A model for the energy-dependent time-lag and rms of the heartbeat oscillations in GRS 1915+105

0.66 MB·

by  Mubashir Hamid Mir

MNRAS000,1–8(0000) Preprint21January2016 CompiledusingMNRASLATEXstylefilev3.0
A model for the energy-dependent time-lag and rms of the
heartbeat oscillations in GRS 1915+105
Mubashir Hamid Mir,1,2⋆ Ranjeev Misra,2 Mayukh Pahari,2
Naseer Iqbal1 and Naveel Ahmad1
1 Department Of Physics, Universityof Kashmir, Srinagar-190006, India
6 2 Inter-UniversityCenterforAstronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune-411007, India
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0
2
21January2016
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a
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ABSTRACT
0 Energy dependent phase lags reveal crucial information about the causal relation be-
2
tween various spectral components and about the nature of the accretion geometry

around the compact objects. The time-lag and the fractional root mean square (rms)
]
E spectra of GRS 1915+105 in its heartbeat oscillation class/ρ state show peculiar be-
H haviouratthefundamentalandharmonicfrequencieswherethelagsatthefundamen-
tal show a turn around at ∼ 10 keV while the lags at the harmonic do not show any
h. turn around at least till ∼ 20 keV. The magnitude of lags are of the order of few sec-
p ondsandhencecannotbeattributedtothelighttraveltimeeffectsorComptonization
- delays. The continuum X-ray spectra can roughly be described by a disk blackbody
o
and a hard X-ray power-law component and from phase resolved spectroscopy it has
r
t been shown that the inner disk radius varies during the oscillation. Here, we propose
s
a thatthereisadelayedresponseoftheinnerdiskradius(DROID)totheaccretionrate
[ such that r (t) ∝ m˙ β(t−τ ). The fluctuating accretion rate drives the oscillations
in d

2 of the inner radius after a time delay τd while the power-law component responds
immediately. We show that in such a scenario a pure sinusoidal oscillation of the ac-
v
cretion rate can explain not only the shape and magnitude of energy dependent rms
0
7 and time-lag spectra at the fundamental but also the next harmonic with just four
8 free parameters.
4
Key words: accretion,accretiondiscs−blackholephysics−X-rays:binaries−X-rays:
0
. individual: GRS 1915+105
1
0
6
1
: 1 INTRODUCTION tions in X−ray intensity by nearly an order of magnitude
v
(Taam, Chen & Swank 1997; Belloni et al 1997).
i GRS 1915+105 exhibits a range of variability in its
X A promising model for the ρ class variability
light curve and power density spectra (PDS), along
is that it is driven by radiation pressure instabil-
r with other peculiar features like near-Eddington ac-
a ity (Lightman & Eardley 1974; Taam & Lin 1984;
cretion rate consistent for 20 years, jet activities −
Taam, Chen & Swank 1997). However, numerical simula-
from steady radio flickering to super-luminal radio
tionsoftheinstabilitysuggeststhatthereisneedtochange
ejections (Mirabel & Rodriguez 1994) and regular tran-
the viscous prescription and take into account a fraction of
sitions into various spectral states in time-scale from
energy dissipating in a corona to explain the overall X-ray
msec to hours (Morgan, Remillard & Greiner 1997;
modulation (e.g. Nayakshin,Rappaport,& Melia 2000;
Muno, Morgan & Remillard 1997; Mirabel & Rodriguez
Janiuk,Czerny, & Siemiginowska 2000; Janiuk & Czerny
1994; Belloni et al 1997). The timing analysis features 14
2005).Phaseresolvedspectroscopyofthevariabilityreveals
variability classes with each having their own peculiar light
a consistent picture where the instability causes the inner
curves, phase/time-lags and coherence (Belloni et al 1997;
disk radius to vary with the luminosity variation, which
Pahari et al. 2013a,b). Among the 14 variability classes,
maybeduetomassejectionfromthesystem(Neilsen et al.
the ρ class shows characteristic limit-cycle variability in
2011, 2012). The spectral modeling at different phases
the time-scale of 50−100 sec with large amplitude oscilla-
show complex components consisting of disk emission and
a power-law with a high energy cutoff.
An alternate approach to phase resolved spectroscopy
⋆ E-mail:[email protected] is to understand the energy dependent phase lag and
(cid:13)c 0000TheAuthors
2 M.H.Mir et al.
8 7
N=1 N=1
N=3 N=3
6 N=6 6 N=6
N=10 N=10
4 5
c) 2 c) 4
e e
s s
ag ( 0 ag ( 3
e-l e-l
m m
Ti -2 Ti 2
-4 1
-6 0
-8 -1
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Energy(keV) Energy(keV)
0.4 0.25
N=1 N=1
N=3 N=3
0.35 N=6 N=6
N=10 N=10
0.2
0.3
ms 0.25 ms 0.15
al r al r
n 0.2 n
o o
cti cti
a a 0.1
Fr 0.15 Fr
0.1
0.05
0.05
0 0
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Energy(keV) Energy(keV)
Figure 1. The predicted time-lag and rms versus energy for the fundamental (left panels) and the next harmonic (right panels) for
differentvaluesofthepower-lawtodiskfluxratioparameterN.Theotherparametersarekeptatsomeconstant fiduciaryvalues.Note
thesensitivityofthepeakofthetime-lagcurveforthefundamentaltoN.
rms spectra of an oscillation. This is particularly use- tioncanprovideinsightintohowthetheinnerradiusvaries
ful for Quasi periodic oscillations (QPO). Such anal- with the accretion rate.
ysis of high frequency QPOs can give crucial infor- As we describe below the time-lag and rms spectra for
mation regarding the size of the emitting region (e.g. theρclassvariabilityisfairlycomplexandourmotivationis
Lee, Misra & Taam2001;Kumar& Misra2014)andforthe to identify the simplest model which can describe not only
low frequency ones the causal connection between spectral the behaviour of the fundamental but also simultaneously
parameters(Misra & Mondal2013).Ingram & van derKlis that of thenext harmonic.
(2015) have considered both energy dependent rms and
phase resolved spectroscopy to argue that the low fre-
quency QPO in the χ state has a geometric origin.
Frequency resolved spectroscopy at the QPO frequency
2 THE DELAYED RESPONSE OF THE INNER
which is related to the energy dependent rms curves
DISK MODEL(DROID)
can provide crucial information regarding the nature of
the oscillation (Axelsson,Done, & Hjalmarsdotter 2014). We assume that the X−ray emission primarily consists of
Janiuk & Czerny (2005) report hard photon time-lags for two components, a soft disk blackbody and a hard X−ray
the ρ class variability which they interpret as the delay for power-lawextendingfrom∼1keVtohighenergiesduetoa
thecorona toadjust toavaryingaccretion rate.Indeedthe corona.WeneglectcontributionsfromtheIronlineemission
magnitude of the time delays of the order of seconds im- and otherreflection features..
plies that it is not due to light travel time effects and in- The blackbodydisk fluxis given by
stead should be associated with time delays between dif-
ferent structural parameters. Since the inner disk radius is F ∝E2 ∞ r dr (1)
known to vary during the oscillation (Neilsen et al. 2012), d Z exp( E )−1
rin kT(r)
it would be interesting to see if energy dependenttime-lags
andthermsspectraofthedifferentharmonicsoftheoscilla- where E is the energy of the photon, r is the radius, rin is
the inner disk radius and T(r) is the surface disk temper-
ature. From the standard disk blackbody model, we have
MNRAS000,1–8(0000)
Energy-dependent Time lag and rms 3
6 β=.30 9 β=.30
β=.50 β=.50
4 β=.70 8 β=.70
β=1.0 β=1.0
7
2
6
0
ec) ec) 5
ag (s -2 ag (s 4
e-l -4 e-l
m m 3
Ti -6 Ti
2
-8
1
-10 0
-12 -1
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Energy(keV) Energy(keV)
0.35 β=.30 0.25 β=.30
β=.50 β=.50
β=.70 β=.70
0.3 β=1.0 β=1.0
0.2
0.25
s s
m m 0.15
al r 0.2 al r
n n
o o
cti 0.15 cti
a a 0.1
Fr Fr
0.1
0.05
0.05
0 0
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Energy(keV) Energy(keV)
Figure 2. The predicted time-lag and rms versus energy for the fundamental (left panels) and the next harmonic (right panels) for
different values of the parameter β which relates the dependence of the inner radius withthe accretion rate. The other parameters are
keptatsomeconstantfiduciaryvalues.Notethesensitivityofthetime-lagcurveforthenextharmonictoβ.
(Frank,King& Raine 2002): andthattheaccretionrateundergoesapuresinusoidalvari-
ation at an angular frequencyω
r −3/4
kTr =kTin(cid:18)rin(cid:19) (2) m˙(t)=m˙o(1+δm˙)=m˙o(1+|δm˙|eiωt) (7)
where Tin ∝ m˙1/4rin−3/4, m˙ is the accretion rate. The which drives the variability of the inner radius, rin(t) =
power-law component is takento be rino(1+δrin) where
Fp ∝m˙ΓE−P (3) δrin=−β|δm˙|eiω(t−τd)+ β(β2+1)|δm˙|2e2iω(t−τd) (8)
where it has been assumed that the normalization of the
power-law depends on the accretion rate (∝ m˙Γ) while for SubstitutingtheaboveinEqn(4),andcollectingtermspro-
simplicity thephoton indexP does not vary. portional to eiωt and e2iωt, for the fundamental oscillation,
The variation of the total flux F = F +F due to we get:
T d p
variations of the accretion rate and inner radius, to second |δm˙|
order is given by δFtf = 1+ Fp (cid:16)Aeiωt+Beiω(t−τd)(cid:17) (9)
1 Fd
δF = (F ) δm˙ +(F ) δr + (F ) δm˙2
t t m˙ t rin in 2 t m˙m˙ and for thenext harmonic:
1
+(F ) δm˙δr + (F ) δr 2 (4) |δm˙|2
t m˙rin in 2 t rinrin in δFth= 1+ Fp (cid:16)A′e2iωt+B′e2iω(t−τd)+C′e2iω(t−τd/2)(cid:17)
Here we denote the normalized variation δX =∆X/X and Fd
define (10)
X ∂Z XY ∂ ∂Z
(Z)X = Z ∂X and (Z)XY = Z ∂X ∂Y (5) The coefficients A,B,A′,B′ and C′ are derived and listed
in Appendix (A). The ratio of the power-law to disk flux
Weassumethattheinnerdiskradiusfollowstheaccre- F /F can beconvenientlywritten as
p d
tion ratewith a time delay definedby:
F
rin(t)∝m˙−β(t−τd) (6) Fpd =NG(E,kTin,P) (11)
MNRAS000,1–8(0000)
4 M.H.Mir et al.
8 p=1.7 7 p=1.7
p=2.0 p=2.0
6 p=2.3 6 p=2.3
p=3.0 p=3.0
4 5
ec) 2 ec) 4
ag (s 0 ag (s 3
e-l e-l
m m
Ti -2 Ti 2
-4 1
-6 0
-8 -1
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Energy(keV) Energy(keV)
0.3 p=1.7 0.18 p=1.7
p=2.0 p=2.0
p=2.3 0.16 p=2.3
0.25 p=3.0 p=3.0
0.14
0.2 0.12
ms ms
al r al r 0.1
n 0.15 n
o o
acti acti 0.08
Fr 0.1 Fr 0.06
0.04
0.05
0.02
0 0
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Energy(keV) Energy(keV)
Figure 3. The predicted time-lag and rms versus energy for the fundamental (left panels) and the next harmonic (right panels) for
differentvaluesofthepower-lawindexp.Theotherparametersarekeptatsomeconstant fiduciaryvalues.
ThefunctionGisgiveninAppendixBandN isaparameter some fiduciary values i.e. T , β, p, ω∗τ , Γ & ∆m˙ at 1.5
in d
which can beobtained from thetime-averaged spectrum. keV, 0.65, 2.7, 2.2 ,0.05 and 0.4 respectively. A salient fea-
The energy dependent rms for the fundamental and tureof themodelis thatthetime-lagincreases with energy
next harmonic are given by 1 |δFf(E)| and 1 |δFh(E)| and then decreases. This occurs because the high energy
√2 t √2 t
while the phase lag with respect to a reference energy photons are dominated by the power-law component which
Eref is given by the phase of δFtf(E)(δFtf(Eref))∗ and varieswiththeaccretionratewithoutanytime-delay.Onthe
δFth(E)(δFth(Eref))∗. otherhand,thelowenergyphotonsfromthediskcomponent
The energy dependent variations δFf and δFh given aremoresensitivetotheaccretionrateratherthantheinner
t t
radius. Thus the low energy photons from the disk and the
by Equations (9) and (10) depend on several parameters
high energy ones from the power-law component have rela-
which can be divided into two groups. The parameters of
tively less time delay as compared to photons with energy
the first group can be estimated from the time averaged
close to T , which depend significantly on the inner radius
spectrumwhicharetheinnerdisktemperatureT ,thehigh in
in
whichisassumedtobedelayedwithrespecttotheaccretion
energyphotonindexP andN whichparametrizes theratio
rate. Thus, the turnover energy depends sensitively on the
ofthepower-lawtodiskflux.Theparametersof thesecond
flux ratio of the power-law and disk components character-
group pertain to the nature of the oscillation and are the
ized by the parameter N. Since the ratio can be estimated
exponents for inner disk radius (β), the power-law flux (Γ)
from the time-averaged spectrum, there is little leverage to
with the accretion rate, ωt which is the phase difference
d
tune the other parameters to get the observed lag turnover
(between the inner disk radius & the accretion rate) and
energy.
anoverallnormalization ofthevariation |δm˙|.Thusin this
simplisticmodel,theenergydependentrmsandtime-lagfor
thefundamentalandnextharmonicarecompletelyspecified In Figure (2), we show the rms and time-lag for vari-
bythesefourparameters. Ofthesethenormalization of the ations for another important parameter β which is the ex-
variability |δm˙| determines the normalization of the rms of ponentspecifyingtherelation betweentheinnerradiusand
thefundamentalwhileitssquaredeterminesthatofthenext theaccretion rate. It is interesting to note thesensitive be-
harmonic. haviour of the time-lag for the next harmonic to β where
Toillustratethedependenceofthepredictedenergyde- small changes in β lead to dramatic qualitative changes i.e.
pendent rms and lag of this model on parameters, we show from hard to soft lags. In Figure (3), are shown the rms
as an example in Figure (1) the predicted curves for differ- andtime-lagvariationsfordifferentphotonindices.Thebe-
ent values of the power-law to disk flux ratio characterized haviouris similar tothat observedfrom theFigure(1) since
by N while the other parameters of the model are fixed at theratio Fp given by Eqn (B1) dependson N and p.
Fd
MNRAS000,1–8(0000)
Energy-dependent Time lag and rms 5
10 23 May 2001 Table 1.Bestfitparametersobtained fromtime-averagedspec-
n) Hz2−1 1 ρ class in GRS 1915+105 tdreupmendfiettnitngtimaned-lafgroamndthrempsrfooprotsheed2m3oMdealyfi2t0ti0n1goobfsethrveateinoenrgoyf
ea 0.1 GRS1915+105duringtheρclass
m
s/
m 0.01
ower (r 10−3 (kkTeVin) Nd p Np N
P
10−4
2 1.47+0.01 220+11.10 2.85+0.02 34+2.3 3.15+.25
0.01 11.14 0.02 2.4 .25
− − − − −
0 β ωτd Γ δM˙
χ
−2 0.73 2.25 0.1 0.47
10−3 0.01 Frequ0e.n1cy (Hz) 1 10 Note: kTin isthe inner disktemperature (keV), p isthe photon
power-law index, Nd is the normalization of the disk blackbody
Figure 4.Power density spectrum shown for the“ρ”variability componentNp isthepower-lawcomponentnormalizationandN
class observed on23 May,2001 (OBS-Id: 60405-01-02-00), fitted isobtainedfromthespectralinformation.N isrelatedtothera-
withLorentiansandbrokenpower-lawalongwithresiduals. tiobetween thenormalizationsofpower-lawanddiskblackbody
(AppendixB). β isthe power-lawvariationof Rin withm˙ ,ωtd
is the phase difference, Γ is the dependence of power-law nor-
malization on m˙ and δm˙ is the amplitude of the accretion rate
2.1 Comparison with observations variability.
The ‘ρ’ variability class shows a range of frequencies from
∼ 10-20 mHz with the occasional presence of multiple har- 20keVforRXTE/PCA,weareunabletoverifythemodel-
monics. Using RXTE/PCA archival data for the white predicted saturation using actual data.
noise-subtracted, rms-normalized power density spectrum While a detailed analysis of the energy dependent rms
observed during ‘ρ’ class on 23 May, 2001 (OBS-Id: 60405- and time-lag for different observations during the ρ class
01-02-00) is shown in the top panel of Figure 4 fitted with variabilitywillbepresentedelsewhere(Miret.al.tobesub-
multipleLorentiansandapower-lawwithabreakfrequency mitted),here we haveused one observation as a typical ex-
at ∼ 0.1 Hz. Residuals of the fit are shown in the lower ample.Wenotethatthequalitativebehaviouroftheenergy
panel.Fromthefitting,thefundamentalandthenextthree dependence is fairly generic. Indeed, the detailed phase re-
harmonics are clearly detected with > 3σ significance at solved spectroscopy of differentobservations also show that
19.5+1.8 mHz, 39.1+2.5 mHz, 58.6+5.4 mHz and 78.2+6.5 theenergy dependenceis similar despitechangesin thefre-
1.2 2.5 4.7 5.3
mHz−respectively. − − − quencyof theoscillations (Neilsen et al. 2011, 2012)
For the above observation, we obtain the time aver-
aged photon spectrum from standard2 data file of PCA.
For data extraction, we use PCU2 only since it is reliably
3 DISCUSSION AND CONCLUSION
on throughout the observation and has thebest calibration
accuracy.UsingPCAresponsesandlatestbackgroundspec- GRS1915+105 showsafascinating clockwork initsρ-class,
tral file, the source spectrum in the energy range 3.0−25.0 reflected in the highly periodic light curves which resemble
keVisfittedwithatwocomponentmodel:multi-colourdisk closelytothehumancardiogram,thusisalsocalledasheart-
blackbody(diskbbinXSpec)modelandasimplepower-law beatstate.Wehavestudiedtheenergy-dependentfrequency
(powerlaw in XSpec), along with a Gaussian component at resolved lags and fractional rms of this variability and they
∼ 6.4keVand allcomponentsaremodified byGalactic ab- show a uniquebehaviour. With the lags of the order of few
sorption(TBabsinXSpec).Thefitgaveaχ2/dof =40.41/40 seconds,theycannotbeattributedtolightcrossingtimeef-
and thebest fit spectral parameters are listed in Table (1). fects or due to Comptonization delays. The interesting fea-
We calculate time-lag spectra using cross spectrum ture is that the lag at the fundamental frequency show a
techniques(Nowak et al. 1999) in different energy bands in phasereversal∼10keVwhileat thenextharmonicit keeps
the range 3.20 − 19.85 keV, where the reference band for increasingwithenergy.Thefractionalrmsamplitudeatthe
lagcalculation ischosentobe3.69 −4.52 keVatbothfun- fundamentalandthenextharmonicarealsonon-monotonic
damentalandharmonicfrequencies.Thermswascalculated with energy.
bycomputingthepowerspectrumfordifferentenergybins. We show, that these qualitative features can be ex-
Each power spectrum was fitted with a broken power-law plained in a model where the power-law component varies
andLorentziansasshowninFig.4.Thenbyintegratingthe directly with the accretion rate while the inner disk radius
fitted components, therms was calculated. Table (1) shows responds after a time delay (DROID).Apart from parame-
thevaluesoftheparametersusedtomodeltheobservedlag tersestimatedfromthetimeaveragedspectrum,theenergy
and fractional rms. As can be seen in Fig.(5), the time-lag dependent rms and time lag for the fundamental as well as
at fundamental (top left panel) shows a reversal around ∼ the next harmonic are specified by only four free parame-
12 keV while for the next harmonic (top right panel) the ters and hence it is remarkable that such a simple picture
time-lagseemstoriseatleastupto∼20keV.However,the can even qualitatively reproducethe overall features.
model-predicted spectrum fits the data well and the exten- Itshouldbeemphasizedthattheanalysisofenergyde-
sionofthefittedmodeltill40keVshowssaturationaround pendent time-lag and rms and the model proposed to ex-
25keV.Becauseofpoorcoherenceandpoorstatisticsabove plainthemarenotdifferentbutcomplimentarytothephase
MNRAS000,1–8(0000)
6 M.H.Mir et al.
6 6
Model
Obs-Lag-Fundamental
4 5
2 4
ag (sec) 0 e Lag 3
me-l -2 Tim 2
Ti
-4 1
-6 0
Model
Obs-Lag-Harmonic
-8 -1
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Energy(keV) Energy(keV)
0.4 0.26
Model Model
Obs-RMS-Fundamental 0.24 Obs-RMS-Harmonic
0.35
0.22
0.3
0.2
ms ms
al r 0.25 al r 0.18
n n
o o
acti 0.2 acti 0.16
Fr Fr 0.14
0.15
0.12
0.1
0.1
0.05 0.08
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
Energy(keV) Energy(keV)
Figure 5.Theobserved andpredicted energydependent time-lagandrmsatthe fundamental (leftpanels) andforthenext harmonic
(rightpanels)observedon23May,2001.TheparametersusedforthemodelarelistedinTable1.Thefourpredictedcurvesaredetermined
byonlyfourfreeparameters
resolved spectroscopy done earlier by Neilsen et al. (2011, (e.g. (Sobolewska & Z˙ycki 2006)). The time-lag measured
2012). Indeed as expected both analysis find that the ba- and modelled in this work is of the order of few seconds.
sic feature of the oscillation is that the inner disk radius Unlessthereflectorissignificantlyfarawayfromthecentral
oscillates. The analysis done here provides another way of object, any time-lag between the continuum and the reflec-
analyzingandunderstandingthedata.Moreover,for obser- tion component (which is of the order sub-millisecs to few
vationsof othertypesof oscillations, energy dependentrms millisecs for a ∼ 10 M black hole) can be neglected for
⊙
andtime-lagmaybemoreeasilyestimatedthanperforming the analysis done here. However, the rms spectra obtained
a complete phaseresolved spectroscopy. from AGN (e.g., MCG 6−30−15) do show significant dip
(Z˙yckiet al. 2010) at the position of 6.4 keV Fe emission
A remarkablefeature of thismodel is that theprimary
line which is solely due to reflection. Therefore, instead of
driver is a pure sinusoidal oscillation in the accretion rate
a powerlaw the combined continuum and reflection compo-
andtheotherharmonicscanbeattributedtothenon-linear
nent needs to be considered, which will effect the results
relationship between the accretion rate and the inner disk
primarily in the Iron line region (Fabian et al. 2009). Such
radius as well as that between the two and the emergent
analysis would necessarily involve complex numerical com-
spectrum. As the accretion rate increases the disk extends
putationwhichispresentlyoutofscopeofthisworkbutcan
inwards, but this does not happen instantaneously. The in-
be considered as an extension of the present idea in future.
ner disk responds after a time-delay that is found for this
Itislikelythatthedeviationofthedatafrom thepredicted
observation to be ∼ 20 secs, which may be noted to be of
model,especiallyfortheshapeoftheenergydependentrms
the order of the viscous timescale. Thus our results present
curves, arises because of the simple spectral model used. A
a scenario which has the potential to be developed into a
morerealisticmodelwouldwarrantfittingthedatastatisti-
physicalinterpretation basedon thenatureof theaccretion
callyandobtainingparametervalueswithconfidencelimits.
disk instability.
It will be interesting to study how the parameters of the
The spectral components used in this model are sim-
model change for different oscillations of the ρ class, espe-
plistic. Instead of a power-law, the high energy component
ciallyasafunctionofthefrequencyoftheoscillations.Such
should be correctly modeled as a thermal Comptonization
astudywillshedlightonstructuralchangesthatoccurdur-
spectrum where the Comptonized flux should depend on
ing these oscillations and will help in obtaining a complete
the accretion rate. This will lead to variations in the spec-
hydrodynamicunderstandingof thephenomenon.
tral index which is considered in this work to be a con-
stant.Moreover,thereshouldalsobeareflectioncomponent Finally,themodelmaybeapplicabletootherQPOsob-
which may play an important part in the QPO formation servedindifferentsystemswherethedisccomponentvaries
MNRAS000,1–8(0000)
Energy-dependent Time lag and rms 7
significantlyduringtheoscillation.Observationsofdifferent where Z ≡E/kTin and the relation Tin ∝m˙1/4rin−3/4 has
black hole systems by the recently launched Indian multi- been used.Its variation is given by
wavelengthsatelliteASTROSATisexpectedtoprovidehigh 1
δF = (F ) δm˙ +(F ) δZ+ (F ) δm˙2
quality eventtagged data which would beideally suited for D D m˙ D Z 2 D m˙m˙
such analysis. 1
+(F ) δm˙δZ+ (F ) δZ2 (A2)
D m˙Z 2 D ZZ
Here and as has been used in the main text, we denote the
normalized variation δX =∆X/X and define
4 ACKNOWLEDGEMENTS
X ∂Z XY ∂ ∂Z
(Z) = and (Z) = (A3)
MHM is highly thankful to IUCAA, Pune for allowing the X Z ∂X XY Z ∂X ∂Y
periodic visit to the institute which has helped in carrying The derivatives can be computed to be (FD)m˙ = 2/3,
out this work and to UGC, New Delhi for granting the re- (FD)m˙m˙ =−2/9,
search fellowship. 8
Z3
(F ) =f(Z)= (A4)
D Z 5
(eZ−1) ∞ y3 dy
Z exp(y) 1
R −
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(F ) =f(Z)( −k(Z))=f(Z)( − ) (A5)
D ZZ 3 3 eZ−1
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657 and (F ) =2f(Z)/3.
D m˙Z
Belloni, T., Mendez, M., King, A. R., van der Klis, M. & van SinceZ dependson T which in turndependson m˙ &
in
paradijs,J.1997,ApJ,488,L109 rin, as Z =αm˙−1/4ri3n/4, we have:
Belloni, T., Mendez, M., King, A. R., van der Klis, M. & van
3 1 3 5 3
paradijs,J.1997,ApJ,479,L145 δZ = δr − δm˙ − (δr )2+ (δm˙)2− δm˙δr (A6)
Belloni,T.,Klein-Volt,M.,Mendez, M.,van der Klis,M.& van 4 in 4 32 in 32 16 in
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Fabian, A. C., Zoghbi, A.,Ross, R. R.,et al. 2009, Nature, 459, accretion rate as Fp ∝m˙ΓE−P and hence
540
Γ(Γ−1)
Frank, J., King, A. R., Raine, D. J., 2002, Accretion Power in δFp =Γδm˙ + 2 (δm˙)2 (A7)
Astrophysics,3rdedn.CambridgeUniv.Press,Cambridge
Thevariation in thetotal fluxF =F +F is
IngramA.,vanderKlisM.,2015,MNRAS,446,3516 t D p
JaniukA.,CzernyB.,SiemiginowskaA.,2000,ApJ,542,L33 F F
δF = DδF + pδF (A8)
Januik,A.,Czerny,B.,2005,MNRAS,205,356 t F D F p
t t
KumarN.,MisraR.,2014,MNRAS,445,2818
Next assuming that the inner disk radius follows the
Lee,H.C.,Misra,R.,Taam,R.E.,2001,ApJ,549,L229
Lightman,A.P.,Eardley,D.M.,1974,ApJ,187,L1 accretion rate with a time delay rin(t) ∝ m˙−β(t−τd) and
thattheaccretionrateundergoesapuresinusoidalvariation
Mirabel,I.F.,Rodriguez,L.F.,1994,Nature,371,46
Misra,R.,Mondal,S.,2013, ApJ,779,71 at an angular frequencyω i.e. δm˙ =|δm˙|eiωt theinner disk
Morgan,E.,Remillard,R. A.,&Greiner,J.,1997,ApJ,482,993 variation is given by
Muno, M. P.,Morgan, E. H., & Remillard,R. A , 1999, ApJ, β(β+1)
527,321 δrin=−β|δm˙|eiω(t−τd)+ 2 |δm˙|2e2iω(t−τd) (A9)
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Pahari,M.,Neilsen,J.,Yadav,J.S.,Misra,R.,Uttley,P.,2013a, δFtf = 1+ Fp (cid:16)Aeiωt+Beiω(t−τd)(cid:17) (A10)
ApJ,778,136 Fd
Pahari,M.,Yadav, J.S., Rodriguez, J.,Misra,R.,et al.,2013b, and for thenext harmonic:
ApJ,778,46
TSoabamole,wRs.kEa,.,ML.inA,.D,Z.˙Nyc.kCi,.,P1.9T84.,,2A0p0J6,,2M8N7,R7A61S,370,405 δFth= 1|δ+m˙F|2p (cid:16)A′e2iωt+B′e2iω(t−τd)+C′e2iω(t−τd/2)(cid:17)
Taam,R.E.,Chin,X.,Swank,J.H.,1997, ApJ,485,L83 Fd
Z˙ycki,T. P.,Ebisawa,K.,Niedzwiecki,A.,Miyakawa,T.,2010, (A11)
PASJ,62,1185 where
2 f(z) F
A= + +Γ p
3 4 F
d
3
B= βf(z)
APPENDIX A: ACCRETION RATE INDUCED 4
SPECTRAL VARIABILITY OF A DISK AND A A′=−1 + f(z) k(z) − 1 + 1Γ(Γ−1)Fp
POWER-LAW COMPONENTS 9 8 (cid:18) 4 3(cid:19) 2 Fd
3 3
The accretion disk fluxcan berewritten as B′= f(z) β2 k(z)−1 −β(β+1)
8 (cid:18) (cid:18)4 (cid:19) (cid:19)
FD ∝E−2/3m˙2/3ZZ∞ expy(5y/)3−1dy (A1) C′ = 136βk(z)f(z)
MNRAS000,1–8(0000)
8 M.H.Mir et al.
APPENDIX B: ESTIMATING THE
POWER-LAW TO DISK FLUX RATIO FROM
TIME-AVERAGED SPECTRUM
In terms of Z =E/kT , the ratio of the power-law to disk
in
fluxcan bewritten as
F
p =NG(Z,p)=NZ−(p+2)f(Z)(eZ −1) (B1)
F
d
From Xspec model fitting of the spectrum we get the nor-
malization of the power-law Fp = NpE−P and that of the
disk emission,
(r /kms) 2
N = in cos(θ) (B2)
d (cid:18)(D/10kpc)(cid:19)
By carefully comparing the flux equations, substituting Z
for E and converting theenergy unit from keV to ergs, one
gets
N = 163πNNdp(kTin)−(p+2)hc2(cid:18)101k05pc(cid:19)−2(cid:16)ekregVs(cid:17)−3 (B3)
or numerically
N ≈1.316×102× Np kTin −(p+2) (B4)
N (cid:18)1keV (cid:19)
d
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theauthor.
MNRAS000,1–8(0000)