Application of the Ghosh & Lamb Relation to the Spin-up/down Behavior in the X-ray Binary Pulsar 4U 1626-67

0.63 MB·

by  Toshihiro Takagi

Publ.Astron.Soc.Japan(2014)00(0),1–13 1
doi:10.1093/pasj/xxx000
Application of the Ghosh & Lamb Relation to the
Spin-up/down Behavior in the X-ray Binary
Pulsar 4U 1626–67
6
1
0 Toshihiro TAKAGI1,2,*, Tatehiro MIHARA2, Mutsumi SUGIZAKI2, Kazuo
2 MAKISHIMA2 and Mikio MORII3

n
a 1DepartmentofPhysics,NihonUniversity,1-8-14Kandasurugadai,Chiyoda-ku,Tokyo
J 101-8308,Japan

9 2MAXIteam,RIKEN,2-1Hirosawa,Wako,Saitama351-0198,Japan
1
3ResearchCenterforStatisticalMachineLearning,InstituteofStatisticalMathematics,10-3

] Midori-cho,Tachikawa,Tokyo190-8562,Japan
E
H ∗E-mail:[email protected]
. Received;Accepted
h
p
- Abstract
o
r We analyzed continuous MAXI/GSC data of the X-ray binary pulsar 4U 1626–67 from 2009
t
s Octoberto2013September, anddetermined thepulseperiod andthepulse-period derivative
a
[ for every60-dinterval by the epochfolding method. Theobtained periods areconsistentwith

those provided by the Fermi/GBM pulsar project. In all the 60-d intervals, the pulsar was
1
v observed to spin up, with the spin-up rate positively correlated with the 2–20 keV flux. We
4 applied the accretion torque model proposed by Ghosh & Lamb (1979, ApJ, 234, 296) to the
9
MAXI/GSC data, as well as the past data including both spin-up and spin-down phases. The
8
Ghosh & Lamb relation was confirmed to successfully explain the observed relation between
4
0 the spin-up/down rate and the flux. By comparing the model-predicted luminosity with the
.
1 observedflux,thesourcedistancewasconstrainedas5–13kpc,whichisconsistentwiththat
0 by Chakrabarty (1998, ApJ, 492, 342). Conversely, if the source distance is assumed, the
6
datacanconstrainthemassandradiusoftheneutronstar,becausetheGhosh&Lambmodel
1
: depends on these parameters. We attempted this idea, and found that an assumed distance
v
of,e.g., 10kpc givesamass inthe rangeof1.81–1.90 solar mass,andaradius of11.4–11.5
i
X
km,althoughtheseresultsarestillsubjecttoconsiderablesystematicuncertaintiesotherthan
r thatinthedistance.
a
Keywords:pulsars:individual(4U1626–67)—stars: neutron—X-rays: binaries
1 Introduction inneredgeoftheaccretiondisk, andacceleratesthepulsarro-
tation untilitfinallyreachesanequilibrium determinedbythe
AnX-raybinarypulsarisasystemconsistingofamagnetized
accretion rateand themagnetic moment of thepulsar. Sofar,
neutron star and a stellar companion. The X-ray emission is
numbersoftheoretical studies havebeenperformedonthein-
powered bygravitational energyofaccretingmatter. Inasys-
teraction between the pulsar magnetosphere and the accretion
temwithalow-masscompanionstar,theaccretionflowiscon-
flows. Along this scenario, Rappaport & Joss (1977), Ghosh
sideredtotakeplaceviaRoche-Lobeoverflow,andtoforman
& Lamb (1979) (GL79 hereafter) and Lovelace et al. (1995)
accretion disk in the vicinity of the pulsar. The angular mo-
(LRB95 hereafter) proposed their accretion models and pre-
mentum of the accreting gas is transferredto the pulsar at the
(cid:13)c 2014.AstronomicalSocietyofJapan.
2 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
sentedequations describing thepulse-period derivative P˙ asa 2008,andfigure1visualizeslong-termbehaviorofthesequan-
functionoftheluminosityL,thepulseperiodP,themassM, tities.Itclearlyshowsthatthesourcemadetransitionstwicebe-
the radius Rand the surface magnetic field B of the neutron tweenthespin-upandthespin-downphases,atMJD 48000
c ∼
star. These models have been examined against observational (1990June)andMJD 54000(2008February),separatedby
∼ ∼
data(e.g. Joss&Rappaport1984;Fingeretal.1996;Reynolds 18years.IneachphaseP˙ werealmostconstant,anditsabsolute
etal.1996;Bildstenetal.1997;Klochkovetal.2009;Sugizaki valueswereverysimilaras P˙ =2 5 10−11 ss−1. These
| | ∼ ×
etal.2015). TheresultsshowthattheobservedP˙–Lrelations period-changebehaviorsuggeststhat4U1626–67isclosetoan
are grossly consistent with the model predictions. However, equilibrium stateinwhich thenettorquetransferfromtheac-
thevalidityofthemodelshasnotyetbeenfullyconfirmed,be- creting matter to the pulsar is approximately zero. At the last
causeitrequireslong-termmonitoringofsomesuitableobjects transition in2008 whenthesourceturnedfromthespin-down
withknownB ,coveringsignificantpulse-periodandluminos- into the spin-up phase, the flux increased by a factor of 2.5
c ∼
itychangeswithasufficientsamplingrate. (Camero-Arranz et al. 2010). These properties, together with
Whenthesemodelsdescribingtheaccretiontorquearebet- the accurate knowledge of Bc, make this object ideal for our
tercalibrated,theobservedP˙–Lrelationscangiveusobserva- study.
tional constraints on M and R, which are very important be- Monitor of All-sky X-ray Image (MAXI; Matsuoka et al.
causetheyaredirectlyconnectedtotheequationofstate(EOS) 2009) is an X-ray all-sky monitor on the International Space
ofnuclearmatter.Sofar,variousobservationalattemptstomea- Station. Since the in-orbit operation started in 2009 August,
sure M and R have been carried out, and generally yielded its main instrument, the GSC (Gas Slit Camera; Mihara et al.
M 1.4M and R 12 km (e.g. reviews by Bhattacharyya 2011; Sugizaki et al. 2011), has been scanning the whole sky
201≃0andO¨⊙zel2013)≃. However, themeasurementsarenotyet every 92 min in the 2–20 keV band. The GSC field of view
accurateenoughtoconstraintheEOS.Furthermore,thevalues typically scans a celestial point source for about 60 s in each
ofM (mainlyfromX-raybinarypulsars)andR(mainlyfrom transit,whichislongenoughtostudythe7.6-spulsationfrom
weakly-magnetized neutronstars)havebeenderivedfromdif- 4U1626–67. Thus,theMAXI/GSCdataareusefultostudythe
ferentpopulationsofneutronstars.Therefore,furtherstudiesof long-termvariationoftheflux,P,andP˙.
theP˙–Lrelationsareexpectedtobevaluable. Inthispaper,weanalyzetheMAXI/GSCdataof4U1626–
4U1626–67isalow-massX-raybinarypulsarfirstdetected 67 from MJD 55110 (2009 October 6) to MJD 56550 (2013
with the Uhuru satellite (Giacconi et al. 1972), and its 7.6-s September15)anddeterminetheflux,P,andP˙.Wethenapply
coherent pulsation was discovered by Rappaport etal.(1977). thespin-up/downmodelsproposedbyGL79andLRB95tothe
Because no period modulation due to orbital motion has been previous and the MAXI/GSC data, to examine whether either
detectedbeyondanupperlimitofa sini 13lt-ms(a isthe model canexplain the observed behavior of 4U1626–67, and
x x
≤
orbitalsemi-majoraxisofthepulsarandiistheorbitalinclina- toevaluatehowthesedatacanconstrainthesourcedistance,as
tion),themassofthecompanionstarisestimatedtobeverylow wellasthemassandradiusoftheneutronstar.
( 0.03 0.09M for11◦ i 36◦;Levineetal.1988). Itis
∼ − ⊙ ≤ ≤
henceclassifiedasanultracompactX-raybinary(vanHaaften
2 Data Analysis
etal.2012). TheBeppoSAX observation revealedacyclotron
resonancescatteringfeatureat 37keV,indicatingthesurface 2.1 X-raylightcurvewithMAXI
∼
magneticfieldofBc=3.2 1012 (1+zg)G,wherezg isthe X-rayeventsof4U1626–67 wereextractedfromall-skyGSC
×
gravitationalredshift,
data, and accumulated over every 60-d interval from MJD
1 55110 (2009 October 6) toMJD 56550 (2013 September 15),
z = 1 2GM −2 1, (1)
g − Rc2 − using the on-demand analysis system provided by the MAXI
(cid:16) (cid:17) team2. We employed the standard regions to extract the on-
representedbythegravitational constantGandthevelocityof
lightc(Orlandinietal.1998).Thefeaturewasconfirmedbythe sourceandbackgroundevents;2◦radiuscircleforthesourcere-
Suzaku observation (Iwakirietal. 2012). The sourcedistance gionandanannuluswithinnerandouterradiiof2◦.1and3◦,re-
spectively,forthebackground.Wefittedalltheobtainedspectra
was estimatedtobe5–13 kpcfromtheoptical fluxbyassum-
withapower-lawmodelwithout absorption. Thefitswereac-
ingthattheeffectiveX-rayalbedooftheaccretiondiskishigh
(>0.9)(Chakrabarty1998). ceptable for all the 60-d intervals, and gave photon indices of
∼Since the discovery of the 7.6-s pulsation in 1977, the pe- 1.0 1.3. Wecalculatedthe2–20keVmodelflux,andpresent
∼
infigure2aitsvariationoverthe4yearsanalyzed,whereerrors
riod of 4U 1626–67 has been repeatedly measured with vari-
ousX-raysatellites(e.g. referencesinChakrabartyetal.1997; refer to 1σ statistical uncertainties. The flux was thus almost
Camero-Arranz et al. 2010). Table 1 summarizes the X-ray
1http://gammaray.nsstc.nasa.gov/batse/pulsar/data/sources/4u1626.html
fluxes, periods and period derivatives observed from 1978 to 2http://maxi.riken.jp/mxondem
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 3
Table1.X-rayflux,periodandperiodderivativeobtainedinpastobservationsof4U1626–67.
Observation Flux Pulsation
Date Period Band Obs. Bolometric∗ Epoch P† P˙ Ref.‡
(MJD) (keV) (10−10ergs−1cm−2) (MJD) (s) (10−11ss−1)
1978Mar 43596 0.7–60 24 3 25 3 43596.7 7.679190(26) 4.55§ 1
± ± −
1979Feb 43928–43946 2–10 5.1 0.3 19 1 43946.0 7.677632(13) 4.9 0.1§ 2
± ± − ±
1983May 45457–45459 2–20 10.1 0.2 19.0 0.4 45458.0 7.671350(1) 13 5 3
± ± − ±
1983Aug 45576 2–10 5.6 0.5 20.3 1.8 45576.9 7.67077(1) 5.65 0.10l 4
± ± − ±
1986Mar 46519–46520 1–20 14 0.4# 22 0.7 46520.0 7.6664220(5) 4.96 0.06§ 5
± ± − ±
1987Mar 46855–48012 1–20 8.89 0.56 14.0 0.9 6
± ±
1988Aug 47400.3 7.6625685(30) 7
1990Apr 47999–48002 2–60 20.6 24.7 48001.1 7.660069(2) 8
∼ ∼
1990Jun 48043–48530 1–20 6.67 0.89 10.9 1.5 6
± ±
1990Aug 48133.5 7.66001(4) 4
1993Aug 49210–49211 0.5–10 2.8 9.9 9
∼ ∼
1996Aug 50301–50306 2–60 6.1 6.5 10
∼ ∼
2000Sep 51803 0.3–10 2.4 8.4 51803.6 7.6726(2) 11
∗∗
2001Aug 52145 0.3–10 1.8∗∗ 6.3 52145.1 7.6736(2) 3.39 0.96l 11
±
2003Jun 52793 0.3–10 1.8∗∗ 6.3 52795.1 7.67514(5) 2.74 0.37l 11
±
2003Aug 52871 0.3–10 1.6∗∗ 5.6 52871.2 7.67544(6) 4.6 1.2l 11
±
2007Jun 54280 15–50 2.8 0.1 5.6 0.2 54280 7.6793 2.9 12
± ± ∼
2007Sep 54370 15–50 2.6 0.1 5.2 0.2 54370 7.6793 2.7 12
± ± ∼
2007Dec 54450 15–50 4.0 0.1 8.0 0.3 54450 7.6793 1.9 12
± ± ∼
2008Jan 54480 15–50 4.6 0.2 9.3 0.4 54480 7.6793 0.18 12
± ± ∼
2008Feb 54510 15–50 4.6 0.1 11.5 0.3 54510 7.6793 0.53 12
± ± ∼ −
2008Mar 54530 2–100 10.1 0.8 11.7 0.9 13
± ±
2008Mar 54550 15–50 5.8 0.1 14.5 0.3 54550 7.6793 2.3 12
± ± ∼ −
2008Jun 54620 15–50 5.9 0.1 14.8 0.2 54620 7.6793 2.7 12
± ± ∼ −
Allerrorsrepresent1-σuncertainties.
Converted0.5–100keVflux,assumingthespectralmodelsinCamero-Arranzetal.(2012).
∗
Valuesinparenthesesare1σerrorinthelastdigit(s).
†
(1) Pravdo et al. (1979) (HEAO 1/A-2), (2) Elsner et al. (1983) (Einstein/MPC), (3) Kii et al. (1986) (Tenma), (4) Mavromatakis (1994)
‡
(EXOSAT/GSPC,ROSAT),(5)Levineetal.(1988)(EXOSAT/ME),(6)Vaughan&Kitamoto(1997)(Ginga/ASM),(7)Shinodaetal.(1990)(Ginga),(8)
Mihara(1995)(Ginga),(9)Angelinietal.(1995)(ASCA),(10)Orlandinietal.(1998)(BeppoSAX),(11)Kraussetal.(2007)(Chandra,XMM-Newton),
(12)Thedatawerereadfromfigure4inCamero-Arranzetal.(2010)(Swift/BAT),(13)Camero-Arranzetal.(2010)(RXTE/PCAspectra).
Averagedvalue.
§lAveragedP˙ calculatedfromP oftheobservationandthepreviousoneinthistable.
#TheerrorwasestimatedbythecountratewithME,usingthevaluesofthecountrateandtheerrorwithGSPC.
Absorption-correctedflux.
∗∗
4 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
constant at 8.6 10−10 ergcm−2s−1 before MJD 56200,
∼ ×
andslightlyincreasedto 9.5 10−10ergcm−2s−1afterthat
∼ ×
time.
2.2 Pulseperiodsandpulse-periodderivativeswith
MAXI
Thepulsartiminganalysisof4U1626–67wascarriedoutwith
theGSCevent dataofrevision 1.5, which have atimeresolu-
tionof50µs.Weextractedeventswithina1◦.5radiusfromthe
sourceposition, andthen applied thebarycentric correction to
their arrivaltimes. Background was notsubtracted inthetim-
ing analysis. Todetermine P and P˙, we employed the epoch
foldingmethod. There,χ2 ofthefoldedpulseprofile,defined
as
χ2 = n (yi−y¯) 2, y¯ = (√1yi)2yi, (2)
Fig.1.Bolometricflux(top),period(middle)andperiodderivative(bottom) Xi=1(cid:20) √yi (cid:21) P (√1yi)2
of4U1626–67obtainedbypastX-rayobservationsfrom1978to2008. In iscalculatedforeachtrialvalueoPfPandP˙,wherenisthenum-
thetoppanel,observedX-rayintensitiesareconvertedtothemodelfluxin berofbinsofthefoldedprofileandy isthenumberofeventsin
i
the0.5–100keVbandassumingthetypicalspectralmodelgivenbyCamero- thei-thbin. Weusedn=32,andconfirmedthattheexposure
Arranzetal.(2012). Filledcirclesandsolidlinesrepresentthepastobser-
is uniform over the 32 bins within 0.5%. The epoch-folding
vationsintable1andBATSE1observations,respectively.
analysiswasperformedforevery60-dinterval,tobecoincident
withthelight-curvetimebinsemployedinsection2.1. Ineach
interval,wesearchedforthevaluesofP andP˙ thatmaximize
χ2. Here,theP andP˙ values measuredwiththeFermi/GBM
pulsarproject3wereusedtoselectthesearchranges,andP˙ was
assumedconstantineachinterval. Figure3showstheobtained
χ2valuesontheP–P˙ plane,employingMJD55290–55350as
atypicalexample. The1-σ errorsofP andP˙ wereestimated
byMonte-Carlosimulations(seeAppendix3).Werepeatedthe
analysis intheenergybandsof2–20keV,2–10keV,2–4keV,
4–10keVand10–20keV,andthenselectedtheresultsofthe2–
10keVbandbecausethemaximumχ2 wasthehighestamong
them.Resultsfromtheotherenergybandswereconsistentwith
these.
Figure 2b and 2c show time variations of the obtained P
and P˙, respectively. The absolute value of P˙ increased with
the flux increase around MJD 56400. We fitted the data in
figure 2b with a liner function, because their distribution ap-
pears quite linear. The best-fit slope was then obtained as
P˙ = 2.87 10−11ss−1.Figure2dshowstheresidualsfrom
h i − ×
thebest-fitline,wheretheresultsoftheFermi/GBMpulsardata
areoverlaid.TheresultsoftheMAXI/GSCandtheFermi/GBM
arefoundtoagreewitheachotherwithintheerrors.
Fig.2.The2–20keV flux(panel a), theperiod(panel b), andtheperiod
derivative(panelc)of4U1626–67,obtainedevery60dwiththeMAXI/GSC 2.3 Estimationofbolometricflux
datafrom2009Octoberto2013September.Panel(d)showsresidualsfrom
thebest-fitlinearfunctiontotheperioddatain(b). Thehorizontal linein In the following sections, we apply the theoretical models of
(c) represents the slope of the line in (b) (−2.87×10−11 ss−1). The pulsarspin-up/down,proposedbyGL79andLRB95,totheob-
Fermi/GBMresultsaresuperposedbyverticallinesegments.
3http://gammaray.nsstc.nasa.gov/gbm/science/
pulsars/lightcurves/4u1626.html
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 5
3 Application of the Ghosh & Lamb relation
As reviewed in section 1, GL79 derived arelation between P˙
and L in accreting X-ray pulsars, assuming that the accreting
mattertransfersitsangularmomentumtothepulsaratthe“outer
transitionzone”,r [equation(A2)].Theequationsweusedare
0
summarizedinAppendix1. Rappaport&Joss(1977)alsopro-
posed analmost equivalent equation. Sincetheir modelequa-
tionincludesanunknownparameter(ξv /v ),ofwhichthere-
r ff
lation to n(ω ) in the GL79 relation is calculable. Therefore,
s
weemploytheGL79relation.
3.1 ModelequationsrelatingP˙ totheflux
AccordingtoGL79,P˙ isexpressedby
Fig.3.Distributionofχ2 ofthefoldedpulseprofiles, shownasafunction P˙ = 5.0 10−5µ3072n(ωs)S1(M)P2L3776 syr−1, (3)
− ×
ofthetrialP andP˙,obtainedfromthe2–10keVMAXI/GSCeventdatain
MJD55290–55350. Therightbarindicatestheχ2 values. Themaximum where µ30 is the magnetic dipole moment µ in units of
χ2is108for31degreesoffreedomatP=7.6777282sandP˙=−2.64× 1030 Gcm3,andL37 isLinunits of1037 ergs−1. Thefunc-
10−11ss−1. tions n(ω ) and S (M) aregiven in Appendix 1, where ω is
s 1 s
thefastnessparameterdefinedbyequation(A6). Ifω isinthe
s
rangeof0 0.9,n(ω )isapproximatedbyequation(A4)within
s
served data including those from the previous measurements −
5%. Sincen(ω )changesfrompositivetonegativedepending
s
and from the MAXI/GSC. For this, we have to estimate the
onω (figure7),P˙ canbecomebothpositive(ω >0.349;spin
s s
bolometric flux F of the individual observations, consider-
bol down)andnegative(ω <0.349;spinup).
s
ing different energy bands used in different observations, and
As shown in equation (A5), S (M) contains the effective
1
employingappropriatespectralmodels.
momentofinertiaI. ItisexpressedasafunctionofM andR,
The energy spectrum of 4U 1626-67 were studied in both dependent on the EOS. We utilize its approximation given by
the spin-up and spin-down phases (table 1), and the changes Lattimer&Schutz(2005),
in the spectral shape between these two phases were reported
I (0.237 0.008)MR2
(e.g.Jainetal.2010;Camero-Arranzetal.2012).Accordingto ≃ ±
Camero-Arranzetal.(2012),thespectrainbothphasescanbe M R −1 M 4 R −4
1+0.42 +0.009 ,
× M 10km M 10km
fittedwithamodelcomposedofablackbodyandacutoffpower " (cid:18) ⊙(cid:19)(cid:16) (cid:17) (cid:18) ⊙(cid:19) (cid:16) (cid:17) #
law,andtheirdifferenceisintheblackbodycomponent,whose (4)
temperatureis 0.5keVinthespin-up phaseand 0.2keV
∼ ∼ whichisapplicableinmostofthemajorEOSmodelsifM/R>
in the spin-down phase. We hence employed these respective 0.07M km−1. ∼
spectralmodelsforthespin-upandspin-downphases,andcon- ⊙
In4U1626–67,thesurfacemagneticfieldisknownfromthe
verted the 2–20 keV MAXI/GSC flux to those in the 0.5–100
cyclotronresonancescatteringfeatureasB =3.2 1012(1+
c
keV band, which we identify with F . Since the power-law ×
bol z )G(section1).Itisconsideredtorepresentthefieldstrength
g
continuum isflat(photonindex 1)below 20keVandex-
∼ ∼ near the magnetic poles. Assuming that the magnetic axis is
ponentially cuts off above 20 keV, the fluxes in the energy
∼ alignedtothepulsarrotationaxis,µattheequatorintheGL79
bandbelow0.5keVandabove100keVarenegligible. Forex-
modelisrepresentedby
ample,theconversionfactorfromthe2–20keVfluxobserved
1
by the MAXI/GSC (in the spin-up phase) to the 0.5–100 keV µ = 2BcR3. (5)
flux is 1.88. In a similar way, we calculated F in the past
bol Ifthesourceemissionisisotropic,LiscalculatedfromF
bol
observations,andpresenttheresultsintable1.
andthedistanceDas
Infigure4,weplottherelationbetweentheobservedP˙ and
L = 4πD2F . (6)
the calculated F , including the past data. It clearly shows bol
bol
theirnegativecorrelation,expectedfromthepulsarspinupdue Sincethepulsaremissionisanisotropic,itisnotexactlycorrect.
totheaccretiontorque.Furthermore,thedatapointsinthespin- Weemploythisapproximationandthendiscusstheeffectlater.
upandspin-downphasesapparentlydefinesawell-definedsin- Combining equations (3), (4), (5), and (6), as well as the
gledependenceonF . expressionforz [equation(1)],weobtainatheoreticalmodel
bol g
6 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
Fig.4.ArelationbetweenP˙ andFbolbytheMAXI/GSCandthepastobservations. Filledcircles,opencirclesandsquaresrepresenttheMAXI/GSCdata,
theSwift/BATdataandtheothersintable1,respectively.Thedashed-dottedhorizontallineindicatesP˙ =0.Thesolidlineisthebestfitmodelcalculatedby
equation(3),assumingadistanceof10kpc.TheparametersareM=1.83M⊙andR=11.4km,withχν2of2.9for37degreesoffreedom. Dashedtwo
linesshowthecasewhenRischangedby±1kmwithMkeptunchanged,whiledottedtwolinesthosewhenMisvariedby±0.3M⊙withRfixedat11.4
km.
equationtodescribetheobservedP˙–Lrelation,includingthree 4,wherethebestfitvalueswereobtainedasM=1.83M and
⊙
unknownparameters,D,M andR. R=11.4km(errorsareconsideredlater).
Tounderstand how themodelcurvedepends onM andR,
weshowinfigure4somepredictionsbyequation(3)whenei-
3.2 Comparisonbetweenthedataandtheory
therM orRisslightly changed. Thus,changes inR(withD
In order to determine the possible parameter ranges of D, M andM fixed)causes parallel displacements ofthe modelwith
andR,wefittedtheobservedP˙–F relationinfigure4with littlechangesinitsslope,whilethoseinM(withDandRfixed)
bol
appearsmainlyinslopechangeswiththe“zero-cross”pointnot
themodelpreparedasabove.Whenerrorsassociatedwithsome
pastmeasurements ofP˙ areunavailable, weassumedanarbi- much affected. In other words, the observed P˙–Fbol relation
trarilysmallvalue(∆P˙ =6 10−16ss−1),becausetheoverall hasessentiallytwodegreesoffreedom,namely,thezero-cross
× pointandtheslope,andtheirjointuseallowsustosimultane-
errorsaredominatedbythoseinF . Thistreatmentwascon-
bol
firmedto little affect the fitting results. Since it is difficult to ously constrain two (in the present case M and R) out of the
constrainallthe threeparameterssimultaneously fromtheP˙– threemodelparameters: theother one(D inthepresentcase)
remainsunconstrained. Below, letus consider physical mean-
F relationalone,wefirstassumedthesourcedistanceDtobe
bol
ingsbehindthismodelbehavior.
somevaluesfrom3to20kpc,andthendeterminedtheallowed
M–Rregionsasafunctionoftheassumeddistance. Asanex- Infigure4,thezero-crosspointatthespin-up/downthresh-
ample,thefittingresultassumingD=10kpcisshowninfigure old represents a torque-equilibrium condition, wherein so-
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 7
calledco-rotationradius,whichisalmostuniquelydetermined whichapproximatelyagreeswiththelocusinfigure5.
by the observed P (with some dependence on M), can be Infigure4, the best-fit reduced chi-squared, χ 2 =2.9 for
ν
equatedwithmagnetosphericradius,ortheouter-transition ra- ν=37degreesoffreedom(DOF),isnotwithintheacceptable
dius r0 in the GL79 model (Appendix 1). At this radius, the range. Onepossiblecauseforthislargeχ2 maybesystematic
gravitationalpressurecalculatedfromL D2Fbol shouldbal- errorsontheobservedfluxes,becausethefluxdatatakenfrom
∝
ance the magnetic pressure, and this condition specifies the various pastresultsmustbesubject tocross-calibrationuncer-
value of µ. By further comparing this µ with the accurately taintiesamongthedifferentinstrumentsemployed. Wethusre-
measuredBc,wecanconstrainRviaequation(5). Asaresult, peatedthemodelfittingbygraduallyincreasingthesystematic
thezero-crosspointbecomesmoresensitivelytoRratherthan errorsinthefluxfrom1%,tofindthatχ 2 1isattainedifthe
ν
∼
toM. Morequantitatively, thetorque-equilibrium conditionin systematicerrorsaresetat5.5%. Thisnumberisquitereason-
equation(A6),ωs=0.349,canbecombinedwithequation(5), able, becausethefluxesoftheCrabnebula measuredbythese
toyieldascalingforthefluxatthetorqueequilibriumas instruments scatter by 10% (Kirschetal. 2005) mostlikely
∼
Fbol∝M−23R5D−2, (7) dthueesetoinusntrcuemrtaeinnttsi.es intheabsolutephotometricsensitivities of
wheredependencesonBcandP wereomitted. Sincethefitχν2 wasfoundtodependlittleonD,wehave
TheslopeoftheP˙–F relationinfigure4meansthecon- decidedtoemploythesystematicerrorof5.5%throughout,and
bol
versionfactorfromanincrementoftheluminosity (andhence calculatedthestatisticallyallowedM–Rregionatthe68%con-
oftheaccretiontorque)tothatintheneuron-starrotation.Thus, fidencelimits (χ2 increment∆χ2<2.3 for2DOF).Infigure
it is inversely proportional to I, so that an increase in M will 5,theobtainedallowedregionisindicatedbyapairofdashed
maketheslopesmaller(intheabsolutevalue). AlargerRwill lines,andtherangesofM andRforrepresentativedistancesof
actinthesamesensethroughI,butthiseffectispartiallycan- 6,7,8,...,13kpcarelistedintable2.
celedbyaninducedincreaseinµ,throughequation(3),which
would make period changes easier. As a result, the slope be-
3.3 Systematicuncertainties
comesmainlydeterminedbyM. Quantitatively, atthehighest
spin-up regime which most accurately specifies the slope, we
Althoughthepresentmodelfittinghasbeenfoundtohaveaca-
canapproximateω 0,andhencen(ωs) constantfromfig- pability ofratheraccuratelyconstraining M andRwhenD is
→ ∼
ure7,torewriteequation(3)as given,theuncertaintyrangeinfigure5(dashedlines)considers
P˙ M−170R−72L76 M−170R−27D172Fbol67 (8) onlystatisticalerrors.Wethereforeneedtoevaluatesystematic
− ∝ ∝ errorsassociatedwithseveralassumptionsandapproximations
when ignoring the higher-order terms in equation (4). This which we have employed. Among them, the approximations
yieldstheslopeas involved in equation (A4) for n(ω ) of the GL79 model, and
s
equation(4)forI,areconsideredtobeaccuratetowithin<5%
dP˙ 10 2 12 1
−dFbol ∝M−7 R−7D 7 Fbol−7. (9) (GL79)and<10%(Lattimer&Schutz2005),respectively.We
henceneglectingtheseeffects,andconsiderbelowmoredomi-
By changing the assumed D, we calculated the best-fit M
nantsourcesofsystematicerrors.
andR,andshowtheirlocusasasolidlineinfigure5,wherethe
One obvious uncertainty is inequation (6),which assumes
mass-radiusrelationsfromrepresentativeEOSs[SLy(Douchin
thatthetime-averagedfluxofapulsarisidenticaltotheaverage
&Haensel2001),APR(Akmaletal.1998)andShen(Shenet
overthewholedirection.Althoughthedifferencebetweenthese
al.1998a;Shenetal.1998b)presentedinYagi&Yunes(2013)]
twoaverageshasnotbeenestimatedin4U1626–67, Basko&
are also shown. In order for the derived M and R fall in the
Sunyaev(1975)examinedthisissueinHerX-1,asimilarlow-
nominal neutron-star parameters range, M =(1.0 2.4) M
and R=8.5 15 km (e.g. Bhattacharyya 2010; O−¨zel 2013⊙), massX-raybinarypulsar,andconcludedthatthedifferenceisat
− most50%. Assumingthattheconditionissimilarin4U1626–
thedistancemustbeD=5 13kpc. Thisisinagoodagree-
− 67,weassignasystematicuncertaintytothefluxupto 50%,
mentwithChakrabarty(1998). Forreference,thelocusinfig- ∼
whichistransferredalmostdirectlytothatinthenormalization
ure5canbeanalyticallycalculatedinthefollowingway.When
factorofequation(3).AnotheruncertaintyinherenttotheGL79
a value of D is given, the measured zero-point flux specifies
modelisthoseintheaccretiongeometry,includingtheexactlo-
M−2/3R5D−2 via equation (7), while the slope in figure 4
specifiesM−10/7R−2/7D12/7Fb−o1l/7viaequation(9).Byelim- caantdiotnheofatnhgele“souatmerotnrgantshiteiopnuzlsoanre’s”rroadtaituisonr0aoxfise,qiutsatimonag(nAe2ti)c,
inating D from these two scalings, and ignoring the weakly
varyingfactorF−1/7,weobtain axis,andtheaccretionplane;weassumedthattherotationand
bol magneticaxesareparallel, andareperpendicular totheaccre-
M R2 (10) tionplane. Alltheseeffectsmayberepresentedeffectivelyby
∝
8 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
4 Application of the Lovelacemodel
As described insection 1anddetailed inAppendix 2, LRB95
developedanother(inasensemoresophisticated)model,tobe
called “Lovelace model” here, to explain therelation between
P˙ andL, assumingmagneticoutflows and/or magneticbreak-
ing. Using the “turnover radius” r [equation (A8)] and the
to
co-rotationradiusr [equation(A9)],themodelpredictsspin-
cr
upwithoutflowswhenr <r ,andprovidesbothspin-upand
to cr
spin-down solutions with magnetic braking by the disk when
r >r . Toexplainboththespin-upandspin-downbehavior
to cr
of4U1626–67withtheLRB95model,r >r musttherefore
to cr
besatisfiedinthespin-downphase. However,inthespin-down
phaseof4U1626–67,wefoundthatr >r isnotsatisfiedun-
to cr
dertheparameterswhichweassumed.Forexample,thevalues
areestimatedtober =1.3 108cmandr =7.0 108cm
to cr
× ×
Fig.5.Variousconstraints ontheneutron starparameters, shown onthe withFbol=9.3 10−10ergcm−2s−1,M=1.73M ,R=11.1
mass-radius plane. Thesolidlinesindicate howthebest-fit valuesofM × ⊙
km, D=6kpc, andαD =0.01. Thus, the Lovelace model
andR,allowedbythedatainfigure4,varyasDischanged. Thecases m
cannotexplainthespin-downphaseof4U1626–67.
offourdifferentvaluesofthenormalizationfactorAareshown. Apairof
dashedlinesrepresent68%errorsfortheA=1.0curve,whiledottedlines
Eventhough theLovelacemodelhastheaboveproblem, it
givecontoursofthesourcedistanceto4U1626–67.Thegraysolidlinesare
mass-radiusrelationspredictedbythreeEOSs;SLy,APRandShen. mightprovideareasonableexplanationtothespin-upbehavior
of 4U 1626–67. Because r <r is satisfied in the spin-up
to cr
Table2.AllowedM–Rregionsoftheneutronstarof4U1626– phaseofthis object, weshould employ the“spin-up with out-
67foranassumeddistancecorrespondingtothedashedlinesin flows” solution by LRB95, which describes the P˙–L relation
figure5 as
Assumeddistance Mass Radius
∗ ∗
αD 0.15
(kpc) (M⊙) (km) P˙ ≈ −4.3×10−5µ300.285 0.1m R60.85
6.0 1.09–1.15 9.03–9.11
0.425 (cid:16) (cid:17)
M −
7.0 1.28–1.34 9.70–9.79 × M I45−1P2L370.85syr−1, (11)
8.0 1.45–1.53 10.3–10.4 (cid:18) ⊙(cid:19)
9.0 1.63–1.71 10.9–11.0
10.0 1.81–1.90 11.4–11.5 whereαistheviscousparameterinShakura&Sunyaev(1973),
11.0 1.98–2.08 11.8–12.0 D is the magnetic diffusivity parameter, R is R in units of
m 6
12.0 2.15–2.26 12.2–12.4 106 cm, and I is I in units of 1045 gcm2. In LRB95, α
45
13.0 2.32–2.43 12.6–12.8 is assumed to be 0.01 to 0.1 and D to be of order of unity.
m
Equation(11)isequivalenttoequation(3),wherethethemajor
68%confidence(2-parameterserrors)limits.
∗
difference is that the factor n(ω ) in the latter is replaced by
s
αD intheformer.
m
uncertaintyinµ .Becauser isproportionaltoµ 4/7andthe
30 0 30 We thus selected the spin-up-phase data from table 1, and
righthandsideofequation(3)toµ 2/7,anuncertaintyinµ
30 30 fitted them with equation (11), over the parameter ranges of
by,e.g.,50%,wouldinducea25%changeinthecoefficientof
M =1.0 2.4M andR=8.5 15km. AresultforD=6
equation(3). − ⊙ −
kpcandαD =0.01isshowninfigure6,inthesameformat
m
Tojointlytakeintoaccountalltheseuncertainties,wehave as figure 4 (but limited to P˙ <0). The fit is far from being
decided to introduce an artificial normalization factor A, and acceptable, with χ 2 =46 for 30 DOF. Changing D or αD
ν m
multiplied it to the right hand side of equation (3). Then, the didnotsolvetheproblem.Thisisnotsurprising,sinceequation
modelfitting was repeated by changing Afrom0.5 to 1.5. In (11) can explain a torque-equilibrium condition (P˙ =0) only
figure5,theallowedM–RregionsforA=0.5,0.8,1.2,and1.5 whenthefluxtendstozero. ThismakeacontrasttotheGL79
arealso drawn. Thus, the uncertainty indeed affects the mass model, and disagrees with the measurements. In conclusion,
determination, butRremainsverywellconstrained aslongas theLRB95modelcannotexplaintheobservedbehavior of4U
Dissomehowdetermined. 1626–67.
PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0 9
served flux. Our distance estimate is also consistent with that
ofChakrabarty(1998), 5–13kpc, whichwasderivedfromthe
opticalfluxassumingthattheeffectiveX-rayalbedooftheac-
cretiondiskis>0.9.
Conversely,∼if the distance D is assumed, the P˙–F re-
bol
lation can fixM and R with relatively small statistical errors.
Table 2 lists the allowed M and R ranges for typical source
distances assumed. Thus, onceD canbedeterminedbysome
othermeans,M andRof4U1626–67canbeconstrainedtoa
rathernarrowrange,whichwouldbeusefultopin-downthenu-
clearEOS.Apointofparticularimportanceisthatthepresent
methodcanprovidetheinformationonR,whichismorevitally
needed than that onM, without not much affectedby various
systematicerrors(figure5).
In order to further increase the reliability of the GL79
method,itisimportanttounderstandthesystematicerrors(sec-
Fig.6.AscatterplotbetweenP˙ andFbolinthespin-upphases,obtained tion3.3). Inthis respect,anapplication oftheGL79modelto
bytheMAXI/GSCandothersatellites, presented inthesamemanneras
BeX-raybinaries byKlus etal.(2014) isworthnoting. They
figure4. ThedashedhorizontallineindicatesP˙ =0. Thesolidlineisthe
comparedthesurfacemagneticfieldofthesepulsarscalculated
bestfitmodelbyequation(11),withM=1.73M⊙,R=11.1km,D=6
kpc,andαDm=0.01.Thefitgoodnessisχν2=46(1400/30)for30DOF. using the GL79 model (=BGL79), with that measured using
cyclotronresonancescatteringfeatures(=B ). Theratiowas
C
5 Discussion foundasBGL79/BC=3 4intwoexamples,GROJ1008–57
−
andA0535+26. ThiscorrespondstoavalueofA 1.5,which
ApplyingtheepochfoldinganalysistotheMAXI/GSCdata,we ≃
isconsistentwiththe50%uncertaintyinAassumedinsection
determinedP,P˙,andtheX-rayfluxof4U1626–67 forevery
3.3.Thus,studyingalargernumberofsourceswouldbeimpor-
60-dintervalfrom2009Octoberto2013September.Thepulsar
tant.
has been spinning up throughout this period, and the spin-up
ratewaspositivelycorrelatedwiththeflux.
On the P˙–F plane (figure 4), these MAXI/GSC results Acknowledgments
bol
wereconfirmedtobefullyconsistentwiththosefromthepast
The authors are grateful to all members of the MAXI team, BATSE
observations (table 1). In fact, the overall data jointly define pulsar team andFermi/GBM pulsar project. This workwas supported
a well defined P˙ vs. Fbol correlation, covering rather evenly by RIKEN Junior Research Associate Program, and the Ministry of
thespin-upandspin-downphasesfromP˙ = 6 10−11ss−1 Education,Culture,Sports,ScienceandTechnology(MEXT),Grant-in-
− ×
to+5 10−11 ss−1. Utilizing thesefavorableconditions, we AidNo.24340041.
×
have successfully shown that the accretion-torque theory by
GL79 can adequately explain the overall P˙–F behavior of
bol Appendix 1 Ghosh & Lamb model
4U 1626–67, while that of LRB95 model cannot explain the
dataevenlimitingthecomparisontothespin-upphase. GL79derivedanequationbetweenP˙ andLinanX-raybinary
pulsar. The accreting matter transfers the angular momentum
Another favorable condition of this object is the accurate
to the pulsar at the “outer transition zone”, r . Equation (11)
knowledge of its surface magnetic field. As a result, the ob- 0
servedP˙–Fbol relation werefound toconstrain, viatheGL79 inGL79denotesr0=0.52rA(0),whererA(0)isthecharacteristic
Alfvenradius.Substitutingthenumbers
model, two of the three unknown parameters; the distance D,
athlleowmeadsstoMtaokfetahneypvuallsuaer,inantdheitnsormadiinuaslRm.asWshanendrMadiaunsdraRngaeres rA(0) = 3.2×108M˙1−772 µ3074 MM −71 cm, (A1)
ofneutronstars,namelyM=(1.0 2.4)M andR=8.5 15 (cid:18) ⊙(cid:19)
− ⊙ − 1
kkmpc.resTpheicstiivselcyo,ntshiestdeinsttawncitehctahnebleimciotnDstra>in3edktpocDde=riv5e−d b1y3 r0 = 1.7×108M˙1−772 µ3074 MM −7 cm, (A2)
(cid:18) ⊙(cid:19)
Ciohraokfr4abUar1t6y2e6t–a6l7.(u1n9d9e7r);thteheassesuamutphtoiorsnaonf∼aIly=ze1dth1e0P4˙5bgehcmav2- whereM˙17istheaccretionrateM˙ inunitsof1017gs−1.
× GL79gave their theoretical P˙–Lrelation [equation (15)in
andM =1.4M ,thenestimatedthemassaccretionratetobe
M˙ >1 1016 g⊙s−1usingasimilarconsiderationtotheGL79 GL79]as
fram∼ew×ork, and compared the expected luminosity to the ob- P˙ = 5.0 10−5µ3072n(ωs)S1(M)P2L3776 syr−1, (A3)
− ×
10 PublicationsoftheAstronomicalSocietyofJapan,(2014),Vol.00,No.0
InLRB95,αisassumedas0.01to0.1andD isorderunity.
m
r hasbasicallythesamenatureasr intheGL79model[equa-
to 0
tion (A2)]. According to r and r , LRB95 demonstrates a
to cr
magneticoutflowcase(r <r )andmagneticbrakingofthe
to cr
diskcase(r >r ),wherer is
to cr cr
r GM 31 1.5 108 M 31P23 cm, (A9)
cr≡ ω 2 ≈ × M
(cid:16) ∗ (cid:17) (cid:18) ⊙(cid:19)
andω =2π/P.
∗
Whenr issmallerthanr ,thepulsarshowsspin-upwith
to cr
magneticoutflow. P˙ equation[equation(18b)inLRB95]con-
sistsM˙ andr .BymovingP totherightside,
to
1
P˙ ≈−5.8×10−5P2M˙17I4−51 MM 2 10r8tocm 12 syr−1.(A10)
Fig.7.Thefunctionn(ωs)withωs. Theapproximationbyequation(A4)is (cid:18) ⊙(cid:19) (cid:16) (cid:17)
effectiveinωs=0−0.9.Thus,n(ωs)becomes0atωs∼0.349,ispositive BydeducingM˙ =LR/GM,
inωs<∼0.349,andnegativeinωs>∼0.349. αD 0.15
P˙ ≈ −4.3×10−5µ300.285 0.1m R60.85
awnhderSe1L(Mis)deafireneddesbcyriLbed=reMs˙p(eGctiMve/lyR)b.yTehqeufautinocntison(1s0n)(aωnsd) × MM −0.425I45−1P(cid:16)2L370(cid:17).85syr−1. (A11)
(cid:18) ⊙(cid:19)
(17)inGL79as
Itisequivalenttoequation(A3). Theindicesarethesame,and
n(ω ) 1.39[1 ω 4.03(1 ω )0.173 0.878 ] thefactorisalmostthesame. ThedifferenceistheαDm term
s s s
≈ − { − − } insteadofthen(ω )term.
×(1−ωs)−1, (A4) When rto is lasrger than rcr, magnetic braking takes place.
3
S1(M) = R667 MM −7I45−1. (A5) Iwtocraknswasorskpifno-rdobwotnh.spin-upandspin-down,althoughitmostly
(cid:18) ⊙(cid:19)
Here,ω istheso-calledfastnessparameter,whichisadimen-
s
sionless parameter, defined as theratio of thepulsar’s angular Appendix3 Errorestimation ofperiod and
frequency to the Keplerian angular frequency of the accreting periodderivative
matter. Thisω isexpressedapproximatelybyequation(16)in
s We usually use the folding method to obtain P and P˙ of the
GL79as
pulse, orbit etc. First we assume a set of P and P˙, then fold
ωs 1.35µ3067 S2(M)P−1L37−37. (A6) thelightcurvewiththem.Ifthereisnoperiodicitytheresultant
≈
folded light curve is flat. If there is a pulsation with P and
Here,S (M)isgivenbyequation(18)inGL79as
2 P˙ it shows apulse shape. Wecalculate χ2 which is asum of
2
3 M −7 squared deviation from the mean, to evaluate the existence of
S2(M) = R6−7 M . (A7) apulse. Wechange P andP˙ tofindthemost-likely P andP˙
(cid:18) ⊙(cid:19) which give themaximum χ2. Wecan drawdistribution ofχ2
Equation(A4)isaccuratetowithin5%for0 ω 0.9. The
≤ s≤ intheP andP˙ plane. Figure3is anexample which weused
behaviorofn(ω )isplottedinfigure7,wherethezerocrossover
s inthis paper toobtain P andP˙ of4U1626–67. Thus wecan
pointisseentotakeplaceatω 0.349.
s∼ deriveP andP˙,however,theirerrorsarenotobvious.
Appendix2 Lovelacemodel
A.3.1 Method1-parameteraandthestandard
LRB95introducedmagneticoutflowsandmagneticbreakingto estimate-
explaintherelationbetweenP˙ andM˙. Aftermanynumerical
The most primitive and straight-forward method for error-
integrationstheyintroducedrto,wheretheangularvelocityωa estimation is to assume that the errors ∆P, ∆P˙ corresponds
of accreting matter reaches the maximum (dω /dr=0). The
a toadifferenceofpulsenumberainthewholetimespanT of
s
mattertransferstheangularmomentumtothepulsaratr=r .
to theobservation.
r isgivenbyequation(16)inLRB95as
to aP2 2aP2
∆P = ∆P˙ = (A12)
rto≈0.91×108 α0D.1m 0.3µ030.57M˙1−70.3 MM −0.15cm.(A8) a is usualTlys less than 1 pulse. EqTusa2tion (A12) are led by as
(cid:16) (cid:17) (cid:18) ⊙(cid:19)