Demographics of triple systems in dense star clusters
Giacomo Fragione, Miguel A. S. Martinez, Kyle Kremer, Sourav Chatterjee, Carl L. Rodriguez, Claire S. Ye, Newlin C. Weatherford, Smadar Naoz, Frederic A. Rasio
DDraft version July 24, 2020
Typeset using L A TEX twocolumn style in AASTeX63
Demographics of triple systems in dense star clusters
Giacomo Fragione,
1, 2
Miguel A. S. Martinez,
1, 2
Kyle Kremer,
1, 2
Sourav Chatterjee, Carl L. Rodriguez, Claire S. Ye,
1, 2
Newlin C. Weatherford,
1, 2
Smadar Naoz,
5, 6 and Frederic A. Rasio
1, 2 Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA Center for Interdisciplinary Exploration & Research in Astrophysics (CIERA), Northwestern University, Evanston, IL 60208, USA Department of Astronomy & Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road, Navy Nagar, Colaba, Mumbai400005, India Astronomy Department, Harvard University, 60 Garden St., Cambridge, MA 02138, USA Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA Mani L. Bhaumik Institute for Theoretical Physics, UCLA, Los Angeles, CA 90095, USA
ABSTRACTDepending on the stellar type, more than ∼
50% and ∼
15% of stars in the field have at least oneand two stellar companions, respectively. Hierarchical systems can be assembled dynamically in densestar clusters, as a result of few-body encounters among stars and/or compact remnants in the clustercore. In this paper, we present the demographics of stellar and compact-object triples formed viabinary–binary encounters in the
CMC Cluster Catalog , a suite of cluster simulations with present-day properties representative of the globular clusters (GCs) observed in the Milky Way. We show howthe initial properties of the host cluster set the typical orbital parameters and formation times of theformed triples. We find that a cluster typically assembles hundreds of triples with at least one blackhole (BH) in the inner binary, while only clusters with sufficiently small virial radii are efficient inproducing triples with no BHs, as a result of the BH-burning process. We show that a typical GCis expected to host tens of triples with at least one luminous component at present day. We discusshow the Lidov-Kozai mechanism can drive the inner binary of the formed triples to high eccentricities,whenever it takes place before the triple is dynamically reprocessed by encountering another clustermember. Some of these systems can reach sufficiently large eccentricities to form a variety of transientsand sources, such as blue stragglers, X-ray binaries, Type Ia Supernovae, Thorne-Zytkow objects, andLIGO/Virgo sources. INTRODUCTIONStellar multiplicity is an omnipresent outcome of thestar-formation process (Duchˆene & Kraus 2013). Morethan ∼
50% of stars are thought to have at least onestellar companion (e.g., Tokovinin 2014a). Tokovinin(2014b) showed that at least ∼
13% of F-type and G-type dwarf stars in the Hipparcos sample live in triplesystems (an inner binary orbited by an outer compan-ion), while Riddle et al. (2015) found a relatively largeabundance of 2+2 quadruples (a binary where the com-ponents are themselves binaries) with Robo-AO, thefirst robotic adaptive optics instrument. Sana et al.(2014) estimated that ∼
80% of O-type stars have atleast one companion and ∼
25% have at least twosuch companions in their sample. Using a large high-resolution radial velocity spectroscopic survey of B-typeand O-type stars, Chini et al. (2012) estimated that at [email protected] least 50-80% of them are multiples. Recently, a blackhole of ∼ (cid:12) has been claimed to live in the triplesystem HR 6819, ∼
300 pc from the Sun (Rivinius et al.2020).In dense star clusters, hierarchical systems of starsand/or compact remnants can form through few-body(particularly binary–binary) encounters in the clusters’dense cores (e.g., Fregeau et al. 2004; Leigh & Geller2013; Antognini & Thompson 2016; Fragione et al.2019d). In this process, one of the two binaries capturesa component of the second binary, with the remainingobject leaving the system. Leigh et al. (2016) estimatedthat the branching ratio of this process can be as highas ∼ a r X i v : . [ a s t r o - ph . GA ] J u l Fragione et al.
GCs represent the ideal environment to study the im-portance of gravitational dynamics in dense stellar sys-tems and how dynamics shape both cluster evolutionand survival (see, e.g., Heggie & Hut 2003). Impor-tantly, frequent dynamical encounters between clustermembers are fundamental in creating and explaining theexistence of a number of exotic populations, such X-raybinaries (e.g., Clark 1975; Verbunt et al. 1984; Heinkeet al. 2005; Ivanova 2013; Giesler et al. 2018; Kremeret al. 2018), radio sources (e.g., Lyne et al. 1987; Sigurds-son & Phinney 1995; Ransom 2008; Ivanova et al. 2008;Fragione et al. 2018; Ye et al. 2019), and gravitationalwave (GW) binaries (e.g., Moody & Sigurdsson 2009;Banerjee et al. 2010; Rodriguez et al. 2015, 2016; Askaret al. 2017; Banerjee 2017; Chatterjee et al. 2017b,a;Hong et al. 2018; Fragione & Kocsis 2018; Samsing &D’Orazio 2018; Rodriguez et al. 2018a; Zevin et al. 2018;Kremer et al. 2019e). However, with the possible ex-ception of Antonini et al. (2016), there have been nocomprehensive studies about the origin of hierarchicalsystems in dense star clusters, and how this depends onclusters’ primordial properties.What makes hierarchical triple and multiple systemsof particular interest is that they can produce exoticobjects, transients, and GW sources over a larger por-tion of the parameter space compared to binaries. Thisadditional portion is enabled by the Lidov-Kozai (LK)mechanism (Lidov 1962; Kozai 1962). In recent years, anumber of authors have shown how hierarchical triplesare efficient in producing GW sources (e.g., Petrovich &Antonini 2017; Hamers et al. 2018; Hoang et al. 2018;Fragione & Kocsis 2019; Liu et al. 2019; Stephan et al.2019; Fragione & Kocsis 2020), tidal disruption events(e.g., Chen et al. 2009; Fragione & Leigh 2018; Fragioneet al. 2019c), white dwarf (WD) mergers (e.g., Toonenet al. 2018; Fragione et al. 2019e), and millisecond pul-sars (e.g., Ford et al. 2000). In this framework, the ec-centricity of the inner binary is not constant, but ratheroscillates between a minimum and a maximum value(determined by the triple initial configuration), due tothe tidal force imposed by the third companion (for a re-view see Naoz 2016). As a result, the inner binary com-ponents may be efficiently driven to sufficiently smallseparations to merge either due to physical collision ordissipation of GWs.In this paper, we study the role of dense star clustersin producing triple systems of all possible componentconfigurations. We use a grid of 148 independent clustersimulations (presented in Kremer et al. 2020) , run using CMC (for
Cluster Monte Carlo ), which covers roughlythe complete range of GCs observed at present day inthe Milky Way. We systematically explore the effectof initial virial radii (and subsequent BH dynamics) onclusters of various masses, metallicities, and locations https://cmc.ciera.northwestern.edu within the Galactic tidal field. We dissect the originof triples assembled in dense star clusters as a functionof the clusters’ initial properties, describe the triple de-mographics, and determine if they can produce exotica,transients, and GW sources.The paper is organized as follows. In Section 2, we de-scribe the numerical method used to evolve our clustermodels. In Section 3, we analyze the origin of triple sys-tems in star clusters, while in Section 4 we describe theirdemographics and general properties. In Section 5, weestimate the transient and GW phenomena as a result ofthe LK mechanism. Finally, in Section 6, we discuss theimplications of our findings and lay out our conclusions. METHODSHere, we summarize the methods we use to evolve ourpopulation of clusters. For a detailed description seeKremer et al. (2020).We use
CMC , a H´enon-type Monte Carlo code (H´enon1971a,b; Joshi et al. 2000, 2001; Fregeau et al. 2003;Chatterjee et al. 2010, 2013; Pattabiraman et al. 2013;Rodriguez et al. 2015).
CMC incorporates the physics rel-evant to both the overall evolution of the cluster proper-ties and the specific evolution of the stars and compactobjects therein.The main process that shapes the evolution of globalproperties of clusters is two-body relaxation (e.g., Heg-gie & Hut 2003). In
CMC , this is implemented by usingthe H´enon orbit-averaged Monte Carlo method (Joshiet al. 2000). To account for the fact that dense star clus-ters are subject to the tidal field of their host galaxy, weadopt an effective tidal mass-loss criterion that matchesthe tidal mass loss found in direct N -body simulations(Chatterjee et al. 2010).Single and binary stars are respectively evolved withthe SSE and
BSE codes (Hurley et al. 2000, 2002; Chat-terjee et al. 2010), with up-to-date prescriptions for neu-tron star (NS) and black hole (BH) formation (Fryer &Kalogera 2001; Vink et al. 2001; Belczynski et al. 2002;Hobbs et al. 2005; Morscher et al. 2015; Rodriguez et al.2016). In particular, two scenarios are considered for NSformation: iron core-collapse supernovae and electron-capture supernovae (Ye et al. 2019). In our simulations,the former receive natal kicks drawn from a Maxwellianwith dispersion σ = 265 km s − , the latter with disper-sion 20 km s − . Updated prescriptions for pulsar forma-tion and evolution are also implemented (see Ye et al.2019, for details). BHs are assumed to be formed withmass fallback and receive natal kicks by sampling fromthe same distribution as core-collapse supernovae NSs,but with kicks reduced in magnitude according to thefractional mass of fallback material (Fryer et al. 2012;Morscher et al. 2015). We also include prescriptionsto account for pulsational-pair instabilities and pair-instability supernovae (Belczynski et al. 2016).Binary–single and binary–binary strong encountersare integrated using Fewbody (Fregeau et al. 2004; emographics of triple systems in dense star clusters
Cluster models
We use a set of 148 independent cluster simulations.We consider different total number of particles (singlestars plus binaries; N = 2 × , 4 × , 8 × ,1 . × , and 3 . × ), initial cluster virial radius( r v / pc = 0 . , , , Z/ Z (cid:12) = 0 . , . , ( R gc / kpc = 2 , , W = 5 (King 1962). Stellar masses are drawn froma canonical Kroupa (2001) initial mass function (IMF)in the range 0 . −
150 M (cid:12) . The primordial stellar bi-nary fraction is fixed to f b = 5%, with secondary massesdrawn from a uniform distribution in mass ratio (e.g.,Duquennoy & Mayor 1991). Binary orbital periods aresampled from a log-uniform distribution (e.g., Duquen-noy & Mayor 1991), with the orbital separations rangingfrom near contact to the hard/soft boundary, while bi-nary eccentricities are drawn from a thermal distribution(e.g., Heggie 1975).Each simulation is evolved to a final time T H = 14Gyr, unless the cluster disrupts or undergoes a colli-sional runaway process (Kremer et al. 2020).Primordial triples are not included in our cluster simu-lations. However, during strong binary–binary encoun-ters, stable hierarchical triple systems can be formed(Rasio et al. 1995). Limitations in CMC currently requirethese triples to be broken artificially at the end of theintegration timestep. Nevertheless, whenever a stabletriple is formed, its properties are logged, including themasses, stellar types, radii, and the semi-major axes andeccentricities for the inner and outer orbits . Since welack information regarding the mutual orientation of thetwo orbits, we sample their argument of periapsis ω , co-sine of the relative inclination cos I , and orbital phasesfrom a uniform distribution (Antonini et al. 2016). Toaverage out over these uncertainties, we realize this pro- Assuming a Milky Way-like galactic potential (e.g., Dehnen &Binney 1998) Note that, since these triple systems are de facto destroyed inthe Monte Carlo simulations, it is possible for the components ofthese triples to subsequently form new triple systems, when inreality they could survive for a significant period of time. cedure 10 times for each triple formed in each of the 148clusters presented in Kremer et al. (2020). DISSECTING THE ORIGIN OF TRIPLESIn this Section, we discuss the relevant formationchannels of triples in star clusters, the characteristicsof their progenitors, the formation times, and the recoilkicks that triple systems are imparted at the momentof formation. We label the inner and outer semi-majoraxes of the formed triples a in and a out , respectively, theinner and outer eccentricities e in and e out , respectively,the mass of the components of the inner binary m and m ( m < m ), the total mass of the inner binary m in = m + m , and the mass of the outer component m . The total mass of the triple is m t = m in + m , whilethe initial relative inclination of the inner and outer orbitis i . We label the remaining object m s (fourth objectleaving the system after the binary–binary interaction).3.1. Progenitors
We find from our simulations that the majority oftriple systems ( ∼ .
2% of the overall triple population)are formed as a result of binary–binary encounters. Ingeneral, the probability of binary–binary encounters is(Binney & Tremaine 2008)Γ bin − bin ∼ n σv disp , (1)where n bin is the density of binaries, σ is the cross-section, and v disp is the velocity dispersion. Since n bin islargest in the core, the typical binary–binary encounteroccurs in the core of dense star clusters. Of all thebinary–binary encounters, the ones that successfully cre-ate triples involve two binaries of quite disparate sizes.Here, the tighter binary ejects a member of the widerbinary and inserts itself, thus creating a stable hierar-chical triple. The replaced object receives a dynamicalrecoil kick and is ejected from the encountering system ,while the captured one becomes the tertiary in the newlyformed triple system.We illustrate in Fig 1 the properties of binaries thatlead to the formation of triple systems in binary–binaryencounters for a cluster with initial number of stars N = 8 × . The other initial cluster parameters are r v = 1 pc, r g = 8 kpc, Z = 0 . (cid:12) . In the top panel,we show the maximum of the semi-major axes ( a b , and a b , ) of the two binaries that undergo the binary–binaryencounter as a function of the minimum of them. Wealso overplot the probability density contours. We findthat the bulk of the interactions that produce a triple in-clude two binaries, of which one is wider than the otherby ∼ In some cases, its recoil velocity would be high enough to eject itfrom the cluster (see Sect 3.4).
Fragione et al. a b, 1 , a b, 2 )/AU)10123 L o g ( m a x ( a b , , a b , ) / A U ) N = 8 × 10 r v = 1 pc r g = 8 kpc Z = 0.1 Z . . . l o g ( T / M y r ) e b, 1 , e b, 2 )0.20.40.60.81.0 m a x ( e b , , e b , ) N = 8 × 10 r v = 1 pc r g = 8 kpc Z = 0.1 Z . . . . l o g ( T / M y r ) max( a b, 1 , a b, 2 ) × m in / m s (AU)10 a o u t ( A U ) N = 8 × 10 r v = 1 pc r g = 8 kpc Z = 0.1 Z l o g ( T / M y r ) Figure 1.
Properties of binaries that lead to the forma-tion of triple systems in binary–binary encounters for a clus-ter with initial number of stars N = 8 × ( r v = 1 pc, r g = 8 kpc, Z = 0 . (cid:12) ). Semi-major axes (top panel), ec-centricities (middle panel), and outer semi-major axis as afunction of the orbital elements of the binaries in the en-counter (bottom panel) are shown. In the top two panels,the solid red lines represent the density contours of 10%,30%, 60%, 90% probability regions. The dashed red line inthe bottom panel represents the x = y line. The color maprepresents log formation time. of the components of a wider binary. We also show inFig 1 (middle panel) the maximum eccentricities of thetwo binaries undergoing the binary–binary encounters( e b , and e b , ) as a function of the minimum of them.Since encounters thermalize the distribution of the ec-centricities of the progenitors (Heggie 1975), most of thebinaries that produce triples are highly eccentric.Since the typical triple-producing binary–binary en-counter involves a tight binary exchanging into a widebinary, we expect the outer semi-major axis distributionof the outer semi-major axis of the triples to be relatedto the orbital elements of the ionized binaries. Fromenergy conservation, m in m a out ∼ m m s max( a b , a b ) , (2)where m s is the mass of the replaced component in thewider binary (Sigurdsson & Phinney 1993). Therefore,the outer semi-major axis of the triple is linearly relatedto the semi-major axis of the wider binary through a out ∼ m in m s max( a b , a b ) . (3)We show this in the bottom panel of Fig 1. As expected,the majority of the systems lie on the x = y line. Triplesthat are outliers with respect to this simple scaling aresystems formed during resonant encounters, where theenergy is redistributed in a more complex way duringmultiple passages and interactions among the four ob-jects (two binaries) involved in the encounter.3.2. Cluster and triple properties
The initial conditions of the parent cluster set the dis-tribution of the orbital elements of the formed triplesystems. We show this in Figure 2, where we plot the cu-mulative distribution functions of inner and outer semi-major axes of triples in clusters of various initial num-bers of stars, virial radii, and metallicities.In the top panel of Figure 2, we illustrate the cumu-lative distribution function of triples in clusters of dif-ferent initial number of stars ( N = 2 × –3 . × )and r v = 2 pc, Z = 0 .
01 Z (cid:12) , r g = 20 kpc. Triples thatform in larger clusters tend to have smaller inner andouter semi-major axes. We find that ∼
50% of thesystems have a in / au (cid:46) (0 . , , , ,
5) and a out / au (cid:46) (1 , , , , × for N = (32 , , , , × , re-spectively. This comes from the fact that binaries thatundergo binary–binary scattering and produce a triplesystem are tighter in more massive clusters. In theseenvironments, stellar densities are typically higher thanin less massive clusters and wide binaries are ionized byencounters with stars and compact objects.We plot in the middle panel of Figure 2 the cumulativedistribution function of triples in clusters of different ini-tial virial radii r v / pc ∈ [0 . ,
4] and N = 8 × , r g =8 kpc, Z = 0 .
01 Z (cid:12) . Triples that form in clusters with emographics of triple systems in dense star clusters C D F N = 2.0 × 10 N = 4.0 × 10 N = 8.0 × 10 N = 1.6 × 10 N = 3.2 × 10 r v = 2 pc Z = 0.01 Z r g = 20 kpc N = 2.0 × 10 N = 4.0 × 10 N = 8.0 × 10 N = 1.6 × 10 N = 3.2 × 10 r v = 2 pc Z = 0.01 Z r g = 20 kpc C D F r v = 0.5 pc r v = 1 pc r v = 2 pc r v = 4 pc N = 8.0 × 10 Z = 0.01 Z r g = 8 kpc r v = 0.5 pc r v = 1 pc r v = 2 pc r v = 4 pc N = 8.0 × 10 Z = 0.01 Z r g = 8 kpc a in (AU)0.00.20.40.60.81.0 C D F Z = 0.01 Z Z = 0.1 Z Z = Z N = 8.0 × 10 r v = 2 pc r g = 20 kpc a out (AU)0.00.20.40.60.81.0 Z = 0.01 Z Z = 0.1 Z Z = Z N = 8.0 × 10 r v = 2 pc r g = 20 kpc Figure 2.
Cumulative distribution functions of inner ( a in ; left panels) and outer ( a out ; right panels) semi-major axis of triplesin clusters of various initial numbers of stars (top panels), virial radii (central panels), and metallicities (bottom panels). larger virial radii tend to have wider inner and outerorbits. We find that ∼
50% of the triple systems have a in / au (cid:46) (0 . , , ,
4) and a out / au (cid:46) (70 , , , r v / pc = (0 . , , , r v typically have ahigher density and velocity dispersion. Thus, the pro-genitor binaries (that later undergo binary–binary en-counter to form triples) have to be more compact inorder to remain bound after encounters with stellar orcompact objects. Finally, in the bottom panel of Figure 2, we plot thecumulative distribution function of triples in clusters ofdifferent initial metallicities Z/ Z (cid:12) = (0 . , . , , with N = 8 × , r v = 2 pc, and r g = 20 kpc. Triplesthat form in higher metallicity clusters tend to havesmaller inner and outer semi-major axes. We find that ∼
50% of the triple systems have a in / au (cid:46) (2 , , . a out / au (cid:46) (400 , , Z/ Z (cid:12) = (0 . , . , Fragione et al. the BH-burning process as BHs in metal-poor clusters.Thus, these clusters typically have higher densities anddispersion velocities. As a result, metal-poor clustersallow wider binaries to form triples compared to metal-rich clusters.3.3.
Cluster properties and formation times
Triple systems are not formed uniformly in time.Rather, they track the evolutionary paths of the par-ent cluster. The clock of a star cluster is essentially setby its half-mass relaxation time (Spitzer 1987) t rh ∼ N / r / (cid:104) m (cid:105) / G / ln Λ , (4)where (cid:104) m (cid:105) is the average mass in the cluster and ln Λis the Coulomb logarithm. As discussed in greater de-tail in Kremer et al. (2019a), the initial cluster size, setby its initial virial radius, is the key parameter whichdetermines the ultimate fate of a cluster and its BHpopulation (“BH-burning“ mechanism). Clusters withsmaller initial r v have shorter relaxation times and havea dynamical clock that runs faster compared to clus-ters born with larger initial virial radius. These clusterscould eject the majority of their BH population overtheir lifetime and appear as core-collapsed clusters.In Figure 3, we plot the formation time ( t form ) oftriples in clusters of various initial numbers of stars,virial radii, and metallicities (same as Figure 2).In the top panel, we show the cumulative distributionfunction of triples in clusters of different initial numbersof stars N ∈ [2 × , . × ], r v = 2 pc, Z = 0 .
01 Z (cid:12) ,and r g = 20 kpc. As expected from Eq. 4, larger starclusters have longer evolutionary timescales. Hence,triples are assembled through binary–binary scatter-ings later compared to smaller clusters. We find that ∼
50% of the triples are assembled at t form (cid:46) . ∼ . t rh ) for N = 2 × , while ∼
50% of thetriples are assembled at t form (cid:46) ∼ . t rh ) for N = 3 . × .We plot in the middle panel of Figure 3 the cumula-tive distribution function of triples in clusters of differ-ent initial virial radii r v / pc ∈ [0 . , N = 8 × , Z = 0 .
01 Z (cid:12) , r g = 8 kpc. As discussed, the initialcluster size sets the dynamical clock of a stellar clus-ter. Among the four represented clusters, the ones with r v = 0 . r v = 1 pc are core-collapsed (see Fig-ure 5 in Kremer et al. 2020). Clusters with small initialvirial radii form most of the triple systems much morequickly than clusters with larger initial sizes.Finally, in the bottom panel of Figure 3, we showthe cumulative distribution function of triples in clus-ters of different initial metallicities Z/ Z (cid:12) ∈ [0 . , N = 8 × , r v = 2 pc, and r g = 20 kpc. Star clusterswith smaller metallicities form more massive BHs thanclusters with higher metallicities (see Figure 1 in Kremeret al. 2020). Therefore, BHs are dynamically processed C D F N = 2.0 × 10 N = 4.0 × 10 N = 8.0 × 10 N = 1.6 × 10 N = 3.2 × 10 r v = 2 pc Z = 0.01 Z r g = 20 kpc C D F r v = 0.5 pc r v = 1 pc r v = 2 pc r v = 4 pc N = 8.0 × 10 Z = 0.01 Z r g = 8 kpc t form (Myr)0.00.20.40.60.81.0 C D F Z = 0.01 Z Z = 0.1 Z Z = Z N = 8.0 × 10 r v = 2 pc r g = 20 kpc Figure 3.
Formation time ( t form ) of triples in clusters of var-ious initial numbers of stars (top panel), virial radii (centralpanel), and metallicities (bottom panel). faster as the dynamical clock of the host cluster runsfaster in the former case, producing triples on shortertimescales. 3.4. Recoils and ejections
Binary–binary exchange encounters impart recoilkicks to any triples they produce. Leigh et al. (2016) emographics of triple systems in dense star clusters m s ) in such an encounter is well described by the dis-tribution f ( v ej , s ) dv ej , s = 3 | E | M v ej , s (cid:16) | E | + M v , s (cid:17) dv ej , s , (5)where M = m s ( m s + m in ) m in (6)and | E | is the total initial energy. From the conservationof linear angular momentum, the recoil velocity of thetriple is v rec = m s m t v ej , s . (7)This recoil kick can be large enough to eject the triplefrom the core (where it will eventually sink back as aresult of dynamical friction) or even from the cluster.We use the data recorded on binary–binary scatteringsthat lead to the formation of stable triple systems duringthe cluster’s lifetime to compute v rec . In Figure 4, weshow the recoil velocity v rec of the triple systems assem-bled in the cores of star clusters of various initial N (forfixed r v = 2 pc, r g = 20 kpc, and Z = 0 .
01 Z (cid:12) ) as a func-tion of the cluster escape speed v esc at the moment offormation. For these clusters, we find that (cid:46)
1% of theformed triples could escape the clusters due to dynam-ical recoil kicks (if they do not encounter other stars orcompact objects). Moreover, the escaping systems tendto be ejected from the cluster at later times, when thecluster escape speed decreases to lower values. Most ofthe triples will not leave the cluster. Rather, they willbe kicked on elongated orbits out of the cluster core.As they are more massive than the average star, theywould sink back to the cluster core on a dynamical fric-tion timescale t df ∼ (cid:104) m (cid:105) m t t rh , (8)where m t = m in + m is the total mass of the triple. DEMOGRAPHICSIn this section, we discuss how the parent cluster ini-tial conditions shape the orbital properties of the formedtriples and describe their demographics.We are interested in triples that are hierarchically sta-ble. While simulating strong encounters inside CMC,triple stability is checked using the stability criteriagiven by Mardling & Aarseth (2001), a out a in R (cid:18) e out , m out m in (cid:19) ≥ . , (9) This assumes that the initial angular momentum is negligible.For a general discussion, see Valtonen & Karttunen (2006). where R (cid:18) e out , m out m in (cid:19) = (cid:20)(cid:18) m out m in (cid:19) e out √ − e out (cid:21) − / × (1 − e out ) (cid:18) − . i ◦ (cid:19) . (10)We subdivide the triple population into four cate-gories, such that the stellar types of the two components k and k of the inner binary (see Hurley et al. 2000)are always k ≤ k : • triples with a main-sequence (MS) star in the innerbinary; • triples with a giant (G) star in the inner binary; • triples with a WD in the inner binary; • triples with a NS or BH in the inner binary.Among the systems with an inner BH-BH binary, wealso consider triples where all the components are BHs,that we label BH-BH-BH.As a general trend, we find that a cluster typically as-sembles hundreds of triples with an inner BH-BH binary(of which ∼ r v ≤ ∼
10 times more systems with an inner MS-MSbinary. Again, this is a natural consequence of the BHburning process (Kremer et al. 2019d), since only clus-ters with small initial virial radii are able to eject mostof their BH population, thus allowing lighter objects tosink to their innermost regions and efficiently producetriples. Moreover, we find that ∼
50% of the overalltriple population from our simulations consists of sys-tems where all the components are BHs. Roughly 10%of the systems take the form of a binary BH with a non-BH third companion and ∼
38% the form of an innerbinary with at least one MS star. Other triples consti-tute the remaining ∼ Gravitational wave captures and mergers duringtriple formation
A handful of triple systems ( ∼ .
1% of the overall pop-ulation) are formed during binary–single encounters as aresult of GW captures (Samsing et al. 2019). In this pro-cess, the single has the chance to pass sufficiently closeto the binary to dissipate some energy via GW radia-tion, thus remaining bound to the binary itself. For alltriples assembled this way in our simulations, we showin Figure 5 the outer mass as a function of the inner
Fragione et al.
10 20 30 40 50 6010 v r e c ( k m / s ) v rec = v esc N = 2.0 × 10
10 20 30 40 5010 v rec = v esc N = 4.0 × 10 l o g ( T / M y r )
20 40 60 8010 v r e c ( k m / s ) v rec = v esc N = 8.0 × 10
30 40 50 60 70 80 90 v esc (km/s)10 v rec = v esc N = 1.6 × 10 l o g ( T / M y r )
40 60 80 100 120 140 v esc (km/s)10 v r e c ( k m / s ) v rec = v esc N = 3.2 × 10 l o g ( T / M y r ) Figure 4.
Recoil velocity v rec of the triple systems assembled in the cores of star clusters of various initial numbers of stars N ( r v = 2 pc, r g = 20 kpc, Z = 0 .
01 Z (cid:12) ), as a function of the cluster escape speed v esc at the moment of formation. The dot-dashedblue line represents v rec = v esc . Color code: log formation time. emographics of triple systems in dense star clusters m + m (M )10 m ( M ) m MS m G m WD m BH Figure 5.
Outer mass as a function of the total mass ofthe inner binary of the triple systems that form through GWcaptures during binary–single encounters. The binary, whichbecomes the inner binary of the triple, is always a binary BH. m (M )10 m m ( M ) star-starWD-starNS-starBH-starBH-BH Figure 6.
Masses of the components ( m and m ) thatmerge during binary–binary encounters that lead to tripleformation. Different colors represent different stellar andcompact object types. binary’s total mass. We find that the binary that in-tervenes in the process, which later becomes the innerbinary of the triple, is always a binary BH. The major-ity of the triples formed through GW captures are madeup of three BHs, while a few systems have a star (eitherMS or G) or WD as the outer companion. We find noGW capture systems with a NS outer companion.During the binary–binary encounters that produce atriple system, two of the objects can pass close enoughto merge. This can occur in multiple ways: collision andmerger of two stars (MS or G), tidal disruption of starsby a compact object, and merger of two compact ob-jects. In Figure 6, we plot for all simulations the massesof the components ( m and m ) that merge during binary–binary encounters which yield triples ( ∼ . ∼
90% and ∼
10% ofthe mergers are with MS or G stars, respectively.In thestandard scenario for triple formation, the tighter binaryejects the single star it replaces, but no ejection occursin this process.4.2.
Stability and softness
We define the softness parameter (Heggie 1975) η ≡ Gm in m a out (cid:104) m (cid:105) v , (11)where (cid:104) m (cid:105) and v disp are the average mass in the clusterand the cluster velocity dispersion, respectively. Triplesthat have η (cid:28) η (cid:29) a out /a in (cid:38)
10, regardless of thecomposition of the inner binary. We also show in Fig-ure 7 the probability distribution function of the soft-ness parameter η (right panels) of all triples formed inthe simulations. We find quite generally that triple pop-ulations have η (cid:29)
1, with only a small tail of soft triplesand a main peak at η ∼ Formation time, inner mass ratio, outer massratio
We show in Figure 8 the probability distribution func-tion of the formation time (left), the inner mass ratio(center), and the outer mass ratio (right) of all the triplepopulations formed in the simulations.As a common trend, we find that triples whose innerbinary has at least one BH typically form on a shortertimescale compared to other triples. This can be under-stood in terms of the BH-burning mechanism (Kremeret al. 2019d). In this process, strong dynamical encoun-ters between the BHs act as an energy source for therest of the cluster. Thus, BHs tend to occupy the in-nermost and densest parts of the cluster, where mostof the binary–binary interactions take place, preventingother components from efficiently segregating there. Asa result, triples whose inner binary does not contain aBH tend to form on longer timescales, when most of theBHs have been processed and have left the cluster.For MS stars, we define the inner mass ratio m MS /m comp as the ratio between the MS star’s mass( m MS ) and that of its companion ( m comp ). If there aretwo MS stars in the inner binary, we define the inner0 Fragione et al. SOFT
MS-MSMS-GMS-WDMS-NSMS-BH P D F MS-MSMS-GMS-WDMS-NSMS-BH SOFT
G-GG-WDG-NSG-BH P D F G-GG-WDG-NSG-BH SOFT
WD-WDWD-NSWD-BH P D F WD-WDWD-NSWD-BH SOFT
NS-NSNS-BHBH-BHBH-BH-BH a out / a in P D F NS-NSNS-BHBH-BHBH-BH-BH
Figure 7.
Probability distribution function of the ratio of outer and inner semi-major axes ( a out /a in ; left) and the softnessparameter ( η ; right) of all triple populations formed in our 148 cluster simulations. Top panels: triples with a MS star plus acompanion in the inner binary. Central-top panels: triples with a G star plus a companion in the inner binary. Central-bottompanels: triples with a WD plus a companion in the inner binary. Bottom panels: triples with a NS or BH plus a companion inthe inner binary. emographics of triple systems in dense star clusters P D F MS-BHMS-NSMS-WDMS-GMS-MS MS-BHMS-NSMS-WDMS-GMS-MS MS-BHMS-NSMS-WDMS-GMS-MS P D F G-BHG-NSG-WDG-G G-BHG-NSG-WDG-G G-BHG-NSG-WDG-G P D F WD-WDWD-NSWD-BH WD-WDWD-NSWD-BH WD-WDWD-NSWD-BH T f (Myr)0.00.20.40.60.81.0 P D F NS-NSNS-BHBH-BHBH-BH-BH inner mass ratio0.00.10.20.30.40.50.6 NS-NSNS-BHBH-BHBH-BH-BH outer mass ratio0.00.10.20.30.40.50.60.70.8 NS-NSNS-BHBH-BHBH-BH-BH
Figure 8.
Probability distribution function of the formation time ( T f ; left), the inner mass ratio (center), and the outer massratio (right) of all the triple populations formed in our 148 cluster simulations. Top panels: triples with a MS star plus acompanion in the inner binary. Central-top panels: triples with a G star plus a companion in the inner binary. Central-bottompanels: triples with a WD plus a companion in the inner binary. Bottom panels: triples with a NS or BH plus a companion inthe inner binary. Fragione et al. r c / r h C l u s t e r m a ss ( M ) N u m b e r o f t r i p l e s Figure 9.
All late-time snapshots (10–13 Gyr) for modelclusters masses and concentrations (colored points). MilkyWay clusters (black points) are taken from Baumgardt &Hilker (2018), with the size of each black point correspondingto the integrated V-band magnitude of each cluster (Harris1996, larger symbols denote clusters that are best observed).Color code: number of triples with at least one luminouscomponent that survive at present day unperturbed in thecluster. mass ratio as m MS , /m MS , , with m MS , > m MS , . Thesame applies to G stars, WDs, NSs, and BHs. The outermass ratio is simply defined as the ratio between the to-tal mass of the inner binary and the mass of the tertiary.Interestingly, we find that the inner mass ratio is usu-ally peaked at ∼
1, unless the system only has one BHin the inner binary. The distribution of outer mass ra-tios is also nearly peaked at ∼
1, except for systemswith an inner binary comprised of a MS-MS, MS-BH,WD-BH, or NS-BH. The secondary peaks at ∼ Survivability
In the dense stellar environment of star clusters, triplesystems may be perturbed through encounters withother passing objects. Such encounters will alter theorbital properties of the triple significantly or even dis-rupt it. This process happens on a typical timescale (Binney & Tremaine 2008; Ivanova et al. 2008) T enc = 8 . × yr P − / , d m − / , M (cid:12) σ −
110 km s − n − pc − ×× (cid:34) m trip , M (cid:12) + (cid:104) m (cid:105) M (cid:12) P / , d m / , M (cid:12) σ
210 km s − (cid:35) − , (12) Quantities x a are expressed with physical units u as x a ,u ≡ x a /u ,so that x a ,u is dimensionless. where P out is the orbital period of the outer orbit and (cid:104) m (cid:105) is the average stellar mass in the cluster.We show in Figure 9 all late-time snapshots (10–13Gyr) for model clusters compared to Milky Way clusters.The latter are taken from Baumgardt & Hilker (2018)and represented such that their size is proportional tothe integrated V-band magnitude of each cluster (Har-ris 1996). Thus, larger symbols denote clusters that arebest observed. In color code, we represent the numberof triples with at least one luminous (observable) com-ponent that survive in the cluster, i.e. triples whoseencounter timescales are long enough to remain unper-turbed. We find that clusters are on average expectedto host tens of luminous triples at present. TRIPLE-ASSISTED MERGERS: TRANSIENTSAND GRAVITATIONAL WAVESIn this section, we discuss the LK mechanism thattakes place in triple systems. We then apply an ana-lytical formalism to compute the maximum eccentricityattained by the triples formed in our simulations (sub-divided as described in Section 4) and to infer the frac-tion of systems that result in a merger, a transient phe-nomenon, or GW emission by the LK mechanism.5.1.
Lidov-Kozai mechanism
A triple system made up of an inner binary that is or-bited by an outer companion undergoes LK oscillationsin eccentricity whenever the initial mutual inclination ofthe inner and outer orbits is in the range 40 ◦ (cid:46) i (cid:46) ◦ (Lidov 1962; Kozai 1962, quadrupole order of approxi-mation). During these cycles, the eccentricity and incli-nation of the inner orbit can experience periodic oscil-lations on a secular quadrupole LK timescale T LK = 815 π m trip M P P in (cid:0) − e (cid:1) / . (13)In the previous equation, P in and P out are the orbital pe-riods of the inner and outer binaries, respectively. Wenote that the exact size of the LK inclination windowdepends also on the physical parameters of the three ob-jects, thus varying from case to case (e.g., Grishin et al.2018). On this typical timescale, the relative inclina-tion of the inner orbit and outer orbit slowly increaseswhile the orbital eccentricity of the inner orbit decreases,and vice versa, conserving angular momentum (see Naoz2016, for a review). The inner eccentricity can reach al-most unity during LK cycles, which is typically achievedin the case i ∼ ◦ .Whenever the outer orbit is eccentric (octupole orderof approximation), the inner eccentricity can reach al-most unity even if the initial inclination lies outside ofthe window ∼ ◦ -140 ◦ (Naoz et al. 2013). This happensover the octupole timescale T oct = 1 (cid:15) T LK , (14) emographics of triple systems in dense star clusters (cid:15) = m − m m + m a in a out e out − e . (15)Nevertheless, LK cycles can be suppressed by addi-tional sources of precession (e.g., Fabrycky & Tremaine2007; Naoz et al. 2013), such as non-dissipative tides,that operate on a timescale (Kiseleva et al. 1998; Eggle-ton & Kiseleva-Eggleton 2001) T tide = 8 a / Gm in ) / (1 − e ) e + e × (cid:20) m m k R + 2 m m k R (cid:21) − , (16)where k , R and k , R are the apsidal motion con-stants and radii of the two stars in the binary (Hut1981), respectively, or general relativistic (GR) preces-sion, that operates on a typical timescale (Peters 1964) T GR = a / c (1 − e )3 G / ( m + m ) / . (17)To compute the maximum eccentricity e max attainedby triples, we use the following equation to find the rootof j min = (cid:112) − e (e.g., Liu et al. 2015)38 × { e + ( j −
1) + (5 − j ) × (cid:20) − (( j − ζ min + e ζ − j cos I ) j (cid:21) − (1 + 4 e − e cos ω ) sin I } + (cid:15) GR ( j − − j − )+ (cid:15) Tide × (cid:18) − j + 3(1 − j ) j − − j + 3(1 − j ) j (cid:19) = 0 . (18)The above equation is derived in a quadrupole approx-imation, but has been shown to remain approximatelyvalid even when the octupole effect is non-negligible(e.g., Anderson et al. 2016, 2017; Liu et al. 2019). Inthe previous equation, e is the initial inner binary ec-centricity, j = (cid:112) − e , ζ min = L ( e = e max ) /L out , and ζ = L ( e = e ) /L out , where L and L out are the angularmomenta of the inner and outer binaries, respectively.The parameters (cid:15) GR = 3 Gm a (1 − e ) / c a m out (19)and (cid:15) Tide = 15 m a (1 − e ) / k Love , ∗ R ∗ a m ∗ m out (20) This assumes that only one of the two objects in the inner binaryraises tides. If both components of the inner binary raise tides, (cid:15)
Tide has a contribution from both components. represent the relative strength of the apsidal precessiondue to GR and tidal bulge of the star . Here, R ∗ and k Love , ∗ are the radius and the Love number of a givenstar, respectively. For a MS star, a good approximationis k Love , ∗ = 0 . T enc (see Eq. 12). Such encounters canreset the triple by altering the orbital properties signifi-cantly. In the case of soft triples, encounters with othercluster members will even tend to disrupt it, on average.Thus, unlike triples in isolation, LK cycles must occuron timescales shorter than the encounter timescale. If T LK < T enc , the inner binary eccentricity can reach highvalues and trigger the interaction, or even the merger,of the components in the inner binary. If T LK > T enc ,LK oscillations could be suppressed by stellar encoun-ters (Antonini et al. 2016).As an example, we show in Figure 10 a comparison be-tween the LK timescale and the encounter timescale fortriples with a MS star plus a companion in the inner bi-nary: MS-MS (top-left panel), MS-G (top-right panel),MS-WD (center-left panel), MS-NS (center-right panel),and MS-BH (bottom-left panel). In each panel, for thesystems that satisfy T LK < T enc , we represent in colorcode the maximum eccentricity attained by triples, com-puted by using Eq. 18.We showed in Section 3 that triple systems experiencea recoil kick as a result of the binary–binary exchangeencounter. The recoil kick can be large enough to ejectthe triple from the core. If not ejected from the cluster,the triple would have a new elongated orbit with peri-center in the cluster core and apocenter in the clusteroutskirts. The triple would eventually sink back to thecore as a result of dynamical friction (Eq. 8). However,the encounter timescale of the triple would be longerthan given by Eq. 12 since it would spend most of itsorbit in regions less dense than the core. To bracketthe uncertainties, we show the results of our LK analy- We do not include precession due to rotational distortion of thestar, which is usually negligible. Fragione et al. T L K ( y r ) MS-MS MS-G e m a x T L K ( y r ) MS-WD T enc (yr)10 MS-NS e m a x T enc (yr)10 T L K ( y r ) MS-BH e m a x Figure 10.
Comparison between the LK timescale ( T LK ) and the encounter timescale ( T enc ) for triples with a MS star plus acompanion in the inner binary: MS-MS (top-left panel); MS-G (top-right panel); MS-WD (center-left panel); MS-NS (center-right panel); MS-BH (bottom-left panel). Color code: maximum eccentricity attained by triples with T LK < T enc , computedusing Eq .18. emographics of triple systems in dense star clusters T enc goes to in-finity (essentially corresponding to a triple ejected fromthe cluster environment; see Section 3.4).5.2. Collision and accretion in triples with amain-sequence star, a giant, or a white dwarf inthe inner binary
During the LK evolution, the inner orbital eccentricityis excited, which can result in crossing of the Rochelimit. Given a binary system with components m i and m j , we define the dimensionless number (Eggleton 1983) µ ji = 0 .
49 ( m j /m i ) / . m j /m i ) / + ln(1 + ( m j /m i ) / ) . (21)Thus, the Roche limit is defined as a Roche , ij ≡ R j µ ji , (22)where R j is the radius of m j . The definition of a Roche , ji is obtained with the substitutions i → j and j → i . Fortriples that comprise of a MS star or a G star in theinner binary, we compute e max from Eq. 18 and defineRoche-lobe overflow to occur whenever (e.g., Stephanet al. 2019) a (1 − e max ) ≤ a Roche . (23)We show in Figure 11 the probability distributionfunction of the ratio of the inner binary’s pericenterduring a LK cycle to the the Roche semi-major axis(Eq. 22), for triples with a MS star or a G star inthe inner binary. The shaded area represents the re-gion where a (1 − e max ) /a Roche ≤
1, where a Roche-lobeoverflow can take place. According to the companion ofthe MS or G star in the inner binary of these triples,the LK cycles can produce either accretion or a physicalmerger. In the case of MS inner binaries, MS-MS andMS-G would likely form blue stragglers and rejuvenatedgiants, MS-WD would form cataclysmic variables, andMS-NS or MS-BH would give birth to X-ray binaries,millisecond pulsars, or Thorne-Zytkow objects. On theother hand, G-G mergers would form rejuvenated gi-ants, while mergers of G with a compact object couldgive birth to ultracompact X-ray binaries (Hurley et al.2000, 2002; Ivanova et al. 2010; Naoz et al. 2016; Peretset al. 2016; Kremer et al. 2018; Fragione et al. 2019c;Kremer et al. 2019c; Stephan et al. 2019). In Figure 11,we also illustrate a comparison of the systems that sat-isfy a (1 − e max ) /a Roche ≤ T enc usingEq. 12 (solid line) and when T enc goes to infinity (dot-ted line). We find that there is not a significant differ-ence between using Eq. 12 to compute T enc and treating T enc as infinite, since for these systems the LK timescaleis typically smaller than the encounter timescale fromEq. 12. We estimate that ∼
35, 43, 38, 32, and 14% of thetriple systems merge with inner MS-MS, MS-G, MS-WD, MS-NS, and MS-BH binaries, respectively, while ∼
12, 38, 16, and 15% of the systems merge for tripleswith inner G-G, G-WD, G-NS, and G-BH binaries, re-spectively. Assuming a GC density ρ GC ∼ .
31 Mpc − (Rodriguez et al. 2015; Rodriguez & Loeb 2018), we es-timate a merger rate of ∼ − –10 − Gpc − yr − forthese populations of triples, consistent with the previ-ous estimated rates in cluster binaries (Kremer et al.2019c) and in field triples (Fragione et al. 2019c).In Figure 12, we plot the probability distribution func-tion of the ratio of the inner binary’s pericenter during aLK cycle to the Roche semi-major axis, for triples witha WD in the inner binary. The shaded area representsthe region where a (1 − e max ) /a Roche ≤
1. The outcomeof the accretion depends on the components of the in-ner binary. WD-WD mergers can lead to Type Ia SNe,while WD-NS and WD-BH mergers can lead to tidaldisruption events and gamma-ray bursts (Hurley et al.2002; Fryer et al. 1999; Perets et al. 2016; Fragione et al.2019e; Leigh et al. 2020).We estimate that ∼ T enc is computedusing Eq. 12 and the case T enc goes to infinity, sincefor these systems the LK timescale is typically smallerthan the encounter timescale from Eq. 12. MergingWDs have masses in the range ∼ . (cid:12) –1 . (cid:12) , whilemerging NSs and BHs have typical masses of ∼ . (cid:12) and ∼
10 M (cid:12) , respectively. Assuming a GC density ρ GC ∼ .
31 Mpc − (Rodriguez et al. 2015; Rodriguez& Loeb 2018), we compute a merger rate of ∼ − Gpc − yr − , consistent with the estimated rate for thiskind of merger in field triples (Fragione et al. 2019e).5.3. Gravitational wave mergers in triples with a whitedwarf, a neutron star, or a black hole in the innerbinary
For triples comprised of an inner binary with two com-pact objects, GW emission becomes relevant. Given abinary of components M and M , semi-major axis a ,and eccentricity e , it would merge through GW emis-sion in isolation on a timescale Peters (1964) T GW = 5256 a c G ( M + M ) M M (1 − e ) / . (24)When LK oscillations are relevant in a triple system,the inner binary would spend a fraction of its time ∝ (1 − e ) / at e ∼ e max , where it loses energy efficientlydue to GW emission. Thus, the GW timescale would bereduced compared to a binary in isolation (e.g., Grishinet al. 2018) T (red)GW = 5256 a c G ( M + M ) M M (1 − e ) . (25)6 Fragione et al. P D F MS-MSRLOF 10 P D F MS-NSRLOF 10 a (1 e max )/ a Roche P D F G-WDRLOF 10 a (1 e max )/ a Roche a (1 e max )/ a Roche
Figure 11.
Probability distribution function of the ratio of the inner binary’s pericenter during a LK cycle to the Rochesemi-major axis (Eq. 22), for triples with a MS or a G star in the inner binary. The shaded area represents the region where aRoche-lobe overflow can take place ( a (1 − e max ) /a Roche ≤ T LK < T enc and dotted linesrepresent the case where T enc goes to infinity. We show in Figure 13 the cumulative distributionfunction of the merger time ( T f + T ( red )GW ) for triples witha WD in the inner binary. If the reduced GW mergertime is shorter than the LK timescale that is required toreach the maximal eccentricity, we use the secular LKtime (Fragione et al. 2019a). We find that ∼ T enc usingEq. 12 and in the case T enc goes to infinity, respectively.Triples with an inner WD-BH and WD-NS binarycould be observed by LISA up to the point of disrup-tion. The GW frequency at disruption is (Fragione Note that this corresponds to circular orbits, but the peak GWfrequency at disruption is similar for arbitrary eccentricities towithin ∼ emographics of triple systems in dense star clusters a (1 e max )/ a Roche P D F WD-WDWD-NSWD-BHRLOF
Figure 12.
Probability distribution function of the ratioof the inner binary’s pericenter during a LK cycle and theRoche semi-major axis, for triples with a WD in the inner bi-nary. The shaded area represents the region where a Roche-lobe overflow can take place ( a (1 − e max ) /a Roche ≤ T LK < T enc and dotted linesrepresent the case T enc goes to infinity. et al. 2019e) f GW = G / ( M + M WD ) / πR / = 0 .
09 Hz (cid:18) M WD M (cid:19) M / , . (cid:12) R − / , km , (26)where R WD ∝ M − / is the WD radius and M is theBH or NS mass . The total characteristic GW strainfor observing the GWs for a duration T obs averaged overbinary and detector orientation is approximately (Rob-son et al. 2019) h c = 8 √ G c M M WD R t D ( T obs f GW ) / = 2 . × − × T . , D −
110 Mpc M . ,
10 M (cid:12) M . , . (cid:12) R − . , km . (27)In Figure 13, we also show the merging systems withan inner binary BH that merge due to the LK mecha-nism. We estimate that ∼ .
1% of triples with a binaryBH as inner binary merge within a Hubble time. We findthere is no significant difference between the cases wherethe tertiary is any kind of object (BH-BH) or a BH (BH-BH-BH), thus implying that the majority of BH mergersdue to the LK mechanism take place in triple systemswhere all the objects are BHs. Moreover, we find nodifference in the merger fractions when computing T enc We introduced the abbreviated notation X ,a = X/a C D F WD-WDWD-NSWD-BHBH-BHBH-BH-BH
Figure 13.
Cumulative distribution function of the mergertime ( T f + T ( red )GW ), for triples with an inner binary comprisedof two compact objects that merge due to the LK mecha-nism. The LK mechanism does not produce NS-NS or BH-NS mergers in our models. using Eq. 12 and in the case T enc goes to infinity, respec-tively. None of the triples with a NS in the inner binarymerge within a Hubble time. The reason is that tripleswith NSs in the inner binary are formed at late times,when most of the BHs have been ejected in the BH-burning process (Kremer et al. 2019d), as shown in Fig-ure 8. Therefore, triple systems likely do not contributeto the rates of NS-NS and BH-NS mergers in clusters,which remain too small to account for LIGO/Virgo ob-servations, as shown in detail by Ye et al. (2020).In order to estimate the local cosmological rate of BH-BH mergers in cluster triple systems, we compute thecumulative merger rate as (e.g., Rodriguez et al. 2015) R ( z ) = (cid:90) z R ( z (cid:48) ) dV c dz (cid:48) (1 + z (cid:48) ) − dz (cid:48) , (28)where dV c /dz is the comoving volume at redshift z and R ( z ) is the comoving (source) merger rate. The comov-ing rate is given by R ( z ) = f × ρ GC × dN ( z ) dt , (29)where ρ GC ∼ .
31 Mpc − (Rodriguez et al. 2015; Ro-driguez & Loeb 2018), f ∼ dN ( z ) /dt is the number ofmergers per unit time at a given redshift. To estimate dN ( z ) /dt , we draw 10 random ages for the host clus-ter, from the metallicity-dependent age distributions ofEl-Badry et al. (2018), where the merger originated andthen compute the effective merger time for each merger.We find that the merger rate for BH triples in star clus-ters is ∼ . − yr − in the local Universe, consistent8 Fragione et al. with Antonini et al. (2016), within the uncertainties. Weleave a detailed calculation and discussion of the impli-cations of BH mergers in triples to a companion paper(Martinez et al., submitted). DISCUSSION AND CONCLUSIONSStellar multiplicity is an omnipresent outcome of thestar-formation process (Duchˆene & Kraus 2013). Morethan ∼
50% and ∼
25% of stars are thought to have atleast one and two stellar companions, respectively. Hi-erarchical systems can also be formed in star clusters(Fregeau et al. 2004; Leigh & Geller 2013). In thesedynamically-active environments, few-body interactionsbetween stars and/or compact remnants can efficientlyassemble hierarchical systems, primarily due to binary–binary encounters. In this process, one of the two bina-ries captures a star in the second wider binary, with thefourth object leaving the system.In this paper, we have presented for the first time thedemographics of triple systems of stars and compact ob-jects assembled in dense star clusters of various masses,concentrations, and metallicities. We have made use ofthe ensemble of cluster simulations presented in Kremeret al. (2020), which covers roughly the complete rangeof GCs observed at present day in the Milky Way.We have demonstrated that triples are efficiently as-sembled in binary–binary encounters that involve twobinaries of quite different sizes. In this process, thetighter binary replaces one of the components in thewider binary. The object that is removed is then ejected,while the captured one becomes the tertiary in the newlyformed triple system. During these binary–binary en-counters, triple formation can lead to GW captures andmergers of stars and compact objects. We have foundthat a cluster typically assembles hundreds of tripleswith an inner BH-BH binary (of which ∼ r v ≤ ∼
10 times more systems with inner MS-MSbinaries. We have also found that ∼
50% of the overalltriple population from our simulations consists of sys-tems where all the components are BHs.Roughly 10% ofthe triples consist of an inner BH-BH binary with witha non-BH tertiary companion, while ∼
38% consist ofan inner binary containing at least one MS star. Othertriples constitute the remaining ∼
2% of the population.We have shown that the initial properties of the hostcluster set the typical orbital parameters and forma-tion times of the assembled triples. Smaller and less-extended clusters form triples faster and with wider in-ner and outer orbits with respect to more massive andconcentrated clusters. We have also found that tripleswhose inner binary comprises at least one BH typically form on a shorter timescale compared to other triples.This is a direct consequence of the BH-burning mecha-nism (Kremer et al. 2019d).We have discussed how the LK mechanism can drivethe inner binary of the formed triples to high eccen-tricities, whenever it takes place before the triple is dy-namically reprocessed by encountering another clustermember. Some of these systems can reach sufficientlylarge eccentricities to form a variety of exotica, tran-sients and GW sources, such as blue stragglers, rejuve-nated giant stars, X-ray binaries, Type Ia Supernovae,Thorne-Zytkow objects, and LIGO/Virgo sources.We have also estimated that the Milky Way’s glob-ular clusters are expected to host tens of triples withat least one luminous component at present day. Dueto their high densities, only one triple star system isknown to exist in GCs (e.g., Prodan & Murray 2012).The system in question, called 4U 1820-30, is locatednear the centre of the GC NGC 6624 and consists of alow-mass X-ray binary with a NS primary and a WDsecondary, in orbit with a period ∼
685 s. There is alsoa large luminosity variation for this system with a pe-riod of ∼
171 days, thought to be due to the presenceof a tertiary companion (Grindlay et al. 1988). An-other confirmed triple system in the GC M4 is madeup of an inner binary comprised of a pulsar (PSR 1620-26) and a white dwarf, orbited by a substellar tertiary(Arzoumanian et al. 1996; Rasio et al. 1995). Thesesystems could be naturally explained by binary–binaryinteractions involving planetary systems in dense stel-lar environments (Kremer et al. 2019b). A few nearbyopen clusters are also known to have comparably highmultiplicity fractions (see e.g., Leigh & Geller 2013, fora more detailed review). The Hyades (Patience et al.1998), Pleiades (Mermilliod et al. 1992; Bouvier et al.1997) and Praesepe (Mermilliod & Mayor 1999; Bouvieret al. 2001) have binary fractions of, respectively, 35%,34% and 40%, and triple fractions of, respectively, 6%,3% and 6%. Notably, the open cluster Taurus-Aurigaappears to have a multiplicity fraction higher than thefield. Kraus et al. (2011) performed a high-resolutionimaging study to characterize the multiple-star popula-tions in Taurus-Auriga. They found that ∼ / / ∼ / / emographics of triple systems in dense star clusters A. TRIPLE SYSTEMS FORMED IN CLUSTER SIMULATIONS
Table A1 . Initial cluster parameters and number of different triples formed. Triples with a main-sequence or a giant plus acompanion in the inner binary. r v (pc) r g (kpc) Z N
MS-MS MS-G MS-WD MS-NS MS-BH G-G G-WD G-NS G-BH1 0.5 2 0.0002 2 ×
150 16 17 0 314 0 0 0 62 0.5 2 0.0002 4 ×
326 12 67 0 1192 1 1 0 63 0.5 2 0.0002 8 ×
180 1 33 13 312 0 0 0 44 † . ×
24 0 0 0 0 0 0 0 05 0.5 2 0.002 2 ×
314 21 21 1 553 0 2 0 06 0.5 2 0.002 4 ×
261 6 78 6 330 0 2 0 57 0.5 2 0.002 8 ×
251 7 66 7 259 0 1 2 08 0.5 2 0.002 1 . ×
201 3 2 0 154 0 0 0 39 0.5 2 0.02 2 ×
283 37 10 4 40 14 1 0 110 0.5 2 0.02 4 ×
298 23 50 5 51 3 10 0 011 0.5 2 0.02 8 ×
342 16 51 9 101 0 0 1 812 0.5 2 0.02 1 . ×
412 10 81 10 109 0 0 0 213 0.5 8 0.0002 2 ×
281 44 44 2 664 3 2 0 1614 0.5 8 0.0002 4 ×
248 2 37 4 866 0 3 0 515 0.5 8 0.0002 8 ×
184 1 41 6 258 0 1 0 116 † . ×
35 0 0 0 0 0 0 0 017 0.5 8 0.002 2 ×
293 10 66 6 358 0 1 0 518 0.5 8 0.002 4 ×
247 4 53 5 350 0 0 0 219 0.5 8 0.002 8 ×
237 7 34 8 230 0 0 0 120 0.5 8 0.002 1 . ×
190 1 1 0 97 0 0 0 121 0.5 8 0.02 2 ×
221 21 38 1 134 3 1 0 322 0.5 8 0.02 4 ×
279 17 38 2 100 0 7 3 923 0.5 8 0.02 8 ×
283 10 53 3 122 0 2 0 224 0.5 8 0.02 1 . ×
349 7 60 4 142 0 2 0 625 0.5 20 0.0002 2 ×
232 6 70 1 600 0 1 0 126 0.5 20 0.0002 4 ×
294 2 21 3 623 0 0 0 327 0.5 20 0.0002 8 ×
168 3 39 8 308 0 0 0 128 † . ×
38 0 0 0 0 0 0 0 0
Table A1 continued Fragione et al.
Table A1 (continued) r v (pc) r g (kpc) Z N
MS-MS MS-G MS-WD MS-NS MS-BH G-G G-WD G-NS G-BH29 0.5 20 0.002 2 ×
298 17 86 3 463 4 5 0 330 0.5 20 0.002 4 ×
272 6 44 2 551 0 2 0 131 0.5 20 0.002 8 ×
187 3 29 1 180 0 1 0 032 0.5 20 0.002 1 . ×
132 5 1 0 160 0 0 0 033 0.5 20 0.02 2 ×
301 25 38 2 84 1 3 0 234 0.5 20 0.02 4 ×
277 5 24 2 59 0 0 0 635 0.5 20 0.02 8 ×
291 13 46 5 100 0 3 0 336 0.5 20 0.02 1 . ×
360 8 62 8 103 0 3 0 237 1 2 0.0002 2 ×
70 2 38 2 259 0 1 0 438 1 2 0.0002 4 ×
81 1 61 3 209 0 3 0 139 1 2 0.0002 8 ×
17 1 2 0 159 0 0 0 040 1 2 0.0002 1 . ×
21 0 0 0 85 0 0 0 141 1 2 0.002 2 ×
85 1 34 0 252 0 3 0 342 1 2 0.002 4 ×
92 4 57 2 370 0 2 0 043 1 2 0.002 8 ×
17 0 0 0 227 0 0 0 044 1 2 0.002 1 . × ×
127 11 34 1 32 5 4 0 946 1 2 0.02 4 ×
157 3 36 1 105 0 4 1 1147 1 2 0.02 8 ×
182 15 44 1 62 0 2 0 348 1 2 0.02 1 . ×
97 4 6 0 107 0 0 0 349 1 8 0.0002 2 ×
97 2 48 1 528 0 2 0 150 1 8 0.0002 4 ×
50 0 35 5 332 0 0 0 051 1 8 0.0002 8 ×
15 0 0 0 150 0 0 0 052 1 8 0.0002 1 . ×
17 2 1 0 88 0 0 0 053 1 8 0.002 2 ×
119 4 41 0 396 0 1 0 254 1 8 0.002 4 ×
30 1 2 0 163 0 1 0 155 1 8 0.002 8 ×
17 0 0 0 137 0 0 0 056 1 8 0.002 1 . ×
15 0 1 0 141 0 0 0 057 1 8 0.02 2 ×
142 16 45 0 76 3 2 0 858 1 8 0.02 4 ×
158 18 46 1 63 0 2 0 459 1 8 0.02 8 ×
159 11 20 0 65 0 1 0 360 1 8 0.02 1 . ×
88 3 2 0 78 0 0 0 361 1 20 0.0002 2 ×
72 2 53 3 309 0 1 0 162 1 20 0.0002 4 ×
82 2 59 9 532 0 0 0 263 1 20 0.0002 8 ×
15 1 0 0 135 0 0 0 064 1 20 0.0002 1 . ×
10 0 0 0 130 0 0 0 065 1 20 0.002 2 ×
104 7 30 0 566 0 1 0 366 1 20 0.002 4 ×
51 1 12 1 331 0 0 0 467 1 20 0.002 8 ×
14 0 1 0 294 0 0 0 168 1 20 0.002 1 . ×
12 1 0 0 91 0 0 0 069 1 20 0.02 2 ×
120 29 57 0 90 2 3 0 270 1 20 0.02 4 ×
123 5 29 1 99 0 0 0 1171 1 20 0.02 8 ×
120 7 10 1 86 0 0 1 272 1 20 0.02 1 . ×
89 0 2 0 40 0 0 0 073 2 2 0.0002 2 ×
46 1 15 0 559 0 0 0 274 2 2 0.0002 4 ×
34 0 20 0 269 0 0 0 175 2 2 0.0002 8 × . × ×
34 3 16 0 218 0 1 0 078 2 2 0.002 4 ×
34 1 13 0 292 0 0 0 079 2 2 0.002 8 × . × ×
35 8 4 0 29 0 0 0 282 2 2 0.02 4 ×
34 5 9 0 26 0 0 0 1
Table A1 continued emographics of triple systems in dense star clusters Table A1 (continued) r v (pc) r g (kpc) Z N
MS-MS MS-G MS-WD MS-NS MS-BH G-G G-WD G-NS G-BH83 2 2 0.02 8 ×
30 0 0 0 18 0 0 0 384 2 2 0.02 1 . ×
37 0 0 0 21 0 0 0 085 2 8 0.0002 2 × × × . × ×
64 6 48 0 390 0 3 0 490 2 8 0.002 4 × × . × ×
27 0 4 0 20 0 0 0 194 2 8 0.02 4 ×
23 0 5 0 20 0 0 0 295 2 8 0.02 8 ×
16 1 1 0 26 1 0 0 296 2 8 0.02 1 . ×
35 1 0 0 20 0 0 0 097 2 20 0.0002 2 ×
14 0 6 0 324 0 0 0 198 2 20 0.0002 4 ×
15 0 2 1 212 0 0 0 099 2 20 0.0002 8 × . × ×
16 0 0 0 231 0 0 0 0102 2 20 0.002 4 ×
11 0 1 0 186 0 0 0 0103 2 20 0.002 8 × . ×
10 1 0 0 70 0 0 0 1105 2 20 0.02 2 ×
29 4 5 0 16 0 0 0 0106 2 20 0.02 4 ×
23 1 0 0 37 0 0 0 1107 2 20 0.02 8 ×
19 0 1 0 20 0 0 0 2108 2 20 0.02 1 . ×
26 1 0 0 75 1 0 0 0109 4 2 0.0002 2 × × × . × × × × . × × × ×
11 0 0 0 11 0 0 0 1120 4 2 0.02 1 . ×
17 0 2 0 10 0 0 0 0121 4 8 0.0002 2 × × × . × × × × . × × × × . ×
14 1 0 0 10 0 0 0 2133 4 20 0.0002 2 × × × . × Table A1 continued Fragione et al.
Table A1 (continued) r v (pc) r g (kpc) Z N
MS-MS MS-G MS-WD MS-NS MS-BH G-G G-WD G-NS G-BH137 4 20 0.002 2 × × × . × × ×
10 0 0 0 5 0 0 0 0143 4 20 0.02 8 ×
10 1 1 0 6 0 0 0 0144 4 20 0.02 1 . ×
13 0 1 0 8 0 0 0 0145 1 20 0.0002 3 . ×
14 0 0 0 26 0 0 0 0146 2 20 0.0002 3 . ×
11 0 0 0 37 0 0 0 0147 1 20 0.02 3 . ×
49 0 0 0 29 0 0 0 0148 2 20 0.02 3 . ×
44 1 3 0 31 0 0 0 1
Note —Models marked with a dagger ( † ) indicates the model was stopped due to onset of collisional runaway (see Kremer et al. 2020, for details). Table A2 . Initial cluster parameters and number of different triples formed. Triples with a white dwarf, neutron star,or a black hole plus a companion in the inner binary. r v (pc) r g (kpc) Z N
WD-WD WD-NS WD-BH NS-NS NS-BH BH-BH BH-BH-BH1 0.5 2 0.0002 2 × ×
18 11 3 0 0 562 3013 0.5 2 0.0002 8 ×
42 16 27 4 0 755 6354 † . × × ×
50 3 10 0 1 422 3697 0.5 2 0.002 8 ×
30 15 8 1 0 662 6108 0.5 2 0.002 1 . × × ×
25 8 1 0 0 263 23411 0.5 2 0.02 8 ×
18 11 8 0 1 291 27112 0.5 2 0.02 1 . ×
18 4 17 1 0 438 41213 0.5 8 0.0002 2 ×
20 12 5 0 2 246 12514 0.5 8 0.0002 4 ×
125 31 75 3 0 692 46715 0.5 8 0.0002 8 ×
34 24 16 3 6 797 69416 † . × ×
41 1 1 0 0 267 23618 0.5 8 0.002 4 ×
16 0 9 0 0 479 43819 0.5 8 0.002 8 ×
24 4 12 2 0 555 52120 0.5 8 0.002 1 . × ×
54 3 3 0 0 82 7222 0.5 8 0.02 4 ×
23 1 20 0 0 150 14123 0.5 8 0.02 8 ×
11 4 41 0 2 328 27824 0.5 8 0.02 1 . × ×
39 5 1 2 0 303 18126 0.5 20 0.0002 4 × ×
33 18 5 1 4 877 73328 † . × ×
38 7 39 0 0 269 21330 0.5 20 0.002 4 ×
29 1 15 0 0 445 37831 0.5 20 0.002 8 ×
13 5 19 0 0 739 68232 0.5 20 0.002 1 . × Table A2 continued emographics of triple systems in dense star clusters Table A2 (continued) r v (pc) r g (kpc) Z N
WD-WD WD-NS WD-BH NS-NS NS-BH BH-BH BH-BH-BH33 0.5 20 0.02 2 × × × . ×
11 2 13 0 0 516 48037 1 2 0.0002 2 ×
18 0 59 0 0 446 29038 1 2 0.0002 4 ×
67 10 5 1 1 628 48539 1 2 0.0002 8 × . × ×
28 0 14 0 1 285 23142 1 2 0.002 4 ×
56 2 41 0 0 620 51743 1 2 0.002 8 × . × × × ×
13 2 10 0 0 308 29448 1 2 0.02 1 . × ×
33 3 6 0 0 486 28450 1 8 0.0002 4 ×
38 4 24 0 1 738 58251 1 8 0.0002 8 × . × ×
22 0 42 0 0 329 27554 1 8 0.002 4 × × . × × ×
13 3 2 0 0 325 31559 1 8 0.02 8 × . × ×
46 3 5 0 2 421 27662 1 20 0.0002 4 ×
40 11 15 1 2 809 52363 1 20 0.0002 8 × . × ×
14 0 26 0 0 415 35266 1 20 0.002 4 × × . × ×
10 0 1 0 0 178 16370 1 20 0.02 4 × × . × × ×
25 3 9 0 0 755 57175 2 2 0.0002 8 × . × × × × . × × × × . × × × Table A2 continued Fragione et al.
Table A2 (continued) r v (pc) r g (kpc) Z N
WD-WD WD-NS WD-BH NS-NS NS-BH BH-BH BH-BH-BH87 2 8 0.0002 8 × . × ×
18 0 3 0 0 398 26890 2 8 0.002 4 × × . × × × × . × × × × . × × × × . × × × × . × × × × . × × × × . × × × × . × × × × . × × × × . × × × × . × × × × . × × × × . × Table A2 continued emographics of triple systems in dense star clusters Table A2 (continued) r v (pc) r g (kpc) Z N
WD-WD WD-NS WD-BH NS-NS NS-BH BH-BH BH-BH-BH141 4 20 0.02 2 × × × . × . × . × . × . × Note —Models marked with a dagger ( † ) indicates the model was stopped due to onset of collisional runaway (see Kremer et al. 2020, fordetails). REFERENCES
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