Supporting evidence for the signature of the innermost stable circular orbit in Rossi X-ray data from 4U1636-536

1.2 MB·English

by  Didier Barret

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2)
Supporting evidence for the signature of the innermost
stable circular orbit in Rossi X-ray data from 4U 1636–536
Didier Barret1, Jean-Francois Olive1, & M. Coleman Miller2 ⋆
1Centre d’Etude Spatiale des Rayonnements, CNRS/UPS, 9Avenue du Colonel Roche, 31028 Toulouse Cedex 04, France
7 2Department of Astronomy, Universityof Maryland, College Park, MD 20742-2421, United States
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Accepted XXX.Received2006XXX;inoriginalformXXX

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ABSTRACT

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1 Analysis of archival Rossi X-ray Timing Explorer (RXTE) data on neutron star
low-mass X-ray binaries has shown that for several sources the quality factor Q of
1
the lowerkilohertzQuasi-PeriodicOscillations(QPO)dropssharplybeyondacertain
v
frequency. This is one possible signature of the approach to the general relativistic
2
innermoststable circularorbit (ISCO), but the implications of such aninterpretation
1
3 forstronggravityanddensematterareimportantenoughthatitisessentialtoexplore
1 alternate explanations. In this spirit, M´endez has recently proposed that Q depends
0 fundamentally onmass accretionrate(as measuredby spectralhardness)ratherthan
7 the frequency of the QPO. Specifically, he has suggested that analysis of multiple
0 sources shows trends in Q similar to those previously reported in individual sources,
/ andhesurmisesthattheISCOthereforedoesnotplayaroleintheobservedsharpdrop
h
p in Q in any source. We test this hypothesis for 4U 1636–536 by measuring precisely
- spectral colors simultaneously with the lower QPO frequency and Q after correction
o for the frequency drift, over a data set spanning eight years of RXTE observations.
r WefindthatinthissourcethereisnocorrelationbetweenQandspectralhardness.In
t
s particular,noapparentchangesinhardnessareobservedwhenQreachesitsmaximum
a
before dropping off. We perform a similar analysis on 4U 1608–522; another source
:
v showing a sharp drop in the quality factor of its lower kHz QPO. We find that for
Xi thissource,positiveandnegativecorrelationsareobservedbetweenspectralhardness,
frequency and Q. Consequently, if we are to searchfor a common explanation for the
r
a sharp drop in the quality factor seen in both sources, the spectral hardness is not a
goodcandidatefortheindependentvariablewhereasthefrequencyremains.Therefore,
we conclude that the ISCO explanation is viable for 4U 1636–536,and thus possibly
for others.
Key words: Accretion - Accretion disk, stars: neutron, stars: X-rays
1 INTRODUCTION withtheseparation betweentheupperandlowerkHzQPO
staying close to the spin frequency νspin or half the spin
Kilohertzquasi-periodicbrightnessoscillations(kHzQPOs)
frequency despite variations in both QPOs (van der Klis
have been observed with the Rossi X-ray Timing Explorer
2006).
(RXTE,Bradt, Swank and Rothschild,1993) from some 25
neutron star low-mass X-ray binary systems (NS LMXBs). There is as yet not a complete consensus about the
TheseQPOsarestrong(fractionalrmsamplitudesareoften physicalprocessesthatgeneratetheseQPOs,letalonetheir
>10% in the 2-60 keV band), sharp (quality factors up to high quality factors which pose severe constraints on all
Q ≡ ν/FWHM ∼ 200), high frequency (commonly 700– existing models (Barret et al. 2005). Suggestions include
1000 Hz, with the highest claimed detection at 1330 Hz for beat-frequency mechanisms (Miller, Lamb & Psaltis 1998),
4U 0614+091; van Straaten et al. 2000), and substantially some manifestation of geodesic frequencies (Stella & Vietri
variable (QPO frequencies in a given source can change by 1999), and resonant interactions (Abramowicz et al. 2003,
hundreds of Hertz). They also commonly come in a pair, Abramowicz et al. 2005). There is, however, broad agree-
ment that the upper kHz QPO frequency is close to the
orbital frequency at some special radius (or the vertical
⋆ E-mail:[email protected] epicyclic frequency in some resonance models, but this is
2 Barret, Olive, Miller
within a few Hertz of the orbital frequency for the relevant Recently, M´endez (2006) compiled data from multiple
stellar spin parameters). By itself, this implies that there sourcesandsuggestedthatitisinfactthespectralhardness
must be an upper limit to the QPO frequency. For very (hismeasureforthemassaccretionrate)thatistheprimary
hard high-density equations of state or very low-mass stars factor in determining Q. Here we test this suggestion with
the limiting frequency could in principle be set by the or- RXTE data on 4U 1636–536. In § 2 we discuss our selec-
bital frequency at the stellar surface, but in realistic cases tionofdataandprocessing algorithms. Wealsopresentour
the maximum will instead be determined by the innermost results,andspecificallyshowthatthereisnoapparentcorre-
stable circular orbit (ISCO). If observational signatures of lationbetweenthequalityfactorofthelowerkHzQPOand
the approach to the ISCO were observed, this would con- the spectral hardness. Therefore, for this source and per-
firm the strong-gravity prediction of unstable orbits, which haps others, there is no evidence that the accretion rate is
hasnoparallel in Newtonian gravity.Itwould also allow us the primary determinant of Q. We discuss the implications
directmeasurementofthemassoftheneutronstar(seedis- in § 3.
cussion in Miller, Lamb, & Psaltis 1998), hence the search
for such signatures is of great importance for fundamental
physicsand astrophysics.
2 DATA ANALYSIS AND RESULTS
It was proposed theoretically (Miller, Lamb, & Psaltis
1998, see also Kluzniak, Michelson, & Wagoner 1990) that We used the data presented in Barret, Olive & Miller
astheradiusthatdeterminestheupperkHzQPOfrequency (2005a,b, 2006). The same analysis scheme applies for the
approachestheISCO(andthusasthelowerpeakapproaches dataselection.WeconsideralldatarecordeduptoSeptem-
theISCOfrequencyminusνspin orνspin/2),thiswillleadto ber 2004. All PCA Science Event files were retrieved from
(1)asymptotingofthefrequencytoalimitingvalue,(2)de- the HEASARC archive. A file represents a temporally con-
crease in theamplitudeof theoscillation, and(3) sharp de- tiguouscollectionofdatafromasinglepointing.571filesare
creaseinthequalityfactorQ≡ν/FWHMoftheoscillation. considered here. They have been filtered for X-ray bursts
Zhang et al. (1998) suggested that the first of these signa- and data gaps. Leahy normalized Fourier power density
tures is apparent in the data from 4U 1820–30, but com- spectra were computed between 1 and 2048 Hz, over 8 sec-
plications in the relation between countrate and frequency ond intervals with a 1 Hzresolution.
(M´endezetal.1999)havemadetheinterpretationofthisre- Inparallel,wehaveanalyzedthePCAStandard2data
sultuncertain(seehowever,e.g.Bloseretal.2000).Morere- (a collection of 129-channel spectra accumulated every 16
cently,analysisofarchivalRXTEdatafrommultiplesources seconds), following standard recipes, using REX 0.31. We
has revealed a sharp drop in Q for the lower QPO with in- filteredthedatausingstandardcriteria:Earthelevationan-
creasing frequency,and thisdrop is qualitatively and quan- gle greater than 10 degrees, pointing offset less than 0.02
titativelyconsistentwithwhatisexpectedfortheapproach degrees,timesincethepeakofthelastSAApassagegreater
to theISCO (Barret, Olive& Miller 2005a,b 2006). than 30 minutes, electron contamination less than 0.1. The
If confirmed,this result is of great fundamentalimpor- background of the PCA has been estimated using pcaback-
tance.Itisthusessentialtoexaminealternateexplanations. est 3.0, and the latest bright background model as rec-
Inparticular,asdiscussedinBarret,Olive,&Miller(2006), ommended for sources brighter than 40 counts/s/PCU. To
there are many factors that collectively determine Q for an avoidanypossible discontinuityneartheloss ofitspropane
oscillation. Theoretical arguments (Miller, Lamb, & Psaltis layer(inMay2000),weexcludePCU0inouranalysis.PCU
1998) as well as recent observational results (Gilfanov & units 2 and 3 provide a good overlap between the Stan-
Revnivtsev 2005) suggest that although the high observed dard 2 and Science Event data (they both provide twice as
amplitudesrequirethattheenergyweseeintheQPOislib- muchdataasPCUunit1forinstance).ForeachObsID,for
eratedatthestellarsurface,thefrequencyandsharpnessof the 2 PCU units, considering only the the top layer (layer
theQPO isdetermined in theaccretion disk.Insuch a pic- 1), we have first generated a response matrix using the lat-
ture,in whichthefrequenciesaregenerated in somespecial est version of pcarsp 10.1. For comparison with previous
annulus of the disk, Q depends on the width of the annu- works, we intend to compute the colors from data recorded
lus, theinward radial drift speed, and the numberof cycles in 4 adjacent energy bands: 3.0-4.5 keV, 4.5-6.4 keV, 6.4-
a given oscillation lasts. As discussed in Barret, Olive, & 9.7 keV, 9.7-16 keV. For each ObIDs, we read out from
Miller(2006), approachtotheISCOcanaffectthefirsttwo theresponse matrix therelativechannelvaluescorrespond-
of these factors in a way that agrees with thedata. ing to these energy boundaries, and we use the FTOOLS
However, it is also possible that other, non-spacetime- chantrans to convert the latter into their absolute channel
related,effectsplayarole,e.g.,plasmaprocessesinthedisk, values, as needed for saextract, called by REX. Since the
corona, or stellar surface, or interaction with the stellar energy boundaries are not exactly equal to theabove exact
magnetic field. Without a detailed model of this type one values defined above and change in time with the detector
cannot rule definitively for or against such ideas. Generi- gains, a correction must applied. Following Barret & Olive
cally, though, one expects that such factors in Q will de- (2002),foreachObsID,wehaveextractedforeachPCUunit
pend fundamentally on the mass accretion rate, whereas aPHAcountspectrum.Byfittingthecountspectrumwith
theISCO-relatedeffectsdependfundamentallyonthespace- apolynomialfunction,onecancomputetheexactnumberof
time.Therefore,astrongcorrelationbetweenQandaproxy countswithintheexactenergybounds.Thisgivesusanav-
for the mass accretion rate would suggest plasma or mag- erage correction factor, which corresponds to the difference
neticfieldinteractions,whereasthelackofsuchacorrelation
combined with the observed dependence of Q on frequency
would argue in favor of an ISCO interpretation. 1 http://heasarc.gsfc.nasa.gov/docs/xte/recipes/rex.html
The innermost stable circular orbit in 4U 1636–536 3
Figure 1.TheLeahynormalizedPowerDensitySpectrumaver- Figure 2.Thequalityfactor-frequencydependency isshownon
agedovertheScienceEventfileisshownatthetop.Inthemiddle the top panel. Each point is obtained by shifting-and-adding all
panel,adynamicalPDSisshown.Theimagehasbeensmoothed 8second PDSover segments of 1024 second duration (see text).
to make the QPO appear more clearly. The bottom two panels Only points of significance greater than 4σ are shown. The rise
representsthesimultaneousevolutionofthesoftandhardcolors ofthequalityfactorwithfrequency, followedbyasharpdrop,is
(HR1andHR2,respectively)ascomputedfromStandard2data. clear,albeitwithlargerscatterthaninBarretetal.(2005a),who
The non complete overlap between the two data sets is because averagedtheQPOoverlongerintegrationtimes(typicalduration
theStandard2data(onlyPCU3dataareconsidered)arefiltered ofanObsID,i.e.4000seconds).Themidandbottompanelsshow
with more stringent criteria than the Science Event data. Note thesoftandhardcolorsmeasuredfromPCU3dataasafunction
thatwhiletheQPOfrequencyvariesfrom730Hzto850Hz,there offrequency. Onlypoints forwhichtheoverlapbetween the Sci-
arenoappearantchanges inboththesoftandhardcolors. enceEventdataandtheStandard2dataislargerthan50%are
shown. There is a slight continuous anticorrelation between soft
colorsandfrequency,butnocorrelationatallbetweenfrequency
between the counts extracted with saextract and the corre- andhardcolor.Afitbyastraightlineyieldsaχ2 of230for256
spondingcountsintheexactenergybands.Finally,another degreesoffreedom.
smaller correction is applied to account for the fact that
when the energy boundaries move, the corresponding effec-
tive areas also change. The light curves in the four energy
lation originates. We wish to study the dependency of the
bands are normalized to the same effective area, which is,
quality factor versus the soft and hard colors. One would
for each ObsID, estimated directly from the response ma-
expect that if the drop is indeed related to a spacetime ef-
trix generated for each PCU. After these two corrections,
fect, it is not primarily dependent on the energy spectrum.
the light curves become all comparable. These corrections
Thisideacan betested withspectral colors, which allow us
areespeciallyimportantbecausetheobservationsspanfrom
to search for subtle spectral variations.
1996 to 2004.
In order to estimate the quality factor of the QPOs,
We have computed two spectral colors: the soft color
onemustfirstcorrectforthefrequencydrifts.Hereweusea
(HR1) defined as the ratio between the 4.5 and 6.4 keV
slidingwindowbasedtechnique.Namely,wegroupasmany
countsand3.0and4.5keVcounts,andthehardcolor(HR2) consecutive 8 second PDS (N 6 64) as needed to detect a
as the ratio between the 9.7 and 16.0 keV counts and 6.4
QPOwithasignificanceaboveathresholdof3σ.Themax-
and9.7 keVcounts.ForagivenScienceEventfile,theend-
imumintegration timeisthen64×8=512seconds.Incase
productofouranalysisatthisfirststagecanbesummarized
ofnodetectionwithinsuchanintervalstartingatT0,anew
in Figure 1 where the averaged PDS over the file is shown,
search starts at T0+32 sec. The QPO frequency in each 8
together with the dynamical power density spectrum and
second PDS is then estimated using a linear interpolation
the soft and hard colors as derived from the Standard 2
between all detected QPO frequencies within a continuous
data.
segment. We identify those files containing a lower kilo-Hz
QPOinthequalityfactor-frequencyplane(seeBarretetal.
2005a for details). We keep those files in which the qual-
2.1 Quality factor and spectral colors against
ity factor recovered after correction for the frequency drift
frequency in 4U 1636–536
is larger than 30, corresponding to a mean QPO frequency
For the sake of clarity, our analysis here is focussed on the largerthan650andsmallerthan950Hz.Wethendividethe
lowerkilo-HzQPO,forwhichwehavearguedthatthedrop data into segments of 1024 seconds, and shift-and-add all 8
of coherence at some critical frequency may be related to second PDS within each segments to a reference frequency
anapproachoftheISCOoftheregionfromwhichtheoscil- whichissettothemeanQPOfrequencyassignedafterinter-
4 Barret, Olive, Miller
Figure3.Thecolor-colordiagramofallthearchivalRXTEdata Figure4.Thevariationofthequalityfactorandhardcolorwith
analyzed for 4U 1636–536. The color is averaged over segments frequency,groupingthedataofFigure2withafrequencybinof
of1024secondduration(redfilledcircles).Burstsandgapshave 20Hz.Thehardcolorisconsistentwithremainingconstantover
been filtered out. In blue, the colors corresponding to the QPO the frequency range sampled by our analysis (a fit yields χ2 of
detections is shown. Only data from PCU 3 are shown, but the 15.2for14d.o.f.).
sameresultsisobtainedwithPCU2data.
polation to the8 second PDS over thecontinuous segment.
Some segments are not complete and we have removed all
those in which the total numberof 8 second PDS shifted is
less than 64 (i.e. 512 seconds). Within each segment, when
thereisanoverlapofatleast50%withtheStandard2data,
wecomputethecorrespondingcolorsHR1andHR2.InFig-
ure 2, we plot the resulting quality factor and spectral col-
ors.Nodramaticchangesineitherthesoftandhardcolorsis
observed when the quality factor of the lower kilo-Hz QPO
shows a clear drop. In particular, over the frequency range
spannedbythelowerkilo-HzQPO,thehardcolorisconsis-
tentwith beingconstant (afitbyastraight lineyieldsaχ2
of295and230for260and256degreesoffreedom(d.o.f.)for
PCU2 and PCU 3 data). Over a narrower frequency range
(780-930 Hz), Di Salvo et al. (2004) obtained very similar
results. There is a negative smooth anticorrelation between
soft color and lower QPO frequency, but nothing notable
around the frequency where the quality factor of the QPO
drops off, around 850 Hz. Figure5.Qualityfactorversushardcolorfor4U1636–536using
In Figure 3, we show the color-color diagram for all grouped data of Figure 2. The 1σ error on the mean hard color
the data considered and overlay the region over which the isrepresentedbythehashedregion.
lower QPO was studied with the present technique. The
lower QPO is detected only over a very delimited region
of the color-color diagram, even though that it samples a withabinof20Hztogetsufficientstatisticsandstillenough
relatively wide range of frequency and quality factor. This points to keep a good description of the overall behavior of
by itself shows that even if subtle correlations are masked the quality fact and hard color with frequency. The results
by the statistics (but see Fig. 4), the drop of the quality are shown in Figure 4. The size of the error bars on hard
factor is not associated with a dramatic spectral change in colorshasbeendecreasedonaveragebyafactorof2.7com-
thissource.Itisinterestingtonotethatwiththeprocedure pared to the data of Figure 2; the mean error on the hard
describedinthispaper,lowerkilo-HzQPOsaredetectedat color is reduced to0.0016. A fitwith aconstant yieldsa χ2
theintersection between thebottom and diagonal branches of 15.2 for 14 d.o.f. for PCU 2 data (15.5 for 13 d.o.f for
of the color-color diagram. The count rate varies from 130 PCU 3 data).
counts/s to 340 counts/s in PCU3. In Figure 5, we plot the quality factor against the log-
Next, we have grouped the data of Figure 2 using arithm of the hard color to enable a comparison with the
weightedaverages(andthe1σerrors)inthefrequencyspace, data presented for 4U 1608–522 in Mendez (2006).
The innermost stable circular orbit in 4U 1636–536 5
2.2 Comparison with 4U 1608–522
Usingthesametechniqueasdescribedabove,wehaverepro-
cessed the data of 4U 1608–522 presented in Barret, Olive
& Miller (2006) focussing on the lower kilo-Hz QPOs. This
enables a direct comparison of two homogenous data sets
obtainedthroughthesameprocessingscheme.Thedataset
usedincludesdatapresentedinMendezetal.(1999),aswell
as observations performed in september 2002, march 2003
and 2004 (proposal numbers 70059, 80406, 90408). As for
4U1636-536,weidentifysegmentscontainingalowerkilo-Hz
QPOinthequalityfactor-frequencyplane.For4U1608–522,
as shown in Barret, Olive & Miller (2006), the lower QPO
so identified spans a frequency range going from ∼ 570 to
∼900Hz.PCU2providesthebestoverlapbetweentheSci-
enceEventandthestandard2dataforthissource.InFig.6,
weshowthequalityfactor,soft andhardcolors against fre-
quency.Duetothelargercountrateof4U1608–522(varying
from150counts/supto480counts/sinPCU2),inorderto
get similar errors on the hard color as the one of 4U 1636–
Figure6.Sameasfig.2butfor4U1608–522.Thequalityfactor-
536, onecanuseshorterintegration timeforestimating the
frequency dependency is shown on the top panel. Each point is
QPO parameters. We have used 8×96 = 768 seconds for
obtained by shifting-and-adding 8 second over segments of 768
4U 1608–522, instead of 1024 seconds for 4U 1636–536. For second duration (chosen such that the mean error on the hard
4U 1608–522, the hard color is clearly not consistent with coloristhesameastheonemeasuredfor4U1636–536inFig.2).
beingconstant:afitbyaconstantyieldsaχ2 of371for103 Only points of significance greater than 4σ are shown. The mid
d.o.f. Negativeand positivecorrelations between hardcolor andbottom panelsshowthesoftandhardcolorsmeasuredfrom
and frequency are observed (the same behavior is seen in PCU2dataasafunctionoffrequency.Onlypointsforwhichthe
other PCA units). overlapbetweentheScienceEventdataandtheStandard2data
islargerthan 50%areshown.Thesametrends areseeninPCU
We then grouped the data shown on Fig. 6 also with a
1and3.
bin of 20 Hz to produce Fig. 7 and 8. The mean error on
hardcolorsfor4U1608–522isreducedto0.0020,i.e.slightly
largerthanthoseoffig.4.Fittingthehardcolorwithacon- sourcebehavior(e.g.accretionrate,Mendez2006),butFig.
stant results in a χ2 of 263 for 16 d.o.f. Restricting the fre- 4 shows that the effect produced is by no means universal,
quency range to above 650 Hz (the minimum frequency in aswe failed to detectit in asimilar way in 4U 1636–536. It
4U 1636–536), yields a χ2 of 147 for 13 d.o.f. Since we are could be that the drop of the quality factor and change in
interestedintheregionwhereQreachesitsmaximumbefore hard color are simply concomitant in 4U 1608–522.
dropping off, one can consider only those points from, say, From this comparison, one can therefore conclude that
100Hzbeforethepeak(around840Hz)uptothelastpoint, if we are to search for a common explanation for the sharp
at ∼930Hzin4U1636–536 and∼900 Hzin 4U1608–522, drop in the quality factor seen in both sources, the hard
respectively.Weobtainaχ2of9for8d.o.fand95for7d.o.f color is not a good candidate for the independent variable
for 4U 1636–536 and 4U 1608–522. Clearly for the latter, whereas the frequency remains.
thehardcolorisnotconsistentwithbeingconstantoverthe
frequency range spanned by the lower QPO, including the
interestingpartwhichencompassestheQdrop-off.Wehave
3 DISCUSSION
verified that the same conclusion is reached, independently
of the integration time used to estimate the QPO parame- The results presented in this paper demonstrate that the
ters. For instance, using 64 seconds for 4U 1608–522 as in drop of the quality factor of the lower kilo-Hz QPO in
Mendezetal.(1999),themeanerroronthehardcolorafter 4U 1636–536 is not accompanied by a significant change
binning is 0.0025 (20% larger than with 768 seconds), and in the energy spectrum of the source, as measured by the
theχ2is164for15d.o.f.LookingatFig.6,itisworthnoting spectral colors. Therefore, in this source, if the quality fac-
that the relationship between the hard color and frequency tor depends fundamentally on mass accretion rate, it must
is revealed to be more complex than, and certainly not as somehow do so in a way that leaves no detectable correla-
smooth as, previously thought, based on measurements ob- tion between quality factor and either spectral measures or
tained with lower statistics (e.g. Mendez et al. (1999)). countrate (see Barret, Olive, & Miller 2005a,b, 2006). This
Comparing Fig 4 and Fig. 6 requires some comments. seems difficult and contrived. In contrast, the strong corre-
First,giventhatthesizeoftheerrorbarsarefullycompara- lation of Q with frequency in the lower peak is as expected
ble,itshows thatifasimilar trendasseen for4U1608–522 if it is largely driven by approach to the ISCO. Together
weretobepresentin4U1636–536, wewouldhaveobserved withtheobservedsteadydropinrmsamplitude,saturation
it. Second, we note that in 4U 1608–522, significant varia- of the QPO frequency with increasing count rate, and the
tions of the hard color (still limited at the ∼5 % level) are quantitative consistency of ISCO models with the Q ver-
observed around the peak of the quality factor-frequency susfrequencycurve(Barret,Olive,&Miller2005a,b,2006),
curve. These variations may reflect some changes in the 4U 1636–536 behaves as expected if the phenomenology is
6 Barret, Olive, Miller
L/LEdd ∼0.02−0.2, and low (Qmax∼10−20) for sources
withL∼LEdd.Asimilar pattern,althoughlessmonotonic,
is shown in his Figure 4, of Q versus hard color. Because
this pattern with luminosity or hard color is similar to the
behavior in individual sources with frequency (Q is low at
lowfrequency,rises toapeak,thendropssharply),M´endez
concludes that it is unlikely that the ISCO plays a role in
any of these systems.
We believe that there is another interpretation. At the
low luminosity end of M´endez’s correlation there is a sin-
gle key source: 4U 0614+091. Figure 1 of Barret, Olive, &
Miller (2006) shows that there is no apparent drop in the
quality factor of the lower QPO up to ∼ 700 Hz. On the
other hand, lower QPOs at frequencies above 700 Hz have
beenreportedwith low Qvalues(vanStraaten etal.2000),
yet without correction for the frequency drift. The status
of 4U 0614+091 is thus different than the status of sources
suchas4U1636–536 forwhich,thankstothenecessary fre-
quency drift correction applied, a clear maximum has been
Figure 7. Same as figure 4 but for 4U 1608–522. The data are observed.Thisisanissuebecauseifthe1330Hzdetectionof
those recorded with PCU 2. Clearly, thanks to the improved theupperQPO is real, one would not expect,for any plau-
statistics, therelationshipbetween thehardcolorandfrequency sible spin frequency of the neutron star, that the frequency
is revealed to be more complex than previously thought for this atwhichQstartsdecreasingtobe∼700Hzorso.Acareful
source(e.g.Mendezetal.1999). re-examination of the 4U 0614+091 data is thusunderway.
At the high luminosity end, the sources all have very
highluminosityindeed,comparabletoEddington.Standard
disk accretion theory (e.g., Shakura & Sunyaev 1973) then
suggests that the disk thickness will be comparable to the
orbital radius, and that as a consequence the inward ra-
dial drift speed (which scales as (h/r)2, where h is thedisk
half-thickness) will belarge as well. Asdiscussed in Barret,
Olive,&Miller(2006), alargeinwardspeedwillnecessarily
decreaseQregardlessofotherfactors. Wenotethatthisef-
fectcanalsobeimportantinreducingthemaximumQfrom
∼200 to ∼100 around a luminosity L∼0.1−0.2LEdd, as
seen by M´endez (2006). It is therefore not surprising that
high luminosity sources have low Q, but it is also not rele-
vant to the evaluation of the behavior of Q with frequency
in much lower luminosity sources.
Asafinalremark,aswehavediscussedpreviously(e.g.
Barret,Olive,&Miller2005a,b,2006),iftheISCOinterpre-
tationiscorrectforourdata,thenonecaninferamassofthe
order of 2 M⊙ for the neutron star. This is consistent with
phase-resolved spectroscopy of 4U 1636–536 at the VLTby
Casares et al.(2006), whoassume plausiblebinaryparame-
Figure 8. Same as figure 5 but for 4U 1608–522. The data are ters (inclination, disk flaring angle, mass of the donor star)
those recorded with PCU 2. The data points are labeled in as-
and infer a mass 1.6−1.9 M⊙ for the neutron star. In ad-
cendingorderwithincreasingfrequency.
dition, a variety of modern models for neutron star matter,
involving hyperons, quarks or normal matter, predict max-
linked to the spacetime and not to the mass accretion rate. imum neutron star masses as large as ∼ 2M⊙ (Jha et al.
Other sources need to be analyzed similarly, but the ISCO 2006, Klahn et al. 2006). Our results thusadd to thegrow-
hypothesisanditsattendantimplications forstronggravity ing evidencefor heavyneutron stars in accreting systems.
and dense matter are still entirely viable.
What, then, could be the explanation for the results
of M´endez (2006), in which he found a correlation between
hardness(oraverageluminosity) andmaximumreported Q
4 CONCLUSIONS
over sources spanning a range of two orders of magnitude
in luminosity? The left panel of his Figure 3 shows that The case for the ISCO is still promising. Analysis of the
the maximum reported quality factor is low for the lowest- typethat we perform in this paperwill beneeded for other
luminositysource(4U0614+091,atL/LEdd ≈6×10−3 and sources,todeterminethestrengthofevidenceinthosecases.
Qmax∼30[Barret,Olive,&Miller2006obtainedQmax ∼50 In addition, focused observations or re-analysis of specific
for this source]), high (Qmax ≈ 100−200) for sources with objects will beuseful, starting with 4U 0614+091.
The innermost stable circular orbit in 4U 1636–536 7
5 ACKNOWLEDGEMENTS Zhang W., Smale A. P., Strohmayer T. E., Swank J. H.,
1998, ApJ, 500, L171
We thank Mariano M´endez for stimulating discussions and
for helpful interaction along the revision of the paper. We
are also grateful to Jean-Luc Atteia for discussions about
statistics related issues, and toDiego Altamirano for initial
discussions. MCM was supported in part by a senior NPP
fellowship at Goddard Space Flight Center. This research
has made use of data obtained from the High Energy As-
trophysics Science Archive Research Center (HEASARC),
provided by NASA’sGoddard Space Flight Center.
We thank an anonymous referee for comments that
helped to improve thecontent of this paper.
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