A Hard Look at the Neutron Stars and Accretion Disks in 4U 1636-53, GX 17+2, and 4U 1705-44 with $\emph{NuSTAR}$

0.64 MB·

by  R. M. Ludlam

DRAFTVERSIONJANUARY10,2017
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
AHARDLOOKATTHENEUTRONSTARSANDACCRETIONDISKSIN4U1636-53,GX17+2,AND4U1705-44WITH
NUSTAR
R.M.LUDLAM1,J.M.MILLER1,M.BACHETTI2,D.BARRET3,4,A.C.BOSTROM1,E.M.CACKETT5,N.DEGENAAR6,7,T.DI
SALVO8,L.NATALUCCI9,J.A.TOMSICK10,F.PAERELS11,M.L.PARKER6
DraftversionJanuary10,2017
ABSTRACT
7 We present NuSTAR observations of neutron star (NS) low-mass X-ray binaries: 4U 1636-53, GX 17+2,
1 and4U1705-44. We observed4U1636-53inthehardstate, with anEddingtonfraction,FEdd, of0.01;GX
0
17+2and4U1705-44wereinthesoftstatewithfractionsof0.57and0.10,respectively.Eachspectrumshows
2
evidenceforarelativisticallybroadenedFeK line. Throughaccretiondiskreflectionmodeling,weconstrain
α
n the radius of the inner disk in 4U 1636-53to be Rin = 1.03±0.03 ISCO (innermoststable circular orbit)
a assumingadimensionlessspinparametera∗ = cJ/GM2 = 0.0,andRin = 1.08±0.06ISCOfora∗ = 0.3
J (errors quoted at 1 σ). This value proves to be model independent. For a∗ = 0.3 and M = 1.4 M⊙, for

6 example,1.08±0.06ISCOtranslatestoaphysicalradiusofR = 10.8±0.6km,andtheneutronstarwould
havetobesmallerthanthisradius(otheroutcomesarepossibleforallowedspinparametersandmasses). For

E] GX17+2,Rin =1.00−1.04ISCOfora∗ =0.0andRin =1.03−1.30ISCOfora∗ =0.3.Fora∗ =0.3and
M =1.4M⊙,Rin =1.03−1.30ISCOtranslatestoR=10.3−13.0km.Theinneraccretiondiskin4U1705-
H 44maybetruncatedjustabovethestellar surface, perhapsbyaboundarylayerormagnetosphere;reflection
. models give a radius of 1.46-1.64 ISCO for a∗ = 0.0 and 1.69-1.93 ISCO for a∗ = 0.3. We discuss the
h
implicationsthatourresultsmayhaveontheequationofstateofultradense,coldmatterandourunderstanding
p
of the innermost accretion flow onto neutron stars with low surface magnetic fields, and systematic errors
-
o relatedtothereflectionmodelsandspacetimemetricaroundlessidealizedneutronstars.
r
t
s
a 1. INTRODUCTION vations. LMXBs consist of a roughly stellar-mass compan-
[ ion that transfers matter onto a compact object via Roche-
Theequationofstate(EOS)ofultradensematterisnotyet
known. Earth laboratoriesare unable to replicate the neces- lobe overflow. Broad iron line profiles have been seen
1
in these accreting systems that harbor a black hole (BH;
v saryenvironmenttoconstraintheEOS.Thus,theonlywayto
e.g. Fabianetal. 1989; Milleretal. 2002, 2007; Reisetal.
4 studymatterundertheseextremeconditionsisthroughobser-
2008, 2009a) or NS (e.g. Bhattacharyya&Strohmayer
7 vationsofneutronstars(NS).TheEOSsetsthemassandra-
2007; Papittoetal. 2008; Cackettetal. 2008, 2009, 2010;
7 diusoftheNS.Theoreticalmass-radiustrackshavebeencom-
diSalvoetal. 2009, 2015; Egronetal. 2013; Milleretal.
1 piledfordifferenttheoreticalmodels(seeLattimer&Prakash
2013) as the primary accreting compact object. LMXBs
0 2016 for a review). Three-bodyinteractions between nucle-
are ideal laboratories for conducting radius measurements
. ons make the radius of the NS the most important quantity
1 through reflection studies since all accretion occurs via the
in determining the EOS because it does not change quickly
0 disk, as opposed to their high-mass X-ray binary (HMXB)
as a functionof mass. Hence, constrainingthe radiusof the
7
counterpartswhichcanaccreteviastellarwinds.
1 NS can, in turn, lead to a determination of the EOS of the
X-raydisklinesareproducedfromanexternalhardX-ray
: cold,densematterunderextremelydensephysicalconditions
v (Lattimer2011). sourceirradiatingthe accretiondisk. Inthe case ofNSs, the
i hardX-ray source couldbe a hotcorona, the stellar surface,
X Low-mass X-ray binaries (LMXBs) are one setting in
or a boundary layer and may be thermal or non-thermal in
which we can attempt to constrain the EOS through obser-
r nature. Regardlessof thenatureofthe hardX-rayemission,
a
1DepartmentofAstronomy,UniversityofMichigan,1085SouthUni- theasymmetricallybroadenedprofileoftheFeKα linegives
versityAve,AnnArbor,MI48109-1107,USA adirectmeasureofthepositionofinnerdisksincetheeffects
2INAF/Osservatorio Astronomico di Cagliari, via della Scienza 5, I- ofgravitationalredshiftandDopplerredshift/boostingonthe
09047Selargius(CA),Italy emission line become stronger closer to the compact object
3UniversitdeToulouse;UPS-OMP;IRAP;Toulouse,France
(Fabianetal.1989).
4CNRS;InstitutdeRechercheenAstrophysiqueetPlantologie; 9Av.
colonelRoche,BP44346,F-31028Toulousecedex4,France The Fe Kα line in NS LMXBs can set an upper limit
5DepartmentofPhysics&Astronomy,WayneStateUniversity,666W. for the radius of NS since the disk must truncate at or be-
HancockSt.,Detroit,MI48201,USA forethesurface(Cackettetal.2008,2010;Reisetal.2009b;
6InstituteofAstronomy,MadingleyRoad,CambridgeCB30HA,UK
Milleretal.2013;Degenaaretal.2015). Sincethemagnetic
7AntonPannekoekInstituteforAstronomy,UniversityofAmsterdam,
field could also truncate the accretion disk, studies of disk
Pastbus94249,1090GEAmsterdam,TheNetherlands
8Dipartimento diFisicaechimica, Universita´ degliStudidiPalermo, reflectioncan also be used to set an upperlimit onthe mag-
viaArchirafi36,90123,Palermo,Italy netic field strength (Cackettetal. 2009; Papittoetal. 2009;
9IstitutoNazionalediAstrofisica,INAF-IAPS,viadelFossodelCava- Milleretal. 2011; Degenaaretal. 2014, 2016; Kingetal.
liere,00133Roma,Italy
2016;Ludlametal.2016).
10SpaceSciences Laboratory, 7GaussWay,University ofCalifornia,
WeanalyzeNuSTAR(Harrisonetal.2013)observationsof
Berkeley,CA94720-7450,USA
11Columbia Astrophysics Laboratory, 550 West 120th Street, New NS LMXBs 4U1636-53, GX 17+2, and 4U1705-44. 4U
York,NY10027,USA 1636-53 and 4U 1705-44 are categorized as atoll sources
2 Ludlametal.
while GX 17+2 is a “Z” source, according to the classifica-
Table1
tion of Hasinger&vanderKlis (1989). Atoll sources have
NuSTARObservationInformation
lowerluminosities(∼ 0.01−0.2LEdd)thanZ sources(see
Doneetal.(2007)forareview). TheZsourceshavesoftX-
rayspectrathatcanbedescribedbytwothermalcomponents Source ObsID Date NetExp(ks) NetRate(cts/s)
4U1636-53 30101024002 06/06/2015 19.2 40
(multicolordiskblackbodyandsingletemperatureblackbody;
GX17+2 30101023002 03/22/2016 23.3 361
Linetal. 2007). Atolls can have soft or hard spectra with
4U1705-44 30101025002 04/22/2016 28.7 174
transitional phases in between. The hard state can be mod-
eledwellbyapowerlawandthermalcomponentwhenneeded
LAOR, KRYLINE) to account for relativistic broadening and
(Linetal.2007).
foundthattheFelinedidnotchangesignificantly. Theycon-
Further,thespectralstatemaybeassociatedwiththelevel
cludethatthelineisbroadenedbymechanismsotherthanjust
of disk truncation (Doneetal. 2007). A study of 4U 1608-
relativistic broadening, though the data are consistent with
52 by Gierlin´ski&Done (2002) found that at low luminos-
a disk remaining at the ISCO if timing parameters trace the
ity the accretion disk was truncated while at high luminos-
massaccretionrateratherthantheinnerradius.
itythe innerradiusofthediskmovedinwards. Pintoreetal.
(2016)foundsimilarbehaviorforSAXJ1748.9-2021. How- 1.2. GX17+2
ever,Sannaetal.(2014)foundthattheinnerdiskdidnotseem
GX17+2isa bursterlocateda maximumdistanceof 13.0
to be correlatedwith the spectral state for 4U 1636-53. The
kpc(Gallowayetal.2008)withaspinfrequencyof293.2Hz
innerdiskradiusofSerpensX-1(SerX-1)also doesnotap-
(Wijnandsetal. 1997). The counterpart of GX 17+2 is not
peartochangemuchforarangeofluminosities(Chiangetal.
knowncurrently. ItmaybeNPSer(Tarenghi&Reina1972)
2016).
or star “A” proposed by Deutschetal. (1999) from optical
4U1636-53wasinthehardstateatthetimeoftheNuSTAR
and IR variability studies. However, Callananetal. (2002)
observation,whileGX17+2and4U1705-44wereinthesoft
proposed that the IR variability could be explained by syn-
state. Wefocusonconstrainingtheinnerdiskradiusinthese
chrotron flares. The system has an inclination of less than
sources and the implications this has on the EOS for ultra-
45◦ (Cackettetal. 2010, 2012). DiSalvoetal. (2000) per-
dense, cold matter. The following sections (§1.1, 1.2, 1.3)
formedthefirstextensivespectralanalysisofGX17+2.They
give a brief introduction about each source. Section 2 pro-
were able to limit the radius of the NS between 10-19 km
videsdetailsontheobservationsofeachsourceandhowthe
based upon comptonization of photons within their spectra.
datawerereduced.Section3presentsouranalysisandresults.
Cackettetal. (2010) found a similarly small limit through
Section4 discusses thoseresultsand we concludein section
modelingtheFereflectionintheaccretiondisk.
5.
1.1. 4U1636-53 1.3. 4U1705-44
4U 1636-53 is a well studied, persistent LMXB that ex- 4U1705-44isapersistentlybrightsourcethatshowsType
hibits Type 1 X-ray bursts and has a spin frequency of 581 1 X-ray bursts (Langmeieretal. 1987) which place it at a
Hz(Zhangetal.1997;Strohmayer&Markwardt2002). The maximum distance of 7.8 kpc (Gallowayetal. 2008). The
source is located a maximum of 6.6 kpc away from in- source has been observed in both the hard and soft states
spection of the Type 1 X-ray bursts (assuming the brightest (Barret&Olive2002;Pirainoetal.2007).BroadFeemission
type-I X-ray bursts hit the Eddington limit; Gallowayetal. hasbeenineachofthesestates. Theinclinationofthesystem
2008). The companion star is a 0.4 M⊙, 18th magni- is between 20◦ − 50◦ (Pirainoetal. 2007). Di Salvo et al.
tude star with an orbital period of 3.8 hours (Pedersenetal. (2009)foundevidenceformultipleemissionlinesforSXVI,
1982; vanParadijsetal. 1990). kHz quasi-periodic oscilla- Ar XVIII, and Ca XIX in addition to iron in observations
tions (QPOs) suggest that 4U 1636-53 may harbor a NS as takenwithXMM-Newton. Theselinesgaveaninnerdiskra-
large as 2 M⊙ (Barretetal. 2006; Casaresetal. 2006). The diusof2.3±0.3ISCO(assuminga∗ =0).Reisetal.(2009b)
systemregularlyundergoesstatetransitionson∼40daytime foundaninnerdiskradiusof1.75+0.17ISCOwhenusingob-
−0.28
intervals(Shihetal.2005). servations obtained with Suzaku. Cackettetal. (2010) used
Reflectionstudiessuggestthissourcehasahighinclination, both XMM-Newton and Suzaku observations and found that
butalackofdipsintheX-raylightcurvelimitstheinclination the inner disk ranged from 1.0-6.5 ISCO. The inner disk of
∼ 70◦ (Franketal. 1987; Casaresetal. 2006; Sannaetal. 4U 1705-44appearsto be truncated in many studies though
2013). Pandeletal. (2008) found the inner disk radius was thedegreeoftruncationvaries.
consistent with the innermost stable circular orbit (ISCO)
when looking at observations taken with XMM-Newton and 2. OBSERVATIONSANDDATAREDUCTION
RXTE while in the transitional and soft state. Cackettetal. The details of each NuSTAR observation are listed in Ta-
(2010)foundlargerinnerdiskradiiat∼8±4ISCO(assuming ble 1. Lightcurves and spectra were created using a 120′′
a∗ = 0) whenapplyingblurredreflectionmodelsto the low circular extraction region centered around the source using
fluxstate,butvaluesconsistentwiththeISCOweremeasured theNUPRODUCTStoolfromNUSTARDASv1.5.1withCALDB
in the soft state. Additionally, Sannaetal. (2013) analyzed 20160421.Abackgroundwasgeneratedandsubtractedusing
the source in the soft and transitional states with two addi- anotherregionofthesamedimensioninaregionawayfrom
tionalobservations. Theymeasuretheinnerdiskradiustobe thesource.Therewereatotalof6Type1X-rayburstspresent
aslargeas4.45ISCOinlowfluxstates,whichissmallerbut inthelightcurvesfor4U1636-53anda singleType1X-ray
consistent with Cackettetal. (2010). Lyuetal. (2014) used burstfor4U1705-44. Wewillreportontheburstsinasepa-
observations from Suzaku, XMM-Newton, and RXTE to see ratepaper. We createdgoodtimeintervals(GTIs)toremove
if the inner disk radius inferred from Fe line changed over ∼10–150saftertheinitialfastrise(dependingonthedura-
fluxstates. Theyusedavailabledisklinemodels(DISKLINE, tionoftheindividualburst)toseparatethesefromthesteady
AHardLookat4U1636-53,GX17+2,and4U1705-44 3
emission. TheseGTIswereappliedduringthegenerationof asfollows: logoftheionizationparameter(log(ξ)), temper-
the spectra for both the FPMA and FPMB. Initial modeling atureoftheincidentblackbodyinkeV(kT),ironabundance
of the persistent emission spectra with a constant, fixed to 1 (A ),reflectionfraction(f ),redshift(z),andnormaliza-
Fe refl
forthe FPMA and allowed to floatfor the FPMB, foundthe tion.TheparametersofRELCONVareasfollows:inneremis-
floatingconstanttobewithin0.95-1.05ineachcase. Wepro- sivity index (q ), outer emissivity index (q ), dimension-
in out
ceededtocombinethetwosourcespectra,backgroundspec- less spin parameter (a∗), inner disk radius in units of ISCO
tra, andancillaryresponsematricesvia ADDASCASPEC. We (Rin), and outer disk radius in units of gravitational radii
useADDRMFtocreateasingleredistributionmatrixfile. The (Rout).
spectra were grouped using GRPPHA to have a minimum of Additionally, we use REFLIONX (Ross&Fabian 2005) to
25countsperbin(Cash1979). testtherobustnessofourresultsobtainedwith RELXILL and
thenatureoftheFeKαlineinourBBREFL. Inthecaseofthe
3. SPECTRALANALYSISANDRESULTS 4U1636-53,weuseaversionofREFLIONX12thathasavari-
WeuseXSPECversion12.8.1(Arnaud1996)inthiswork. ablehighenergycut-off.Weconvolvethismodelwiththerel-
Allerrorsarequotedat1σ (68%)confidencelevelandwere ativisticblurringkernelKERRCONV(Brenneman&Reynolds
calculated from Monte Carlo Markov Chain (MCMC) of 2006). Thisprovidesacompletelymodelindependentcheck
length 100,000. We perform fits over the energy range in to values obtained with RELXILL. In the case of GX 17+2
which the spectrum is not background dominated for each and 4U 1705-44,we have a version of REFLIONX13 that has
source. Weaccountfortheneutralcolumnhydrogendensity beenmodifiedtoassumeablackbodyisilluminatingthedisk.
alongthelineofsightviaTBABS(Wilmsetal.2000).Theso- Thisreflectionmodeldoesnotrepresentanindependentcheck
larabundancewassettoWILM(Wilmsetal.2000)andVERN onBBREFL sincetheyarederivedfromthesameparentcode
crosssections(Verneretal.1996)wereused. (Ross&Fabian1993). Itcan,however,beusedtoverifythat
We experimented with different phenomenological one, the Fe K line is dynamicallybroadenedand nota result of
α
two, and three-component models to describe the spectral gas broadening. This is because the REFLIONX model con-
continua. None of the sources required three components. tainsadditionalphysicsthataccountsforgaseffects. Ithasa
The continuumof the spectrum obtained from 4U 1636−53 broaderrangeofelements,chargestates,andionizationwhile
waswelldescribedwithacut-offpower-lawmodel. Thecut- itaccountsforthe localradiationfieldateachpoint. Hence,
offenergymayreflecttheelectrontemperatureofthecorona, ifweareabletoobtainsimilarvaluesforvariousparameters,
andthesimplecontinuumcomponentmayonlybeanapprox- thenthelineprofileisdynamicalinorigin.
imationofaregiondominatedbyComptonization.Thespec- A few reasonable conditions were enforced when making
tral continua of GX 17+2 and 4U 1705−44 were well de- fitswithRELXILLandRELCONV. First,wetietheouteremis-
scribed by a combination of disk blackbody (Mitsudaetal. sivityindex,q totheinneremissivityindex,q ,tocreatea
out in
1984) and simple blackbody components. It is possible to constantemissivityindex.Next,wefixthespinparameter,a∗
infer radii from both disk blackbody and simple blackbody (where a∗ = cJ/GM2), in the model RELCONV to 0.0 and
components; however, it is also clear that scattering can 0.3inthesubsequentfitssinceNSinLMXBshavea∗ ≤ 0.3
harden emergent spectra and cause falsely small radii to be (Milleretal.2011;Gallowayetal.2008). Thisdoesnothin-
inferred(e.g.,Londonetal.1986;Shimura&Takahara1995; der our estimate of the inner radiussince the position of the
Merlonietal.2000).Lowtemperature,optically-thickComp- ISCO is relatively constant for low spin parameters (correc-
tonizationcansometimesbeapproximatedasablackbody;it tions for frame-dragging for a∗ < 0.3 give errors ≪ 10%;
is likely that our simple blackbody component accounts for Milleretal. 1998). Further, the outer disk radius has been
thisprocessintheboundarylayer(e.g.,Gilfanovetal.2003). fixedto 990 R (whereR = GM/c2). In the case thatwe
g g
Giventhemanycomplexitiesandpossibledistortions,wedo use KERRCONV, we tie the emissivity indices to create one
notlooktothecontinuaforrobustphysicalinferences;rather, emissivityindexandfixtheouterdiskradiusto400ISCO.
weutilizediskreflectionforthispurpose.
To properly describe the reflection features and relativis- 3.1. 4U1636-53
tic effects present in the data for 4U 1636-53 we employ
As noted above, initial fits were performed from 3.0-50.0
the model RELXILL (Garc´ıaetal. 2014). This model self-
keV with an absorbed cut-off power law to model the con-
consistentlyaccountsforX-rayreflectionandrelativisticray
tinuumemission. Althoughthecontinuumiswelldescribed,
tracing for a power law irradiating an accretion disk while
this model gives a poor fit (χ2/dof = 2435.47/876)since
properly taking into account the inclination in the reflection
itdoesnotaccountforthereflectionpresentwithinthespec-
from the disk. The parameters of this model are as fol-
trum. AbroadFeK linecentered∼ 7keVwitharedwing
lows: inner emissivity (q ), outer emissivity (q ), break α
in out extendingtolowerenergiesandbluewingextendingupto9
radius (R ) between the two emissivities, spin parame-
ter (a∗ =brceJa/kGM2), inclination if the disk (i), inner radius keV,implyingahighinclination,canbeseeninthetoppanel
ofFigure1. The additionof a thermalcomponentimproved
of the disk (R ) in units of ISCO, outer radius of the disk
in theoverallfit,butthetemperaturewasunfeasiblylowforthe
(R ),redshift(z),photonindexofthepowerlaw(Γ),logof
out sensitivityofNuSTAR(0.18±0.02keV)andthenormaliza-
theionizationparameter(log(ξ)),ironabundance(A ),cut
offenergyofthepowerlaw(Ecut),reflectionfractionF(ferefl), tion was notphysicallypossible (3.6+−233.4.2×107), hence we
andnormalization. excludeditfromsubsequentmodeling.
We apply BBREFL (Ballantyne 2004) to GX 17+2 and 4U WeperformedtwofitswithTBABS*RELXILLforthediffer-
1705-44 in order to model the emergent reflection emission entspinvalues.Eachoftheseprovideasignificantlybetterfit
from a disk assuming an irradiating blackbody continuum, (28σ improvement)overtheabsorbedcut-offpowerlawand
andconvolveitwithRELCONV(Dauseretal.2010). Theiron
abundanceinBBREFLcomesintwoflavors: solarabundance 12https://www-xray.ast.cam.ac.uk/∼mlparker/reflionx models/reflionx hc.mod
and twice solar abundance. The parameters of BBREFL are 13http://www-xray.ast.cam.ac.uk/∼mlparker/reflionx models/reflionx bb.mod
4 Ludlametal.
Table2 Table3
4U1636RelxillModelingforDifferentSpinParameters 4U1636ReflionxModelingforDifferentSpinParameters
Component Parameter RELXILL Component Parameter REFLIONX
TBABS NH(1022)† 0.3 0.3 TBABS NH(1022)† 0.3 0.3
RELXILL q 2.25±0.05 2.19±0.04 CUTOFFPL Γ 1.74±0.01 1.75±0.01
a†∗ 0.0 0.3 Ecut(keV) 20.7±0.3 20.6±0.3
i(◦) 78.2±1.67 78.5±1.22 norm 0.20±0.01 0.20±0.01
Rin(ISCO) 1.03±0.03 1.08±0.06 KERRCONV q 2.33±0.04 2.28±0.05
Rout(Rg)† 990 990 a†∗ 0.0 0.3
Rin(km) 12.4±0.4 10.8±0.6 i(◦) 78.6±1.2 78.3±1.2
z† 0.0 0.0 Rin(ISCO) 1.02±0.02 1.08±0.06
Γ 1.74±0.01 1.74±0.01 Rout(ISCO)† 400 400
log(ξ) 3.3±0.1 3.26±0.06 Rin(km) 12.2±0.2 10.8±0.6
AFe 4.9±0.1 4.8±0.1 REFLIONX ξ 1800±600 1100±500
Ecut(keV) 20.5±0.3 20.5±0.3 AFe 4.4±0.5 4.6±0.3
frefl 0.42±0.05 0.40±0.04 frefl 0.6±0.2 0.6±0.3
norm(10−3) 2.21±0.06 2.22±0.03 z† 0 0
Funabs,3.0−50.0keV 8.9±0.2 8.9±0.1 norm(10−7) 2.7±0.9 6.8±4.0
Lunabs,3.0−50.0keV 4.6±0.1 4.63±0.06 Funabs,3.0−50.0keV 9±3 9±5
Funabs,0.5−50.0keV 12.8±0.3 12.8±0.2 Lunabs,3.0−50.0keV 5±2 5±3
Lunabs,0.5−50.0keV 6.7±0.2 6.67±0.09 Funabs,0.5−50.0keV 14±5 14±8
χ2ν(dof) 1.06(862) 1.06(862) Lunabs,0.5−50.0keV 7±2 7±4
†=fixed χ2(dof) 1.25(862) 1.23(862)
ν
†=fixed
Note.—Errorsarequotedat1σconfidencelevel.Theabsorptioncolumn
densitywasfixedtotheDickey&Lockman(1990)valueandgiveninunits Note.—Errorsarequotedat1σconfidencelevel.Theabsorptioncolumn
ofcm−2.Thespinparameterispeggedatanupperlimitof0.3.Theinner densitywasfixedtotheDickey&Lockman(1990)valueandgiveninunits
diskradiusinunitsofkmassumesaNSmassof1.4M⊙.Fluxisgivenin ofcm−2.Thespinparameterispeggedatanupperlimitof0.3.The
unitsof10−10ergscm−2s−1.Luminosityiscalculatedatamaximumof REFLIONXmodelusedhasavariablehighenergycutoffwhichwetiedto
6.6kpcandgiveninunitsof1036ergss−1. thecutoffenergyofthecutoffpowerlawusedtomodelthecontinuum
emission.Additionally,wetiedthephotonindexbetweenREFLIONXand
CUTOFFPL.TheinnerdiskradiusinunitsofkmassumesaNSmassof1.4
can be seen in Table 2. The ionization parameter is reason- M⊙.Fluxisgiveninunitsof10−10ergscm−2s−1.Luminosityis
calculatedatamaximumof6.6kpcandgiveninunitsof1036ergss−1.
ableforaccretingsourcesacrossallfits. Theironabundance
islarge,butfixingittolowervaluesdoesnotchangetheinner
diskradius.Regardlessoftheironabundance,eachfitconsis- blackbody component. This gives a poor fit (χ2/dof =
tently givesa small innerdisk radiusof Rin = 1.03±0.03 4289.59/670)becausethereflectionspectrumisnotyetmod-
ISCOfora∗ =0.0andRin =1.08±0.06ISCOfora∗ =0.3. eled. TheFeKα emissioncanbeseeninthemiddlepanelin
Thefitreturnsahighinclinationof76.5◦−79.9◦foreachspin Figure1. The redwing extendsdownto ∼ 4 keV while the
value. Thephotonindexis1.74±0.01withacutoffenergy bluewingdropsaround∼7keV.
at ∼ 20.5 keV. The best fit spectrum can be seen in Figure We use BBREFL to model the emergent reflection emis-
2. Figure 3 shows the goodness-of-fit versus the inner disk sion and convolveit with RELCONV. The overallmodel we
radius(toppanel). usedwasTBABS*(DISKBB+RELCONV*BBREFL). Thismodel
Tocheckthatthevaluesobtainedwith our RELXILL mod- provides a better fit with χ2/dof = 793.2/665 (33σ im-
eling of 4U 1636-53 are not model dependent, we apply provement for the highest χ2). Parameters and values can
ν
the model REFLIONX to characterizethe emergentreflection beseeninTable4. Figure2showsthebestfitspectrum. For
spectrum arising from an incidentcut-off power law contin- a∗ =0.0,theinnerdiskradiusistightlyconstrainedtoRin =
uum. We blur the emergent reflection with the relativistic 1.00−1.02 ISCO for A = 1.0 and R = 1.00−1.04
Fe in
convolution model KERRCONV. This version of REFLIONX ISCOforAFe =2.0. Forthehighervalueofspin,a∗ =0.3,
hasanadjustablecut-offenergywhichwetiewiththecut-off R =1.15+0.15ISCOforA =1.0andR =1.10±0.07
powerlawmodelusedtomodelthecontinuumemission.This in −0.08 Fe in
ISCO for A = 2.0. The emissivity index is high in each
Fe
model, however, is angle averaged, unlike RELXILL, which case(raytracingassumingeithera hotspotata modestlati-
properly takes into account the inclination of the disk when
tude, ora heatedequatorialregion,bothpredictq = 3.0; D.
tracingeachphotonfromthedisk. Wilkins,priv.comm.).Theinclinationliesbetween25◦−38◦
TheresultingfitcanbeseeninTable3foraspinof0.0and
forallfits. Figure3showsthechangeingoodness-of-fitver-
0.3. Theinnerdiskradiusisconsistentwiththevaluesfound
sustheinnerdiskradius(middlepanel).
in the RELXILL modeling. Additionally, the photon index, To test the origin of the Fe K line, we use a version of
α
highenergycut-off,emissivityindex,spinparameter,ioniza-
REFLIONX that assumes a blackbody is illuminating the ac-
tion,andinclinationarewithinerrorforthevaluesfoundwith
cretiondisk. Table4showsparametersandvaluesusingthis
RELXILL. Thisdemonstratesthatthe innerdisk radiusmea- model. The overallfit is worse, but still requiresa small in-
surementisrobust.
ner disk radiusof R = 1.00−1.02 ISCO for both values
in
ofspin. Theinclinationandionizationisconsistentwiththe
3.2. GX17+2
valuesfoundinthepreviousfitswithBBREFL. Theemissivity
Initial fits were performedfrom 3.0-30.0keV with an ab- index is close to simple expectations. Regardless, the small
sorbedsingletemperatureblackbodyandamulti-temperature inner radii measured with REFLIONX imply that dynamical
AHardLookat4U1636-53,GX17+2,and4U1705-44 5
Table4
GX17+2ReflectionModeling
Component Parameter BBREFL REFLIONX
TBABS NH(1022)† 0.9 0.9 0.9 0.9 0.9 0.9
DISKBB Tin 1.93±0.04 1.992±0.001 1.93±0.05 1.92±0.06 1.92±0.01 1.92±0.01
norm 26±1 23.8±0.1 26±2 26±2 26.8±0.4 26.7±0.4
BLACKBODY kT ... ... ... ... 2.86±0.01 2.86±0.01
norm(10−2) ... ... ... ... 2.9±0.1 2.9±0.1
RELCONV q 8.1+−01..80 3.7+−40..73 6.0±3.0 4.5±1.5 3.5±0.1 3.2±0.1
a†∗ 0.0 0.3 0.0 0.3 0.0 0.3
i(◦) 35.1+0.2 30+8 30±5 30±4 25.9±0.2 25.8±0.2
−0.6 −2
Rin(ISCO) 1.00+0.02 1.15+−00..1058 1.02±0.02 1.10±0.07 1.01±0.01 1.01±0.01
Rout(Rg)† 990 990 990 990 990 990
Rin(km) 12.0+0.2 11.5+−10..58 12.2±0.2 11.0±0.7 12.1±0.1 10.1±0.1
BBREFL log(ξ) 2.47+−00..0071 2.45±0.01 2.33+−00..0091 2.40±0.06 ... ...
kT (keV) 3.15+0.01 3.12+0.04 3.03+0.05 3.03±0.02 ... ...
−0.07 −0.01 −0.01
A† 1.0 1.0 2.0 2.0 ... ...
Fe
frefl 1.69+−00..0647 1.52+−00..6071 1.2+−00..24 1.27+−00..0540 ... ...
z† 0 0 0 0 ... ...
norm(10−26) 8.83+0.60 8.61+0.02 13.0+1.0 13.1+0.08 ... ...
−0.01 −0.36 −0.03 −0.01
REFLIONX ξ ... ... ... ... 240±20 230±7
AFe ... ... ... ... 2.1±0.4 2.1±0.4
frefl ... ... ... ... 0.9±0.1 0.9±0.1
z† ... ... ... ... 0 0
norm ... ... ... ... 1.7±0.2 1.8±0.1
Funabs,3.0−30.0keV 7.3+−00..63 7.30+−00..0341 7.3+−00..86 7.3±0.1 7.3±0.9 7.3±0.5
Lunabs,3.0−30.0keV 1.5±0.1 1.48+−00..0016 1.5+−00..21 1.48±0.02 1.5±0.2 1.5±0.1
Funabs,0.5−30.0keV 10.7+−00..84 10.7+−00..14 10.7+−10..28 10.7±0.8 11.0±1.3 11.0±0.7
Lunabs,0.5−30.0keV 2.2+−00..21 2.16+−00..0018 2.2±0.2 2.2±0.2 2.2±0.3 2.2±0.1
χ2(dof) 1.06(665) 1.13(665) 1.19(665) 1.18(665) 1.28(664) 1.26(664)
ν
†=fixed
Note.—Errorsarequotedat1σconfidencelevel.TheabsorptioncolumndensitywasfixedtotheDickey&Lockman(1990)valueandgiveninunitsofcm−2.
Limbbrighteningwasassumed.Theemissivityindexwaspeggedatthehardlimitof3.0.TheREFLIONXmodelusedhasbeenmodifiedforablackbody
illuminatingtheaccretiondisk.TheinclinationforREFLIONXwaspeggedwithinthelimitsfoundwiththeBBREFLduetothemodelbeingangleaveragedand
notabletoconstraintheinclination.TheionizationparameterwasalsopeggedwithintheBBREFLvaluessinceitwasunconstrainedonitsown.Theblackbody
temperatureinREFLIONXwastiedtothetemperatureoftheblackbodyusedtomodelthecontinuumemission.Theemissivityindexwaspeggedatthehard
limitof3.0.TheinnerdiskradiusinunitsofkmassumesaNSmassof1.4M⊙.Fluxisgiveninunitsof10−9ergscm−2s−1.Luminosityiscalculatedata
maximumof13.0kpcandgiveninunitsof1038ergss−1.
broadeningis dominantin shaping the iron line, and any at- a blackbodyilluminating the disk. The resulting fits can be
mosphericeffectsaresecondary. seeninTable5. Eventhoughtheoverallfitisslightlyworse,
thediskstillrequiresatruncationofR =1.21±0.18ISCO
in
3.3. 4U1705-44 for a∗ = 0.0 and Rin = 1.37±0.27 ISCO for a∗ = 0.3.
Initial fits were performedfrom 3.0-30.0keV with an ab- Neither model strongly requires that the disk extends to the
sorbedsingletemperatureblackbodyandamulti-temperature ISCO,thoughtheISCOiswithin2sigmaofthenominalbest-
blackbody component. This gives a poor fit (χ2/dof = fit valuesfoundwith REFLIONX. Thisconfirmsthatdynam-
4504.9/670)sincethestrongdiskreflectionspectrumhasnot ical broadeningis the dominantmechanismfor the iron line
been modeled. See the bottom panel of Figure 1 for the Fe shape.
K line. SincethediskdoesnotextenddowntotheISCO,wecalcu-
α
We use BBREFL to model the emergent re- latetheextentofa possibleboundarylayerandplaceanup-
flection emission and convolve it with RELCONV perlimitonthemagneticfieldstrengthsincetheseareplausi-
(Dauseretal. 2010). The overall model we used was blescenariosfordisktruncation. Popham&Sunyaev(2001)
TBABS*(DISKBB+RELCONV*BBREFL). This provides a layoutthe Newtonianframeworkforboundarylayer behav-
better overall fit (34σ improvement for the highest χ2). ior for differentmass accretion rates. We estimate the mass
ν
ParametersandvaluescanbeseeninTable5. Theinnerdisk accretionfor4U1705-44tobe(3.4±0.4)×10−9M⊙ yr−1
radius is slightly truncated prior to the neutron star surface from the 0.5-30.0 keV unabsorbed luminosity and using an
between1.46-1.64ISCOfora∗ = 0.0forbothvaluesofiron efficiency of η = 0.2 (Sibgatullin&Sunyaev 2000). Using
abundance.Fora∗ =0.3,theinnerdiskradiusistruncatedat Equation(25)inPopham&Sunyaev(2001),weestimatethat
1.69-1.93ISCO.Theinclinationisbetween24.0◦−26.1◦for theboundarylayerextendsoutto∼ 1.2ISCO(assuming1.4
all fits. The change in goodness-of-fitversus the inner disk M⊙&a∗ =0.0).Additionalfactors,suchasspinandviscos-
radiuscanbeseeninthebottompanelofFigure3. ity in the layer, can extend this region to be consistent with
Again,wetesttheoriginofourFeK lineinthistruncated thetruncationoftheinnerdisk.
α
disk with the version REFLIONX that has been modified for Ifthediskisimpededbythemagnetosphere,wecanplace
6 Ludlametal.
Table5
4U1705-44ReflectionModeling
Component Parameter BBREFL REFLIONX
TBABS NH(1022)† 0.7 0.7 0.7 0.7 0.7 0.7
DISKBB Tin 2.13±0.01 2.13±0.01 2.00±0.01 2.00±0.01 1.95±0.02 1.95±0.03
norm 9.65±0.02 9.6±0.1 11.48±0.02 11.60±0.01 12.3±0.5 12.4±0.6
BLACKBODY kT ... ... ... ... 2.54±0.02 2.54±0.03
norm(10−2) ... ... ... ... 1.15±0.05 1.15±0.08
RELCONV q 3.1±0.1 3.1±0.1 3.2±0.1 3.2±0.1 2.8±0.3 2.7±0.3
a†∗ 0.0 0.3 0.0 0.3 0.0 0.3
i(◦) 24.4±0.4 24.4±0.4 24.6±0.5 25.6±0.5 13±11 16±6
Rin(ISCO) 1.54±0.08 1.82±0.09 1.50±0.07 1.78±0.09 1.21±0.18 1.37±0.27
Rout(Rg)† 990 990 990 990 990 990
Rin(km) 18.4±1.0 18.2±0.9 18.0±0.8 17.8±0.9 14.5±2.2 13.7±2.7
BBREFL log(ξ) 2.67±0.02 2.66±0.02 2.74±0.04 2.74±0.04 ... ...
kT (keV) 2.83±0.01 2.84±0.02 2.67±0.01 2.67±0.01 ... ...
A† 1.0 1.0 2.0 2.0 ... ...
Fe
frefl 2.0±0.1 2.1±0.2 0.74±0.02 0.74±0.02 ... ...
z† 0 0 0 0 ... ...
norm(10−26) 1.24±0.06 1.24±0.08 2.1±0.2 2.1±0.2 ... ...
REFLIONX ξ ... ... ... ... 430±20 430±20
AFe ... ... ... ... 2.0±0.4 2.0±0.6
frefl ... ... ... ... 0.72±0.06 0.72±0.08
z† ... ... ... ... 0 0
norm ... ... ... ... 0.57±0.04 0.60±0.05
Funabs,3.0−30.0keV 3.4±0.2 3.4±0.2 3.4±0.3 3.4±0.3 3.4±0.3 3.4±0.4
Lunabs,3.0−30.0keV 2.5±0.1 2.5±0.2 2.5±0.2 2.5±0.2 2.5±0.2 2.5±0.3
Funabs,0.5−30.0keV 5.0±0.2 5.0±0.3 5.0±0.5 5.0±0.5 5.2±0.5 5.2±0.6
Lunabs,0.5−30.0keV 3.7±0.2 3.7±0.2 3.7±0.3 3.7±0.3 3.8±0.3 3.8±0.5
χ2(dof) 1.16(665) 1.16(665) 1.19(665) 1.19(665) 1.21(664) 1.21(664)
ν
†=fixed
Note.—Errorsarequotedat1σconfidencelevel.TheabsorptioncolumndensitywasfixedtotheDickey&Lockman(1990)valueandgiveninunitsofcm−2.
Limbbrighteningwasassumed.TheREFLIONXmodelusedhasbeenmodifiedforablackbodyilluminatingtheaccretiondisk.TheinclinationforREFLIONX
waspeggedwithinthelimitsfoundwiththeBBREFLduetothemodelbeingangleaveragedandnotabletoconstraintheinclination.Theionizationparameter
wasalsopeggedwithintheBBREFLvaluessinceitwasunconstrainedonitsown.TheblackbodytemperatureinREFLIONXwastiedtothetemperatureofthe
blackbodyusedtomodelthecontinuumemission.InnerdiskradiusinunitsofkmassumesaNSmassof1.4M⊙.Fluxisgiveninunitsof10−9ergscm−2
s−1.Luminosityiscalculatedatamaximumof7.8kpcandgiveninunitsof1037ergss−1.
an upper limit on the strength of the field using the upper disks.Differentdiskreflectionspectraareconsideredforeach
limit of Rin = 9.8 Rg. Assuming a mass of 1.4 M⊙, tak- sourceinordertoexaminehowinferredradiidependonmod-
ingthedistancetobe7.8kpc,andusingtheunabsorbedflux elingassumptions.Forplausiblecombinationsofneutronstar
from0.5-30.0keVof5.2×10−9 ergcm−2 s−1 asthebolo- masses and dimensionless angularmomenta, our results im-
metric flux, we can determine the magnetic dipole moment, plythatdisksextendtoanISCO,thatneutronstarsaresmaller
µ, fromEquation(1)takenfromCackettetal. (2009). Ifwe than their ISCO, and the results begin to place meaningful
make the same assumptions about geometry (i.e. k = 1, constraintsonneutronstarradii. Inthissection,weconsider
A
f = 1) and use an accretion efficiency of η = 0.2, then the results within the contextof the neutronstar equationof
ang
µ ≃ 2.2×1026Gcm3. Thiscorrespondstoamagneticfield state, implications for the inner accretion flow onto neutron
strengthofB ≃4.3×108GatthemagneticpolesforaNSof stars,andevaluatepossiblesystematicerrorsandavenuesfor
10km. Moreover,ifweassumeadifferentconversionfactor improvementinfuturestudies.
k = 0.5(Longetal.2005)thenthemagneticfieldstrength
atAthepoleswouldbeB ≃1.5×109G.Notethatthemagnetic 4.1. NeutronStarRadiusConstraints
field strengthat the pole is twice as strongas at the equator. 4U 1636-53is foundto have an innerdisk radiusof 1.00-
However,thetype1X-rayburstthatoccurredduringtheob- 1.03ISCOfora∗ = 0.0and1.02-1.14ISCOfora∗ = 0.3as
servationmeanmaterialisstillreachingthesurfaceoftheNS constrained from RELXILL modeling of the reflection spec-
andnopulsationshavebeenseen. trum.Weappliedanotherreflectionmodel,REFLIONX,totest
the robustness of our measurement. The resulting fit gave a
4. DISCUSSION nearly identical inner disk measurement. For a∗ = 0.3 and
Using NuSTAR, we have taken a hard look at three well- 1.4M⊙,1.08±0.06ISCOtranslatesto10.8±0.6km.Forthe
known neutron star X-ray binaries. Our observations cap- lowervalueofa∗ =0.0and1.4M⊙,1.03±0.03ISCOtrans-
tured 4U 1636−53 in the hard state, while GX 17+2 and latesto12.4±0.4km.Thissmallinnerdiskradiusisinagree-
4U1705−44werecaughtinsoftstates. OwingtoNuSTAR’s ment with previous measurements for this source that were
broad bandpass and its ability to measure robust spectra at consistent with the ISCO (Pandeletal. 2008; Cackettetal.
high flux levels, we are able to constrain different proper- 2010; Sannaetal. 2013). The high inclination of 4U 1636-
tiesofthesesourcesthroughmodelingofreflectionfromtheir 53 is consistent with previousreflection studies (Franketal.
AHardLookat4U1636-53,GX17+2,and4U1705-44 7
5
1.1 105 (a) 4U 1636−53
(a) 4U 1636−53
V−1 104
e
1.1 nts k
u
Co 1000
Ratio 1.05 100
1.4
o 1.2
1 Rati
1
5 10 20
95 Energy (keV)
0.3 4 5 E6nergy (keV7) 8 9 10 106
(b) GX 17+2
1
1. 105
(b) GX 17+2 eV−1 104
nts k
ou 1000
C
5
1.0 100
o 10
ati
R
o 1.2
1 ati
R
1
5 10 20
Energy (keV)
95 106
0.3 4 5 6 7 8 9 10
Energy (keV) (c) 4U 1705−44
105
5
1
1. V−1 104
(c) 4U 1705−44 unts ke 1000
o
C
1
1. 100
10
Ratio 1.05 1.4
o 1.2
ati
R
1
1 0.8
5 10 20
Energy (keV)
Figure2. (a)4U1636-53spectrumfitfrom3.0-50.0keVwithRELXILL(red
95 dashline)forthefitinTable2.Thepanelbelowshowstheratioofthedatato
0.3 4 5 E6nergy (keV7) 8 9 10 themodel. (b)GX17+2spectrumfitfrom3.0-30.0keVwithDISKBB(blue
dot-dashline)andBBREFL(reddashline)forthefitinTable4. Thepanel
Figure1. Ratioofthedatatothecontinuum modelintheFeKbandfor belowshowstheratioofthedatatothemodel. (c)4U1705-44spectrumfit
NuSTARobservationof4U1636-53,GX17+2,and4U1705-44. Theiron from3.0-30.0keVwithDISKBB(bluedot-dashline)andBBREFL(reddash
lineregionfrom5-8keVwasignoredtopreventthefeaturefromskewingthe line)forthefitinTable5.Thepanelbelowshowstheratioofthedatatothe
fit.Thedatawererebinnedforplottingpurposes.(a)Asimplecut-offpower model.Thedatawererebinnedforplottingpurposes.
lawwasfitovertheenergiesof3.0-5.0keVand8.0-50.0keV.Forpanels(b)
and(c), asimplediskblackbodyandsingletemperature blackbodywasfit
overtheenergiesof3.0-5.0keVand8.0-30.0keV. a∗ = 0.0 and 1.4 M⊙, 1.00 − 1.02 ISCO translates to
12.0−12.2 km. For the higher value of a∗ = 0.3 and 1.4
1987; Casaresetal. 2006; Cackettetal. 2010; Sannaetal. M⊙, 1.03−1.30ISCO translatesto 10.3−13 km. The in-
2013). clination was found to be 25◦ − 38◦. Cackettetal. (2010)
Similarly, the inner disk radius in GX 17+2 is tightly foundsimilarlysmallinnerdiskradiiandlowinclinationfor
constrained to 1.00-1.02 ISCO for a∗ = 0.0 and 1.03-1.30 thissystem.
ISCO for a∗ = 0.3 byusing theblackbodyreflectionmodel Thus, in the most sensitive and robust spectra of 4U
BBREFL. We founda tighter constrainton the inner disk ra- 1636−53 and GX 17+2 yet obtained, the innermost extent
diusof1.00-1.01ISCOwhenweappliedaREFLIONX model of the accretion disk is found to be close to the ISCO, with
thatwasmodifiedforablackbodyilluminatingthedisk. For consequences for the neutron star radii. There may still be
8 Ludlametal.
ceedtoexaminetheimplicationsoftheresultsfortheEOSof
ultradensematterinamoregeneralizedway. Ratherthanas-
sumingspecificvaluesa = cJ/GM2, weneedtodetermine
thelikelyrangeofagiventhemeasuredspinfrequenciesfor
thesesources.
40
4U 1636-53 and GX 17+2 have known rotation frequen-
(a) 4U 1636-53
35 cies: ν1636−53 = 518.0 Hz (Gallowayetal. 2008) and
ν17+2 = 293.2Hz (Wijnandsetal. 1997), respectively. The
30 total angular momentum, J, can be obtained from the spin
frequency assuming a reasonable range of mass and radius
25 for a neutron star and a solid sphere (J = 2MR2ω where
5
20 ω = 2πνspin). For 4U 1636−53,ν = 518Hz then implies
0.09±0.05 < a1636−53 < 0.44±0.23, andν = 293.2Hz
15 implies0.05±0.04 < a17+2 < 0.25±0.13. These values
assume a mass range of 1.3 ≤ MNS/M⊙ ≤ 2.1, consistent
10
with the range of masses that have been measured directly
(Jacobyetal. 2005; Demorestetal. 2010; Freireetal. 2011;
5
Kiziltanetal.2013). Thelowerlimitoftheradiusrangewas
80 determined by where causality approximately intersects the
(b) GX 17+2 largestmeasureNS. The upperlimitofthe radiusrangewas
70 limitedbybreakup(a∗ =0.7).
The radius of the ISCO around a compact object in units
60
of gravitational radii is depends on its spin Bardeenetal.
50 (1972); the ranges above therefore enable a translation
to gravitational radii and then into kilometers. Figure
2∆χ40 4 plots these ranges in the mass versus radius plane
used to characterize the EOS. Several equations of states
30
from Akmaletal. (1998), Lattimer&Prakash (2001), and
20 Horowitz&Piekarewicz (2001) are also plotted, as well as
knownNS masses frompulsartiming methodsandbinaries.
10 The regions allowed by our models must be considered up-
perlimitsontheradiusofneutronstars,sincetheneutronstar
20
canbesmallerthanitsISCO.Diskreflectionisnotyetableto
18 (c) 4U 1705-44 ruleoutplausibleEOS;however,deeperX-rayspectraand/or
mass measurements in these systems can greatly reduce the
16
allowedregionsinthismass–radiusplane.
14
4.2. ImplicationsofDiskTruncationin4U1705−44
12
In 4U 1636−53 and GX 17+2, the inner disk appears to
10
extendto the ISCO, and we cannotplace any constraintson
8 the magnetic field in these sources. In contrast, 4U 1705-
6 44 hasan innerdisk radiusof 1.46-1.64ISCO fora∗ = 0.0
and 1.69−1.93 ISCO for a∗ = 0.3. For a spin of 0.0 and
4
stellarmassof1.4M , 1.46-1.64ISCOtranslatesinto17.5-
⊙
2 19.7km. Aspinof0.3andstellarmassof1.4M ,1.69-1.93
⊙
ISCOtranslatesinto16.9-19.3km.
1.0 1.2 1.4 1.6 1.8 2.0 Thesimilaritybetweentheresultsofourfitsandpriorwork
R (ISCO)
in suggests that such modeling is converging on a relatively
consistent set of physical constraints. A truncated disk has
beenindicatedin4U1705−44inseveralpriorinvestigations
(diSalvoetal. 2009, 2015; Reisetal. 2009b; Egronetal.
Figure3. The change in goodness-of-fit versus inner disk radius for the 2013; Cackettetal. 2010, 2012). Our results closely match
NuSTARobservationsof4U1636-53,GX17+2,and4U1705-44takenover thoseofReisetal.(2009b);thatworkreportedthediskof4U
50evenlyspacedstepsgeneratedwithXSPEC“steppar”. Theinnerdiskra- 1705−44wastruncatedabovethestellarsurfacewithagapof
diuswasheldconstantateachstepwhiletheotherparameterswerefreeto
∼3.5km.Ourmodelsfindthattheinclinationoftheinnerdisk
adjust.Thebluedashedlineshowsthe68%confidencelevel.(a)4U1636-53
fitcorrespondingtothefirstcolumninTable2.(b)GX17+2fitcorrespond- isbetween24.0◦−26.1◦;thisisagainlargelyconsistentwith
ingtothefifthcolumninTable4. (c)4U1705-44fitcorrespondingtothe previousreflectionstudies(20◦−50◦,Pirainoetal.2007;≤
firstcolumninTable5. 35◦, Reisetal. 2009b; D’A`ıetal. 2010; Cackettetal. 2010,
aboundarylayerpresentonthesurfaceoftheNS,butinthis 2012;Egronetal.2013).DiSalvoetal.(2015)findaslightly
caseitwouldhavetobequitesmall. Giventhat1ISCOcorre- largerinclinationthanwedoof43◦±5◦.
spondsto12km,fora∗ =0and1.4M⊙,andusingthefidu- Disksaroundneutronstarscanbetruncatedbyaboundary
cialneutronstarradiusof10km,theboundarylayerwouldbe layer, or by magnetic pressure. It is also possible that the
about∼2km.Theinnerdiskradiiarenotstronglydependent innerdiskmayevaporateatlowaccretionrates,qualitatively
uponthereflectionmodelsthatareutilized.Wethereforepro- similartotheexpectedtruncationofdisksaroundblackholes
AHardLookat4U1636-53,GX17+2,and4U1705-44 9
3.0
NL4
2.5 Causality PA1 4U 1636-53
ENG
AP4
2.0 J1614-2230
SQM3
Z271 J1903+0327
SQM1
1.5 J1909-3744
GS1
1.0
0.5
)
⊙
M
( 3.0
ss NL4
a
M 2.5 PA1
Causality ENG GX 17+2
AP4
2.0 J1614-2230
SQM3
Z271 J1903+0327
SQM1
1.5 J1909-3744
GS1
1.0
0.5
nucleons nucleons+exotic strange quark matter
5 10 15 20
Radius (km)
Figure4. Constraints onthecold, ultradensematterequationofstatefromFeKα reflectionmodelingtodeterminetheinnerdiskradius, assumingthatthe
stellarsurfaceistruncating thedisk. Thegrayregionisexcluded bycausality (thespeedofsoundmustbelessthanthespeedoflight). Thecurvelabeled
NL4isfromAkmaletal.(1998)andZ271isfromHorowitz&Piekarewicz(2001). Allothermass-radiuscurvesarelabeledasinLattimer&Prakash(2001).
Theshadedregionsfor4U1636-53andGX17+2correspondtotheallowedvaluesformassandradiusgivenaspinfrequencyof518.0Hz(Gallowayetal.
2008)and293.2Hz(Wijnandsetal.1997),respectively. Forreasonablevaluesformassandradius,aspinfrequencyof518.0Hzrelatestoaspinparameter
of0.09±0.05 < a∗ < 0.44±0.23and293.2Hzgives0.05±0.04 < a∗ < 0.25±0.13. Thehatchedarearepresentstheerrorsonthespinparameter.
ThedashedlinesineachpanelrepresentthesolidareaconstraintsfromtheNSintheotherpanel. Theyellowhorizontallinesarethemeasuredmassesforthe
NSsPSRJ1614-2230(M = 1.97±0.04M⊙;Demorestetal.2010),PSR1903+0327(M = 1.667±0.021M⊙;Freireetal.2011)andPSRJ1909-3744
(M =1.438±0.024M⊙;Jacobyetal.2005).TheorangeregionrepresentsmassrangefoundforaNSinadoubleNSsystem(M =1.33−1.55M⊙;see
Kiziltanetal.2013forareview).
atlowmassaccretionrates. Evidenceofthismaybeseenin duringtheobservation,suggestingthatsomematerialwasstill
HETEJ1900.1-2455(Papittoetal.2013). reachingthesurface,like4U1705−44.
In Section 3.3, we estimated that the boundary layer
could push out to 1.2 ISCO based on the arguments in
4.3. InneraccretionflowsandM˙
Popham&Sunyaev (2001). This is smaller than the radius
implied by most of our disk reflection models, but the pre- It is expected that the inner radius of an accretion disk is
dictedextentoftheboundarylayercanbeincreasedforspe- partly set by the mass accretion rate through the disk (see,
cific combinations of radiative efficiency and stellar spin. e.g., Gonza´lezMart´ınez-Pa´ısetal. 2014). Indeed, as noted
D’A`ıetal. (2010) estimated that the boundary layer in 4U above,severalmechanismsmighttruncatetheinneraccretion
1705−44 extended out to ∼ 2 RNS in the soft state. Our diskaroundaneutronstar,buteachtruncationmechanismbe-
estimateisconsistentwiththispictureassumingaNSradius comesmore effectiveat lower mass accretion rates. Indeed,
of10km. theradialextentoftheinnerdiskmaybeanimportantfactor
Iftheneutronstarmagnetosphereistruncatingthedisk,we indeterminingthephenomenamanifestedindifferentpartsof
placeanupperlimitonthemagneticfieldstrengthatthepoles the “Z” and “atoll” tracks followed by many persistent neu-
tobeB ≤ 0.4−1.5×109 (seeSection3.3). Arecentstudy tron star X-ray binaries, and even the position of the source
byKingetal.(2016)alsofoundatruncateddisksurrounding alongsuchtracks.
thewell-knownneutronstar X-raybinaryandpulsarAqlX- The sensitive spectra that we have obtained with NuSTAR
1. An inner disk radius of R = 15 ± 3 R = 2.88 ± have permitted particularly strong radius constraints. Prior
in g
0.58ISCOwasmeasuredviadiskreflection. Ifthediskwas NuSTARobservationsofneutronstarshavealsomeasuredin-
nottruncatedbyaboundarylayerandinsteadbytheneutron ner disk radii. It is now pragmatic to consider what a mod-
star magnetic field, an upper limit of B < 5 ± 2 × 108 G estcollectionofrobustspectralconstraintsimpliesaboutthe
results(Kingetal.2016). AqlX-1hada Type1X-rayburst evolutionoftheinneraccretiondiskaroundneutronstarsasa
functionofthemassaccretionrate. Forconsistency,wecare-
10 Ludlametal.
fully replicated the models considered in prior work within For 4U 1636−53, both RELXILL and
XSPEC, and extrapolated the models to a common energy RELCONV×REFLIONX return small and formally consistent
range (0.5–50 keV). We then use the maximum distance to innerdiskradii. DifferentrunswithRELCONV×BBREFL and
eachsourcetoconvertunabsorbedfluxestoluminosities,and RELCONV×REFLIONX all return small radii consistent with
dividedby an Eddingtonlimitof 3.8×1038 ergss−1 as per R ≤1.1ISCOinfitstoGX17+2.Insomecases,theerrors
in
Kuulkersetal.(2003)toobtaintheEddingtonfraction. are formally different at the 1σ level of confidence, but the
Table7liststhekeydatathatresultfromthisexercise.Fig- valuesareconsistentovermoreconservativeranges. Inthese
ure5plotstheinnerdiskradiusversusEddingtonfractionfor twocases,themodelsareincloseagreementandsuggestthat
asetofeightneutronstarsobservedwithNuSTAR.Acrossal- theradiusvaluesthatemergecanbetakenseriously. For4U
mosttwoordersofmagnitudeinEddingtonfraction–imply- 1705−44, a greater range of inner disk radii emerges with
inganequivalentrangeinmassaccretionrateiftheefficiency different models and parameter selections. The most recent
is independent– the innerdisk appearsto remainveryclose model, REFLIONX, returns radii that are nearly consistent
totheISCO.Themostobviousexceptiontotheoveralltrend with the ISCO at 1σ; other models return values as large as
isAqlX-1;however,thissourceisknowntobeapulsarandit R = 1.8ISCO, andthe majorityofmodels– andthe best
in
mayhavea slightlyhighermagneticfield thanothersources overallfits – measurea truncateddisk. Deeper observations
inthesample. of 4U 1705−44may be requiredto obtain a more definitive
Cackettetal. (2010) found similar results when looking pictureofthatsource.
at the inner disk radius dependence on Eddington fraction Theemissivityprofileoftheaccretiondiskencodesthege-
for a sample of 10 neutron stars that were observed with ometryoftheemittingsourceandspacetimeoftheinnermost
XMM-Newton and Suzaku. Chiangetal. (2016) recently environment. Anemissivityindexofq = 3isexpectedfora
looked at Suzaku observations of Ser X-1 over a range of point source emitter in a (nearly) Schwarzschild spacetime,
flux states and found that the inner disk radius changes lit- and different plausible geometries for neutron stars (bound-
tle between 0.2− 0.6 LEdd. Disk reflection studies under- ary layers, hotspots) appear to producethe same emissivity
taken with NuSTAR have the advantages of a broad, contin- profile (D. Wilkins, priv. comm.). Both families of reflec-
uous bandpass that enables better characterizationof the di- tion modelsprefer a flatter emissivity in fits to the spectrum
rectcontinuumandstrongerconstraintsonthetotalreflection of 4U 1636−53. This may be the result of a moreextended
spectrum, and spectroscopy that is not distorted by photon coronainthehardstate.FitstothespectrumofGX17+2with
pile-up(whichcanfalselyskewtheshapeoftheline:Ngetal. BBREFL generally prefer a much steeper index, but fits with
2010;Milleretal.2010). the more recent REFLIONX model are only slightly steeper
than q = 3. This is broadly consistent with the direct con-
4.4. Potentialsystematicerrorsandmodelingissues
tinuum, which suggests that a boundary layer or hot spot is
Wehaveobtainedspectrawithveryhighstatisticalquality, likely irradiating the disk. The models for 4U 1705−44are
and thereforespectral fitting results with small statistical er- allbroadlyconsistentwithq =3.
rors. Systematicerrorsarelikelytobecomparableorlarger, Theabundanceofironaffectsthelocalstrengthofreflection
and should be considered. In practice, the most important relativetothedirectcontinuum,nottheshapeoftheline.Itis
sourcesofsystematicerrorsaredifficulttoquantify,butthey theshapeofthediskreflectionspectrum–andparticularlythe
areimportanttomention. shapeoftheFeKemissionline–thatisusedtoinfertheinner
All measures of black hole spin that utilize the accretion radiusoftheaccretiondisk,andtotherebysetanupperlimit
disk, and all limits on neutron star radii that utilize the disk ontheradiusoftheneutronstar. Theabundanceofirondoes
(includingdisk reflectionandQPOs), assume thatgasin the notdirectlyaffectourradiusmeasurements.However,itisin-
disk orbits as test particles would orbit. If real fluid disks terestingtoconsiderthisparameterandwhetherornotitisac-
pushslightlyinwardoftheISCOdefinedfortestparticles,it curatelydetermined.Bothfamiliesofreflectionmodelsprefer
amountstoasystematicerroronsuchmeasurements.Thereis an iron overabundanceof 4.5–5.0relative to solar values, in
noastrophysicaltestofthisassumption,butnumericalsimu- fitsto4U1636−53. Itisunlikelythattheabundanceofiron
lationscanpotentiallyprovidesomeinsights. Explorationsof isthathighinthelowmasscompanionstarin4U1636−53,
disksaroundblackholessuggestthataccretiondisksdogen- butitispossiblethatthismeasurementcorrectlydescribesthe
erallyobeythetestparticleISCO(Reynolds&Fabian2008). atmosphereoftheaccretiondisk. There,theionizationstruc-
Similarsimulationshavenotbeenperformedinthepresence turemayskewtherelativeabundancesofdifferentelementsto
of a neutronstar, and the influenceof a boundarylayer may valuesthatdonotreflecttheoverallabundanceswithintheac-
also inducesmall changes. New simulationscan help to ad- cretionflow.FitstothespectraofGX17+2and4U1705−44
dresssuchsystematics. areconsistentwithironabundancesof1.0–2.0timessolarval-
Apart from the reaction of the accretion disk to the po- ues,butthesefitsalsoreturnlowerionizationparameters,po-
tential, it is possible that our assumption of a Kerr metric tentiallyconsistentwithionizationaffectingabundancemea-
is itself a source of systematic error. The neutron star that surements. It is also possible that the enhanced abundances
wehavestudiedmayhaveasmallquadrupolemoments,and maythetheresultofeffectsinespeciallydensegas. Inthese
thiscouldchangetheeffectivepotentialinwhichthediskor- cases, the abundance would increase to replicate the contin-
bits. Onetreatmentsuggeststhatsystematicerrorsrelatedto uumfor a lower density disk allowed by the atomic data set
quadrupole-induced deviations from a pure Kerr metric are withinthecurrentreflectionmodels(Garc´ıaetal.2016).
≤10% even for a dimensionless spin parameter of a = 0.3 Finally,wesimplynotethattheionizationparametersmea-
(Sibgatullin&Sunyaev1998). sured in different fits to each source spectrum are com-
Theblurredreflectionmodelsthemselvescanpotentiallybe parable to the values seen for other NS reflection studies,
sourcesofsystematicerror. Someancillaryparametersofthe 2.3 < log(ξ) < 4.0 (Cackettetal. 2010; Milleretal. 2013;
reflection models require a note of caution. We now turn to Degenaaretal.2015).
theseissues.