Bright Giant Pulses from the Crab Nebula Pulsar: Statistical Properties, Pulse Broadening and Scattering due to the Nebula

0.41 MB·English

by  N. D. Ramesh Bhat

DraftversionFebruary2,2008
PreprinttypesetusingLATEXstyleemulateapjv.16/07/00
BRIGHT GIANT PULSES FROM THE CRAB NEBULA PULSAR: STATISTICAL PROPERTIES,
PULSE BROADENING AND SCATTERING DUE TO THE NEBULA
N. D. Ramesh Bhat1, Steven J. Tingay2, and Haydon S. Knight1
1CentreforAstrophysics&Supercomputing, SwinburneUniversity,Hawthorn,Victoria3122,Australia
2DepartmentofImagingandAppliedPhysics,CurtinUniversity,Bentley,WesternAustralia,Australia
Draft version February 2, 2008
ABSTRACT
We report observations of Crab giant pulses made with the Australia Telescope Compact Array and
a baseband recorder system, made simultaneously at two frequencies, 1300 and 1470 MHz. These
observations were sensitive to pulses with amplitudes >∼3 kJy and widths >∼0.5 µs. Our analysis led to
the detection of more than 700 such bright giant pulses over 3 hours, and using this large sample we
8 investigatetheiramplitude,width,arrivaltimeandenergydistributions. Thebrightestpulsedetectedin
0 ourdatahasapeakamplitude of∼45kJyandawidthof∼ 0.5µs,andthereforeaninferredbrightness
0 temperature of∼1035 K.Thedurationofgiant-pulseemissionistypically∼1µs,howeveritcanalsobe
2
as longas 10µs. The pulse shape ata hightime resolution(128ns)showsrichdiversityandcomplexity

n in structure and is marked by an unusually low degree of scattering. We discuss possible implications
a for scattering due to the nebula, and for underlying structures and electron densities.
J
Subject headings: pulsars: general – pulsars: individual (Crab pulsar) – ISM: structure – ISM:
2
individual (Crab Nebula) – scattering

]
h 1. introduction 2003). Given such extreme short durations, giant pulses
p are best studied using data from baseband observations.
- The Crab Nebula pulsar B0531+21 is well known
Such data allow coherent dedispersion of voltage samples
o for its emission of giant radio pulses, and was origi-
r nally discovered through the detection of such pulses to remove the deleterious effect of interstellar dispersion,
t therebyyieldingmoreaccuratedescriptionsofpulsestruc-
s (Staelin & Reifenstein 1968). These sporadic, large-
a ture, shape and amplitude. However, the bulk of the ob-
amplitude and short-duration bursts can be hundreds or
[ servational studies of the Crab until the early 2000s were
thousandsoftimesmoreenergeticthanregularpulses(e.g.
carried out using traditional filterbank or spectrometer
1 Lundgren et al. 1995), and they remain one of the most
v enigmatic aspects of pulsar radio emission. While several data, due to limitations of data throughput and comput-
4 ing. With the adventofwide-bandwidthrecordersandaf-
other pulsars are now known to emit giant pulses (e.g.
3 fordable high-power computing, these limitations are be-
Cognard et al. 1996; Johnston et al. 2004; Knight et al.
3 ing gradually overcome. As a result, baseband observa-
2006), only 3 objects, viz. the Crab and the millisec-
0 tionsareincreasinglyemployedingiantpulsestudies(e.g.
ond pulsars B1937+21 and J1823–3021A, generate giant
1. pulses numerous enough to allow detailed studies of their Soglasnov et al. 2004; Popov & Stappers 2007).
0 In this paper, we report our observations of Crab giant
characteristics.
8 pulsesmadeusingtheAustraliaTelescopeCompactArray
Observations so far have unravelled several fundamen-
0 (ATCA)andthebasebandrecorderrecentlydevelopedfor
tal properties of giant pulse emission. It is now fairly
: theAustralianLongBaselineArray. Wedetectedover700
v well established that the fluctuations in their ampli-
i tudes are due to changes in the coherence of the radio giant pulses with amplitudes >∼3 kJy and widths ∼0.5 to
X 10 µs, and using this large sample we investigate aspects
emission (Lundgren et al. 1995), and that they are su-
r such as the pulse amplitude, width, arrival time and en-
a perpositions of extremely narrow nanosecond structures
ergy distributions, as well as details of the pulse structure
(Hankins et al. 2003). Observations also suggest that the
and shape. Our observations show a much lower degree
emission is broadband, extending over several hundreds
of scattering than reported before and we discuss possible
of MHz (Sallmen et al. 1999; Popov et al. 2006). Another
implications for scattering due to the nebula.
important characteristic of giant pulses is their tendency
tooriginateinaverynarrowphasewindowofregularradio
2. observations and data processing
emission; and for the Crab, these windows even coincide
withthephasesofthehigh-energy(frominfraredtoγ-ray) 2.1. Observations with the ATCA
emission (Moffett & Hankins 1996). Finally, the distribu-
Observations of the Crab pulsar were made in January
tionofgiantpulseenergiesisknowntofollowapower-law
2006. Although the array offers maximal sensitivity for
form (Argyle & Gower 1972; Lundgren et al. 1995), much
pulsar observations in its tied-array mode, our observing
in contrast to the Gaussian or exponential distribution
set-up was designed also to meet certain technical goals
typical of regular pulses (e.g. Hesse & Wielebinski 1974).
in addition to pulsar observations, and offered a sensitiv-
Of all the giant-pulse–emitting pulsars, the Crab is the
ity equivalent to that of a single ATCA antenna. In this
most well-studied. It is known to emit pulses as ener-
set-up, the array was configured to record two indepen-
getic as ∼ 104 times regular pulses (Cordes et al. 2004)
dent IF (intermediate frequency) signals, each 32 MHz in
andshowsstructurespersistingdownto2ns,withinferred
bandwidth,fromasingleantenna. Attheantenna,two64-
brightness temperatures as high as 1037 K (Hankins et al.
MHz bands were Nyquist-sampled at 128 MHz, with four
1
2 Bhat et al.
operations, such as coherent dedispersion, detection and
search for giant pulses, were performed at the Swinburne
University of Technology Supercomputing Facility.
2.2. Data Decoding and Dedispersion
Datasamplesfromboththepolarisationchannelsofthe
two IF bands were organised into a single data stream
while recording, and were subsequently decoded into sep-
arateIFdatastreamsbeforede-dispersionandfurtherpro-
cessing. As our processing software is designed to handle
a single data stream at a given time, this meant two sep-
arate passes for processing each short block of data. The
unpackeddatafromthisstagewerethendedispersedusing
thecoherentdispersionremovaltechniqueoriginallydevel-
opedbyHankins(1971). Theprocedureswehaveadopted
are similar to those described in Knight et al. (2006) and
Bhat et al.(2007). VoltagesamplesarefirstFouriertrans-
formed to the frequency domain and the spectra are di-
vided into a series of sub-bands. Each sub-band is then
multipliedbyaninverseresponsefilter(kernel)fortheISM
(e.g.Hankins & Rickett1975;van Straten2003),andthen
Fouriertransformedbackto the time domainto construct
a time series with a time resolution coarser than the orig-
inal data. By splitting the input signal into several sub-
bands (four in our case), the dispersive smearing is essen-
tiallyreducedtothatofanindividualsub-band. Thisalso
means the procedure uses shorter transforms than single-
channel coherent dedispersion and is consequently a more
computationallyefficientmethod. Thecoherentfilterbank
stream data obtained in this manner are then square-law
detected, corrected for dispersive delays between the sub-
bands, and summed in polarisation to construct a single
coherently dedispersed time series for the entire band.
2.3. Search for Giant Pulses
Following the procedures of decoding and dedispersion,
Fig. 1.— Example plots illustrating our giant pulse detection. weperformedarigoroussearchforgiantpulseswithineach
Top: Dedispersed time series of a short data segment around the 10sblockofdata. Ourpulsedetectionprocedureinvolved
giant pulse. Middle panels: Diagnostic plots from the giant pulse progressive smoothing of time series with matched filters
search;plotsofsignal-to-noiseratiovstrialdispersionmeasureand
of widths ranging from 0.5 to 16 µs in steps of 0.5 µs,
timeresolution,anddedispersedpulsesoverfour8-MHzsub-bands
ofthe32MHzrecorderbandwidth. Bottom: Thededispersedpulse and identifying the intensity samples that exceed a set
overatimewindowof33ms(i.e. onerotationperiod). threshold (e.g. 12σ for the 0.5 µs smoothing time). In
additiontoperformingdedispersionattheCrab’snominal
DM of 56.75 pc cm−3, we also performed this procedure
bitspervoltagesample,fortwoorthogonallinearpolarisa- overalargenumberofadjacentDMvalues (typicallyover
tions. These data were convertedto circular polarisations a DM range ≈ 0.5 pc cm−3, in steps of 0.001 pc cm−3).
◦
byinsertinga90 phaseshifterinthesignalpath. Thesig- For each DM and the matched filter width, we computed
nalswerethendigitallyfilteredtoprovidetwo-bitsampled thesignal-to-noiseratio(S/N)ofthepulseamplitudeover
voltagetimeseriesforthe32MHzbandwidth. Thisresults a short stretch of data centred at the pulse maximum.
inanaggregatedatarateof512megabitspersecond. The Fromthis analysis,diagnosticplotsaregeneratedforeach
system provided flexibility to record either two separate candidate giant pulse as shown in Fig. 1. These plots are
frequency bands from a single antenna, or two identical subjectedtoa carefulhumanscrutinyto discriminatereal
frequency bands from two different antennas. The pulsar giant pulses from spurious signals.
observations were made over a total duration of 3 hours;
thefirsthalfofwhichusedtwo32-MHzbands(dualpolar- 2.4. Summary of Detections
isation) centred at frequencies 1300 and 1470 MHz, while
We adopted a rather conservative threshold of 12σ in
theremaininghalfusedtwoidenticalfrequencybandscen-
order to account for possible departures of noise statis-
tred at 1300 MHz from two different antennas. This ac-
tics from those expected of pure white noise and also to
cumulated a total data volume of 5.5 terabytes, but split
limitthenumberofcandidatesignalstoareasonablenum-
intoalargenumberofshortblocksof10sdurationeachto
ber. With this threshold and the 0.5 µs final time reso-
facilitate further analysis and processing. The data were
lution (∆t) employed in our analysis, a pulse will need
stored onto disks for offline processing and all processing
to be >∼3 kJy in amplitude to enable a clear detection
Bright Giant Pulses 3
(see § 2.5). We detected 706 giant pulses from our 3-hr
long observation, of which 413 are from the first half of
observations that were made simultaneously at 1300 and
1470MHz. However,only 70%of these 413 were detected
in both frequency bands. This fraction is similar to that
reported by Sallmen et al. (1999) from their observations
at 600 and 1400 MHz, although quite different fractions
have been reportedfrom observationsat widely separated
frequencies (Kostyuk et al.2003) andat lowerfrequencies
(Popov et al. 2006). This probably suggests that not all
pulsesareentirelybroadband,acharacteristicpresumably
intrinsic to the giant-pulse emission phenomenon. Our
sample includes pulses with widths rangingfrom0.5to 10
µs. Given ∆t = 0.5 µs, any shorter-duration pulses that
maybepresentinourdatawillbesmoothedtoaneffective
resolution1 of 0.5 or 1 µs. Moreover, pulses broader than
10 µs are likely to be weaker than our detection thresh-
old. The strongest pulse in our observations is 0.5 µs in
duration and has a S/N of 217. Fig. 2.—Histogramsofgiant-pulseamplitudesat1300and1470
MHz (in red and green respectively) constructed from our sample
2.5. System Sensitivity and Flux Calibration ofgiantpulses. Estimatesforthebestfitslope(β)are−2.33±0.14
and−2.20±0.18at1300and1470MHzrespectively,andareshown
The Crab Nebula is a fairly bright and extended source asshort-dashedandlong-dashedlines. Thebrightestpulsedetected
in the radio sky, with a flux density of ∼955ν−0.27 Jy inourdata(at1300MHz)hasapeakamplitudeof∼45kJy.
(Bietenholz et al.(1997);whereνisthefrequencyinGHz)
and a characteristic diameter of ∼5′.5. Thus, in general,
As a result, oftentimes measurements tend to underesti-
there will be a significant contributionfrom the nebula to
matethetrueamplitudes. ObservationsofLundgren et al.
thesystemnoise(S),dependingonthefrequencyofobser-
(1995)andCordes et al.(2004)(at430MHz)weremarked
vationandthe coverageofthenebulawithinthetelescope
by significant instrumental and dispersion smearing (315
beam. However, in our case, the problem is much simpli-
and 152 µs respectively) and were more prone to scintil-
fied as the nebula is unresolvedby a single antenna of the
array (half-power beam width ≈ 33′ at our observing fre- lation and scattering given their observing frequencies <∼1
GHz. On the other hand, smearing due to instrument or
quencies). In fact, the entire nebular region occupies only
residual dispersion is negligibly small for baseband obser-
asmallfraction(≈3%)ofthe beamsolidangle. Measure-
vations, and consequently our data enable a better and
ments madein parallelwiththe observationsyieldsystem
more accurate characterisation of the pulse amplitudes.
temperatures (T ) of 114 ± 3 and 103 ± 2 K respec-
sys
Fig. 2 shows the distributions of pulse amplitudes for our
tively at 1300 and 1470 MHz. Thus, assuming a nominal
gain(G) of≈ 0.1KJy−1fora single ATCA antennaatL- data.
The cumulative energy distribution provides a mean-
band, these measurements translate to system equivalent
ingfulwayofcharacterizingthefrequencyofoccurrenceof
flux densities (S =T /G) of 1140 and 1030 Jy respec-
sys sys
GPs. FollowingKnight et al. (2006), we define this distri-
tively at 1300 and 1470 MHz. Scaling for our processing
butionintermsoftheprobabilityofapulsehavingenergy
parameters and the recording bandwidth (∆B), and also
greater than E , and can be expressed as
accountingforthelossofS/Nduetoour2-bitdigitisation, 0
theseestimateswillcorrespondtosystemnoiseof250and P(E >E )=KEα (1)
0 0
227 Jy at the 1-σ level2. In other words, minimum de-
Figure3showssuchadistributionforourdata. Thepulse
tectable pulse amplitudes of 3 and 2.7 kJy respectively at
energyisestimatedbyintegratingtheamplitudebinsover
1300 and 1470 MHz for a 12-σ threshold.
the extent of emission. The uncertainties in the energy
estimates depend on the pulse strength and width, and
3. statistical properties of giant pulses
may range from ∼0.5% for strong and narrow pulses, to
3.1. Pulse Amplitudes and Energies asmuchas∼35%forweak,broadpulses. Suchlargeuncer-
taintiesatlow energies,andpossiblemodulations inpulse
An important distinguishing characteristic of giant
amplitudes due to scintillation, may probably explain a
pulses (GPs) is their power-law distribution of ampli-
gradualflattening seenat low energies. In any case, much
tudes and energies. The amplitude determination can be
inagreementwiththe earlierwork,noevidenceis seenfor
potentially influenced by factors such as smearing due to
a high-energy cut-off, suggesting that exceedingly bright
instrumentandanyresidualdispersion,aswellasexternal
andenergeticpulsesarepotentiallyobservableoverlonger
effects suchasscatteringandscintillationdue to the ISM.
durations of observation.
1
Excluding the segments near the low and high ends of
In practice, pulses that are narrower than 0.5 µs will be detected
thedistributionwherethebehaviourtendstodepartfrom
as either 0.5 µs wide (most flux in one sample) or 1 µs wide(pulse
comes injustontheborderoftwosamples)inouranalysis. a power-law, we obtain a best-fit value of −1.59 ± 0.01
2 for α (where the error is purely formal) and K = 3.97.
S=ηNσTsysG−1(npol∆B∆t)−1/2,whereη isthelossofS/Ndue Interestingly, this slope is comparable to the published
to digitisation, Nσ is the detection threshold in units of σ, npol is values for other GP-emitters such as PSRs J0218+4232
thenumberofpolarisationssummed.
4 Bhat et al.
0.01 ∼1000 times brighter than the brightest pulse detected
in the Arecibo observations of Cordes et al. (2004). To
the best of our knowledge, this marks the brightest pulse
everrecordedfromtheCrabpulsarwithinthe L-bandfre-
0.001 quency range (i.e. ∼1–2 GHz). Detection of such exces-
sively bright giant pulses offers a promising technique for
finding pulsars in external galaxies (Johnston & Romani
2003; Cordes et al. 2004).
0.0001 3.2. Pulse Widths
There have been very few attempts of characterising
giant-pulse widths. The first analysis by Lundgren et al.
(1995) was limited by their coarse time sampling (205
µs) and was further constrained by a severe dispersion
22 4 6 8 2200 40 60 80
smearing (70 µs). More recent work of Popov & Stappers
(2007) had a much higher time resolution (4.1 µs). As
Fig. 3.—Cumulativedistributionofgiant-pulseenergiesat1300
MHz. The slope estimates of the linear segments below and above our observations were made at a higher frequency than
an apparent break near ∼10kJy µs are –1.4and –1.9 respectively, the above(hence the effect ofscattering is reduced)and a
whiletheoverallslopeis–1.6. finaltime resolution(0.5 µs)that is eight-foldbetter than
Popov & Stappers (2007), our analysis allows a more ac-
curate characterizationof giant-pulse widths.
(Knight et al.2006),B0540–69(Johnston et al.2004)and
Our observations show that many giant pulses tend to
B1937+21 (Cognard et al. 1996; Soglasnov et al. 2004).
have a significant structure at 0.5 µs resolution, ranging
However, it is known from the work of Moffett (1997)
from a simple narrow spike to severalclosely spaced com-
thatasingleslopedoesnotaccuratelydescribetheenergy
ponentswithinafewµs. FollowingLundgren et al.(1995)
distribution at 1400 MHz and that there occurs a break
andPopov & Stappers(2007),weadoptthenotionof“ef-
aroundanenergyof∼2kJyµs,wheretheslopeαchanges
fectivepulsewidth”W ,whichisessentiallytheaveraging
from −3 to −1.8. Given that this energy corresponds to e
time that yields the maximum S/N. In most cases, it is a
the detection threshold of our GP search (see § 2.5), our
close representation of true pulse width, although it may
slope estimate of −1.6 is in general agreement with Mof-
be slightly biased to stronger component(s) in the case
fett’s work. RecentworkofPopov & Stappers(2007)sug-
of highly structured pulses. The histogram of widths ob-
gests that the slope estimate depends on the pulse width,
tainedin this manneris shownin Fig.4. The distribution
evolvingfrom−1.7to−3.2whengoingfromtheirshortest
is highly skewed, with measured widths ranging from 0.5
(4 µs) to longest (65 µs) GPs. Their analysis also shows
to 10 µs and with a clear peak at 1 µs. Giant pulses of
a clear evidence for a break where the power-law tends
widths larger than 10 µs are absent in our data; however,
to steepen (their Fig. 2). Such a break is also apparent
this may be a selectioneffect as suchpulses may verywell
in our data, as the slope tends to steepen near ∼10 kJy
be below our detection threshold. Our analysis confirms
µs. Adoptingthisasthebreakpoint,weobtainslopeesti-
a general tendency for stronger pulses to be narrower, an
mates of −1.37 ± 0.01and −1.88 ± 0.02for the segments
observation also noted by Popov & Stappers (2007).
below and above it (K = 3.72 and 4.42 respectively).
Our distributionofgiant-pulsewidths canbe compared
The pulse energy distribution also serves as a useful
with a similar plot obtained for GPs from the millisecond
guideforestimatingtheratesofoccurrenceofgiantpulses.
pulsar PSR B1937+21 by Soglasnov et al. (2004). Simi-
Forinstance,onthebasisofourobservations,apulsethat
is about 50 times more energetic than regular pulses3 can larities in the two distributions are quite striking, despite
the fact that GPs from PSR B1937+21 are intrinsically
be expected roughly once in 8500 pulse periods, i.e., ap-
proximately once every 5 minutes. The most energetic narrower (We<∼15 ns) and show little structure. Thus, it
appearsthatexponential-taileddistributionsareprobably
pulse detected in our data has an estimated energy of 66
kJy µs; i.e. an energy per µs that is almost ∼ 105 times
larger than that of regular pulses.
The brightest pulse detected in our data has an esti-
matedpeak flux density (S )of∼ 45kJyandaneffective
ν
width of 0.5 µs. The equivalent brightness temperature
(T ) is given by (based on the light-travelsize and ignor-
b
ing relativistic dilation),
2
S D
ν
T = , (2)
b
(cid:18)2k (cid:19)(cid:18)νδt(cid:19)
B
where ν is the frequency of observation, δt is the pulse
width, k is the Boltzmann constant and D is the Earth-
B
pulsar distance. The inferred brightness temperature for
our strongest pulse is therefore ∼ 1035 K. This is almost Fig. 4.— Left: Distributions of giant-pulse widths at 1300 and
1470 MHz (in red and green respectively). Right: Plot of pulse
3 amplitudes vs the effective pulse widths. An apparent low cut-off
Assuming mean pulse energies estimated from parameters in the at 0.5 µs and a discretization in pulse width are due to the 0.5 µs
publishedliterature(Manchester etal.2005). finaltimeresolutionadopted forourprocessing.
Bright Giant Pulses 5
common characteristics of giant-pulse widths. Studies of
more pulsars are necessary to confirm such a conjecture.
1000
3.3. Arrival Times and Phases
OneofthemoststrikingpropertiesofCrabGPsistheir 500
occurrencewithinwell-defined,relativelynarrowlongitude
40 42 44 46 48 50
rangesofregularradioemission. Thisisalsotruewiththe
0
millisecond pulsars PSRs B1937+21and J0218+4232,ex-
0 20 40 60 80 100
cept that in the case of the former, GPs tend to occur
near the trailing edges of the main pulse and interpulse
(Soglasnov et al. 2004), and in the latter case, they oc- 1000
cur near the two minima of the integrated pulse profile
(Knight et al. 2006). In order to determine the pulse ar-
rivaltimes,weusethepulsar’sspin-downmodelalongwith 500
theTEMPOsoftwarepackagetoobtaintheanticipatedpulse
40 42 44 46 48 50
arrival phases. As most pulses in our data are very nar-
row, our data allow precise determinations of the arrival 0
times andphases. Pulse times-of-arrival(TOAs)obtained 0 20 40 60 80 100
in this manner are plotted in Fig. 5.
Fig. 6.—Abrightgiantpulseat1300and1470MHz,wherethe
The total range of longitudes where GPs occur (in the
mainpulse region)is approximately±200µs,or±2◦.2 in plotted time window corresponds to ≈0.003 cycles of the pulsar’s
rotationperiod. Thispulsepairisdisplayedatatimeresolutionof
angular units, with an RMS of 84 µs (0◦.9). Thus, a vast 128nsandillustratessignificantdifferencesseeninpulsewidthand
majority of GPs (75%) occur within a narrower window structuraldetailsbetweenthetwofrequencies.
of ±100 µs. As contributions from the dispersive and in-
strumental smearing are negligible in our case, this RMS
excellent agreement with the published values in the lit-
corresponds essentially to the intrinsic pulse-phase jitter
erature (Popov & Stappers (2007): 84% and 16% respec-
and can be compared to a value of 90 µs estimated by
tively at 1200 MHz; Kostyuk et al. (2003): 86% and 14%
Lundgren et al.(1995)fortheirdataat800MHz. Theplot
respectively at 2228 MHz).
of joint statistics oftiming residuals andpulse amplitudes
showsnoobviouscorrelationexceptforageneraltendency
4. pulse broadening and scattering due to the
forstrongerpulsestooriginatewithinnarrowerphasewin-
nebular plasma
dows. A similar property was also noted by Cordes et al.
(2004) in their Arecibo observations at a frequency of 430 4.1. Pulse Shape and Estimation of Scattering
MHz. Finally,avastmajorityofGPstendtooccurwithin
Figure 6 shows an example of a bright giant pulse from
the main pulse window − 87% at the main pulse region
our data. At a time resolution of 128 ns, the pulse is
andthe remainder13%inthe interpulseregion. This isin
resolvedintomultiplenarrowcomponents,andthisreadily
confirmsthebasicpictureofGPscomprisingfinestructure
on veryshort timescales. Discerning such fine structure is
(a) however limited by time resolution and the smearing due
5000 to multipath scattering. This pulse pair also exemplifies
the differences seen in pulse structure between 1300 and
1470MHz,adetailedanalysisofwhichisbeyondthescope
of the present work. Our pulse shapes can be compared
4000
to observations of Sallmen et al. (1999) ∼10 yr ago (their
Fig. 1),anditisstrikingthatourobservationsaremarked
by a much lower degree of scattering.
3000 In order to estimate the pulse broadening due to inter-
stellar scattering (e.g. Williamson 1972; Cordes & Lazio
2001), we adopt the CLEAN-based deconvolution ap-
proach developed by Bhat et al. (2003). Unlike the tradi-
2000
tionalfrequency-extrapolationapproach(e.g.Lo¨hmer et al.
2001;Kuzmin et al.2002),thismethodmakesnoprioras-
sumption of the intrinsic pulse shape, and thus offers a
1000 more robust means of determining the underlying pulse
broadening function (PBF). The procedure involves de-
convolving the measured pulse shape in a manner quite
similar to the CLEAN algorithm used in synthesis imag-
0
-200 0 200 ing, while searching for the best-fit PBF and recovering
the intrinsic pulse shape. It relies on a set of figures of
Fig. 5.—Arrivaltimesofgiantpulsesdetectedinthemainpulse meritthataredefinedintermsofpositivityandsymmetry
and interpulse regions (the red and green symbols respectively) at of the resultant deconvolved pulse and some parameters
1300 MHz (left), along with their statistics (right, top) and joint
characterizing the noise statistics in order to determine
statistics withpulseamplitudes(right, bottom).
6 Bhat et al.
measuring the time delay between the pulse arrival times
900
at 1300 and 1470 MHz. This value is in excellent agree-
800
mentwiththatreportedintheJodrellBank(JB)monthly
700
ephemeris (Lyne 1982) on the nearest date of our observ-
600 ing, but it is significantly lower than that measured near
500 the observing epochs of Sallmen et al. (1999).
400 25 30 35 40 Thus, over a 10-yr time span between Sallmen et al’s
300 and our observations, the pulsar’s DM decreased by 0.09
200 pc cm−3while the pulse broadening decreased by a fac-
tor of 5. Given the strong frequency dependence of pulse
100
broadening(τ ∝ν−x)andthe observationalevidence for
0 d
the scaling index x changing overthe time (Kuzmin et al.
0 5 10 15 20 25 30 35 40 45 50 55 60 65
2002; Bhat et al. 2007), it is more meaningful to adopt a
frequency independent parameter such as the scattering
Fig. 7.— A bright giant pulse from observations at 1300 MHz.
measure (SM) for comparison purposes. This parameter
The time window corresponds to 0.002 cycles of the pulse period.
The reconstructed pulse (green) and the best-fit pulse-broadening quantifiesthetotalscatteringalongthelineofsight(LOS)
function (inset) from our deconvolution procedure are also shown. and is defined as the LOS integral of C2, which is the
Thee−1pointofthisone-sidedexponentialgivesanestimateofthe spectral coefficient of the wavenumber spnectrum of elec-
pulse-broadeningtime,forwhichourmeasuredvalueis0.8±0.4µs.
tron density irregularities. It can be related to τ via the
d
relationτ ≈1.1W SM6/5ν−xD,whereν is inGHz, Dis
d τ
the best-fit PBF. For the purpose of our analysis, we as- in kpc, and W is a geometric factor that depends on the
τ
sume the simplest and most commonly used form for the LOS-distributionofscatteringmaterial(Cordes & Rickett
PBF that corresponds to a thin-slab scattering screen ge- 1998). Assuming W = 1, we can estimate the effective
τ
ometry. ThefunctionalformforsuchaPBFisaone-sided SM for a uniform medium, and for the measurements of
exponential (e.g. Williamson 1972) and is given by Sallmen et al’s and ours, we obtain values of 1.3×10−2
1 −t and 3× 10−3 kpc m−20/3respectively. That is, the SM
G(t)= exp U(t), (3) changedby0.01kpc m−20/3whentheDMchangedby0.09
(cid:18)τd(cid:19) (cid:18)τd (cid:19) pc cm−3, or a factor 4 decrease in the scattering strength
where U(t) is the unit step function, U(< 0) = 0,U(≥ associated with a DM change of only 0.16%.
0) = 1. Fig. 7 shows the best-fit PBF and the recon- AcloserlookattheJBephemerisrevealsquiteasystem-
structed pulse obtained in this manner. The e−1 point of atic decrease in the Crab’s DM variations between 1996
this PBF is the scattering time τd, for which our decon- and2006,withthelowestDMrecordednearmid-2005and
volution procedure yields an estimate of 0.8±0.4 µs. a reversal of the trend in early 2006. A plot of SM esti-
This measurement of a low scattering is further sup- matesfortheavailableτ measurementsduringthisperiod
d
ported by the following observational facts. First, a pulse isshowninFig.8alongwiththeDMmeasurementsatrel-
broadeningof∼0.8µswillimplyrapidintensitydecorrela- evantepochs. Theseplotsillustrateagradualreductionin
tions infrequencyoncharacteristicscales∼1/2πτd ∼ 0.1 the Crab’s DM and scattering over this 10-yr time span.
to 0.3MHz andthis is confirmedby ouranalysis. Second, Such variations are too large and smooth to be caused by
as seen in Fig. 4, a large number of pulses detected in our refractivescintillationeffectsintheISM,andaretherefore
datahavewidths∼1µs,justaboutwhatwewouldexpect indicative of a nebular origin.
given the ∼0.8 µs broadening due to scattering and the The material within the Crab Nebula, specifically the
0.5 µs time resolution. perturbed thermal plasma associated with it, has often
been advanced as the source of excessive scattering and
4.2. Scattering due to the Nebula
anomalous DM variations on several occasions. The most
Our measurement of τ can be compared to that re- remarkableobservationsinsupportofthis arethe anoma-
d
ported by Sallmen et al. (1999) based on their observa- lous scattering recorded in 1974-1975 (Lyne & Thorne
tions in 1996. Their value of 95 ± 5 µs at 600 MHz 1975; Isaacman & Rankin 1977) and the reflection event
would scale to 4.3 µs at 1300 MHz, assuming a canon- in 1997 (Backer et al. 2000; Lyne et al. 2001). The scat-
ical ν−4 dependence expected from scattering due to a tering event in 1974 was especially noted for its extreme
turbulentplasmascreen. Thisisfivetimeslargerthanour activity, where the pulsar’s DM rose by 0.07 pc cm−3and
measurement. A somewhat smaller value is expected on the scattering increased by an order an magnitude over a
the basis of observations of Kuzmin et al. (2002) in the time span of several months. Isaacman & Rankin (1977)
year 2000. However, our measurement is consistent with ascribed this to a two-component scattering-screenmodel
that of Bhat et al. (2007) at 200 MHz, if we extrapolate wherethe variablescreenassociatedwiththe nebula gives
their measurements using their revised frequency scaling risetosuchrapidchangesinbothDMandscattering. The
of ν−3.5±0.2. anomalous dispersion event in 1997 was seen as discrete
As described in § 2.3, our GP detection procedure in- moving echoes of the pulse, and was interpreted as reflec-
cludesdeterminationofthebestDMbyperformingdedis- tionsfromanionizedshellintheouterpartsofthenebula
persion over many trial values around its nominal value. by Lyne et al. (2001), and in terms of variable optics of a
As most pulses in our data are very narrow, this proce- triangularprismlocatedintheinterfaceofthe nebulaand
dureallowsaprecisedeterminationoftheCrab’strueDM the supernova ejecta by Backer et al. (2000).
at our observing epoch. An estimate of 56.751± 0.001 Fig. 8 reveals a similar change in DM and a large but
pc cm−3obtained in this manner is further confirmed by
Bright Giant Pulses 7
parameteris essentiallya productofnormalizedvariances
(at small and large scales) and other terms such as the
outer scale and filling factor. The electron density n is
e,cn
unknown, but assuming a nominal value of 1 cm−3, we
getF =55,whichismuchlargerthanthattypicalofthe
cn
Galactic spiral arms (F ∼10).
ThelargevaluesofδSMandF indeedconfirmthema-
cn
terialwithinthenebulaasthesourceofexcessivelystrong
scattering. While the scattering and dispersion events of
1974 and 1997 were interpreted in terms of a single large
structure with electron density ∼1,500 cm−3, the long-
term systematic variations in dispersion and scattering as
MJD - 50000 shown in Fig. 8 can be interpreted in terms of a scenario
whereby the nebular segment of the LOS is populated by
many smaller structures of much lower densities. A varia-
tioninthenumberdensityofsuchstructuresmaythenac-
countfortheobservedchangesinDMandSM.Astheneb-
ula is thought to comprise fine structure on many length
scales, perhaps even on scales much finer than the fila-
mentary structure suggested by optical observations (e.g.
Hester et al. 1995), such a picture seems quite plausible.
For the sake of simplicity, if we model the measured
changes in scattering and dispersion to arise from N such
structuresofsizeδs anddensityn ,theresultantcon-
sc e,sc
MJD - 50000 tributions to SM and DM are given by
Fig. 8.— Measurements of DM and SM of the Crab pul- i=N
sar from observations over the past 10 years (1996 to 2006). δSM ≡ SM =NC F n2 δs (5)
The SM values (upper panel) are estimated from the published sc,i SM sc e,sc sc
pulse-broadening measurements and decorrelation bandwidths – Xi=1
1996.4: Sallmenetal.(1999);2000.6: Kostyuketal.(2003);2001.5: i=N
Kuzminetal. (2002); 2002.1, 2002.4: Cordesetal. (2004); 2003.9:
δDM ≡ DM =Nn δs (6)
Popovetal. (2006); 2005.6: Bhatetal. (2007); 2006.0: our obser- sc,i e,sc sc
vations. TheDMmeasurementsarefromtheJodrellBankmonthly Xi=1
ephemeris (lower panel); the thick solid circles correspond to the Followingequations(4)–(6)andtheconstraintF n =
cn e,cn
observingepochs oftheSMmeasurements.
55, the electron density of such a structure can be esti-
mated as
δSM 1
lessdramaticchangeinscattering. Adirectinterpretation n = (7)
e,sc
ofsuchobservationscanbe made alongthe linesofavari- (cid:18)55CSM (cid:19)(cid:18)Nδssc (cid:19)
able scatteringscreenmodel,aswediscussindetailinthe Thus,withjusttwodirectmeasurementsalone(δDMand
following section. δSM), it is hard to constrain all 3 free parameters of the
model. However, assuming a reasonable value for δs ∼
sc
4.3. Implications for the Nebular Structure and Densities 10−5 pc (i.e. an order of magnitude smaller than that
implied by the 1997 reflection event), we estimate n ∼
The observed changes in DM and SM can be used to e,sc
100cm−3forN∼100. Thisisalmostanorderofmagnitude
constrain the combination of the electron density and the
smaller than the densities required to produce the reflec-
“fluctuation parameter” in the nebular region, denoted
tion event. Indeed, several different combinations of size
as n and F respectively. Following Cordes & Lazio
e,cn cn
and density are possible; nonetheless, the underlying pic-
(2003) and Bhat et al. (2004), the measured decrements
ture is the presence of many moderately-dense structures
in DM and SM can be expressed as δDM = n δs
and δSM = C2 δs, where δsis the size of thee,cnneb- in the nebular region, with a filling factor ∼ 10−3.
n,cn We note that the electron density estimated above, es-
ular scattering region and C2 is the equivalent turbu-
n,cn timates of the electron temperature in the Crab nebula
lent intensity (assuming a uniform distribution of scat-
(Temim et al. 2006; Hester et al. 1995; Davidson 1979)
tering material). The parameters F and C2 can be
cn n,cn thatyieldvaluesbetween6000and16000K,andthesizeof
related as (Taylor & Cordes 1993; Cordes & Lazio 2002) theCrabnebula(approximately6′=3pc,foradistanceof
Cn2,cn = CSMFcnn2e,cn, where CSM = [3(2π)1/3]−1Ku 2 kpc), imply that the nebula should produce a minimal
foraKolmogorovspectrum,andK =10.2m−20/3cm6 to optical depth to free-free absorption at GHz frequencies
u
yield SM in units of kpc m−20/3. The above expressions and a significant optical depth at lower frequencies. We
canbecombinedtoyieldtheratioofδSMandδDM,and estimate, basedonthese parameters,that a free-free opti-
is given by cal depth (τff) of 0.007 at 1 GHz may be possible, along
lines of sight that include the scattering structures dis-
δSM
=C F n (4) cussed above. For radio sources behind the Crab nebula,
SM cn e,cn
δDM
this would give a decrease in the observed flux density,
Thus our measurements of δSM= 0.01 kpc m−20/3 and comparedtotheintrinsicfluxdensity,of0.7%duetofree-
δDM=0.09pc cm−3yieldF n =55. Thefluctuation freeabsorptionatthisfrequency. Atlowerfrequenciesthis
cn e,cn
8 Bhat et al.
decrease would be more significant: 3% at 500 MHz, 12% Acknowledgements: The ATCA is part of the Australia
at 250 MHz, and 60% at 100 MHz. Telescope, which is funded by CSIRO for operation as a
If a radio interferometer operating at these frequencies National Facility by ATNF. Data processing was carried
couldachieveanangularresolutionhighenoughtoresolve out at Swinburne University’s Supercomputing Facility.
the Crab nebula and detect compact sources of emission We thank Matthew Bailes and Simon Johnston for fruit-
behind the nebula, it may be possible to survey the free- fuldiscussions,andWillemvanStratenandJorisVerbiest
freeabsorptionduetothenebulaalongmanylinesofsight, foracriticalreadingofthemanuscript. Wealsothankthe
probing the structures that are producing the scattering refereeforacriticalreviewandseveralinspiringcomments
of the pulsar emission. The upcoming future instruments andsuggestionsthat helpedimprovethe presentationand
suchastheextendedLOFARtelescopeinTheNetherlands clarityofthepaper. ThisworkissupportedbytheMNRF
andEurope,ortheMurchisonWide-fieldArray(MWA)in researchgrant to Swinburne University of Technology.
WesternAustraliamaybeabletoundertakesuchasurvey.
5. summary and conclusions
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