Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://jet.sao.ru/hq/grb/team/vvs/Ozel_et_al.pdf Äàòà èçìåíåíèÿ: Fri Dec 17 12:13:49 2010 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 01:48:37 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: accretion disk

Draft version November 9, 2010
A Preprint typ eset using L TEX style emulateap j v. 03/07/07

THE BLACK HOLE MASS DISTRIBUTION IN THE GALAXY
¨ Feryal Ozel1 , Dimitrios Psaltis1 , Ramesh Narayan2 , Jeffrey E. McClintock3
2 1 Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721 Institute for Theory and Computation, Harvard University, 60 Garden St., Cambridge, MA 02138 and 3 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138 Draft version November 9, 2010

arXiv:1006.2834v2 [astro-ph.GA] 8 Nov 2010

ABSTRACT We use dynamical mass measurements of 16 black holes in transient low-mass X-ray binaries to infer the stellar black hole mass distribution in the parent population. We find that the observations are best described by a narrow mass distribution at 7.8 ± 1.2 M . We identify a selection effect related to the choice of targets for optical follow-ups that results in a flux-limited sample. We demonstrate, however, that this selection effect does not introduce a bias in the observed distribution and cannot explain the absence of black holes in the 2 - 5 M mass range. On the high mass end, we argue that the rapid decline in the inferred distribution may be the result of the particular evolutionary channel followed by low-mass X-ray binaries. This is consistent with the presence of high-mass black holes in the persistent, high-mass X-ray binary sources. If the paucity of low-mass black holes is caused by a sudden decrease of the supernova explosion energy with increasing progenitor mass, this would have observable implications for ongoing transient surveys that target core-collapse supernovae. Our results also have significant implications for the calculation of event rates from the coalescence of black hole binaries for gravitational wave detectors. Subject headings: X-rays: binaries -- black hole physics

1. INTRODUCTION

The distribution of stellar black hole masses in the Galaxy is intricately related to the population and evolution of massive stars, the energetics and dynamics of supernova explosions, and the dividing line between neutron stars and black holes. Inferring this distribution from observations helps address a number of outstanding questions in these a r ea s . Observations of mass losing evolved stars are one of the direct ways of studying the end stages of stellar evolution (Smartt 2009). Similarly, supernova and transient surveys (e.g., Rau et al. 2009 and references therein) provide increasingly larger data samples that have revealed a range of supernova properties and energetics that is wider than previously anticipated. Studies of the end products of supernova explosions, namely of the black hole and neutron star mass distributions, provide a different and complementary approach to constraining models of massive stellar populations, evolution, and explosions (Timmes, Woosley, & Weaver 1996; Fryer 1999; Fryer et al. 2002; Woosley, Heger, & Weaver 2002; Zhang, Woosley, & Heger 2008). The mass distribution of stellar black holes is a key ingredient in the calculation of the event rates for gravitational wave observatories such as LIGO, GEO500, and Virgo (Abadie et al. 2010). In addition, inferring this distribution may shed light on the origins and properties of pulsar-black hole binaries (Narayan, Piran, & Shemi 1991), which are believed to be some of the best laboratories for strong-field gravity tests (Wex & Kopeikin 1999). During the last two decades, the masses of a large number of black holes in X-ray binaries have been measured (Remillard & McClintock 2006; McClintock & Remillard 2006), providing a sample suitable for the statistical inference of their parent mass distribution. Using this sample, we find in this article strong evidence for a very narrow distribution of masses for black holes in transient low-mass X-ray binaries. In particular, we show that there is a paucity of black holes with masses 2 - 5 M . Our results confirm and strengthen an earlier finding of Bailyn et al. (1998), who argued for a low-mass gap based on a limited sample of black holes. We explore in detail below known selection biases and demonstrate that the low-mass gap cannot be attributed to the observational selection of targets. We further argue that the lack of high-mass black holes in low-mass X-ray binaries may result from a particular evolutionary path that leads to their formation.
2. ACCRETING STELLAR-MASS BLACK HOLES AND BLACK HOLE CANDIDATES

In Tables 1 and 2, we present basic data for all 23 confirmed black-hole X-ray binaries (Remillard & McClintock 2006). For the purpose at hand, we divide these systems into three groups: eight transient systems with orbital periods exceeding one day; nine shorter-period transient systems for which the measurement of black hole mass is problematic; and six systems with persistent X-ray sources and massive O/B-type secondaries. In these tables, we provide the following data for most of the systems: Galactic coordinates, maximum X-ray intensity, orbital period, a distance estimate, mass function, mass ratio, inclination angle, and black hole mass. Our primary focus is on the 17 transient systems listed in Tables 1 and 2. We summarize in Table 3 data for thirty-two additional transient systems that are believed to contain black hole primaries based on the spectral and timing properties of their X-ray sources. For each of these systems, we give both


2 celestial and Galactic coordinates and the maximum X-ray intensity that has been reported.
2.1. The Twenty-Three Black Hole Binaries

We now survey the constraints that have been placed on the masses of the 23 confirmed stellar black holes via dynamical measurements. In determining the masses of black holes in X-ray binaries, the mass function f (M ) M sin3 i Porb K 3 = 2 G (1 + q )2 (1)

is the most important and secure observable. The orbital period Porb and the half-amplitude of the velocity curve of the secondary star K can, in most cases, be determined precisely and accurately (Remillard & McClintock 2006; Charles & Coe 2006). These two quantities define the value of the mass function, which is an absolute lower limit on the mass of the compact ob ject: i.e., M f (M ). The mass function relates the black hole mass M , the orbital inclination angle i, and the mass ratio q M2 /M , where M2 is the mass of the secondary star. Values of the mass function, the inclination, and the mass ratio for the 23 established black hole binaries are given in Table 2. The mass-ratio estimates were obtained or derived from Orosz (2003) and Narayan & McClinto ck (2005), with a few refinements based on the references cited in the table. Determining the black hole mass M , the quantity of interest, is challenging because in many cases it is difficult to obtain secure constraints on the inclination and the mass ratio. Table 1 separates the 23 systems into three groups based on their X-ray behavior: the persistent sources (PS), the long-period transients (LPT), and the short-period transients (SPT). This is important for understanding possible selection effects, which we explore in Section 4. Table 2, on the other hand, separates sources into three different groups based on the amount of data available on their mass ratios and inclinations. This impacts the inference of the masses of individual sources, which we discuss in Section 3. These tables sparely give for each source a few key references, which are supplemented by some additional references in the discussion that follows.
2.1.1. The Persistent Sources

These systems contain O/B-type secondaries and are persistently X-ray bright. Five of them have relatively massive black-hole primaries, M > 10 M . However, the mass of LMC X-3 is presently poorly constrained: 4 M M 11M (Cowley 1992). In the case of Cyg X-1, a very wide range of masses down to 5 M has recently been considered (Caballero-Nieves et al. 2009). However, recent VLBA observations have shown that the distance exceeds 1.5 kpc (M. Reid, private communication), which firmly establishes M > 8 M . There are two caveats on the mass constraints on IC 10 X-1 and NGC 300-1: These results are based on less-reliable emission-line radial velocities and assume specific lower bounds on the masses of the secondary stars.
2.1.2. The Long Period Transient Sources

Nearly all transient black hole sources, which are fed by Roche-lobe overflow, exhibit long periods of deep quiescence during which the spectra of their secondary stars are prominent. For all eight of these long-period systems (Porb > 1 d) there is strong evidence that the masses of their black hole primaries exceed 6 M . We comment on the two weakest cases. (1) GX 339-4 has never reached a deep enough quiescent state to reveal its photospheric absorption lines, and its mass function was determined via the Bowen emission lines (Hynes et al. 2003). Furthermore, this system does not exhibit ellipsoidal light curves that allow its inclination to be constrained. For a defense of the M = 6 M lower limit see Munoz-Darias et al. (2008). (2) GRS 1915+105: The pioneering mass measurement of this system by Greiner et al. (2001) requires confirmation. Their spectroscopic orbital period disagrees by 8% with a more recent photometric determination of the period (Neil et al. 2007). Again, there are no well-behaved ellipsoidal light curves that can be used to constrain the inclination, which in this case is inferred from a kinematic model of the relativistic radio jets (Mirabel & Rodr´ uez 1999). ig
2.1.3. The Short Period Transient Sources

Significant constraints have been placed on the black hole primaries of only three of the nine short period systems (Porb < 1 d). It is difficult to obtain reliable inclination constraints for these systems because studies in quiescence of their small, late-type secondaries are compromised by the presence of a relatively strong and variable component of non-stellar light (e.g., Zurita et al. 2003; Cantrell et al. 2008), which is continually present in all of these systems. A measurement of the inclination angle has been obtained only for A0620­00, the prototype system, which is two magnitudes brighter than the other short period transients. The study by Cantrell et al. (2010), a tour de force that makes use of 32 photometric data sets spanning 30 years, relies on 10 data sets obtained when the source was in a "passive" quiescent state (Cantrell et al. 2008). As indicated in Table 2, the authors constrain the mass to be M = 6.6 ± 0.25 M . Meanwhile, the masses of three other systems (XTE J1118+480, Nova Oph 1977 and GS 2000+251) have by dint of their large mass functions been shown to exceed 6 M (Table 2). The large mass function of XTE J1859+226 (Table 2) suggests that it also should be included in this group; however, this result is unreliable, having only been presented in an IAU Circular (Filippenko & Silverman 2001), and the orbital period is uncertain (Zurita et al. 2002). We exclude this source from further considerations on deriving the black-hole mass distribution. Among the 23 systems in Table 2, arguably the best candidate for hosting a low-mass black hole is GRO J0422+32. However, by briefly discussing recent attempts to determine the system's inclination and mass, we show that current results are presently unreliable: Beekman et al. (1997) constrained the inclination to lie in the range i = 10 - 31


3
TABLE 1 Properties of Twenty-Three Black Hole Binaries Coordinate Name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 1354-64 1543-47 1550-564 1655-40 1659-487 1819.3-2525 1915+105 2023+338 0422+32 0620-003 1009-45 1118+480 1124-683 1650-500 1705-250 1859+226 2000+251 0020+593 0055-377 0133+305 0538-641 0540-697 1956+350 Common Name or Prefixa (GS) (4U) (XTEJ) (GROJ) GX 339-4 V4641 Sgr (GRS) (GS) (GROJ) (A) (GRS) (XTEJ) Nova Mus 91 (XTEJ) Nova Oph 77 (XTEJ) (GS) IC 10 X-1 NGC 300-1 M33 X-7 LMC X-3 LMC X-1 Cyg X-1 Typeb LPT LPT LPT LPT LPT LPT LPT LPT SPT SPT SPT SPT SPT SPT SPT SPT SPT PS PS PS PS PS PS l (deg) 310.0 330.9 325.9 345.0 338.9 6.8 45.4 73.1 166.0 210.0 275.9 157.6 295.3 336.7 358.2 54.1 63.4 ··· ··· ··· ··· ··· ··· b (deg) -2.8 +5.4 -1.8 +2.5 -4.3 -4.8 -0.2 -2.1 -12.0 -6.5 +9.4 +62.3 -7.1 -3.4 +9.1 +8.6 -3.0 ··· ··· ··· ··· ··· ··· Max. Int. (Crab) 0.12 15 7.0 3.9 1.1 13 3.7 20 3 50 0.8 0.04 3 0.6 3.6 1.5 11 0.00006 0.00004 0.00002 0.06 0.03 2.3 Porb (hr) 61.1 26.8 37.0 62.9 42.1 67.6 739 155.3 5.1 7.8 6.8 4.1 10.4 7.7 12.5 9.2d 8.3 34.9 32.3 82.9 40.9 93.8 134.4 D (kpc) > 25 7.5 ± 0.5 4.4 ± 0.5 3.2 ± 0.5 9±3 9.9 ± 2.4 9±3 2.39 ± 0. 2±1 1.06 ± 0. 3.82 ± 0. 1.7 ± 0.1 5.89 ± 0. 2.6 ± 0.7 8.6 ± 2.1 8±3 2.7 ± 0.7 ··· ··· ··· ··· ··· ··· Referencesc (distance) 1 2 3 4 5 6 7 8 9,10 11 10 12 10 13 14 10 14 ··· ··· ··· ··· ··· ···

14 12 27 26

a The entries in parentheses are prefixes to the co ordinate names that identify the discovery X-ray mission. b PS = p ersistent source; LPT = long p erio d transient; SPT = short p erio d transient. c References: 1. Casares et al. 2009; 2. J. Orosz, private communication; 3. Orosz et al. 2010; 4. Hjellming & Rup en 1995; 5.

and co ss that is not constrained. Gelino and Harrison (2003) present the strongest evidence for a low mass; they conclude i > 43 and M < 4.92 M . However, in a more recent Keck K -band study Reynolds et al. (2007) found that "No clear ellipsoidal modulation is present in the light curve..." and concluded, "...that previous infrared-based attempts to constrain the mass of the putative black hole in this system are prone to considerable uncertainty." Thus, it appears that a far more comprehensive photometric study (cf. Cantrell et al. 2010) is required in order to obtain a firm mass constraint. There are six systems in Table 2 with black holes of indeterminate mass. The prospects for measuring or usefully constraining the masses of two of them, LMC X-3 and XTE J1859-226, are bright: Work on the former is almost complete (J. Orosz, private communication), and the additional data required to confirm the orbital period and large mass function of the latter are obtainable. That leaves four systems (GRO J0422+32, GRS 1009­45, Nova Mus 1991 and XTE J1650­500) with black hole masses that are only very weakly constrained: M > f (M ) 1 - 3 M . As indicated above and as Cantrell et al. (2010) have shown, it will be very challenging to place stronger and reliable constraints on the masses of these black holes.
2.2. Thirty-two Transient Black-Hole Candidates

Hynes et al. 2004; 6. Orosz et al. 2001; 7. Fender et al. 1999; 8. Miller-Jones et al. 2009; 9. Webb et al. 2000; 10. Hynes et al. 2005; 11. Cantrell et al. 2010; 12. Gelino et al. 2006; 13. Homan et al. 2006; 14. Barret et al. 1996 d Unconfirmed and uncertain value; see text. ncluded M > 9 M . Webb et al. (2000) concluded i < 45 and M > 2.2 M with a maximum ma

Data for 32 X-ray transient systems are given in Table 3. These systems lack radial velocity data, and most even lack an optical counterpart. Thus, presently, there are no dynamical constraints on the masses of their compact primaries, which are believed to be black holes because they share certain characteristic X-ray properties with the 23 established black holes (McClintock & Remillard 2006). As indicated in Table 3, the primary source of information about these systems is the catalogue of Liu et al. (2007) and references therein. For additional information and references on many of these systems, see Table 4.3 and text in McClintock and Remillard (2006).
3. BLACK HOLE MASS MEASUREMENTS AND CONSTRAINTS

In this section, we use the measurements of the mass functions, as well as any available constraints on the mass ratios and inclinations for the black holes in low-mass X-ray binaries shown in Table 2 in order to place quantitative constraints on the individual black hole masses. In particular, our aim is to derive the likelihood Pi (data|M ), which measure the chance of obtaining the particular set of data shown in Table 2 for the i-th source if that source had mass M. We divide the sources into three categories based on the amount and quality of information regarding their mass ratios and inclinations: (i) For six sources, the mass ratios and the inclinations are tightly constrained, leading to well-determined black hole masses. In this case, the probability distribution can be described as a Gaussian Pi (data|M ) = Ci exp -(M - M0,i )2 2 2M ,i (2)


4 with a mean M0,i and a standard deviation M ,i . In this and the following expressions, Ci is a proper normalization constant such that Pi (data|M )dM = 1.
0

(3)

This category includes A0620-003, 4U 1543-47, XTE J1550-564, GRO J1655-40, V4641 Sgr, and GS 2023+338. (ii) For the sources in the second category, there is only a measurement of the mass function and constraints on the mass ratio q . Here, we assume a Gaussian probability distribution over the mass function with a mean f0,i and a standard deviation 0,i . For the mass ratio, we adopt a uniform distribution dq (4) qmax - qmin between the minimum and maximum allowed mass ratios, qmin and qmax , respectively. For each value of the mass ratio, the lack of eclipses implies a maximum value of the inclination, i.e., a minimum value of cos i, such that P (q )dq = (cos i)min = 0.462 q 1+q
1/3

.

(5)

Assuming a uniform distribution over cos i sub ject to this constraint, i.e., P (cos i|q )d(cos i) = yields
qm
ax

d(cos i) , 1 - (cos i)min d(cos i) 1 - (cos i)m

(cos i)min cos i 1, - M sin3 i/(1 + q )2 ] 2 2f ,i
2

(6)

1

Pi (data|M ) = Ci
qm
in

dq
(cos i)
m in

ex p -
in

[f

0,i

.

(7)

The following nine sources belong to this category: GROJ 0422+32, GRS 1009-45, XTE J1118+480, Nova Mus 91, MS 1354-64, XTE J1650-500, GX 339-4, Nova Oph 77, and GS 2000+251. (iii) This last category includes only GRS 1915+105, for which the mass function and the inclination have been measured, and the mass ratio has been constrained (see the discussion in Section 2 about the inclination measurement). In this case, we calculate the probability distribution over mass using equation (7), supplemented by a Gaussian distribution over inclination
qm
ax

1

Pi (data|M ) = Ci
qm
in

dq
(cos i)
m in

d(cos i) 1 - (cos i)m

ex p -
in

[f

0,i

- M sin3 i/(1 + q )2 ]2 (i - i0 )2 - 2 2 2f ,i 2i

.

(8)

Figure 1 shows the likelihoods Pi (data|M ) for the 16 sources in the above three categories. The top panel includes sources in categories (i) and (iii), while the bottom panel shows those in category (ii). A clustering of the observed black hole masses between 6 - 10 M is already evident from Figure 1. In the next section, we will carry out a formal Bayesian analysis to determine the parameters of the underlying mass distribution that is consistent with the observed Pi (data|M ) shown here. If the likelihood for each source was narrow enough such that there was little or no overlap between them, then adding the likelihoods for the entire sample and coarsely binning the resulting distribution would provide a good estimate of the underlying mass distribution. Even though this condition is not entirely satisfied here, especially at the high mass end, we nevertheless show in Figure 2 this approximate mass distribution to get a sense of its gross properties.
4. THE INTRINSIC DISTRIBUTION OF BLACK HOLE MASSES

In this section, we use a parametric form of the black-hole and 3 in order to determine its parameters. We will first cons cut-off given by exp(Mc /Mscale ) P (M ; Mscale , Mc ) = Mscale

mass distribution and the data discussed in Sections 2 ider an exponentially decaying mass distribution with a exp(-M /M 0,
scale

), M >M M M

c c

.

(9)

This choice of the mass distribution is motivated by theoretical expectations based on the energetics of supernova explosions, as well as the density profiles and mass distributions of pre-supernova stars. The typical value of the mass scale is expected to lie in the range Mscale 5.5 - 9 M (as we infer from the various figures in Fryer & Kalogera 2001), whereas the cutoff mass is simply expected to be the maximum neutron star mass. Our goal is to find the values of the mass scale Mscale in the exponential and the cut-off mass Mc that maximize a properly defined likelihood and to estimate their uncertainties. We will show below that the particular choice of the functional form of the mass distribution does not affect the main conclusions of the paper. In Section 3, we calculated, for each observed black hole, the probability Pi (data|M ), which measures the chance of making a particular observation if the black hole has mass M . What we want to calculate here is the probability P (Mscale , Mc |data), which measures the likelihood of the parameters of the black hole mass distribution, given the observations. Using Bayes' theorem, we can write this as P (M
scale

, Mc |data) = C2 P (data|M

scale

, Mc )P (M

scale

)P (Mc ) ,

(10)


5

Fig. 1.-- The likelihoods Pi (data|M ) for the 16 sources in low-mass X-ray binaries that have been securely identified as black holes. The top panel includes sources in categories (i) and (iii), while the bottom panel shows those in category (ii). The categories are based on the amount of information available on the mass ratios and inclinations of the black hole binaries and are discussed in more detail in the text.

where C2 is the normalization constant and P (Mscale ) and P (Mc ) cut-off mass. We assume a flat prior over the mass scale between i.e., Mscale 0, 1 , 0
are the priors over the values of the mass scale and Mscale = 0 and a maximum value Mscale = Mmax , 0
scale

Mm > Mmax .

ax

The upper limit Mmax is imposed mostly for computational reasons and does not affect the results. We also adopt a similar prior over the cut-off mass between the maximum neutron-star mass, which we set to 2 M , and the minimum well-established mass measurement for a black hole. As will be evident from the results, the particular choice of this

02

02

501+5191

833+3202

4-933XG

51

51

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)oM( ssaM eloH-kcalB

74-3451

5252-3.9181

465-0551

46-4531

084+8111

300-0260 052-5071

01

01

152+0002

54-9001

5

5

04-5561 23+2240

0

6.1 4.1 2.1 0.1 8.0 6.0 4.0 2.0 0.0 02.0
386-4211

51.0

01.0

50.0

doohilekiL doohilekiL

005-0561

(11)


6
1
Fig. 2.-- The solid line shows the sum of likelihoods for the mass measurements of the 16 black holes in low-mass X-ray binaries. Note that because of the high-mass wings of the individual likelihoods, the shape of their sum is artificial at the high mass end. The dashed and dotted lines show the exponential and Gaussian distributions, respectively, with parameters that best fit the data (see §4). TABLE 2 Dynamical Data for Twenty-Three Black Hole Binaries Coordinate Name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0620-003 1543-47 1550-564 1655-40 1819.3-2525 2023+338 0422+32 1009-45 1118+480 1124-683 1354-64 1650-500 1659-487 1705-250 2000+251 1915+105 1859+226 0020+593 0055-377 0133+305 0538-641 0540-697 1956+350 Common Name or Prefixa (A) (4U) (XTEJ) (GROJ) V4641 Sgr (GS) (GROJ) (GRS) (XTEJ) Nova Mus 91 (GS) (XTEJ) GX 339-4 Nova Oph 77 (GS) (GRS) (XTEJ) IC 10 X-1 NGC 300-1 M33 X-7 LMC X-3 LMC X-1 Cyg X-1 2. 0. 7. 2. 3. 6. 1. 3. 6. 3. 5. 2. 5. 4. 5. 9. 7. 7. 2. 0. 2. 0. 0. f (M ) (M ) 76 ± 0.01 25 ± 0.01 73 ± 0.40 73 ± 0.09 13 ± 0.13 08 ± 0.06 19 ± 0.02 17 ± 0.12 1 ± 0.3 01 ± 0.15 73 ± 0.29 73 ± 0.56 8 ± 0.5 86 ± 0.13 01 ± 0.12 5 ± 3.0 4 ± 1.1d 64 ± 1.26 6 ± 0.3 46 ± 0.07 3 ± 0.3 886 ± 0.037 251 ± 0.007 q -q i (deg) 51.0 ± 20.7 ± 74.7 ± 70.2 ± 75 ± 2 55 ± 4 ··· ··· ··· ··· ··· ··· ··· ··· ··· 66±2 ··· ··· ··· ··· ··· ··· ··· 0. 1. 3. 1. 9 5 8 9 M (M ) 6.6 ± 0. 9.4 ± 1. 9.1 ± 0. 6.3 ± 0. 7.1 ± 0. 12 ± 2 ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· > 20 > 10 15.65 ± ··· 10.91 ± >8 25 0 6 27 3 References
b

q

mi n

max

0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. · · · · · · ·

056­0.064 25­0.31 0­0.040 37­0.42 42­0.45 056­0.063 076­0.31 12­0.16 035­0.044 11­0.21 08­0.15 0­0.2c 0­0.4c 0­0.053 035­0.053 025­0.091 ·· ·· ·· ·· ·· ·· ··

a The entries in parentheses are prefixes to the co ordinate names that identify the discovery X-ray mission. b References: 1. Neilsen et al. 2008; 2. Cantrell et al. 2010; 3. Orosz 2003; 4. Orosz et al. 2010; 5. Greene et al. 2001; 6.

Charles & Co e 2006; 7. Filipp enko et al. 1995; 8. Filipp enko et al. 1999; 9. McClinto ck et al. 2001; 10. Gelino et al. 2006; 11. Orosz et al. 1996; 12. Casares et al. 2009; 13. Orosz et al. 2004; 14. Hynes et al. 2003; 15. Munoz-Darias et al. 2005; 16. ~ Greiner et al. 2001; 17. Neil et al. 2007; 18. Harlaftis & Greiner 2004; 19. Fender et al. 1999; 20. Filipp enko & Chorno ck 2001; 21. Zurita et al. 2002; 22. Prestwich et al. 2007; 23. Silverman & Filipp enko 2008; 24. Crowther et al. 2010; 25. Orosz et al. 2007; 26. Cowley 1992; 27. Orosz et al. 2009; 28. Caballero-Nieves et al. 2009; 29. M. Reid, private communication. c Estimated range based on extreme values observed for systems with comparable orbital p erio ds. d Unconfirmed and uncertain value; see text.

52

02

)oM( ssaM eloH-kcalB

51

01

5

10.0

1.0
1.45 1.54 1,2 3 4 3,5 3 6 3,7 3,8 3,9,10 3,11 12 13 14,15 3,6 3,6 16,17,18,19 20,21 22,23 24 25 26 27 28,29

sdoohilekiL fo muS


7
01
Fig. 3.-- The parameters of an exponential black hole mass distribution with a low-mass cutoff. The cut-off mass is well above theoretical expectations, indicating a sizable gap between neutron-star and black hole masses. Furthermore, the mass scale in the exponential is significantly smaller than theoretical expectations.

range does not affect the measured parameters. We also repeated this analysis with logarithmic priors in the two parameters and found that the results are insensistive to the choice of priors. In equation (10), the quantity P (data|Mscale , Mc ) measures the chance that we make a particular set of observations for the ensemble of black holes, given the values of the parameters of the mass distribution. We need now to estimate this quantity, given the likelihoods for the individual sources. We will assume that each measurement is independent of all the others, so that P (data|M
scale

, Mc ) =
i

dM Pi (data|M )P (M ; M

scale

, Mc ) .

Combining this last equation with equation (10) we obtain P (M
scale

, Mc |data) = C P (M

scale

)P (Mc )
i

dM Pi (data|M )P (M ; M

where C is the overall normalization constant. We show in Figure 3 the 68% and 95% confidence contours of the mass scale and cut-off mass that best describe the observations and compare them to the theoretical expectation. The lack of black holes below 5 M and the rapid decline of the exponential distribution at the high mass end are both remarkable (see the dashed line in Fig. 2). The latter result is not at odds with the relatively high mass of GRS 1915+105 because of the wide and shallow mass probability distribution of this source. In order to explore whether the small number of sources with very well-determined masses dominate this result, we did the following test. We repeated the calculation using the mass functions and constraints on the mass ratios for all sources but ignoring any information on the inclinations of the binary systems. This, in effect, is equivalent to treating all sources in category (i) using the formalism we applied to sources in category (ii), integrating over all possible values of inclination. Figure 4 shows the 68% and 95% confidence contours of the parameters of the exponential distribution in this test case. Although the allowed range of values is increased, as expected, the low-mass gap and the discrepancy with the theoretically expected mass scale remain robust. The narrowness of the mass distribution implied by the above results motivated us to explore different functional forms of the underlying distribution, and in particular, a Gaussian function. The Gaussian function here serves as a phenomenological two-parameter description of a narrow distribution and is not necessarily motivated by theory. We show in Figure 5 the parameters of such a Gaussian distribution that best describes the observations. The masses of all 16 black holes are consistent with a narrow distribution at 7.8 ± 1.2 M . This result is in agreement with an earlier, more limited, study by Bailyn et al. (1998).

01

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noitatcepxE sratS nortueN %86
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laciteroehT 6 M ssaM ffo-tuC %59 4 2 0 8 6 4 2 )0M(
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(12)

scale

, Mc ) ,

(13)


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TABLE 3 Thirty-Two Transient Black-Hole Candidates Source Name 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1A 1524-61 4U 1630-47 XTE J1652-453 IGR J17098-3628 SAX J1711.6-3808 GRO J1719-24 XTE J1720-318 IGR J17269-4737 GRS 1730-312 GRS 1737-31 GRS 1739-278 KS J1739-304 1E 1740.7-2942 1A 1742-289 H1743-322 XTE J1748-288 IGR J17497-2821 SLX 1746-331 Swift J1752-223 Swift J1753.5-0127 XTE J1755-324 4U 1755-33 GRS 1758-258 SAX J1805.5-2031 XTE J1817-330 XTE J1818-245 Swift J1842.5-1124 EXO 1846-031 IGR J18539+0727 XTE J1856+053 XTE J1908+094 XTE J2012+381 RA(J2000) 15 16 16 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 19 20 28 34 52 09 11 19 19 26 33 40 42 42 43 45 46 48 49 49 52 53 55 58 01 05 17 18 42 46 53 56 08 12 17. 01. 20. 45. 37. 36. 58. 49. 52. 09 40. 44. 54. 37. 15. 05. 38. 48. 15. 28. 28. 40. 12. 34 43. 24. 17. 39. 54 39 53. 37. 2 61 33 93 1 93 994 28 3 03 0 83 0 57 06 037 3 10 29 6 0 40 54 4 33 8 077 71 Dec(J2000) -61 52 58 -47 23 34.8 -45 20 39.6 -36 27 58.2 -38 07 05.7 -25 01 03.4 -31 45 01.25 -47 38 24.9 -31 12 25 -31 02 24 -27 44 52.7 -30 30 51 -29 44 42.6 -29 01 07 -32 14 01.1 -28 28 25.8 -28 21 17.37 -33 12 26 -22 20 32.78 -01 27 06.22 -32 28 39 -33 48 27 -25 44 36.1 -20 30 48 -33 01 07.8 -24 32 18.0 -11 25 00.6 -03 07 21 +07 27 +05 19 48 +09 23 04.90 +38 11 01.1 l(deg) 320.3 336.9 340.5 349.6 348.6 359.9 354.6