Table Of ContentPubl.Astron.Soc.Japan(2014)00(0),1–13 1
doi:10.1093/pasj/xxx000
Evidence for a 36 ks Phase Modulation in the
Hard X-ray Pulses from the Magnetar 1E
1547.0−5408
6
1
0 Kazuo MAKISHIMA,1,2,3 Teruaki ENOTO,4,5,6 Hiroaki MURAKAMI,1 Yoshihiro
2
FURUTA,1 Toshio NAKANO,1,2 Makoto SASANO,1 and Kazuhiro NAKAZAWA1,2
b
e 1 DepartmentofPhysics,TheUniversityofTokyo,7-3-1Hongo,Bunkyo-ku,Tokyo113-0033,
F
Japan
0 2 ResearchCenterfortheEarlyUniverse,TheUniversityofTokyo,
1
7-3-1Hongo,Bunkyo-ku,Tokyo113-0033,Japan
] 3 MAXITeam,TheInstituteofPhysicsandChemicalResearch(RIKEN),
E
2-1Hirosawa,Wako,Saitama351-0198,Japan
H
4 HighEnergyAstrophysicsLaboratory,RIKENNishinaCenter,
h. 2-1Hirosawa,Wako,Saitama351-0198,Japan
p 5 TheHakubiCenterforAdvancedResearch,KyotoUniversity,Kyoto606-8302,Japan
o- 6 DepartmentofAstronomy,KyotoUniversity,Kitashirakawa-Oiwake-cho,Sakyo-ku,Kyoto
r 606-8502,Japan
t
s
a ∗E-mail:[email protected]
[
Received;Accepted
2
v
Abstract
4
8 The Suzaku data of the highly variable magnetar 1E 1547.0−5408, obtained during the 2009
6
Januaryactivity,werereanalyzed. The2.07spulsation,detectedinthe15–40keVHXDdata,
7
0 was found to exhibit phase modulation, which can be modeled by a sinusoid with a period of
. 36.0+4.5 ks and an amplitude of 0.52±0.14 s. While the effect is also seen in the 10–14 keV
1 −2.5
0 XIS data, the modulation amplitude decreased towards lower energies, becoming consistent
5
with 0 below 4 keV. After the case of 4U 0142+61, this makes the 2nd example of this kind
1
of behavior detected from magnetars. The effect can be interpreted as a manifestation of
:
v
torque-free precession of this magnetar, which is suggested to be prolately deformed under
i
X thepresenceofstrongtoroidalfieldof∼1016 G.
r
a Key words: Stars:individual:1E 1547.0−5408— Stars:magnetars — Stars:magnetic field —
Stars:neutron
1 Introduction acteristictwo-componentX-rayspectra,consistingofthelong-
observed soft component and the mysterious hard component
Magnetars(e.g.,Thompson&Duncan1995;Mereghetti2008)
(Kuiperetal.2006;denHartogetal.2006;Enotoetal.2010b).
are considered to be magnetically-powered isolated neutron
stars (NSs) having ultra-strong dipole magnetic fields, Bd = Magnetarsarethoughttoharborevenhighertoroidal(orin-
1014−15 G, as derived from their pulse periods P and period ternal) magnetic fields Bt (e.g., Thompson & Duncan 1995;
derivatives P˙. These values of Bd exceed so-called critical Harding&Lai2006;Braithwaite2009;Dall’Ossoetal.2009).
field,4.4×1013G,abovewhichtheLandaulevelseparationsof Thesefieldsmaybegeneratedandamplifiedintheprogenitor’s
electrons exceed their rest-massenergy. Magnetars emitchar- core,duringitsfinalgravitationalcollapseinvolvingstrongdif-
(cid:13)c 2014.AstronomicalSocietyofJapan.
2 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
ferentialrotation(e.g.,Takiwakietal.2009). Somemodelsex- effectisinterpretedasamanifestationoffreeprecessionandT
plaintheproduction ofenergeticparticlesinmagnetarsinvok- isidentifiedwiththeassociatedslipperiod,equation(2)yields
ingtoroidalmagneticfieldsandtheassociatedtwistedmagnetic ǫ=1.6×10−4,andhence equation (1)implies Bt ∼1016 G.
configuration(e.g.,Beloborodovetal.2013). ThisprovidesoneofthefirstdirectestimationsofBtofamag-
Observationally, thepresenceofintenseBt issupportedby netar.
the recently discovered low-Bd magnetars, SGR 0418+5729
(Rea et al. 2013), SGR1822−1606 (Rea et al. 2012), and
As seenabove, theessential assumptions madebyMEA14
3XMMJ185246.6+003317 (Reaetal.2014);theirburstactiv-
tointerprettheobservedphasemodulationof4U0142+61are;
itysuggeststhattheirinternalfieldsaremuchstrongerthantheir
ǫ6=0, α6=0, and the broken axial symmetry of the hard X-
externallyobservablefieldsofBd=6×1012 G,2.7×1013 G,
and<1.4×1013G,respectively.Thepossibleprotoncyclotron rayemissionpatternaroundxˆ3. Iftheseassumptionsgenerally
holdinobjectsofthisclass,weshouldobservethesameeffects
featureinSGR0418+5729,whichsuggestsalocalsurfacefield
from other magnetars as well. Ifdetected from a fair number
of > 2×1014 G (Tiengo et al. 2013), provides another sup-
ofthem,theeffectwillprovidesuchimportantinformationas;
port,becauseapartofsuchintenseinternalfieldswouldemerge
fromtheNSsurfacetoformlocalmultipoles. Thus,thestrong howBtisdistributedamongthem,whetherBdandBtarecor-
related,andwhetherBtactuallydecreases(aspostulatedinthe
Bt mayberegardedasoneofthemostimportantpropertiesof
magnetar scenario itself; Thompson & Duncan 1995) towards
magnetars.However,itremaineddifficulttomoredirectlyesti-
objectswithlargercharacteristicages.Theresultswillalsopro-
mateBt,becausetoroidalfieldsareintrinsicallyconfinedinside
videvaluable clues totheorigin ofthehardX-raycomponent
theNSandinvisiblefromoutside.
(Kuiperetal.2006;denHartogetal.2006),tothenatureofthe
Afterpioneeringworksby,e.g.,Carlini&Treves(1989)and
intriguingspectralevolutionofmagnetarsdiscoveredbyEnoto
Heyl & Hernquist (2002), the way around this difficulty was
et al. (2010b), and to the ultimate origin of these mysterious
exploredbyMakisihmaetal. (2014; hereafterMEA14)inthe
objects(e.g.,Vink&Kuiper2006;Nakanoetal.2015). Thus,
followingway. WhenanNShasBt∼1016 G,isitexpectedto examiningothermagnetarsforthehardX-rayphasemodulation
bedeformedintoaprolateshape(Ioka 2001;Cutler2002;Iokai
isextremelyimportantandurgent.
&Sasaki2004),toadegreeof
ǫ≡(I1−I3)/I3∼1×10−4(Bt/1016G)2 (1) In the present paper, we have chosen the highly variable
where I3 is the moment of inertia around the NS’s symmetry magnetar 1E 1547.0−5408 asour 2nd target. Itisthe fastest-
axis xˆ3, and I1 that around axes orthogonal to xˆ3. Under the rotatingmagnetarwithP =2.07s,possiblyassociatedwiththe
absenceofexternaltorque,suchanNShasaconstantrotation supernovaremnantG327.24−0.13(Gelfand&Gaensler2007).
periodProt≡2πI3/Laroundxˆ3,andaconstantprecessionpe- The observed value of P˙ = 2.4×10−11s s−1 (Kuiper et al.
riod Pprec≡2πI1/L around the angular momentum vector L~ 2012), together with P, yields Bd =2.2×1014 G and avery
(withL≡|L~|)whichisfixedtotheinertialframe. Thesetwo young characteristic age of τ ≡ P/2P˙ = 1.4 kyr. On 2009
January 22, a strong bursting activity was detected from 1E
periods, which were degenerate when spherical, thus become
different just by ǫ. If xˆ3 is tilted from L~ by a non-zero wob- 1547.0−5408 with Swift (Gronwall et al. 2009). This activ-
itywascontinuouslycoveredbyRXTE,INTEGRAL,andSwift
blingangleα,theNSissaidtoexhibit freeprecession, which
observations(Kuiperetal.2012). Thesemonitoringstudiesre-
is the most basicbehavior of atorque-freerigid body with an
vealed an increase in the total flux at the burst onset by more
axialsymmetry(Landau&Lifshitz1976;Butikov2006). This
than an order of magnitude, followed by a rapid decline, and
conceptshouldbedistinguished fromthemorefamiliarforced
complexchangesinthespin-downcharacteristicsalongtheac-
precessiondrivenbyexternaltorque,asseeninspinningtops.
tivityevolution.
Suppose that a precessing NS has a radiation pattern that
breaksaxialsymmetryaroundxˆ3.Then,asarguedbyMEA14,
theobservedpulseswillbephase-modulatedatthebeatperiod
As already reported by Enoto et al. (2010a) (hereafter
betweenProtandPprec,whichiscalledslipperiodandisgiven Paper I) and Enoto et al. (2012), this object was observed by
as
Suzakuduringits2009Januaryoutburst,andyieldedbyfarthe
T ≡Pprec/ǫ=(1/Prot−1/Pprec)−1. (2) strongesthardX-ray(>10keV)signalsamongsome10mag-
netarssofarobservedwith Suzaku. Although this sourcewas
Using the Suzaku X-ray observatory (Mitsuda et al. 2007), observed with Suzaku again on 2010 August, these follow-up
MEA14infactfoundthattheP=8.687spulsationofthespec- data are not utilized here, since the source had faded by that
tralhardcomponentofthemagnetar4U0142+61issubjectto timeinhardX-raystoalevelcomparabletotheHXDdetection
phase modulation with aperiod of T =55±4 ks. When this limit(Iwahashietal.2013).
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 3
2 Observation to the normalization of Brazier (1994), which we adopt here,
should obey achi-square distribution with 2ndegrees offree-
As described in Paper I and Enoto et al. (2012), the present
dom.Thehighernoisetoleranceofthistechniqueisrealizedby
Suzaku observation of 1E 1547.0−5408 was performed from
discardingthepowerinallharmonicshigherthanisspecified.
2009 January 28 21:34 (UT) to January 29 21:32 (UT), for
Figure1ashowsperiodogramscalculatedinthiswayusing
a gross duration of 86 ks, with the target placed at the HXD
the 2–10 keV XIS0 data and the 12–70 keV HXD-PIN data,
nominal position. This was a Target-of-Opportunity observa-
both including backgrounds. The harmonic number was cho-
tion in response to the activity onset, and this epoch falls on
sentoben=2,consideringthedouble-peakedpulseprofileof
“segment 3” (from 2009 January 22 to February 1) of Kuiper
this source (e.g., Paper I). These energy bands, as well as the
etal.(2012). AmongthethreeoperatingcamerasoftheX-ray
trial period range (2.0713–2.0730 s), arethe sameas those in
Imaging Spectrometer (XIS: Koyama et al. 2007), XIS0 was
PaperI,whiletheemployedperiodsearchstep,1×10−6s,is4
operatedinthetimingmode(P-summode),witha7.8mstime
timesfiner. Theresultsreconfirmessentialfeaturesoffigure3
resolution. The Hard X-ray Detector (HXD: Takahashi et al.
ofPaperI,includingthebest-estimateperiodof
2007)wasoperatedinthestandardmode,yieldingatimereso-
lutionof61µs. Furtherdetailsoftheobservationarefoundin
P0s=2,072.135±0.005ms (3)
PaperI.
Below,wemainlyutilizethedatafromXIS0andHXD-PIN referringtotheXISresult,andthesidelobepatterns.TheHXD
processedasdescribedinPaperI,inwhicheachdetectedevent periodogrampeak,
istaggedwiththeenergyandthearrivaltime.ThosefromXIS1
andXIS3hadtoolowatimeresolution(2.0s)toresolvethepul- P0h=2,072.147±0.005ms, (4)
sation,andsignalstatisticswereratherlowinHXD-GSO.The is consistent with equation (3) within errors, although it is
netexposuresachievedwithXIS0was42.6ks,whilethatwith slightlydisplacedtolongerperiods. Thisbehaviorisalsoseen
the HXD was 33.5 ks. After subtracting the background and in figure 3 of Paper I. The errors in P0s and P0h refer to the
applying dead-time corrections, the object was detected with range where pulse-phase differences amount to ±10% of one
HXD-PIN at a 15–70 keV count rate of 0.174±0.004 c s−1 pulsecyclewhenaccumulatedoverthegrossdurationof86ks.
(PaperI),anda1–10keVrateof5.5±0.1cs−1withXIS0. These errors approximately agree with widths of the peaks in
Asdescribed inPaperI,13short-burst candidates werere- theperiodograms.
moved, and arrival times of the XIS0 and HXD-PIN events
At P0h, the 12–70 keV HXD periodogram in figure 1a
were photon-by-photon converted to those to be measured at reachesthepeakheightofX=28.21,whereZ2isabbreviated
2
thesolar-systembarycenter. Theconversionutilizedthesource
toX. Intermsofchi-squaredistribution with 2n=4degrees
position of (α2000,δ2000)=(15h50m54s.11,54◦18′23.′′7), as offreedom, thechanceprobability offinding X values higher
wellasthespacecraftorbitalinformation. thanthisisfoundasPh=1.1×10−4;here,thesubscriptspec-
2
ifiestheharmonicnumbern,andthesuperscript“h”meansthe
HXD.Thisagreeswiththeprobablity of∼1×10−4 obtained
3 Data analysisand results
inPaperIin10–70keVwiththechi-squaretechnique.
3.1 ReconfirmationofPaperI
In figure 1a, the XIS and HXD periodograms both show a
Since the present paper re-analyzes the same data set of 1E number of side lobes. Such a structure at a period P′ often
1547.0−5408 as used in Paper I, let us begin with reproduc- arisesduetobeatbetweenthepulsationatP0(representingP0s
ing the timing results of that publication. While Paper I uti- orP0h)andacertainlongerperiodT.Basedonthisassumption,
lizedtheconventionalchi-squaremethodtosearchforthepul- we calculated the required period as T =|(1/P0−1/P′)−1|,
sations, here we instead employ the Z2 technique (Buccheri and show it in green at the top of figure 1. In addition, the
n
et al. 1983; Brazier 1994) which is free from the ambiguity required values of T (inunits ofks) of major XIS0and HXD
of phase-bin number per pulse cycle and is generally less af- sidelobesaregivenexplicitlyinthefigure. Thus,manyofthe
fectedbythePoissonerrors. ForeachtrialpulseperiodP,this sidelobes (particularlyofXIS0)infigure1acanbeexplained
techniqueeffectivelycalculatesthefoldedpulseprofileusinga as a beat of P0 with T =5−6 ks or T =10−12 ks, which
sufficientlylargebinnumber(inpracticenotspecified). Then, agree with the Suzaku’s orbital period Porb =5.6 ks or twice
the Fourier power of the profile is summed up to a specified this value, respectively. However, the HXD-PIN periodogram
harmonicnumbern(n=1,2,...), andnormalizedinacertain showsseveralunexplainedpeaks;atT=24.8ksand38.7kson
mannertoreflectthedatastatistics. Thisprocessyieldsthesta- theleftsideofP0,aswellasatT =51.8ks,21.1ks,and18.1
tistical quantity calledZ2, which evaluates thesignificance of ks on theright side. Thesestructures, alsoseen infigure3of
n
periodicity atP. Ifnointrinsic periodicity ispresentatP ex- PaperI,suggestthatthehardX-raypulsescouldbeaffectedby
ceptPoissonianfluctuations,theZ2 valuecalculatedaccording somelongperiodicityintherangeofT =18−52ks(5–14hr).
n
4 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
3.2 DetailedanalysisoftheHXDdata Suzaku HXD on relatively short time scales (Kokubun et al.
2007), the pulse phase is seen to vary clearly in figure 2a, by
Below,weanalyzetheHXD-PINdatainfurtherdetails,select-
up to about half a pulse cycle. On the other hand, we do not
inganenergyrangeof15–40keV.Thelower energyboundis
noticeinfigure2aparticularsecular changesinthepulsepro-
thus raisedfrom 12keV usedin figure1a to 15 keV, to avoid
file,pulsedflux,orpulsephase. Thisdoesnotcontradicttothe
increasedsoftthermalnoise.Likewise,theupperenergybound
discoverybyKuiperetal.(2012)oftheemergingnewcompo-
is lowered from 70 keV to 40 keV, to reduce the contribution
nentinthehardX-raypulseprofile,becausethegrowthofthat
fromthenonX-raybackground.
componentisestimatedtobeatmost∼10%acrossthepresent
Suzakupointing, asjudgedfromthehistoryofthehardX-ray
3.2.1 EvolutionofhardX-raypulseprofiles
pulsedflux(figure9ofKuiperetal.2012).
TheZ2periodogramcalculatedfromthe15–40keVHXD-PIN
2
dataisshowninfigure1binblack. Exceptsomesmalldiffer-
3.2.2 Periodicphasemodulation
ences,itissimilartothe12–70keVresultsshowninbrownin
Wenextneedtoknowwhetherthepulse-phasefluctuationsseen
panel(a). Thepeakvalue,alsoobtainedattheperiodofequa-
in figure 2a are periodic or not. This requires a finer time
tion(4),becomes
division by increasing J, but individual pulse profiles would
X0≡(X)A=0=22.06, (5) then lose statistics. Therefore, we chose another way; the
data accumulation was repeated with J =20 and K =5, but
whichissomewhatsmallerthanthatin12–70keV.
the results are presented in figure 3a in the form of five light
In the 15–40 keV periodogram, the period range of T =
curves,{C(j,k);j=0,..,J =19}usingjasabscissa.Forpre-
18−52 ks, where the unexplained side lobes were noticed in
sentation, 3-point running averages were again applied over
figure1a,isapproximately preserved(although detailedofthe
(j−1,j,j+1). Thus,thebackground-inclusive countratesin
side lobes changed). Since this period range does not contain
thefivepulsephasesmostlyvariedratherperiodically,by2to3
particularartificialperiodicitiesoftheobservationsuchasdata
cyclesacrossthe86ksgrosspointing, although theamplitude
gapsorbackgroundvariations,someproperties(e.g.,amplitude,
iscomparabletothestatisticalerrors.
phase,orpulseshapes)ofthe2.07shardX-raypulsationcould
WecalculateddiscreteFourierpowerspectraofthefivelight
besubjecttoslowvariationswithperiodsinthisrange. Infact,
curves in figure 3a, before taking the running averages, and
Kuiper etal.(2012) found that the pulse shapes werechange-
summedthemtoobtainasinglepowerspectrumwhichisshown
ableduringtheir“segment3”whichcoversthepresentSuzaku
by a red solid line in figure 3c. It indeed reconfirms the sug-
observation,thatanewintermediatepeakinthe>4keVpulse
gestedpowerenhancement,atwavenumbersofq=2andq=3,
profileemerged,andthatthepulsedhardX-rayfluxincreased
withthecorrespondingperiodrangeof
meantimealthoughthetotal4–27keVfluxwasrapidlydeclin-
ing. 86/3=29ks to 86/2=43ks. (6)
To examine the evolution of the hard X-ray pulse proper-
Theweightedmeanofq=2andq=3overthepowerspectrum
ties, we divided the whole observation into consecutive time
infigure3cbecomes
segments of approximately equal lengths, denoted as j =
0,1,...,J−1(j=0atthebeginningandj=J attheend),and
q=2.45, or T =35.1ks, (7)
sortedtheentire15–40keVHXD-PINevents(includingback-
grounds) into atwo-dimensional arrayC(j,k). Here, the col- whichisfullyconsistentwiththeimplication ofthesidelobes
umnk=0,1..,K−1representsthepulsephase(i.e.,thephoton infigure1.
arrivaltimesmoduloP0h).Thus,aset{C(j,k);k=0,..,K−1} Asa“control”study, figure3bpresentsanothersetoflight
givesthepulseprofileinthej-thtimesegment. Weemployed curves,calculatedinthesamewaybutpurposelychangingthe
J =6 and K =12, and show in figure 2a the obtained time- folding periodfromP0 toadummyvalue of2.0s. Itsbehav-
dividedsixpulseprofilesasafunctionofk.There,runningav- ior is clearly more random, without particularly favored peri-
erageswereappliedto3consecutivedatapoints(k−1,k,k+1). ods.Indeed,these5controllightcurvesyieldedthebluepower
Thus, the pulse amplitude changes with time, and the pulse spectruminfigure3c,whichisconsistentwithwhitenoise.
shape is also variable; it is relatively flat-topped with a single Toevaluatethesignificanceoftheq=2andq=3enhance-
deepminimum, with anoccasional shallower secondary mini- mentsintheredpower spectrumoffigure3c,werepeatedthe
mum at about half acycle offset. Such a short-term variation same control study 20 times, by changing the dummy period
in the pulse shape was observed from 4U 0142+61 (Enoto et around2.0s. Thederivedensemble-averagedpowerspectrum,
al. 2011), as well as from 1E 1547−5408 itself (Kuiper et al. shown in figure 3c by a dashed black line, is approximately
2012). white(withhintsofrednoiseatq=1andbackgroundvariation
More importantly, thanks to the high sensitivity of the atq=4correspondingto5Porb).Aroundthismean,individual
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 5
powerspectraexhibiteda1σscatterby∼±0.5. Therefore,the panel (a) shows a two-dimensional color map of the value of
q=2andq=3excess(above1.0)intheredpowerspectrumis X ≡Z2 (the maximum as P is varied) obtained on each grid
2
estimatedtobe2.4and1.6sigmaeffects,andtheirsumimplies point(A,φ),foraparticularcaseofT=36.0ks.Itsprojections
a∼3sigmaexcess. ontheφandAaxesarepresentedinpanels(b)and(c),respec-
Sincethefivelightcurvesinfigure3adonotvaryinphase, tively,wheremultipletracesindicatecrosssectionsofpanel(a)
thesuggestedlongperiodicity isunlikely toresultfrompulse- atvariouspositions.WhilethevalueofX0inequation(5)isre-
amplitude variations, but considered to reflectthe pulse-phase producedatA=0inpanel(c),aclearpeakisseenatA∼0.52
jittering found in figure 2a. Thus, the hard X-ray pulsation sandφ∼270◦. ThisvalueofAagreeswellwithwhatcanbe
is inferredto exhibit periodic phase modulation. To reinforce readfromfigure2b. Furthermore,thevalueofφ,togetherwith
this inference,weaccumulated thepulseprofilesin6separate equation(8),predicts∆t=+A(themaximumdelay)atabout
phases of an assumed periodicity of T =36 ks, of which the thebeginning(t∼0)oftheT=36ksphase,againinagreement
choiceisjustifiedlater. Theresultispresentedinfigure2b;in withfigure2b. Althoughfigure2aapparentlysuggestsarather
thiscase,therowjofthearrayC(j,k)denotesthepulsephase, longperiodicity (e.g.,T >100ks),thisisprobably abeatbe-
i.e., the arrival times modulo T, rather than the arrival times tweentheT =36ksmodulationandtheNyquistperiodtherein
themselves.Itvisualizeaperiodicandapproximatelysinusoidal (1×86/6=29ks),whichshouldappearat140ks.
pulse-phasemodulation,withanamplitudeof∆k∼±3.5bins In panels (a)-(c) of figure 4, the peak significance reaches
(±0.3pulsecycleor±0.6s)whichisconsistentwiththeinfer- Xpk=45.60;here,Xpk(T)denotesthepeakvalueofXfound
encefromfigure2a. ona(A,φ)planeforagivenvalueofT. Panel(d)offigure4
shows how this Xpk(T) depends on the assumed modulation
3.2.3 Demodulationanalysis periodT.Overtheentire(P,T,A,φ)spacesurveyed,thehigh-
The above semi-quantitative analysis justifies us to apply the est value of X, to be called the grand-maximum and denoted
“demodulation analysis”, developed by MEA14, to the HXD Xmax,isthusfoundas
data, to more rigorously quantify the pulse-phase modula-
tion. Specifically, we consider that the hard X-ray pulsa- Xmax≡Xpk(36.0ks)=45.60, (9)
tionof1E1547.0−5408 issomehowphasemodulated, likein
4U 0142+61, at a period T in the range of equation (6). We forasetofmodulationparametersgivenas
hencehenceassumethatthepeaktimingtofeach2.07spulse
isdisplacedsinusoidallyby A=0.52±0.14s, φ=276◦±15◦, T =36.0+−42..55 ks. (10)
∆t=Asin(2πt/T−φ), (8)
Here,theerrorsarerepresentedinthesamewayasinMEA14,
i.e., by the standard deviations of Gaussians fitted to the dis-
where A and φ are the amplitude and initial phase of the as-
tributions (aboveuniformbackgrounds)infigure4. Thisvalue
sumedmodulation,respectively.
ofT,alreadyemployedtoproducefigure2b,agreeswellwith
As formulated in MEA14, such effects can be removed
equation(7).
by shifting the arrival times of individual HXD photons back
by −∆t. This procedure was previously employed, e.g., by Asshowninfigure1binred,thedemodulationwithequation
Koyamaetal.(1989)tosearchfororbitalmotionsintheanoma- (10)haschangedtheHXD-PINperiodogramsignificantly. The
lous X-ray pulsar 1E 2259+586, and by Terada et al. (2008) peak at P0h has become much higher, just from X0 to Xmax.
to remove the orbital effects in the fastest-spinning accreting Furthermore, the Porb-unrelated side lobes have diminished;
white dwarf binary, AE Aqr, and successfully detect its hard powers therein are thought to have been restored to the main
X-ray pulsations. We hence applied these time displacements peak.
totheindividual HXDphotons, andre-calculatedtheZ2 peri- Infigure2c,weagaincalculatedthe15–40keVfoldedpulse
2
odograms to see whether the pulse significance changes. The profiles in the same 6 time segments as in figure 2a, but af-
pulse period was scanned over a range of P = 2.072130− ter applying the timecorrections by equation (8)toindividual
2.072152(withastep1µsec),i.e.,thejointerrorrangeofequa- photons usingtheparametersinequation (10). Thus, thecon-
tion (3) and equation (4). The triplet (T,A,φ) describing the secutive6profileshaveindeedlinedupintheirphases.Asmall
phase modulation was scanned over A=0−0.80 sec(with a subpeak, justpriortotheminimum,isseeninatleast4outof
0.05secstep),φ=0−360◦ (3◦−10◦ step),andT =10−80 the6profiles.Theseresultsreinforcethevalidityofthedemod-
ks(withastepof0.2to0.5ks).Theharmonicnumbern=2is ulationanalysis. Figure2dreproduces,anddirectlycompares,
retained. thetime-averaged15–40keVpulseprofilesbeforeandafterthe
The results of this demodulation analysis are presented in demodulation. Thus,thepulseamplitudehasbeennearlydou-
figure4,justinthesameformatasfigure2ofMEA14. There, bled.
6 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
3.2.4 Significanceofthephasemodulation forachi-squaredistributionofanydegreeoffreedom. Thus,a
Although the phase demodulation procedure largely increased valueofXmaxwhichisequaltoorlargerthanequation(9)will
theHXDpulseamplitude(andhencethepulsesignificance),we appearwithachanceprobabilityof
needtoexaminewhetherthisincreaseisstatisticallysignificant,
f(45.60)=(3±1)×10−3, (12)
ormerelyduetoachancesuperpositionofPoissonnoiseofthe
dataonthealreadyexistingpulseperiodicityatP0h. inagoodagreementwith theimplication ofpanel(a). Asno-
The nominal chance probability of finding values of X ≥ ticedinAppendix, thisestimateisconsideredratherconserva-
Xmax of equation (9) is extremely small, P2h = 3.0×10−9. tive.
However, we cannot use this face value to evaluate the statis- Considering that MEA14 used n = 4, it may be interest-
tical significance of the detected effect, for the following two ing to examine what happens if different values of n are em-
reasons. ployed. We hence repeated the same analysis with n=1, 3,
1. This P2h must be multiplied by the total number of inde- and 4, to find Xmax =25.64, 47.06, and 49.92, respectively.
pendent trials involved in the computation of figure 4, but The optimum parameters were not much different from those
this quantity is not easy to estimate. The total number of inequation(10). Althoughthelargerharmonicnumberssome-
parametersearchsteps,actuallyemployed,wouldconsider- what increased Xmax from equation (9), the effect is not sig-
ablyoverestimatethis number,becausethestepswerecho- nificant considering the increased degree of freedom. In fact,
sen, rather arbitrarily, to be fine enough not to miss the X the implied chance occurrence probability, P1h =2.8×10−6,
peaks. Ph =1.8×10−8, and Ph =4.2×10−8, areall much higher
3 4
2. WhenevaluatingthesignificanceofXmax,weneedtocon- thanP2h obtained above,inagreementwiththedouble-peaked
siderthefactthattheHXDsignalswerealreadypulsing,be- pulseprofiles.Therefore,weretainn=2below.
forethedemodulationprocedure,atP0h(PaperI)withasig- Inshort,wecanexclude,at>99.6%confidence,thepossi-
nificanceofX0≫n. Thevariationsinthepulseamplitude, bility that Xmax of equation (9) in figure 4 arised via chance
profiles,andphase(figure2)wouldfurthercontributetoX. superposition of the photon count fluctuations on top of the
hardX-raypulsation ofwhich theparameters varywith white
These two problems can be combined into a single question:
noise. Thisconfidencelevelismorestringentthanthecaseof
when we scan over P, A, φ and T, like in figure 4, what is
4U0142+61,whereitwas>99%(MEA14).
theoverallchanceprobabilityfortheXstatisticstoincreaseby
chancefromX0toXmaxorhigherwhenthesourceispulsating
withvariableparametersbutwithoutanyexcessvariationatT 3.3 AnalysisoftheXISdata
abovethewhitenoise?
The two-component spectral model, fitted in Paper I to the
Toanswertheaboveinquiry,weconducted1,000simulation
XIS+HXDspectrumof1E1547.0−5408,indicatesthatthesoft
runsasdetailedinAppendix, usingtheactual15–40keVdata
thermal component and the hard power-law component cross
themselves,butpurposelyrandomizing,eachtime,thephaseof
overat∼5keV.Therefore,thehardestendoftheXISenergy
the assumed 36 ks modulation: any white noise would not be
band,withitsP-sumtimeresolution,isexpectedtoprovidein-
affectedbysuchrandomization. Figure5ashowsadistribution
dependent information on the hard component, and allows us
ofthegrand-maximumvalueXmax obtainedinthissimulation
tocrosschecktheHXD-PINresults.(Thiswasnotfeasiblewith
study. Itisconsidered torepresenttheprobability distribution
4U0142+61,sincetheXISwasnotoperatedintheP-summode
ofXmax,tobefoundinthesamedemodulationanalysisofthe
inthatobservation.) Tolimitthesoft-componentcontributionto
HXDdataoverthesame(P,T,A,φ)spaceasfigure4,butun-
<10%,wechooseanenergyrangeof10–14keV,andperform
der the absence of power enhancement (above a white noise)
thesameanalysisasabove. Althoughthisenergyrange,which
atanyT examined. There,valuesexceedingequation(9)were
isinfactcoveredbytheXIS,isseldomutilizedinspectralstud-
actually found in three out of the 1,000 runs, namely, with a
ies mainlybecause ofthecalibration uncertainty therein, such
probabilityof0.3%.
problemsdonotaffecttimingstudies.
Figure 5b gives the upper integral probability associated
Figure 6 show the results of this demodulation analysis on
withfigure5a. IntherangeofXmax>35, itisapproximated
the10–14keVXIS0data,presentedinthesamewayasfigure4.
byanexponentialas
Panels(a),(b),and(c)allrefertotheconditionofT =36.0ks
f(>Xmax)=f0exp −Xmax−40 (11) afterequation(10).Onthe(A,φ)plane,aclearpeakwith
(cid:18) g0 (cid:19)
Xpk(36ks)=27.64 (13)
with f0 =0.038±0.005 and g0 =2.2±0.2, where Xmax =
40.0 was chosen as a reference pivot point. The value of g0 is seenatasimilar position tofigure4a. Morespecifically, A
is consistent with the exponential factor, 2.0, that is expected andφcanbeconstrainedas
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 7
◦ ◦
A=0.46±0.11s, φ=260 ±17 . (14) disappearsinthe1–4keVrange.
Taking the 7–10 keV case as an example, we again calcu-
These values agree, within errors, with the corresponding re-
latedthe12-binfoldedpulseprofilesfor6phasesofT =36ks,
sultsinequation(10).Ifthemodulationparametersarefixedto
inasimilarmannertofigure2b.Theresult,showninfigure8d,
thebest-fitvaluesofequation(10)asdeterminedwiththeHXD,
againvisualizesthesinusoidalphasemodulation,clearlywitha
weinsteadfind
smalleramplitudethaninfigure2b. Theamplitudecanberead
Xpk(36ks)=17.08. (15) as∆k/K∼±2/12=±0.17pulsephase,orA=0.17P0h=0.35
s, in a good agreement with figure 8a. In this case, the pulse
Thedemodulationwithequation(14)haseffectsonthe10–
peak is more clearly traced than the valley. Furthermore, the
14 keV periodograms as shown in figure 7a. While the value
modulation phase is somewhat shifted from equation (10) to
ofX∼5foundbeforethedemodulation(black)iswithinran- φ∼360◦.
domfluctuations, aclearpeak,ofwhichtheheightisgivenby
Combining figure 8 with equation (10) and equation (14),
equation(13),appearedataperiodthatisinfullagreementwith
theenergydependenceofAisvisualizedinfigure9.There,the
equation(3)andequation(4).
dashed curverepresentsthefractional contribution ofthehard
Figure 7b shows 10–14 keV XIS0 pulse profiles folded at
componentatthatenergy,namely
P0s, obtained using the demodulation parameters of equation
(14)andpresentedinthesamecolorsasinfigure2d.Again,the A=A0×Fh/(Fs+Fh) (16)
demodulation proceduresignificantly increasedthepulsefrac-
tion. The red pulse profile is similar to that in figure 2d, al- whereA0=0.5sisafiducialamplitudeatthehardestenergy
thoughtherelativedepthsofthetwopulseminimaarereversed. limit, while Fs and Fh are the intensities of the soft and hard
Forreference,1–10keVXIS0pulseprofileisalsoshown. components, respectively, according to the spectral decompo-
sition (ModelB)inPaperI.Interesting enough, this empirical
TheT-dependence of Xpk is shown infigure6d. Thus, in
quantity behaves verysimilarlytotheobservedenergydepen-
additiontoasmallhumpatT =36−38ks,weobservemany
otherpeaks,somebeingmuchhigher. Amongthem,thesharp denceofA,suggesting thatthephasemodulation islimitedto
thehardX-raycomponent.
featureatT ∼11ksislikelytobecausedbybackgroundvari-
ations, because it is close to 2Porb. Besides, the plot reveals Givenfigure9,weattemptedtodividethe15–40keVHXD-
a prominent peak at T = 44 ks, which is probably due to a PINbandinto finerranges,ortoextend theanalysis tohigher
chance superposition of Poissonfluctuations ontop of the de- energiesbeyond40keVupto∼70keV.However,duetolim-
modulated pulsation. As a result, we cannot claim the pres- ited statistics, the data provided no significant information on
enceoftheT =36ksperiodicitybasedontheXISdataalone. thepossibleenergydependenceoftheHXDphasemodulation.
Nevertheless, theXISresultssignificantly reinforcetheHXD- Likewise,theHXD-PINdatabelow15keV(downto 10keV)
detected phasemodulation: sincethepulsation isinsignificant had so poor statistics that they did not allow us to cross con-
intherawperiodogram(figure7a),theprobabilityoffindinga firm the 10–14 keV XIS results. All what can be said is that
peakofXpk=17.08ofequation(15)atthesame(P,T,A,φ) the pulse-phase modulation in the HXD-PIN signal is consis-
pointasspecifiedbytheHXDresult,canbedirectlyevaluated tent with being energyindependent overthe10–70 keVrange
withasimplechi-squaredistributiontobePs=1.9×10−3. If withinratherlargeuncertainties.
2
weinsteaduseequation (13),theprobability furtherdecreases
toPs=1.5×10−5.
2 3.5 AnalysisoftheHXD-GSOdata
AsdescribedinPaperI,thesignalfrom1E1547.0−5408 was
3.4 Energydependenceofthemodulationamplitude
marginallydetectedinthe60–100keVrangewithHXD-GSO.
Nowthatthe36ksphasemodulationinthehardX-raypulses, Therefore, we analyzed the HXD-GSO data invarious energy
firstdetectedwiththeHXD,hasbeenreinforcedbythe10–14 ranges from 50 to 200 keV, but the data gave significant evi-
keVXISdata,ournexttaskistoexaminehowthisphenomenon denceforneitherthepulsation,noritsphasemodulation. This
behavestowardslowerenergies. Inthesamewayasabove,we isnotsurprising,becauseHXD-GSOhasaconsiderablyhigher
henceanalyzedtheXIS0datainthreelowerenergies;7–10keV backgroundthanHXD-PIN(Kokubunetal.2007).
where the spectralhard component is still dominant, 4–7 keV Even though the HXD-GSO data provide no information
wheretheycompete,and1–4keVdominatedbythesoftcom- on the source pulsation, they allow us to perform an impor-
ponent. Figure8shows theresults ofthis softX-rayanalysis, tant control study. Wehence purposely chose a veryhigh en-
namely,scansinAinwhichT =36.0ksisfixedwhileφisal- ergy range, 300–307 keV, which is dominated by background
lowedtotakeanyvalue. Thus,the36ksmodulationamplitude and has nearly the same counting rate, 0.46 ct s−1, as the
clearydecreasestowards lowerenergies, andtheeffectalmost background-inclusive 15–40 keV HXD-PIN rate (0.44 ct s−1;
8 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
figure2). Thegreentraceinfigure4dshows thederivedXpk generallyspindown, presumablyemitting magnetic dipolera-
behavior inthishighenergyband. Thus,thebaselinevalueof diation. This will cause parabolic changes in thepulse phase,
Xpk ∼10 is much lower than that in 15–40 keV, Xpk ∼22, which could mimic a periodic phase modulation. From Entry
due to the absence of the pulsation. In addition, no peaks No.3ofTable4inKuiperetal.(2012),theaveragespin-down
are observed around T ∼36 ks. Since these HXD-GSO data rate of 1E 1547.0−5408 is considered to be P˙ ∼2×10−11s
were acquired by the same instrument as the HXD-PIN data, s−1aroundthepresentobservation. Althoughthiswouldcause
at completely the same time, and under the same background a phase difference of only ∼0.1 pulse cycle across the 86 ks
conditions,theresultarguesagainstinstrumentaloriginsofthe length ofthepresentdata, therestill remainsapossibility that
T =36ksperiodicityinthe15–40keVpulsephases. theinstantaneousP˙ wasconsiderablyhigher.
Withtheaboveinmind,theZ2analysiswasrepeatedonthe
2
15–40keVHXD-PINdata,incorporatingthistimetheP˙ term.
4 Discussion
WedefinedtheperiodP0asthevalueatthemiddlepointofthe
Through an extensive reanalysis of the Suzaku data of 1E observation, and scanned it over the same joint error range as
1547.0−5408 acquired in the 2009 outburst, we obtained ev- definedbyequation(3)andequation(4).Theanalysiswasfirst
idence that the 2.07 s hard X-ray pulsation detected with the conducted assuming no phase modulation (A=0). However,
HXD is subject to phase modulation with a period of T =36 thepulsesignificancedidnotincreasesignificantly;instead,the
ks (figure 4). Following the first detection from 4U 0142+61 X valuedecreasedfromequation(5)as|P˙|isincreased. Thus,
(P0=8.69s,T =55ks,andA∼0.75s;MEA14),thismakes alooselimitwasobtainedas−1.5×10−10<P˙ <3.2×10−10
1E 1547.0−5408 a 2nd magnetar exhibiting this kind of evi- s s−1. Wenext fixed T =36 ks, and examined the parameter
dence. region around equation (10). However, the pulse significance
didnotincrease,either. Theobtainedlimitisagainuninterest-
ing, −1.1×10−10 <P˙ <0.8×10−10 ss−1. Inshort,period
4.1 Trivialpossibilities
derivativescannotexplaintheobservedpulse-phasemodulation
4.1.1 Instrumentalorigin effects.
The detected effect should be first examined for instrumen-
tal or observational artifacts. In section 3.2.4, we excluded 4.1.3 Non-periodicorigin
the statistical fluctuation origin at a 99.6% confidence level. AnotherpossibilityisthatthehardX-raypulseprofile(includ-
Furthermore, the HXD-GSO data analysis (section 3.5) indi- ingitsphase)isfluctuatingrandomlyforsomeunspecifiedrea-
cates that the HXD instrument does not suffer unexpected ar- sons,andthevariabilitypowerhappenedtobehighatT∼36ks.
tifacts (e.g., background variations) that can explain away the Tohelpexaminationofthiscase,thewavenumberq≡86ks/T
Xpk maximum at T = 36 ks. The value of T = 36.0 ks isdrawninblueatthetopoffigure4d.Thus,thesignificantex-
=6.4Porb is not inharmonic resonancewith the Suzaku’sor- cessinXpk≡Z22aboveequation(5),e.g.,Xpk>30,isconfined
bitalperiod,either. toarathernarrowrangeofthewavenumberasq=2.4±0.5,in
Inthis respect, ofparticular importance is thecrossconfir- agoodagreementwithfigure3candequation(7).Furthermore,
mation ofthe phasemodulation with the10–14 keV XISdata wehave alreadyconfirmedin section 3.2.2 that thepower en-
(figure 6). This further rules out the possibility of the effect hancementsatq=2andq=3infigure3care3sigmaeffects.
being due to some artifact specific to the HXD. Furthermore, Thesefactsaltogethermeanthatthephenomenonshouldbere-
the clear decrease of A towards lower energies (figure 9) ex- gardedasperiodicratherthanrandom.
cludesanyinstrumentalorartificialoriginspecifictotheoverall
There is additional evidence arguing against random phase
Suzakumission(e.g.,unexpectedlylargedriftsinthespacecraft
fluctuations,atleast10<T<30ksisconcerned.Inthisrange,
clock),becausesucheffectsshouldbeindependentofthepho-
the15–40keVXpk valuesinfigure4darecomparabletoX0,
tonenergy.Wearehenceleftwithcelestialinterpretations.
andareallfoundatA<0.15s,implyingthatthepulsesignifi-
cancediminishesbydemodulationwithT inthisrange. Overa
4.1.2 Periodderivatives longerperiodrangeof30<T <50ks,Xpkinfigure4disob-
Basedonfigure2b,figure3a,andfigure3c,wehavesofarre- tainedatA=0.5−0.6s,asasimplecontinuationfromequation
gardedthephasemodulationasperiodic,andperformedthede- (10). AtT >50ks,thevaluesofAbecomeratherpoorlycon-
modulationanalysistoarriveatthemodulationperiodofequa- strainedbecauseoftheinsufficientdatalengthcomparedtoT.
tion(10).However,thetotaldataspan,86ks,containsonly2.4 Nevertheless, the pulse-phase modulation is not likely to con-
cycles ofthe36ks oscillation. Hence, wecould bemistaking tainsignificantpowerattheselongperiods,either,becausethe
someothereffectsfortheperiodicphasechanges. power spectrum in figure 3c shown in red is not enhanced at
One possibility is the period derivative, because magnetars q=1abovethePoissoniancontribution. Therefore,thepulse-
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 9
phasemodulationisunlikelytohave“pinknoise”natureinclud- component,becauseitispresumablythermalemissionfromthe
ing secular changes, which would usually contain significant NSsurface,andshouldbesymmetricaroundthemagneticaxis
poweratq=1 alongwhichtheheatwillflow.
To assess the above issue yet from another aspect, we ap- If,incontrast,theemissionpatternbreakssymmetryaround
pliedthedemodulation analysisseparatelytothefirstandsec- xˆ3,theobservedsignalswoulddependnotonlyonthedirection
ondhalvesofthe15–40keVHXDdata. Theprocedureisthe ofxˆ3asseenfromus,butalsoontheNS’srotationanglearound
sameasinsubsection 3.2.3, exceptthatT =36.0kswaskept xˆ3relativetotheL~-xˆ3plane. Sincethisrelativeanglechanges
fixed because of the shorter data length. Then, the first half withtheslipperiodT ofequation(2),thepulsephasebecomes
gave the significance peak with Xmax ≡ Xpk(36) = 32.4 at modulated at T (MEA14). With a natural identification of T
A=0.58±0.18 sandφ=272◦±18◦,whilethesecondhalf with themodulation periodof 36ks, this casecanexplain the
yieldedXmax=22.7atA=0.42±0.17sandφ=293◦±30◦. behaviorofthehardX-raycomponent,andhenceequation(2)
In terms of the parameter (A,φ), the two halves agree within yields
eprurlosres-pwhiatsheemacohduoltahteiro,nanisdcwonitshideeqruedatitoonha(1v0e)p.eTrshiusste,dthtehr3o6ugkhs |ǫ|=|∆I/I|=P0/T = 0.58+−00..0047 ×10−4. (17)
(cid:0) (cid:1)
thepresentobservation.
4.2.2 Doubleperiodicity
Finally,figure10showsthe15–40keVpulseprofilesinthe
Iftheaboveinterpretation iscorrect,weshouldbeabletoob-
1stand2ndhalvesofthedata,demodulatedusingthecommon
serveperiodicity also atProt =Pprec/(1+ǫ). Fromequation
setofparametersgivenbyequation(10). Infact,wediscarded
(17),itshouldappearasaprogradeperiodat
the initial 14 ks of the 1st half and the final 14 ks of the 2nd
half,tomakethembothrepresentexactlythetimeperiodof36 P+ =2,072.027±0.014ms (18)
rot
ks. Withinthephotoncountfluctuations, thetwoprofileagree
ifthedeformationisprolate(ǫ>0),andaretrogradeperiodat
witheachother, with χ2=0.78forν =12(calculated before
applying the running average). Thus, the pulse profiles from P− =2,072.267±0.014ms (19)
rot
the1stand2ndmodulationcycles,whendemodulatedusingthe
if oblate (ǫ<0), although they could appear simultaneously.
commonsetofparameters,becomeconsistentwitheachother.
Ofthem,P+ isalreadyvisibleinfigure1b. Nevertheless,the
Inconclusion,wecanregardthephasevariationsinthehard rot
issue deserves further examination, because the periodicity at
X-raypulsesasperiodic,inspiteoftheratherlimiteddataspan.
Prot would also be subject to the phase modulation at T, and
thesignificancewouldchangebythedemodulationprocedure.
4.2 Interpretationasfreeprecession Figure11showsalltheperiodograms obtained whilecom-
puting figure 4, superposed onto a single plot. In addition to
4.2.1 Freeprecession
themainpeakcoincidentwithequation(4),theresultsthusre-
Letushereafterlimitourinterpretationtoperiodiccelestialef-
vealclearenhancementsat∼2,072.03msand∼2,072.25ms.
fects. Thesimplestof themis binarymotion ofthemagnetar: SincetheycanbereadilyidentifiedwithP+ andP−,respec-
rot rot
indeed,A=0.5sandT =36kscanbeexplainediftheputa- tively, our expectation has been confirmed. Furthermore, the
tive companion star has amass of the order of 0.1 solar mass P+ peakissomewhatmoresignificant,inagreementwithour
rot
(dependingontheinclination). However,evenputtingasidethe
assumption that the object has aprolate form. Tomorequan-
astrophysicalrealityofsuchabinary,theabsenceofthephase
titativelyassesthesignificanceofthesetwoenhancementsand
modulationinsoftX-rays(figure9)clearlyrulesoutthisinter-
theirdifferencesremainsourfuturestudy.
pretation,justasin4U0142+61.
Fromtheabovearguments,freeprecession(Carlini&Treves
4.3 Comparisonwith4U0142+41
1989;Heyl&Hernquist2002;MEA14)describedinsection1
remains the most promising possibility. In this scenario, we IntermsoffreeprecessionofanaxisymmetricNS,letuscom-
identifyxˆ3withthedipolemagneticaxis,assumethatitistilted pare 1E 1547.0−5408 and 4U 0142+41 (MEA14). The most
from L~ (i.e., α6=0), and consider that the NS is deformed to importantfeaturecommontothemisthesimilardegreeofde-
have a non-zero value of ǫ = ∆I/I. Then, Prot and Pprec, formation,ǫ=1.6×10−4 in4U0142+41andǫ=0.6×10−4
whichweredegeneratewhenǫ=0,splitintoslightlydifferent 1E 1547.0−5408 [equation (17)]. This suggests that the de-
two values according to equation (1). Of them, Pprec around formation in the two objects has a common origin. Another
L~ isobservedasthepulseperiodofequation (3)andequation resemblance between them is the decrease of the pulse-phase
(4), i.e., Pprec =P0, while Prot around xˆ3 cannot bedetected modulation amplitude towards lower energies; at least in 1E
whentheemissionpatternissymmetricaroundxˆ3evenifǫ6=0 1547.0−5408, theeffectisconsistent with beingabsentinthe
andα6=0both hold. Thiscaseisthought toapply tothesoft softspectralcomponent. Asalreadydescribedinsection4.2.1,
10 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
theseproperties canbeinterpretedbyconsideringthatthesoft canexplainthelargemodulationamplitudeof1E1547.0−5408
componentisemittedsymmetricallyaroundthestellarmagnetic reachingA∼T/4.Althoughtheactualconfiguration(e.g.,con-
axis,whilethehardcomponentbreaksthatsymmetry.Thisdif- tribution from the other pole) would not be as simple as this,
ferencesuggests that the two spectral components aredistinct more detailed modeling of the emission pattern is beyond the
notonlyintheirspectralshapes,butalsointheiremissionpat- scopeofthepresentpaper.
ternsand/oremissionregions. HowabouttheremainingitemsNo.2andNo.3?Inthe2009
Whilethetwosourcesthusshowsimilarbehavior,theyalso dataof4U0142+61,thehardX-raypulseprofilewasricherin
exhibitthefollowingthreedifferences,whichmustbeexplained fine structures, consisting of three peaks separated by a quar-
inordertofurthersolidifytheinterpretation. ter cycle (figure 1f of MEA14). This is considered to be the
1. WhilewemeasuredA/P0=0.08±0.04in4U0142+61,the reason why n = 4 was needed. Moreover, such a complex
ratio is much higher, A/P0 =0.25±0.07 (approximately pulse profile of 4U 0142+61 is thought to have been strongly
±1/4pulsecycle)in1E1547.0−4508 inenergiesabove10 smearedoutbytheprecession,andhencethehardX-raypulse
keV(figure9). wasdifficulttodetectwithoutdemodulationatleastinthe2009
2. Whilen=4wasneededin4U0142+61,n=2isoptimum Suzakuobservation.Incontrast,thepulseprofilesofthepresent
in1E1547.0−45408(section3.2.4). sourcearebasicallydouble-peaked. Thisisconsideredtomake
3. The hard X-ray pulsation of 4U 0142+61 in the 2009 n=2appropriateforthepresentsource,andmaketheprofile
Suzaku observation was detected only after the demodula- lessaffectedbythe phasemodulation. Inaddition, theeffects
tion,whereasthatof1E1547.0−5408issignificantevenbe- in 1E 1547.0−5408 can be approximated as a sort of pulse-
foresuchcorrections(PaperI). profile change (figure12), with the “dawn” and “dusk” inten-
sitieschangingacrosstheslipperiod, ratherthanacontinuous
Let us consider item No.1 first. In 4U 0142+61, the sinu-
phaseshift.Thispropertyisthoughttobeanotherfactorthathas
soidalhardX-raymodulationwithA/P0=0.08wasattributed
madethepulseeasiertodetectwithoutdemodulation. Thus,the
to positional (or directional) deviations of the X-ray emission
differencesNo.2andNo.3canbeexplaininaqualitativeway.
region (or direction) from the magnetic poles (or from xˆ3).
There,theemissionpatternwasimplicitlyandwithoutparticu-
larreasonassumedtobepencil-beamlike. Incontrast,thecase
4.4 Relationstootherneutronstars
of 1E 1547.0−5408 can be better explained by assuming that
theemissionreachesus(forsimplicity,fromasinglepole)ina Sofar,afairnumberofreportshavebeenmadeonpossiblede-
fan-beam-likepattern,asillustratedinfigure12.Thebeamwill tections of precession from NSs, including accretion-powered
pointtoustwiceperPprec=2.07s,whenthepoleappearsfrom objects (e.g., Staubert et al. 2009) and rapidly rotating NSs
therimoftheNS,tobecalled“dawn”,anddisappearsbehind (e.g, Shabanova et al. 2001; Chukwude et al. 2003; Haberl et
it, tobecalled“dusk”. This explains thedouble-peaked pulse al.2006).However,asdiscussedinMEA14,suchabehaviorin
profiles(figure2,figure7),andthe∼±1/4cycleoffsetofthe accretion-poweredsystemsshouldberegardedasforcedpreces-
hardX-raypulsepeaksfromthoseinthesoftX-rays(figure7b). sion, becauseof thestrongaccretion torque. Thefast-rotating
Asalreadydescribedinsection1andsection4.2.1,wefur- NSscanexhibitfreeprecession,becausetheymustbedeformed
ther assumethat the fan-beam emission pattern is asymmetric intooblateshapesbycentrifugalforce.However,theprecession
aroundxˆ3. Then, atacertaintimet0 ofthe36ksslipperiod, wouldbedampedonrathershorttimescales(Cutler2002),be-
thesignalwillbestrongeratthe“dusk”pulsephasethanatthe causeanoblatebodyattainsitsminimumenergyatα=0under
“dawn”phase,asillustratedinfigure12a;thepulsepeakisde- aconstant L~. Therefore,thesereports remaingenerally rather
layed by ∼ 1/4 cycle from the average, just as in the red to unconvincing.
orange modulation phases in figure 2b. At t∼t0+T/4, the In contrast to these ordinary NSs, magnetars are expected
configuration will become as in figure 12b, so that the inten- tomoreubiquitously exhibitthefreeprecession, becausetheir
sitywillbesimilarbetween“dawn”and“dusk”;thepulse-peak strong Bt will deform them into prolate shapes (Ioka 2001;
phase will occur at about “noon” phase with zero delay, as in Cutler 2002; Iokai &Sasaki 2004); their centrifugal deforma-
theyellowprofileinfigure2b.Then,att∼t0+T/2,thesignal tion (oblate) is negligible, e.g., |ǫ| < 10−6, because of their
willbecomestrongestatthe“dawn”pulsephase,asshownby ratherlong pulseperiods, P =2−11s. Actually, thepresent
figure12c;thepulsepeakprecedstheaverageby∼Pprec/4,as data may favor the prolate (ǫ>0) case(subsection 4.2.2). In
representedby thegreento blueprofiles in figure2b. Finally, suchaprolatemagnetar,thefreeprecessionwillspontaneously
att∼t0+3T/4,thepulseamplitudebecomeslow,becausethe develop, becauseinternalenergydissipation processes,withL~
emissionregionisbehindthehorizonwhenthebeampointsto- conserved, will increase α on reasonable time scales (Cutler
wardsus;thismaycorrespondtothepurpleprofileinfigure12b. 2002). (As mentioned in MEA14, decay in α by emission of
Thus,thefan-beamconfigurationwithabrokenaxialsymmetry gravitationalwavesisnegligible.) Therefore,thetwonecessary
