadena, CA 91125, USA

Department of Astronomy and Astrophysics, University of Toronto, 50 Saint
George Street, Toronto, ON M5S 3H4, Canada

Department of Physics, University of Toronto, 60 Saint George Street, Toronto,
ON M5S 1A7, Canada

Canadian Institute for Advanced Research, CIFAR Program in Cosmology and
Gravity, Toronto, ON, M5G 1Z8

Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121, Bonn,
Germany

Department of Physics, Carnegie Mellon University, 5000 Forbes Ave. Pittsburgh
PA 15213 USA

Thoth Technology Inc. 33387 Highway 17, Deep River, Ontario K0J 1P0, Canada

Department of Physics and Astronomy, York University, 4700 Keele St, Toronto,
Ontario M3J 1P3, Canada

David A. Dunlap Department of Astronomy and Astrophysics, University of
Toronto, 50 Saint George Street, Toronto, ON M5S 3H4, Canada

Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50
Saint George Street, Toronto, ON M5S 3H4, Canada

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 MHz frequency
range, with the dedispersed data binned by 4 in both time and frequency
(giving $10.24{\rm\,\mu s}$ time resolution and $1.56{\rm\,MHz}$ frequency
resolution). The red marks to the left of the images indicate noisy frequency
channels (determined from the channel standard deviation before gain
correction). The top panels show frequency-averaged pulse profiles (in units
of kJy), while the side panels show the pulse fluence spectra integrated over
the full 0.84 ms (in units of ${\rm kJy\,ms}$). Panels $a$–$f$ and $j$–$l$ are
the brightest pulses in 2015 and 2018, respectively (ordered by descending
brightness), while panels $g$–$i$ are three further 2015 pulses chosen for
their particular profiles. Note that the faint curved feature visible in
panels $a$–$f$, starting at $\sim\\!0.4$ms at 800 MHz and leaving the panels
at $\sim\\!650\,$MHz, is a reduction artefact (it represents incorrectly
dedispersed leakage from neighbouring channels).

We observed the Crab pulsar (PSR B0531+21) with the 46m antenna at the
Algonquin Radio Observatory (ARO) in 2015 July and 2018 April, for 14 hours
total integration, using a receiver covering the 400–800 MHz band. We
coherently dedispersed the signal, searched for giant pulses, and created
background-subtracted, time-resolved pulse spectra (see Methods).

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

In particular, from the figure, as well as from a categorisation of the 148
brightest pulses by eye (see Methods), we find that most pulses are ‘regular’
(e.g., panels $a$–$e$ and $l$ of Figure 1), showing only a smooth variation
with frequency, and a scattering tail that becomes increasingly long towards
lower frequency (with the scattering time longer in 2018 than in 2015). In
these, the scattering has likely blended together the short time-scale
variations seen at higher frequencies[1, 2, 3] (1–43 GHz), where one often
sees multiple micro-bursts over a few to a few tens of micro-second, with sub-
structure down 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 multiple indepedent giant pulses
occurring in a single rotation, as also seen at higher frequencies,[3] but
here partially blended together due to the scattering.

More unusal are the pulses that differ in their frequency spectrum.
Specifically, some show ‘partial’ emission (e.g., $i$), where the pulse is
bright in only a sub-band of frequencies rather than throughout the band,
while three others show ‘banded’ spectra ($f$, $h$, and $j$). Possibly similar
spectral structure at frequencies comparable to ours has been reported for
giant pulses from the millisecond pulsars M28A[7] and PSR B1937+21,[8] but
what is different for the three banded pulses we observe is that the bands
appear to vary over the scattering tail, shifting in frequency or changing in
relative strength.

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

At first glance, the drifting bands are reminiscent of a surprising earlier
observation of the Crab, at high frequency, where it was found that pulses in
one particular pulse component showed bands with a typical spacing
proportional to frequency, $\Delta\nu\simeq 0.06\nu$, which often drift upward
over the few-$\rm\mu{}s$ duration of the pulse.[3] It would be tempting to
associate our banded pulses with these, especially as our observed band
spacing follows the same scaling relation. Nevertheless, the association seems
unlikely, because the banded spectra we see are all from giant pulses
associated with the Crab’s main pulse component, for which banded spectra have
never been seen at high frequencies. In contrast, the component for which they
were seen, the so-called high-frequency interpulse, is, as its name suggests,
not visible at our relatively low frequencies.

A physical clue comes from the fact that the drifts and other variations in
the banded spectra occur on the same timescale as the scattering, of a few ms,
which is much longer than the duration of giant pulses. Thus, the variation
must be related to the fact that as time proceeds, one sees radiation from the
pulse that has followed increasingly longer paths towards us, having been bent
slightly by matter in the Crab nebula.

Because the scattering timescale is much longer than the inverse of the band
spacing, however, it is unlikely that the scattering itself causes the bands.
Given this, as well as the fact that the banding patterns are not shared
between neighbouring pulses while in at least one case they are seen in two
telescopes simultaneously, we conclude that the banding is not due to
instrumental or scattering effects, but intrinsic to the pulsar emission (for
details, see methods). Indeed, banding itself seems not unexpected given the
behaviour seen at high frequency, where giant pulses in the main pulse
component often show several bright nano-shots spaced closely together. If
sometimes the broadband nano-shots are spaced by a few to a few tens of ns,
this would at lower frequencies naturally lead to ‘partial’ and ‘banded’
spectra with structure on the scale of 10–100 MHz.

Figure 2:  The model geometry: ‘S’ is a blob of radiating plasma moving at an
angle $\theta_{r}$ with respect to the observer ‘R’, with a velocity $\vec{v}$
sufficiently relativistic that its emission is strongly beamed forward in the
observer’s frame (as indicated by the shading). At the start of the burst, the
observer sees emission that travelled directly to the observer, but in the
scattering tail the emission took a less direct path, initially leaving the
source at an angle $\delta$ relative to the direct line of sight, before being
bent towards the observer in the Crab Nebula (at about 1 pc from the pulsar).
If, as drawn, the angles are in the same direction, then as one proceeds along
the scattering tail to larger $\delta$, one will see a steadily increasing
blueshift.

If the banding in the spectra is indeed intrinsic to the emission, what could
cause the changes during the scattering tail? Inspired by the fact that in
pulse $j$ the whole pattern moves in concert, we suggest that as the viewpoint
shifts slightly during the scattering tail, one sees a difference in Doppler
shift. Specifically, we envision a situation like that depicted in Figure 2,
where a group of charged particles moves relativistically, with a high Lorentz
factor $\gamma$. Any emission from these particles will be beamed forward,
within a beam of size $\sim\\!1/\gamma$. For that beam to be observable, it
must include the line of sight, i.e., the particles should move at an angle
$\theta_{r}\,\hbox to0.0pt{\raise 1.72218pt\hbox{$<$}\hss}{\lower
3.44444pt\hbox{$\sim$}}\,1/\gamma$ relative to the line of sight. The
scattering screen will then bend light towards us which was originally emitted
in a slightly different angle, either closer or further away from the
direction the particles are moving in and thus with a different projected
velocity.

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

$D=\frac{1}{\gamma(1-\beta\cos\theta_{r})}\simeq\frac{2\gamma}{1+\gamma^{2}\theta_{r}^{2}}$
(1)

where $\beta=v/c$ and $\gamma=1/\sqrt{1-\beta^{2}}$ as per convention. The
approximate equality is for highly relativistic motion, with $\gamma\gg 1$,
for which $\beta\simeq 1-1/2\gamma^{2}$ and $\cos\theta_{r}\simeq
1-\frac{1}{2}\theta_{r}^{2}$ (the latter because $\theta_{r}$ has to be small,
$\,\hbox to0.0pt{\raise 1.72218pt\hbox{$<$}\hss}{\lower
3.44444pt\hbox{$\sim$}}\,1/\gamma$).

The above implies that for a fractional change in the viewing angle from
$\theta_{r}$ to $\theta_{r}+\delta$, as resolved by our scattering screen, the
resulting change in observed frequency $f_{r}$ will be,

$\frac{\Delta f_{r}}{f_{r}}=\frac{\Delta
D}{D}=\frac{-\gamma^{2}(2\theta_{r}\delta+\delta^{2})}{1+\gamma^{2}(\theta_{r}+\delta)^{2}}.$
(2)

For an observer in the center of the beam, with
$\theta_{r}\ll\delta<1/\gamma$, one expects $\Delta
f_{r}/f_{r}\approx-\gamma^{2}\delta^{2}$, indicating a downward drift. For the
statistically more likely case depicted in Figure 2, with an observer at the
edge of the beam, i.e., with $\theta_{r}\simeq 1/\gamma$ , one finds $\Delta
f_{r}/f_{r}\approx\gamma\delta$ (where the sign would be opposite if
scattering was in the other direction, and cases where $\delta$ and
$\theta_{r}$ are not aligned on the sky would be intermediate).

The scattering angle $\delta$ can be estimated using the known geometry for
the Crab pulsar. The distance of the screen is about $d\simeq 1{\rm\,pc}$ from
the pulsar[9, 10]. The scattering time measured from the dynamic spectrum of
the 2018 drifting pulse is approximately $\tau\simeq 0.5\,$ms at 450 MHz.
Since from the geometry,

$c\tau=\frac{d}{\cos\delta}-d\simeq\frac{d}{2}\delta^{2},$ (3)

we infer $\delta^{2}\simeq 1\times 10^{-11}$ and thus $\delta\simeq
0.6\,$arcsec. Given that we observe about a 5% change in observed frequency,
this implies $\gamma\simeq 0.05/\delta\simeq 10^{4}$.

Figure 3:  Comparison of pulse $j$ with the drifting bands with a simple model
in which the initial profile is shifted in frequency and attenuated as time
progresses. The panels show the observed dynamic spectrum from Figure 1, the
approximate model, and the residuals from the model. For the model, we started
with the observed spectrum, averaged over the first 24 time samples
($61{\rm\,\mu s}$) of the start of the pulse. For subsequent time samples, we
Doppler shifted this profile by a factor $D=1+a\sqrt{(t-t_{0})/\tau_{\rm
ref}}$, where we arbitrarily fixed $\tau_{\rm ref}=250{\rm\,\mu{}s}$ to make
the scale factor $a$ easy to interpret and then chose $a=0.028$ to give a
decent approximation to the data. For the intensity evolution, we assumed that
the scattering efficiency followed the usual exponential decay,
$\exp(-(t-t_{0})/\tau(f))$, where $\tau(f)=\tau_{\rm ref}(f/f_{\rm ref})^{-4}$
is the scattering time scaled such that it matches our arbitrary reference
time $\tau_{\rm{}ref}$ at some reference frequency, which we adjusted to
$f_{\rm ref}=465{\rm\,MHz}$ to roughly match the data. For completeness, we
also multiplied the intensities with the Doppler boosting factor, $D^{3}$.
Given that scattering screen is unlikely to be as simple as assumed, the model
reproduces the observations remarkably well.

If the above holds, then as a function of time, the Doppler shift should
roughly vary as $\Delta D(t)/D\propto\delta(t)\propto\sqrt{t}$. This scaling
with the square root of time seems consistent from what one approximately sees
for pulse $j$ in Fig. 1.

To explore this further, we compared the evolution of the observed spectrum
with a simple model that starts with the initial spectrum but shifts and dims
it over time, assuming the $\sqrt{t}$ scaling above for the shifts and the
standard exponential decay[11] for the dimming, in which the decay timescale
scales as $\tau\propto{}1/f_{r}^{4}$. The free parameters are the shifting and
dimming scale factors, which we roughly adjust to best match the data. As one
can see in Fig. 3, the model can reproduce the observations quite well.

In our scenario, different giant pulses would be emitted in different
directions relative to the line of sight, and thus bands could drift up, down,
or not at all. Furthermore, for pulse $j$, we implicitly assumed that the
scattering is asymmetric around the line of sight, dominated by the side in
which pulse $j$ was emitted. From observations of other pulsars, we know
scattering geometries vary as the pulsar moves and probes different parts of
the scattering screen[12]. For the Crab, similarly, variations in scattering
geometry on timescales of weeks are suggested by echoes and changes in
scattering time[13, 14]. At times when the screens is more symmetric, one
would expect to observe both upward and downward-drifting features
simultaneously. Since the structure of the scattering screen could be quite
complex and the scattering strength is frequency dependent, an uneven
distribution of material on different sides of the screen could result in the
overall drift direction being frequency dependent. Indeed, this may help
understand the observed dynamic spectra of pulse $f$ (Fig. 1), which appears
to have downward drifts at short delay, especially in the top half of the band
and upwards shifts at longer delays in the bottom of the band.

We note that while our observations provide direct evidence that small changes
in viewing direction during the scattering time result in different spectra,
it is evidence only for highly relativistic motion if the shifts are
interpreted as Doppler shifts. In principle, it may be possible to get similar
shifts in a model in which the banding results from interference of a few
pulses that arise in different locations, as long as those locations are
sufficiently far apart that their separation changes by 5% as seen from the
screen. This may be possible without relativistic motion if a single strong
nano-shot causes echoes in the pulsar wind (see Methods).

For the case that the plasma that emits the giant pulse moves highly
relativistically, a prediction is that together with the frequency drift,
there also is a Doppler boost in intensity, scaling as $\propto{}D^{3}$. Thus,
independent of any banding, pulses which drift upward should be slightly
brighter in the scattering tail than the average, while those that drift
downward should be fainter. We looked for this effect for pulse $j$, and
indeed find that its scattering tail is slightly brighter than for other
pulses, but unfortunately the evidence is only marginal (see Methods).

It should be possible to find more conclusive evidence at lower frequencies,
where the scattering time is longer[15, 16, 17], corresponding to larger
changes in viewing geometry (one generically expects $\tau\propto 1/f^{4}$ and
thus $\delta\propto 1/f^{2}$, which yields $\delta\simeq 6{\rm\,arcsec}$ at
150 MHz and $\delta\simeq 40{\rm\,arcsec}$ at 60 MHz). Indeed, once the
scattering angle $\delta$ becomes larger than the beam opening angle
$1/\gamma$, one should sometimes see a suppression of the scattering tail. For
a given pulse, the exact effect would depend on geometry and the pulse’s value
of $\gamma$, but generically one predicts that individual pulses will no
longer share the same scattering profile but rather show a range of behaviour
around the average.

Taking our model results at face value, our inference of highly relativistic
motion provides a strong constraint on the emission mechanism, since for
different models vastly different values and distributions of $\gamma$ are
expected, ranging anywhere from $\gamma\sim 1$ to $\gamma\sim 10^{7}$ (e.g.,
refs 3, 18, 19, 20, 21, 22, 23, 4). A model that seems to match particularly
well is a recently proposed one[4] in which the giant pulses are produced by
pair plasma blobs created in reconnection events outside of the light
cylinder, moving relativistically through ambient magnetic structures that
causes the electrons and positrons to bunch up and emit coherent GHz radio
emission via the free electron laser mechanism[5]. In this picture, for the
parameters of the Crab pulsar, the plasma has to move with $\gamma\,\hbox
to0.0pt{\raise 1.72218pt\hbox{$>$}\hss}{\lower
3.44444pt\hbox{$\sim$}}\,10^{3}$. More generally, a large $\gamma$ value makes
it easier to understand the high apparent brightness temperatures of the
nanoshots.

The Crab’s giant pulses are exceeded in brightness temperature only by the
Fast Radio Bursts (FRBs). Since FRBs share quite a number of properties with
giant pulses, it may be that they also arise from plasma that moves highly
relativistically[24]. Repeating FRBs may be the closest analogues. While those
often show multiple burst substructure referred to as the ‘downward march’[25]
which given its persistent direction is unlikely to be related, it may still
be worthwile to look carefully at FRBs with long scattering tails. Most
promising may be those for which significant scattering has already been
inferred, such as FRB 121102[26], FRB 181017.J1705+68[25], and FRB
190117.J2207+17[27] (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 would be particularly valuable.

## References

  * [1] Hankins, T. H., Kern, J. S., Weatherall, J. C. & Eilek, J. A.  Nanosecond radio bursts from strong plasma turbulence in the Crab pulsar.  _Nature_ 422, 141–143 (2003).
  * [2] Hankins, T. H. & Eilek, J. A.  Radio Emission Signatures in the Crab Pulsar.  _ApJ_ 670, 693–701 (2007).  0708.2505.
  * [3] Eilek, J. A. & Hankins, T. H.  Radio emission physics in the Crab pulsar.  _Journal of Plasma Physics_ 82, 635820302 (2016).  1604.02472.
  * [4] Lyutikov, M.  Coherent emission in pulsars, magnetars and Fast Radio Bursts: reconnection-driven free electron laser.  _arXiv e-prints_ arXiv:2102.07010 (2021).  2102.07010.
  * [5] Colgate, S. A. & Noerdlinger, P. D.  Coherent Emission from Expanding Supernova Shells.  _ApJ_ 165, 509 (1971).
  * [6] Metzger, B. D., Margalit, B. & Sironi, L.  Fast radio bursts as synchrotron maser emission from decelerating relativistic blast waves.  _MNRAS_ 485, 4091–4106 (2019).  1902.01866.
  * [7] Bilous, A. V., Pennucci, T. T., Demorest, P. & Ransom, S. M.  A Broadband Radio Study of the Average Profile and Giant Pulses from PSR B1821-24A.  _ApJ_ 803, 83 (2015).  1412.7629.
  * [8] McKee, J. W. _et al._ A detailed study of giant pulses from PSR B1937+21 using the Large European Array for Pulsars.  _MNRAS_ 483, 4784–4802 (2019).  1811.02856.
  * [9] Lawrence, S. S. _et al._ Three-Dimensional Fabry-Perot Imaging Spectroscopy of the Crab Nebula, Cassiopeia A, and Nova GK Persei.  _AJ_ 109, 2635 (1995).
  * [10] Martin, T., Milisavljevic, D. & Drissen, L.  3D mapping of the Crab Nebula with SITELLE. I. Deconvolution and kinematic reconstruction.  _MNRAS_ (2021).  2101.02709.
  * [11] Lang, K. R.  Interstellar Scintillation of Pulsar Radiation.  _ApJ_ 164, 249 (1971).
  * [12] Hill, A. S. _et al._ Deflection of Pulsar Signal Reveals Compact Structures in the Galaxy.  _ApJ_ 619, L171–L174 (2005).  astro-ph/0411752.
  * [13] McKee, J. W., Lyne, A. G., Stappers, B. W., Bassa, C. G. & Jordan, C. A.  Temporal variations in scattering and dispersion measure in the Crab Pulsar and their effect on timing precision.  _MNRAS_ 479, 4216–4224 (2018).  1806.10486.
  * [14] Driessen, L. N., Janssen, G. H., Bassa, C. G., Stappers, B. W. & Stinebring, D. R.  Scattering features and variability of the Crab pulsar.  _MNRAS_ 483, 1224–1232 (2019).  1811.10395.
  * [15] Popov, M. V. _et al._ Instantaneous radio spectra of giant pulses from the crab pulsar from decimeter to decameter wavelengths.  _Astronomy Reports_ 50, 562–568 (2006).  astro-ph/0606025.
  * [16] Karuppusamy, R., Stappers, B. W. & Lee, K. J.  Crab giant pulses at low frequencies.  _A &A_ 538, A7 (2012).
  * [17] Eftekhari, T., Stovall, K., Dowell, J., Schinzel, F. K. & Taylor, G. B.  A Low frequency Survey of Giant Pulses from the Crab Pulsar.  _ApJ_ 829, 62 (2016).  1607.08612.
  * [18] Petrova, S. A.  On the origin of giant pulses in radio pulsars.  _A &A_ 424, 227–236 (2004).
  * [19] Istomin, Y. N.  Origin of Giant Radio Pulses.  In Camilo, F. & Gaensler, B. M. (eds.) _Young Neutron Stars and Their Environments_ , vol. 218, 369 (2004).
  * [20] Lyutikov, M.  On generation of Crab giant pulses.  _MNRAS_ 381, 1190–1196 (2007).  0705.2530.
  * [21] Lyubarsky, Y.  Radio emission of the Crab and Crab-like pulsars.  _MNRAS_ 483, 1731–1736 (2019).  1811.11122.
  * [22] Philippov, A., Uzdensky, D. A., Spitkovsky, A. & Cerutti, B.  Pulsar Radio Emission Mechanism: Radio Nanoshots as a Low-frequency Afterglow of Relativistic Magnetic Reconnection.  _ApJ_ 876, L6 (2019).  1902.07730.
  * [23] Machabeli, G., Chkheidze, N. & Malov, I.  Energy accumulation mechanism in pulsar magnetospheric plasma eigen-waves and formation of Giant Radio Pulses.  _Astroph. & Space Sci._ 364, 40 (2019).  1805.04848.
  * [24] Lu, W., Kumar, P. & Zhang, B.  A unified picture of Galactic and cosmological fast radio bursts.  _MNRAS_ 498, 1397–1405 (2020).  2005.06736.
  * [25] CHIME/FRB Collaboration _et al._ CHIME/FRB Discovery of Eight New Repeating Fast Radio Burst Sources.  _ApJ_ 885, L24 (2019).  1908.03507.
  * [26] Josephy, A. _et al._ CHIME/FRB Detection of the Original Repeating Fast Radio Burst Source FRB 121102.  _ApJ_ 882, L18 (2019).  1906.11305.
  * [27] Fonseca, E. _et al._ Nine New Repeating Fast Radio Burst Sources from CHIME/FRB.  _ApJ_ 891, L6 (2020).  2001.03595.
  * [28] Loken, C. _et al._ SciNet: Lessons Learned from Building a Power-efficient Top-20 System and Data Centre.  In _Journal of Physics Conference Series_ , vol. 256 of _Journal of Physics Conference Series_ , 012026 (2010).
  * [29] Ponce, M. _et al._ Deploying a Top-100 Supercomputer for Large Parallel Workloads: the Niagara Supercomputer.  _arXiv e-prints_ arXiv:1907.13600 (2019).  1907.13600.
  * [30] Bandura, K. _et al._ ICE-Based Custom Full-Mesh Network for the CHIME High Bandwidth Radio Astronomy Correlator.  _Journal of Astronomical Instrumentation_ 5, 1641004 (2016).  1608.04347.
  * [31] Whitney, A., Kettenis, M., Phillips, C. & Sekido, M.  VLBI Data Interchange Format (VDIF) (invited).  In _8th International e-VLBI Workshop_ , 42 (2009).
  * [32] Lyne, A. G., Pritchard, R. S. & Graham Smith, F.  23 years of Crab pulsar rotational history.  _MNRAS_ 265, 1003–1012 (1993).
  * [33] Hobbs, G. B., Edwards, R. T. & Manchester, R. N.  TEMPO2, a new pulsar-timing package - I. An overview.  _MNRAS_ 369, 655–672 (2006).  astro-ph/0603381.
  * [34] Hankins, T. H., Eilek, J. A. & Jones, G.  The Crab Pulsar at Centimeter Wavelengths. II. Single Pulses.  _ApJ_ 833, 47 (2016).  1608.08881.
  * [35] Cordes, J. M., Bhat, N. D. R., Hankins, T. H., McLaughlin, M. A. & Kern, J.  The Brightest Pulses in the Universe: Multifrequency Observations of the Crab Pulsar’s Giant Pulses.  _ApJ_ 612, 375–388 (2004).  astro-ph/0304495.
  * [36] Allen, C. W.  _Astrophysical quantities_ (Athlone Press, London, 1973).
  * [37] Bietenholz, M. F. _et al._ The Radio Spectral Index of the Crab Nebula.  _ApJ_ 490, 291–301 (1997).  astro-ph/9707195.
  * [38] Vandenberg, N. R.  Meter-wavelength VLBI. IV. Temporal and spatial scattering of the Crab nebula pulsar’s radiation.  _ApJ_ 209, 578–583 (1976).
  * [39] Rudnitskii, A. G., Karuppusamy, R., Popov, M. V. & Soglasnov, V. A.  Studies of cosmic plasma using radioastron VLBI observations of giant pulses of the pulsar B0531+21.  _Astronomy Reports_ 60, 211–219 (2016).

A. B. thanks Thierry Serafin Nadeau and Rebecca Lin for help with the data
reduction. We also thank Laura Newburgh and Andre Renard for their critical
contributions to the ARO 46m front-end and back-end systems, respectively, the
latter and Tom Landecker for conducting the overlapping DRAO observations,
Sasha Philippov, Maxim Lyutikov, Wadiasingh Zorawar and Sterl Phinney for
discussions about the theoretical interpretation, and all members of the
scintillometry group for lively discussions. U.-L. P. receives support from
Ontario Research Fund—Research Excellence Program (ORF-RE), Natural Sciences
and Engineering Research Council of Canada (NSERC) [funding reference number
RGPIN-2019-067, CRD 523638-201, 555585-20], Canadian Institute for Advanced
Research (CIFAR), Canadian Foundation for Innovation (CFI), Simons Foundation,
Thoth Technology Inc, which owns and operates ARO, and Alexander von Humboldt
Foundation. W. L. was supported by the David and Ellen Lee Fellowship at
Caltech. Computations were performed on the Niagara supercomputer at the
SciNet HPC Consortium [28, 29]. SciNet is funded by: the Canada Foundation for
Innovation; the Government of Ontario; Ontario Research Fund - Research
Excellence; and the University of Toronto.

A. B. analyzed the data, discovered the giant pulses with drifting bands, and
created all of the figures. A. B., H.-H. L., D. L., and M. H. v. K wrote the
majority of the text. R. M., W. L., and U.-L. P., helped interpret the
results. J. B. P. designed the feed used for these observations, B. Q. mounted
it and ensured the telescope and instrumentation performed properly, while K.
V. designed, built and installed the analog chain.

The authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to A. B. (email:
a.bij@queensu.ca).

##  0.1 Data recording and reduction.

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.

We observed the Crab pulsar with the 46m antenna at the Algonquin Radio
Observatory (ARO) on 2015 July 23 and 24 (12.5 hours spread over 6 scans), and
2018 April 25 (1.5 hours over 2 scans). Data are taken at 400-800 MHz with two
linear polarizations. The raw voltage streams (baseband) are processed using a
CHIME acquisition board[30], which digitizes the signals, passes them through
a polyphase filter, channelizes them to 1024 frequency channels, and then
records these in standard VDIF format[31]. Different receivers were used for
the observation at 2015 and 2018, which had significantly different frequency
responses (see below).

The observation performed on 2015 July 24 partially overlaps with an
observations of Crab pulsar at the John A. Galt Telescope at the Dominion
Radio Astrophysical Observatory (DRAO). These observations were originally
taken for very long baseline interferometry with ARO, but here we will use
them only to rule out instrumental effects at ARO. They covered the same
frequency band and were taken using a similar receiver and backend.

To search for giant pulses, we first coherently dedispersed the data to a
reference frequency of 800 MHz, using a dispersion measure (DM) of
$56.7708{\rm\,pc/cm^{3}}$ for 2015 and $56.7562{\rm\,pc/cm^{3}}$ for 2018,
obtained by using the closest DM value from the Jodrell Bank ephemeris[32] as
an initial guess and then optimized by adjusting the 4th significant figure
such that the rising edge of the pulse is aligned, by-eye, in frequency. Next,
we sum the power from all frequency channels and both polarizations, apply a
rolling boxcar window of size 1.3 ms, and flag any peaks more than 5$\sigma$
above the noise. We calculate the rotational phase with the TEMPO2
package[33], using the Jodrell Bank ephemeris[32]. However, while our clocks
were tied to a maser, they had small random offsets at the start of each
integration, and hence only the relative phases are useful within each
continuous integration.

Inspecting the data, we noticed substantial differences between the frequency-
dependent gains of the two polarizations, in particular in the 2015 data. To
help calibrate the differential gains, we calculated mean net fluxes in each
polarization using bright pulses, using that the intrinsic polarization of
giant pulses is quite random[34], and hence an average should serve as an
unpolarized source. We choose the brightest 1000 pulses for each observation
scan, subtracting off the off-pulse average, aligning and then averaging them
to get the frequency profile. We then use a smoothed profile as our relative
gain, dividing all our intensities by them.

We show the result in Figure 4. One sees that in the first section 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 strong drop at the bottom of the band, while no such drop is seen in
the X polarization.

To flux-calibrate the gain-corrected data, we use the Crab nebula itself,
which contributes significantly to the off-pulse emission. To get its net
contribution, we subtract the average off-pulse signal from pointings towards
other pulsars in the same run from the off-pulse signal from our pointings
towards the Crab. We then use that the nebular flux spectrum of the nebula is
$F_{\rm neb}(f)\simeq 955{\rm\,Jy}(f/{\rm 1\,GHz})^{-0.27}$ (as in ref. 35, we
take the normalization from ref. 36 and slope from ref. 37), so that the flux
of a giant pulse can be estimated as from the measured (gain-corrected)
intensities by,

$F_{\rm GP}(f)=F_{\rm neb}(f)\,\frac{I_{\rm GP}(f)}{I_{\rm neb}(f)}.$ (4)

Figure 5:  Categorized giant pulses as a function of time and phase. Marker styles reflect the different pulse categories (as labelled; see Table 1), while marker sizes scale with the peak flux. Breaks on the time axis are used to remove periods between scans, with the large break between 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 of the first scan of each year. The top and bottom panels show phase windows around the main and interpulse components, respectively. Note that a constant phase offset has been applied in each scan to account for inaccuracies in the absolute clock times.  Table 1:  Giant pulse categorization. The 148 giant pulses with the highest S/N are categorized in different groups, with $N$ the number of pulses that have a given feature, fraction the relative occurence rate, and IP is the fraction that occurred in the interpulse phase. Note that ‘Drifting’ is a sub-category of ‘Banded’.  |  | Fraction | IP
---|---|---|---
Feature | $N$ | (%) | (%)
All | 148 |  100 | 10
Regular | 129 | 87 | 10
Multi-peak |  009 |  06 |  00
Partial |  007 |  05 | 30
Banded |  003 |  02 |  00
Drifting |  002 |  01.3 |  00

##  0.2 Categorization of the brighter pulses.

We classified all 148 selected giant pulses by eye, using the categories in
the main text. We summarize the results in Figure 5 and Table 1. From Figure
5, one sees that there are no obvious trends or clustering of pulse features
with time and phase.

We note that all three banded pulses are very bright, with the drifting pulses
$j$ and $f$ among the brightest of the pulses we detect. This suggests our
classification may be biased against faint pulses with banded structure. It
would be interesting to verify whether this is indeed the case, but we leave
that to future work since here we are concerned with understanding the
individual banded pulses themselves.

In contrast, numerous multi-peak pulses are found among the relatively fainter
pulses. This can be understood from the fact that in any given pair, at least
one is likely to be faint, as faint pulses are much more common, and a faint
additional pulse is easier to detect if the other pulse in the pair is faint
as well. Also here, we leave a more rigorous analysis for future work.

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

##  0.3 Excluding instrumental origins of banding and drifting.

Figure 6: Raw dynamic spectra of the giant pulse with drifting bands binned by
64 in time (163.84 $\mu$s) and 4 in frequency (1.5625 MHz). One sees that
there are no drop outs or broad-band RFI events that could, after
dedispersion, lead to bands in the frequency spectra. The structure around 475
MHz is low-level radio-frequency interference, which makes that part of the
spectrum somewhat less reliable.

Figure 7:  Comparison of giant pulses showing banding in their spectra with
neighbouring pulses. _Left:_ The three giant pulses with banding; _Middle:_
Next brightest single pulse within 5 s of each banded pulse; _Right:_ Average
of giant pulses in a 30 s windows centered on the banded pulse. All spectra
were constructed in the same way as those in Fig. 1. One sees that the banding
does not persist beyond the banded pulses.  Figure 8:  Dynamic spectra of
banded giant pulse $h$ in Figure 1 observed at two telescopes, ARO (_Left_)
and DRAO (_Right_). The time and frequency resolution are the same as in
Figure 1, and the data were reduced the same way, except without gain and flux
calibration. While the DRAO data suffer from more artefacts (mostly because
some channels are missing, causing the striated horizontal pattern), it is
clear that they show the same frequency bands seen at ARO, thus confirming
those are not instrumental in origin.

In principle, it might be possible to produce apparent banding due to non-
linearities, drop-outs, etc., in ways that might not 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, we ruled out that the
banding and drifting effect is due to saturation, by verifying that the raw
VDIF data for the time around the upper drifting pulse. We find that both the
real and imaginary parts of the channel samples, of both polarizations and for
all samples, have probability distributions that are approximately the same
before, during, and after the sweep of the pulse (see Fig. 6). Other non-
linear effects also seem unlikely, as the banding is also seen where the pulse
is weaker and possible non-linear effect thus less strong: at high frequency
(for pulses $f$ and $h$; which arrives before the brighter lower-frequency
signal due to dispersion), and during the scattering tail (which arrives after
the bright peak). We also looked for drop-outs in time, which in the
dedispersed time stream might show up as bands111 Any “bands” induced by time
variability in the observed data would drift upwards slightly in the
dedispersed data. Given the dispersion measure of $\sim\\!57{\rm pc/cm^{3}}$
of the Crab, however, the expected drift rate of $\sim\\!0.19{\rm\,MHz/ms}$ at
$450\,$MHz is far smaller than the drift rates shown by some of our banded
pulses.. We found no such drop-outs (see Fig. 6). Finally, since some giant
pulses are strongly polarized and the intensity spectra might be affected by
polarization leakage, we confirmed that the spacing of none of the bands
matches that expected from the known Faraday rotation.

Beyond the individual time segments, we also checked for possible longer term
problems with the instrumental response using data recorded close in time to
each banded pulse. In Figure 7, we compare the dynamic spectra of the three
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 that the neighbouring pulses do not share the bands,
indicating that the bands are intrinsic to each pulse.

The strongest evidence against an instrumental origin of the banded feature is
that one banded pulse ($h$ in Fig. 1) is simultaneously observed at DRAO and
ARO (while the ARO timestamp has some uncertainty due clocks not being
synchronized with GPS, it is easily sufficient to identify a given Crab
cycle). In Fig. 8, we show the burst as detected at the two telescopes (after
de-dispersion and subtraction of the off-pulse, but without gain correction).
One sees that the banding is similar in both.

##  0.4 Excluding interplanetary and interstellar scintillation as the cause
of banding and drifting.

The light from the pulsar is bent due to electron density fluctuations in the
surrounding Crab nebula, causing delays seen in the scattering tail. On the
way to us, the light also encounters electron density fluctuations in the
interstellar medium and in the solar wind, which cause interstellar and
interplanetary scintillation, respectively.

Interplanetary scintillation (IPS) generally causes only weak modulation in
our frequency band, but since we observed an unusual event, we will
nevertheless consider whether a strong IPS event could reproduce the drifting
bands that we observed. The typical timescale on which IPS modulates flux is
$\Delta t\sim{\rm few~{}s}$. Given a typical velocity in the solar wind of
$v\sim 500{\rm\,km/s}$, the size of the scattering involved is thus $L=v\Delta
t\sim 10^{3}{\rm\,km}$, which corresponds to scattering angles of
$\theta=L/\lambda\sim 0.1{\rm arcsec}$ (for wavelength $\lambda=0.5{\rm\,m}$
[600 MHz]). For a typical distance to a density perturbation $d\sim
1{\rm\,AU}$, the expected scattering time is then
$\tau=\frac{1}{2}d\theta^{2}/c\sim 0.1\,$ns, which corresponds to a de-
correlation bandwith $\Delta\nu=1/2\pi\tau\sim 500{\rm\,MHz}$. The latter is
larger than our observation bandwidth and hence IPS would appear as a
modulation in time, affecting all frequencies simultaneously.

Since the pulse is dispersed in the interstellar medium, the IPS time
modulation would appear as bands in frequency after de-dispersion. However,
like the instrumental effects discussed above, the bands would not drift
noticeably. Moreover, due to the proximity of the source of interplanetary
scintillation, the scintillation pattern seen by two telescopes separated by
more than the size $L\sim 10^{3}{\rm\,km}$ should not be the same, yet we
detected one banded pulse simultaneously at ARO and DRAO (which are separated
by about $3000{\rm\,km}$). Thus, we conclude that the banding we observe is
not introduced by IPS.

The Crab pulsar is known to be affected by interstellar scintillation.
Interpolating from the observed angular broadening of the Crab in Very Long
Baseline Interferometry[38, 39], we infer a scattering angle $\theta\sim
1.5\,$mas at 600 MHz for interstellar scintillation. For a screen roughly
halfway to the pulsar, at $d_{\rm screen}\simeq 1{\rm\,kpc}$, this implies a
scattering time $\tau=\frac{1}{2}d_{\rm eff}\theta^{2}/c\simeq 5{\rm\,\mu s}$
(where the effective distance $d_{\rm eff}=d_{\rm screen}d_{\rm psr}/(d_{\rm
psr}-d_{\rm screen})\simeq 2\,$kpc). Given this, the de-correlation bandwidth
is $\Delta\nu=1/2\pi\tau\simeq 30{\rm\,kHz}$. Since this value is much smaller
than the $390\,$kHz width of our frequency channels, it would be unlikely to
be observable, let alone able to explain the observed $\sim\\!15{\rm\,MHz}$
width of the observed banding.

In addition, no frequency modulation would be expected if the Crab pulsar,
broadened by nebular scattering, is not a point source to the interstellar
screen. Since the physical size of the interstellar scattering region is
roughy $L=\theta d_{\rm screen}\simeq 1.5$ AU, it will resolve any source at
the distance of the Crab that is larger than $(\lambda/L)(d_{\rm psr}-d_{\rm
screen})\simeq 0.5\,$mAU. This can be compared with the size of the nebular
scattering region: as seen from the pulsar, we inferred from the observed
scattering time that it subtends an angle $\delta\simeq 0.6{\rm\,arcsec}$ (see
main text), which, at $\sim\\!1{\rm\,pc}$ from the pulsar, implies a
scattering disk size of $\sim\\!0.6{\rm\,AU}$. Thus, as was already noted in
ref. 38, the scattering region in the nebula is too large for there to be any
scintillation due to the interstellar medium.

##  0.5 Details of the Model for the Drifting Bands

In the main text, we presented a simple model in which we took a smoothed
version of the spectrum of pulse $j$ in the first $61{\rm\,\mu s}$, and
evolved that by applying a time-varying Doppler shift and scattering
efficiency (parameters in caption of Fig. 3). In Fig. 9, we compare the
spectra at different time slices.One again sees that the model reproduces the
data quite well, especially considering the simplicity of the model.

Figure 9:  Spectra of pulse $j$ as a function of time. Overdrawn is our model,
which is based on a smoothed version of the first (top) spectrum, evolved in
frequency and time as described in the main text and in Fig. 3.  Figure 10:
Comparison of pulse intensity throughout the scattering tail in various sub
bands, with flux normalized to peak flux and shown in a logarithmic scale to
ensure that exponential decay with time looks linear. The curves correspond to
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). Pulse $j$ appears to dim
slightly more slowly than the comparison pulses, as is expected in our model
due to increasing Doppler boosting.

In our proposed Doppler-shift interpretation, we expect as a function of time
not just a shift in frequency by the change in Doppler factor $\Delta D/D$,
but also a boost in intensity by a factor $(\Delta D/D)^{3}$. Since pulse $j$
is drifting up, this boost factor will increase during the tail, and hence the
scattering tail should be slightly brighter, by about 15%, than would be the
case for a pulse for which the Doppler factor did not vary. To see if this is
indeed the case, we compare in Fig. 10 the brightness in the scattering tail
of the drifting pulse $j$ with that of the brightest other pulse within 5 s as
well as that of the average of all pulses within 30 s (see Fig. 7. One sees
that the scattering tail of pulse $j$ does appear slightly boosted, as
expected by the model. Also shown is the model brightness, which reproduces
the observed brightness well. If one were to remove the Doppler boosting from
the model, it would fall closer to the profiles of the other pulses.

We note, however, that the significance of the difference is limited, with the
differences between pulse $j$ and the comparisons not much larger than those
between pulse $j$ and the model. A stronger test may be possible at lower
frequency, where the range in scattering angle, and hence in Doppler shift,
would be larger.

##  0.6 Implication of the Drifting Bands.

The drifts we observe in the banded structure of some pulses almost certainly
reflect the change in viewing geometry that occurs during the scattering tail.
So far, we have assumed that the source of the burst intrinsically emitted a
banded spectrum, and that the drift corresponded to a change in Doppler shift,
from which we derived that the source had to move highly relativistically.

Here, we discuss what happens if one relaxes the assumption that the source
produces bands intrinsically, instead requiring that the banding reflects
interference between multiple nanoshots, which do not necessarily originate
from the same region but are causally connected. In this scenario, the banding
in frequency comes from interference between multiple nanoshots, and the width
of the bands thus reflects the inverse of the arrival time separation between
different nanoshots.

In this picture, during the scattering tail, the nanoshots are viewed at an
angle $\delta$ away from the direct LOS, resulting in a change of path
difference $\Delta x_{\perp}\delta$ between nanoshots, where $\Delta
x_{\perp}$ is the distance between the nanoshots perpendicular to the LOS
(i.e., projected on the sky). As the interference pattern of multiple
nanoshots moves by about half of the pattern spacing in the scattering tail,
$\Delta x_{\perp}\delta\simeq 0.5\lambda$. Given our observation wavelength
$\lambda\simeq 0.5{\rm\,m}$ and $\delta\simeq 0.6{\rm\,arcsec}$ (see Main
text), we infer a separation $\Delta x_{\perp}\sim 100{\rm\,km}$.

Given the typical observed frequency width of the bands of 50 MHz, in the
observer’s frame the nanoshots need to be separated in time by about 20 ns.
Since at the speed of light this corresponds to only 6 m, for the nanoshots to
be causally connected, the signal must appear to move superluminally on the
sky, by a factor of roughly $10^{4}$. This is possible if the shots represent
bursts arising from a single source that emits bursts (or causes bursts to be
emitted) as it travels almost towards the observer at $\gamma\sim 10^{4}$.

The above recovers the conclusions in the main text, perhaps not surprising
since neither mathematically nor physically can one distinguish between a blob
moving relativistically that is emitting a banded spectrum or multiple
nanoshots. An implication in either case then is that the physical separation
along the line of sight is a factor $\gamma$ larger than that on the sky, or a
factor $\gamma^{2}$ relative to the observed time delay, i.e., about
$10^{8}\times 20\,\hbox{lt-ns}\sim 2\,\hbox{lt-s}$ (or about 400 times the
light cylinder radius $cP/2\pi\simeq 1600{\rm\,km}$).

Instead of requiring the emission region of different nanoshots to be
separated by $\sim 100\,$km, alternatively, we could also consider nanoshots
to be echos of a single impulse. Here, the picture would be that the light
emitted at an angle $\delta^{\prime}$ away from the LOS encounters a large
overdensity region of pulsar wind $d^{\prime}$ away from the emission region,
which scatters it. Similar to the calculation above, to explain the frequency
widths of the bands, the scattered images, referred to as echoes, have to be
delayed by $\tau^{\prime}=\frac{1}{2}d^{\prime}\delta^{\prime 2}/c\sim
20{\rm\,ns}$. In order for the nebular screen to resolve the different echoes
in the scattering tail, they have to be separated by
$d^{\prime}\delta^{\prime}\sim 100{\rm\,km}$ during the first scattering. This
yields $d^{\prime}\sim 2\,\hbox{lt-s}$ and $\delta^{\prime}\sim
10^{-4}{\rm\,rad}=20{\rm\,arcsec}$.

It is not obvious whether it is possible to produce such echoes in the pulsar
wind, especially given the observation that giant pulses are composed of
multiple nanoshots also at high frequency, where scattering would be less
effective. However, there are observational differences between this scenario
and the Doppler shift scenario proposed in the main text. In particular, in
this scenario, Doppler boosting does not play a role and hence one does not
expect variations in the scattering tails of individual burst. This unlike for
the Doppler shift scenario, where especially at low frequencies one expects to
occasionally observe a strongly boosted tail or a drop in flux corresponding
to the scattered light coming from outside the relativistic beam.