Dynamical Interactions and the Black Hole Merger Rate of the Universe Binary black holes can form efficiently in dense young stellar clusters, such as the progenitors of globular clusters, via a combination of gravitational segregation and cluster evaporation. We use simple analytic arguments supported by detailed N-body simulations to determine how frequently black holes born in a single stellar cluster should form binaries, be ejected from the cluster, and merge through the emission of gravitational radiation. We then convolve this ``transfer function'' relating cluster formation to black hole mergers with (i) the distribution of observed cluster masses and (ii) the star formation history of the universe, assuming that a significant fraction gcl of star formation occurs in clusters and that a significant fraction gcand of clusters undergo this segregation and evaporation process. We predict future ground--based gravitational wave (GW) detectors could observe ~500 (gcl/0.5) (gcand/0.1) double black hole mergers per year, and the presently operating LIGO interferometer would have a chance (50%) at detecting a merger during its first full year of science data. [astro-ph/0701887] Dynamical Interactions and the Black Hole Merger Rate of the Universe Dynamical Interactions and the Black Hole Merger Rate of the Universe Ryan M. O’Leary roleary@cfa.harvard.edu Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138 Richard O’Shaughnessy oshaughn@northwestern.edu Frederic A. Rasio rasio@northwestern.edu Northwestern University, Department of Physics and Astronomy, 2132 Tech Drive, Evanston, IL 60208 (August 16, 2007) Abstract Binary black holes can form efficiently in dense young stellar clusters, such as the progenitors of globular clusters, via a combination of gravitational segregation and cluster evaporation. We use simple analytic arguments supported by detailed N 𝑁 N -body simulations to determine how frequently black holes born in a single stellar cluster should form binaries, be ejected from the cluster, and merge through the emission of gravitational radiation. We then convolve this “transfer function” relating cluster formation to black hole mergers with (i) the distribution of observed cluster masses and (ii) the star formation history of the universe, assuming that a significant fraction g cl subscript 𝑔 cl {g_{\rm cl}} of star formation occurs in clusters and that a significant fraction g evap subscript 𝑔 evap {g_{\rm evap}} of clusters undergo this segregation and evaporation process. pacs: 04.30.Db, 95.55.Ym, and 97.60.Lf Given our understanding of how isolated binary stars evolve, noninteracting stellar systems should produce relatively few double black hole (BH-BH) binaries tight enough to merge through the emission of GW within the age of the universe Belczynski et al. ( 2006 ); O’Shaughnessy et al. ( 2005 ); Belczynski et al. ( 2002 ) . Portegies Zwart and McMillan ( 2000 ) demonstrated that interactions between black holes (BHs) in dense cluster environments could produce merging BH-BH binaries much more efficiently than through the evolution of isolated binaries. Thermal equilibrium with the surrounding stellar cluser is only restored when so few BHs remain that the subcluster’s internal timescale once again becomes commensurate with the cluster’s interaction timescale. This process of segregation, decoupling, and evaporation has been extensively studied theoretically Kulkarni et al. ( 1993 ); Sigurdsson and Hernquist ( 1993 ) and numerically, both in full N 𝑁 N -body simulations ( N ≈ 10 3 − 10 5 𝑁 superscript 10 3 superscript 10 5 N\approx 10^{3}-10^{5} ) Merritt et al. ( 2004 ); Portegies Zwart and McMillan ( 2000 ); Portegies Zwart et al. ( 2006 ); Khalisi et al. ( 2006 ) and in using approximations in larger N 𝑁 N systems Gurkan et al. Unless suppressed strongly (i.e., g cl ​ g evap ≪ 10 − 2 much-less-than subscript 𝑔 cl subscript 𝑔 evap superscript 10 2 {g_{\rm cl}}{g_{\rm evap}}\ll 10^{-2} ), the BH-BH merger rate due to clusters will significantly exceed the average rate densities expected from isolated stellar evolution, ≈ 10 − 2 ​ Mpc − 3 ​ Myr − 1 absent superscript 10 2 superscript Mpc 3 superscript Myr 1 \approx 10^{-2}{\rm Mpc}^{-3}{\rm Myr}^{-1} Belczynski et al. ( 2002 ); O’Shaughnessy et al. ( 2007 ) . For example, in an optimistic case ( g cl = g evap = 1 subscript 𝑔 cl subscript 𝑔 evap 1 {g_{\rm cl}}={g_{\rm evap}}=1 ), the initial LIGO network could detect roughly 10 10 10 events per year, based on an estimated network range to 14 ​ M ⊙ + 14 ​ M ⊙ 14 subscript 𝑀 direct-product 14 subscript 𝑀 direct-product 14M_{\odot}+14M_{\odot} binaries of 125 ​ M ​ p ​ c 125 M p c 125{\rm Mpc} ; the “enhanced LIGO” upgrade, with roughly twice the sensitivity, should see roughly 2 15 / 6 superscript 2 2 2 2 The factor relating the initial and advanced LIGO detection rates is not precisely geometrical (i.e., 20 15 / 6 superscript 20 15 6 20^{15/6} ) due to cosmological redshift of the emitted gravitational waves out of LIGO’s sensitive band, as well as cosmological volume factors influencing the scale of the light cone near z ≈ 0.5 𝑧 0.5 z\approx 0.5 Flanagan and Hughes ( 1998 ) ; see Eqs. (13-20) of O’Leary et al. for details O’Leary et al. ( 2006 ) . Gravitational wave observatories provide useful information precisely because g cl subscript 𝑔 cl {g_{\rm cl}} and g evap subscript 𝑔 evap {g_{\rm evap}} are so weakly constrained electromagnetically. On the other hand, gravitational segregation should at best compete with and more likely occur more slowly than “infant mortality,” the tendency of roughly 70 − 90 % 70 percent 90 70-90\% of young clusters to disrupt within their first ∼ 10 ​ Myr similar-to absent 10 Myr \sim 10\,{\rm Myr} due to photoionization- and SN -driven gas ejection Lada and Lada ( 2003 ); Fall et al. ( 2005 ); Lamers et al. ( 2005 ); Gieles et al. ( 2005 ); Kroupa and Boily ( 2002 ); Bastian and Goodwin ( 2006 ) .

ISBN 0-387-95436-8 (see §18); doi:10.1007/0-387-21625-1_19. - ↑ Ayres T.R. 1984. Capella HL. In Cool stars, stellar systems, and the Sun, edited by Sallie L. Baliunas and Lee Hartmann, Berlin/Heidelberg, Springer-Verlag, 1984, Lecture Notes in Physics, 193, pp. 202–204. - ↑ Famaey B. et al. 2007. The Hyades stream: an evaporated cluster or an intrusion from the inner disk? Astronomy and Astrophysics 461, #3, 957–962. Several other stars in the same visual field have been catalogued as companions but are physically unrelated. == Nomenclature == α Aurigae (Latinised to Alpha Aurigae) is the star system's Bayer designation. It also has the Flamsteed designation 13 Aurigae. It is listed in several multiple star catalogues as ADS 3841, CCDM J05168+4559, and WDS J05167+4600. As a relatively nearby star system, Capella is listed in the Gliese-Jahreiss Catalogue with designations GJ 194 for the bright pair of giants and GJ 195 for the faint pair of red dwarfs. The traditional name Capella is Latin for (small) female goat; the alternative name Capra was more commonly used in classical times. Almost simultaneously, British astronomer Hugh Newall had observed its composite spectrum with a four prism spectroscope attached to a 25-inch (64 cm) telescope at Cambridge in July 1899, concluding that it was a binary star system. Many observers tried to discern the component stars without success. Known as "The Interferometrist's Friend", it was first resolved interferometrically in 1919 by John Anderson and Francis Pease at Mount Wilson Observatory, who published an orbit in 1920 based on their observations. This was the first interferometric measurement of any object outside the Solar System. A high-precision orbit was published in 1994 based on observations by the Mark III Stellar Interferometer, again at Mount Wilson Observatory. Capella also became the first astronomical object to be imaged by a separate element optical interferometer when it was imaged by the Cambridge Optical Aperture Synthesis Telescope in September 1995. In 1914, Finnish astronomer Ragnar Furuhjelm observed that the spectroscopic binary had a faint companion star, which, as its proper motion was similar to that of the spectroscopic binary, was probably physically bound to it. In February 1936, Carl L. Stearns observed that this companion appeared to be double itself; this was confirmed in September that year by Gerard Kuiper. This pair are designated Capella H and L. === X-ray source === Two Aerobee-Hi rocket flights on September 20, 1962, and March 15, 1963, detected and confirmed an X-ray source in Auriga at RA 05h 09m Dec +45°, identified as Capella. A major milestone in stellar X-ray astronomy happened on April 5, 1974, with the detection of the strongest emission of X-rays up to that time from Capella, measured at more than 10,000 times the x-ray luminosity of the Sun. A rocket flight on that date briefly calibrated its attitude control system when a star sensor pointed the payload axis at Capella. During this period, X-rays in the range 0.2–1.6 keV were detected by an X-ray reflector system co-aligned with the star sensor. The X-ray luminosity (Lx) of ~1024 W (1031 erg/s) is four orders of magnitude above the Sun's X-ray luminosity. Capella's X-rays are thought to be primarily from the corona of the most massive star. Capella is ROSAT X-ray source 1RXS J051642.2+460001. described a scale model of the system where Capella A was represented by spheres 13 and 7 inches across, separated by ten feet. The red dwarfs were then each 0.7 inch across and they were separated by 420 feet. At this scale, the two pairs are 21 miles apart. === Capella A === Capella A consists of two yellow evolved stars that have been calculated to orbit each other every 104.02128±0.00016 days, with a semimajor axis of 111.11±0.10 million km (0.74272±0.00069 AU), roughly the distance between Venus and the Sun. The pair is not an eclipsing binary—that is, as seen from Earth, neither star passes in front of the other. The MK spectral types of the two stars have been measured a number of times, and they are both consistently assigned a luminosity class of III indicating a giant star. The composite spectrum appears to be dominated by the primary star due to its sharper absorption lines; the lines from the secondary are broadened and blurred by its rapid rotation. The composite spectral class is given as approximately G3III, but with a specific mention of features due to a cooler component. The most recent specific published types are K0III and G1III, although older values are still widely quoted such as G5IIIe + G0III from the Bright Star Catalogue or G8III + G0III by Eggen. Capella Aa has slowed until it is rotationally locked to the orbital period, although theory predicts that it should still be rotating more quickly from a starting point of a rapidly-spinning main sequence A star. Capella has long been suspected to be slightly variable. Its amplitude of about 0.1 magnitudes means that it may at times be brighter or fainter than Rigel, Betelgeuse and Vega, which are also variable. The system has been classified as an RS Canum Venaticorum variable, a class of binary stars with active chromospheres that cause huge starspots, but it is still only listed as a suspected variable in the General Catalogue of Variable Stars. Unusually for RS CVn systems, the hotter star, Capella Ab, has the more active atmosphere because it is located in the Hertzsprung gap—a stage where it is changing its angular momentum and deepening its convection zone. The active atmospheres and closeness of these stars means that they are among the brightest X-ray sources in the sky. However the X-ray emission is due to stable coronal structures and not eruptive flaring activity. Coronal loops larger than the Sun and with temperatures of several million kelvin are likely to be responsible for the majority of the X-rays.

One of the oldest problems in physics is that of calculating the motion of N particles under a specified mutual force: the N-body problem. Much is known about this problem if the specified force is non-relativistic gravity, and considerable progress has been made by considering the problem in one spatial dimension. Here, I review what is known about the relativistic gravitational N-body problem. Reduction to one spatial dimension has the feature of the absence of gravitational radiation, thereby allowing for a clear comparison between the physics of one-dimensional relativistic and non-relativistic self-gravitating systems. After describing how to obtain a relativistic theory of gravity coupled to N point particles, I discuss in turn the two-body, three-body, four-body, and N-body problems. Quite general exact solutions can be obtained for the two-body problem, unlike the situation in general relativity in three spatial dimensions for which only highly specified solutions exist. The three-body problem exhibits mild forms of chaos, and provides one of the first theoretical settings in which relativistic chaos can be studied. For N≥4, other interesting features emerge. Relativistic self-gravitating systems have a number of interesting problems awaiting further investigation, providing us with a new frontier for exploring relativistic many-body systems.