Black hole spin will have a large impact on searches for gravitational waves
with advanced detectors. While only a few stellar mass black hole spins have
been measured using X-ray techniques, gravitational wave detectors have the
capacity to greatly increase the statistics of black hole spin measurements. We
show what we might learn from these measurements and how the black hole spin
values are influenced by their formation channels.Comment: 4 pages, 2 figures, pre-GW150914 detection, refereed and accepted
  contribution to proceedings of 11th Edoardo Amaldi Conference on
  Gravitational Waves, June 2015, Gwangju, Kore

Nielsen, Alex B.

Journal of Physics : Conference Series

English

arXiv

Black hole spin will have a large impact on searches for gravitational waves with advanced detectors. While only a few stellar mass black hole spins have been measured using X-ray techniques, gravitational wave detectors have the capacity to greatly increase the statistics of black hole spin measurements. We show what we might learn from these measurements and how the black hole spin values are influenced by their formation channels

Nielsen, A.

MPG.PuRe

On the distribution of stellar-sized black hole spins

Alex B. Nielsen

Crossref

Journal of Physics Conference Series

On the distribution of stellar-sized black hole spinsAlex B. NielsenMax Planck Institut für Gravitationsphysik, Callinstrasse 38, D-30167 Hannover, Germanyand Leibniz Universität Hannover, Welfengarten 1-A, D-30167 Hannover, GermanyAbstract. Black hole spin will have a large impact on searches for gravitational waves withadvanced detectors. While only a few stellar mass black hole spins have been measured using X-ray techniques, gravitational wave detectors have the capacity to greatly increase the statisticsof black hole spin measurements. We show what we might learn from these measurements andhow the black hole spin values are influenced by their formation channels.1. IntroductionGravitational wave observations will be sensitive to the spins of black holes [3]. Simulated searchpipelines indicate that accounting for black hole spin can substantially improve the sensitivityof gravitational wave searches for binaries containing one [2] or two [15] black holes. However,this improvement comes at a cost of increased computational resources and is only realised ifthe spins of black holes are indeed significant. It is therefore necessary to balance the increasedcost with the expectation that black holes will indeed have large spins. Here we discuss thatevidence.Truly isolated black holes are expected to satisfy the Kerr bound, a∗ ≡ cJ/GM2 < 1, whereJ and M denote the angular momentum and mass of the black hole and c and G the speed oflight and Newton’s gravitational constant respectively. This bound is manifested in the formulafor the spatial coordinate of the event horizon in the Kerr solution, r = M +M√1 − a2∗.The value of a∗ can formally be computed for any object that has a mass and angularmomentum and it is worth recalling that the Kerr bound is only a bound on black holes. Manynon-compact objects such as the Earth, the Sun and extremely rapidly rotating massive starslike VFTS 102[4] do not satisfy this bound, whilst compact objects like the rapidly rotatingmillisecond pulsar PSR J1748-2446ad [6] and the near-extremal black hole candidate CygnusX-1[5] do. This is described in Table 1 where the spin values for the solid objects assume theyare constant density spheres.While the exterior spacetimes of these objects are all approximately vacuum, axisymmetricand stationary as required by the Kerr solution, except for Cygnus X-1 they do not describeblack holes (do not contain horizons) and so can have a different multipole structure to the Kerrsolution and are not constrained by the Kerr bound.Things are even more extreme for elementary particles. The quantum spin of an elementaryparticle can be related to an asymptotic classical angular momentum in the sense of the Einstein-de Haas effect. In this way we can simply calculate values for the mass, specific angularmomentum and charge of elementary particles. An electron’s mass is 2.3 × 10−66secs and itsspecific angular momentum is 6.4 × 10−22 so its a∗ value is a whopping 2.8 × 1044. (In factthings are even more extreme in the context of the Kerr-Newman solution as the electron has a11th Edoardo Amaldi Conference on Gravitational Waves (AMALDI 11) IOP PublishingJournal of Physics: Conference Series 716 (2016) 012002 doi:10.1088/1742-6596/716/1/012002Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd 1Figure 1. Masses and spins for 10 black holeswith approximate error bars. The three high-mass-X-ray-binary systems, LMC X-1, CygnusX-1 and M33 X-7 are indicated by names abovethe line and in red online.Figure 2. Distribution (upper bound) ofspins for black holes in NSBH binaries, withmasses greater than 2M, assuming NS natalspins of 0, BH natal spins of 0.5 and mergingin 15 billion years. Adapted from [14].square root charge of 4.6 × 10−45 seconds). The reason for the violation of the Kerr bound inthese cases is that elementary particles typically have Planckian values of spin, but not of mass.Of the Standard Model elementary particles, only the Higg’s boson satisfies the Kerr boundbecause its charge and spin are thought to be zero.Object Mass [s] J/M [s] a∗Earth 1.5 × 10−11 1.3 × 10−8 895Sun 4.9 × 10−6 6.1 × 10−6 1.2VFTS 102 1.2 × 10−4 9.3 × 10−3 75PSR J1748-2446ad 6.9 × 10−6 2.9 × 10−6 0.4Cygnus X-1 7.30 × 10−5 7.23 × 10−5 0.99Table 1. Approximate values of mass and specific angular momentum for the Earth, Sun, arapidly spinning massive star VFTS 102, a rapidly spinning neutron star PSR J1748-2446ad anda rapidly spinning black hole Cygnus X-1. For ease of comparison, both the mass and specificangular momentum values are given in seconds.2. X-ray observationsX-ray observations of accretion disks have been able to measure the spins of around 10 stellarmass black holes [11]. These are displayed in Fig 1.Of these most are Low Mass X-ray Binary (LMXB) sources and are unlikely to form thedouble compact object systems necessary to be seen by the current generation of ground-basedgravitational wave detectors. Three of the systems (LMC X-3, M33 X-7 and Cygnus X-1) areHigh Mass X-ray Binaries (HMXB) where the companion to the black hole is a massive star,with mass greater than ∼ 10M and these systems do have a chance to form either neutronstar-black hole binaries or binary black holes systems. These systems in themselves are nottargets for current ground-based GW observatories as they are many millions of years awayfrom merging but it is interesting to note that all three of these HMXBs have black holes withlarge spins a∗ > 0.85 and this suggests that the population of black holes in compact binarysystems might be dominated by black holes with large spins. The probability of obtaining threevalues all above 0.7 from a flat distribution is only 3% .11th Edoardo Amaldi Conference on Gravitational Waves (AMALDI 11) IOP PublishingJournal of Physics: Conference Series 716 (2016) 012002 doi:10.1088/1742-6596/716/1/0120022On the other hand, observations of neutron star pulsars suggest that the spins of neutronstars in compact binaries are very low a∗ < 0.05. If this is indeed the case it implies thatthe formation channels of these objects are very different in terms of their angular momentumevolution. Neutron stars may be born with low spins or they may spin down rapidly after birth.The observation of young slowly spinning pulsars, such as the Crab pulsar rotating at only∼ 30Hz with a spin down rate of ∼ 0.01Hz per year, 1000 years after its birth, constrains modelsof rapid spin-down for neutron stars.3. Before collapse - tidal lockingTo form a compact binary system requires two large heavy stars in a binary. Single massivestars are typically slowly rotating at the end of their lives dues to stellar wind losses and notdifferentially rotating due to angular momentum redistribution by magnetic torques [1]. Butstars in binaries can spin up, increasing their rotational angular momentum by mass transfer [10].The spin rate of both stars can be affected by tides and mass transfer [10]. Tidal interactions willtend to circularize the orbits of massive stars[7]. Massive stars in binaries can have rotationalvelocities ∼ 200km/s [10].Close binary systems are expected to be tidally locked [8]. The tidal locking timescale isbetween 100 and 10,000 years for helium core stars before they turn into black holes [12]. Thetidal locking is more efficient on the larger object the closer to equal mass the bodies, perhapsgiving an explanation why BHs in LMXBs have a wide range of spins, but predominantly highspins in HMXBs. It has been shown that there is a correlation between the black hole massand orbital period in X-ray transients that may be partly explained by tidal locking [9]. Forsimplicity here we will assume that the moment of inertia for each star is given by kmr2 wherek is a constant (equal to 2/5 for solid balls of uniform density). The relationship between radiusand mass is often assumed to be r ∼ mα where typical values of α for massive stars are 0.5 to0.8. Then we find that the dimensionless spin parameter a∗ for an object with mass m1 is givenbya∗ = k√c2rGm(ra)3/2(m1m)2α−1√m1m+m2m. (1)where a denotes the semi-major axis of the orbit. Note that the three HMXB systems wouldhave a∗ > 1 if they were tidally locked stars at their current mass and separation values.Observed rotation rates of massive stars in binaries tend to be significantly higher than thetidal locking rate [10], since a 10 − 10 binary would need a separation of 10r to be tidallylocked if the rotational velocity is ∼ 300km/s, corresponding to a∗ ∼ 100. Stars that are furtherapart will have a lower rotational velocity. A separation of 1000 solar radii would have a tidallylocked velocity of ∼ 300m/s, since a∗ ∼ a−3/2.4. During collapse - core collapse and supernovaeDifferent scenarios have been proposed for how collapse proceeds for massive stars. Thesedistinguish between failed core collapse (collapsar) with no explosion and successful core collapse(supernova) where the outer envelope is partially ejected [17]. Direct collapse may be favouredfor heavy systems as the energy flux from the core is not enough to expel the outer shell. If astar collapses directly to a black hole, without shedding any matter and conserving its mass andits angular momentum, then its a∗ value would be conserved.For failed core collapse models, O’Connor and Ott [13] find a linear relationship betweeninitial specific angular momentum of the star and final black hole spin parameter (their fig 11).The relation is approximately j = 2.75a∗. The specific angular momentum before collapse, j,11th Edoardo Amaldi Conference on Gravitational Waves (AMALDI 11) IOP PublishingJournal of Physics: Conference Series 716 (2016) 012002 doi:10.1088/1742-6596/716/1/0120023can be calculated using the tidal lock model above, to obtain a final a∗ after collapse of,a∗ = 155k(ra)3/2( mm)2α+0.5. (2)It should be noted that for rapidly spinning black holes, centrifugal support will tend to preventthe black hole from accreting large amounts of matter during a collapse scenario. If an initial,near extremal black hole forms, subsequent shells with a∗ > 1 will not be accreted but expelled[9]. In this case, as in the successful core collapse, detailed modeling is required to derive finalmass and spin values for the remnant.5. After collapse - common envelope phaseThe distribution of upper limits on the final BH spins, assuming natal spins of zero for neutronstars and 0.5 for black holes, for merging NSBH systems in [14] is shown in Fig. 2. This showsthat if accretion in the common envelope phase does behave as a thin disk, then the final spinsof BHs in NSBH binaries may be quite large, with most values falling in the range 0.8 to 1.0.The distribution in Fig. 2 actually contains two distinct populations with different formationchannels. Most of the high spin systems are actually light black holes that formed initially aftercollapse as neutron stars and subsequently accreted sufficient material during common envelopeto form black holes, where the threshold for black hole formation is set at 2M. For these lightsystems, the accretion of a small amount of matter is enough to spin them up considerably.The other distribution consists of objects that collapsed to black holes before the commonenvelope phase. These systems are typically more massive and hence do not spin up as muchduring common envelope.6. ConclusionThe limited X-ray evidence suggests that black holes in compact binaries may have high spins.We have discussed several mechanisms that could lead to these high spins, including tidal lockingof massive progenitor stars, rapid collapse to black holes and common envelope accretion. Thereare still large uncertainties of all these processes, although they tend to reinforce one another.Measuring the spins of a large number of black holes with gravitational wave detectors will helpto constrain some of these model uncertainties.AcknowledgmentsThis work was supported by the Max Planck Society. The author would like to thank RichardO’Shaughnessy for helpful conversations and sharing data for Fig. 2.References[1] E. Abdikamalov, S. Gossan, A. M. DeMaio and C. D. Ott, arXiv:1311.3678 [astro-ph.SR].[2] T. Dal Canton et al., Phys. Rev. D 90 (2014) 8, 082004 [arXiv:1405.6731 [gr-qc]].[3] C. Cutler et al., Phys. Rev. Lett. 70 (1993) 2984 [astro-ph/9208005].[4] P. L. Dufton et al., Astrophys. J. 743 (2011) L22 [arXiv:1111.0157 [astro-ph.SR]].[5] L. Gou, J. E. McClintock, M. J. Reid, J. A. Orosz, J. F. Steiner, R. Narayan, J. Xiang and R. A. Remillardet al., Astrophys. J. 742 (2011) 85 [arXiv:1106.3690 [astro-ph.HE]].[6] J. W. T. Hessels, S. M. Ransom, I. H. Stairs, P. C. C. Freire, V. M. Kaspi and F. Camilo, Science 311(2006) 1901 [astro-ph/0601337].[7] J. R. Hurley, C. A. Tout and O. R. Pols, Mon. Not. Roy. Astron. Soc. 329 (2002) 897 [astro-ph/0201220].[8] K. R. Lang, Essential Astrophysics, Springer-Verlag, Berlin (2013)[9] C. H. Lee, G. E. Brown and R. A. M. J. Wijers, Astrophys. J. 575 (2002) 996 doi:10.1086/341349 [astro-ph/0109538].[10] S. E. de Mink, N. Langer, R. G. Izzard, H. Sana and A. de Koter, Astrophys. J. 764 (2013) 166[arXiv:1211.3742 [astro-ph.SR]].11th Edoardo Amaldi Conference on Gravitational Waves (AMALDI 11) IOP PublishingJournal of Physics: Conference Series 716 (2016) 012002 doi:10.1088/1742-6596/716/1/0120024[11] J. E. McClintock et al., Class. Quant. Grav. 28 (2011) 114009 [arXiv:1101.0811 [astro-ph.HE]].[12] E. M. Mendez, MNRAS 413, (2011) 183 arXiv:1011.6385 [astro-ph.HE].[13] E. O’Connor and C. D. Ott, Astrophys. J. 730 (2011) 70 [arXiv:1010.5550 [astro-ph.HE]].[14] R. W. O’Shaughnessy, J. Kaplan, V. Kalogera and K. Belczynski, Astrophys. J. 632 (2005) 1035 [astro-ph/0503219].[15] S. Privitera et al., Phys. Rev. D 89 (2014) 2, 024003 [arXiv:1310.5633 [gr-qc]].[16] H. Sana, A. de Koter, S. E. de Mink, P. R. Dunstall, C. J. Evans, V. Henault-Brunet, J. M. Apellaniz andO. H. Ramirez-Agudelo et al., arXiv:1209.4638 [astro-ph.SR].[17] S. E. Woosley, Astrophys. J. 405 (1993) 273.11th Edoardo Amaldi Conference on Gravitational Waves (AMALDI 11) IOP PublishingJournal of Physics: Conference Series 716 (2016) 012002 doi:10.1088/1742-6596/716/1/0120025

Black hole spin will have a large impact on searches for gravitational waves with advanced detectors. While only a few stellar mass black hole spins have been measured using X- ray techniques, gravitational wave detectors have the capacity to greatly increase the statistics of black hole spin measurements. We show what we might learn from these measurements and how the black hole spin values are influenced by their formation channels

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This content has been downloaded from IOPscience. Please scroll down to see the full text.Download details:IP Address: 194.95.158.13This content was downloaded on 18/11/2016 at 08:42Please note that terms and conditions apply.You may also be interested in:Perturbative evolution of particle orbits around Kerr black holesRamón López-Alemán, Gaurav Khanna and Jorge PullinRXTE Observations of GRS 1915+105J. Greiner, E.H. Morgan and R.A. RemillardNext-to-next-to-leading order spin--orbit effects in the gravitational wave flux and orbitalphasing of compact binariesAlejandro Bohé, Sylvain Marsat and Luc BlanchetElectromagnetic self-forces and generalized Killing fieldsAbraham I HarteOn the distribution of stellar-sized black hole spinsView the table of contents for this issue, or go to the journal homepage for more2016 J. Phys.: Conf. Ser. 716 012002(http://iopscience.iop.org/1742-6596/716/1/012002)Home Search Collections Journals About Contact us My IOPscienceOn the distribution of stellar-sized black hole spinsAlex B. NielsenMax Planck Institut für Gravitationsphysik, Callinstrasse 38, D-30167 Hannover, Germanyand Leibniz Universität Hannover, Welfengarten 1-A, D-30167 Hannover, GermanyAbstract. Black hole spin will have a large impact on searches for gravitational waves withadvanced detectors. While only a few stellar mass black hole spins have been measured using X-ray techniques, gravitational wave detectors have the capacity to greatly increase the statisticsof black hole spin measurements. We show what we might learn from these measurements andhow the black hole spin values are influenced by their formation channels.1. IntroductionGravitational wave observations will be sensitive to the spins of black holes [3]. Simulated searchpipelines indicate that accounting for black hole spin can substantially improve the sensitivityof gravitational wave searches for binaries containing one [2] or two [15] black holes. However,this improvement comes at a cost of increased computational resources and is only realised ifthe spins of black holes are indeed significant. It is therefore necessary to balance the increasedcost with the expectation that black holes will indeed have large spins. Here we discuss thatevidence.Truly isolated black holes are expected to satisfy the Kerr bound, a∗ ≡ cJ/GM2 < 1, whereJ and M denote the angular momentum and mass of the black hole and c and G the speed oflight and Newton’s gravitational constant respectively. This bound is manifested in the formulafor the spatial coordinate of the event horizon in the Kerr solution, r = M +M√1 − a2∗.The value of a∗ can formally be computed for any object that has a mass and angularmomentum and it is worth recalling that the Kerr bound is only a bound on black holes. Manynon-compact objects such as the Earth, the Sun and extremely rapidly rotating massive starslike VFTS 102[4] do not satisfy this bound, whilst compact objects like the rapidly rotatingmillisecond pulsar PSR J1748-2446ad [6] and the near-extremal black hole candidate CygnusX-1[5] do. This is described in Table 1 where the spin values for the solid objects assume theyare constant density spheres.While the exterior spacetimes of these objects are all approximately vacuum, axisymmetricand stationary as required by the Kerr solution, except for Cygnus X-1 they do not describeblack holes (do not contain horizons) and so can have a different multipole structure to the Kerrsolution and are not constrained by the Kerr bound.Things are even more extreme for elementary particles. The quantum spin of an elementaryparticle can be related to an asymptotic classical angular momentum in the sense of the Einstein-de Haas effect. In this way we can simply calculate values for the mass, specific angularmomentum and charge of elementary particles. An electron’s mass is 2.3 × 10−66secs and itsspecific angular momentum is 6.4 × 10−22 so its a∗ value is a whopping 2.8 × 1044. (In factthings are even more extreme in the context of the Kerr-Newman solution as the electron has a11th Edoardo Amaldi Conference on Gravitational Waves (AMALDI 11) IOP PublishingJournal of Physics: Conference Series 716 (2016) 012002 doi:10.1088/1742-6596/716/1/012002Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd 1Figure 1. Masses and spins for 10 black holeswith approximate error bars. The three high-mass-X-ray-binary systems, LMC X-1, CygnusX-1 and M33 X-7 are indicated by names abovethe line and in red online.Figure 2. Distribution (upper bound) ofspins for black holes in NSBH binaries, withmasses greater than 2M, assuming NS natalspins of 0, BH natal spins of 0.5 and mergingin 15 billion years. Adapted from [14].square root charge of 4.6 × 10−45 seconds). The reason for the violation of the Kerr bound inthese cases is that elementary particles typically have Planckian values of spin, but not of mass.Of the Standard Model elementary particles, only the Higg’s boson satisfies the Kerr boundbecause its charge and spin are thought to be zero.Object Mass [s] J/M [s] a∗Earth 1.5 × 10−11 1.3 × 10−8 895Sun 4.9 × 10−6 6.1 × 10−6 1.2VFTS 102 1.2 × 10−4 9.3 × 10−3 75PSR J1748-2446ad 6.9 × 10−6 2.9 × 10−6 0.4Cygnus X-1 7.30 × 10−5 7.23 × 10−5 0.99Table 1. Approximate values of mass and specific angular momentum for the Earth, Sun, arapidly spinning massive star VFTS 102, a rapidly spinning neutron star PSR J1748-2446ad anda rapidly spinning black hole Cygnus X-1. For ease of comparison, both the mass and specificangular momentum values are given in seconds.2. X-ray observationsX-ray observations of accretion disks have been able to measure the spins of around 10 stellarmass black holes [11]. These are displayed in Fig 1.Of these most are Low Mass X-ray Binary (LMXB) sources and are unlikely to form thedouble compact object systems necessary to be seen by the current generation of ground-basedgravitational wave detectors. Three of the systems (LMC X-3, M33 X-7 and Cygnus X-1) areHigh Mass X-ray Binaries (HMXB) where the companion to the black hole is a massive star,with mass greater than ∼ 10M and these systems do have a chance to form either neutronstar-black hole binaries or binary black holes systems. These systems in themselves are nottargets for current ground-based GW observatories as they are many millions of years awayfrom merging but it is interesting to note that all three of these HMXBs have black holes withlarge spins a∗ > 0.85 and this suggests that the population of black holes in compact binarysystems might be dominated by black holes with large spins. The probability of obtaining threevalues all above 0.7 from a flat distribution is only 3% .11th Edoardo Amaldi Conference on Gravitational Waves (AMALDI 11) IOP PublishingJournal of Physics: Conference Series 716 (2016) 012002 doi:10.1088/1742-6596/716/1/0120022On the other hand, observations of neutron star pulsars suggest that the spins of neutronstars in compact binaries are very low a∗ < 0.05. If this is indeed the case it implies thatthe formation channels of these objects are very different in terms of their angular momentumevolution. Neutron stars may be born with low spins or they may spin down rapidly after birth.The observation of young slowly spinning pulsars, such as the Crab pulsar rotating at only∼ 30Hz with a spin down rate of ∼ 0.01Hz per year, 1000 years after its birth, constrains modelsof rapid spin-down for neutron stars.3. Before collapse - tidal lockingTo form a compact binary system requires two large heavy stars in a binary. Single massivestars are typically slowly rotating at the end of their lives dues to stellar wind losses and notdifferentially rotating due to angular momentum redistribution by magnetic torques [1]. Butstars in binaries can spin up, increasing their rotational angular momentum by mass transfer [10].The spin rate of both stars can be affected by tides and mass transfer [10]. Tidal interactions willtend to circularize the orbits of massive stars[7]. Massive stars in binaries can have rotationalvelocities ∼ 200km/s [10].Close binary systems are expected to be tidally locked [8]. The tidal locking timescale isbetween 100 and 10,000 years for helium core stars before they turn into black holes [12]. Thetidal locking is more efficient on the larger object the closer to equal mass the bodies, perhapsgiving an explanation why BHs in LMXBs have a wide range of spins, but predominantly highspins in HMXBs. It has been shown that there is a correlation between the black hole massand orbital period in X-ray transients that may be partly explained by tidal locking [9]. Forsimplicity here we will assume that the moment of inertia for each star is given by kmr2 wherek is a constant (equal to 2/5 for solid balls of uniform density). The relationship between radiusand mass is often assumed to be r ∼ mα where typical values of α for massive stars are 0.5 to0.8. Then we find that the dimensionless spin parameter a∗ for an object with mass m1 is givenbya∗ = k√c2rGm(ra)3/2(m1m)2α−1√m1m+m2m. (1)where a denotes the semi-major axis of the orbit. Note that the three HMXB systems wouldhave a∗ > 1 if they were tidally locked stars at their current mass and separation values.Observed rotation rates of massive stars in binaries tend to be significantly higher than thetidal locking rate [10], since a 10 − 10 binary would need a separation of 10r to be tidallylocked if the rotational velocity is ∼ 300km/s, corresponding to a∗ ∼ 100. Stars that are furtherapart will have a lower rotational velocity. A separation of 1000 solar radii would have a tidallylocked velocity of ∼ 300m/s, since a∗ ∼ a−3/2.4. During collapse - core collapse and supernovaeDifferent scenarios have been proposed for how collapse proceeds for massive stars. Thesedistinguish between failed core collapse (collapsar) with no explosion and successful core collapse(supernova) where the outer envelope is partially ejected [17]. Direct collapse may be favouredfor heavy systems as the energy flux from the core is not enough to expel the outer shell. If astar collapses directly to a black hole, without shedding any matter and conserving its mass andits angular momentum, then its a∗ value would be conserved.For failed core collapse models, O’Connor and Ott [13] find a linear relationship betweeninitial specific angular momentum of the star and final black hole spin parameter (their fig 11).The relation is approximately j = 2.75a∗. The specific angular momentum before collapse, j,11th Edoardo Amaldi Conference on Gravitational Waves (AMALDI 11) IOP PublishingJournal of Physics: Conference Series 716 (2016) 012002 doi:10.1088/1742-6596/716/1/0120023can be calculated using the tidal lock model above, to obtain a final a∗ after collapse of,a∗ = 155k(ra)3/2( mm)2α+0.5. (2)It should be noted that for rapidly spinning black holes, centrifugal support will tend to preventthe black hole from accreting large amounts of matter during a collapse scenario. If an initial,near extremal black hole forms, subsequent shells with a∗ > 1 will not be accreted but expelled[9]. In this case, as in the successful core collapse, detailed modeling is required to derive finalmass and spin values for the remnant.5. After collapse - common envelope phaseThe distribution of upper limits on the final BH spins, assuming natal spins of zero for neutronstars and 0.5 for black holes, for merging NSBH systems in [14] is shown in Fig. 2. This showsthat if accretion in the common envelope phase does behave as a thin disk, then the final spinsof BHs in NSBH binaries may be quite large, with most values falling in the range 0.8 to 1.0.The distribution in Fig. 2 actually contains two distinct populations with different formationchannels. Most of the high spin systems are actually light black holes that formed initially aftercollapse as neutron stars and subsequently accreted sufficient material during common envelopeto form black holes, where the threshold for black hole formation is set at 2M. For these lightsystems, the accretion of a small amount of matter is enough to spin them up considerably.The other distribution consists of objects that collapsed to black holes before the commonenvelope phase. These systems are typically more massive and hence do not spin up as muchduring common envelope.6. ConclusionThe limited X-ray evidence suggests that black holes in compact binaries may have high spins.We have discussed several mechanisms that could lead to these high spins, including tidal lockingof massive progenitor stars, rapid collapse to black holes and common envelope accretion. Thereare still large uncertainties of all these processes, although they tend to reinforce one another.Measuring the spins of a large number of black holes with gravitational wave detectors will helpto constrain some of these model uncertainties.AcknowledgmentsThis work was supported by the Max Planck Society. The author would like to thank RichardO’Shaughnessy for helpful conversations and sharing data for Fig. 2.References[1] E. Abdikamalov, S. Gossan, A. M. DeMaio and C. D. Ott, arXiv:1311.3678 [astro-ph.SR].[2] T. Dal Canton et al., Phys. Rev. D 90 (2014) 8, 082004 [arXiv:1405.6731 [gr-qc]].[3] C. Cutler et al., Phys. Rev. Lett. 70 (1993) 2984 [astro-ph/9208005].[4] P. L. Dufton et al., Astrophys. J. 743 (2011) L22 [arXiv:1111.0157 [astro-ph.SR]].[5] L. Gou, J. E. McClintock, M. J. Reid, J. A. Orosz, J. F. Steiner, R. Narayan, J. Xiang and R. A. Remillardet al., Astrophys. J. 742 (2011) 85 [arXiv:1106.3690 [astro-ph.HE]].[6] J. W. T. Hessels, S. M. Ransom, I. H. Stairs, P. C. C. Freire, V. M. Kaspi and F. Camilo, Science 311(2006) 1901 [astro-ph/0601337].[7] J. R. Hurley, C. A. Tout and O. R. Pols, Mon. Not. Roy. Astron. Soc. 329 (2002) 897 [astro-ph/0201220].[8] K. R. Lang, Essential Astrophysics, Springer-Verlag, Berlin (2013)[9] C. H. Lee, G. E. Brown and R. A. M. J. Wijers, Astrophys. J. 575 (2002) 996 doi:10.1086/341349 [astro-ph/0109538].[10] S. E. de Mink, N. Langer, R. G. Izzard, H. Sana and A. de Koter, Astrophys. J. 764 (2013) 166[arXiv:1211.3742 [astro-ph.SR]].11th Edoardo Amaldi Conference on Gravitational Waves (AMALDI 11) IOP PublishingJournal of Physics: Conference Series 716 (2016) 012002 doi:10.1088/1742-6596/716/1/0120024[11] J. E. McClintock et al., Class. Quant. Grav. 28 (2011) 114009 [arXiv:1101.0811 [astro-ph.HE]].[12] E. M. Mendez, MNRAS 413, (2011) 183 arXiv:1011.6385 [astro-ph.HE].[13] E. O’Connor and C. D. Ott, Astrophys. J. 730 (2011) 70 [arXiv:1010.5550 [astro-ph.HE]].[14] R. W. O’Shaughnessy, J. Kaplan, V. Kalogera and K. Belczynski, Astrophys. J. 632 (2005) 1035 [astro-ph/0503219].[15] S. Privitera et al., Phys. Rev. D 89 (2014) 2, 024003 [arXiv:1310.5633 [gr-qc]].[16] H. Sana, A. de Koter, S. E. de Mink, P. R. Dunstall, C. J. Evans, V. Henault-Brunet, J. M. Apellaniz andO. H. Ramirez-Agudelo et al., arXiv:1209.4638 [astro-ph.SR].[17] S. E. Woosley, Astrophys. J. 405 (1993) 273.11th Edoardo Amaldi Conference on Gravitational Waves (AMALDI 11) IOP PublishingJournal of Physics: Conference Series 716 (2016) 012002 doi:10.1088/1742-6596/716/1/0120025

On the distribution of stellar-sized black hole spins

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