The fan beam model for the pulse evolution of PSR J0737-3039B

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by  Lab Saha

MNRAS000,1–9(2016) Preprint24January2017 CompiledusingMNRASLATEXstylefilev3.0
The fan beam model for the pulse evolution of PSR
J0737-3039B
⋆
L. Saha and J. Dyks
Nicoulas Copernicus Astronomical Center,Polish Academy of Sciences, Rabian´ska 8, 87-100, Torun, Poland
7
1
0 AcceptedXXX.ReceivedYYY;inoriginalformZZZ
2

n
ABSTRACT
a
J AverageradiopulseprofileofapulsarBinadoublepulsarsystemPSRJ0737-3039A/B
exhibits an interesting behaviour. During the observation period between 2003 and
3
2 2009, the profile evolves from a single-peakedto a double-peaked form, following dis-
appearance in 2008 indicating that the geodetic precession of the pulsar is a possible

] origin of such behaviour. The known pulsar beam models can be used to determine
E
the geometryofPSRJ0737-3039Binthe contextofthe precession.We study howthe
H fan-beamgeometryperforms in explainingthe observedvariationsof the radioprofile
. morphology. It is shown that the fan beam can successfully reproduce the observed
h
evolutionofthe pulse width, andshould be consideredas a serious alternativefor the
p
conal-like models.
-
o
Key words: pulsars:general– pulsars: individual ( PSR J0737 – 3039)
r
t
s
a
[

1 1 INTRODUCTION measure the geometry of B (Breton et al. 2008). Based on
v theobservedradioprofilefrompulsarA,therelativisticpre-
PSR J0737-3039A/B is a system of two neutron stars in
4 cession ofpulsarB’sspinaxisaroundthetotalangularmo-
37 amhPigahrklyesrrealadtiiovistteilcescoorpbeit(dBiuscrogvayereedt ailn. 22000033; Lbyynetheet a6l4. mentum was estimated to be 4◦.77+−00◦◦..6665/yr (Breton et al.
2008).
6 2004). It is an ideal example of eclipsing double pulsar sys-
It has been found that the observed pulse profile of A
0 tem which provides a unique opportunity to test theories
. of general relativity (Krameret al. 2006). The system con- is quite stable. However, pulse shape and the intensity of
1
theemission from pulsarBvarywith orbitallongitude.Ra-
sists of two pulsars, one with a short spin period of 23 ms
0
dio observation by Parkes telescope system during obser-
7 (hereafterpulsar“A”)andthesecondonewithalongerspin
vation made between 2003–2005, showed two bright phases
1 periodof2.8s(hereafterpulsar“B”).Thesetwopulsarsorbit
◦
centered around longitude of ∼ 210 (hereafter bp1) and
: eachotherina2.4hrcycle.Theorbitalinclinationwascal-
v culatedtobe88.◦7(Kramer et al.2006;Ransom et al.2004) ∼ 280◦ (hereafterbp2).Inaddition,inbp2thepulseprofile
i
X and89.◦7(Coles et al.2005)throughtwodifferentmethods of B was observed to evolve from a single-peaked to a dou-
ble form (Burgay et al. 2005). The presence of two bright
r establishing that the orbital plane is almost edge-on to our
a line of sight (hereafter los), which is an ideal geometry to phases is also established by radio observations with the
100 m Green Bank Telescope (GBT) made during 2003–
form an eclipsing binary system. Radio observation of this
2009(Perera et al.2010,hereafterPMK10).Althoughthere
system displaysaroughly 30seclipseof pulsarAbypulsar
was some overlap between the observations made by these
Bwhen theemission from pulsarAis obscuredbythefast-
twotelescopes,thebehaviouroftheobservedpulseprofileis
spinninginnermagnetosphereofpulsarB(Lyneet al.2004;
quite different. In this overlapping time interval, pulse pro-
Lyutikov& Thompson2005).Thetwopulsarsareseparated
byadistanceofabout9×105kmintheirorbitsasmeasured files from the GBT system were single-peaked in both the
bright phases. However, the profiles then got broader and
byradio timinganalysis (Lyneet al. 2004).
splitintoadoubleform,alongwithareductionoftheirflux
From the prescription given in Barker & O’Connell
◦ densities. Finally, thepulses disappeared in March 2008.
(1975), the precession rate is obtained as 5 .0734 ±
0◦.0007/yr(Breton et al.2008).Followingtheobservational To explain the observed pulse profile and to determine
results of the ∼ 30 s eclipse of A by B, a simple geometric the geometry of precessing pulsars, the traditional choice
model was usedto explain theobserved morphology andto of emission model is a circular hollow-cone beam (Kramer
1998) as initially used for PSR B1913+16. The profile of
thispulsar,whenmappedontotheplaneofspinandpreces-
⋆ E-mail:[email protected] sionphase,createsapatternthatcanbeinterpretedwithan
(cid:13)c 2016TheAuthors
2 L. Saha and J. Dyks
hourglass-shapedbeam(Weisberg & Taylor2002).However,
Clifton & Weisberg (2008) have shown that for a specific
choiceofmodelparameterstheconalbeamcanalsoproduce
such an hourglass-shaped pattern.Similarly, in order toex-
plaintheobservedsecularvariationofthepulseprofilefrom
the PSR J0737-3039B and its disappearance in 2008, sim-
plecircularhollow-conebeamandellipticalhorseshoebeam
models are used (PMK10). Lomiashvili & Lyutikov (2014)
have studied the secular and orbital visibility of pulsar B,
byconsideringtheeffectofthemagnetizedwindfrompulsar
A on the magnetosphere of pulsar B. The basic framework
of all of these studieswas theconal beam model.
In this paper, we attempt to understand the ob-
served changes in the pulse profile of pulsar B within
the framework of the fan beam model (Michel 1987;
Dykset al. 2010; Wang et al. 2014), which is success-
ful in explaining various observational data, e.g. double
notches in pulsar profiles, bifurcated emission components
(Dyks& Rudak 2012), the rate of the radius to frequency
mapping (Karastergiou & Johnston 2007; Chen & Wang
2014; Dyks& Rudak2015),and localised distortions of po-
larisation angle curve(Dykset al. 2016).
Withregardtotheprecession-drivenchangeofprofiles,
a slow quasi-linear evolution of the peak-to-peak separa-
tion had been shown to be typical of the fan beam model Figure1.Skyprojectionofdipolarmagneticfieldlines(dotted)
(Fig. 17 in sect. 6.3.3 of Dykset al. 2010), as opposed to viewedfromabovethepolarregion.Twothicklinesegmentsthat
followafixedmagnetic azimuth representfanbeams emitted by
the conal model, in which the passage through different re-
plasmastreams.Thelineof sightistraversingthebeams attwo
gions of a cone usually implies nonlinear changes. Based
different places (e.g., a and b). The angular distance between
on this a fan beam was predicted for the main pulse of these two points, as measured around Ω~, is the observed pulse
PSR J1906+0746, and confirmed to be like that two years
widthW.
later(Desvignes et al.2012).TheinterpulseemissionofPSR
J1906+0746, however, was mapped into a non-elongated
patch. free parameters, two cases are considered in which one of
The paper is organized as follows: In Section 2, we set the beams (with the index ‘1’) is located in a specific way.
up the formulae to calculate the profile width W based on In an asymmetric case, the beam 1 is oriented meridion-
thefan-beammodel.Section3describesthepeak-separation ally (either at φm,1 = 0 or φm,1 = π). In a symmetric
datausedinthemodelling.ResultsarepresentedinSection case, both beams are equidistant from the main meridian
4 and discussed in Section 5. (φm,1 =−φm,2).Thelineofsightiscuttingthroughthefan
beamsatamagneticcolatitudeθ ,whichcanbecalculated
m
from thefollowing equation:
2 MODEL DESCRIPTION
cosζ(t)=cos(π−φ )sinαsinθ +cosαcosθ , (3)
m m m
We consider a coordinate system as described in Kramer
where α is the angle between the magnetic axis ~µ and
(1998) (see Fig. 4 therein). The angle between pulsar spin the pulsar spin axis Ω~ (for details see the Appendix A in
axis and theline of sight, ζ(t), can be written as
Dyks& Pierbattista 2015).
The pulse longitude, φ, for that point where the mag-
cosζ(t)=cosλcosi+sinλsinicosΦ(t), (1) neticazimuthφm andζ(t)intersecteachotherattimetcan
thenbe written as
where λ is the angle between orbital momentum and spin
cosθ −cosαcosζ(t)
axis,iistheangleofinclinationoftheorbitalplanenormal cosφ(t)= m . (4)
sinαsinζ(t)
with respect to the line of sight and Φ(t) is the precession
phasewhich is definedas The pulse width W is defined here as the separa-
tion between the peaks of observed components, and the
peaks are associated with the locations of the fan beams
Φ(t)=Ωp (T0−t), (2)
in Fig. 1. In the conal-beam model, W can be written as
whereΩpistheprecessionrateandT0istheinitialreference 2φ(θm,α,ζ),whereθmisconsideredasahalfofafixedopen-
timedefinedasthetimewhenspinaxisofthepulsarcomes ing angle of the conal-beam. In the fan beam model, how-
closest totheline of sight. ever, W is determined by the beams’ magnetic azimuths:
It is assumed that the radio emission pattern of W(t)=|φ2(φm2)−φ1(φm1)|,wherethepulselongitudesφ1
J0737−3039B consists of two fan-shaped beams, each one and φ2 are calculated from equation 4.
following a fixed magnetic azimuth φm,i with i ∈ {1,2} For different choices of α, λ, T0, and φm2, we calculate
(thick-line sections in Fig. 1). To minimise the number of the width as a function of time and compare it to the ob-
MNRAS000,1–9(2016)
Fan beam model for PSR J0737 – 3039B 3
Figure2.EvolutionofPSRJ0737-3039Bprofileshownforbothradio-brightorbitalphases:bp1(left)andbp2(right).Itisacompilation
ofpublisheddatafromGBT(820MHz,Pereraetal.2010)andtheParkestelescope(1390MHz,Burgayetal.2005).Latetimeprofiles
(MJD>53800 are aligned according to the leading component. The early time data are aligned differently for the two orbital phases:
inthecaseofbp1,accordingtothebrightestpeakintheprofile;inthecaseofbp2,tomatchtheoverallon-pulsewindow.Verticallines
areaddedforreference.Thezeropointofspinphaseaxisisarbitrary.Datacourtesy:B.Perera,M.McLaughlin(GBT)andM.Burgay,
R.Manchester(Parkes).
◦ ◦
served pulse profile of pulsar B. The range of α and λ is traverse case (ζ < α, φm1 = π, 90 < φm2 < 180 , see
(0,π), whereas T0 covers full precession period. Fig. 1 for definition of the magnetic azimuth φm). The pa-
◦
rameters are considered to have grid size of 0. 6 for α and
Narrow two-peaked profilescan only beobserved when ◦
λ,1 for φm2, and 1 yrfor T0.
both streams extend either towards the rotational equator,
or away from it (i.e., down or up in Fig. 1). Therefore, to When the pulsar is viewed in the equatorward region
probe the full range of φm2 we consider two different cases: (sinζ(t)>sinα)eachmagneticazimuthiscutbythesight-
◦
the outer-traverse case (φm1 = 0, 0 < φm2 < 90 ) with line one time per rotation period P, and no constraints on
the sightline detecting the beams at ζ > α, and the inner- therangeofradio-brightcolatitudesareimposed,i.e.,θm =
min
MNRAS000,1–9(2016)
4 L. Saha and J. Dyks
0andθm =π.Forcircumpolarviewing,however,thesight-
max
line can cut through each B-field line twice, so two streams
canproducefour-peakedpulseprofile.Inprinciplesuchmod-
elsmayberejectedashavingtoomanypeaks,however,since
one solution for θ is usually much larger than the other,
m
and the radio emissivity is likely decreasing with θ , it is
m
also possible to ignore the far value. We choose this last
option, i.e., whenever the beam’s magnetic azimuth is cut
twice per P,only thesmaller value of θ is used.
m
Since the pulsar B has not been detected after March
2008, we reject all models which predict detectable flux in
the years 2009–2016. This is done in the following way. For
each model with a given λ and T0, we first determine the
rangeofviewingangleζ whichcorrespondstotheperiod of
B’svisibility (2003–2008). Thenwecheckifanyζ from this
rangeoccursin theyears2009-2016. Ifitdoes, themodelis
◦
rejected. In all the following calculations, we use i = 88 .7
and Ωp =4.◦8/yr. Figure 3. χ2 map for the fan beam model of the pulse width
dataonpulsarB.Theχ2contoursarefortheconfidencelevelsof
1σ (solidline), 2σ (dotted line)and 3σ (dashed line).The‘plus’
signon the leftmarks the minimum χ2 value, which should not
3 OBSERVATIONAL DATA
be considered as a meaningful best-fit solution. The result was
Observations of PSR J0737−3039B have been described obtained for the outer traverse through the asymmetric beam
bothinBurgay et al.(2005)andPMK10,andwereproduce (φm1=0).
them in Fig. 2. Unfortunately, only the late time data on
peak separation from PMK10 can be interpreted uniquely, accuracybetterthan2σ.Thepronouncedbayofhighχ2 on
and consistently for both bp1 and bp2. Therefore, as the
theright-handsideis causedbythecondition ofinvisibility
main modelling target we use the GBT data recorded be-
of pulsar B in the years 2009-2016, which resulted in the
tween December 24, 2003, and June 20, 2009 (PMK10). As
rejectionofsomegood-precisionfits.Withouttheinvisibility
can be seen in Fig. 2, thepulse profile was single-peaked in condition,the2σ and3σ contoursofχ2 havetheshapeofa
late2005.Intheearly2006thetwo-peakedpulseprofileap-
undistortedrhombus.
pearsandthepulsewidthincreasesovertimewhiletheflux
The observations constrain both the observed width
isdecreasing.Finally,theradiosignalofpulsarBdisappears
Wobs(t) and the roughly constant rate of width change
at theendofMarch 2008. Thischaracteristic hasbeenseen
(dW/dt)obs atanymomentwhenthepulsarBisdetectable.
for both thephases, bp1, and bp2.
ThehighcapabilityofthemodeltoadjusttobothWobsand
We use the data on the pulse width, i.e., the peak-to-
(dW/dt)obs can bemost easily understoodin thecase of an
peak separation of components, as determined in PMK10, ◦
orthogonal precession (λ=90 , middle horizontal region of
although we do not use the same errors. The changes of
Fig. 3).
W, illustrated in Fig. 6 of PMK10, can be well described
Sincetheobservedprofileisonlyafewdegreeswide,our
as linear except their dispersion around the linear trend is
sightline must traverse the beam close to the dipole axis.
muchlarger than expectedfor theGaussian distribution.A
Stated otherwise, the sightline’s smallest angular distance
likely reason for this is the low time resolution of the data
fromthedipoleaxis,asmeasuredwithinagivenpulsarspin
(512or256binsacrossthespinperiod).Therearesometimes
period, i.e., the impact angle β =ζ−α, must be generally
only four data points within each profile component, which small(β≪α∼ζ).1Thepathofasightlinetraversethrough
isfittedwithathree-parameterGaussianatonlyonedegree
thebeammaythenbeconsideredastraightlineorthogonal
of freedom. The number of flux measurements which are
tothemainmeridian,andthepulsewidthW canberoughly
involvedinthefitting,isthenmuchsmallerthanthenumber
approximated by:
ofthoseusedtocalculatetheirone-sigmaerror(severaltens
of data points within the noisy off-pulse region). When the W ≈βtanφm2/sinζ ≈(ζ(t)−α)tanφm2/sinα. (5)
errors of W are rescaled by a factor of 1.7, then 68 per
Note that the small difference between ζ and α has been
cent of data points is within the reach of the linear trend.
neglected in the denominator which takes into account the
Therefore, in thefollowing, we assume that 68% confidence
‘small circle’ effect. Accordingly,thederivativeof W is
errors(hereaftercalled1σ)arelargerbyafactorof1.7than
those publishedin PMK10. dW ≈ −Ωptanφm2sinλsinisinΦ(t). (6)
dt sinαsinζ(t)
4 RESULTS 1 Special cases, like those with φm1 ≈ φm2 can give small W
even at large β, however, these cases constitute a marginal pe-
Fig. 3 presents the 1σ, 2σ and 3σ contours of χ2 on the
ripheryofthewholeparameter space, andusuallydonotmatch
(α,λ) plane, as calculated for the asymmetric outer case dW/dt.Typically,withinthatpartoftheparameterspacewhich
◦
(φm1 = 0, φm2 < 90 ). It can be seen that for a very large canpossiblyreproducethedata,theimpactangleβ needs tobe
range of α and λ the model can reproduce the data with small.
MNRAS000,1–9(2016)
Fan beam model for PSR J0737 – 3039B 5
Figure 4.Themapofχ2 onthe(φm2,T0)plane,calculatedfor
Figure5.Nominally-bestfitofthefanbeammodeltothepulse
the samecase, andthe sameconfidence levelsas intheprevious
width data taken with GBT between Dec 24, 2003 and Jun 20,
figure.
2009 (PMK10). Solid line presents the asymmetric beam case
markedwiththecrossinFig.3(φm1=0).
◦
In the case of the orthogonal precession (λ ≈ 90 ), the
near-orthogonal orbit inclination i implies, through eq. (1),
that |Φ(t)| ≈ |ζ(t)| and eq. (6) reduces to |dW/dt| ≈
|Ωptanφm2|/sinα.Thusforanyαthemodelcanreproduce
an arbitrary value of |dW/dt| through adjustment of the
azimuthal beam separation (φm2). In this orthogonal case
◦
(λ ∼ i ∼ 90 ), the width given by eq. (5) is simply equal
to W ≈ |[Ωp(T0−t)−α]tanφm2|/sinα and for any set of
(α,φm2,t) the width can be reproduced by adjustment of
theimpact angle through T0.
In the corners of Fig. 3, no combination of model pa-
rameterscanreproducethedata.Thiscanbeunderstoodby
consideringthecaseofλ≪1radandα≪1rad.Thedipo-
lar magnetosphere is then viewed roughly at a right angle
withrespecttothedipoleaxis(ζ ∼θ ∼90◦)whichimplies
m
W ≈φm2anddW/dt≈0.Thus,forthesmallprecessionan-
gle the precession cannot ensure as steep variation of W as
observed.Atafixedα,thevalueofλneedstobesufficiently
large for the observed dW/dt to appear at some precession
Figure 6. The viewing angle ζ as a function of time for the
phase.Theparameters(T0,φm2)arethenadjustedtoallow modelsshowninthepreviousfigure(solidline:φm1 =0,dashed
the model to match the data. Beyond the rhombus region line:φm2=−φm1).Twoverticallinesaredrawntoindicatethe
thisis not possible. regionwherethepeak-separationdataisavailablefromtheGBT.
Thesmall-angleprecession (λ≪1rad)canonlyrepro-
ducethedatawhenα≈90◦ (bottomandtopcornersofthe
rhombus).Thisisbecausethebeammaynowbeviewedar- bands have the width of a half precession period and their
bitrarily close tothedipole axis, sothetiny wiggling of the bordersdonotdependonφm2 becausetheevolution ofζ is
magnetosphere can change the small value of β by a large fully determined by λ,i and T0 (eq. 1).
factor. Solid line in Figure 5 presents the ‘best’ fit curve of
It needs to be emphasized that the symmetric rhom- W(t) for the asymmetric outer case (φm1 =0, φm2 <90◦).
busshapeof thelow-χ2 contoursis characteristic of theor- Thisfitmayonlybenominallyconsideredasabestsolution,
thogonal orbit inclination with respect to the line of sight becauseitisnotstatisticallysignificantincomparisontothe
(i ≈ 90◦). For i 6= 90◦, the rhombus gets transformed into other models within the low χ2 region. The parameters of
an elongated parallelogram. this fit are: α = 4◦.2, λ = 82◦.8, φm2 = 13◦.0 and T0 =
The confidence contours (68, 95 and 99.7%) are also 2006 yrs, thus giving χ2/dof = 0.9. An independent fit of
shownonthe(φm2,T0)map(Fig.4).AgaintheobservedW the symmetric beam (φm1 = −φm2, dashed line) provides
◦
can be reproduced with reasonable precision within a very similar parameters, thistime with α=6 .6. The dipole tilt
large part of parameter space. There are horizontal bands seemstohavesuspiciouslysmallvalue,however,bynomeans
visible, in which T0 gives no acceptable solution. In these should these parameters be considered unique. Satisfactory
bandsζ changeswith timeinawrong direction,causingW data reproduction can beachieved for any point within the
to decrease with time, in contrast to the observations. The rhombus-shaped part of the χ2 map in Fig. 3. An example
MNRAS000,1–9(2016)
6 L. Saha and J. Dyks
Figure 7. A fit to the bp1 data taking a random point within
therhombus-shapedpartoftheχ2 map.
Figure8.B’sreappearancetimefortheouterasymmetriccaseof
Fig.3.Notethattheinnertraversecasewouldproduceamirror-
with randomly selected α and λ is shown in Fig. 7. It has reflectedimage,withtheearlyreappearancetimeandthemissing
χ2/dof =0.95. rhombuscornerlocatedonthelefthandside.
Precessional changes of ζ that correspond to the
nominally-bestsolutionsofFig.5arepresentedinFig.6.In
(2017-2078). When moving from left to right in Fig. 8,
both cases the line of sight approaches the magnetic dipole
(whichwascalculated for theoutertraversecase),t moves
axisandthenretreats.Thisbehaviourisnotuniversal:other r
toearliertimes.Fortheinnertraverse(whenbeamsarede-
solutions(forparametersatotherlocationswithintherhom-
tectableatζ <α,notshown)thet contourslookasamir-
bus) provide excellent data reproduction with the sightline r
rorreflection of Fig. 8, i.e., t increases from theleft-corner
passingfarintotheregionontheothersideofthemagnetic r
invisibility bay rightwards.
axis.
It may appear disturbing that possible values of the
When the fan beams are displaced to the other side of
◦ reappearance time, i.e. those which allow for good quality
the magnetic pole (inner traverse case, i.e.: φm1 = 180 ,
◦ ◦ datamatch,spanbasicallythefullprecessionperiod(Fig.8),
90 < φm2 < 180 ), the results are analogous to those de-
scribed above. The χ2 map on the (α,λ) plane is mirror whereasthepossible precession phaseT0 isexcludedwithin
reflected, i.e., the right-hand side bay in the rhombus of a half of Pprec (Fig. 4).2 The variations of W(t) have a
Fig. 3 appears in the left-hand side corner (i.e. at α<45◦, sinusoid-likeform,withthe‘wavelength’ofthesinusoidfixed
and λ∼90◦).The low-χ2 bandson the(φm2,T0) pla∼neget rbeyatphpeeavraaluneceoifsPtporehco.rTizhoentoanlllyywshaiyftttoheenssiunruesosoidoninerTo0r(lwaittehr
mirror-reflected and shift vertically by a half of precession
a vertical adjustment to match the already-observed data).
dpaertiao)d. (all values of T0 are therefore consistent with the SothereisacorrespondencebetweenT0andtr,butahalfof
The pulse width data for the other bright phase (bp2) theT0 values,seeminglyneededtoexplainalltr,ismissing.
Thisapparentparadoxiscausedbythemathematicalprop-
haveessentially thesame characteras thebp1data. There-
erties of a sinusoid. Let us first consider t coincident with
fore, the same analysis applied for bp2 gave similar results, r
the last data point (middle 2008). That is, consider a case
i.e., the good data match within the large part of parame-
inwhichimmediately afterthemaximumobservedwidthis
ter space. For brevity, we skip the presentation of the bp2
reached (W =Wmax), the width starts to decrease with no
results.
disappearanceoftheBpulsar.Suchbehaviourwouldbere-
producedwiththemaximumoftheW(t)sinusoidplacedat
4.1 Reappearance time tr.However,ifthesupposedreappearanceisnowdelayedby
some ∆t (at which time the decreasing W again takes on
r
Instead of a firm prediction for the reappearance time tr, the same maximum-observed value as in the middle 2008),
onlyamapoftheparameter-dependenttr canbeobtained. itisenoughtoshiftthesinusoidhorizontallybyatwotimes
Fig.8presentscontoursoffixedtr onthe(α,λ)map,calcu- smaller time interval (∆T0 = ∆tr/2) to both pass through
latedforthecasepresentedinFig.3.Foranypairof(α,λ), W(tr)=Wmax and match the real data points. A time dis-
values of φm2 and T0 are kept at their best-fit magnitudes. placementoftheW(t)curveproducestwotimeslargerdelay
Foreachsuchmodel,therangeofviewingangles(ζvmisin,ζvmisax) of tr. This is because the same values on both sides of the
wasdeterminedbasedonthevisibilityofthepulsarBinthe
years2003-2008.Thent hasbeendeterminedasthesoonest
r
moment when ζ(t) returnsback intothevisibility range. 2 Thishalf-periodvoidinthe allowedT0 occurs ifthegeometry
As can be seen in Fig. 8, possible reappearance times isconstrained tothe outer traversecase (|φm,1|<90◦, |φm,2|<
(calculated for model parameters which can ensure good 90◦).Figures4and8werebothcalculatedfortheoutertraverse
data match), range across most of the precession period geometry.
MNRAS000,1–9(2016)
Fan beam model for PSR J0737 – 3039B 7
sinusoid’speakaretwicemoredistantfromeachotherthan the pulsar may appear in future. Such reappearance, along
from the phase of the peak. The nearly-half-period-long in- with new measurements of W and dW/dt, would constrain
terval of T0 in which the solution gives the correct sign of some model parameters, provided the same radio beam is
dW/dt (see the bands in Fig. 4) is therefore sufficient to exposed to usagain.
ensureall possible reappearance times. The ambiguity in well-fitting parameters tempted us
to include the early data on PSR J0737−3039B from
Burgay et al. (2005, see Fig. 2 therein). Unfortunately, for
4.2 Anticipated model constraints from future
bothobservationalandtheoreticalreasons,suchanalysisap-
observations
pears tobe inconclusive.
It is interesting to estimate what constraints on the model There are the following observational problems. First,
parameters will become available when PSR J0727−3039B thebp1profilesfrom theperiod Jun 2003 –Nov 2004 (bot-
reappearsatacertainmomentinfuture.Tostudytheprob- tom left corner of Fig. 2) cannot be unambiguously decom-
lem we supplemented the GBT data on W with a set of posedintotwoseparateGaussian components.Second,it is
artificialdatapoints,createdinthefollowing way:theorig- hardtodecideifthebp1componentsmergeoranew(third)
inalGBTdatafromFig.5werereversedintimeandshifted componentisemerging,assuggestedinBurgay et al.(2005).
byanarbitrarytimeinterval.Thiswasdonebecausethepre- Third,theearly bp2profilesseem toexhibitdifferentshape
cessionchangestheviewingangleζ inasinusoid-likeway,so evolution, with a nearly fixed peak-to-peak separation, but
a given range of monotonically changing W is expected to theleadingcomponentdecreasinginstrength.Thismayin-
appeartwiceperprecessionperiod,inareversedtimeorder. dicate that our sightline probes widely different beam por-
WeaddednodatapointswithW beyondtherangeobserved tions in bp1 and bp2, because of orbital-dependent distor-
so far, although the future, more sensitive radio telescopes tions of magnetosphere (see Lomiashvili & Lyutikov 2014),
may be capableto detect larger range of W. or that thebeam shapeitself dependson orbital phase.
Suchexperimentshowsthattheconstrainingcapability On thetheoretical side, thesimplest model suchas the
of additional data is moderate. Only the precession phase one used in the previous section meets serious difficulties
(T0)istightlyconstraineddowntoaboutayear.Thedipole withsimultaneousreproductionofboththeearly(2003-Nov
tilt α can be limited with a precision of a few degrees, only 2004)andlatetimeevolution(2006-2008).Theunresolvedor
when theB pulsar reappears close to the year 2036. In this singleshapeofprofilein2005canbeinterpretedasanover-
specific case the low-χ2 contour on the (α,λ) plane is a lapping emission from two nearby streams (Fig. 9c), how-
narrow vertical stripe at α ∼ 90◦. However for t depart- ever,thelargeandroughlyconstantpeak-to-peakseparation
r
ing from 2036, the stripe bulges left or right on the (α,λ) ofbp2componentsintheearlyperiodcannotbeunderstood
plane, assuming an arc shape. At the same time the lim- ifthefanbeammodelislimitedtosuchtwobeamsonly.The
its on α quickly become much less precise. For t ∼ 2078 conalmodelisevenless likelytoexplainthis,becauseofits
r
or t ∼ 2017, the value of α (and λ) is basically uncon- lackofflexibilitywhichresultsfromtheconalbeamsymme-
r
strained. This effect is similar to the well-known α-ζ corel- try. The fan beam model, on the other hand, still lacks the
lation in the polarisation angle fitting, which produces the physical framework and offers some flexibility in choosing
arc-shaped χ2 contours on the (α,ζ) plane. Regardless of thebeam geometry.
the actual reappearance time, the additional data do not Animportantquestioniswhetherthebeamresponsible
provide any constraints on the precession angle λ, and the fortheearly-periodprofileislocatedonthesamesideofthe
azimuthal separation of beams φm,2. magneticaxisasthelate-seasonbeam,i.e.,whetherourline
Itisinterestingtoaskifthefuturedataallowsustodis- ofsighthaspassedoverthemagneticpolein2005(bymov-
criminatebetweentheconal-likeandthefan-beam(patchy) ing meridionally, eg. from the outer- to inner-traverse side)
models. In principle, the conal model seems to predict defi- As is the case in Fig. 9, the system of fan beams is pos-
nitereappearancetimes(eg.2035inPerera et al.2010;2034, sibly different (asymmetric) on both sides of the magnetic
2043, and 2066 in Lomiashvili & Lyutikov 2014), so if the pole(equatorwardvspoleward)whichleaveslotsofflexibil-
pulsarreappearsatadifferenttime,onlythefanbeammodel ity for the fan beam model to reproduce different pulse be-
willstayconsistent.However,the“conal”modelhasrecently haviourintheearlyandlateperiod.Unfortunately,through
gained much flexibility: the emission ring became a part of the modelling of the late-time data from PMK10, it is not
an ellipse with an adjustable eccentricity and an adjustable possible to distinguish between these scenarios. When the
circumpolar azimuth of its major axis. fan-beam model of Sect. 2 is used, these late-time data can
bewellreproducedbysolutionsofbothtypes,namelythose
forwhichoursightlinemovesacrossthepole,butalsothose
with the sightline retreating back after a close approach to
5 DISCUSSION
thepole (without thepole crossing).
In terms of probability,the fan-shaped beam appears to be The modelling freedom is even larger than that, be-
asuccessfulmodelforthelate-timeevolutionofthepulsarB cause the physical origin of profiles’ double form is not
profile.Thereisnoneedforfine-tuning,instead,thereexists known. Aside from the polar-tube-related conal interpreta-
a large space of parameter values which can reproduce the tions, multiple fan-beam types are possible, with various
observedroughlylinearchangesW.However,thisflexibility transverse emissivity profiles, and with different arrange-
preventsuniquedeterminationofpulsarandbeamgeometry. mentof fan-shapedsubbeams(cartoon examples areshown
This ambiguity also holds for the question of reappearance in Fig. 9). Divergence of the magnetic field suggests that
time of pulsar B. Since the model parameters are not yet beamsbecomewiderat largerdistancefrom thedipoleaxis
fullyconstrained,themodelcanadjusttoanydateonwhich (Fig. 9a), and this effect is visible in the three dimensional
MNRAS000,1–9(2016)
8 L. Saha and J. Dyks
Figure9.Varioustypesofsky-projectedfanbeams.Dottedlines
mark the dipolar B-field, and the central dot is the magnetic
Figure 10. A cartoon comparing the horseshoe beam fitted by
dipoleaxis.a)Singlebeamwhichgetswiderawayfromtheaxis,
Lomiashvili&Lyutikov(2014;greyarc)toahypotheticalsystem
as caused by the spreading B-field. b) Single beam of a fixed
oftwofanbeams(solidlinecontours).Becauseofroughlysimilar
width.c)Systemoftwofanbeamswhichdivergealongwiththe
location within the magnetosphere, the beams are suspected to
spreadingB-fieldandoverlapnearthedipoleaxis.d)Bifurcated
havesimilarorbitalvisibility.Sinceopensidesofthebeamspoint
beamwithlobesconvergingawayfromthedipoleaxis.e)Bifur-
in opposite directions, the observed increase in W requires an
cated beam with a fixed distance between its lobes. The latter
oppositeprecessiondirectionforeachbeamtype.
two beams (dand e) are typical of the X-modecurvature radia-
tion.Caseecorrespondstoafixedradiusofcurvatureofelectron
trajectory.
type of a beam (see Fig. 3 in Gil et al. 2004). The angu-
larseparation of emission lobes in thebeam is proportional
simulationsofWangetal.(2014,seefigs.5-12therein).How- to ρ−1/3 (eq. 3 in Dyks & Rudak 2012), so in the dipolar
ever,thefanbeamobservedinPSRJ1906+0746(Desvignes field it slowly decreases with the distance from the dipole
et al. 2012) does not show much broadening. This is not axis, as marked in Fig. 9d. If the changes of ρ along the
surprising, because the local magnetospheric emissivity de- particle trajectory are negligible, the beam 9e is expected.
pends on the emitted spectrum and the latter is sensitive In the macroscopic case the split-fan form may result from
totheenergydistributionofemittingparticles.Theemitted the aforementioned transverse density profile of the stream
intensity is then dependent on the acceleration and cooling (Fig.12inDyks& Rudak2015).Ashallowbifurcationcould
history of the radiating charges, which may be different at possibly appear also for a stream in which most plasma is
differentlaterallocationsintheemission region.Inthecase gyratingat a preferred pitch angle.
of the curvature radiation at standard conditions (charge Givenalltheseuncertainties(thelimitedtimespanand
Lorentz factor γ ∼102, curvatureradius of their trajectory qualityofdata,thelargespaceofwellfittingparameters,the
ρ∼107cm)thecurvatureradiationspectrumbarelyreaches modelflexibilityintermsof thenumberand location of fan
the observed ∼1 GHz band. Therefore, any transverse dif- beams, and the uncertain fan beam origin and geometry),
ferencesinaccelerationorcoolingcaninfluencetheobserved moredataforalargerintervalofprecessionphaseisrequired
width of the fan beam in a latitude-dependent way. If the toanswerthequestiononwhetherthefanbeamscanexplain
emission region has the form of a dense curved stream of the apparently different pulse behaviour observed in 2003
particles (such as shown in Fig. 12 of Dyks& Rudak 2015) and 2007.
thentheassociation ofplasmadensitywiththeemittedfre- This paper has made use of the ‘static-shape’ dipo-
quency (also present in the radius-to-frequency mapping) lar magnetic field, undistorted by the pulsar A’s wind.
will move the site of strong radio emission to the center of Such field geometry cannot be used to model the visibil-
the stream (at increasing altitude). It is for such observa- ity of PSR J0737−3039B in specific orbital phase intervals.
tionalandtheoreticalreasons,thatthelatitude-dependence Lomiashvili & Lyutikov (2014) carefully modelled the sys-
of the fan-beam width should be considered mostly uncon- temwith awind-distorted field.They showthattheorbital
strained, or at least not only determined by the spread of visibilitycanbereproducedwithinauniqueintervalofpre-
thedipolarmagneticfield.Afixed-widthexampleofsuchan cessionphase,althoughtheprecessiondirectionneedstobe
arbitrary beam is presented in Fig. 9b. opposite to that predicted by the general relativity. Their
Doublestructureoftheobservedprofilemayhavevari- result is based on a beam of a modified-cone type: it is a
ousreasons.Itmaybecausedbyemissionfromtwo,mostly piece of elliptic arc resembling a horseshoe, with its open
independent,nearbystreams,producingasystemconsisting sidedirectedtowardsthemagneticaxis.Letusconsiderthe
oftwofanbeams(Fig.9c).Alternatively,itmayresultfrom case in which the horseshoe beam (grey contour in Fig. 10)
asinglestreamwhichitselfemitsasplit-fanbeam(Fig.9d). is replaced with a V-shaped fan beam system considered
In this case the bifurcation may have either the micro- or in this paper (solid line contours in Fig. 10). Let the fan
macroscopic origin. The curvatureradiation in the extraor- beam be located within a similar region of magnetosphere
dinary mode is a known mechanism which produces this as the horseshoe beam (on the same side of the magnetic
MNRAS000,1–9(2016)
Fan beam model for PSR J0737 – 3039B 9
axis, within asimilar intervalof magneticazimuthsand co- Coles,W.A.,McLaughlin,M.A.,Rickett,B.J.,Lyne,A.G.,&
latitudes),exceptthebeam’sopensideisnowpointingaway Bhat,N.D.R.2005,ApJ,623,392
from thedipoleaxis.Becauseofthesimilar magnetospheric Desvignes, G.,Kramer,M.,Cognard, I.,et al.2012, inProceed-
location,thefanbeamshouldstillbecapableofreproducing ings of the International Astronomical Union, Vol. 8, Neu-
tron Stars and Pulsars: Challenges and Opportunities after
theorbitalvisibility.However,sincetheopensideofthefan
80years,199–202
beam is pointing away from the dipole axis, the observed
Dyks,J.,&Pierbattista,M.2015,MNRAS,454,2216
increase in W can only be reproduced for an opposite pre-
Dyks,J.,&Rudak,B.2012,MNRAS,420,3403
cession direction (consistent with the general relativity). It
—.2015, MNRAS,446,2505
is therefore reasonable to expect that the V-shaped beam Dyks,J.,Rudak,B.,&Demorest,P.2010,MNRAS,401,1781
pointingtowardsthedipoleaxis(anddirectingitsopenside Dyks, J., Serylak, M., Oslowski, S., et al. 2016, ArXiv e-prints,
away from it) can reproduce the bright orbital phases in a arXiv:1608.00770
morenaturalwaythantheconal-likehorseshoebeam(i.e.,in Gil,J.,Lyubarsky,Y.,&Melikidze,G.I.2004,ApJ,600,872
consistency with thegeneral relativity). Karastergiou,A.,&Johnston, S.2007,MNRAS,380,1678
Lomiashvili & Lyutikov(2014)haveshownthatthera- Kramer,M.1998,ApJ,509,856
Kramer,M.,Stairs,I.H.,Manchester,R.N.,etal.2006,Science,
dio beam geometry is not very sensitive to the wind dis-
314,97
tortionofthemagnetosphere:parametersoftheirhorseshoe
Lomiashvili,D.,&Lyutikov,M.2014,MNRAS,441,690
beam are consistent with those found by PMK10 for the
Lyne, A. G., Burgay, M., Kramer,M., et al. 2004, Science, 303,
staticshape(pure)dipole.Thissuggeststhatourinferences
1153
ontheperformanceofthefanbeamgeometryshouldprevail Lyutikov,M.,&Thompson,C.2005, ApJ,634,1223
in thecase of thewind-distorted magnetic field. Manchester, R. N., Kramer, M., Stairs, I. H., et al. 2010, ApJ,
710,1694
Michel,F.C.1987,ApJ,322,822
Perera, B. B. P., McLaughlin, M. A., Kramer, M., et al. 2010,
6 SUMMARY
ApJ,721,1193(PMK10)
Ransom, S. M., Kaspi, V. M., Ramachandran, R., et al. 2004,
This study shows that the late-time secular changes of
ApJ,609,L71
pulsedemission fromPSRJ0737-3039B canbeexplainedin
Wang,H.G.,Pi,F.P.,Zheng,X.P.,etal.2014,ApJ,789,73
theframework ofthefan-beam model.Theestimated space
Weisberg,J.M.,&Taylor,J.H.2002,ApJ,576,942
of acceptable model parameters is quite large as compared
to that for the conal-beam model. However, at least some
parameters should become more tightly constrained when Thispaper has been typeset fromaTEX/LATEX fileprepared by
theauthor.
the information about the reappearance of the pulse and
more data on W is providedby futureobservations.
It is suggested here that the radio beam of PSR
J0737−3039 consists of elongated patches with mostly
azimuthal orientation. The same or similar geometry
has appeared valid for two other precessing objects
(Manchester et al.2010;Desvignes et al.2012).Itisremark-
ablethatpulsarswithpatchyorfanbeamsmakeupforsome
50% ofprecessingobjects, forwhich beammappinghasap-
peared feasible so far.
ACKNOWLEDGEMENTS
The authors thank B. Perera and M. McLaughlin for the
GBT data, as well as M. Burgay and R. Manchester for
the Parkes data on PSR J0737−3039B. We also thank
A. Frankowski and our referee for useful comments. This
work was supported by the National Science Centre grant
DEC-2011/02/A/ST9/00256.
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