Modeling the Optical-X-ray Accretion Lag in LMC X-3: Insights Into Black-Hole Accretion Physics

7.5 MB·

by  James F. Steiner

Draftversion January23,2014
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
MODELING THE OPTICAL – X-RAY ACCRETION LAG IN LMC X–3: INSIGHTS INTO BLACK-HOLE
ACCRETION PHYSICS
James F. Steiner1†, Jeffrey E. McClintock1, Jerome A. Orosz2, Michelle M. Buxton3, Charles D. Bailyn3,
Ronald A. Remillard4, and Erin Kara5
Draft version January 23, 2014
ABSTRACT
4 TheX-raypersistenceandcharacteristicallysoftspectrumofthe blackholeX-raybinaryLMCX–3
1 makethis sourceatouchstoneforpenetrating studies ofaccretionphysics. We analyzea rich,10-year
0
collectionofoptical/infrared(OIR)time-seriesdatainconjunctionwithallavailablecontemporaneous
2
X-ray data collected by the ASM and PCA detectors aboard the Rossi X-ray Timing Explorer. A

n cross-correlation analysis reveals an X-ray lag of 2 weeks. Motivated by this result, we develop
≈
a a model that reproduces the complex OIR light curves of LMC X–3. The model is comprised of
J threecomponentsofemission: stellarlight;accretionluminosityfromthe outerdiskinferredfromthe

2 time-lagged X-ray emission; and light from the X-ray-heated star and outer disk. Using the model,
2 we filter a strong noise component out of the ellipsoidal light curves and derive an improved orbital
period for the system. Concerning accretion physics, we find that the local viscous timescale in the

] disk increases with the local mass accretion rate; this in turn implies that the viscosity parameter
E
α decreases with increasing luminosity. Finally, we find that X-ray heating is a strong function of
H
X-rayluminosity below 50%of the Eddingtonlimit, while abovethis limit X-rayheating is heavily
≈
. suppressed. We ascribe this behavior to the strong dependence of the flaring in the disk upon X-
h
ray luminosity, concluding that for luminosities above 50% of Eddington, the star lies fully in the
p ≈
shadow of the disk.
-
o Subject headings: black hole physics — accretion, accretion disks — X-rays: binaries
r
t
s
a 1. INTRODUCTION both chose the source to benchmark the performance of
[ accretion-disk spectral models. Likewise, our group has
LMC X–3 was one of the first extragalactic X-ray

1 sources to be discovered (Leong et al. 1971). It was usedthe extensive thermal-state data available for LMC
X–3tofirmlyestablishthe existenceofaconstantinner-
v subsequently identified with a B main-sequence star
disk radius in black hole binary systems (Steiner et al.
9 (Jones et al. 1974; Johnston et al. 1979). With the dis-
2010).
2 covery that the star is in a 1.7-day binary orbit with a
Recently, Smale & Boyd (2012) have identified two
5 massive dark companion, M & 2.3 M⊙ (Cowley et al.
long-duration“low”phases,eachlastingseveralmonths,
5 1983), LMC X-3 became the first example of an extra-
during which the system was in a hard state. This
. galactic stellar-mass black hole.
1 unusual behavior is likely related to the well-known
LMC X–3 is an unusual system straddling the bound-
0 variability of the source on long timescales. Initially,
ary between transient and wind-fed black hole binary
4 Cowley et al. suggested a superorbital X-ray period
systems (e.g., Soria et al. 2001). Its global characteris-
1
of 198 (or possibly 99) days, but subsequent
: tics, however, show that its mass transfer is governed ∼ ∼
v by Roche-lobe overflow, which places it with the tran- studies of both optical and X-ray variability find a
Xi sients. Although it exhibits extreme X-ray variability, range of superorbital periodicities extending from ∼
100 500 days (Brocksopp et al. 2001; Wen et al. 2006;
spanning roughly three orders of magnitude in luminos-
−
r Kotze & Charles 2012). There is no stable long period
a ity, LMC X–3 is unique among the transients because,
in this source, and the superorbital variability appears
with a few notable exceptions (e.g. Smale & Boyd 2012;
not to be attributable to the precession of a warped
Wilms et al. 2001), it remains in a disk-dominated ther-
disk or related at all to the orbital dynamics of the sys-
mal state (approximately 90% of the time).
tem. Instead,thelong-termvariabilityislikelyproduced
In fact, the dominance of the soft thermal component
by changes in the mass accretion rate (Brocksopp et al.
and the relative simplicity of LMC X–3’s spectrum are
2001; hereafter B01).
thereasonsKubota et al.(2010)andStraub et al.(2011)
Several groups have used the extensive data available
[email protected] for LMC X–3 to search for connections between X-ray
1Harvard-Smithsonian Center for Astrophysics, 60 Garden variabilityandopticalvariability. Notably,B01findthat
Street, Cambridge,MA02138. theX-rayandopticallightcurvesarecorrelated,withan
2DepartmentofAstronomy,SanDiegoStateUniversity,5500
X-raylagof5–10days,whereasCowley et al.(1991)find
CampanileDrive,SanDiego,CA92182-1221.
3Astronomy Department, Yale University, P.O. Box 208101, a somewhat longer 20-day lag.
∼
NewHaven,CT06520-8101. A complementary body of work has focused on ex-
4MIT Kavli Institute for Astrophysics and Space Research, amining very fast (. 10s) variability in the hard state
MIT,70VassarStreet,Cambridge,MA02139.
5Department of Astronomy, Cambridge University, Mading- for other black-hole binary systems (e.g., Gandhi et al.
leyRoad,Cambridge,CB30HA,UK. 2010;Durant et al.2011). In these systems, a surprising
†HubbleFellow. subsecond anticorrelation between X-ray and optical is
2 Steiner et al.
commonly observed that is unlikely to be related to X- 15
ray reprocessing. Rather, it appears that this behavior g/s) 10
is related to the presence of a jet. In the case of LMC 1810 5
X–3,nojetisexpected,particularlyforthethermalstate ot (
d 0
relevant here. M-
-5
In the standard α-disk theory of accretion 16.5
(Shakura & Sunyaev 1973), one naturally expects B
17.0
the signalsbetweenanytwowavebandsto be correlated
with a time delay that corresponds to the viscous time 16.5
for a parcel of gas to travel from the outer emission V 17.0
region(longer wavelength band) in the disk to the inner 17.5
16.5
(shorter wavelength band). Let us consider an annular I 17.0
region with an outer radius R and a gap in radius ∆R
17.5
between the two zones that are of interest in this paper, 16.5
17.0
namely, the regions of optical and X-ray emission. Since J 17.5
18.0
the smaller radius in this case is negligible compared to
R, the viscous time for gas with viscosity ν to traverse 4000 4500 5000 5500 6000
MJD-50000
the distance ∆R R is simply,
∼ Fig. 1.—SMARTSOIRandRXTE X-ray(ASM+PCA)light
t =R2/ν. (1) curves. The light curve amplitude (top) is expressed in units of
visc m˙X, as discussed in § 2.1. The red curve shows our adopted,
In this paper, we link the optical variability of LMC smoothedlightcurve.
X–3 to its X-ray variability by using a wealth of optical derive reliable estimates of flux and construct an X-ray
andX-raydata. Morespecifically, weemploytwoexten- light curve.
sive optical / infrared (OIR) data sets and, in the X-ray TheSMARTSOIRlightcurvesandthecontemporane-
band, we combine data collected daily by RXTE ’s All- ousASM+PCAX-raylightcurveareshowninFigure1.
Sky Monitor (ASM) with high-sensitivity data obtained
inmanyhundredsofpointedobservationsusingRXTE’s 2.1. Unifying PCA and ASM Data Sets
ProportionalCounter Array (PCA).
In order to most fully capture the X-ray variability of
2. DATA LMCX–3,wecombinetheASMandPCAdatasetsinto
a single, unified light curve. Given the complementary
Our principal OIR data set was obtained using the
cadence and sensitivity of the two instruments, this ap-
ANDICAM (Bloom et al. 2004) on the SMARTS 1.3 m
proach results in a light curve that for our purposes is
telescope(Subasavage et al.2010). ANDICAMisadual-
distinctly superior to using only one of the light curves
channelimager, which we used to obtain pairs of images
singly. Themethodweadopthereforcombiningthelight
ofLMCX–3inB,V,I,orJ. Thedatawerereducedand
curvesdoesnotcriticallyaffectourfinalresults;e.g.,the
calibratedwithrespecttoseveralfieldstars,asdescribed
characterofourresultsareunaffectedifweusetheASM
in Orosz et al. (2014).
data alone.
The ANDICAM data were supplemented by B and V
We choose to standardize the two data sets by deter-
data kindly supplied by C. Brocksopp and described in
mining for each ASM and PCA observation the mass
B01. These data are a subset from a larger sample ob-
accretion rate onto the black hole, m˙ . We make this
X
tainedusingthe0.91mDutchTelescopeoftheEuropean
choice because LMC X–3 is nearly always in the ther-
Southern Observatory between 1993 and 1999 during a
malstate,astateinwhichthestableinner-radiusresults
total of 16 different observing runs. Our full data set
in a constant efficiency (depending only on the spin pa-
is derived from 470, 514, 476, and 440 SMARTS obser-
rameter),andm˙ isthereforesimplyproportionaltothe
X
vations in B,V,I, and J, respectively, and 356 and 685
bolometricluminosityemittedbythedisk. Themassac-
additionalESO observationsin B and V (B01). We em-
cretionratedeterminedfromPCAspectralfitscanthebe
ploy just 75% of the B01 data set, because we exclu-
comparedto the 2–12keVPCAflux,providingascaling
∼
sively consider those data collected after 1996 March 7
relationshipthatisthenusedtodeterminem˙ valuesfor
X
(MJD 50149) when RXTE operations entered a ma-
the 2–12 keV flux measurements from ASM data, which
≥
ture phase.
cannotbefitforbolometricdiskluminosity. Fortunately,
Our RXTE X-ray data set is derived from both ASM
whenLMCX–3doesoccasionallyenterthehardstateits
and PCA observations. The calibrated ASM data using
luminosity is so low that m˙ 0 is a quite reasonable
X
the full bandwidth (2–12 keV) and 1-day time bins were ≈
approximationfor the purposes of this study.
retrievedfromthewebsitesupportedbytheASM/RXTE
We first determine m˙ for the PCA data by fit-
X
team7. The PCA data, derived from 1598 observations ting the 1600 PCU-2 spectra to a standard disk
made using the PCU-2 detector, were all reduced and model th∼at accommodates the presence of a weak
analyzed following closely our standard methods (e.g., Compton component. The complete spectral model
Steiner et al. 2010), but in this case the data were par- in XSPEC notation (Arnaud 1996) takes the form:
titioned in somewhat finer time bins, using continuous tbabs (simpl kerrbb2). The disk component ker-
exposure intervals with a mean of 2 ks (and range 300s rbb2(×McClinto⊗ck et al.2006;Li et al.2005)fitsform˙
X
to 5000s). Our goalhere is relativelymodest, namely,to using fixed – and here, approximate – values for black-
hole mass, inclination and distance. simpl models the
7 http://xte.mit.edu/ASM lc.html Comptonpower law (Steiner et al. 2009) and tbabs the
Accretion Lags in LMC X–3 3
With the unified X-ray light curve now in hand (i.e.,
12 m˙ (t)), we use the unsmoothed data and examine the
X
relationship between X-ray and OIR variability. Specif-
n) 10 ically, we assess the correlated variability between m˙ X
o
ati and the OIR bands as a function of time lag using the
aliz 8 discrete correlationfunction (DCF) of Edelson & Krolik
m (1988), with all observation times referred to the Solar
nor 6 Systembarycenter. ADCFisderivedfromapairoftime
y
ar series,Ai andBj,byfirstestablishingasetoflagsτ and
bitr 4 cross-fluxesΥ,whicharecalculatedacrossallpairedtime
ar differences amongst the data:
dot ( 2
M- ∆tij =ti tj, (2)
−
0 (A A¯)(B B¯)
i j
Υ = − − , (3)
0 10 20 30 40 50 60 ij σ σ
A B
F (mCrab)
X
where A¯andB¯ arethe means of seriesA and B, respec-
Fig.2.— Spectral fitting results for the ∼1600 PCA spectra tively. Similarly, σ and σ are the standard deviations
A B
showingm˙X asafunctionof2–12keVX-rayflux. Thelowessfit of A and B. The DCF for time lag τ is then
weadoptisoverlaidinred. Resultsfor6%ofthefullPCAsample
areomitted fromthe figureeither because thespectral fitusedto
1
derive m˙X was statistically poor or it provided no constraint on DCF(τ)= Υij, (4)
m˙X. N(τ)
∆Xtij∈τ
low-energy absorption (Wilms et al. 2000). Each PCA where N(τ) is the number of ∆t elements in the bin τ.
ij
spectrum thus delivers a value of the accretion rate8 At We convert the OIR data into fluxes and then stan-
the same time, we compute the observed PCA flux F dardizealldatasetstounityvarianceandzeromeanand
X
from2–12keVrelativetothestandardfluxoftheCrabin then compute a DCF between each OIR light curve and
the same band (Toor & Seward 1974). As illustrated in the (unsmoothed) m˙ time series ( 2.1), binning the
X
§
Figure 2, the relationshipbetween m˙ andF obtained time lagdatainto 1dayintervals. Giventhis sampling
X X
in this way is well defined and tightly constrained9. andgivendatatha∼tspanadecade,correlatedvariability
We likewise converted the 2–12 keV ASM count rates isdiscernibleovertimescalesrangingfromdaysto years.
intofluxesrelativetotheCrab(1Crab=75ASMcs−1) AsshowninFigure3,theX-raytimelagforthestrongest
and then interpolated the relationship derived for the featuresinthecrosscorrelationofm˙ withtheOIRflux
X
PCA in order to establish the relation between flux and ineachindividualbandis 2weeks. Thisvalueisinter-
≈
m˙ fortheASM.(ToaccommodatenegativeASMcount mediate between those found by B01 and Cowley et al.
X
rates,weusedanextrapolationofthelow-fluxPCAdata (1991). A possible explanation for the differences is the
rather than adopting a floor value for m˙ .) lag’s luminosity dependence discussed in 6.
X
§
Having established a common scale for the two detec- The lag of the X-ray flux is naturally explained as a
tors,wethenusedthelowessnonparametricsmoothing consequence of the viscous time delay for a density per-
algorithm(Cleveland 1979) to derive a representationof turbation to propagate from the outer disk, the location
thetrue,underlyinglightcurve(whichisatallpointslo- of peak OIR emission, to the vicinity of the black hole
cally determined). Because the errors are grossly differ- where the X-rays are produced.
ent for the PCA and ASM data, we achieve appropriate Why are the cross-correlationpeaks so broad? An ex-
weighting via Monte-Carlo randomization: A lowess amination of the X-ray autocorrelation (Figure 4) sug-
curve fit is derived for each of 1000 random realizations gests an answer. The autocorrelation timescale over
and the median curve is taken as the most representa- whichthe X-raysourcebrightensordims, severalweeks,
tivefit. Here,weusedathird-orderlowesscurvefitand is similar to the typical lag time of 2 weeks. This is
≈
adoptedawindowlengthofapproximatelyonemonth,a notsurprisinggiventhat the timescale forthe inner disk
choicemotivatedbytheX-rayautocorrelationtime( 3). tochangefromfainttobright–the autocorrelationtime
§
The precise settings used in fitting the data only affect – should be commensurate with the time required for a
our results cosmetically. For example, with a window parcel of gas to travel from the outer disk to the center,
length of 3 days, we obtained the same - though noisier i.e., the viscous time.
- results. On longer timescales, first at 100 days and 160
∼
days, other features appear in the autocorrelation func-
3. CORRELATEDVARIABILITY tion (Figure 4). As evidenced by alternating posi-
tive/negative harmonic peaks, the anti-correlation fea-
8 The other fit parameter of the disk component, spin, is al- ture at 160 days, e.g., is likely to be at least partially
lowedtovary,butisdistributedwithmodestscatteraboutafixed related to radiative or mechanical feedback in the sys-
value. A separate paper is forthcoming on the spin of LMC X–3 tem, whereby periods of strong accretion inhibit the ac-
(Steineretal.2014).
cretion flow onto the outer disk. This is also evident in
9 Roughly 0.7% of the data, 11 data points, fall > 5σ below
the strong anti-correlation signature between OIR and
the curve. These exceptional spectra show an unusually strong
Compton component, more than an order of magnitude brighter X-raybands at 120dlag(where the opticalbehavior
thanistypical forthedisk-dominatedstatesofLMCX–3. now follows afte∼r t−he X-ray).
4 Steiner et al.
OIR-dominant
0.6 X-ray heating
0.4 B
0.2
0.0 X-ray-dominant
-0.2
-0.4
0.6
0.4 V Fig. 5.—Aschematicrepresentationofourmodelforthebinary
n 0.2 system. The companion star on the left is shown heated by the
o 0.0 central X-raysource at high latitudes and shielded by the disk in
ati -0.2 theequatorialregion. Theouterandinnerregionsofthedisk(not
el -0.4 toscale)producerespectivelyOIRandX-rayemission.
r
or 0.6
C 0.4 I
- 0.2 Ourpurposeis to modelthe complexOIRlightcurves
s
s 0.0 of LMC X–3 as a blend of three components of emission
o
Cr --00..42 attributable to (1) the companion star; (2) a multitem-
0000....0246 J pacdoneennrdascitesdutefarorerrstwayhhnjeeecrthrmerianoXltLr-doMrinasCykes;mXaai–nrse3ds,ioru(en3np)lrhioketchereeems,sraaeednsgy.itohon(etWshreeienrisnstoehneueodrcdeenivsso.ikt)-
-0.2 Ourmodel for the system is shownschematically inFig-
-0.4
ure 5. We approximate the net emission F(t) in a given
-200 -100 0 100 200 OIR band i as a sum of the three components:
Time Lag (d)
F (t)=X (t)+S (t)+H (t), (5)
i i i i
Fig.3.—Cross-correlationsbetweenX-rayandfourOIRbands.
whereX (t)istheOIRdiskemissionwhichrelatestothe
The vertical dotted line at 15 days is meant to guide the eye in i
assessingthe≈2weekX-raylag. Random realizationshave been later X-ray emission (or m˙ X) at time t+tvisc; Si is the
used to assess the error for each cross-correlation, which is the direct flux from the star plus any steady component of
sourceofscruffonthecurves. disk light; and H is the reprocessed emission from the
i
outer disk and companion star.
0.8
4.1. Stellar Component
The simplest term, S , is a constant plus sinusoidal
i
n 0.6 terms that approximate the asymmetric ellipsoidal vari-
o
ti ability of the tidally distorted star:
a
l
e
orr Si(t)≡Ci+ci[sin(4πφ−π/2)+ǫe(φ)], (6)
oc 0.4 where phase φ (t t0)/P and
t ≡ −
u
A 1, sin(2πφ+π/4)>0,
y) e(φ)≡max(0,sin(4πφ+π/2))× 1, sin(2πφ+π/4) 0
a 0.2 (cid:26)− ≤
r (7)
-
X allowforthedifferenceintheamplitudesofthetwomin-
(
ot ima. Bothǫandci,thenormalizationsrespectivelyofthe
d asymmetricandsinusoidalterms,arefreeparameters,as
M- 0.0 are t , C , and P.
0 i
4.2. Disk Component
-0.2 The connection of the term X (t), the variable OIR
i
1 10 100 1000 componentofdiskemission,tom˙ X isobvious. However,
m˙ was derived for the inner X-ray-emitting portion of
Time Lag (d) X
the disk, which is located far from the optical-emitting
Fig.4.— X-ray autocorrelation. The typical ≈ 20-day “pulse- region, and the mass transfer rates may be different in
width” response of the X-ray light-curve shows that significant thetworegions. Therefore,inordertocompute theOIR
changes inthe accretion flow requireseveral weeks tomaterialize.
disk emission from m˙ , our model must include a de-
Thistimescaleisresponsibleforthebreadthofthelagfeaturesin X
Fig.3. Errorsareshownbutaregenerallycomparabletoorsmaller scription of how matter flows from the outer disk to the
thanthelinewidth. inner region. This is established via a transfer function
which maps between X-ray and OIR regimes. We envi-
In the next section, we develop a model for the sion the disk described with such a model as a series of
accretion-driven, 2 week time lag in the context of concentric rings. The accretion flow for each individual
≈
a simplistic physical picture of the disk-star system in ring is approximately steady-state, but the steady-state
LMC X–3. solution varies from ring to ring. The order of the rings
are fixed, but they are allowed to stretch or compress in
4. THEACCRETIONMODEL width during inflow.
Accretion Lags in LMC X–3 5
Meanwhile, the observed,viscosity-inducedtime delay 5. RESULTS
oftheX-rayfluxrelativetotheopticalwillhavesomede-
The model has been implemented in python and
pendence on the mass supply of the individual rings. In
applied at once to the composite OIR data set.
ourgenericprescription,weincorporatethis dependence
All fits are computed using the Markov Chain
as a free power-law scaling on the time delay, which de-
Monte-Carlo (MCMC) routine emcee-hammer
pends on m˙ , a choice that is explained further in Sec-
X (Foreman-Mackey et al. 2013) in a Bayesian for-
tion6. We likewiseassume thatthe opticalemissionhas
malism. Because our model is an oversimplification of
a power-law dependence on m˙ :
X complex real processes, we expect the quality of the fits
to be limited by systematic effects. We therefore adopt
X (t) a m˙ (t+δt(t))/1018g s−1 β, (8)
i i X 10%and 20%(systematic)errorbarsfor the opticaland
≡ ×
IR data, respectively, and we ignore the much smaller
with a and β as free(cid:2)parametersandδt(t) is t(cid:3)he viscous
i measurement errors. These round values were chosen
lag at time t, defined as
by making a preliminary fit to the data and assessing
ψ the rms fit residuals. Because the uncertainties were
m˙ (t+δt(t))
δt(t) ∆ X , (9) estimated in this utilitarian way, any goodness-of-fit
i
≡ (cid:18) <m˙ X > (cid:19) statistics -such as χ2/ν - should be interpreted with
reserve.
where ∆ and ψ are free parameters. A slightly modi-
i We use a flat prior (i.e., uniform weighting) on the
fied prescription for the accretion emission is considered
period; for all other parameters, we use informed pri-
in Appendix B. Finally, we allow for a potential scaling
ors whenever one is evident. Otherwise, we default to
error in our conversion from X-ray flux to m˙ by intro-
X flat priors on the logarithm of normalization terms (i.e.,
ducingafloatingoffseta whichmodifies thezero-point
X scale-independent weighting), and flat priors on shape
of m˙ . For simplicity, rather than introducing a new
X parameters. The fitting results, which are based on
variable, m˙ will be understood to contain an implicit
X our analysis of all the OIR data, are presented in Ta-
free zero-point, which is determined in the fit.
ble 1. MCMC directly computes the posterior probabil-
ity distribution of a model’s parameters,but it does not
4.3. Reprocessed Emission specifically optimize the fit quality. Therefore, the best-
The X-ray heating by the central source, which oc- fitting value of χ2/ν given in Table 1 has been obtained
curs on a timescale of seconds, is treated as instanta- via other standard optimization methods (e.g., Leven-
neous. That is,the heating termH (t) respondsdirectly bergMarquardt,downhill-simplex,etc.),whiletakingthe
i
to changes in m˙ (t) (zero time lag). We arbitrarily pa- MCMC results as a starting point.
X
rameterize the dependence of H (t) on m˙ as a broken To achieve our fits via emcee-hammer, we used 500
i X
power law. This choice allows for variable shielding by walkerswith 15000steps apiece for a total of 7.5 million
the disk as the X-ray luminosity varies. MCMC samples. Our analysis and results are based on
The heating term contains two elements: The pri- the final 2.5 million samples. Convergence has been di-
mary term, which has no orbital phase dependence, de- agnosedfollowingGelman & Rubin(1992)usingastrin-
scribesX-rayheatingoftheouterdisk;andthesecondary gent criterion of R˜ < 1.1. For our analysis, across all
term, Z(t), describes X-ray heating of the companion parameters, 1.049<R˜ <1.085.
star. Z(t) depends on phase and is maximum at su- In Figure 6, we show for the V-band light curve (the
perior conjunction of the star. We assume that Z(t) bandwiththemostdata)thebest-fitmodelbrokendown
varies sinusoidally and that its normalization constant intoitscomponentcontributions. Thefiguremakesclear
hi = h is the same for all of the OIR bands. This ap- that the total flux is dominated by the constant compo-
proximation is valid because for the B-type companion nent of stellar light, while the disk and X-ray-heating
star (Cowley et al. 1983) the OIR bands all lie in the component contributions are at most 1/2 and 1/4
Rayleigh-Jeans part of the spectrum. as large, respectively (as determined∼in the fit).∼The
maximumcontributionofX-rayheatingtothetotalstel-
Hi(t)≡hci(1+Z(t))( m(˙m˙γbX1rea(kt))γmm1˙˙,bXre(atk) γ2, mm˙˙XX((tt))<>mm˙˙bbrreeaakk,,ltcaiio/rnCfliutox≈tish5eh−ms˙t1γbe01rlel%aakr(∼flSuexc0t.1ido6un%e4.t.o1M),eelalwinphwsiolheiidletah,letvhaaersiycamobnimltirteiytbruiics-
term is a factor 10 smaller still. B01 conclude that
(cid:16) (cid:17) (10)
∼
where the optical variability of LMC X-3 is due to viscous disk
emissionratherthanX-rayreprocessing,i.e., thatrepro-
Z(t)=η/2(1+sin(2πφ π/2)), (11)
− cessing is of secondary importance. Our results support
so that Z(t) varies between 0 and η each orbit, where η, B01’s conclusion.
h, γ , γ , and m˙ are fit parameters. Our fits indicate that as the X-ray flux varies over its
1 2 break
Ourfullmodelconsistsofatotalof27freeparameters full range that the fraction of stellar light to the total
(plus two additional calibration parameters to align the light in both the B and V bands is 80% 10%. A
∼ ±
B01 and SMARTS datasets). There are 16 “chromatic” consistent value of 70% 90% has been derived inde-
≈ −
parameters that contain a subscript i (4 each per OIR pendently for a wide range of X-ray luminosities using
band)andthere are11“gray”parametersthatareinde- spectroscopic data (Orosz et al. 2014).
pendent of wavelength (e.g., the orbital period P). Cor- The value of the scaling index β = 1.31 0.08 (90%
±
relations amongst the fit parameters are discussed and confidence), which relates the OIR flux to the time-
shown in Appendix A. lagged disk emission m˙ X (Section 4.2), may indicate
6 Steiner et al.
that the rates of inflow at the outer and inner radii processingregion,andthatthiseffectdominatesoverthe
are not matched. If so, the implication is that mass is self-shadowingbythedisk. Above 50%L ,theshad-
Edd
∼
lost from the body of the disk, especially at high values owingdominatesandtheheatingsignaldwindlesrapidly.
of m˙ . This scenario is readily explained by the ac- For the shadowing to be substantial without produc-
X
tion of disk winds (e.g., Miller et al. 2006; Luketic et al. ing more OIR emission than is observed, the flare in the
2010; Ponti et al. 2012; Neilsen & Homan 2012). Alter- disk’s scale height must move inward as the X-ray lu-
natively, as discussed in 6, a standing shock from the minosity increases. That is, we envision a flared funnel-
§
accretion stream could be responsible for the mismatch like regioninthe inner disk that contractsinaroundthe
in rates. black hole as luminosity increases and as the flaring be-
From the flux densities computed in Table 1, very comes more pronounced. The OIR emission from the
rough constraints can be placed on the temperature of reprocessing region remains modest, even at the highest
thestarandoftheOIR-emittingregionofthedisk. How- luminosities,becausethereprocessingexcessisrelegated
ever, such calculations are fundamentally limited by the toshorterwavelengthsandemittedfromwithinadimin-
accuracy of the absolute OIR flux calibration, which is ishedareaofthedisk. Apredictionofthispictureisthat
marred by 20–30% zero-point calibration differences be- the pattern of reprocessing shifting to smaller area and
tween SMARTS and B01 data sets. We rely on the shorter wavelength at growing luminosity should be ob-
SMARTScalibrationasourstandardratherthanthatof servableinoptical/UVforthoselow-inclinationsystems
B01 because the former calibration is based on a larger which show substantial luminosity variability.
sample of standard stars that were observed more fre- As viewed from the X-ray source, the star subtends
quently. an angle of 37◦ (ignoring disk obscuration). The cor-
∼
In Figure 7, we show two versions of the OIR light responding fraction of the X-ray heating signal that is
curves folded on LMC X–3’s orbital period: On the left reprocessed in the face of the star is obtained from the
arethelightcurvesintheiroriginalform,andontheright fit: η/(1+η) 40%(Equations10,11). The reprocessed
≈
are the filtered light curves produced by removing the emission from the star can be distinguished from the
two nonstellar contributions, i.e., the m˙ -induced com- reprocessed disk emission because the former is modu-
X
ponent X(t) and the reprocessed X-ray emission H(t). lated at the orbital period. For simplicity, the fraction
Plainly,the ellipsoidalvariabilityismuchmoreapparent ofthe totalreprocessedemissioncontributedby the star
in the filtered light curves. is treated as a constant, free parameter.
Because of the long baseline and abundance of data,
6.2. Wavelength-Independence of the Lag
the signal evident from the filtered light curves (using
thecompletemodel)allowsonetodeterminethe1.7-day In canonical thin accretion disk theory
orbital period to the remarkable precision of two-tenths (Shakura & Sunyaev1973),thewavelengthofpeakemis-
of a second. (Prior to this work, the best determination sionfromthediskscalesasλ (R) R3/4,whereasfor
max
of the period has been P = 1.7048089 1.1 10−6 d irradiation-dominated disks the peak∝wavelength scales
± ×
(Song et al. 2010); our result is in good accordance with roughly as λ (R) R3/7 (Chiang & Goldreich 1997).
max
theirs.) Theorbitalphase,however,conformstoitsprior On this basis, one ex∝pects a factor 2 variation in the
and is otherwise unconstrained. The quality of our pe- viscous time lag acrossthe OIR ban∼ds. However,as evi-
riod determination is illustrated in Figure 8 where it is dentfromTable1andfrominspectionofFig.3,theOIR
shown to compare favorably with an independent deter- time lag is essentially independent of wavelength from
mination based on three decades of spectroscopic veloc- the B-band ( 4400 ˚A) to the J-band ( 12,500 ˚A)10.
ity data. As indicated in the figure, the two data sets Theconstan≈cyoftheviscoustimelag≈withwavelength
jointlydeterminethe periodtoaprecisionof90millisec- implies that the OIR signal arises from a single radius
onds (P =1.704808 1 10−6 d). in the disk. This result is incompatible with any simple
± ×
The raw light curves, by comparison, cannot even de- disktheoryforwhichtemperaturefallsoffmonotonically
liver a unique orbital period because of strong aliasing. with radius, unless one imagines locating the radius of
A Lomb-Scargle (Scargle 1982) search shows that the OIR emission at the disk’s outer edge. However, the
strongestfalse periodis favored9:1overthe true period. outer edge is obviously ruled by the data because the
Moreover,theuncertaintyinthe perioddeterminationis entire disk would then be so hot that it would outshine
a factor 3 worse than achieved using the light curve the star.
∼
model.
6.3. Shocks and Hotspots in the Outer Disk
6. DISCUSSION Instead,theconstancyofthetimelagwithwavelength
6.1. Reprocessing and Self-Shadowing is naturally explained by the presence of a “hot spot”
or similar structure within the outer disk that is bright
The intensity of the X-ray heating component in the
enoughtodominatetheOIRlagsignal. Dopplermodula-
OIR bands is a surprisingly strong function of X-ray lu-
minosity, scaling as H m˙ 1.9, up to a critical value of tiontomographicmapsofaccretingbinarysystemsshow
i ∝ X clear evidence for such features in other sources (e.g.,
m˙ 6.2 1018 g/s (roughly 50% of the Eddington
X ≈ × ∼ Steeghs 2004; Calvelo et al. 2009; Kupfer et al. 2013).
limit). Above this luminosity, the heating signal drops
Anexcellentexampleisprovidedbythemapoftheblack
off rapidly as m˙ −3.4 such that at the maximum X-ray
X hole binary A0620–00, which reveals bright spots em-
luminosity ( L ) its intensity is a factor of 10 below
Edd bedded in a hot “ring” in the disk (dominated by two
≈
its peak. A likely explanation for the initial superlin-
ear rise with luminosity is that the flaring of the disk 10 Thewavelength-independence ofthelaghasbeenverifiedby
at higher luminosity increases the solid angle of the re- examiningsub-intervalsofthedataaswell.
Accretion Lags in LMC X–3 7
1997 1998 1999 2008 2009 2010 2011
3 //
Full Model Disk Emission Ellipsoidal Modulation
Steady Flux X-ray Heating
)
s
t
i
n
u 2

y
r
a
r
t
i
b 0.2
r
(a 1 0.1
x 0
u
-0.1
l
F

-1.00 //
g -0.5 BV IJ BS M0 1ARTS
a
m 0.0
∆ 0.5
1.0
//
200 400 600 800 10001200 4400460048005000520054005600
HJD - 2,450,000.5
Fig. 6.— The model of the observed light curve and its three components for the band with the most data, V. The composite model
isshownindarkgreen; the accretion emissionderivedusingtheX-raydata inred; andthe ellipsoidalandX-ray-heatingcomponents are
shown in blue and purple, respectively. The inset is a blowup showing the ellipsoidal and X-ray heating terms for several orbital cycles.
Thefitresidualsareshowninthelowerpanels.
large crescent structures; Neilsen et al. 2008). The ring analternativeexplanationforthesuperlinearscalingbe-
is locatedat45%or60%ofthe L1 radius(R ) depend- tween OIR and X-ray emission (β = 1.3), which in Sec-
L1
ingonwhether oneassumes,respectively,aKeplerianor tion5we attributedto disk winds. Here,where the OIR
ballistic trajectory for the gas stream. This range in ra- emission originates in a shock, we instead envision that
diusexactlybracketsthecircularizationradiusofthesys- this nonlinear relationship could plausibly result from
tem, the radius at which the angular momentum of the changes in the structure or strength of the shock caused
tidal stream matches the angular momentum of the gas by variations in m˙ .
s
in the disk. The circularization radius R is located at
C
50%RL1forA0620–00,whilethetruncationradiusofthe 6.4. Estimating α-viscosity
outer disk is predicted to be located at 80 90% R
L1
∼ − Thecoolingtime forthe shock-heatedgasis negligibly
(Frank et al. 2002). A schematic diagram of a system
like A0620–00 is shown in Figure 9. short compared to the viscous timescale. Therefore, the
stability and brightness of these structures are related
The temperatures of these bright spots and rings are
to the instantaneous value of m˙ . The ring itself must
poorly constrained in quiescent black-hole binaries, and s
they are essentially unconstrained in active systems. In be relatively narrow owing to the prompt cooling. The
cooledgasrelaxestothediskprofile,andthenitproceeds
order to crudely gauge the temperature of such a spot,
inward to the center on the viscous timescale.
welooktotherecentworkbyKupfer et al.(2013)where
for a cataclysmic variable (CV) system they have mea- The outermost disk is always dominated by gas pres-
sure so that the scale height at each radius is given,
suredT 30,000K.Suchhotspotsareattributed
hotspot
≈ roughly, by the ratio of the sound speed in the mid-
to shocks produced by the tidal stream of gas from the
companion star impinging on the disk. A strong shock plane, cs = kT/µmp, to the Keplerian velocity in the
(Mach number > 10) can result in up to a hundred- disk. Using the α disk approximation for disk viscosity
fold jump in temMperature in the post-shock gas. (Shakura &pSunyaev 1973),
If the mass flow rate in the stream m˙ is sufficiently
s
large,a shockthat traversesinwardfrom Rout will even- ν =αcsz, (12)
tuallystallatthecircularizationradiusR . Forsimplic-
C where z is the scale height of the disk. Then,
ity, we assume that the flow is purely ballistic, i.e., that
the velocity is unaffected by m˙ , and that the hotspot
s R3/2
luminosity increases as m˙s increases. We now mention ν αc2 , (13)
≡ s√GM
8 Steiner et al.
TABLE 1
Best-Fitting Model
Param. Global B V I J Priora
P (d) 1.704805±3×10−6 ··· ··· ··· ··· Flat
β 1.31±0.08 ··· ··· ··· ··· N(1,0.5)
m˙break/max(m˙X) 0.545+−00..001076 ··· ··· ··· ··· Log
ψ 0.24+0.03 ··· ··· ··· ··· Flat
−0.04
aX(1018 gs−1) 0.08+−00..1013 ··· ··· ··· ··· Flat
t0−T0,Cowleyb (d) 0.004+−00..001127 ··· ··· ··· ··· N(0,0.01)
∆i (d) ··· 15.5±1.1 15.4±0.8 16.4±1.1 16.+−13.. Flat
ai(µJy) ··· 17±3 15±3 9.4+−21..16 5.0+−10..38 Log
ci(µJy) ··· 62±6 46±4 22±4 9+−53 Log
Ci(µJy) ··· 670±11 547±8 285±6 146±4 Log
∆mBc 0.293±0.012 ··· ··· ··· ··· Flat
∆mVc 0.216+−00..000182 ··· ··· ··· ··· Flat
log (h) −2.3+0.2 ··· ··· ··· ··· Logonh
10 −0.3
γ1 1.9±0.3 ··· ··· ··· ··· N(1,0.5)
γ2 −3.4+−01..70 ··· ··· ··· ··· Flat
η 0.7+0.3 ··· ··· ··· ··· Flat
−0.2
log (ǫ) −1.1+0.4 ··· ··· ··· ··· Flatonǫ
10 −0.6
χ2/νd 2426/2894
Note. —Bestfitmodelandassociated90%credibleintervals. Ofthe29parameters,16arenormalization
termstothecomponentfluxes.
a Theshapeofthepriorassumed. “Flat”indicatesuniformweighting,and“log”referstouniformweighting
onthelogarithmoftheparameter. N(µ,σ)describesaGaussiancentereduponµwithstandarddeviationσ.
b T0,Cowley =2445278.66HJD(Cowleyetal.1983).
c Zero-pointdifferencesbetweenthedatasetsofB01andSMARTS.
d Thegoodness-of-fitiscalculatedusing10%systematicuncertaintyontheopticalmodeland20%systematic
uncertaintiesontheIR;thesmallmeasurementerrorsonthedatacontributenegligiblyandwereignored.
16.5 B 1.0
this work
17 Spectroscopic Period (Song et al. 2010)
17.5 0.8 P = 1.7048080 d ± 0.09 s
16.5 V
m.)
17 or
17.5 y n 0.6
mag 16.5 I bitrar
171.75 J Density (ar 0.4
17 0.2
17.5

18
0.0
18.5
1.704790 1.704795 1.704800 1.704805 1.704810 1.704815 1.704820
-2 -1 0 1 2
Period (days)
Phase
Fig. 8.— Our result for the orbital period alongside the defini-
Fig.7.— (left) OIR light curves folded on the orbital period.
tivespectroscopicdetermination(Songetal.2010). Thecombined
A zero-point offset has been applied to bring the B01 data (open
resultisshowninred.
symbols) into alignment with the SMARTSdata (filled symbols).
Theshiftisroughly0.3and0.2maginB andV respectively(see is strongly illuminated; and (2) m˙ alone determines the
Table 1). (right) OIR light curves obtained after removing the s
X-rayand OIRemission. Building onthese assumptions
heatinganddisk-emissiontermsH(t)andX(t). Errorbarsreflect
an estimated 20% uncertainty on the accretion “noise” which has and our measured value of the time lag, we now derive
beensubtracted. Alowesssmoothingofthelightcurveisoverlaid a rough estimate of α in the outer disk. We first con-
onthedataineachpaneltoguidetheeye. sider the conditions in the disk and the location of the
OIR-emitting hotspots.
and the viscous timescale can be written as
The strong X-ray irradiation of the disk will modify
µm
t √GMR p . (14) its structure at large distances from the central source,
visc
≈ αkT i.e., R > 1010 cm (Hartmann 1998). Meanwhile, the
(cid:16) (cid:17)
In what follows, recall that we are approximating the midplane temperature, being less sensitive to irradia-
evolving disk as a superposition of independent, steady- tion, will tend to follow the viscous dissipation profile,
state solutions (Section 4.2). We make two other as- T R−3/4. However,thetemperatureoftheirradiated
m
∝
sumptions,whichareeminentlyreasonable: (1)Thedisk surface layer of the disk will fall off much more slowly,
Accretion Lags in LMC X–3 9
the ratio of gas-to-magnetic pressures, P /P (a net
gas mag
quantity usually referred to as “plasma β”). Given the
scaling, α P /P , if P is sufficiently insensitive
mag gas mag
∝
to the accretion flow in the outer disk, then one would
expect that as the mass transfer rate (m˙ ) varies, when
m˙ increases, P will too. A more specific prediction
gas
Fig.9.—TheOIRlagsignalisdominatedbyabrightring(yel- would be beyond scope of this work, but we note qual-
low)intheouter diskatwhichtheaccretionstream(dashed line) itatively that this effect very naturally gives rise to an
shocks into the disk. This ringmay coincide withthe circulariza- inverse relationship between α and mass accretion rate,
tionradius,wheretheangularmomentumoftheaccretionstream
as required by our fit.
(dashedline)matches thedisk’sKeplerianangularmomentum.
as Ts R−3/7 (Chiang & Goldreich 1997), or even as 7. CONCLUSIONS
∝
slowly as T R−1/3 (Fukue 1992). To our knowledge,
s ∝ Motivated by a roughly two-week time-lagged corre-
the vertical temperature gradient in strongly irradiated
lation between the OIR and X-ray light curves of LMC
diskshasnotbeengivenseriousconsideration. Generally,
X–3, we develop a new method of analysis that is appli-
T andT inthe outerdisk,determinedfromirradiation
s m cabletoactiveX-raybinarysystems. WemodeltheOIR
andviscousdissipation,respectively,arecomparableand
emission deterministically as a combination of accretion
within a factor of a few of one another, and it is reason-
emission, X-ray reprocessing, and stellar emission. The
able to approximate the vertical temperature profile as
components driven by accretion are computed using the
isothermal(e.g., Cunningham1976; King1998). For the
X-ray light curve. The model allows accretion signals to
averagemass accretionrate ontoLMC X–3,we estimate
be filtered out of the OIR light curves, thereby isolating
for any reasonable set of disk parameters that to within
the stellar component of light.
a factor 3 the midplane temperature T 105 K at
∼ C ∼ The utility of this technique is demonstrated for LMC
R .
C X–3, a system which exhibits large-amplitude, but sim-
Forthistemperatureandradius,usingEquations12-14
ple,variabilityinitsperennialthermalstate. Wedemon-
we find to order of magnitude that α 0.5. Our result
∼ stratethatthismethodimprovestheobservabilityofthe
agrees well with other measurements of α, particularly
stellar ellipsoidal light curves and leads to an improved
those obtained for CVs (King et al. 2007, andreferences
determination of the orbital period. Furthermore, the
therein). Themostreliableoftheseothermeasurements,
method allowsone to disentanglethe OIR componentof
like our estimate for LMC X-3, were derived for irradi-
disk emission from the reprocessed X-ray emission. The
ated outer disks.
time lags in the system are independent of wavelength,
which indicates that they originate from a hot ring at
6.5. α Scales with Luminosity
a single, fixed radius in the disk. We identify the ra-
A surprising outcome of our model and analysis is the dius of this ring as the circularization radius where the
positive luminosity scaling of the time lag (ψ 0.25; tidal stream from LMC X–3’s B-star companion meets
≈
Eq. 9). Recalling that t m˙ ψ, Eq. 14 gives αt the disk, inducing a bright shock.
T−1 m˙ −1/4. For a convsistcan∝t vaXlue of α, one expevcitssct∝o By interpreting our results through the lens of α-
∝ X disk models, we estimate the viscosity in the outer disk:
findβ = 0.25,whereaswefindapositivevalueofβ. As
− Based upon the average properties of the disk, we esti-
abottomline,wefindthattheviscosityparametervaries
mate α 0.5 to order magnitude. Furthermore, we un-
with luminosity (or equivalently, mass accretion rate) as ∼
expectedlyfindthatαdiminishesasluminosityincreases
fαac∝t,ma˙lX−re1/a2d.yAbneeinnvseurgsegessctaeldinfgorsuLcMhCasXw–e3pbreadseicdtphuarseilny (α ∝ m˙ X−1/2). We speculate that this result may be re-
latedto changesinP /P relatedto evolutionin the
on changes in its disk spectra as the luminosity varies gas mag
mass accretion rate in the outer disk (e.g, Bai & Stone
(Straub et al. 2011).
2011).
If this scalingis relatedto the increase in temperature
which results from a higher m˙ , then α may be lower
X
in the inner disk11. However, GRMHD simulations in- It is a pleasure to thank Poshak Gandhi, Chris Done,
dicate that α increases as radius decreases, at least for Ramesh Narayan, Andy Fabian, and Joey Neilsen for
the innermost disk where MHD turbulence is dominant helpful discussions which have improved this work, as
(Penna et al. 2013). Of course, there is no reason to ex- wellastheanonymousreferee,forher/hishelpfulreview.
pectviscositytobeconstantoverthedisk. Inparticular, WethankCatherineBrocksoppformakingherB01data
the viscosity may depend on other factors such as the setavailabletous,andXueningBaiforhisinputondisk
disk’s density, temperature or magnetization. theory. Support for JFS has been provided by NASA
Bai & Stone (2011) provide one possible explanation Hubble Fellowship grant HST-HF-51315.01.
for the scaling relation we find relating α and luminos-
ity. These authorsfind aninversescalingbetween αand
11 On the other hand, this dependence cannot be purely due
to an inherent scaling with temperature, i.e. α ∝ T−2, without the α-disk interpretation breaking down. A weaker scaling with
temperaturewouldbecompatible.
10 Steiner et al.
TABLE 2
Best Fit with Model2
Param. Global B V I J Prior
P (d) 1.704805±3×10−6 ··· ··· ··· ··· Flat
β 1.29±0.08 ··· ··· ··· ··· N(1,0.5)
m˙break/max(m˙X) 0.546+−00..001026 ··· ··· ··· ··· Log
log10(a2) 0.1+−00..23 ··· ··· ··· ··· Flatona2
a3 0.4±0.4 ··· ··· ··· ··· Flat
ψ 0.31+0.04 ··· ··· ··· ··· Flat
−0.05
aX(1018 gs−1) 0.10±0.09 ··· ··· ··· ··· Flat
t0−T0,Cowley (d) 0.001±0.014 ··· ··· ··· ··· N(0,0.01)
∆i (d) ··· 12.6±1.0 12.6±0.8 15.4±1.2 11.2±1.5 Flat
ai(µJy) ··· 17+−43 16±3 9.9±1.9 5.2±1.1 Log
ci(µJy) ··· 61±6 47±4 22±4 10±4 Log
Ci(µJy) ··· 677±10 551.+−69 287±6 148±4 Log
∆mB 0.293±0.012 ··· ··· ··· ··· Flat
∆mV 0.215±0.010 ··· ··· ··· ··· Flat
log (h) −2.6+0.3 ··· ··· ··· ··· Logonh
10 −0.4
γ1 2.3±0.4 ··· ··· ··· ··· N(1,0.5)
γ2 −3.5±0.9 ··· ··· ··· ··· Flat
η 0.8±0.3 ··· ··· ··· ··· Flat
log (ǫ) −1.1+0.3 ··· ··· ··· ··· Flatonǫ
10 −0.7
χ2/ν 2395/2892
Note. —Bestfitmodelandassociated90%credibleintervals. AllnotesfromTable1arealsoapplicablehere.
APPENDIX
CORRELATIONS WITHIN THEFIT
Our model has many free parameters ( 30), and it is therefore nontrivial to assess parameter degeneracies in the
≈
model. In Figure 10 we present pairwise correlations from a randomly-selected subsample of the MCMC chains for
the principal fit parametersof the model. Parameterswhich are independently determined for the various OIR bands
are shown together in color.
Few trends are strongly evident. The strongest is an anticorrelationbetween γ , the index which relates the degree
1
of X-ray heating to luminosity, and the normalization to that scaling, h (Eq. 10). Also evident is a much weaker and
positive correlation between γ and h.
2
Next-mostprominentis ananticorrelationbetweenβ andthe disk emissionstrengtha , wherethe effectisstrongest
i
for shorter wavelengths. While the wavelength dependence of a given pair of model parameters does not necessarily
indicate a degeneracy in the model, it can nevertheless be revealing. In particular, strongly positive trends with
wavelength are evident and expected amongst parameters that characterize the disk brightness and stellar brightness
(i.e.,panelswithpairsofa ,c ,andC ). Conversely,thewavelengthindependenceoftheaccretionlag,∆ ,animportant
i i i i
result from this work, is evident in Figure 10.
CONSIDERING NON-STATIONARY BEHAVIOR
One drawback to our approach is that although we model dynamic evolution of the accretion-disk system, we have
adopted a structure rooted in a steady-state formalism. We now consider an improved treatment of non-stationary
behavior with our model. Descriptively, we better allow for compression and expansion of the individual annuli by
introducing a term that allows for stretching action by adjacent rings. Specifically, we modify Equation 8 by allowing
X-rayheatingtodependonthetimederivativeoftheinner-diskaccretionrate,m¨ . Withthismodification,Equation8
X
becomes
m˙ (t+δt(t)) m¨ (t+δt(t)) m˙ (t+δt(t)) a3 β
X X X
X (t) a +a , (B1)
i ≡ i× 1018g s−1 2 1018g s−2 × 1018g s−1
(cid:20) (cid:18) (cid:19) (cid:21)
where a and a are additional free parameters used to determine radial expansion or compression of the flow. This
2 3
alternateformulationhasonlyaminoraffectonthefitstotheOIRlightcurves;thefittingresultsaregiveninTable2.
Thequalityofthefitisonlymoderatelyimproved,∆χ2 =29fortwoadditionalfreeparameters. Theprincipalchange
is that the (average) time lag inferred is smaller by 20% compared to the principal model, which is perhaps a
≈
consequence ofa phase shift induced when mixing the m˙ -lightcurve with its time derivative. Detailed consideration
X
ofnon-stationarydiskmodelingwithintheparadigmofabifurcated,two-stateaccretionflow,asappliedtoLMCX–3,
can be found in Cambier & Smith (2013).
REFERENCES
Arnaud,K.A.1996,inAstronomicalSocietyofthePacific Bai,X.-N.,&Stone,J.M.2011, ApJ,736,144
ConferenceSeries,Vol.101,AstronomicalDataAnalysis
SoftwareandSystemsV,ed.G.H.Jacoby&J.Barnes,17