Identification of QPO frequency of GRS 1915+105 as the relativistic dynamic frequency of a truncated accretion disk
aa r X i v : . [ a s t r o - ph . H E ] J a n Draft version January 22, 2020
Preprint typeset using L A TEX style emulateapj v. 01/23/15
IDENTIFICATION OF QPO FREQUENCY OF GRS 1915+105 AS THE RELATIVISTIC DYNAMICFREQUENCY OF A TRUNCATED ACCRETION DISK
Ranjeev Misra , Divya Rawat ∗ , J S Yadav , Pankaj Jain Inter-University Center for Astronomy and Astrophysics, Ganeshkhind, Pune 411007, India andDepartment of physics, IIT Kanpur, Kanpur, Uttar Pradesh 208016, India
Draft version January 22, 2020
ABSTRACTWe have analyzed
AstroSat observations of the galactic micro-quasar system GRS 1915+105, whenthe system exhibited C-type Quasi-periodic Oscillations (QPOs) in the frequency range of 3.4-5.4Hz. The broad band spectra (1-50 keV) obtained from simultaneous LAXPC and SXT can be welldescribed by a dominant relativistic truncated accretion disk along with thermal Comptonization andreflection. We find that while the QPO frequency depends on the inner radii with a large scatter,a much tighter correlation is obtained when both the inner radii and accretion rate of the disk aretaken into account. In fact, the frequency varies just as the dynamic frequency (i.e. the inverse of thesound crossing time) as predicted decades ago by the relativistic standard accretion disk theory for ablack hole with spin parameter of ∼ .
9. We show that this identification has been possible due tothe simultaneous broad band spectral coverage with temporal information as obtained from
AstroSat . Keywords: accretion, accretion disks — black hole physics — X-rays: binaries — X-rays: individual:GRS 1915+105 INTRODUCTIONFor a test particle orbiting a black hole, there arethree characteristic frequencies depending on the radius(Stella et al. 1999a,b). The first is the Keplerian fre-quency which is the inverse of the time period of the or-bit. There is the periastron precession frequency whichis the Keplerian frequency minus the Epicyclic one andrelates to how an orbit will precess in General Relativ-ity. There is also the Lense-Thirring precession frequencywhich is related to the wobbling of the orbit out of theplane which arises only in General Relativity when theblack hole is spinning.Apart from these three relativistic test particle fre-quencies, there are two other frequencies related tothe two characteristic speeds in an accretion disk, thesound ( c s ( r )) and the radial inflow speeds ( v r ( r )) where r is the radial distance. The dynamical frequency isthe inverse of the sound crossing time i.e. f dyn ∼ c s ( r ) /r . In the standard thin relativistic accretion disk(Shakura & Sunyaev 1973; Novikov & Thorne 1973) thesound speed is c s ( r ) = h ( r )( GM/r ) / A − B C − / D / E / (1)where M is the mass of the black hole. The relativisticterms A , B , C , D , E are functions of r and the black holespin patrameter, a . They asymptotically tend to unityin the Newtonian limit i.e. when r tends to infinity. Thescale height in the inner regions of the disk is given by h ( r, a ) ∼ cm ˙ M A B − C / D − E − L (2)where ˙ M is the accretion rate in units of 10 grams/second. The relativistic term L is a function of r and a and arises due to the relativistic phenomenon ofthe existence of a last stable orbit at which the disk flowno longer depends on viscosity and the height vanishes. [email protected] Thus f dyn ˙ M = N r/r g ) − . ( M/ . M ⊙ ) − × A B − D − . E − . L (3)where r g = GM/c is the Gravitational radius and themass of the black hole, M has been scaled by 12 . M ⊙ ,which is reported black hole mass for the source GRS1915+105 (Reid et al. 2014). N is a factor of order unityto incorporate the assumptions made in the standard ac-cretion disk theory especially in the radiative transferequation. It should be emphasised that A,B,D,E and Lare functions of radii and are important for small radii, r < r g . Thus the functional form of f dyn significantlydeviates from its Newtonian dependence of ∝ r − . inthis regime. Note that f dyn does not depend on theunknown turbulent viscosity parameter α of the stan-dard disk theory in contrast to the viscous time-scale τ visc ∼ r/v r , where v r is the radial inflow velocity of thedisk. τ visc is an order of magnitude higher than the dy-namical time-scale and depends inversely on both α andaccretion rate squared.X-ray binaries show variability on a wide range of time-scales which include broad band noise and nearly pe-riodic oscillations termed as Quasi-periodic Oscillations(QPOs) (van der Klis 2005). For systems harbouringblack holes, the QPO frequency ranges from milli-Hertzto hundreds of Hertz prompting classification in very low(milli-Hz), low ( Hz) and high frequency QPOs ( ∼ Misra et al. the phenomenon should also be varying. An attractivecandidate for this radius is the truncation or inner ra-dius of a standard disk beyond which there is a hotinner flow (Shapiro et al. 1976; Narayan & McClintock2008). Since the characteristic time-scales depend onGeneral Relativistic corrections, identification of a QPOfrequency with one, opens the exciting possibility of test-ing the theory in the strong field regime.However, as discussed below it has proved to be dif-ficult to make reliable and independent estimate of theinner disk radius. Indirect schemes have been employedto identify the QPO frequencies. For example, taking ad-vantage of the different radial dependencies of the char-acteristic frequencies, correlation between frequencies ofdifferent QPOs or breaks in the broad band noise havebeen used to identify the QPO frequencies (Psaltis et al.1999; Belloni et al. 2002; Stella et al. 1999a,b). Thismethod depends on the relatively rare detection of morethan two QPOs at the same time (Motta et al. 2014).Another method has been to use the correlation of theQPO frequency with some other features, such as thehigh energy spectral index, as proxy for a character-istic radius (Titarchuk & Osherovich 1999). However,since the dependence of the high energy spectral indexwith radius is model dependent and sensitive to assump-tions of the unknown viscosity, the best one can ob-tain are empirical scaling relations, which have proveduseful to compare between different black hole systems(Titarchuk & Fiorito 2004).The inner radius of an accretion disk can be measuredby fitting the spectra of these sources with a truncatedaccretion disk model (Muno et al. 1999; Sobczak et al.2000). Till recently, the detection of QPOs in black holesystems have been done by the Proportional Counter Ar-ray (PCA) on board the Rossi X-ray Timing Experiment(RXTE) observatory. Since the radius has to be mea-sured strictly simultaneously with the QPO, the spectralanalysis needed to be restricted to data obtained fromRXTE. However, the PCA had a relatively poor spectralresolution and its effective energy range was from 3 to20 keV, while the typical maximum colour temperatureof the disk is around 1 keV. Moreover, since the spectraldata was restricted in the energy range, simple modelshad to be used to fit the spectra. This limited energyrange led to severe systematic uncertainties in the innerdisk radius with some values being unphysically small.Moreover, the results were sometimes contradictory likeQPO frequency increasing with radius for one systemwhile decreasing for another (Sobczak et al. 2000). Nev-ertheless, correlations have been observed between thefrequency and the radius which have been used as evi-dence for some models, although there were large scatterin the estimated values (Mikles et al. 2009). A criticallimitation of these earlier works was that these analy-sis were not sensitive enough to test the variation of theQPO frequency with accretion rate, since that requiresbroadband data.The Large Area X-ray Proportional Counter (LAXPC)(Yadav et al. 2016; Agrawal et al. 2017) and the Soft X-ray Telescope (SXT) (Singh et al. 2016, 2017) on boardthe Indian Space Observatory AstroSat (Agrawal 2017)is ideally suited to study correlation between the QPOfrequency and the disc inner radius. The high time pre-cision and the large area of LAXPC provide timing and spectral information in the 4 - 50 keV band, the SXT pro-vides simultaneous spectral coverage in the low 1.0 - 5.0keV band. As reported by Rawat et al. (2019), AstroSatobserved the black hole system GRS 1915+105 from 28 th March 2017 18:03:19 till 29 th March 2017 19:54:07 whenthe source transited from a relatively steady state called χ class, through an intermediate state (IMS), to a flaringstate (heartbeat state(HS)) where large amplitude oscil-lations are seen. All through the observation, the sourceexhibited C-type QPOs in the frequency range 3.4-5.4Hz.In this work, we examine the spectral evolution of thesource during this observation and supplement the re-sults with observations made two days later on 1 st April2017, when the source shows both χ and ’heartbeat state’with QPOs in the same frequency range. It was fortu-itous that the source was undergoing a transition andshowed a QPO all through enabling us to study the spec-tral properties of the source and correlate them with thevarying QPO frequency. DATA ANALYSIS2.1.
Timing Analysis
Rawat et al. (2019) split the 28 th March 2017 obser-vation into several segments and presented the timingproperties for each of them. From the Power DensitySpectra (PDS) they obtained the QPO frequencies forsegments corresponding to the χ , intermediate and heart-beat states. Following Rawat et al. (2019) we do thesame analysis for the AstroSat
LAXPC observation ofGRS 1915+105 during 1 st April 2017 00:03:21 till 1 st April 14:38:01. The data was analyzed for the two unitsLAXPC 10 and LAXPC 20, using the LaxpcSoft . Likethe earlier observation, the source exhibited χ and heart-beat states which were divided into 6 segments (2 for χ and 4 for the heartbeat state). Lightcurves for represen-tative segments for a χ and heartbeat state are shownin top panel of Figure 1 and the corresponding PDS areshown in the bottom panel. The PDS were fitted us-ing lorentzian functions and the QPO frequency witherror was estimated using the same technique given inRawat et al. (2019). Thus combining the two observa-tions we have a total of 16 segments ( 5 for χ , 3 forintermediate and 8 for the heartbeat states) for whichthe QPO frequency has been estimated and tabulated inthe first column of Table 1.2.2. Spectral Analysis
For each of the 16 segments, simultaneous spectral datawas obtained from LAXPC 10, 20 and SXT. The LAXPCspectra, background and response files were generatedusing the LaxpcSoft . For SXT data reduction recent arfand rmf files are used, details of which are given at As-troSat website . The SXT spectra were extracted from asource region of 12 arcmins and the standard backgroundspectra were used for all spectra.The SXT (energy range 1.0 - 5.0 keV) and LAXPC 10and 20 ( energy range 4 - 50 keV) spectra of each dataset was analyzed together using the X-ray spectra fittingsoftware XSPEC (Arnaud 1996) using its inbuilt mod-els. During the spectral fitting gain variation for SXT http://astrosat-ssc.iucaa.in/?q=laxpcData http://astrosat-ssc.iucaa.in/?q=sxtData dentification of QPO frequencies of GRS 1915+105 R a t e ( c oun t s / s e c ) Time (sec) st April 2017 χ state R a t e ( c oun t s / s e c ) Time (sec) st April 2017Heartbeat state −4 −3 P ( f ) * f st April 2017QPO~4.05 Hz χ Frequency (Hz) 10 −4 −3 P ( f ) * f st April 2017QPO~5.13 Hz χ Frequency (Hz)
Figure 1.
Top panel shows the 2.0 sec binned 1000 sec lightcurves of χ class and heartbeat state. The corresponding PDS in 0.2-20.0 Hzrange are shown in bottom panels. LAXPC10 & LAXPC20 are used for lightcurve and PDS extraction here. −4 −3 k e V ( P ho t on s c m − s − k e V − ) th March 2017 χ class
102 5 20−202 χ Energy (keV) 10 −4 −3 k e V ( P ho t on s c m − s − k e V − ) th March 2017IMS
102 5 20−202 χ Energy (keV) 10 −4 −3 k e V ( P ho t on s c m − s − k e V − ) th March 2017HS
102 5 20−202 χ Energy (keV)
Figure 2.
Spectral fitting including residuals are shown for χ state, Intermediate state and heartbeat state. was taken into account by using the gain fit command inXSPEC. The offset value obtained ranged from 1.4 to 2.4eV. Additional systematic error of 3% was included. Totake into account possible uncertainties in the effectivearea of the instruments a variable constant was includedto the LAXPC 10 and 20 spectra relative to SXT, whosevalues ranged from 0.81 to 0.92.The spectra were fitted using the relativistic diskmodel, “kerrd” (Ebisawa et al. 2003), and the convo-lution model “simpl” (Steiner et al. 2009) to take intoaccount the Comptonization of the disk photons in theinner flow. The accretion rate and the inner radius ofthe disk were estimated from the best fit values obtainedfrom the “kerrd” model. The mass of the black hole,distance to the source and inclination angle of the disk were taken to be 12 . M ⊙ , 8 . o (Reid et al.2014) respectively. The colour factor was fixed to 1 . .
98 (Blum et al. 2009), as the spectralfitting was found to be insensitive to its value. For thekerrdisk the emissivity index for both the inner andouter parts of the disk was fixed at 1.8 (Blum et al.2009). The rest frame energy of the iron line was fixedat 6 . Misra et al. an appropriate factor of 1.235, since for “kerrd”, theradius is measured in r g , while for the kerrdisk it is inunits of the radius of marginal stability. Absorptionby intervening matter was modelled using “tbabs”(Wilms et al. 2000) with a column density fixed at4 × cm − (Blum et al. 2009). Representative spectrafor a χ , intermediate and heartbeat state are shown infigure 2. Note that for the spectra of heartbeat state theiron line component is insignificant. Table 1 lists thebest fit values of the parameters which are the accretionrate, inner disk radius, the fraction scattered into theComptonizing medium, the index of the Comptonizedspectrum and flux in the line emission. RESULTSThe upper left panel of Figure 3 shows the variationof the QPO frequency with radius where a broad anti-correlation is visible, however, it is difficult to quantifythe dependence because of the significant scatter. In-deed, the scatter suggests that the QPO frequency de-pends not only on the inner radius but also on someother parameter. The upper right and bottom panels ofFigure 3 show the variation of the frequency with theaccretion rate and the accretion rate with inner radii ,where again there seems to be a correlation but with alarge scatter. However, if one considers the frequency todepend both on the radius and the accretion rate andin particular if it is of the form ∝ ˙ M F ( R in ) then thecorrelation is significantly better. This is illustrated inleft panel of Figure 4 where the QPO frequency dividedby the accretion rate is plotted against the inner radius.More pertinently the variation is the same as predictedby the standard accretion model for the dynamic fre-quency (Equation 3) represented by lines for differentvalues of the spin parameter a . Note that the predictedfunctional form depends only on a and the normalizationfactor N which should be of order unity. While a formalfit gives a = 0 . ± . a = .
91 and a = .
99 to illustratethe constraints the data imposes on a.It is interesting to note that the best fit value ofthe black hole spin parameter obtained here is a ∼ . ± .
002 which is consistent with a = 0 . ± . . ± .
002 and N = 0 . ± . χ ∼ DISCUSSION AND SUMMARYThe result obtained in this paper relies on the accuracyof some measured and theoretically estimated quantities.Future improvement on the estimate of these would refinethe fitting presented here and can provide a robust valueof the spin parameter. These include the uncertaintiesin (a) the estimated distance to the source, mass of theblack hole and the inclination angle used; (b) the effec-tive area and response of the LAXPC and SXT detectors,and (c) the theoretically estimated colour factor, espe-cially since this was done for a non-spinning black hole(Shimura & Takahara 1995). Note that most of theseuncertainties are independent of each other and will giverise to a secular shift in the radii and accretion rate.The analysis has been done by fixing the neutral col-umn density value, nH at 4 × cm − as obtained byBlum et al 2009 using Suzaku data. If we instead al-low it to vary its value ranges from 3 . × cm − to4 × cm − for different orbits with a typical error of0 . × cm − . Since we expect the column density notto vary during the course of the observation we have usedthe value obtained by Blum et al. (2009). If instead weuse the average value obtained from the present observa-tion, i.e. we fix it to 3 . × cm − , we get qualitativelysimilar results, with the best fit values for the spin pa-rameter and normalization to be a = 0 . ± .
002 andN = 0 . ± . r ∼ r g ) the radial functional form of Equation3, is approximately 1 /r instead of 1/ r . due to its de-pendence on the relativistic terms, A,B,D,E and L. Thismeans that the QPO frequency is roughly proportionalto ˙ M /R in , which in turn is proportional to the disk flux.Thus, an approximate dependence of the QPO frequencyon total flux (if the disk component dominates) is ex-pected in this scenario. Moreover, the spectral indexof the Comptonization component may also depends on dentification of QPO frequencies of GRS 1915+105 Table 1
Spectral Parameters for GRS 1915+105 in 1.0-50.0 keV energy range
Exposure time State QPO frequency Accretion rate Inner radius Fraction scatter Gamma Flux in line emission χ / Dof(sec) (Hz) 10 gm s − (R g ) 10 − photons cm − s − th March χ class 3 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . χ class 3 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . χ class 3 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . IMS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . IMS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . IMS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . – 419.1/4101209 HS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . – 519.6/4371211 HS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . – 492.7/4361213 HS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . – 539.4/4391216 HS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . – 490.5/4371 st April χ class 4 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . χ class 4 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . HS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . – 438.1/4071175 HS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . – 519.5/4391260 HS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . – 458.9/439762 HS . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . – 459.4/411 Note . — Here, IMS and HS stand for intermediate and Heartbeat states respectively. Q P O f r e qu e n cy ( H z ) Radius (r g ) χ classIMSHS 3.5 4 4.5 5 5.5 0.6 0.65 0.7 0.75 0.8 0.85 Q P O f r e qu e n cy ( H z ) Mdot (10 g/s) χ classIMSHS 0.6 0.65 0.7 0.75 0.8 0.85 2.5 3 3.5 4 4.5 5 5.5 6 M do t ( g / s ) Radius (r g ) χ classIMSHS Figure 3.
The variation of QPO frequencies with inner disk radii and Accretion rate are shown in upper left and upper right panelsrespectively. The bottom panel shows the variation of the accretion rate with inner disk radii.
Misra et al. Q P O f r e qu e n cy ( H z ) / M do t ( g / s ) Radius (r g ) χ classIMSHSa=0.97, N=0.22a=0.91, N=0.35a=0.99, N=0.17 R a d i u s ( R g ) Radius (R g ) using model (simpl*kerrd) with SXT(simpl*diskbb) with SXT(simpl*diskbb) without SXT(powerlaw+diskbb) with SXTY=X Figure 4.
Left panel shows variation of QPO frequency divided by the accretion rate with inner disk radii. The lines represent the f dyn ˙ M as predicted by the relativistic standard accretion disk model (Equation 3) for dimensionless spin parameter a = 0 . ± .
002 (best fitwith N = 0 . ± .
01 and reduced χ ∼ a = 0 .
91 ( N = 0 .
35) and a = 0 .
99 ( N = 0 . dentification of QPO frequencies of GRS 1915+105 AstroSat to study the spectral properties of GRS1915+105 with the QPO frequency. We find that the fre-quency depends on the accretion rate and inner radiusof the disk, just as it was predicted for the dynamicalfrequency of a relativistic accretion disk. Thus, we iden-tify the QPO frequency as the inverse of the sound crosstime from the inner disk radius where strong GeneralRelativistic effects dominate. ACKNOWLEDGMENTWe thank the referee for constructive comments. Thisresearch has used the data of AstroSat mission of theIndian Space Research Organisation (ISRO), archivedat the Indian Space Science Data Centre. The authorswould like to acknowledge the support from the LAXPCPayload Operation Center (POC) and SXT POC at theTIFR, Mumbai for providing support in data reduction.REFERENCES
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