# The Spin of the Near-Extreme Kerr Black Hole GRS 1915+105

**ABSTRACT** Based on a spectral analysis of the X-ray continuum that employs a fully

relativistic accretion-disk model, we conclude that the compact primary of the

binary X-ray source GRS 1915+105 is a rapidly-rotating Kerr black hole. We find

a lower limit on the dimensionless spin parameter of a* greater than 0.98. Our

result is robust in the sense that it is independent of the details of the data

analysis and insensitive to the uncertainties in the mass and distance of the

black hole. Furthermore, our accretion-disk model includes an advanced

treatment of spectral hardening. Our data selection relies on a rigorous and

quantitative definition of the thermal state of black hole binaries, which we

used to screen all of the available RXTE and ASCA data for the thermal state of

GRS 1915+105. In addition, we focus on those data for which the accretion disk

luminosity is less than 30% of the Eddington luminosity. We argue that these

low-luminosity data are most appropriate for the thin alpha-disk model that we

employ. We assume that there is zero torque at the inner edge of the disk, as

is likely when the disk is thin, although we show that the presence of a

significant torque does not affect our results. Our model and the model of the

relativistic jets observed for this source constrain the distance and black

hole mass and could thus be tested by determining a VLBA parallax distance and

improving the measurement of the mass function. Finally, we comment on the

significance of our results for relativistic-jet and core-collapse models, and

for the detection of gravitational waves.

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**ABSTRACT:**Context: Change of sign of the LNRF-velocity gradient has been found for accretion discs orbiting rapidly rotating Kerr black holes with spin a>0.9953 for Keplerian discs and a>0.99979 for marginally stable thick discs. Such a "humpy" LNRF-velocity profiles occur just above the marginally stable circular geodesic of the black hole spacetimes. Aims: Aschenbach (2004) has identified the maximal rate of change of the orbital velocity within the "humpy" profile with a locally defined critical frequency of disc oscillations, but it has been done in a coordinate-dependent form that should be corrected. Methods: We define the critical "humpy" frequency νh in general relativistic, coordinate independent form, and relate the frequency defined in the LNRF to the distant observers. At radius of its definition, the resulting "humpy" frequency νh is compared to the radial νr and vertical νv epicyclic frequencies and the orbital frequency of the discs. We focus our attention to Keplerian thin discs and perfect-fluid slender tori where the approximation of oscillations with epicyclic frequencies is acceptable. Results: In the case of Keplerian discs, we show that the epicyclic resonance radii r3{:1} and r4{:1} (with ν_v{:}ν_r=3{:}1, 4{:}1) are located in vicinity of the "humpy" radius rh where efficient triggering of oscillations with frequencies νh could be expected. Asymptotically (for 1-a<10-4) the ratio of the epicyclic and Keplerian frequencies and the humpy frequency is nearly constant, i.e., almost independent of a, being for the radial epicyclic frequency ν_r{:}νh ˜ 3{:}2. In the case of thick discs, the situation is more complex due to dependence on distribution of the specific angular momentum ℓ determining the disc properties. For ℓ=const. tori and 1-a<10-6 the frequency ratios of the humpy frequency and the orbital and epicyclic frequencies are again nearly constant and independent of both a and ℓ being for the radial epicyclic frequency ν_r{:}νh close to 4. In the limiting case of very slender tori (ℓ˜ℓms) the epicyclic resonance radius r4{:1}˜ rh for all the relevant interval of 1-a<2× 10-4. Conclusions: .The hypothetical "humpy" oscillations could be related to the QPO resonant phenomena between the epicyclic oscillations in both the thin discs and marginally stable tori giving interesting predictions that have to be compared with QPO observations in nearly extreme Kerr black hole candidate systems. Generally, more than two observable oscillations are predicted.Astronomy and Astrophysics 03/2007; 463(3):807-816. · 5.08 Impact Factor - SourceAvailable from: Shahar Hadar[Show abstract] [Hide abstract]

**ABSTRACT:**Massive objects orbiting a near-extreme Kerr black hole quickly plunge into the horizon after passing the innermost stable circular orbit. The plunge trajectory is shown to be related by a conformal map to a circular orbit. Conformal symmetry of the near-horizon region is then used to compute the gravitational radiation produced during the plunge phase.03/2014; - SourceAvailable from: Henry Cao[Show abstract] [Hide abstract]

**ABSTRACT:**Everything from the smallest particle to the grand universe is constructed by Torque Grids. The grand structure of the universe is made up of infinite hierarchical Torque Grids; this theory falsifies Big Bang Theory (BBT) and Black Hole Theory. A Torque Grid is 10 -25 times smaller than an atom, and our universal Torque Grid size is 4.98 * 10 26 m. The Universe is timeless. The configuration of Spiral Arm Galaxy can also be explained by Unified Field Theory.International Journal of Physics. 12/2013; 1(1):162-170.

Page 1

arXiv:astro-ph/0606076v2 16 Aug 2006

The Spin of the Near-Extreme Kerr Black Hole GRS 1915+105

Jeffrey E. McClintock1, Rebecca Shafee2, Ramesh Narayan1, Ronald A. Remillard3,

Shane W. Davis4, Li-Xin Li5

ABSTRACT

Based on a spectral analysis of the X-ray continuum that employs a fully

relativistic accretion-disk model, we conclude that the compact primary of the

binary X-ray source GRS 1915+105 is a rapidly-rotating Kerr black hole. We

find a lower limit on the dimensionless spin parameter of a∗> 0.98. Our result is

robust in the sense that it is independent of the details of the data analysis and

insensitive to the uncertainties in the mass and distance of the black hole. Fur-

thermore, our accretion-disk model includes an advanced treatment of spectral

hardening. Our data selection relies on a rigorous and quantitative definition of

the thermal state of black hole binaries, which we used to screen all of the avail-

able RXTE and ASCA data for the thermal state of GRS 1915+105. In addition,

we focus on those data for which the accretion disk luminosity is less than 30% of

the Eddington luminosity. We argue that these low-luminosity data are most ap-

propriate for the thin α-disk model that we employ. We assume that there is zero

torque at the inner edge of the disk, as is likely when the disk is thin, although

we show that the presence of a significant torque does not affect our results. Our

model and the model of the relativistic jets observed for this source constrain the

distance and black hole mass and could thus be tested by determining a VLBA

parallax distance and improving the measurement of the mass function. Finally,

we comment on the significance of our results for relativistic-jet and core-collapse

models, and for the detection of gravitational waves.

Subject headings: X-ray: stars — accretion, accretion disks — black hole physics

— stars: individual (GRS 1915+105)

1Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138

2Harvard University, Department of Physics, 17 Oxford Street, Cambridge, MA 02138

3Kavli Center for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge,

MA 02139

4Department of Physics, University of California, Santa Barbara, CA 93106

5Max-Planck-Institut f¨ ur Astrophysik, Karl-Schwarzschild-Str. 1, Postfach 1317, 85741 Garching, Ger-

many

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1.Introduction

GRS 1915+105 has unique and striking properties that sharply distinguish it from the

40 known binaries that are believed to contain a stellar-mass black hole (Remillard & Mc-

Clintock 2006, hereafter RM06). It is the most reliable source of highly relativistic radio

jets in the Galaxy (Mirabel & Rodr´iguez 1994; Fender et al. 1999; Miller-Jones et al. 2006),

and it is the prototype of the microquasars (Mirabel & Rodr´iguez 1999). GRS 1915+105

(hereafter GRS1915) frequently displays extraordinary X-ray variability that is not mimicked

by any other black hole system (e.g., Belloni et al. 2000; Klein-Wolt et al. 2002). Its black

hole (BH) primary is unique in displaying a constellation of high-frequency QPOs (HFQ-

POs), namely, 41 Hz, 67 Hz, 113 Hz and 166 Hz. The 67 Hz QPO is atypically coherent

(Q ≡ ν/∆ν ∼ 20) and relatively strong (rms > 1%) compared to the HFQPOs observed

for six other accreting BHs (Morgan et al. 1997; McClintock & Remillard 2006, hereafter

MR06). Among the 17 transient and ephemeral systems that contain a dynamically con-

firmed BH (RM06), GRS1915 is unique in having remained active for more than a decade

since its discovery during outburst in 1992 (MR06). GRS1915 has an orbital period of 33.5

days and is the widest of the BH binaries (BHBs), and it likely contains the most massive

stellar BH (Greiner et al. 2001; Harlaftis & Greiner 2004; RM06).

Zhang et al. (1997) first argued that the relativistic jets and extraordinary X-ray behav-

ior of GRS1915 are due to the high spin of its BH primary. In their approximate analysis,

they found that both GRS1915 and GRO J1655–40 had high spins, a∗> 0.9 (a∗= cJ/GM2,

where M and J are the mass and angular momentum of the BH; a∗= 0 for a Schwarzschild

hole and a∗= 1 for an extreme Kerr hole). Subsequently, Gierlin´ ski et al. (2001) estimated

the spin of GRO J1655–40 and LMC X–3. Recently, we have firmly established the method-

ology pioneered by Zhang et al. and Gierlin´ ski et al. by constructing relativistic accretion

disk models (Li et al. 2005; Davis et al. 2005) and by modeling in detail the effects of spectral

hardening (Davis et al. 2005, 2006). We have made these analysis tools publicly available via

XSPEC (kerrbb and bhspec; Arnaud 1996). Using this modern methodology, spins have now

been estimated for several stellar-mass BHs, most notably: GRO J1655–40 and 4U 1543–47

(Shafee et al. 2006, hereafter S06), GRS1915 (Middleton et al. 2006), and LMC X-3 (Davis

et al. 2006).

All of the plausibly reliable estimates of BH spin to date, including the present work,

depend on fits to the X-ray continuum and measurements of the X-ray luminosity, coupled

with optical measurements of BH mass, orbital inclination, and distance (e.g., S06). In this

paper, we show that GRS1915 does indeed harbor a rapidly-spinning Kerr BH as suggested

by Zhang et al. (1997). However, in the case of GRO J1655–40 the results obtained by

ourselves and others show that the spin of this BH is modest (a∗∼ 0.75; S06; Gierlin´ ski et

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al. 2001) and much lower than the value (a∗∼ 0.93) suggested by Zhang et al. The high

spin reported herein for GRS1915 contradicts the modest spin value (a∗∼ 0.7) reported by

Middleton et al. (2006), and we discuss this inconsistency in detail in §5.3.

Our spin estimates are based on an analysis of the “thermal state” of BHBs (MR06)

whose remarkably simple properties have been recognized for decades. Basic principles of

physics predict that accreting BHs should radiate thermal emission from the inner accretion

disk, and a multi-temperature model of a thin accretion disk was published shortly after

the launch of Uhuru (Pringle & Rees 1972; Shakura & Sunyaev 1973; Novikov & Thorne

1973; Lynden-Bell & Pringle 1974). A nonrelativistic approximation to this model, now

referred to as diskbb in XSPEC (Arnaud 1996) was first implemented and used extensively

by Mitsuda et al. (1984) and Makishima et al. (1986). The two parameters of the model are

the temperature Tinand radius Rinof the inner edge of the accretion disk. In their review

on BHBs, Tanaka and Lewin (1995) show for a few BHBs (see their Fig. 3.14) that as the

thermal disk flux varies by 1–2 orders of magnitude the value of Rinremains constant to

within ? 20%. This striking result prompted Tanaka & Lewin to comment that Rin, which

was typically found to be ∼ tens of kilometers, must be related to the radius of the innermost

stable circular orbit (RISCO). The stability of Rinhas by now been observed in great detail

for many BHBs (e.g., Ebisawa et al. 1994; Sobczak et al. 1999; Sobczak et al. 2000; Park

et al. 2004). Further strong evidence for a thermal disk interpretation is provided by plots

of the observed disk flux versus apparent temperature, which track the expected L ∝ T4

relation for a constant inner disk radius (Gierlin´ ski & Done 2004; Kubota & Done 2004).

Spin can be determined because it has a profound impact on the behavior and properties

of a BH. Quantitatively and specifically, consider two BHs with the same mass M, one a

Schwarzschild hole and the other an extreme Kerr hole. For the Kerr hole, the radius of

the ISCO is six times smaller and the binding energy at the ISCO seven times greater than

for the Schwarzschild hole. Relative to the spinless BH, the much deeper gravity well of

the extreme Kerr hole hardens the X-ray spectrum and greatly increases its efficiency for

converting accreted rest mass into radiant energy. The continuum fitting approach that we

use is based on measuring spectral shape (hardness) and luminosity (efficiency).

This paper is organized as follows. In §§2–4 we discuss respectively the selection, re-

duction, and analysis of the data. In §5 we present our results for GRS1915 and compare

them with those of Middleton et al. (2006), and we present a table summary of the spins

of GRS1915 and three other BHs. The discussion topics in §6 include a description of our

methodology, our rationale for favoring low-luminosity data, the natal origin of BH spin, the

significance of measuring BH spin, and a proposed test of our model. In §7 we offer our

conclusions.

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2.Data Selection

Our primary resource is the huge and growing archive of data on GRS1915 that has been

obtained during the past decade using the large-area PCA detector on board the Rossi X-ray

Timing Explorer (RXTE; Swank 1998). Many BHBs have by now been observed hundreds

of times, but none has been observed more often than GRS1915. The net inventory of

RXTE pointed observations on this source from 1996 to the present now totals 4.7 Ms,

which corresponds to 1311 pointed observations each of duration 1–10 ks.

The unique properties of GRS1915 and the great volume of perplexing data present a

serious challenge: Is it possible to identify extended periods of time when GRS1915 was in a

genuine pacific state dominated by thermal emission, and can one use these data to obtain

a reliable estimate of spin? We believe we have answered “yes” to this challenge by using

a quantitative definition of the “thermal state” that is based on our exhaustive studies of

many BHBs and BH candidates (MR06; RM06). For a discussion of BH states, see MR06

and RM06, and for precise definitions of the three outburst states – including the thermal

state – see Table 2 in RM06. For complete overviews on the evolution and energetics of BH

states for six canonical BHBs (i.e., excluding GRS1915), see §5 in RM06.

In the thermal state (formerly high/soft state and “thermal dominant” state; MR06),

which is the only state relevant to this work, the flux is dominated by blackbody-like emission

from the inner accretion disk, QPOs are absent or very weak, and the rms variability is also

weak. Quantitatively, the thermal state is defined by two timing criteria and one spectral

criterion applied over the energy band 2–20 keV (MR06; RM06): (1) QPOs are absent or

very weak: amplitude < 0.005%; (2) the power continuum level integrated over 0.1–10 Hz is

< 0.075 rms; and (3) the fraction of the emission contributed by the accretion disk component

fDexceeds 75% of the total emission.

We now turn to describing how we screened the RXTE data archive for GRS1915 and

identified 20 observations as belonging strictly to the thermal state. As a starting point, all

of these individual PCA observations of GRS1915 that were publicly available as of 2005

January 1 were organized into 640 data segments, where we sometimes combined brief ob-

servations that occurred within an interval of several hours. We then screened for temporal

variability, and 338 relatively “steady” observations were identified for which the rms fluc-

tuations in the count rate divided by the mean count rate was < 16% using 1-s time bins.

Next, a hardness ratio (HR = 8.6 − 18.0 keV / 5.0 − 8.6 keV) was computed for each of

these 338 observations using the scheme of Muno et al. (2001) to normalize the PCA count

rates for several epochs with different PCA gain settings. We then selected a gross sample

of 85 observations that displayed the softest spectra (HR < 0.30). At this point, we strictly

applied the three criteria, which define the thermal state. Applying the timing criteria (1)

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and (2) stated above (i.e., QPO amplitude < 0.005% and rms continuum power < 0.075

rms), left us with 47 candidate observations. Finally, based on a decomposition of the spec-

trum into thermal and nonthermal components, which is described in the following section,

we obtained our sample of 20 observations that additionally meets criterion (3) given above,

namely, that the thermal disk component contributes fD> 75% of the total 2–20 keV flux.

It is this final sample of 20 strictly thermal-state observations that is the focus of this work.

A catalog of the 20 RXTE observations, which span a time interval of 7.5 years, is given in

Table 1.

Finally, we screened the 11 archival observations of GRS1915 obtained by the Advanced

Satellite for Cosmology and Astrophysics (ASCA) and identified two appropriate thermal-

state observations. These two observations, which were made on 1994 September 27 and 1999

April 15, are also cataloged in Table 1. In selecting these data we only applied the spectral

criterion (number 3) mentioned above and applied it only over the observed bandpass of

1.2-10 keV. The limited count rates (Table 1) did not allow us to exercise the two timing

criteria. Because of these limitations, we are somewhat less certain that these observations

correspond to the true thermal state than is the case for the RXTE observations.

3.Data Reduction

In our spectral analysis of the RXTE data, we only include pulse-height spectra from

PCU-2 because it is almost always operating and because fits to the simple power-law (PL)

spectrum of the Crab show that this is the best calibrated proportional counter unit (PCU).

Data reduction tools from HEASOFT version 5.2 were used to screen the event files and

spectra. Data were taken in the “Standard 2 mode,” which provides coverage of the PCA

bandpass every 16 s. Data from all Xe gas layers of PCU-2 were added to make the spectra.

Background spectra were obtained using the tool pcabackest and the latest “bright source”

background model. Background spectra were subtracted from the total spectra using the

tool mathpha. Redistribution matrix files and ancillary response files were freshly generated

individually for each PCU layer and combined into a single response file using the tool pcarsp.

In fitting each of the 20 pulse-height spectra (§4), we used response files that were targeted

to the time of each GRS1915 observation.

It is well known that fits to PCA spectra of the Crab Nebula reveal residuals as large

as 1%, and we therefore added the customary systematic error of 1% to all PCU energy

channels using the tool grppha (e.g., Sobczak et al. 2000). Because large fit-residuals are

often found below 3 keV, which cannot be accounted for by any plausible spectral feature,

and because the spectrum becomes background-dominated and the calibration less certain

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above 25 keV, we restricted our spectral analysis to the 3–25 keV band, which is customary

for analysis of PCA spectra obtained after the gain change of 1999 March. We used this

same 3–25 keV band even for the 14 PCA data sets that were obtained prior to 1999 March.

Table 1. Observations of GRS 1915+105

Mission

(Detector)

Obs.

No.

Date (UT)a

(yymmdd)

MJDExposure

(s)

Count Rateb

(counts s−1)

ASCA

(GIS2)

RXTE

(PCU2)

1

2

1

2

3c

4c

5

6

7

8

9

10

11

12

13

14c

15

16

17c

18

19

20c

940927

990415

960605

960607

960703

960703

970819

970819

970819

970819

971111

971209

971211

980220

980220

980329

991014

011024

030101

031029

031103

031124

48988.1

51283.9

50239.5

50241.4

50267.4

50267.5

50679.2

50679.3

50679.4

50679.5

50763.2

50791.2

50793.4

50804.9

50864.9

50901.7

51465.6

52206.6

52640.4

52941.6

52946.6

52967.5

6019

7153

10768

10960

3424

2944

2176

2608

3328

1488

10432

4544

2368

5472

1520

2768

5824

4384

3184

4128

2496

4240

98.3

143.3

2197.5

2356.1

1402.1

1367.0

4953.9

4686.1

5117.6

4941.5

4532.1

5181.7

4284.8

5426.6

2726.2

1282.3

4976.6

4285.4

1704.0

4445.6

4594.9

1675.1

aStart time of observation. MJD = JD – 2,400,000.5.

bPCA (full bandwidth): counts s−1per PCU; 1 Crab = 2500 cts s−1per PCU.

cKey low-luminosity observations (see §4.2.1 & §6.1).

All PCA count rates for the 20 pulse-height spectra were corrected for dead time. For all

normal events (i.e., good events, rejected events and events in the propane layer) we adopted

a dead time of τN = 8.83 µs, and for Very Long Events we adopted τVLE1 = 59 µs for

setting = 1 and τVLE1= 138 µs for setting = 2. The true event rate corrected for dead time

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divided by the observed rate is then ≡ Rcorr/R = 1.0−(RN×τN+RVLEi×τVLEi), where the

index i refers to the VLE setting for a given observation. The dead time corrections ranged

from 1.016 to 1.080.

As in S06, we again found it necessary to correct the effective area of the PCA despite

a recent official correction (Jahoda et al. 2006), which was made using a nominal and ap-

proximate spectrum of the Crab nebula (Zombeck et al. 1990). We have chosen to correct

our 3–25 keV fluxes to the most definitive Crab spectrum available, namely, the PL index

(Γ = 2.10 ± 0.03) and normalization (A = 9.7 ± 1.0 ph cm−2s−1) given by Toor & Seward

(1974) and the hydrogen column given by Willingale et al. (2001), which implies a 3.0–25.0

keV flux of 2.64 × 10−8erg s−1cm2. We consider the old Toor and Seward results more

reliable than the current but preliminary results that are summarized in Kirsch et al. (2005).

We made these corrections to the effective area as follows: We selected 25 Crab ob-

servations distributed over the 7.5 years spanned by the 20 RXTE observations. The Crab

pulse-height spectra were corrected for dead time and joined with their response files in

the same manner as described above for the GRS1915 spectra. The Crab spectra were fit-

ted over the range 3–25 keV using a simple PL model with the hydrogen column fixed at

NH= 3.45×1021cm−2(Willingale et al. 2001), and the energy flux was computed over this

same interval. The fluxes so computed systematically exceeded the Crab flux quoted above

by the factor 1.091±0.013 (rms). Therefore, the fluxes we obtained from the analysis of the

20 spectra (§4) were all corrected downward by the reciprocal factor 0.917.

For the two ASCA spectra (Table 1), we analyzed only the data from the GIS2 detec-

tor; the calibration of the GIS3 detector, in particular its gain correction, is less certain. We

ignored the data from the SIS detectors because GRS1915 is bright and the pileup effects

are troublesome (Kotani et al. 2000), which makes the SIS data less suitable for fitting the

broad continuum spectrum that is of interest here. Starting with the unscreened ASCA data

files obtained from the HEASARC, we followed as closely as possible the data reduction

procedures and criteria mentioned in Kotani et al. (2000). The GIS events for each detector

were summed within a radius of 6′centered on the source position, and the response function

of the X-ray telescope (Serlemitsos et al. 1995) was applied. Background was not subtracted

for this bright source. A gain correction based on the instrumental gold M-edge was ap-

plied. A systematic error of 2% was added to each energy channel to account for calibration

uncertainties, and the standard dead time corrections were applied. No correction to the

effective area is required because the GIS effective-area calibrations were based on the Toor

& Seward (1974) spectrum of the Crab (Makishima et al. 1996).

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4.Data Analysis

All of the data analysis and model fitting was performed using HEASOFT version 5.2

and XSPEC version 12.2 (Arnaud 1996) except for the model bhspec (see below), which

requires XSPEC version 11.3. We first consider the most conventional analysis of all 22

data sets (i.e., 20 RXTE plus two ASCA) using the simple multi-temperature disk black-

body model diskbb and then describe three successive analyses of these data sets using our

relativistic disk model.

In all the RXTE spectral fits described herein, we fixed the value of the hydrogen column

density at NH= 4.0 × 1022cm−2. This value is consistent with the values determined from

an analysis of the ASCA GIS data for GRS1915 by Ebisawa et al. (1998), who found that

NHwas “always within the range 3.5 − 4.1 × 1022cm−2,” and by ourselves for observations

#1 and #2, respectively: NH= (3.30±0.04)×1022cm−2and NH= (3.75±0.04)×1022cm−2

(§4.1). Our adopted value of NHis also in reasonable agreement with the BeppoSAX value

determined by Feroci et al. (1999), NH∼ 5.6 × 1022cm−2, and with radio and millimeter

determinations of the interstellar column, NH= (3.5 ± 0.3) × 1022cm−2(Chapuis & Corbel

2004).

In the following subsections, we discuss in detail the analysis of the RXTE data over

the energy range 3–25 keV. All of these fits required a nonthermal “tail” component of

emission plus two additional weak line and edge components, which are described below.

On the other hand, the ASCA GIS pulse-height spectra, which were analyzed over the

energy range 1.2–8 keV required neither a tail component nor the edge components. Apart

from these simplifications, the only difference between the analysis of the ASCA data and

the RXTE data is that in the former case we allowed NH to vary freely. Because of the

restricted bandpass of ASCA and the limitations associated with screening these data (§2),

we consider the ASCA results somewhat less reliable than the RXTE results, although in

the case of GRO J1655–40 we found good agreement between the two, most notably in the

case of one simultaneous observation (S06).

4.1.Nonrelativistic Disk Blackbody plus Simple Power-law Model

A basic, conventional model consisting of only three principal components, namely,

a multi-temperature disk blackbody (diskbb), a simple PL model (power), and interstellar

absorption (phabs) with NHfixed at 4.0×1022(MR06) consistently gave unacceptably poor

fits to the RXTE data. In the usual way, we added two additional components, a Gaussian

line gaussian and a broad Fe absorption edge (smedge; e.g., Ebisawa et al. 1994; Sobczak et

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al. 1999, 2000; Park et al. 2004; MR06). In applying the line component, we followed closely

the results obtained from high-resolution ASCA SIS observations of GRS1915. Specifically,

in a pair of GRS1915 SIS spectra, Kotani et al. (2000) found a complex of several, relatively-

narrow absorption features that extend from ∼ 6.4−8.3 keV; for both spectra, the equivalent

width of the total complex is EW ≈ 0.13 keV. Accordingly, given the limited resolution of

the PCA (≈ 18% at 6 keV), we added to our basic model a broad absorption line with a fixed

width of 0.5 keV, which we bounded to lie between 6.3 keV and 7.5 keV. Then, by adding

an additional broad Fe absorption component (smedge) with an edge energy restricted to

the range 6.9–9.0 keV, we were able to obtain good fits to all 20 RXTE spectra. We note

that Kotani et al. also used a sharp absorption edge component in their model, and we used

such a feature in some cases (see §4.2.4).

Using the model described above, we obtained the values of the parameters and fluxes

plotted in Figure 1. There are a total of 8 fit parameters: The disk blackbody temperature

Tinand its normalization constant K, the PL index Γ and its normalization constant, the

smedge optical depth τSand the smedge edge energy ES, the central energy of the Gaussian

absorption line EFeand the intensity of the line NFe. All the fit parameters, except for the

PL normalization parameter, are shown in Figure 1. Also shown is the equivalent width EW

of the Gaussian line, the 2–20 keV disk and PL fluxes (FDand FPL, respectively), and the

ratio of these fluxes fD, which is a key quantity used in the selection of these thermal-state

data (§2).

Finally, we briefly summarize our ASCA GIS2 fit results. For observation #1 (Table 1),

we find kTin= 1.66±0.03 keV, K = 126.1±9.3 and χ2

we find kTin= 1.91±0.03 keV, K = 137.8±8.1 and χ2

for both observations are quoted above. A Gaussian absorption line with a central energy of

6.85 ± 0.04 keV and an equivalent width of 0.11 keV was included in the fit to observation

#1, but was not required or included for observation #2. Neither a smedge component nor

a PL or other tail component of emission was included in these 1.2-8.0 keV fits.

ν= 1.08 for 98 dof. For observation #2

ν= 0.94 for 159 dof. The values of NH

4.2.Relativistic Analysis

As in S06, we estimate a∗ by fitting the thermal component of the X-ray continuum

using a fully relativistic model of a thin accretion disk around a Kerr BH (Li et al. 2005).

The model, which is available in XSPEC under the name kerrbb, includes all relativistic

effects, such as frame dragging, Doppler boosting, gravitational redshift, and light bending.

It also includes self-irradiation of the disk (“returning radiation”) and the effects of limb

darkening. A limitation of kerrbb is that one of its three key fit parameters, namely, the

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spectral hardening factor f that relates the color temperature T and the effective temperature

Teff of the disk emission (f = T/Teff ; Shimura & Takahara 1995; Merloni et al. 2000) is

treated as a constant.

Because of this limitation of kerrbb our work is based on a second, complementary

relativistic disk model called bhspec, which has also been implemented in XSPEC (Davis et

al. 2005, hereafter D05; Davis et al. 2006, hereafter D06). It does not include the effects

of returning radiation, but it provides state-of-the-art capability for computing the spectral

hardening factor f. The code bhspec is based on non-LTE atmosphere models within an α-

viscosity prescription (D05; Shakura & Sunyaev 1973), has just two principal fit parameters

(spin and mass accretion rate), and can be used directly to fit for a∗ (D06). As we now

describe, our approach is to combine the functionalities of bhspec and kerrb into a single

code that we call kerrbb2.

The use of this hybrid code kerrbb2 marks an important difference in methodology

between our earlier work (S06) and the present one. As discussed in S06, kerrbb has three fit

parameters — a∗, f and the mass accretion rate˙M — only two of which can be determined at

one time. In S06, we fitted for f and˙M with a∗fixed, and we also computed the Eddington-

scaled luminosity, l ≡ L/LEdd[LEdd= 1.3×1038M erg s−1and L = L(a∗,˙M), e.g., Shapiro

& Teukolsky 1984] . We then plotted f versus l and graphically compared the fit results to

a model calculation of f versus l performed using bhspec. Finally, by varying the assumed

value of a∗, we determined our estimate of the spin parameter. In the present work, this

procedure has been streamlined using kerrbb2, which we now describe.

The code kerrbb2 is a modified version of kerrbb that contains a pair of look-up tables

for f corresponding to two values of the viscosity parameter: α = 0.01, 0.1. The entries in

the tables were computed using bhspec. The two tables give f versus l for a wide range of

the spin parameter, 0 < a∗< 0.9999. The computations of f versus l were done using the

appropriate, corresponding response matrices and energy ranges used in fitting the spectra

with kerrbb. Thus, kerrbb and the subroutine/table computed using bhspec now allow us

to directly fit for a∗ and l ≡ L/LEdd while retaining the special features of kerrbb (e.g.,

returning radiation). This hybrid code kerrbb2 is used exclusively in all of the data analysis

described herein.

In order to estimate the BH spin by fitting the broadband X-ray spectrum, one must

input known values of the mass M of the BH, the distance D to the binary, and the inclination

i of the black-hole spin axis, which for GRS1915 we take to be the inclination of the non-

precessing and stable jets (Fender et al. 1999; Dhawan et al. 2000b). For GRS1915, we adopt

the following values for these three parameters: M = 14.0 ± 4.4 M⊙(Harlaftis & Greiner

2004), D = 11.0 kpc and i = 66◦± 2◦with D < 11.2 ± 0.8 kpc (Fender et al. 1999). In this

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section we use the nominal values of these parameters, and in §5.2 we examine the effects

on a∗of allowing these parameters to vary.

In all of the relativistic model fits described below, we used precisely the same ancillary

components with the same constraints that we used in our nonrelativistic analysis (§4.1),

namely, the 0.5 keV-wide Gaussian absorption line and the broad absorption component

(smedge). Furthermore, for all of the results presented below, we switched on limb darkening

(lflag = 1) and returning radiation effects (rflag = 1). We set the torque at the inner boundary

of the accretion disk to zero, fixed the normalization to 1 (as appropriate when M, i, and D

are held fixed), allowed the mass accretion rate to vary freely, and fitted directly for the spin

parameter a∗. In the following subsections, we describe our analysis of the 20 RXTE and

two ASCA spectra using kerrbb2 in which we applied in turn three different models for the

tail component, namely, a simple PL model, a thermal Comptonization model, and a simple

PL model plus an exponential cutoff at lower energies.

4.2.1. Relativistic Disk plus Simple Power-law Model

We now consider our baseline analysis of the 20 RXTE PCA pulse-height spectra us-

ing our relativistic disk model kerrbb2 in conjunction with a simple power-law component

power. Following precisely the prescription we used in our nonrelativistic analysis (§4.1), we

added two additional components, a broad Fe absorption line with a fixed width of 0.5 keV

(Kotani et al. 2000) and a broad Fe absorption edge (e.g., Ebisawa et al. 1994). These two

conventional and incidental features, which are required in order to obtain a good fit, are

subject to exactly the same constraints as before (§4.1). As stated earlier, these fits were

done over the energy range 3–25 keV, and the column density was fixed to NH= 4.0×1022.

As before (§4.1), there are a total of 8 fit parameters, 6 of which are identical to those

described previously: the PL index Γ and its normalization constant, the smedge optical

depth τSand the smedge edge energy ES, and the central energy of the Gaussian absorption

line EFeand the intensity of the line NFe. Of course, the two principal fit parameters are now

a∗and˙M in place of the temperature and disk normalization constant, which are returned by

diskbb. The analysis was done for all 20 RXTE observations for both values of the viscosity

parameter.

The fit results are summarized in Figure 2 (α = 0.01) and Figure 3 (α = 0.1) in precisely

the same format used in displaying the diskbb results in Figure 1. That is, the structure of

these figures (e.g., the order of parameters and the ranges over which the parameters are

displayed) is identical to the structure of Figure 1, which summarizes the results of our

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nonrelativistic analysis (§4.1). There are two important differences to note between Figures

2 & 3 and Figure 1. First, the obvious difference is that a∗and˙M are now displayed in place

of Tinand K. Secondly, in Figures 2 and 3, the value of the disk fraction fDin the top panel

is in the range fD∼ 0.9−1.0. This is generally significantly greater than the corresponding

values of fDshown in Figure 1, which occasionally dip down to fD≈ 0.75. Thus kerrbb2 is

able to accommodate a larger fraction of the total flux than diskbb or, correspondingly, the

model for the tail component is less important when fitting with kerrbb2.

The data points for five of the observations in Figures 2 and 3 are enclosed by blue

circles. These are the five lowest-luminosity observations (L/LEdd< 0.3). They are critically

important for our determination of the spin of GRS1915, as we explain in §6.1 and the

Appendix. For four of these observations the values of chi-square are relatively high. As

we show in §4.2.4, the addition of a minor feature to the spectral model allows us to obtain

good fits (χ2

the two important parameters, a∗and ˙M.

ν≈ 1) to these four crucial spectra without significantly affecting the values of

Finally, we briefly summarize our ASCA GIS2 results for the case α = 0.01.

observation #1 (Table 1), we find a∗ = 0.988 ± 0.003,

NH = (3.39 ± 0.04) × 1021cm−2and χ2

a∗= 0.957 ± 0.005, ˙M= (3.66 ± 0.14) × 1018g s−1, NH= (3.99 ± 0.04) × 1021cm−2and

χ2

and an equivalent width of 0.21 keV was included in the fit to observation #1, but was not

required or included for observation #2. No PL or other tail component of emission was

included in these 1.2-8.0 keV fits.

For

˙M = (1.40 ± 0.08) × 1018g s−1,

ν= 1.25 for 95 dof. For observation #2 we find

ν= 0.82 for 159 dof. A Gaussian absorption line with a central energy of 6.77 ± 0.05 keV

4.2.2. Relativistic Disk plus Comptonization Model

In the analysis of the RXTE observations described above in §4.1 and §4.2.1, we found

that the PL component sometimes makes a modest contribution to the total flux at energies

below ∼ 5 keV. We question whether this contribution from the PL is physically reason-

able, since the PL is believed to be produced by Comptonization of the soft disk photons

by a scattering corona. In order to check if this PL flux affects our results, we next fitted

the tail component of emission using a more physically-motivated model for which the disk

component dominates more strongly below several keV. Namely, we used a thermal Comp-

tonization model (comptt) in place of the simple PL component (Titarchuk 1994; Hua &

Titarchuk 1995). A drawback of comptt is its complexity; it has four principal parameters:

the temperature of the soft input photons T0, the coronal plasma temperature Tcor, the

optical depth of the corona τC, and a normalization parameter.

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In determining the spin, we considered three fixed values of T0(§5.1) that are centered

on 2 keV, which is the nominal value of the disk temperature determined in §4.1. As we

show in §5.1, this choice is completely unimportant. We also considered two values of the

coronal temperature, Tcor= 30 keV and Tcor= 50 keV, and we found that this choice is also

unimportant. For the purposes of the discussion at hand, we adopt the values T0= 2.0 keV

and Tcor = 50 keV. Thus, we are left with two fit parameters, τC and the normalization

constant. When fitting with no constraints on τC, we found that the parameter sometimes

ran away to unphysically low values (? 0.01). We therefore set a hard lower bound on

the optical depth: τC> 0.4 (for Tcor= 50 keV). This bound is based on the values of the

photon index determined in §4.1 (Γ ? 4) and a simple calculation that relies on the Zeldovich

approximation as described in §7.5 of Rybicki & Lightman (1979). Finally, we set comptt’s

geometry switch to −1, thereby selecting disk geometry and interpolated values of the β

parameter. Our results for the fitting parameters and other quantities are summarized in

Figure 4 for α = 0.01 only. The structure of this figure is identical with that of Figure 2

except that Γ is replaced by τCand the PL flux FPLis replaced by the 2–20 keV flux in the

comptt component FC.

4.2.3.Relativistic Disk plus Cutoff Power-law Model

Modeling the tail component using the thermal Comptonization model is an effective

way to check on the effects of PL flux below ∼ 5 keV (§4.2.2). However, this model is quite

complex. Therefore, we now consider a simpler model that allows us to cut off the flux at

low energy in an ad hoc way, namely, a simple PL model (§4.2.1) that is cutoff at lower

energies by an exponential (expabs*power in XSPEC). This model has three parameters, the

two standard PL parameters (§4.2.1) plus a cutoff parameter Ec. In §5.1 we consider three

plausible choices for the cutoff energy (Ec = 8,10 & 12 keV), but for now we consider

only the central value, Ec= 10 keV. The fit results for this simple model are summarized

in Figure 5, which is strictly identical in structure to Figure 2. The results shown are for

α = 0.01.

4.2.4. Introduction of a Sharp Absorption Edge

Five values of chi-square in Figure 2 (observation nos. 3, 4, 12, 14 & 17) are rela-

tively high, χ2

dof). Furthermore, these same observations give similarly high values of chi-square for the

Comptonization model (Fig. 4) and the cutoff PL model (Fig. 5) as well. These particular

ν? 1.5, and the fit to observation no. 14 is unacceptably high, χ2

ν= 3.9 (44

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observations are important because four of them are low-luminosity observations (§4.2.1,

§6.1, Appendix). In an effort to improve the fits for these five observations, we followed

the lead of Kotani et al. (2000; §4.1). Specifically, we added to our spectral model a sharp

edge feature (edge in XSPEC), which we bounded to lie in the range 8–13 keV, and we then

refitted these five PHA spectra. The results are summarized in Figure 6, where the new

parameters and fluxes are plotted as red open circles and the small black data points have

been copied from Figure 2. Apart from the new fit results, Figure 6 differs from Figure 2 in

that it includes a pair of additional panels displaying the parameters of the edge component,

EEd and τEd. Note in Figure 6 that the optical depth of the edge component is modest,

τEd≈ 0.2, and that the addition of this feature significantly reduces the optical depth of the

smedge component. Figure 6 contains two important messages. First, with the addition of

the edge component all of the five fits are now good (χ2

and ˙M are scarcely affected by the inclusion of the sharp edge (see §5.5, Fig. 6). Finally, we

found that the sharp edge gave the same improvements in chi-square and the same degree

of stability in the values of a∗and ˙M as well when applied to the Comptonization (§4.2.2)

and cutoff PL models (§4.2.3).

ν≈ 1). Secondly, the values of a∗

4.3. Critique of the Different Analysis Approaches

The disk fraction fD, which is the ratio of the 2–20 keV thermal disk flux to the flux

in the tail component (PL, Compton, or cutoff PL) is an important parameter and it is

therefore displayed in the top panels in Figures 1–5. Note that the value of fDin Figure 1

never dips below 0.75 for any of the 20 observations, which is a principal selection criterion

that we used in selecting these data (§2) via the nonrelativistic analysis (§4.1). The typical

value is ≈ 90%, although for two observations fDdoes fall below 80%. In the case of the

relativistic analyses using the PL tail model, the values of fDare significantly higher with

typical values ? 95% and with few values below 90% (Figures 2, 3 and 5). The comptt

tail model consistently gives the highest values of fD, which approach 100%. In §4.2.2, we

expressed some reservations about the simple PL component’s contribution to the total flux

at low energies. However, as we show in §5, our results for the PL model agree well with the

results obtained for the other two tail models.

A careful comparison of Figures 1–5 shows that the Gaussian line parameters (EFe,

NFe), the line’s equivalent width (EW), and the smedge parameters (ES and τS) change

very little whether the disk is modeled with diskbb or with kerrbb2 and whether the model

for the tail component is a simple PL, a Comptonized plasma, or a cutoff PL. This strongly

indicates that these ancillary parameters, which are required in order to obtain a good fit,

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are quite unimportant. Furthermore, the Gaussian and smedge components are relatively

weak: the Gaussian line has an EW ≈ 0.2 keV, comparable to the ≈ 0.13 keV value reported

by Kotani et al. (2000), and the optical depth of the smedge component is moderate, τS∼ 2

(for comparison, see Ebisawa et al. 1994; Sobczak et al. 1999, 2000; Park et al. 2004).

Finally, if one considers the principal relativistic fit parameters – a∗and ˙M – plotted in

Figures 2–5, one sees that the corresponding values of these parameters from figure to figure

are little affected by the choice of model for the tail component (i.e., PL, Compton, or cutoff

PL) or by the inclusion of a sharp absorption edge (§4.2.4, Fig. 6). Thus, we conclude that

our results are robust to the details of the analysis – that is, they depend weakly on the line

and edge parameters, and they depend weakly as well on the choice of the model for the tail

component of emission.

5.Results

In this section, we present our results in the form of plots of the dimensionless spin

parameter a∗versus the dimensionless luminosity l ≡ L/LLedd. The Eddington-scaled lumi-

nosity l is computed from the two kerrbb2 fit parameters a∗and ˙M and the BH mass M

(§4.2). In this section we consider in turn the following topics: (1) Our results for the spin of

GRS1915; (2) the effects of varying M, i and D; (3) a comparison of our results with those

of Middleton et al. (2006); (4) the effects of returning radiation and torque; (5) a lower limit

on the spin parameter of a∗> 0.98; and (6) a comparison of this limit with the spins of three

other sources.

An important point should be mentioned at the outset. The model that we employ to

fit the continuum spectrum of GRS1915 is physically consistent only if (i) the accretion disk

is in an optically thick thermal state, and (ii) the disk is geometrically thin in the vertical

direction. Through the stringent data selection described earlier we have ensured the first

requirement, but the second criterion requires a further restriction of the data. In § 6.1 we

make use of a Newtonian analysis to estimate the disk thickness, and in the Appendix we

describe a fully relativistic analysis. Based on these two analyses, we show that the accretion

disk will be thin at all radii, with a height to radius ratio less than 0.1, only if the accretion

luminosity is less than 30% of the Eddington luminosity. Only five observations with RXTE

and one observation with ASCA satisfy this restriction, and we therefore focus most of our

attention on these particular data sets (though we present detailed results for all 22 sets).

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