IC10~X-1/NGC300~X-1: the very immediate progenitors of BH-BH binaries
aa r X i v : . [ a s t r o - ph ] D ec Draft version November 15, 2018
Preprint typeset using L A TEX style emulateapj v. 04/03/99
IC10 X-1/NGC300 X-1: THE VERY IMMEDIATE PROGENITORS OF BH-BH BINARIES
Tomasz Bulik , , Krzysztof Belczynski , , Andrea Prestwich Astronomical Observatory, University of Warsaw, Al. Ujazdowskie 4, 00-478 Warsaw, Poland Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland Center for Gravitational Wave Astronomy, University of Texas at Brownsville, Brownsville, TX 78520, USA Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, [email protected], [email protected], [email protected]
Draft version November 15, 2018
ABSTRACTWe investigate the future evolution of two extragalactic X-ray binaries: IC10 X-1 and NGC300 X-1.Each of them consists of a high mass BH ( ∼ −
30 M ⊙ ) accreting from a massive WR star companion( ∼ >
20 M ⊙ ), and both are located in low metallicity galaxies. We analyze the current state of the systemsand demonstrate that both systems will very quickly ( ∼ < . ∼ ∼
15 M ⊙ ). The formation of BH-BH systemseems unavoidable, as (i) WR companions are well within their Roche lobes and they do not expand so noRoche lobe overflow is expected, (ii) even intense WR wind mass loss does not remove sufficient mass toprohibit the formation of the second BH, (ii) even if BH receives the large natal kick, the systems are veryclosely bound and are almost impossible to disrupt. As there are two such immediate BH-BH progenitorsystems within 2 Mpc and as the current gravitational wave instruments LIGO/VIRGO (initial stage)can detect such massive BH-BH mergers out to ∼
200 Mpc, the empirically estimated detection rateof such inspirals is R = 3 . +8 . − . at the 99% confidence level. If there is no detection in the currentLIGO/VIRGO data (unreleased year of s Subject headings: binaries: close — black hole physics — gravitational waves — stars: evolution INTRODUCTION
The interferometric gravitational wave observatoriesLIGO and VIRGO have already reached their design sen-sitivities and both are undergoing further improvements toreach the advanced sensitivity stage. The most promisingsources of gravitational waves that these experiments arelooking for are coalescences of compact objects. Amongthese most attention has been paid to double neutron starsystems (NS-NS). There is an observational evidence oftheir existence and their merger rates seem to warrantdetection with the advanced interferometric experiments.The double black hole binaries (BH-BH) and black holeneutron star binaries (BH-NS) have received less atten-tion in the rate prediction calculations. There are severalreasons for that: the direct detection of such systems inelectromagnetic domain is difficult. From the theoreticalpoint of view formation of such systems is not easy, asthey have to pass through an unstable mass transfer phasewhich is not easy to survive for typical black hole massesof around 10M ⊙ (Belczynski et al. 2007). However, it wasshown recently that in low metallicity environment this ob-stacle can be overcome and that formation rates of binaryblack holes can be quite large (Belczynski et al. 2010a).The recent advances in X-ray instrumentation allowa study of X-ray binaries in the Local Group galaxies.IC10 X-1 has already been discovered in the ROSAT databy Brandt et al (1997). Bauer & Brandt (2004) have foundan X-ray variability of IC10 X-1 in a short Chandra ob-servation. Clark & Crowther (2004) analyzed the possibleoptical counterparts of IC10 X-1(Crowther et al. 2003), and argued that it is a 35M ⊙ WNE star. Subsequentlonger Chandra observation lead to the discovery of X-ray periodicity (Prestwich et al. 2006). Prestwich et al.(2007) analyzed the X-ray and optical data of IC10 X-1,and found that it contains a black hole of a mass at least23M ⊙ in a binary with a ≈ ⊙ companion. This re-sult has been recently confirmed by Silverman & Filipenko(2008), who measured precisely the amplitude of the radialvelocity of the companion. NGC300 X-1 a system similarto IC10 X-1 is the binary NGC300 X-1 (Crowther et al.2007). Crowther et al. (2010) measured precisely the ra-dial velocity amplitude in NGC300 X-1 and showed that itcontains a 20M ⊙ black hole accreting from a 26M ⊙ Wolf-Rayet (WR, or naked helium) star. The orbital periods inboth systems are similar.Apparently, black holes of stellar origin can reach muchlarger mass than previously thought. Although such highmass BHs are already fully explained by current evolu-tionary models (Belczynski et al. 2010b). Additionally,these most massive stellar BHs can be found in binarieswith very massive companions. In this paper we analyzethe future binary evolution of IC10 X-1 and NGC300 X-1using the
StarTrack binary evolution code and we demon-strate that, regardless of evolutionary uncertainties, thesesystems will form close high mass BH-BH binaries. SuchBH-BH binaries formed at low metallicity were alreadypredicted, on theoretical grounds, to be the first detectablesources for gravitational radiation instruments like LIGOand VIRGO (Belczynski et al. 2010a).1 MODEL
Evolutionary code
Our population synthesis code,
StarTrack , was initiallydeveloped to study double compact object mergers in thecontext of GRB progenitors (Belczynski, Bulik & Rudak2002b) and gravitational-wave inspiral sources (Belczyn-ski, Kalogera, & Bulik 2002a). In recent years
StarTrack has undergone major updates and revisions in the physi-cal treatment of various binary evolution phases, and es-pecially the mass transfer phases. The new version hasalready been tested and calibrated against observationsand detailed binary mass transfer calculations (Belczynskiet al. 2008a), and has been used in various applications(e.g., Belczynski & Taam 2004; Belczynski et al. 2004;Belczynski, Bulik & Ruiter 2005; Belczynski et al. 2006;Belczynski et al. 2007). The physics updates that aremost important for compact object formation and evo-lution include: a full numerical approach for the orbitalevolution due to tidal interactions, calibrated using highmass X-ray binaries and open cluster observations, a de-tailed treatment of mass transfer episodes fully calibratedagainst detailed calculations with a stellar evolution code,updated treatment of mass transfer and common envelopephases, and the latest determination of the natal kick ve-locity distribution for neutron stars (Hobbs et al. 2005).The kicks for black holes are decreased proportionally tothe amount of fall back during core-collapse/supernova ex-plosion. For most massive stars ( M zams ∼ >
40 M ⊙ formingmassive black holes without a supernova explosion (seeFryer & Kalogera 2001) we assume no natal kick.The most recent update, employed in this study, con-cerns wind mass loss from massive stars. In particular in-terest here are mass loss rates from massive naked heliumstars. For WR stars we adopt( dM/dt ) = 10 − L . (cid:18) ZZ ⊙ (cid:19) m M ⊙ yr − (1)which is a combination of the Hamann & Koesterke (1998)wind rate estimate that takes into account WR windclumping (reduced winds), and Vink & de Koter (2005)wind Z -dependence who estimated m = 0 .
86 for WR stars.Using the above estimate, along with other updated massloss rates, we were able to recover masses of most massiveblack holes in different galaxies (Belczynski et al. 2010b).The other helium star properties (e.g., radii, luminositiesand lifetimes) are adopted from Hurley, Pols & Tout (2000)who employed detailed evolutionary calculations for WRstars presented later by Pols & Dewi (2002). For the fulldescription of the population synthesis code we refer thereader to Belczynski et al. (2008a).2.2.
Host galaxy metallicity
IC10 is a barred irregular galaxy in the Local Group atthe distance between 600 and 800 kpc (Saha et al. 1996;Sakai, Madore & Freeman, 1999). It is undergoing a veryrapid star formation and has a very high number of WolfRayet stars. IC10 has low metallicity: Lequeux et al.(1997) estimates it to be Z = 0 . Z ⊙ , but later studiesby Massey et al. (2007) place it somewhere between thevalues for LMC and SMC. Thus we conservatively adoptthe value of Z = 0 . Z ⊙ . IC10 is a galaxy with a large star formation rate and it contains more than a hundred WRstars (Massey & Holmes, 2002)The metallicity of the NGC300 galaxy has been mea-sured by Urbaneja et al. (2005). The galaxy exhibits somemetallicity gradients and at the location of NGC300 X-1 it is log( O/H ) + 12 ≈ .
44 which corresponds to the Z = 0 . Z ⊙ (Crowther et al. 2010). RESULTS
The Future Evolution of IC10 X-1
At present IC10 X-1 has an orbital period 34 .
93 hr, blackhole mass is estimated to be 23 −
33 M ⊙ , while its compan-ion is a helium star of a mass 17 −
35 M ⊙ . The system isan eclipsing X-ray source with X-ray luminosity of 2 × erg s − an the lower limit on inclination was placed at 78deg (Silverman & Filipenko 2008).The fate of the WR star is mainly set by the wind massloss. We consider three cases for the IC10 X-1 describingthe current state of the systems: case (a) where the BHmass is 23 M ⊙ and the the WR star mass is 17 M ⊙ ; case (b) where the BH mass is 28 M ⊙ while the WR star massis 25 M ⊙ ; and case (c) where the BH mass is 33 M ⊙ whilethe WR star mass is 35 M ⊙ . The evolution of the WRstars with the metallicity Z = 0 . Z ⊙ (IC10) is followed(see the top panel of Figure 1). The wind mass loss ratedepends strongly on the initial mass of the star and onits metallicity. In each case we assume that the measuredmass corresponds to the initial, unevolved state of the WRstar. In the case (a) the 17 M ⊙ WR star looses 3 M ⊙ , inthe case (b) the 25 M ⊙ it looses 5 M ⊙ , while in the case (c) the 35 M ⊙ star looses 6 M ⊙ over its entire lifetime. Due tothe mass loss from the WR component the orbit expandsslightly (by ∼ ⊙ ) and the period increases to ∼
40 hr.Throughout its evolution the massive WR star, for anyvalue of the adopted mass, is always well within its Rochelobe. For the intermediate mass of 25 M ⊙ , the radius ofWR component is R WR ∼ − ⊙ while the Roche loberadius is R roche ∼ > ⊙ . The Roche lobe volume fillingfactor is of the order of f fill ≡ ( R WR /R roche ) ∼ .
01. TheWR component is filling only 1% of its Roche lobe andthere is no chance of Roche lobe overflow in this system,therefore the only mass transfer proceeds via stellar windas calculated in our model.The WR star eventually undergoes the core-collapse.The WR star, in each case, is so massive that the initialstar mass must have been above M zams = 40 M ⊙ (heliumstar mass is on average about 1 / ⊙ forthe case (a) through 18 M ⊙ for the case (b) to 26 M ⊙ inthe case (c) . Since there is no (or almost no) mass loss inthe core-collapse, the BH most likely receives no (or onlysmall) natal kick.In all cases the binary survives the formation of a sec-ond BH, and thus the IC10 X-1 evolution leads to theformation of a BH-BH binary. The mass loss in neutri-nos induced a small eccentricity e ≈ .
04, however theorbit remains mostly unchanged. The chirp mass of thenewly formed binary varies from 15 M ⊙ for the model (a) through 20 M ⊙ for model (b) and up to 26 M ⊙ in the case (c) . In each case the binary mass and the size of the orbitmake it merge in relatively short time: 2 . (a) , 1 . (b) and 1 . (c) .3.2. The Future Evolution of NGC3000 X-1
NGC300 X-1 has a period of 32 . ⊙ which impliesthe mass of the BH to be 20 M ⊙ . However, if other starscontribute to the measured optical flux than the WR starmass can be 15 M ⊙ and the implied BH mass is 14 . ⊙ .We will refer to the latter estimate as case (a) while theformer one with more massive BH will be denoted as case (b) . The evolution of the WR star at the metallicity ap-propriate for its location within NGC300 ( Z = 0 . Z ⊙ ) iscalculated. The results are shown in the bottom panelof Figure 1. In the case (a) the 15 M ⊙ WR star looses3 M ⊙ , and forms an 11 M ⊙ BH. In the case (b) the WRstar looses 8 M ⊙ in the wind, and forms a 16 M ⊙ BH.Note that mass loss in the case of NGC300 X-1 is rela-tively higher than that of IC10 X-1, as the former systemhost galaxy has more metal rich stars and therefore windmass loss is more efficient. System orbit expands slightly(by ∼ − ⊙ ) and the period increases to ∼ −
47 hrfor case (a) and (b) , respectively. Throughout its evolu-tion the massive WR star ( R WR ∼ − ⊙ ), for any valueof the adopted mass, is always well within its Roche lobe( R roche ∼ > ⊙ ). The Roche lobe volume filling factor isvery small ( f fill ∼ .
01) and there is no chance for Rochelobe overflow.At the time of core collapse the WR component is mas-sive enough to form a BH through direct collapse. Themass of the second BH is 11 M ⊙ (a) or 16 M ⊙ (b) . Thesystem acquires small eccentricity ( e ≈ . . (a) and 3 . (b) . The chirp mass of the BH-BH binary is between 11 M ⊙ (a) and 15 M ⊙ (b) . ThusNGC300 X-1 is another example of a system that willevolve to form a merging BH-BH.3.3. Estimate of the coalescence rate
The Chandra or XMM sensitivity to detect X-ray bi-naries like IC10 X-1 or NGC300 X-1 extends to distancesbeyond the Local Group. However the sensitivity to fullyanalyze such binaries is limited by the possibility of ob-taining spectroscopic orbits. This in turn is limited bythe brightness and spectral properties of the WR stars.The absolute brightness of a massive WR star is about M v ≈ −
5, and a spectroscopic orbit can be measuredfor stars with the apparent magnitude down to m v ≈ ≈ r s , and only a fraction Ω s of the sky has been searchedfor such systems therefore we can estimate the volume inwhich they are detectable as V s = Ω s r s / s = 4 π . This is a conserva-tive assumption, i.e. it underestimate the formation rate of binary BHs, since we overestimate the volume surveyedfor such binaries so far. The lifetime of the IC10 X-1 inthe X-ray bright phase (accretion from intense wind ofWR companion) is not longer than t IC ≈ . t NGC ≈ . ρ IC ≈ V − s t − IC (2) ρ NGC ≈ V − s t − NGC (3) ρ = ρ IC + ρ NGC (4)Assuming that the star formation rate is constant andnoting that the merger times of the BH-BH binaries de-scribed above are significantly smaller than the Hubbletime this is also the estimate of the current BH-BH mergerrate density. A detailed calculation, presented in the Ap-pendix, leads to the estimate of the merger rate density: ρ = 0 . +0 . − . Mpc − Myr − at the 90% confidence level.In order to estimate the detection rate in current grav-itational wave detectors we must take into account thedifferent chirp masses of the two binaries. In order to beconservative we will only consider the low mass cases: (c) for IC10 X-1 with M IC chirp = 15 M ⊙ and (b) for NGC300 X-1 with M NGC chirp = 11 M ⊙ .We assume that the current LIGO sensitivity allows itto detect a NS-NS binary with a chirp mass of 1 . ⊙ to a distance of r NSNS = 18 Mpc. This is the sky aver-aged horizon. The sensitivity range depends on the chirpmass M chirp of a given binary and scales as M / chirp . Thedetection rate of BH-BH inspirals is a sum of the ratesoriginating in IC10 X-1 and in NGC300 X-1 like binaries: R IC = 4 π r NSNS ρ IC M IC chirp . ⊙ ! / (5) R NGC = 4 π r NSNS ρ NGC M NGC chirp . ⊙ ! / (6) R = R IC + R NGC (7)We present the probability density distribution of the to-tal rate R as well as the contributions from each type ofbinary in Figure 2. We have calculated the confidence in-tervals for the total rate, and they are R = 3 . +2 . − . at the68% confidence level, R = 3 . +4 . − . at the 90% confidencelevel, and R = 3 . +8 . − . at the 99% confidence level. DISCUSSION
Potential Caveats
There are two crucial points that lead to determinationof the mass of the BH in both of the systems: (i) the es-timate of the mass of the companion WR star, and (ii)the estimate of its orbital velocity. In the case of (i), theWR star mass, we have used the lowest estimate consis-tent with observations in both cases. Higher masses ofthe WR star imply higher BH masses and only increasethe estimate of the coalescence rate. Thus the uncertaintyfrom the WR mass estimate is not crucial. The secondpoint (ii), the estimate of the orbital velocity may how-ever be overestimated by a systematic effect. It has beensuggested by van Kerkwijk (1993) that the wind could behighly ionized except for the region shadowed by the star.This ionized wind model was originally developed in thecontext of Cyg X-3. In this model large velocities of thelines in the spectrum in this model originate from the windrather then from the orbital velocity of the WR star. Suchan effect would lead to overestimate of the orbital veloc-ity and overestimate of the mass of the mass of the BH.Such an ionized wind model predicts blueshifted spectraat the moment of the X-ray eclipse, and can be distin-guished by simultaneous X-ray and optical observations.Such observations are not yet available for the binariesconsidered here. Moreover, such a model would lead toa different shape of the radial velocity curve, so detailedtime resolved spectroscopy of the system might be help-ful. One needs to note that such an ionized model may bedifficult to apply to IC10-X-1 and NGC300 X-1 because oftheir much longer orbital periods than the orbital periodof Cyg X-3 which is 4.8h.Our conclusion on the lack of ongoing or future Rochelobe overflow is based on the fact that stellar models in-dicate that massive WR stars are very compact R WR ∼ − ⊙ (e.g., Pols & Dewi 2002) while both systems underconsideration are relatively wide, with Roche lobe radii ofWR components of the order of R roche ∼ > − ⊙ . How-ever, it was claimed that some specific physical conditions(iron opacity peak) within WR stars may lead to the ra-dial inflation of the outer atmosphere (e.g., Ishii, Ueno &Kato 1999). This effect was recently examined in detail byPetrovic, Pols & Langer (2006) and it was found that forsub-solar metallicities and realistic calculations (with windmass loss) the WR stars with mass ∼ −
30 M ⊙ remaincompact ( < ⊙ ). Radii of massive WR stars are verydifficult to be observationally determined even if bolomet-ric luminosity and temperature is known (e.g., Hammanet al. 1995; Moffat & Marchenko 1996). Due to strong op-tically thick winds WR stars appear larger than they mostlikely are. For example, the radius of WR component inNGC300 X-1 was observationally determined ∼ − ⊙ (Crowther et al. 2010), while stellar models indicate muchsmaller value.We have assumed that the currently observed WR massis its initial mass. If in fact, the WR star was more mas-sive initially and if by now it has gone through part of itsevolution (as it most likely did) it means that the mass lossfrom now on will be lower (less time left till core-collapse)than we have estimated. Therefore, if anything, we un-derestimate the final mass of WR component and the BHmass it will form. Our mass loss rates are on the highside and thus we may be additionally underestimating themass of the second BH. For example, in case of IC10 X-1the mass loss is estimated at the level of ∼ − M ⊙ yr − (Crowther & Clark 2004) while we employ the wind massloss rates in the range 1 − × − M ⊙ yr − dependingon the adopted mass of WR component (see Fig. 1). Thisis even more clear in the case of NGC300 X-1 for whichmass loss rate is determined to be ∼ × − M ⊙ yr − (Crowther et al. 2010) while we employ much higher rates ∼ × − M ⊙ yr − .Based on high WR star mass in both systems, we havefollowed Fryer & Kalogera (2001) to infer that second BHsin IC10 X-1 and NGC300 X-1 will form through directcollapse of entire star to a BH. Since there is no mass loss in such a case we have assumed zero natal kick in bothcases. This holds true if the natal kicks are connectedwith asymmetry in mass ejection during supernova explo-sion. This has some observational support in the fact thatmost massive BHs in our Galaxy, that are believed to haveformed through direct collapse, show no sign of natal kick(Mirabel & Rodrigues 2003; Dhawan et al. 2007; Martin etal. 2010). If, for some reason, these massive BHs received akick as large as observed for Galactic NSs ( ∼ −
300 kms − ; e.g., Hobbs et al. 2005) both systems would surviveand still form close (but rather eccentric) BH-BH binarieswith coalescence time below Hubble time. The relative or-bital velocity in the pre-explosion binary for both IC10 X-1and NGC300X-1 is ≈ − . Since even large natalkick is not likely to exceed the orbital velocity the binariesare expected to survive (e.g., Kalogera 1996).The estimate of the initial LIGO detection rate is veryhigh and it is actually quite conservative. Including themore accurate fraction of the sky that has been surveyed,leads to a decrease of Ω s and increase of the expected rate.The estimated range r s to obtain spectroscopic orbits ofsuch binaries has been calculated neglecting the poten-tial effects of extinction. Decreasing this range increasesthe expected coalescence rate density. The expected ratedepends on the inverse of assumed lifetime in the X-rayphase, and we have chosen the maximum possible valuesfor the WR lifetime in equation (4).Both IC10 X-1 and NGC300 X-1 are very young sys-tems with current age shorter than t evol ≈
10 Myr (it is anevolutionary lifetime of a M ZAMS = 20 M ⊙ star). There-fore, both systems were formed very recently in Universewith rather low star formation. The merger time for bothsystems was estimated at ∼ − − z ≈ ≈
10 Gyr ago; e.g., Strolger et al. 2004). Sincewe have assumed that the star formation rate (SFR) isconstant as a function of redshift at the level that corre-sponds to the current low SFR (the very young age of bothbinaries) we have again underestimated the formation effi-ciency of similar binaries and the predicted detection ratesare lower limits with respect to the SFR. Moreover, themedian metallicity of galaxies was lower a few billion yearsago when the currently merging systems have formed (e.g.,Pei, Fall & Hauser 1999; Young & Fryer 2007). This couldalso bring the current coalescence rate up as it appearsthat the formation of binaries with massive BHs occurs inlow metallicity host galaxies.On the other hand it is obvious that our arguments arebased on just two objects and therefore are subject to smallnumber statistics and one should treat our results only asan indication of possibility of existence of a large popula-tion of merging massive BH-BH binaries. In summary, thevalue of the rate presented here should be considered as alower limit, while keeping in mind that our arguments andcalculations are based only on two objects.4.2.
Conclusions
We have demonstrated that the future evolution of thebinaries IC10 X-1 and NGC300 X-1 will lead to the for-mation of BH-BH binaries with a merger time of 2 − . − Myr − , which impliesthe detection rate of their coalescences by current LIGOand VIRGO of 3 .
36 yr − . This means that such a mergershould be found in the already gathered LIGO data! Theabsence of a massive BH-BH inspiral signal in the s ∼ − ) that Belczynski et al. (2010a) excluded thismodel as unrealistic in anticipation on no detection in thejust accomplished s s APPENDIX
Let us assume that the formation rate per unit time per unit volume of massive X-ray binaries similar to IC10 X-1 is ρ . Then the expected number of such binaries in a volume V, given that they are X-ray active for a time T, is λ = ρV T .The probability of observing one object is then given by the Poisson distribution: P (1 | λ ) = λe − λ (1)We are interested in measurement of ρ , thus we can use the Bayes theorem to obtain the probability of the rate given asingle observation P ( λ |
1) = P (1 | λ ) P ( λ ) /P (1) (2)where P ( λ ) is the prior probability, and P (1) can be treated as the normalization of the resulting probability distribution.We assume a flat prior P ( ρ ) = const. Given the observed volume V = π R s , where R s ≈ M pc , and the time T IC =0 . M yrs , we obtain the probability distribution of ρ IC , the formation rate of IC10 like systems: dPdρ IC = A ρe − Aρ (3)where A = V T IC . Analogously for NGC300 X-1 like systems we have the probability distribution of the formation rate: dPdρ NGC = B ρe − Bρ (4)where B = V T
NGC , and T NGC = 0 . ρ = ρ IC + ρN GC dPdρ = Z Z dρ IC dρ NGC δ ( ρ − ( ρ IC + ρN GC dPdρ IC dPdρ NGC . (5)A simple calculation yields: dPdρ = A B ( A − B ) (cid:2) e − ρB ( ρ ( A − B ) − − e − ρA ( ρ ( B − A ) − (cid:3) (6). We identify the formation rate with the coalescence rate, which can be justified since the coalescence time of each binaryis smaller than the Hubble time. For model B the rates obtained by Belczynski et al. (2010a) are much lower ( ∼ .
05 yr − ) and are consistent with no detection in s In order to find the probability distribution of the LIGO detection rate coming from each of the binary we notice thatthe probability density of the detection rate is connected with the merger rate via R IC = V GWIC ρ IC , where V GWIC = 4 π r NSNS (cid:18) M IC . M ⊙ (cid:19) / (7)is the volume in which BH-BH from IC10 X-1 like binaries are detectable, r NSNS = 18Mpc is the current sky averagedsensitivity distance for detection by LIGO of binaries with the chirp mass of 1 . M ⊙ and M IC is the chirp mass ofthe BH-BH binary that will form from IC10 X-1. For the chirp mass we conservatively assume the minimum value M IC = 15 M ⊙ . The probability density of the detection rate R IC is dPd R IC = dPdρ IC dρ IC d R IC . (8)Analogously the LIGO probability density of detection rate of BH-BH binaries originating in NGC300 X-1 like binaries is dPd R NGC = dPdρ NGC dρ NGC d R NGC , (9)and we use M NGC = 11 M ⊙ for the calculation. A simple calculation shows that the probability distribution densitiesare dPd R IC = a ρe − bρ (10) dPd R ρ NGC = b ρe − bρ (11)where a = 0 . . R = R IC + R ρ NGC follows along the same lines as above: dPd R = Z Z d R IC dρ NGC δ ( R − ( R IC + R NGC )) dPd R IC dPd R NGC . (12)leads to the result: dPd R = a b ( a − b ) (cid:2) e −R b ( calR ( a − b ) − − e −R a ( calR ( b − a ) − (cid:3) (13). REFERENCESAbadie et al. 2010, CQG, 27, 173001Bauer, F., & Brandt, W. 2004, ApJ, 601, L67Belczynski, K., Kalogera, V., & Bulik, T. 2002a, ApJ, 572, 407Belczynski, K., Bulik, T., & Rudak, B. 2002b, ApJ, 571, 394Belczynski, K., Kalogera, V., Zezas, A., & Fabbiano, G. 2004, ApJ,601, L147Belczynski, K., & Taam, R. 2004, ApJ, 616, 1159Belczynski, K., Taam, R., Kalogera, V., Rasio, F., & Bulik, T. 2007,ApJ, 662, 504Belczynski, K., Bulik, T., & Ruiter, A. 2005, ApJ, 629, 915Belczynski, K., Perna, R., Bulik, T., Kalogera, V., Ivanova, N., &Lamb, D.Q. 2006, ApJ, 648, 1110Belczynski, K., Taam, R., Kalogera, V., Rasio, F., & Bulik, T. 2007,ApJ, 662, 504Belczynski, K., Kalogera, V., Rasio, F., Taam, R., Zezas, A., Bulik,T., Maccarone, T., & Ivanova, N. 2008a, ApJS, 174, 223Belczynski, K., Taam, R., Rantsiou, E., & van der Sluys, M. 2008b,ApJ, 682, 474Belczynski, K., Dominik, M., Bulik, T., O’Shaughnessy, R., Fryer,C., & Holz, D. 2010a, ApJ 715, L138Belczynski, K., Bulik, T., Fryer, C., Ruiter, A., Valsecchi, F., Vink,J., & Hurley, J. 2010b, ApJ, 714, 1217Brandt, W., Ward, M., Fabian, A., & Hodge, P. 1997, MNRAS, 291,709Bulik, T., Belczynski, K., & Rudak, B. 2004, A&A, 415, 407Clark, J., & Crowther, P. 2004, A&A, 414, L45Crowther, P., Carpano, S., Hadfield, L., & Pollock, A. 2007, A&A,469, L31Crowther, P., Drissen, L., Abbott, J., Royer, P., & Smartt, S. 2003,A&A, 404, 483Crowther, P. A.; Barnard, R.; Carpano, S.; Clark, J. S.; Dhillon, V.S.; Pollock, A. M. T., 2010, MNRAS, 403, 41Dewi, J., & Pols, O. 2003, MNRAS, 344, 629 Dhawan, V., et al. 2007, ApJ, 668, 430Fryer, C., & Kalogera, V. 2001, ApJ, 554, 548Hamann, W.-R., Koesterke, L., & Wessolowski, U. 1995, A&A, 299,151Hamann, W.-R., & Koesterke, L. 1998, A&A, 335, 1003Hobbs, G., Lorimer, D., Lyne, A., & Kramer, M. 2005, MNRAS, 360,974Humphreys, R, & Davidson, K. 1994, PASP, 106, 1025Hurley, J., Pols, O., & Tout, C. 2000, MNRAS, 315, 543Ishii, M., Ueno, M., & Kato., M. 1999, PASJ, 51, 417Ivanova, N., Belczynski, K., Kalogera, V., Rasio, F., & Taam, R. E.2003, ApJ, 592,Kalogera, V. 1996, ApJ, 471, 352Kudritzki, R., & Reimers, D. 1978, A&A, 70, 227Lequeux, J., Peimbert, M., Rayo, J., Serrano, A., & Torres-Peimbert,S. 1979, A&A, 80, 155Lorimer, D. 2005, NATO ASIB Proc. 210: The ElectromagneticSpectrum of Neutron Stars, 161Martin, R., Tout, C., & Pringle, J. 2010, MNRAS, 410, 1514Massey, P., & Holmes, S. 2002, ApJ, 580, L35Massey, P., McNeill, R., Olsen, K., Hodge, P., Blaha, C., Jacoby, G.,Smith, R., & Strong, S. 2007, AJ, 134, 2474Mirabel, F., & Rodrigues, I. 2003, Science, 300, 1119Moffat, A., & Marchenko, S. 1996, A&A, 305, L29Nieuwenhuijzen, H., & de Jager, C. 1990, A&A, 231, 134Nugis, T., & Lamers, H. 2000, A&A, 360, 227Orosz, J., et al. 2007, Nature, 449, 872Panter B., et al. 2008, MNRAS, 391, 1117Pei, Y., Fall, M., & Hauser, M. 1999, ApJ, 522, 604Petrovic, J., Pols, O., & Langer, N. 2006, A&A, 450, 219Pols, O., & Dewi, J. 2002, PASP, 19, 233
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Fig. 1.—
The mass loss rates from WR stars as a function of age in the two systems considered in the paper. The top panel correspondsto IC10 X-1 and the bottom panel shows the case of NGC300 X-1. Each line is labeled by the initial mass of the star on the left hand side.On the right hand side we show the final mass of the star, and the mass of the compact object (BH) formed as a result of the evolution.
Fig. 2.—
The probability distributions of the total detection rate in LIGO - thick solid line, along with the contributions from each of thebinaries: IC10 X-1 and NGC300 X-1 - thin solid lines. The dashed lines denote the levels and intervals corresponding to the ranges containing68%, 90%, and 99% of probability. The empirical estimate of the detection rate of BH-BH binaries for the initial LIGO is R = 3 . +8 . − .92