The Geometry of PSR B0031-07

0.86 MB·English

by  J. M. Smits

Astronomy&Astrophysicsmanuscriptno.6292 c ESO2008
(cid:13)
February5,2008
−
The geometry of PSR B0031 07
J.M.Smits1,D.Mitra2,B.W.Stappers3,4,J.Kuijpers1,P.Weltevrede4,A.Jessner5,andY.Gupta2
1 DepartmentofAstrophysics,RadboudUniversity,Nijmegen,TheNetherlands
2 NationalCenterforRadioAstrophysics,Pune,India
3 ASTRON,Dwingeloo,TheNetherlands
4 AstronomicalInstitute‘AntonPannekoek’,Amsterdam,TheNetherlands
7 5 Max-Planck-Insitutfu¨rRadioastronomy,Bonn
0
0 Received¡date¿/Accepted¡date¿
2
ABSTRACT
n
a Context.PSRB0031 07 iswellknown toexhibit threedifferent modesofdriftingsub-pulses(mode A,Band C).Ithasrecently
J −
beenshown that inamultifrequency observation, consisting of2700 pulses, alldriftmodeswerevisibleatlow frequencies, while

1 at4.85GHzonlymode-Adriftornon-driftingemissionwasdetected.ThissuggeststhatmodesAandBareemittedinsub-beams,
3 rotatingatafixeddistancefromthemagneticaxis,withthemode-Bsub-beamsbeingclosertothemagneticaxisthanthemode-A
sub-beams.Diffuseemissionbetweenthesub-beamscanaccountforthenon-driftingemission.

1 Aims.Using the results of an analysis of simultaneous multifrequency observations of PSR B0031 07, we set out to construct a
v geometricalmodelthatincludesemissionfrombothsub-beamsanddiffuseemissionanddescribesth−eregionsoftheradioemission
3 ofPSRB0031 07ateachemissionfrequencyfordriftmodesAandB.
−
9 Methods.Basedontheverticalspacingbetweendriftbands,wehavedeterminedthedriftmodeofeachsequenceofdrift.Torestrict
8 themodel,wecalculatedaveragepolarisationandintensitycharacteristicsforeachdriftmodeandateachfrequency.
1 Results.The model reproduces the observed polarisation and intensity characteristics, suggesting that diffuse emission plays an
0 importantroleintheemissionpropertiesofPSRB0031 07.Themodelfurthersuggeststhattheemissionheightsofthispulsarrange
−
7 fromafewkilometerstoalittleover10kilometersabovethepulsarsurface.Wealsofindthattherelationshipsbetweenheightand
0 frequencyofemissionthatfollowfromcurvatureradiationandfromplasma-frequencyemissioncouldnotbeusedtoreproducethe
/ observedfrequencydependenceofthewidthoftheaverageintensityprofiles.
h
p Keywords.Stars:neutron–(Stars:)pulsars:general–(Stars:)pulsars:individual(B0031-07)
-
o
r
t 1. Introduction currence of mode C, they did not attempt to include this drift
s
a modeintheirmodel.
: Pulsar B0031 07 is well known for its three modes of drift- Here we analyse new multifrequency observations of
v −
ing sub-pulses. They are called mode A, B and C and are
i PSR B0031 07, obtained with the Giant Metrewave Radio
X characterised by their values for P3 of 12, 7 and 4 times Telescope, th−e Westerbork Synthesis Radio Telescope and the
r the pulsar period, respectively (Hugueninetal. 1970). This EffelsbergRadioTelescopesimultaneously.Intotal,theobserva-
a pulsar has been thoroughly studied at low observing fre- tionscontain136000pulsesspreadover7differentfrequencies.
quencies (Hugueninetal. 1970; Krishnamohan 1980; Wright
Fromthese observationswe attemptto restrictthe geometryof
1981; Vivekanand 1995; Vivekanand&Joshi 1997, 1999;
thispulsarandcreateamodelthatreproducesagreatnumberof
Joshi&Vivekanand 2000), but only rarely at an observing
itsobservedcharacteristics.
frequency above 1GHz (Wright&Fowler 1981; Kuzminetal.
InSection2weexplainhowtheobservationshavebeenob-
1986; Izvekovaetal. 1993). Recently, Smitsetal. (2005) anal- tained, how the differentmodes of drift have been determined,
ysed simultaneous multifrequency observations from both the
and which furtheranalyses have been carried out. In Section 3
WesterborkSynthesisRadioTelescopeandtheEffelsbergRadio
wepresentourresults,whichisfollowedbythemodellingofthe
Telescope and detected all three drift modes at 325MHz, but
geometryof the emission of PSR B0031 07in Section 4. The
only detected drift mode A at 4.85GHz. The pulses that were −
discussion and conclusionsfollow in Sections 5 and 6, respec-
classified as mode B or C at low frequency only showed non-
tively.
drifting emission at high frequency.On the basis of their find-
ings,theysuggestageometricalmodelwheremodesAandBat
a given frequency are emitted in two concentric rings around 2. Method
the magnetic axis with mode B being nested inside mode A.
2.1.Definitions
This nested configuration is preserved across frequency with
the higher frequency arising closer to the stellar surface com- To describe the observational drift of sub-pulses we use three
pared to the lower one, consistent with the well known radius- parameters, which are defined as follows: P is the spacing, at
3
to-frequencymappingoperatingin pulsars. Due to the rare oc- thesamepulsephase,betweendriftbandsinunitsofpulsarpe-
riods(P );thisisthe“vertical”spacingwhentheindividualra-
1
Send offprint requests to: J.M. Smits e-mail: Roy.Smits@ dio profilesobtainedduringone stellar rotationare plotted one
manchester.ac.uk abovetheother(stacked).P istheintervalbetweensuccessive
2
2 J.M.Smitsetal.:ThegeometryofPSRB0031 07
−
Table1.ListofknownparametersofPSRB0031 07.Allvalues setting non-overlappingfilter banks for each sub-array and al-
−
arefromTayloretal.(1993). lowsthe16MHzbandwidthtobedividedbetweenthedifferent
radiofrequencybandsofobservations,withoutanyoverlapbe-
Parameter Value tweensignalsfromthedifferentbands,whilestillpreservingthe
P 0.94295s time alignmentof the data from the different frequencybands.
1
P˙ 4.083 10 16 The 243 and 610MHz observations involved polarimetry and
−
DM 10.89p·ccm 3 the observationshave been corrected for Faraday rotation, dis-
−
S400 95mJy persionandparallacticangle.Variationsintherotationmeasure,
S1400 11mJy duetointrinsicfluctuationsoftheionosphereduringthelength
EB˙surf 61..3911·0130111erGg/s of one observation causes a deviation in the position angle of
less than 5 at an observationfrequency of 243MHz. Towards
· ◦
higherfrequencies,thisdeviationdecreaseswith λ2. To correct
foraninstrumentalpolarisationeffect,thepolarisationcalibrator
sub-pulseswithinthesamepulse,givenindegrees,and∆φ,the PSR B1929+10was observedatseveralparallacticangles, and
sub-pulsephasedrift,isthefractionofpulseperiodoverwhich
wasusedtocalibratethedatausingthetechniquedescribedby
a sub-pulse drifts in one pulse period, given in /P . Note that
◦ 1 Mitraetal.(2005).Inmostcasesthecorruptiondueto leakage
P = P ∆φ.
2 3 wasfoundtobesmall,andcouldbecorrecteduptoanaccuracy
×
These parameters are often thought to be associated with of5%.
beams of emission (sub-beams), rotating at a fixed distance The WSRT observations were made at a frequency of
around the magnetic axis. When each sub-pulse that is seen to 840MHz with a bandwidth of 80MHz and recorded using the
be drifting in consecutive pulses (which is observed as a drift pulsar back-end, PuMa (Vouˆteetal. 2002). The signals from
band)isduetoemissionfromanindividualsub-beam,thenP3is 14 telescopes were added with appropriate delays resulting in
therotationtimeofthisconfigurationdividedbythenumberof a gain of 1.2K/Jy. The system temperature at this frequency
sub-beams.However,whenthesub-beamsrotatefastenough,it is around T =150K. The Effelsberg observations were made
sys
becomespossiblethatmultiplesub-beamscontributetoonedrift at a frequencyof 4.85GHz and a bandwidth of 500MHz. The
band.Thisiscalledaliasingandishardtodetect.Theobserved gainis1.5K/Jyandthesystemtemperatureatthisfrequencyis
P3 isthennolongertherotationtimedividedbythenumberof Tsys=27K.Thepolarisationcalibrationisdescribedinchapter4
sub-beams,butdependsonthedegreeofaliasing. ofHoensbroech(1999).TheWSRTobservationshavebeencor-
rectedforFaradayrotation,dispersionandforanyinstrumental
polarisationeffectsusingaproceduredescribedintheAppendix
2.2.Observations
ofEdwards&Stappers(2004).
Here we present observations of PSR B0031 07, which were A 50-Hz signal present in the Effelsberg observation has
−
obtained with the Giant Metrewave Radio Telescope (GMRT), been removedby Fourier transformingthe entire sequence, re-
the Westerbork Synthesis Radio Telescope (WSRT) and the movingthe50Hz peakandFouriertransformingback.Table 2
Effelsberg Radio Telescope (EFF). They comprise 7 different liststhedetailsofalltheobservations.Alltheobservationswere
frequencies. A great part of the observations are simultaneous alignedbycorrelatinglongsequencesofpulseswithpulsesfrom
atdifferentfrequenciesandinclude4hoursinwhichradioemis- the observationat607MHz thatwere obtainedsimultaneously.
sionfromPSRB0031 07isobservedat5frequenciessimulta- Thisprocedureremovesanydelayinarrivaltimebetweenpulses
−
neously. atdifferentfrequenciesduetoretardationandaberration.Theac-
The 150, 243, 325, 610 and 1167 MHz observations were curacyofthisalignmentiswithin1ms.
conductedusingtheGMRTlocatedinPune,India.TheGMRT
is a multi-element aperture synthesis telescope (Swarupetal.
2.3.Findingthedriftsequences
1991)consistingof30antennasdistributedovera25-kmdiam-
eter areawhich canbe configuredas a single dish (Guptaetal. Smitsetal. (2005) have shown that for at least one of the drift
2000). The antennas each have a gain of 0.3K/Jy. The system modes the value for P is the same for both high and low fre-
3
temperatureforthedifferentfrequenciesareTs1y5s7 =482K,Ts2y4s3 quencyobservations.Thisisconsistentwiththeconceptthatthe
=177K,T325 =108K,T607 =92KandT1167 =76K.Thesig- emission of pulsarsoriginatesfromsub-beamsof particlesthat
sys sys sys
nalsfromtheantennasat150,243,325and610MHzhavecir- rotatearoundthemagneticaxisduetotheforce-freemotionof
cularpolarisation,whileat1167MHzthesignalsarelinearlypo- the particlesin the strong magnetic field. Here, we have inves-
larised.Atanyfrequency,orthogonallypolarisedcomplexvolt- tigated the values of P for both mode A and mode B over a
3
agesarriveat the samplerfromeach of the antennas.The volt- largerfrequencyrange,usingthe samemethodasdescribedby
age signals are subsequently sampled at the Nyquist rate and Smitsetal.(2005).Fromthe observationswe chose1500con-
processedthrougha digitalreceiversystemconsistingofacor- secutivepulsesforwhichthesub-pulseswerevisibleinatleast
relator,theGMRTarraycombiner(GAC)andapulsarback-end 4 frequencies. We then tested for each drift sequence whether
forGAC.Thesignalsselectedbytheuserareaddedinphaseand the value of P was the same at each frequency. We can con-
3
fedtothepulsarback-end.Thepulsarback-endcomputesboth firm that in these 1500 pulses, within errors, P remains con-
3
theauto-andcross-polarisedpower,whichwasthenrecordedat stant for mode A over the frequencies243, 607, 840MHz and
a samplingrate of 0.512ms. We have carriedoutsimultaneous 4.85GHz.Furthermore,weunexpectedlyfoundmultipledetec-
observationat243,607and1167MHzbysplittingthewholear- tions of mode-B drift at 4.85GHz, whereas Smitsetal. (2005)
rayinto3differentsub-arrays.Wehaveusedaschemewherein foundnomode-Bdriftin2700pulsesat4.85GHz,norin5350
thedigitalsub-arraycombiner,whichdoestheincoherentaddi- pulses at an evenlower frequencyof 1.41GHz. Still, we could
tionofthemulti-channelbasebanddatafromdifferentantennas, notaccuratelymeasure P at 4.85GHz when the pulsar was in
3
isprogrammedtoblankthedataforaselectedsetoffrequency modeB. We didconfirmthatinthismode P remainsconstant
3
channels for antennas from a given sub-array. This is akin to overthefrequencies243,607and840MHz.Wethenmadethe
J.M.Smitsetal.:ThegeometryofPSRB0031 07 3
−
Table2.DetailsoftheobservationsofPSRB0031 07
−
Date Starttime(UT) Telescope Freq. Timeres. Bandwidth Numberof
(saftermidnight) (MHz) (ms) (MHz) Pulses
02-09-2004 62616 GMRT 325 0.512 16 2678
03-09-2004 69092 GMRT 243 0.512 6.25 14607
03-09-2004 69092 GMRT 607 0.512 9.75 14607
03-09-2004 72150 WSRT 840 0.8192 80 16140
03-09-2004 75847 EFF 4850 0.9206 500 31042
03-09-2004 80087 GMRT 1167 1.024 16 10145
04-09-2004 00554 GMRT 243 1.024 6.25 4409
04-09-2004 00554 GMRT 607 1.024 9.75 4409
07-09-2004 69522 GMRT 607 0.512 9.75 4718
07-09-2004 71441 WSRT 840 0.8192 80 16781
07-09-2004 73690 GMRT 243 0.512 6.25 14394
24-08-2005 43212 GMRT 157 0.512 16 2229
assumptionthatthereisindeedonefundamentalperiodassoci-
ated with the vertical spacing between drift bands, such as the
rotation speed of sub-beams around the magnetic axis, that is
the same at each frequency at each pulse phase. With this as-
sumption, we determined the drift mode of each pulse for all
oftheobservations.Thiswasachievedbyvisuallyfindingdrift
sequences at the frequency that showed this drift most clearly.
Wheneverpossible,these sequenceswere then inspectedat the
remainingavailablefrequenciestoimprovetheexactbeginning
and end of the sequences. Finally, P was calculated for each
3
sequenceandclassifiedaseithermodeA,B,orC.Eventhough
the time span of the observations is about 10 hours, we only
detected 4 short drift sequences with a mode-C drift, none of
themweredetectedat4.85GHz.Becauseofthelownumberof
pulseswhichwereinamode-Cdrift,welimitedtheanalysisto
sequencesofpulseswhichwereinmode-Aormode-Bdrift.
2.4.Averagepolarisationprofiles
Fig.1. Contour plot of the LRFS of PSR B0031 07 at
−
Once the driftclass of all the sequenceswas known,all pulses 0.325GHz as a function of pulse phase during a sequence of
inadriftclasswereaveragedtogetherforeachfrequencysepa- mode-Bdrift.Theleftpanelshowsthepowerspectrumaveraged
rately.Thisresultedinaveragemode-AandBpolarisationpro- overphase.Theupperpanelshowsthepoweraveragedoverfre-
filesandpositionanglesweepsateachfrequency.Wealsomea- quency.
sured the widths of each of these profiles at 10% of the peak
values.
2D time series. The resulting Longitude-Resolved Fluctuation
Spectrum(LRFS)wasthenaveragedoverapulsephasewindow
2.5.Frequencydependenceofthefractionaldriftintensity
containing the pulsar on pulse, giving a Longitude-Averaged
Smitsetal.(2005)havesuggestedthatthemode-Aandmode-B FluctuationSpectrum(LAFS).AcontourplotofaLRFS,aswell
emissionoriginatefromdifferentregionsofthemagnetosphere. as the LAFS, is shownin Fig. 1. The driftintensity was calcu-
Thismeansthatthelineofsightintersectionwiththesub-beams latedbyintegratingtheLAFSoverasmallfrequencyrangecon-
is different for both drift modes (see their figure 4 and 5). In taining the reciprocalof the P3 value of the sequenceand then
particular, this causes the line of sight to miss the center of subtractingthe integrationof theLAFS overa small frequency
the ring ofmode-Bsub-beamsat highfrequencyandonly cuts rangecontainingnoperiodicities.Bydividingthedriftintensity
through a diffuse componentsurrounding the sub-beams. As a by the total intensity in the LAFS of the pulsar signal we ob-
result we do not see any drift bands at high frequency when- tainedthefractionaldriftintensity.Thisfractionaldriftintensity
ever the low-frequencyemission reveals a mode-B drift. If the was calculated for all sequences of drift at each observed fre-
lineofsightintersectionwiththesub-beamsindeeddetermines quency. When the line of sight starts to miss the center of the
howclearlyadriftpatterncanbeobserved,thenthedriftpattern ring of sub-beams and only cut through diffuse emission, then
can provide informationabout the geometryof the sub-beams. thefractionaldriftintensityshouldbecomelow.
To measure how far the line of sight intersection is away from
the center of the sub-beams in a drift sequence, we calculated
2.6.Widthofaveragedriftprofiles
the fractional drift intensity as follows. First we took the am-
plitude at each pulse phase of each pulse in the sequence and The geometricalmodelthat will be constructedto describe the
stackedthemontopofeachothertoproducea2Dtimeseriesof single pulse emission from PSR B0031 07 consists of both
−
thesametypeasshownin Fig.5.We thencalculatedthe abso- driftingandnon-driftingemission.Fromthemodelwecancal-
lutevaluesoftheFouriertransformateachpulsarphaseofthis culate at each frequency and for each drift mode the width of
4 J.M.Smitsetal.:ThegeometryofPSRB0031 07
−
theaverageprofileresultingfromthedriftingemissiononly.In
ordertousethiswidthtotesthowwellthemodeldescribesthe
observedemission,weneededtodeterminethewidthoftheav-
erage profiles resulting from the drifting emission only, in the
observations.Thiswasdoneforeachfrequencyasfollows.For
each driftsequence,we determinedthe averagedriftprofileby
integrating the power in the LRFS at each pulse phase over a
smallfrequencyrange,containingthereciprocaloftheP value
3
of the sequence. For each drift mode these profiles were aver-
agedtogetherandtheirwidthsweremeasured.
2.7.FrequencydependenceofP
2
ThechangeofP withobservationfrequencyreflectsthechange
2
ofthesizeoftheemissionzonewithrespecttothetotalpathof
thelineofsightintersectionatdifferentheightsabovethepulsar
surface.Itthereforelimitstheradius-to-frequencymappingthat
canbe usedfor anymodel. P is oftencalculatedbyaveraging
2
autocorrelationsofsinglepulses,containingmorethanonesub-
pulse.ForPSRB0031 07thismethodfailsathighfrequencies
−
where there are no pulses containingmore than one sub-pulse.
WethereforecalculatedP ,foreachdriftsequence,astheprod-
2
uct of P and the sub-pulse phase drift (∆φ) of that sequence.
3
The phase drift was measured by cross-correlatingconsecutive
pulsesofeachdriftsequenceandaveragingtheresultingcross-
correlations. The peaks of the average cross-correlations were
fitted with a quadratic polynomial, of which the phase shift of
themaximumwastakenasthephasedrift.Foreachdriftmode
andobservationfrequencythevaluesforP wereaveragedover
2
manydriftsequencestoobtainthefrequencydependenceofP
2
foreachdriftmode.ThevaluesforP arenotaffectedbyapos-
2
siblealiasing.
J.M.Smitsetal.:ThegeometryofPSRB0031 07 5
−
Table3.The10%-widthsoftheaverageprofilesfromallpulses Table 5. The 10%-widths of the average drift profiles for drift
and from the drift modes A and B from the observations of mode A and B for 7 differentfrequencies. A ‘ ’ indicates that
−
PSRB0031 07at7differentfrequencies. therewasnotenoughsignaltomeasureawidth.
−
Frequency width(deg) widthmodeA widthmodeB Frequency 10%-widthof 10%widthof
(MHz) (deg) (deg) (MHz) modeA(deg) modeB(deg)
157 45.7 1.8 44 5 46.3 1.4 157 28 2 35.0 1.5
± ± ± ± ±
243 42.3 0.6 39.5 1.5 42.3 0.6 243 27 5 35.2 0.9
± ± ± ± ±
325 40.66 0.15 39.4 0.3 40.30 0.12 325 33.4 0.4 35.6 0.4
± ± ± ± ±
607 39.8 1.1 32 4 39.3 0.9 607 27 3 30 4
± ± ± ± ±
840 40.9 0.6 32.2 0.8 43.0 0.9 840 12.2 0.9 10 4
± ± ± ± ±
1167 36 3 25 6 38 5 1167 9 3 3.1 1.1
± ± ± ± ±
4850 33.7 0.8 23.6 0.6 32.8 0.9 4850 16.4 0.9
± ± ± ± −
Table4.AveragefractionaldriftintensityofdriftmodeAandB Table6.TheaveragevaluesofP fordriftmodesAandBfor7
2
forfourdifferentfrequencies. differentfrequencies.A‘ ’indicatesthattherewasnotenough
−
signaltomeasureP .
2
Frequency Fractional Fractional
(MHz) driftintensity driftintensity Frequency P ofmodeA P ofmodeB
2 2
inmodeA inmodeB (MHz) (deg) (deg)
243 0.14 0.05 0.20 0.05 157 18.9 1.2 19.2 0.7
607 0.07±0.02 0.11±0.03 243 24 ± 4 18.7 ± 1.6
840 0.07±0.02 0.04±0.02 325 19.8 ± 1.0 18.7 ± 0.6
4850 0.07±0.02 0.02±0.01 607 21 ± 3 18.7 ± 0.6
± ± ± ±
840 10 3 14.0 1.9
± ±
1167 14 5 12 3
3. Results 4850 14 ± 2 ±
± −
The(polarisation)profilesof PSR B0031 07for7 frequencies
−
are shown in Fig. 2. No polarisationwas recordedfor the 157,
325and1167MHz observations.The157and325MHz obser-
vations were not simultaneous with the other observationsand
werealignedtohavethecenteroftheprofilesatthesamepulse
phase as the center of the 243 and 607MHz profiles, respec-
tively. The widths at 10% of the maximum were measured for
alltheintensityprofiles.TheyareshowninTable3.Theeffects
ofdispersionarenegligibleandhavenosignificantcontribution
tothemeasuredwidth.Theerrorsquotedhereandelsewherein
thepaperare1-sigmaerrors.
Fig. 3 showsthe fractionaldriftintensity in driftsequences
for5differentfrequenciesoveradurationof4hoursand15min-
utes. Due to interference the drift intensity could not be deter-
mined in some parts of the observations.These parts and parts
wherenopulseswereobservedaremarkedinFig.3ashatched
areas. The average fractional drift intensities of drift modes A
andBforfourofthefrequenciesarelistedinTable4.Theerrors
wereestimatedbyusingthermsoftheregionintheLAFSthat
doesnotcontainthe reciprocalof the P value.The 1167MHz
3
observationisnotlisted becausethisobservationdoesnotcon-
tain enough drift sequences to give a reliable average value.
Thewidthsoftheaveragedriftprofiles,i.e.ofonlythedrifting
component,foreachdriftmodeateachfrequencyareshownin
Table5.At4.85GHztherewasnotenoughsignalinthemode-B
profiletomeasureawidth. Table6liststheaveragevaluesofP
2
fordriftmodesAandBfor7differentobservationfrequencies.
Notethattheobservationsat157and325MHzwerenotpartof
thesimultaneousobservations.
6 J.M.Smitsetal.:ThegeometryofPSRB0031 07
−
Fig.2.Averagepolarisationprofilesfromalltheobservations.Fromlefttorightaretheaverageprofilesofallpulses,ofpulsesthat
are inmodeA and ofpulsesthatarein modeB. The solidline is the intensityandthe dashedanddottedlinesarethe linear and
circularpolarisation,respectively.Onlythe243,607,840,1167and4850MHzprofileswereobtainedsimultaneously.Allprofiles
arebinnedtothelowesttimeresolutionof1.024ms.ThemethodforalignmentisdescribedinSection2.2
J.M.Smitsetal.:ThegeometryofPSRB0031 07 7
−
Fig.3. Fractionaldriftintensity in driftsequencesat 5 differentfrequencies.The gray columnsrepresentsequencesof pulses for
whichadriftintensitycouldbedetected.Light-grayindicatesmode-Adrift,whereasdark-grayindicatesmode-Bdrift.Thehatched
areasindicateregionsduringwhichnopulseswereobserved,orthequalityoftheobservationwasnotgoodenoughtodetermine
thedriftintensity.Thelowergraphisthecontinuationoftheuppergraph.Thetotaldurationis4hoursand15minutes.
8 J.M.Smitsetal.:ThegeometryofPSRB0031 07
−
B emitting where γ is the Lorentzfactor of the secondaryplasma, c is the
field line speed of light and r is the radius of curvature at the emis-
cr
sion point. Following the derivation from Gangadhara (2004),
P(r, θ )
the dipolargeometryof the magnetic field dictateshow the ra-
diusofcurvatureisrelatedtothedistancebetweentheemission
pointandthecenterofthestar,r,andthepolarangletothemag-
neticaxis,θ,as
r
r [5+3cos(2θ)]23
r = (3)
cr sinθ3√2[3+cos(2θ)]
z
Foranglessmallerthan10 thiscanbelinearizedwithinanerror
θ ◦
radius of star of1.5%togive
χ 4
r = r/θ. (4)
cr
3
In the second model, the plasma-frequency model, we as-
sumethattheradiationisemittedatthelocalrelativisticplasma
frequency,giveninSIby
e2n
ω2 = γ 1 , (5)
R rp h i− ǫ m
0 e
Fig.4.Sketchofa fieldline asa functionofdistancealongthe
wherenisthesumofelectronandpositrondensityandeandm
e
magneticaxis(z) and distance fromthe magneticaxis(R)near
are the electronchargeand mass. We furtherassume thatthere
the pulsar, to show the definitionof the angle χ. Point P on an
isadistinctregionaroundthemagneticaxiswhereshotsofpar-
emittingfieldlineisdefinedbyit’spolarcoordinatesr andθ.χ
ticles can create a secondary pair plasma with a fixed particle
is defined as the angle between the magnetic axis and the foot
density which only depends on altitude. The frequency of oc-
pointofthefieldlineonthesurfaceofthestar.
currence of these shots is assumed to decrease as a Gaussian
withincreasingdistancefromthemaximumintensityfieldlines.
Thus, the intensity of the emission falls off with distance from
4. Modellingthegeometry themaximallyemittingfieldlines,whiletheparticledensityof
theshotsandthefrequencyofemissionremainsconstantatcon-
Here, we set out to find a geometry that can reproducea great
stantemissionheight.Since,alongtheopenfieldlines,theden-
number of the observed features of this pulsar, which includes
sityofrelativisticallymovingsecondarypairplasmafallsoffap-
the positionangle sweep (withoutthe orthogonalmode jumps)
proximatelywiththecubeofthedistancetothecenterofthestar,
andthefrequencydependencesofthewidthoftheaverageinten-
inadipolarfield,onaccountoffluxconservation,thefrequency
sity profile,the widthoftheaveragedriftprofile,thefractional drops with distance to the power 3. This leads to the follow-
driftintensityandP ,fordriftmodesAandB. 2
2 ingrelationshipbetweenthealtitudeandfrequencyofemission
Wefirsttrytolimitthegeometricalpossibilitiesbyfittingthe
(Ruderman&Sutherland1975):
widths of the observedaverage profiles to three differentmod-
els that restrict the relationship between height and frequency 2
ω −3
of emission. All models assume that the intensity of the radio r= (6)
ω !
emission decreases with distance from the maximally emitting 0
ringoffieldlinesasaGaussiangivenby
Finally, we try a version of an empirical relationship, as
givenbyThorsett(1991):
1 χ χ 2
waIx=hieseraexnpId−itsh2eth feooiχ−ntwtep0no!siintty,,oχn itshethseurafnagceleobfetthweeesntarthoefmthaegnfiee(t1lidc) rw=hichh0is1s+im ωiωla0r!−to32th,eplasma-frequencymodel,buthasan e(7x-)
line from which emission can be observed (see Fig. 4), χ0 is tra distance parameter h0 that allows for a minimum emission
the angle between the magnetic axis and the foot point on the height.Forallthreemodelswesimulatedtheintensityprofileat
surfaceofthestarofthemaximallyemittingfieldlineandχw is 7differentfrequencies,correspondingtothe7frequenciesofthe
theanglebetweenthefootpointsonthesurfaceofthestarofthe observations,fordifferentvaluesoftheparametersofthemod-
twofieldlinesforwhichtheintensityhasdroppedtoe−21 times els.Theseparametersandthelimitswithinwhichtheywerevar-
themaximumintensity. ied,arelistedinTable7.γ andω werevariedlogarithmically,
0
Inthefirstmodel,thecurvatureradiationmodel,weassume the other parameters linearly. We then calculated the widths at
thatthefrequencyofemissionisequaltothecharacteristicfre- 10% of the maximum.From these widths and the widths from
quency for curvature radiation, given by ((Jackson 1999), Eq. themode-Aandmode-Bintensityprofilesfromtheobservations
14.81) wecalculatedthereducedchi-squareforeachsetofparameters
foreachdriftmode.Wethenminimizedthesechi-squarestofit
3 c the values for the parameters. Initial values for α and β were
ω= γ3 , (2)
2 r ! found by fitting the rotating vector modelfrom Radhakrishnan
cr
J.M.Smitsetal.:ThegeometryofPSRB0031 07 9
−
Table 7. Parameters and their ranges of variation used to fit gesting a minimum emission height ranging from 1 to 50km
thecurvature-radiationmodel(top),theplasma-frequencymodel abovethestellarsurfaceandamaximumemissionheightrang-
(middle)andtheempiricalmodel(bottom). ingfrom10to100kmabovethestellarsurface,wherethelow-
est values for the chi-square always correspond to the lowest
Parametersforthe lowerlimit upperlimit emissionheights.We didnotallowemission heightsbelowthe
curvatureradiation stellar surface, but such values would give low values for the
model reducedchi-square as well. For comparison,we tried to obtain
α 0.1◦ 6◦ emissionheightsbymodellingtheemissionfordifferentvalues
β 0.1◦ 6◦ oftheparametersα,χ0andχw(βwaskeptequaltoα)andmak-
γ 10 2000 ing no assumptions about the relationship between height and
χ 0.3 0.92
0 ◦ ◦ frequencyofemission.Foreachfrequency,theemissionheight
χ 0.06 0.57
w ◦ ◦ was changed in such a way that the width of the profile from
Parametersforthe lowerlimit upperlimit
theobservationwouldbeequaltothewidthoftheprofilefrom
plasma-frequency
themodel.Thisalwaysresultedinsmallemissionheightdiffer-
model
encesbetween frequencies,varyingfrom a few kilometersto a
α 0.1 6
◦ ◦ few tens of kilometers. Also, for each set of parameters it was
β 0.1 6
◦ ◦ possibletochoosevaluesforh andω forwhichtheempirical
ω 0.01GHz 1000GHz 0 0
0
χ 0.3 0.92 modelwouldfitwelltotherelationshipobtainedbetweenheight
0 ◦ ◦
χ 0.06 0.57 andfrequencyofemission.
w ◦ ◦
Parametersforthe lowerlimit upperlimit Based on these results we proceededto make a more com-
empiricalmodel plexmodel,includingtwodriftmodesofsub-pulseemissionsu-
α 0.1 6 perposedonlow-intensitydiffuseemission,assumingtheempir-
◦ ◦
β 0.1◦ 6◦ icalmodelforobtainingtheemissionheightforeachfrequency.
h0 10km 150km TheparametersofthismodelarelistedinTable8,whereR⋆ is
ω0 0.01GHz 1000GHz theradiusofthepulsar,Dw istheanglebetweenthefootpoints
χ 0.3 0.92
0 ◦ ◦ on the surface of the star of the two field lines for which the
χ 0.06 0.57
w ◦ ◦ intensity from the diffuse emission has dropped to 60% of the
maximum intensity, D is the intensity of the diffuse emission
I
dividedby the intensity ofthe sub-pulseemission atthe center
& Cooke (Radhakrishnan&Cooke 1969) to the position angle field line, N is the number of sub-pulses and P is the rota-
sub 3
from all the observations for which polarisation was recorded. tion time of the carousel divided by N . The superscripts A
sub
AscanbeseeninthepanelsofFig.2,theposition-anglesweep and B refer to the value for that parameter in drift mode A or
isverystraight,apartfromorthogonalmodejumpswhichcauses B. Note that we only changed the values of χ and P to de-
0 3
differencesintheposition-anglesweepatdifferentfrequencies. scribeachangeindriftmode.Therotationtimeofthecarousel
After excluding the pulsar phases that contain an orthogonal and the number of sub-beams were derived from the observed
modejump,eachsweepcanbereproducedwiththerotatingvec- verticalspacingbetweendriftbands,assumingthattheobserved
tormodelfromRadhakrishnan&Cooke,whenαandβarekept sub-pulsesare not aliased. If we are in fact observingan alias,
equalandintherangeof0.1◦to6◦. then the rotation time of the carouselas well as the numberof
Once an initial relationship between the altitude and fre- sub-pulses become less. Table 8 also lists the values of these
quency of emission was found, we proceeded to include both parametersthatwere obtainedafter fitting themto describethe
diffuse emission and emission fromrotating sub-beamsto pro- aforementionedfeatures of the observations. The values for h
0
ducesinglepulsesin twodifferentdriftmodesforanumberof andω inTable8leadtoemissionheightsrangingfrom2.3km,
0
frequencies. The parameters of this model were then fitted to at4.85GHz,to13.6kmat0.157GHzabovethestellar surface.
reproduce the position angle sweep and the frequency depen- Regardlessoftherelationshipbetweenheightandfrequencyof
dencesofthewidthoftheaverageintensityprofile,thewidthof emission, these low emission heights are needed to obtain the
theaveragedriftprofile,thefractionaldriftintensityandP2,for frequency dependence of P2, which depends on the fractional
driftmodesAandB.Also,weusedtheresultfromSmitsetal. changeofemissionheight.ThevalueforχA correspondstothe
0
(2005)thatshowsthatfordriftmodeA atlowfrequencies,the last open field line, which for a small value of α is given by
lineofsightintersectionjustpenetratesthemaximallyemitting χ = √R 2π/P c.
lof ⋆ 1
fieldlines,causingtheobserveddoublecomponentintheupper- Fig.5 showsa grayscale plotof thesingle pulsesfromthe
leftpaneloftheirFigure8. observationsontheleftandofthesinglepulsesfromthemodel
ontheright,forboththehighestandsmallestfrequenciesofthe
simultaneous observations. It demonstrates that the model can
4.1.ResultsoftheModelling
indeedreproducebothmodeAandmodeBdriftsandalsothat
For both the curvature-radiation model and the plasma- themodeBdriftdisappearsathighfrequency.Italsoshowsthe
frequencymodel we did not find any sets of values for the pa- changeinprofilewidth.Wefurthercalculatedthewidthsofthe
rameters to describe the frequency dependence of the profile averageintensityprofilesandthevaluesfor P fromthemodel
2
widthscorrectly.Thebestsetofparametersgaveareducedchi- for bothdriftmodesat all frequencies.For each frequency,the
square above300and above15 forthe curvature-radiationand signal-to-noiseratiowasadjustedtobethesameasthesignal-to-
theplasma-frequencymodel,respectively.Incontrast,wefound noiseratioofthedata.Thewidthsofboththeaverageintensity
thattheempiricalmodel,whichincludesanextraparameter,de- profiles and the average driftprofiles from both the model and
scribed the data with a reducedchi-square rangingfrom 0.7 to fromtheobservationsfordriftmodeAandBateachfrequency
3formanydifferentvaluesfortheparameters.Characteristicto are shownin Fig. 6. The averagefractionaldriftintensity from
these sets of parameters were low values for h and ω , sug- boththe modelandthe observationsfordriftmodeA andB at
0 0
10 J.M.Smitsetal.:ThegeometryofPSRB0031 07
−
Table8.Optimisedparametervaluesinafullgeometricalmodel
Values of P2
for the emission of PSR B0031 07, describing two modes of
− 24
drifting sub-pulses superposed on low-intensity diffuse emis- Observations
sion,assumingaradius-to-frequencymappingbasedontheem- 22 Model
20
piricalmodelofThorsett(1991).
g) 18
e
d 16
Parameter Value Parameter Value P2 ( 14
P1 0.9429s χA0 0.85◦ 12
R⋆ 10km χB0 0.81◦ 10
α 1.83◦ χw 0.13◦ 8
β 1.83 D 0.22 0.1 1
◦ w ◦
h 11km D 0.001 Frequency (GHz)
0 I
ω 1.206GHz N 9
0 sub Fig.8. Value of P from the observations (solid) and from the
PA 13s 2
3 model(dashed)ateachfrequency.Notethatthepresenceofnon-
PB 7s
3 drifting emission affects the method used to measure P . (See
2
discussionformoredetails.)
Fig.5.Grayscaleplotsofsinglepulsesattwofrequenciesfrom
thesimultaneousobservations(leftpanels)andfromthemodel
(right panels). The upper plots are at 243MHz and the bottom
Fig.9. Two close ups of the model of the emission zone of
plots are at 4.85GHz. The first 50 pulses are in drift mode A, PSRB0031 07.Theverticallineistherotationaxisofthepul-
thefollowing5pulsesarenullsandtheremainingpulsesarein −
sar. The 7 circles indicate the line of sight trajectories corre-
drift mode B. The pulses in the model were aligned with the
spondingtothe7observedfrequencies.Theemissionzonecon-
pulsesfromthe243-MHzobservation.Thisresultsintheoffset sists of 9 sub-beamssurroundedby diffuse emission, shownas
betweensinglepulsesfromtheobservationandfromthemodel semi-transparent.Bothimagesaretoscale.Thetopimageshows
at4.85GHz.Thereissomeinterferencevisibleinthepulses60 thelocationofthesub-beamsduringmode-Adrift.Thebottom
to68oftheobservationat4.85GHz. imageshowsthelocationofthesub-beamsduringmode-Bdrift,
whichlieslightlyclosertothemagneticaxis.
each frequencyare shownin Fig. 7. Fig. 8 shows the valuesof
P from boththe modeland fromthe observationsat each fre-
2
quency. Since the difference in P between driftmodes due to
2
theirdifferentvaluesforχ isfarlessthantheerroronthemea-
0
suredvaluesfor P ,thevaluesof P fordriftmodeAanddrift
2 2
modeBwereaveragedtogether.Imagesofthemodelcanbeseen
inFig.9.