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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
Page 12
– 12 –
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.
Page 13
– 13 –
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
Page 14
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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,
Page 15
– 15 –
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).
Page 16
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5.1. Spin versus Luminosity for GRS 1915+105
All the results given in this subsection assume the nominal values of the optically-
determined input parameters given in §4.2: M = 14.0 M⊙, i = 66◦, and D = 11.0 kpc
(see §4.2). In the following, we show the results of fitting for the spin parameter using
three different tail models in turn – simple PL, thermal Comptonization and cutoff PL – in
conjunction with our relativistic disk model kerrbb2.
Figure 7 summarizes our fit results (§4.2.1) obtained using our baseline PL tail model
(MR06; RM06). The spin parameter is shown plotted versus the Eddington-scaled luminosity
l. The results for all 20 RXTE and 2 ASCA observations (Table 1) are included in this figure.
The results are shown for two value of the viscosity parameter, α = 0.01 and α = 0.1. All
of the analyses reported herein were computed for both values of α ; however, for low
luminosities (l ? 0.3), which are strongly favored in this work (see §6.1 and the Appendix),
the spin estimates are quite insensitive to the value of α (e.g., Fig. 7), and we therefore
generally show results for only α = 0.01. Error bars are included in Figure 7, although they
are generally too small to be apparent.
The principal result of this paper is captured in the set of six lowest-luminosity data
points (5 RXTE and 1 ASCA) in Figure 7, namely that the spin-parameter estimate is very
nearly unity for l ? 0.3. For the group of four data points at intermediate luminosities,
0.3 ? l ? 0.45, the estimated value of the spin parameter is somewhat depressed, especially
for α = 0.1. At high luminosities, l ? 0.65, the spin estimate is severely depressed and seen
to decrease significantly with increasing l. As we discuss in §6.1 and the Appendix, there
are good reasons to focus only on those data that correspond to l < 0.3. We thus conclude
that GRS1915 has a spin parameter close to the maximal Kerr value of a∗= 1.
We now consider the effects of replacing the simple PL model for the tail component
of emission with a thermal Comptonization model, comptt. As explained in §4.2.2, we con-
sidered this model because we had reservations about the behavior of the simple PL model
at low energies. Additionally, a Comptonization model is more physically motivated and
offers a point of comparison with other studies of spin that exclusively use a supplementary
Comptonization model (e.g., Middleton et al. 2006, D06). The fitted parameters of this
model are displayed in Figure 8. As discussed in §4.2.2, in fitting the data using this com-
ponent, we fixed the thermal temperature of the soft seed photons at three different trial
temperatures – T0= 1.5, 2.0 and 2.5 keV – where the central value was determined from
our nonrelativistic analysis (§4.1; Fig. 1). Figure 8 shows a∗versus l for the three values of
T0where it is immediately obvious that the results obtained using comptt do not depend on
the temperature of the seed photons over the range considered. Error bars are suppressed,
but in all cases their extent is less than the height of the plotting symbols. The results,
Page 17
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which are shown for α = 0.01, can be seen to be nearly identical to the results obtained
using the PL component for α = 0.01 (Fig. 7). Again, for l ? 0.3 we find that a∗≈ 1 and
for intermediate luminosities the spin is slightly depressed (a∗≈ 0.98). As before, the spin
drops very significantly at high luminosities (l ? 0.65).
Next, we consider the results for the cutoff PL model which, like the thermal Comp-
tonization model, contributes negligibly to the flux at low energies. Relative to the Comp-
tonization model, its chief advantage is its greater simplicity, and its disadvantage is its lack
of physical motivation (§4.2). The results are summarized in Figure 9 for α = 0.01 and
for the three values of the break energy mentioned in §4.2.2. The error bars, which do not
exceed the size of the plotting symbols, are suppressed. As shown in Figure 9, the results
are essentially independent of the choice of cutoff energy. Furthermore, the results for the
cutoff PL model at both low and intermediate luminosities are nearly identical to the results
obtained with the thermal Comptonization model (Fig. 8) and with the simple PL model
for α = 0.01 (Fig. 7).
Finally, in Figure 10 we show superposed the results obtained using all three tail models
for α = 0.01. This figure clearly demonstrates the robustness of our principal result, namely,
that the very high spin of GRS1915 does not depend in any significant way on the model used
to resolve the relativistic disk component from the faint, adulterating non-disk component.
Furthermore, in §4.2 we have demonstrated that the minor fitting components, the Gaussian
line and the smedge, operate the same in all the fits and are therefore incidental to the results
that we have obtained for the spin parameter (Figs. 7–10).
5.2.Effects of Varying M, i and D on the Spin of GRS 1915+105
Under the assumption of an intrinsically symmetric jet ejection, Fender et al. (1999)
place an upper limit on the distance to GRS 1915+105 of D = 11.2 ± 0.8 kpc. Further,
Fender et al. treat as realistic only distances that are in the range 9–12 kpc, as indicated
by the entries in their Table 2. We follow their lead. As shown in their table, the kinematic
jet model associates with each distance a unique value of the jet inclination (e.g., adopting
the MERLIN values, D = 11 kpc corresponds to i = 66◦), which we take as the spin axis of
the BH and the accretion disk (§4.2). In turn, each value of i is associated with a definite
value of the BH mass via the dynamical results for GRS1915 (Greiner et al. 2001; Harlaftis
& Greiner 2004). Thus, we have a correlated triplet of numbers D, i and M, which are
given in Table 2 for five values of D. In Figure 11a, we show the effects of varying D from
11–12.5 kpc. The cases D = 9 − 10 kpc are not shown because these values drive a∗toward
higher values and we are interested here in highlighting the lowest values of a∗. Furthermore,
Page 18
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as discussed in §6.4, our fit results indicate that the distance to GRS1915 is unlikely to be
less than 9–10 kpc. In Figure 11a and Table 2, we also include D = 12.5 kpc because this
extreme distance was adopted by Middleton et al. (2006).
Table 2. Parameters for GRS 1915+105
Distancea
(kpc)
Inclinationa
(degrees)
Massb
(M⊙)
9.0
10.0
11.0
12.0
12.5c,d
61.5
63.9
66.0
67.8
68.6
15.5
14.6
14.0
13.5
13.3
aFender et al. 1999.
bBased on Porb, K2and M2from Harliftis & Greiner 2004.
cAdopted by Middleton et. al. 2006.
dIntrinsic jet velocity > c.
In addition to the uncertainty in the distance, the dynamically-determined value of the
BH mass carries its own sizable uncertainty, M = 14.0±4.4 M⊙, because the radial velocity
amplitude of the secondary is known only to a precision of 11% (Greiner et al. 2001; Harlaftis
& Greiner 2004). The effects on the spin due to this uncertainty in the mass are shown in
Figure 11b. As indicated in the figure, the smallest mass, M = 9.6, gives the lowest values of
spin. Limiting our consideration to L/LEdd< 0.3 (§6.1, Appendix), Figure 11ab shows that
for most allowable distances and masses the spin parameter is nearly unity (see §5.5). Finally,
on a separate and incidental matter, we note that our RXTE results are also insensitive to
our adopted value of NHover the full range indicated (§4) because the absorbing column is
already ∼ 85% transmitting at the PCA’s detector threshold energy of 3 keV.
5.3.Comparison with the Results of Middleton et al. for GRS 1915+105
For the nominal 11 kpc distance that we adopt (§5.2), Middleton et al. (2006) report a
single, moderate value of the spin parameter of a∗∼ 0.8 (or a∗∼ 0.7 for their adopted dis-
tance of 12.5 kpc). As we make clear in §5.5, the M06 value of a∗∼ 0.8 is very much less than
the value we find: a∗∼ 0.98 − 0.99. M06 and we used precisely the same publicly-available
relativistic accretion disk models (i.e., kerrbb and bhspec). The key difference between the
two studies is in the methodology of data selection. M06 used a quite different approach
Page 19
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that yielded a restricted data sample comprised solely of high-luminosity observations. As
we conclude below, our results are in fact in reasonable agreement with Middleton et al. in
this high-luminosity regime, which we argue is unreliable for the determination of spin (§6.1,
Appendix).
Both M06 and we agree completely on the necessity of selecting spectra that mini-
mize the nonthermal component and that are dominated by disk emission. However, M06’s
methodologies for selecting such spectra were quite different from ours (§2). One difference is
that M06 based their initial selection on the state classifications defined by Belloni et al. 2000
(see also Belloni et al. 1997, and Belloni 2004), which were devised primarily to study disk-jet
coupling via a unified model of X-ray states and radio jets. We, on the other hand, used
quantitative state definitions that are centered on physical models of X-ray states (MR05;
MR06). These latter state definitions have been applied more widely to many BH binaries
including GROJ 1655–40, 4U 1543–47, XTE J1550–564, H 1743–322, XTE J1859+226, and
GX 339–4 (MR05; MR06). A second difference is that we screened all the available data and
identified 22 observations that are strictly thermal-state data. This yielded a total of 89 ks
of RXTE data and 13 ks of ASCA data (Table 1) compared with the much smaller collection
of data considered by M06. It is this larger data sample that allowed us to identify several
crucial observations at low luminosities (L/LEdd< 0.3) that are completely absent in M06’s
data sample.
In the end, M06 fitted jointly three representative 16-s observations in order to determine
a single spin estimate with the nominal value of a∗= 0.82 (for D = 11 kpc). This single
spin value and the three corresponding luminosities are indicated in Figure 12 by the three
filled triangles, which are connected by a solid line. As shown, these three luminosities
range from L/LEdd = 0.40 − 1.45. Note that MR06’s low value of spin, a∗ ∼ 0.82, is in
reasonable accord with the value of ≈ 0.88 that we find (Fig. 12) for a luminosity of ≈ 80%
of LEdd using the comptt model, which is analogous to the thcomp tail model that M06
used. Furthermore, their somewhat lower spin value, which is an average over a wide range
of luminosity, may be largely due to the inclusion of an observation at super-Eddington
luminosity (L/LEdd≈ 1.45; Fig. 12). Note also that even their mid-luminosity observation
with L/LEdd≈ 0.85 (Fig. 12) corresponds to the effective Eddington luminosity for thin-disk
geometry (§6.1, Appendix). As already mentioned, and discussed in further detail in §6.1 and
the Appendix, the continuum spectral models used by M06 and us are not self-consistent
and become progressively less reliable at higher luminosities. In fact, all three 16-second
observations of M06 correspond to l > 0.3 and are thus in a regime where a number of the
physical assumptions which underly the spectral models are likely to break down.
Page 20
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5.4. Effects of Returning Radiation and Nonzero Torque at the ISCO
All of our results include the self-irradiation of the disk as a result of light deflection
(assuming that the disk is infinitely thin, see Li et al. 2005), which we refer to as returning
radiation. The effects on our results of turning off the returning radiation (rflag = 0) is
shown in Figure 13. As indicated in the figure, the returning radiation boosts the luminosity
of the disk by several percent, but has no significant effect on the spin parameter. The
returning radiation feature is not included in bhspec, the relativistic disk model we used
to compute tables of the spectral hardening factor, which were incorporated into kerrbb
via a subroutine to create kerrbb2 (§4.2). Both kerrbb2 and bhspec can be used directly to
determine the principal fit parameters a∗and˙M. We made a thorough comparison of the fit
results obtained using the two models for GRS1915 (and for 4U 1543–47 and GRO J1655–40
as well). For the purposes of this comparison only, we switched the returning radiation off
for kerrbb2 (rflag = 0). We found that the two models gave very comparable results for a∗
versus l.
Throughout this paper we assume that there is no torque acting at the inner edge of
the disk. This assumption is in agreement with the classic and current literature on thin-
disk accretion, which advocates the use of a zero-torque boundary condition (Shakura &
Sunyaev 1973; Novikov & Thorne 1973; Afshordi & Paczy´ nski 2003; Li 2003). However, as
discussed in §6.1, a torque may be present near the ISCO, especially in the case of thicker,
higher-luminosity disks. Our model kerrbb2 is quite general and capable of handling positive
torques of any magnitude with the dimensionless torque parameter ηTdefined as the ratio
of the power generated by the torque to the gravitational binding energy of the accreted gas
(Li et al. 2005). As illustrated in Figure 14, the spin parameter decreases with increasing
torque. In the presence of sizable torques, the spin parameter of GRS1915 is significantly
depressed at high luminosities, but it is scarcely affected at low luminosities.
5.5.Summary of Results for GRS 1915+105 and Three Additional Sources
In Table 3 we summarize the average values of the spin of GRS1915 returned by kerrbb2
for L/LEdd< 0.3 that are based on the nominal values of M, i and D (§4.2) and on α = 0.01.
The fit results are given for each of the three tail models (§4.2). The quantities displayed are
the Gaussian-weighted mean value of the spin a∗and the standard deviation for N = 5. As
indicated by comparing the two lines in the table, the inclusion of a sharp absorption edge
in the spectral model (§4.2.4) has a negligible effect on the value of the spin parameter.
Page 21
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Table 3: Fitted values of spin for L/LEdd< 0.3
Object Model
Comptt
mean
Power Law
mean
Cutoff Power Law
meanst. dev.st. dev.st. dev.
GRS 1915+105a
GRS 1915+105b
0.998
0.998
0.001
0.001
0.997
0.995
0.001
0.002
0.997
0.996
0.001
0.001
aSharp absorption edge excluded from the fit; see §4.2.4.
bSharp absorption edge included in the fit; see §4.2.4.
The formal and precise values of a∗in Table 3 for all three models (with and without the
edge) are consistent with the physical limit on the Kerr parameter of a∗= 0.998 computed
by Thorne (1974). We consider this agreement accidental given the likely uncertainties in the
idealized thin-disk model and the model for spectral hardening, the systematic uncertainties
in the data, and the uncertainties in M, i and D. Nevertheless, the results in Table 3 indicate
a very high value for the spin parameter of a∗? 0.99.
We now consider somewhat lower values of spin that cropped up during our analysis.
We restrict our discussion to L/LLedd < 0.3 (§6.1, Appendix). For example, for the first
ASCA observation (Table 1), we find a∗ = 0.988 ± 0.003 for nominal values of M, i and
D (§4.2.1). Figure 11a and 11b show respectively the effects of changing D and M on a∗.
Considering distance, the spin is lowest for D = 12.5 kpc: a∗ = 0.991 ± 0.006 (weighted
mean for three observations). Considering mass, the spin is lowest for M = 9.6 M⊙(single
observation with a∗= 0.987±0.001). Finally, the values of a∗are slightly less if one considers
the case α = 0.1 (Fig. 7). Based on these and other considerations and the results in Table 3,
we adopt a∗> 0.98 as a lower limit on the spin parameter.
This lower limit of 0.98 for the spin of GRS1915 and our previously estimated values of
a∗for GRO J1655–40 and 4U 1543–47 (S06) are summarized in Table 4. We obtained very
similar estimates of a∗for the latter two sources using our revised code kerrbb2 (§4.2), and
we will report on this work in a later paper. Also given in Table 4 is an upper limit on a∗
for LMC X-3 obtained by D06. Here we provide conservative estimates of spin, which are
based on the considerations given above for GRS1915 and the full range of variation of a∗
considered for GRO J1655–40 and 4U 1543–47 in R06. The spin estimates given in Table 4
are our bottom-line results.
Page 22
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Table 4. Spin estimates for four sources
Black HoleMissiona∗
GRS 1915+105
GRO J1655–40
4U 1543–47
LMC X–3
RXTE/ASCA
RXTE/ASCA
RXTE
RXTE/BeppoSAX
> 0.98
0.65-0.75
0.75-0.85
< 0.26a
aDavis et al. 2006.
It is important to emphasize that the spins of GRO J1655–40 and 4U 1543–47, although
sizable, are effectively very much less than that of GRS1915, which in turn is significantly
less than the theoretical maximum value of a∗= 1. The implications of the extreme spin of
GRS1915 are not immediately apparent if one considers the parameter a∗alone. Therefore it
is instructive to consider such related dimensionless parameters as the radius of the innermost
stable circular orbit (ISCO) ξ, the binding energy per unit mass at the ISCO η, and the
Keplerian frequency at the ISCO ωK, which are all monotonic functions of a∗(e.g., Shapiro
& Teukolsky 1984). These three quantities are defined and plotted versus a∗in Figure 15,
which also shows for the four BHs in question the values of these quantities for our nominal
estimates of spin. In this approximate comparison, note that both the nominal Keplerian
frequency and the binding energy at the ISCO for GRO J1655–40 and 4U 1543–47 are only
half the values indicated for GRS1915.
6. Discussion
There are four avenues for measuring spin – continuum fitting, high-frequency QPOs,
the Fe K line, and polarimetry (RM06). Because spin is such a critical parameter it is
important to attempt to measure it by as many of these methods as possible, as this will
provide arguably the best possible check on our results. The best current method, continuum
fitting, has the drawback that its application requires accurate estimates of BH mass M,
disk inclination i, and distance D. In contrast, observations of HFQPOs require knowledge
of only M to provide a spin estimate, and once the correct model is known this method
is likely to offer the most reliable measurements of spin. Presently, however, the leading
model of HFQPOs, which was initially proposed by Abramowicz & Kluz´ niak (2001), does
not provide a useful constraint on a∗ for GRS1915 because of the wide range of possible
resonances and the sizable uncertainty in the BH mass (T¨ or¨ ok et al. 2005). Another HFQPO
model, on the other hand, predicts the precise value a∗= 0.99616 for the spin of GRS1915
(Aschenbach 2004). Broadened iron lines do not even require M, although knowledge of i
Page 23
– 23 –
is useful in order to avoid having to include that parameter in the fit. However, there are
serious sources of uncertainty in the model, including the placement of the continuum, the
model of the fluorescing source, and the ionization state of the disk (Reynolds & Nowak
2003). Furthermore, in the case of GRS1915, the line is seldom seen and has provided poor
constraints on the models, and no estimate of spin has been given (Martocchia et al. 2002,
2004; Miller et al. 2004). Polarimetry appears promising because the polarization features
of BH disk radiation can be affected strongly by GR effects (Lightman & Shapiro 1976;
Connors et al. 1980; Dovˇ ciak et al. 2004). Unfortunately, however, there have been no such
observations of BHBs, and there are no mission opportunities on the horizon. In short, the
HFQPO and Fe-line methods are not well enough developed to provide dependable results,
and the required polarimetry data are not available, whereas the continuum method, despite
its limitations, is already delivering results.
The methodology of the continuum-fitting approach is straightforward and transpar-
ent. Its foundation is (1) the simplicity of the rigorously defined thermal state (§1 & §2),
which matches very closely the predictions of the classic thin disk models (§1), and (2) the
vast amount of X-ray spectral data contained in the NASA/GSFC HEASARC archives for
missions ranging from Ginga and RXTE to Chandra and XMM-Newton.
In the thin disk model, there is an axisymmetric radiatively-efficient accretion flow in
which, for a given BH mass M, mass accretion rate ˙M and BH spin parameter a∗, one can
calculate very accurately the total luminosity of the disk, Ldisk= η˙Mc2. The parameter η,
which measures the radiative efficiency of the disk, is a function only of a∗ (see Fig. 15).
We can also calculate precisely the local radiative flux Fdisk(R) emitted at radius R by each
surface of the disk. Moreover, the accreting gas is optically thick, and the emission is thermal
and blackbody-like, making it straightforward to compute the spectrum of the emission.
Most importantly, the inner edge of a thin disk is located very close to the innermost stable
circular orbit (ISCO) of the BH spacetime, whose radius RISCO(in gravitational units) is a
function only of the spin of the BH: RISCO/(GM/c2) = ξ(a∗), where ξ(a∗) is a monotonically
decreasing function of a∗(see Fig. 15). Thus, if one measures the radius of the disk inner
edge, and if one also has an estimate of the mass M of the BH, then one can immediately
obtain a∗. This is the principle behind our method of estimating BH spin, which was first
described by Zhang et al. (1997).
There is one principal difficulty in applying this method. At the high disk temperatures
typically found in BHB disks (Tin∼ 107K), the spectral hardening factor f (§4.2) is expected
to deviate substantially from unity. It is therefore important to have a reliable estimate of f.
Until recently, the only estimate available was that from Shimura & Takahara (1995), whose
seminal but limited study was rather approximate. Within the last year, D05 along with
Page 24
– 24 –
Davis & Hubeny (2006) have computed more accurate disk atmosphere models including
metal opacities and have obtained reliable estimates of f as a function of the disk luminosity
and inclination. The use of a rigorous and modern estimate of f is absolutely essential for the
successful application of this method of estimating BH spin, and it is only now that such an
estimate has become available. Nevertheless, even at the lower luminosities we favor (§6.1),
the vertical structure of real magnetohydrodynamical (MHD) disks may differ in detail from
our models (see §3 of D06 for details). However, preliminary investigations which incorporate
the results of MHD simulations suggest only small changes (∆f/f ? 15%).
6.1. Rationale for Reliance on Low-Luminosity Data
In S06, we argued that the method employed in that paper as well as the present paper to
estimate BH spin is most reliable at low disk luminosities. The argument has been amplified
by D06 (see their §3.1). The main reason to distrust high luminosity data is that the disk is
likely to be vertically thick, whereas the model explicitly assumes a thin disk. The detailed
general relativistic ray tracing used in kerrbb, kerrbb2 and bhspec assumes a razor-thin disk
whose surface is exactly at z = 0. So long as the disk thickness H is much less than the local
disk radius R, we expect only small errors to result from the idealized geometry assumed
in the model. However, as H/R increases we expect various geometrical effects to creep in.
Although it is hard to be quantitative, it is reasonable to think that the errors will become
non-negligible once H/R > 0.1.
Another important assumption in the models is that there is no torque applied at the
inner edge of the disk (§5.4). Krolik (1999) and Gammie (1999) argued that magnetic fields
would be amplified near the inner edge of the disk, where the gas begins to free-fall into the
BH, and that these fields would apply a torque on the disk. The torque will enhance the
energy dissipation near the ISCO and lead to a modification in the profile of the disk flux
F(R). If this effect is strong enough it will introduce a large error in the BH spin estimate.
Interestingly, Li (2004) finds that, in some cases, a strong magnetic field connecting the disk
and the BH may actually move the inner edge of the disk out and cause a reduction in the
luminosity. Afshordi & Paczy´ nski (2003) suggested that the torques are likely to scale as
some positive power of H/R and therefore will be unimportant in very thin disks (see their
Figs. 17 and 18). This topic is still under debate and is likely to be settled only with detailed
models.
To make progress on this question, one approach is to work with the viscous hydro-
dynamic disk equations, including pressure and radial dynamics (as in Narayan, Kato &
Honma 1997 and Afshordi & Paczy´ nski 2003), and to calculate the viscous stress at the
Page 25
– 25 –
sonic radius and the rate of viscous energy dissipation as a function of radius for various
disk thicknesses. Within the limitations of the α-viscosity prescription, this will provide a
clean answer to whether or not the torque at the inner edge is important for thin disks.
Our preliminary analysis appears to support Afshordi & Paczy´ nski’s (2003) assertion that
the torque is unimportant for thin disks. A more detailed, and ultimately more rigorous,
approach is to carry out 3D numerical MHD simulations of realistic thin disks, including
radiative cooling to keep the disk thin. The only work to date involves non-radiative thick
disks and is not yet very useful.
Regardless of the current uncertainty over the magnitude of the torque at the disk inner
edge, we note that at low luminosities (when the disk is thin, see below) the effects on the
spin parameter of even a sizable torque is quite small (see § 5.4).
Another effect that becomes important when the disk is vertically thick is radial advec-
tion of energy (Abramowicz et al. 1988, 1995; Narayan & Yi 1994, 1995). The more energy
advection there is in the disk, the less energy is radiated to infinity, and the larger is the
deviation of F(R) from the idealized thin disk profile assumed in the model. Thus, it is
safest to work with disks that have negligible radial advection, i.e., disks with H/R ≪ 1.
Let us now estimate the disk thickness H and the ratio H/R for a Newtonian thin
accretion disk. The flux emitted by a thin accretion disk around a BH with zero torque at
the inner edge is given by (e.g., Frank et al. 2002)
F(R) =3GM˙M
8πR3
?
1 −
?Rin
R
?1/2?
,(1)
where M is the mass of the BH,˙M is the mass accretion rate, R is the cylindrical radius, and
Rinis the radius of the inner edge of the disk. Let us define the Eddington mass accretion
rate by equating the disk luminosity to the Eddington luminosity,
GM˙MEdd
2Rin
= LEdd≡4πGMc
κ
,i.e.,
˙MEdd=8πcRin
κ
, (2)
where κ is the opacity of the gas. Correspondingly, let us define the Eddington-scaled mass
accretion rate by
˙ m ≡
˙M
˙MEdd
. (3)
We now rewrite the disk flux F(R) in terms of ˙ m and calculate the vertical acceleration due
to radiation pressure,
grad(R) =F(R)κ
c
=3GM ˙ m
R2
??Rin
R
?
−
?Rin
R
?3/2?
.(4)
Page 26
– 26 –
In equilibrium, the radiative acceleration must be balanced by the vertical component of
gravity, which for simplicity we can write as gz(R,z) = GMz/R3. We then find
H
R≈ 3 ˙ m
??Rin
R
?
−
?Rin
R
?3/2?
. (5)
Figure 16 shows H/R as a function of R for various choices of the accretion rate: from below,
˙ m = 0.1, 0.2, ..., 1.2. If we wish to have H/R < 0.1 at all R, then we see that we are limited
to ˙ m ? 0.25, i.e., to Eddington-scaled disk luminosities l ? 0.25. The Appendix describes
a more accurate estimate of the disk height that is calculated for a general relativistic disk
around a Kerr BH. Results are shown in Figure 17. According to that analysis, in order to
have H/R ? 0.1 at all radii, we must restrict our attention to luminosities l ? 0.3. This is
the limit we employ throughout the paper.
If we consider the exact expression for the Newtonian vertical gravity
gz(R,z) =
GMz
(R2+ z2)3/2, (6)
rather than the approximation GMz/R3, then we find that the maximum value of the vertical
gravitational acceleration (which is achieved at z = R/√2) is
(gz)max=
2
3√3
GM
R2. (7)
For any accretion rate greater than about 85% of Eddington, or log(L/LEdd) > −0.06, one
finds that some parts of the disk produce too much radiation to be balanced even by the
maximum vertical gravity (gz)max. Radiation pressure will then cause material to be blown
away from the disk. This critical luminosity is clearly related to the Eddington limit; the
slightly different numerical value, i.e., 85% instead of 100% of the canonical Eddington limit,
is the result of the different geometry of a disk compared to the spherical geometry that one
usually considers. (See Nityananda & Narayan 1982 for a discussion of geometry effects on
the Eddington limit.)
We showed in § 5 that, by focusing on data corresponding to L/LEdd? 0.3, we obtain
very consistent results for the spin parameter of GRS1915, independent of the details of the
spectral model we employ. We also found that the results begin to deviate as we go to higher
luminosities, suggesting that as the disk thickens one or more of the effects described in this
subsection becomes important. It is interesting that the deviations are not random but
very systematic, e.g., the estimate of a∗decreases smoothly and monotonically as L/LEdd
increases. This signature could conceivably be used to identify which of our assumptions
breaks down as the luminosity increases. Detailed viscous disk models with varying disk
thickness might be able to shed some light on this issue.
Page 27
– 27 –
6.2. The Spins of Stellar Black Holes are Chiefly Natal
King and Kolb (1999) provide a global evolutionary and observational argument that
neither significant spinup nor spindown is likely to occur during the lifetime of any BH
binary and hence that BH primaries essentially retain the spin rates that they had at birth.
In the particular case of 4U 1543–47, based on its present accretion rate and modest age
(? 1 Gyr), we argued that the spin of its BH (§5.7) is likewise chiefly natal (S06). The
fast spin reported herein for GRS1915, a∗> 0.98, is almost certainly a natal spin because
the alternative, achieving this spin gradually via accretion torques, would require almost
doubling the mass of the BH (see below). Such a large increment in BH mass is unlikely to
have occurred during the evolution of GRS1915 or any BH binary simply because systems
with initially low- or moderate-mass secondaries (i.e., M ? a few M⊙) obviously cannot
supply the required mass, and systems with high-mass secondaries have lifetimes that are
too short to effect the required mass transfer. We now consider the exceptional case of
GRS1915 in more detail.
GRS1915 presently has a low-mass secondary, M2= 0.81±0.53 M⊙(Harliftis & Greiner
2004) and the most massive primary and longest period of any BH binary (§1). There is a
great deal of uncertainty in evolutionary models for GRS1915 and for all BH binaries. The
specific evolutionary model of Belczynski & Bulik (2002) for GRS1915 argues for a small
transfer of mass to the primary and negligible spin up, which is in agreement with most
generic models (e.g., King & Kolb 1999). An evolutionary model of GRS1915 that links this
source to the ultraluminous X-ray sources implies the most extreme mass transfer and spin
up (Podsiadlowski et al. 2003). These authors argue that the initial secondary mass could
have been as high as 6 M⊙and the BH primary could have accreted as much as ∼ 4 M⊙(see
also Lee et al. 2002). Even for this extreme scenario, the predicted spin up due to accretion
torques is modest. Based on a precise calculation that ignores returning radiation (§5.5), we
find that the transfer of 4 M⊙onto a 10 M⊙natal black hole with zero initial spin yields a final
spin of only ∼ 0.77, which is far less than our limit of a∗> 0.98 (§5.5, Fig. 15). Likewise, to
achieve a final spin of a∗> 0.98 would require an initial spin of a∗> 0.75. Furthermore, if one
includes the effects of returning radiation, then the accretion is less efficient in spinning up
the hole and a somewhat larger natal spin is required. Again neglecting returning radiation,
a 10 M⊙BH that is spun up by accretion torques from a∗= 0 to a∗= 0.98 would have a
final mass of 19.3 M⊙; since some of the rest mass energy is radiated away, the total rest
mass accreted in such a spin up event would be 10.7 M⊙. We thus conclude that the extreme
spin of GRS1915 was likely imparted to the BH primary during the process of its formation.
The generation of large spins is central to GRB models. Natal spins of a∗ ∼ 0.8, in
agreement with our observations, were predicted for GRO J1655–40 and 4U 1543–47 by
Page 28
– 28 –
Lee et al. (2002). The extreme spin of GRS1915, a∗> 0.98, is an expected consequence of
collapsar models (§6.3).
6.3.Significance of Measuring Black Hole Spin
The properties of a BH are completely defined by specifying just two parameters, its
mass M and its dimensionless spin parameter a∗. Furthermore, a BH’s mass simply sup-
plies a physical scale, whereas its spin fundamentally changes the geometry of space-time.
Accordingly, in order to model the ways in which an accreting BH can interact with its envi-
ronment, one must know its spin. For example, consider one of the most intriguing unsolved
problems in astrophysics, namely, the connection between BH spin and relativistic jets that
are commonly observed for both supermassive and stellar-mass BHs and that are so promi-
nent in the case of GRS1915 (e.g., Mirabel & Rodr´iguez 1999). For many years, scientists
have speculated that these jets are powered by BH spin via a Penrose-like process associated
with magnetic fields (e.g., Blandford & Znajek 1977; Hawley & Balbus 2002; Meier 2003;
McKinney & Gammie 2004). However, these ideas will remain mere speculation until suf-
ficient data on BH spins have been amassed and the models can be tested and confirmed.
This provides strong motivation for measuring the spins of accreting BHs.
The strong evidence for natal spins – particularly in the case of GRS1915(§6.2) – is
obviously of major significance in building core-collapse models for SN and GRBs (Woosley
1993; MacFadyen & Woosley 1999; Woosley & Heger 2006). For example, one of the greatest
uncertainties in GRB modeling is whether one can arrive at the core collapse stage with
sufficient angular momentum to make a disk around a BH. The spins of GRO J1655–40 and
4U 1543–47 – and especially GRS1915 – provide strong evidence for the high natal rotation
rates of BHs and thus provide strong support for the collapsar model of “long-soft” GRBs.
The continuing development of gravitational wave astronomy is central to the explo-
ration of BHs, and knowledge of BH spin is fundamentally important to this effort. To
detect the faint coalescence signal for two inspiralling BHs, one must compute the expected
waveform and use it to filter the data. Our spin estimates for GRO J1655-40 and 4U 1543–47
(R06) motivated the first such waveform computation that includes the effects of spin (Cam-
panelli et al. 2006), and our results reported here for GRS1915 present a further challenge
to the waveform modelers.
Page 29
– 29 –
6.4. An Observational Test of the Spin and Jet Models for GRS 1915+105
As mentioned in §5.2, our fit results for the five low-luminosity observations (l < 0.3;
see §4.2.1, Table 1) indicate that the distance to GRS1915 is unlikely to be less than about
9–10 kpc. This result is based on the abrupt and dramatic rise in χ2that occurs for lesser
distances. For the nominal 14.0M⊙value of BH mass and D = 11.0, 10.5, 10.0, 9.5 and
9.0 kpc, the respective values of χ2
13.3, 43.4 (obs. no. 3); 0.5, 1.4, 2.7, 14.2, 43.8 (obs. no. 4); 0.6, 2.0, 6.3, 23.1, 263.6; (obs.
no. 14); 1.0, 1.1, 5.6, 26.8, 77.1 (obs. no. 17); and 0.7, 0.7, 1.4, 12.1, 45.4 (obs. no. 20). This
abrupt rise in χ2indicates that we have reached the limit of our table model (a∗= 0.9999)
and that the fit is demanding unphysical values of a∗ > 1. In Figure 18a, this distance
lower limit, which is a function of BH mass, is indicated by the long slant line labeled “spin
model.” For each assumed value of mass, and hence inclination and distance (see §5.2), the
limiting value plotted in Figure 18a is an average result for the five low-luminosity points at
a 99% level of confidence (∆χ2= 6.6). (The results are very insensitive to the binary mass
ratio, which we have held fixed at its nominal observed value; Harlaftis & Greiner 2004.) To
the right of the vertical line labeled “jet model,” the intrinsic velocity of the radio jet exceeds
the velocity of light (§5.2; Fender et al. 1999). The region below the nearly horizontal line is
disallowed by the jet model and the 1-σ lower limit on the mass function (§5.2; Greiner et
al. 2001). Thus, taken together, the spin and jet models plus the value of the mass function
predict that the distance and BH mass of GRS1915 lie within the triangular region shown
in the figure.
νfor each low-luminosity observation are 0.6, 0.6, 2.6,
Six model-dependent estimates of the distance to GRS1915 are summarized in Figure
18b. Some estimates disagree, others are very uncertain, and none provides a convincing test
of the constraints summarized in Figure 18a. We believe that it should be possible to obtain
a model-independent VLBA parallax distance that is precise to ∼ 10% and to reduce the
uncertainty by a factor of two in the radial velocity amplitude K of the secondary, which
would significantly improve the accuracy of the mass function. Such improvements in the
observational constraints will provide a powerful test of the spin and jet models for GRS1915.
7. Conclusions
Using a rigorous and quantitative definition of the thermal state of a black hole binary
(§2), we screened all the available RXTE PCA and ASCA GIS data and identified a total of 22
observations of GRS 1915+105 that are free of QPOs and strong timing noise and for which
the thermal disk component of emission contributes > 75 % of the total 2–20 keV flux. We
then fitted the 22 disk-dominated spectra using principally a model of a thin accretion disk in
Page 30
– 30 –
the Kerr metric that includes all relativistic effects plus an advanced treatment of the spectral
hardening factor f (§4.2). The spectral fitting of the 22 spectra was repeated a number of
times using three different models for the nonthermal tail component of emission and two
different values of the viscosity parameter (§4). The results for the key relativistic parameters
– the spin a∗and the mass accretion rate ˙M – were shown to be quite independent of any
details of the analysis and insensitive to the uncertainties in the independently-determined
input parameters, namely, the mass, inclination and distance of the black hole (§5).
On theoretical grounds, we argue that the spin parameter can be determined most re-
liably at lower luminosities (§6.1, Appendix). Our relativistic disk model assumes a disk
that is thin and torque-free at its inner edge. Higher luminosities are problematic because
they likely lead to disk thickening and nonzero torques near the ISCO. Based on theoret-
ical arguments, we propose a limit on the disk thickness and a corresponding limit on the
disk luminosity, L/LEdd < 0.3, below which one can obtain reliable estimates of the spin
parameter. Adopting this criterion, we obtain our principal conclusion: GRS 1915+105 is
a rapidly-rotating BH with a lower limit on its spin parameter of a∗ > 0.98. Finally, we
propose an observational test of our spin model.
We thank Keith Arnaud for help in implementing models in XSPEC and the following
people for helpful discussions and encouragement: Stan Woosley, Alexandar Heger, Gerry
Brown, Vicky Kalogera, Cole Miller, and Paul Gorenstein. We also thank an anonymous
referee for helpful comments and a thorough reading of our paper. This research has made
use of data obtained from the High Energy Astrophysics Science Archive Research Center
(HEASARC), provided by NASA’s Goddard Space Flight Center. This work was supported
in part by NASA grant NNG 05GB31G and NSF grant AST 0307433.
A. Vertical Thickness of a Thin Accretion Disk around a Kerr Black Hole
Following Page & Thorne (1974), we define the following functions for later use,
A = 1 + a2
C = 1 − 3x−2+ 2a∗x−3,
∗x−4+ 2a2
∗x−6,B = 1 + a∗x−3,
D = 1 − 2x−2+ a2
(A1)
∗x−4, (A2)
where a∗ ≡ a/cRg is the dimensionless spin of the black hole BH, Rg ≡ GM/c2is the
gravitational radius of the BH of mass M, and x ≡ (R/Rg)1/2. Note, D vanishes on the
horizon of the BH.
On the equatorial plane of the BH, the lapse function and the angular velocity of frame
Page 31
– 31 –
dragging are
χ =
?D
A
?1/2
,ω =2a∗R2
gc
R3A
. (A3)
The angular velocity of a thin Keplerian disk at radius R is
ΩD=
?GM
R3
?1/21
B, (A4)
and the rotational 3-velocity of the disk relative to the locally nonrotating frame is
vφ=A1/2
χ
(ΩD− ω)R . (A5)
The 4-velocity of the disk particle is then
Ua=Γ
χ
??∂
∂t
?a
+ ΩD
?∂
∂φ
?a?
, (A6)
where Γ =
normalization condition UaUa= −1.
The relative acceleration between two neighboring particles moving on geodesics with a
small separation vector Xais given by the geodesic deviation equation (Wald 1984),
?1 − v2
φ/c2?−1/2= B/C1/2is the Lorentz factor. The 4-velocity satisfies the
ga= −R
a
cbdXbUcUd, (A7)
where R
The acceleration is measured in the rest frame of the particles. For a particle above the
equatorial plane at a small height z and corotating with the disk, we have Xa= zea
ea
zis a normalized unit vector orthogonal to the equatorial plane. Combining this with the
Uagiven in equation (A6) and the Riemann tensor of the Kerr spacetime, we can calculate
the relative acceleration,
a
cbdis the Riemann tensor of the spacetime and Uais the four-velocity of the geodesic.
z, where
ga= −gzea
z,gz= ξGMz
R3
,(A8)
where1
ξ =1
C
?1 − 4a∗x−3+ 3a2
∗x−4?
.(A9)
1Our result differs from eq. (5.7.2) of Novikov & Thorne (1973). After intensive examination, we believe
that their formula is incorrect.
Page 32
– 32 –
For a disk that is radiation-dominated (at least at the photosphere), the equilibrium in
the vertical direction is determined by
Fκ
c
≈ gz|z=H, (A10)
where F = F(R) is the radiation flux density of the disk (measured by an observer corotating
with the disk) and κ is the disk opacity. The flux density F has been derived by Page &
Thorne (1974) and is given by
F =3GM˙M
8πR3f0, (A11)
where f0= (2R2/3Rg)f and the expression for f is given in equation (15n) of Page & Thorne
(1974). Our choice of f0instead of f is based on the fact that, unlike f, f0is dimensionless.
Note that at R = RISCO(the innermost stable circular orbit) we have f0= 0, and that as
R → ∞ we have f0→ 1.
By equations (A8)–(A10), the scale-height of the disk is
H
R≈3κ˙M
8πRc
f0
ξ
. (A12)
Following the analysis of the Newtonian case (§ 6.1), we define the Eddington luminosity by
LEdd=4πGMc
κ
= ε˙MEddc2,i.e.,
˙MEdd=4πcRg
κε
, (A13)
except that here ε = ε(a∗) is the radiative efficiency of the relativistic disk (see Page &
Thorne 1974). With the above definition of ˙MEdd, we have L/LEdd= ˙M/˙MEdd≡ ˙ m, where
L is the luminosity of the disk. Then, equation (A12) can be recast into
H
R≈3 ˙ m
2ε
f0
x2ξ.
(A14)
It can be shown that this expression for H/R simplifies to equation (5) in the Newtonian
limit. Note that H/R does not depend on the value of the opacity κ.
Since the ratio H/R = 0 at R = RISCOand also as R → ∞, it must have a maximum
at some finite R > RISCO. It turns out that for any given value of ˙ m, the maximum value
of H/R is very insensitive to variation in a∗(though the radius at which this maximum is
reached varies by a large factor). Examples of H/R as a function of the disk radius are
shown in Figure 17 for two choices of the BH spin, a∗= 0, 0.998. Notice how the two sets of
curves agree very closely as far as their maxima are concerned. Therefore, regardless of the
value of a∗, if we wish to have (H/R)max? 0.1, we require the dimensionless disk luminosity
l ≡ L/LEddto be ? 0.3.
Page 33
– 33 –
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Fig. 1.— Detailed results from fitting all 20 RXTE observations of GRS1915 in the thermal
state with a model consisting of diskbb, a power-law, a Gaussian absorption line and a
smedge component over the energy range 3–25 keV (44 degrees of freedom). The data points
circled in blue correspond to the crucial low-luminosity observations (§5, §6.1, Appendix).
The horizontal dashed line in the bottom panel is drawn at χ2
the panels show: the ratio of the disk to total flux fD, the two diskbb fitting parameters,
the disk inner temperature Tin(keV) and the normalization constant K, the disk flux FD
(10−7erg cm−2s−1) and the power-law flux FPL(10−8erg cm−2s−1), the power-law photon
index Γ, the central energy of the Gaussian absorption line EFe(keV), the intensity of the
line NFe(photons cm−2s−1times 100) and the equivalent width of the line EWFe (keV),
the smedge edge energy ES(keV) and the smedge optical depth τS, and finally the value of
reduced chi-square. See §4.1 for further details.
Fig. 2.— Analogous to Figure 1, but with the non-relativistic disk model diskbb replaced
by our relativistic disk model kerrbb2 (44 dof). The fits were done for a viscosity parameter
α = 0.01. This figure is identical in structure with Figure 1 except that Tinand K are here
replaced by two parameters of kerrbb2, namely, the BH spin parameter a∗ and the mass
accretion rate ˙M (1018gs−1).
ν= 1. From top to bottom,
Fig. 3.— This figure is identical to Figure 2, except that the fits were computed for a
viscosity parameter α = 0.1.
Fig. 4.— Results of fitting the 20 RXTE observations of GRS1915 in the thermal state with
a model consisting of kerrbb2, a thermal Comptonization component comptt, a Gaussian
absorption line and a smedge component (43 dof). The panels are the same as in Figures 2
and 3 except that FPLand Γ are replaced by the flux in the comptt component FCand the
optical depth of the Comptonizing corona τC. See §4.2.2 for other details.
Fig. 5.— Results of fitting the 20 RXTE observations of GRS1915 in the thermal state
with a model consisting of kerrbb2, a cutoff power-law component expabs*power, a Gaussian
absorption line and a smedge component (43 dof). See §4.2.3 for other details.
Fig. 6.— Fit results for a model including a sharp absorption edge for the five observations
in Figures 2–5 that have χ2
to those plotted in Figure 2, and the results of including the sharp edge in the fits are plotted
as open red circles. Note the pair of panels near the bottom displaying the parameters of
the edge component.
ν? 1.5. The small black data points with error bars are identical
Fig. 7.— Spin parameter a∗versus the Eddington-scaled luminosity L/LEddfor all 22 RXTE
and ASCA observations of GRS1915 in the thermal state for two values of the viscosity
Page 39
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parameter α. The tail emission is modeled as a simple power law. For reasons discussed
in § 6.1 and the Appendix, the results are most trustworthy for L/LEdd? 0.3; this limit is
indicated here and below by the vertical dotted line. Data in this regime consistently give
a very high estimate of the spin parameter of GRS1915, a∗→ 1, independent of α or any
other details.
Fig. 8.— Same as Figure 7 except that the tail emission is modeled as a Comptonized plasma
and only the results for α = 0.01 are shown. Results are displayed for three values of T0, the
temperature of the seed photons.
Fig. 9.— Same as Figure 7 except that the tail emission is modeled as a cutoff PL, and
α = 0.01 only. The results are shown for three values of the cutoff energy Ec.
Fig. 10.— Direct comparison of the results displayed in Figures 7–9 for the three different
tail models, for α = 0.01 only. Note how very similar the results are, which shows that the
results are not sensitive to the details of the spectral model used to fit the high-energy tail
component in the spectrum.
Fig. 11.— (a) Effects on the spin estimate of GRS1915 as a result of varying the distance
D to the source over the range 11.0–12.5 kpc. The mass of the BH M and the inclination i
are correlated with D, as explained in §5.2. The results for D = 9 kpc and D = 10 kpc are
not shown for reasons that are given in §5.2. (b) Effects of varying the BH mass M over its
allowed range, keeping D fixed at 11.0 kpc and i fixed at 66.0o.
Fig. 12.— The single spin estimate obtained by M06, which is here referred to a distance
D = 11 kpc, is indicated by the three blue triangles that are connected by a dashed line.
Our results, which are based on the comptt tail model for T0= 2.0 keV and α = 0.01, are
shown as red circles (see Fig. 8).
Fig. 13.— Illustrates the effect of including the returning radiation in the model. The
primary effect is to shift the estimated Eddington-scaled luminosities to higher values. There
is very little effect on the estimates of BH spin a∗.
Fig. 14.— Illustrates the effect of including a nonzero torque at the inner edge of the disk.
Fig. 15.— The behavior of three dimensionless quantities that depend only on the BH spin
parameter: (a) The radius of the ISCO in gravitational units, (b) the specific binding energy
at the ISCO, and (c) the Keplerian orbital frequency at the ISCO. The filled data points
correspond to nominal estimates of the spins of the four BHs (see Table 4): from left to
right, LMC X-3 (a∗ = 0.20), GRO J1655–40 (a∗ = 0.70), 4U 1543–47 (a∗ = 0.80), and
GRS1915 (a∗= 0.99). The horizontal lines in each panel indicate the values of each of the
Page 40
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three quantities in question that correspond to the following key values of spin: a∗ = 0
(short-dashed line), a∗= 1 (long-dashed line), and a∗= 0.998 (dotted line; Thorne 1974).
Fig. 16.— Ratio of disk thickness H to radius R, plotted against R/Rin, for Eddington-
scaled values of luminosity l = ˙ m in steps of ∆l = 0.1 (from l = 0.1 to l = 1.2 upward). The
results are for a Newtonian disk in which Rinis the radius of the inner edge. The horizontal
dashed line corresponds to H/R = 0.1. (See Fig. 17 for the relativistic case.)
Fig. 17.— Ratio of disk thickness H to radius R for a relativistic disk around a Kerr black
hole, plotted against R/Rg, for Eddington-scaled values of luminosity l = ˙ m in steps of
∆l = 0.1 (from l = 0.1 to l = 1.2 upward). The inner radius of the disk is at the innermost
stable circular orbit. The thick lines correspond to a non-rotating black hole (a∗= 0) and
the thin lines to a maximally rotating black hole (a∗= 0.998). The horizontal dashed line
corresponds to H/R = 0.1. It is anticipated that the disk spectral models employed in this
paper (diskbb, kerrbb2, bhspec) are most reliable when H/R ? 0.1, which corresponds to the
luminosity limit l ? 0.3. (See Fig. 16 for the Newtonian case.)
Fig. 18.— (a) The allowed values of BH mass and distance for GRS1915 fall within the
triangular region indicated (see text). (b) A summary of model-dependent distance estimates
for GRS1915. The two relatively precise and disparate estimates at the bottom of the
figure require comment: The one labeled “Radio/IR: lobes” is based on identifying a pair of
extended IRAS sources as the regions where the jets of GRS1915 impact the ISM (Kaiser
et al. 2004). The other estimate labeled “Near-IR systemic velocity + 21 cm” is based on a
systemic velocity of γ = −3±10 km s−1and the Galactic rotation curve (Greiner et al. 2001).
This latter estimate ignores the potentially sizable and unknown uncertainty associated with
a possible peculiar component of radial velocity as well as any kick velocity that may have
been imparted to the system during the formation of the BH (e.g., Jonker & Nelemans 2004).
References: (1) Rodr´iguez et al. 1995; (2) Dhawan et al. 2000a; (3) Dhawan et al. 2000b; (4)
Chapuis & Corbel 2004; (5) Kaiser et al. 2004; (6) Greiner et al. 2001.
Page 41
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5 101520
2.0
4.0
2.0
4.0
7.0
8.0
0.1
0.2
-4.0
-8.0
6.4
7.2
2.0
4.0
4.0
8.0
4.0
8.0
100.0
200.0
2.0
2.2
0.8
1.0
Fig. 1.—
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5 101520
2.0
4.0
2.0
4.0
7.0
8.0
0.1
0.2
-4.0
-8.0
6.4
7.2
2.0
4.0
4.0
8.0
4.0
8.0
10.0
20.0
0.8
1.0
0.8
1.0
Fig. 2.—
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5 1015 20
2.0
4.0
2.0
4.0
7.0
8.0
0.1
0.2
-4.0
-8.0
6.4
7.2
2.0
4.0
4.0
8.0
4.0
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10.0
20.0
0.8
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Fig. 3.—
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5 10 1520
2.0
4.0
2.0
4.0
7.0
8.0
0.1
0.2
-4.0
-8.0
6.4
7.2
1.0
2.0
4.0
8.0
4.0
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10.0
20.0
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Fig. 4.—
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5 101520
2.0
4.0
2.0
4.0
7.0
8.0
0.1
0.2
-4.0
-8.0
6.4
7.2
2.0
4.0
4.0
8.0
4.0
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10.0
20.0
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Fig. 5.—
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2.0
4.0
9.0
0.4
12.0
0.2
2.0
4.0
7.0
8.0
0.1
0.2
-4.0
-8.0
6.4
7.2
2.0
4.0
4.0
8.0
4.0
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20.0
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1.0
Fig. 6.—
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Fig. 7.—
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Fig. 8.—
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Fig. 9.—
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Fig. 10.—
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