# Topology of black hole binary–single interactions

Topology of black hole binary–single interactions Abstract We present a study on how the outcomes of binary–single interactions involving three black holes (BHs) distribute as a function of the initial conditions; a distribution we refer to as the topology. Using a N-body code that includes BH finite sizes and gravitational wave (GW) emission in the equation of motion (EOM), we perform more than a million binary–single interactions to explore the topology of both the Newtonian limit and the limit at which general relativistic (GR) effects start to become important. From these interactions, we are able to describe exactly under which conditions BH collisions and eccentric GW capture mergers form, as well as how GR in general modifies the Newtonian topology. This study is performed on both large- and microtopological scales. We further describe how the inclusion of GW emission in the EOM naturally leads to scenarios where the binary–single system undergoes two successive GW mergers. gravitation, gravitational waves, stars: kinematics and dynamics 1 INTRODUCTION Dynamical interactions of black holes (BHs) in dense stellar environments are likely to play an important role for the assembly of binary black hole (BBH) mergers observable by gravitational wave (GW) interferometers such as LIGO (e.g. Abbott et al. 2016c; Antonini & Rasio 2016; Rodriguez, Chatterjee & Rasio 2016). Major effort is especially being devoted to understand the formation and evolution of BHs in globular cluster (GC) systems (e.g. Samsing, MacLeod & Ramirez-Ruiz 2014; Rodriguez, Chatterjee & Rasio 2016; Wang et al. 2016; Askar et al. 2017; Samsing & Ramirez-Ruiz 2017; Samsing, MacLeod & Ramirez-Ruiz 2018), where the derived BBH merger rate interestingly seems to be comparable with that inferred from the current sample of LIGO detections (e.g. Abbott et al. 2016a,b,d, 2017). Related few-body studies are therefore often focused on deriving cross-sections and rates assuming the orbital and angular momentum initial conditions (ICs) follow an isotropic distribution at infinity, which would be the case for most dense stellar systems (e.g. Hut & Bahcall 1983; Gültekin, Miller & Hamilton 2006; Samsing, MacLeod & Ramirez-Ruiz 2014; Antognini & Thompson 2016; Samsing, MacLeod & Ramirez-Ruiz 2017; Samsing & Ramirez-Ruiz 2017). However, not all few-body interactions take place under isotropic conditions. Dynamical interactions of BHs in active galactic nuclei (AGNs) discs provide one example (e.g. Bellovary et al. 2016; Bartos et al. 2017; McKernan et al. 2017; Stone, Metzger & Haiman 2017), where gas torques force objects to move and interact in approximately the same orbital plane as the disc itself (e.g. Ostriker 1983; Syer, Clarke & Rees 1991; Goodman & Tan 2004; Levin 2007; Kocsis, Yunes & Loeb 2011; McKernan et al. 2011; Just et al. 2012). Although the number of BHs in AGN discs is currently unknown, it has interestingly been pointed out that if BHs do exist in such environments, then they are likely to pile up and undergo frequent interactions in migration-traps similar to those found in planetary discs (e.g. Horn et al. 2012; McKernan et al. 2012, 2017; Bellovary et al. 2016). If the BHs have different masses, then the corresponding difference in migration time can also lead to interactions throughout the disc (e.g. McKernan et al. 2017). Disc-like systems can also form as a result of vector resonant relaxation (e.g. Kocsis & Tremaine 2011, 2015; Roupas, Kocsis & Tremaine 2017). Motivated by these recent ideas, we explore in this paper the dynamics and outcomes of co-planar binary–single interactions. For this, we perform a detailed study of the binary–single phase-space, a representation we broadly refer to as the binary–single topology in analogy with Hut (1983a). Our work should partly be considered as a continuation of that by Hut (1983a), where the topology of co-planar binary–single interactions in the Newtonian point-particle limit was studied. In our work, we expand on that study by including general relativity (GR) corrections in the N-body equation of motion (EOM) as well as BH finite sizes. These corrections break the scale-free nature of the Newtonian point-particle binary–single problem (e.g. Hut 1983a; Hut & Bahcall 1983), leading to several new effects including the formation of GW captures and BH collisions (Samsing et al. 2014). Parts of our presented study describe highly idealized situations, however, we see this as a necessary first step, as the three-body problem with GR effects is largely unexplored for anisotropic distributions; a proper understanding of the BBH outcome distribution and how it relates to the binary–single ICs is crucial for connecting observations of GW events to their astrophysical origin. We further introduce a new unexplored dynamical pathway for generating two successive GW merger events originating from the same binary–single interaction. Here, the first GW merger happens during the interaction while the three BHs are still in a bound state (e.g. Samsing et al. 2014), where the second GW merger happens between the BH formed in the first merger and the remaining bound BH. Interestingly, we find the time span from the first to the second GW merger in co-planar interactions to occasionally be short enough for LIGO to observe both GW mergers. This suggests that AGN discs might be a likely birthplace for such double GW merger events. The paper is organized as follows. In Section 2, we introduce the binary–single interaction channel, together with the ICs and outcomes that are relevant for this study. After this follows in Section 3, a detailed study of the binary–single topology for both the classical and the relativistic limit. In Section 4, we explore in greater detail how the inclusion of GW emission in the N-body EOM affects the resultant topology and range of outcomes. Results from zoom-in simulations performed to study microtopological structures are given in Section 5. Finally, in Section 6, we present our proposed channel for generating double GW merger events through binary–single interactions. Conclusions are given in Section 7. 2 BH BINARY–SINGLE INTERACTIONS In this work, we study interactions between a circular binary and an incoming single, and how the corresponding outcomes distribute as a function of the ICs. Throughout the paper, we assume the three objects to be identical BHs each with a mass m = 10 M⊙, and corresponding Schwarzschild radius Rs. All interactions are performed with our N-body code presented in Samsing et al. (2017), which includes GR corrections using the post-Newtonian (PN) formalism (e.g. Blanchet 2014). To approach the problem in a systematic way, we have for this paper chosen to sorely focus on how the loss of energy and angular momentum through GW emission affects the outcome distributions. Our implementation of GR effects is therefore at the 2.5 PN level, as the lower order terms conserve energy (Blanchet 2014). With the insight provided in this primary study, we will in upcoming work systematically study how each term in the PN expansion affects the results. In this section, we first summarize a few dynamical properties of binary–single interactions with GW emission and finite sizes included in the EOM. We then list the ICs relevant for our considered binary–single study together with the range of possible outcomes. The last part contains a description of what we refer to as the topology. We refer the reader to Hut (1983a) for further details. 2.1 Three-body energetics and interactions The range of possible outcomes is broadly determined by the orbital velocity of the initial binary, vorb, relative to the velocity of the incoming single at infinity, v∞. In the equal mass case, one finds that the initial orbital binary–single energy is negative for v < 1 and positive for v > 1, where $$v \equiv v_{\infty }/(v_{\rm orb}\sqrt{3})$$, as further described in Hut & Bahcall (1983). These two limits are usually referred to as the hard-binary (HB) limit and the soft-binary (SB) limit, respectively (e.g. Heggie 1975). We describe a few characteristics of these two limits below. We will argue that only the HB limit is relevant for our study. 2.1.1 SB limit In the SB limit, the binary–single interaction is always prompt (Hut 1983b), any dynamical corrections to the N-body EOM will therefore rarely have a chance to affect the standard Newtonian distribution of outcomes. As a result, the inclusion of GW emission, as well as collisions and tidal interactions, does not play a significant role in this limit (e.g. Samsing et al. 2014). The topology of the SB limit was studied by Hut (1983a), and will therefore not be discussed further in this paper. 2.1.2 HB limit In the HB limit, the binary–single system either undergoes a direct interaction (DI) or a resonant interaction (RI), depending on the exact ICs (e.g. Samsing et al. 2014). For equal mass interactions, these two interaction types appear statistically in near equal numbers. However, the RI channel has a much higher probability than the DI channel of entering a state where GR effects and finite sizes can affect the three-body dynamics, as described in the following. A DI is characterized by a duration time that is similar to the orbital time of the initial binary, which makes it somewhat similar to an SB limit interaction. The kinematics of a DI is often such that the incoming single promptly pairs up with one of the binary members through a sling-shot interaction that ejects the remaining member. The short nature of this interaction type implies that GR effects and collisions rarely have a chance to affect the outcome. An RI lasts instead from a few to several thousand of initial orbital times (e.g. Hut & Bahcall 1983). The inclusion of GR effects and finite sizes in the EOM therefore have a relative high chance of affecting the dynamics during this type of interaction. How this in particular leads to an increase in the number of BH mergers can be understood as follows. As envisioned by Samsing et al. (2014), an RI can be described as a series of intermediate states (IMS), each of which is characterized by an IMS binary with a bound single. Due to pair-wise semichaotic exchanges of energy and angular momentum during the resonating evolution, each IMS binary has a finite probability for being formed with a high eccentricity, even when the initial binary is circular. If the eccentricity is high enough, GR will affect the EOM of the IMS binary (e.g. Peters 1964), which most notably leads to an inspiral driven by GW emission that is followed by a prompt merger (Gültekin et al. 2006; Samsing et al. 2014). For even higher eccentricities, the IMS binary members will instead undergo a prompt collision (e.g. Samsing et al. 2017, 2018). 2.2 Initial conditions The IC parameters that are relevant for our presented binary–single study are listed and described in the following. An accompanying illustration is shown in Fig. 1. Figure 1. View largeDownload slide ICs and notations for binary–single scatterings. The black dots represent the three interacting BHs, where the indices ‘1’ and ‘2’ denote the initial target binary members, and ‘3’ the incoming single. The impact parameter b and the relative velocity v are both defined at infinity. The angle f denotes the orbital phase of the binary at the time the distance between the binary COM and the incoming single is exactly 20 × a0, where a0 is the initial binary SMA. The angle γ denotes the angle of the incoming single at infinity relative to the initial binary orbital plane. In this work, we study how the outcome of a binary–single interaction depends on f, b for fixed values of v, γ. The ICs are further described in Section 2.2. Figure 1. View largeDownload slide ICs and notations for binary–single scatterings. The black dots represent the three interacting BHs, where the indices ‘1’ and ‘2’ denote the initial target binary members, and ‘3’ the incoming single. The impact parameter b and the relative velocity v are both defined at infinity. The angle f denotes the orbital phase of the binary at the time the distance between the binary COM and the incoming single is exactly 20 × a0, where a0 is the initial binary SMA. The angle γ denotes the angle of the incoming single at infinity relative to the initial binary orbital plane. In this work, we study how the outcome of a binary–single interaction depends on f, b for fixed values of v, γ. The ICs are further described in Section 2.2. Semimajor axis (a0): The semimajor axis of the initial target binary is denoted by a0. As described later, both the classical and the relativistic limit can be studied by simply varying the value of a0. Velocity (v): We denote the rescaled relative velocity between the initial binary COM and the incoming single at infinity by v, as described in Section 2.1. In this work, we set v = 0.01 for all our scatterings, however, none of our conclusions depend on the exact value of v, as long as v ≪ 1. We note that Hut (1983a) only considered interactions for which v ≥ 0.5, our study is therefore the first to explore the ‘ultra’ HB limit. Binary phase (f): The orbital phase of the binary when the single is exactly 20 × a0 away from the binary COM is denoted by f. The value of f is defined relative to an axis that is parallel to the velocity vector of the incoming single at infinity. Impact parameter (b): We define the rescaled impact parameter at infinity by b ≡ b∞/bc, where b∞ is the physical value and bc = a0/v. The value of b can be positive or negative, where a positive (negative) b here corresponds to that the angular momentum vectors of the binary and the incoming single, respectively, have the same (opposite) sign. The initial three-body angular momentum in the co-planar case will therefore equal zero for $$b = -\sqrt{3/4} \approx - 0.87$$ (e.g. Hut 1983a). We further note that the pericentre distance of the single w.r.t. the binary COM, rp, relates to b as rp/a0 = 3b2/8. Orbital plane angle (γ): The angle between the orbital planes of the initial binary and the incoming single is denoted by γ. In this notation, a co-planar interaction will have γ = 0. 2.3 Outcomes and kinematical properties The set of binary–single outcomes and kinematical properties we keep track of in this work is listed and described below. Further details on defining and identifying outcomes in N-body simulations can be found in, e.g. Fregeau et al. (2004) and Samsing et al. (2014, 2017). 2.3.1 Outcomes Binary-single (BS[ij]): The most common endstate is a binary with an unbound single. We denote this endstate by BS[ij], where ‘BS’ is short for ‘binary-single’, and [ij] refers to the object pair in the final binary. Throughout the paper, we denote the two initial binary members by ‘1’ and ‘2’, and the incoming single by ‘3’. GW inspiral (GW): A close passage between any two of the three interacting objects occasionally leads to a GW capture, where the two binary members undergo an inspiral through the emission of GWs (e.g. Gültekin et al. 2006; Samsing et al. 2014). We generally refer to this endstate by a ‘GW inspiral’, but will in our figures simply use ‘GW’ to keep the notation short. As described in Samsing et al. (2018), the probability for a three-body interaction to form a GW inspiral is approximately PGW ≈ 0.04 × [(m/10 M⊙)/(a0/10−2 au)]5/7. For m ∼ 10 M⊙, GR is therefore expected to significantly alter the outcome distribution for a0 ≲ 10−4 au. Collision (Coll): If two objects pass each other at a distance smaller than the sum of their radii without undergoing a GW inspiral first, then the outcome is denoted a ‘collision’, or in short by ‘Coll’. In our case, the collisional distance is simply 2Rs. Following Samsing et al. (2018), the probability for a three-body interaction to undergo a collision is approximately PColl ≈ 0.002 × [(m/10 M⊙)/(a0/10−2 au)]. Note here that $$P_{\rm GW}/P_{\rm Coll} \propto a_{0}^{2/7}$$, i.e. the relative importance of GR increases with a0. Very long interaction (VLint): A finite fraction of the interactions can in theory last infinitely long time. This happens generally when the single in an IMS is sent out on a nearly unbound orbit. In this work, we denote an interaction that did not finish within 2500 × Torb by ‘VLint’, where Torb is the orbital time of the initial target binary – with the only exception of the results shown in Fig. 2, where the limit is set to 1000 × Torb. Figure 2. View largeDownload slide Topological map of binary–single endstates. The figure shows the distribution of endstates derived from equal mass BH binary–single interactions, as a function of the binary phase f and impact parameter b, for BH mass m = 10 M⊙, initial SMA a0 = 10−2 au, relative velocity v = 0.01, and orbital plane angle γ = 0.0. We generally refer to this distribution as the binary–single endstate topology, as described in Section 2.4. The large-scale patterns and structural regions seen in the figure clearly illustrate that the outcome of a binary–single interaction is not completely random, although the three-body problem is known to be chaotic. In this work, we use topological mappings, similar to the one shown here, as a tool to gain further insight into the binary–single problem with finite size effects and GR corrections included in the three-body EOM. Figure 2. View largeDownload slide Topological map of binary–single endstates. The figure shows the distribution of endstates derived from equal mass BH binary–single interactions, as a function of the binary phase f and impact parameter b, for BH mass m = 10 M⊙, initial SMA a0 = 10−2 au, relative velocity v = 0.01, and orbital plane angle γ = 0.0. We generally refer to this distribution as the binary–single endstate topology, as described in Section 2.4. The large-scale patterns and structural regions seen in the figure clearly illustrate that the outcome of a binary–single interaction is not completely random, although the three-body problem is known to be chaotic. In this work, we use topological mappings, similar to the one shown here, as a tool to gain further insight into the binary–single problem with finite size effects and GR corrections included in the three-body EOM. 2.3.2 Kinematical properties Number of IMS (NIMS): For each interaction, we count the number of times the binary–single system splits into an IMS. The total number from initial interaction to final outcome is what we denote NIMS. Minimum distance (Rmin): For all interactions ending as a BS[ij] endstate, we save the minimum distance any two of the three objects undergo during the interaction. This distance is denoted by Rmin. Binary–single energy (EBS): At the end of each interaction, we calculate the orbital energy of the single and the COM of the remaining pair. We denote this energy by EBS. In this notation, a BS[ij] endstate will have EBS > 0, where collisions and GW inspirals generally have EBS < 0, as they preferentially form during the interaction. 2.4 Topological mappings The graphical representation of the distribution of binary–single outcomes as a function of the ICs is what we loosely refer to as the topology, in analogy with Hut (1983a). As an example, Fig. 2 shows the distribution of binary–single endstates as a function of f, b, for a0 = 10−2 au and γ = 0, derived on a 1000 × 1000 grid. In our terminology, we refer to this distribution as the endstate topology, or the endstate-map. Building on the work of Hut (1983a), we focus in this study on topological maps derived from varying f and b, while keeping v and γ fixed. 3 TOPOLOGY OF BINARY–SINGLE INTERACTIONS In this section, we explore the binary–single topology on large scales. For this, we first study the classical limit, which represents the idealized case where the objects are all point-like particles interacting through Newtonian gravity only. After this follows, a study of the relativistic limit, in which GR corrections and finite size effects start to affect the classical topology. In the last part, we briefly explore how the topological maps change for varying γ. For all scatterings performed in this section, we keep finite sizes and GW emission in the EOM for consistency; the classical and the relativistic limits are therefore studied by simply varying a0. We note that f conveniently happens to approximately be the phase of the binary as the incoming single enters the binary. For example, at f ≈ 0, b ≈ 0, we find that all three objects are approximately on a line as the single enters. In our following descriptions, we will therefore simply refer to f instead of the exact binary phase as the single enters the binary. Furthermore, we often refer to just a single value of f, however, due to the rotational symmetry of the binary, results are similar for any phase f + nπ, where n is an integer. 3.1 Classical limit If the three interacting objects are point-like and only Newtonian gravity is included in the EOM, the system is said to be scale-free, as the topology and the relative number of each endstate do not depend on the SMA a0 and mass m, as long as v remains constant (Hut 1983a; Hut & Bahcall 1983). Although such an idealized system does not exist in nature, one approaches this classical scale-free limit as a0 increases relative to the radius of the interacting objects. In this section, we study the topology for a0 = 1 au, which provides a reasonable representation of the classical limit. Results are shown in Fig. 3, and discussed below. Figure 3. View largeDownload slide Topological maps derived from binary–single interactions with a0 = 1 au, v = 0.01, and γ = 0.0. Top left: endstates. Top right: total orbital energy between the endstate binary COM and the remaining single, log (EBS). Bottom left: number of times the binary–single system splits into an IMS before reaching an endstate, NIMS. Bottom right: minimum distance any two of the three objects undergo throughout an interaction, Rmin, in units of the initial SMA a0, log (Rmin/a0). These maps are used to study the classical limit, as described in Section 3.1. Further descriptions of outcomes, endstates, and ICs are found in Section 2. Figure 3. View largeDownload slide Topological maps derived from binary–single interactions with a0 = 1 au, v = 0.01, and γ = 0.0. Top left: endstates. Top right: total orbital energy between the endstate binary COM and the remaining single, log (EBS). Bottom left: number of times the binary–single system splits into an IMS before reaching an endstate, NIMS. Bottom right: minimum distance any two of the three objects undergo throughout an interaction, Rmin, in units of the initial SMA a0, log (Rmin/a0). These maps are used to study the classical limit, as described in Section 3.1. Further descriptions of outcomes, endstates, and ICs are found in Section 2. 3.1.1 Endstate topology The endstate-map (Fig. 3, upper left) shows a clear large-scale topological pattern. We find this to emerge from the distribution of two different small-scale topological pattern types, which we in the following refer to as the homogeneous-type and the random-type. The homogeneous-type is characterized by that neighbouring points share the same endstate, whereas the random-type is characterized by that neighbouring points have a semirandom distribution of endstates. As seen on the figure, the random-type forms one fully connected large-scale structure, which divides the homogeneous-type into a few disconnected islands. This is a feature of the ultra HB limit for which v ≪ 1, as the random-type structure starts to separate into isolated parts as v approaches 1 (e.g. Hut 1983a). Comparing the NIMS-map (Fig. 3, lower left) with the endstate-map shows that the random-type generally originates from RIs and the homogeneous-type from DIs. This is a consequence of the chaotic nature of RIs, which makes the outcomes very sensitive to the ICs. Neighbouring points on the f, b grid in RI regions are therefore likely to have a semirandom distribution of endstates. In contrast, DIs are always prompt. Small variations to the ICs will therefore not have a chance to undergo any significant growth during this type of interaction. Neighbouring points on the f, b grid in DI regions therefore often have similar endstates. The EBS-map (Fig. 3, upper right) relates closely to the other maps, as it is the orbital energy of the single w.r.t. the COM of the first binary that forms (endstate binary or IMS binary), denoted here by E1BS, that determines if the interaction proceeds as a DI or an RI. By definition, E1BS > 0 leads to a DI, in which case EBS = E1BS, whereas E1BS < 0 leads to an RI. As the distribution of E1BS seems to vary smoothly across the f, b space, we conclude that the boarders separating DIs from RIs – or equivalently the coastline of the DI islands – are simply the path for which E1BS = 0. As E1BS approaches zero along these boarders, the associated binary–single interaction time correspondingly approaches infinity, which explains why the ‘VLint’ endstate generally appears between DI and RI regions. As seen, there is a clear asymmetry in the large-scale topology along the b-axis, despite corresponding symmetry in the positional configuration of the three objects. This asymmetry results from the way the binary rotates relative to the incoming single, where b > 0 and b < 0 correspond to prograde and retrograde motion, respectively. For v ≪ 1, the incoming single almost has the same velocity as the binary members as it passes the binary. In the case of prograde orbits, this leads to situations where the three objects almost appear stationary w.r.t. each other. As a result, for certain combinations of f, b the incoming single does not directly encounter any of the binary members, which give rise to the large wave-like BS[12] structures seen for b > 0. In contrast, negative values of b always result in encounters between the single and a binary member due to the retrograde motion. A comparison with the high velocity results shown in Hut (1983a) confirms that the relative orbital motion indeed is the origin of the b-axis asymmetry. For example, the results in Hut (1983a) clearly show how the topology along the b-axis becomes increasingly symmetric as v increases, or in other words, as the velocity of the binary members become less important. 3.1.2 Kinematical examples Having described a few topological relations in the sections above, we now turn to a more detailed kinematical analysis. We note that an analytical solution for describing the dynamics of the SB limit is possible (e.g. Hut 1983b), where a statistical approach has been shown to provide valuable insight in the HB limit (e.g. Samsing et al. 2014). We divide our analysis below into the study of the topology around two orbital configurations: Configuration A (f ≈ 0) and Configuration B (f ≈ π/2). All results presented below are found by visually inspecting a representative set of three-body interactions. Configuration A: We first consider the change in topology for varying b at f ≈ 0. For this configuration, we refer to the binary member that is closest to the incoming single as it enters by ‘1’, the remaining member by ‘2’, and thereby the incoming single by ‘3’ (this notation matches the endstate labels on our figures for f ≈ 0 + π). This configurative labelling is also used in Fig. 4. Figure 4. View largeDownload slide Schematic illustrations showing the dynamics of two examples each representing an exchange interaction through a DI, where the left (E1) leads to BS[13], and the right (E2) to BS[23]. The grey large dots show the three objects at their initial positions, whereas the large black dots show the final configuration. The grey dotted circle illustrates the initial orbit of the two binary members, ‘1’ and ‘2’, before the incoming object ‘3’ enters the system. A simplified version of the orbital paths from initial to final state is shown with thin black lines. These two examples are further described in Section 3.1.2. Figure 4. View largeDownload slide Schematic illustrations showing the dynamics of two examples each representing an exchange interaction through a DI, where the left (E1) leads to BS[13], and the right (E2) to BS[23]. The grey large dots show the three objects at their initial positions, whereas the large black dots show the final configuration. The grey dotted circle illustrates the initial orbit of the two binary members, ‘1’ and ‘2’, before the incoming object ‘3’ enters the system. A simplified version of the orbital paths from initial to final state is shown with thin black lines. These two examples are further described in Section 3.1.2. Starting at b ≈ 0, we see that the topology changes rather abruptly from a BS[23] DI region to a BS[13] DI region. The kinematics leading to these two neighbouring regions is as follows. For slightly negative values of b, object 3 first encounters 1 from below which leads to an upwards deflection of 3 and a downwards deflection of 1. Along its new orbit 1 then interacts with 2, which sends 1 up towards 3 while 2 is being ejected downwards. As a result, [13] escapes as a binary. This interaction is illustrated to the left (labelled ‘E1’) in Fig. 4. For slightly positive b, object 3 instead encounters 1 from above, which gives rise to an interaction where the two objects almost perfectly replace each other, which leads to 1 flying to the left and 3 moving upwards. Object 1 then interacts with 2 from above, which sends 1 downwards and 2 upwards where it pairs up with 3. As a result, [23] escapes as a binary. This interaction is illustrated to the right (labelled ‘E2’) in Fig. 4. At b ≈ −1.5, we see a region that is somewhat similar to the one at b ≈ 0, but with the two BS[23] and BS[13] regions switched. For the region at b ≲ −1.5, we find that 3 first interacts with 2 from below after which the binary has rotated ≈π/2. This interaction sends 2 downwards and 3 up towards 1. 1 and 3 then undergo a near parabolic hook-interaction, which sends 3 back towards 2 by ejecting 1 further upwards. As a result, [23] escapes as a binary. For the region at b ≳ −1.5, the first interaction between 3 and 2 is simply so strong that 2 is immediately kicked out by 3 which then takes its place in the binary. As a result, [13] ends up as a binary. Configuration B: We now consider the change in topology for varying b near f ≈ π/2. For this configuration, we refer to the binary member that moves towards the incoming single by ‘1’, the remaining member by ‘2’, and thereby the single by ‘3’. This configuration is similar to that shown in Fig. 4, but for the binary rotated clockwise by π/2. We first notice that for any value of b ≈ 0 ± 0.5, the endstate will always be of the same type, in contrast to Configuration A considered above. The kinematics near b ≈ 0 are as follows. As object 3 enters the binary, it first accelerates 1 in the upwards direction along its orbital tangent. Object 3 then interacts with 2 from below, which results in 2 being ejected downwards and 3 being sent up towards 1. As a result, [13] escapes as a binary. This basically happens for all b for which 3 enters the binary on an orbit that goes in between 1 and 2. Across larger variations of b, significant differences in the topology appear. For example, interactions near b ≈ 1.5 all evolve as an RI, whereas interactions near b ≈ −1.5 instead evolve as a DI. We find the kinematical differences to be as follows. At b ≈ −1.5, object 3 no longer moves in between 1 and 2, but instead passes the binary from below. As it passes, 3 and 1 first interact, pulling 1 out of its circular motion along an orbit pointing towards the initial motion of 3. Object 3 then interacts with 2 which sends 3 back towards 1 by ejecting 2 in the opposite direction. As a result, [13] escapes as a binary. At b ≈ 1.5, the kinematics is similar to that of b ≈ 0, however, instead of 3 ejecting 2 into an unbound orbit, 2 remains bound after which the interaction then proceeds as an RI. 3.1.3 Close encounter topology The Rmin-map (Fig. 3, lower right) shows a new structural pattern that is characterized by connected wavy bands. These bands map the closest distance any two of the three objects undergo before the system splits into its first binary–single state (BS[ij] or IMS). For example, the band that intersects f ≈ 0, b ≈ 0 and rises until f ≈ π/2, b ≈ 1.0 originates from the first close encounter between objects 1 and 3, where ‘1’ here refers to the binary member that is closest to the incoming single ‘3’ at f ≈ 0, b ≈ 0. Since the trajectory of 3 is not perturbed significantly before it first encounters 1, the value of b for which 1 and 3 undergo a close encounter is therefore approximately that of the shadow image of 1 along an axis parallel to b. The first part of the considered band therefore takes the shape of a sinusoidal curve, which explains why it first rises and then peaks at f ≈ π/2. However, when it goes back towards b ≈ 0 near f ≈ π the sinusoidal curve is strongly perturbed, because object 2 comes in between 1 and 3, which leads to deflections of 3 before it encounters 1. For velocities v > 1, object 3 is hardly deflected, and the bands will therefore trace out a near perfect sinusoidal form. The Rmin-map also shows a population of low Rmin values inside the RI regions. This population originates from IMS binaries that are formed with high eccentricity and corresponding small pericentre distance. 3.2 Relativistic limit We now study the topology of the relativistic limit, where finite sizes and GR corrections start to play a role. To this end, we perform binary–single scatterings with the initial SMA set to a0 = 10−4 au, as this is the scale for which GR corrections are expected to significantly change the outcome distributions (see Section 2.3.1). This unrealistic low value of a0 has only been chosen for illustrative purposes; however, the following results do apply for any a0 – in fact, it is the wide binaries that dominate the rate of relativistic outcomes, due to their larger interaction cross-section (Samsing et al. 2014). Results are shown in Fig. 5, and discussed below. Figure 5. View largeDownload slide Topological maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstates. Bottom plot: number of times the binary–single system splits into an IMS before reaching an endstate, NIMS. These maps are used to study the relativistic limit, as described in Section 3.2. Figure 5. View largeDownload slide Topological maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstates. Bottom plot: number of times the binary–single system splits into an IMS before reaching an endstate, NIMS. These maps are used to study the relativistic limit, as described in Section 3.2. 3.2.1 BH collisions Looking at the endstate-map (Fig. 5, top plot) and the NIMS-map (Fig. 5, bottom plot), we see that BH collisions form through both DIs and RIs. These two collision populations are described in the following. The DI-collisions trace the wavy Rmin bands shown in Fig. 3, with only small differences due to the modified dynamics resulting from including GW emission in the EOM. The bands broaden as the value of b increases. This is due to the corresponding decrease in the relative velocity between the binary member in question and the incoming single, which leads to an increase in the gravitational focusing, and thereby to the seen broadening in b. As the value of b decreases, the relative velocity increases, which makes the gravitational focusing less efficient. The orbit of the incoming single therefore has to be relatively fine-tuned to collide with a binary member, which explains the thinner bands at negative b. The RI-collisions originate from the population of IMS binaries that are formed with a pericentre distance that is smaller than the collisional distance 2Rs. Despite the apparent randomness of the RI regions, the collision endstates do not seem to be completely randomly distributed in these regions. For example, the relative number of collisions clearly decreases for b ≳ 2. This partially results from the corresponding increase in the total angular momentum, which naturally leads to fewer low angular momentum orbits. Further insight into the collision endstates distribution is provided in Fig. 6, which shows what pair that collides. Besides bands originating from collisions with the incoming single (Coll[13], Coll[23]), we interestingly see a band passing through f ≈ 0.7, b ≈ −0.5 that is formed by collisions of the binary members themselves (Coll[12]). We find that the corresponding dynamics is such that the incoming single, during its first passage, simply perturbs the two binary members into a prompt collisional orbit. Although this seems like a fine-tuned problem, the corresponding bands appear relative broad, which indicates a non-negligible formation probability. Figure 6. View largeDownload slide Topological map of collision and GW inspiral endstates derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. This map provides information about what object pair that collides or undergoes a GW inspiral. As seen, the red colours show GW inspirals, where the grey colours show collisions. This map is discussed in Section 3.2. Figure 6. View largeDownload slide Topological map of collision and GW inspiral endstates derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. This map provides information about what object pair that collides or undergoes a GW inspiral. As seen, the red colours show GW inspirals, where the grey colours show collisions. This map is discussed in Section 3.2. 3.2.2 GW inspirals A clear population of GW inspirals is seen near the boarders of the wavy collision bands. The origin of this inspiral population is not surprising, as the collision boarders are characterized by near collisional orbits. When GW emission is included in the EOM, these orbits therefore naturally result in strong GW emission, which then leads to the observed populations of GW inspirals. Just as the population of collisions in the wavy bands forms through DIs, so do the GW inspirals forming along the bands, as seen in the bottom plot in Fig. 5. We note that there are a few collisional bands located in the DI regions with no GW inspirals forming along their edges. We find here that the dynamics leading to the prompt collision at the same time ejects the third BH into an unbound orbit. The endstates populating these bands are therefore characterized by a collision with an unbound companion. This explains why no GW inspirals form along their edges, as the binary–single system splits up into a BS[i,j] state before the GW inspiral is complete. The two binary objects will of course still undergo the GW inspiral, but it will happen without a bound companion. Moving away from the wave-like bands, we see how GW inspirals clearly form throughout the RI regions. We note here that collisions and GW inspirals do not have to follow the same distribution, as the formation of a GW inspiral is not related to a particular value of the IMS binary pericentre distance unlike a collision. Instead, the formation of a GW inspiral relates to how much energy it can lose through GW emission during the orbital time it is isolated from the bound single (Samsing et al. 2014). As a result, GW inspirals can form from IMS binaries with both small and large pericentre distances, depending on the corresponding isolation time, and can therefore appear in regions where collisions are relative rare. For example, in the upper plot in Fig. 5, there are many GW inspirals forming for b > 2.5 but only relative few collisions. A more detailed picture is provided in Fig. 6, which also shows what pair that undergoes the GW inspiral. 3.3 Out-of-plane interactions For completeness, we here study the change in topology for varying orbital-plane angle γ. Endstate-maps for γ = π/4 and γ = π/2 are shown in Fig. 7, and briefly discussed below. Figure 7. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au and varying γ. Top plot: endstate topology for γ = π/4. Bottom plot: endstate topology for γ = π/2. The corresponding endstate topology for γ = 0 is shown in the top of Fig. 5. Figure 7. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au and varying γ. Top plot: endstate topology for γ = π/4. Bottom plot: endstate topology for γ = π/2. The corresponding endstate topology for γ = 0 is shown in the top of Fig. 5. The most notable difference between the co-planar case (γ = 0) and our presented out-of-plane cases (γ = π/4, π/2) is a breakup of the connected wave-like collision and GW inspiral bands seen for γ = 0. To understand this change, we note that the incoming single can only undergo a prompt collision or a prompt GW inspiral near regions where its orbital plane intersects with the binary orbit. When γ = 0, the incoming single intersects for sinusoidal variations in f, b, however, when γ > 0, the orbital plane of the single is only able to intersect the binary orbit at maximum two points, namely at {−π/2, +π/2}. This correspondingly results in isolated regions in the topological map instead of connected bands. As with the co-planar case, the DI collision regions for γ > 0 are surrounded by GW inspirals that form through a prompt capture. However, a large population of both DI and RI collisions and GW inspirals also form beyond these DI regions. In fact, we find that the total number of collisions and GW inspirals is not strongly dependent on γ, although the distribution of orbital parameters at merger might depend on γ due to the corresponding change of angular momentum. We save such a study for future work. 4 THE ROLE OF GW EMISSION We now perform a slightly more controlled study on how the inclusion of GW emission in the EOM affects the topology on different scales. For this, we explore two different topological regions, with and without GW emission included in the EOM. For all interactions, we set a0 = 10−4 au and γ = 0 to have GW emission effects stand out clearly. Results are discussed below. 4.1 Large-scale effects We start by analysing how the inclusion of GW emission affects the large-scale topology. Results are shown in Fig. 8. Figure 8. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstate-map derived without the inclusion of GW emission in the N-body EOM. Bottom plot: endstate-map derived with the inclusion of GW emission in the N-body EOM. Results are discussed in Section 4.1. Figure 8. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstate-map derived without the inclusion of GW emission in the N-body EOM. Bottom plot: endstate-map derived with the inclusion of GW emission in the N-body EOM. Results are discussed in Section 4.1. When GW emission is not included in the EOM (top plot), we see that the collision regions simply appear superimposed on the classical endstate-map, with a distribution that follows the Rmin distribution as expected. Other than occupying part of the map, the inclusion of collisions does not affect the large-scale topological structure, as the EOM is unchanged. When GW emission is included (bottom plot), we see that this not only leads to the formation of GW inspirals, but also a slight modification of the topology. For example, at f ≈ 0.75, b ≈ 0.75, where the collision band intersects the RI region in the classical map, the inclusion of GW emission clearly smoothes out the transition region, by turning a part of the BS[1,3] endstates formed in the DI region into a semirandom mix of outcomes. Generally, one finds that the largest topological changes indeed are found near regions where collision endstates, DIs, and RIs intersect. This results from two factors: the GW emission is strong (the objects are close to colliding) and the orbital energy E1BS is near zero (the boarder between a DI and an RI is characterized by E1BS = 0). This combination makes it possible for the GW emission to change a classical DI into an RI, by emitting the orbital energy out of the system that otherwise would have ejected the single into an unbound orbit. In fact, in Samsing et al. (2017), it was briefly discussed if this way of ‘capturing’ the single into an RI through GW emission would be able to change the relative frequency of endstates. We have here shown and argued that this indeed happens, but only in a very small part of the available phase-space. The effect is therefore negligible from a statistical perspective. 4.2 Small-scale effects We now study how the topology changes on small scales as GW emission is included in the EOM. Results are shown in Fig. 9. Figure 9. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstate-map derived without the inclusion of GW emission in the N-body EOM. Bottom plot: endstate-map derived with the inclusion of GW emission in the N-body EOM. Results are discussed in Section 4.2. Figure 9. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstate-map derived without the inclusion of GW emission in the N-body EOM. Bottom plot: endstate-map derived with the inclusion of GW emission in the N-body EOM. Results are discussed in Section 4.2. In the top plot, where GW emission is not included, we first notice that the considered RI region has a surprisingly rich and complex microtopology. This will be studied in greater detail in Section 5. As in the large-scale example, we see here how the collision regions simply appear as sharp shadows superimposed on top of the classical map. We now turn to the bottom plot where GW emission is included. As seen, the inclusion drastically changes the topology close to the classical collision bands. Not only is a clear population of GW inspirals forming along the bands, but the GW emission further results in band-like structures that alter between collisions and GW inspirals. To understand the kinematics leading to this alternation, we visually inspected a set of binary–single interactions for f ≈ π/2 and varying b. For this configuration, we here denote the binary member moving away from the incoming single by ‘2’, the remaining member by ‘1’, and thereby the incoming single by ‘3’. We find the kinematics to be as follows. First, the large collisional band at the bottom (b ≈ 1.15) is formed by 2 directly colliding with 3 as it enters the binary, as also described in Section 3.1.3. Above this is an inspiral band (b ≈ 1.3), which we find consists of interactions where 3 instead of directly colliding with 2, slightly grazes it, which then leads to a prompt GW inspiral between the two. The subsequent collision band (b ≈ 1.4) is formed by 3 first being captured into a relative wide low angular momentum orbit by 2, after which 3 and 2 collide at their first pericentre passage along their new orbit. The following narrow BS[23] band (b ≈ 1.42) arises from a rather fine-tuned dynamical configuration where 1 becomes kinematically unbound just before 2 and 3 undergo a collision or a GW inspiral. This band is therefore sensitive to the tidal threshold used for labelling a BS[ij] endstate, and is therefore not a fundamental feature. The next few bands form in a similar way to the previous collision band, only with 2 and 3 changing between undergoing a collision or a GW inspiral. To conclude, these topological changes all relate to the capture of object 3 into a bound orbit partly through GW emission. As the capture happens at low angular momentum, and angular momentum further is radiated away during the capture, the system evolves in a state that is likely to result in either a prompt collision or a GW inspiral. 5 TOPOLOGICAL MICROSTRUCTURES We now explore the topology of the microstructures that surprisingly seem to appear within RI regions. For this, we perform scatterings for a0 = 1 au, and γ = 0, with collisions and GW emission included in the EOM. However, we do note that our considered small-scale topology is likely to change slightly if additional PN terms are added to the EOM, we therefore limit ourselves to general conclusions. Results from two selected zoom-in regions, referred to as ‘Zoom Region 1’ and ‘Zoom Region 2’, respectively, are shown in Fig. 10, and discussed below. Figure 10. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3 with a0 = 1 au and γ = 0. Top plot: zoom in on the centre of one of the larger RI regions. Bottom plot: zoom in on the boundary between one of the larger RI regions and its neighbouring DI region. As seen in both plots, the considered regions do not just consist of a semirandom distribution of endstates as it appears to on larger scales. Instead, the regions have in fact a rich variety of microtopological structures. The origin of these structures is discussed in Section 5. Figure 10. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3 with a0 = 1 au and γ = 0. Top plot: zoom in on the centre of one of the larger RI regions. Bottom plot: zoom in on the boundary between one of the larger RI regions and its neighbouring DI region. As seen in both plots, the considered regions do not just consist of a semirandom distribution of endstates as it appears to on larger scales. Instead, the regions have in fact a rich variety of microtopological structures. The origin of these structures is discussed in Section 5. 5.1 Zoom Region 1 The top plot shows a zoom in on one of the larger RI regions. As seen, the region does not only consist of semirandomly distributed endstates, as it appears to on large scales, but in fact hosts clear microtopological structures of various kinds. To gain insight into the dynamics of this region, we visually investigated a representative set of binary–single interactions. From this exercise, we found that most of the interactions near the centre (f ≈ 1.3, b ≈ 1.2) start out in a similar way, where the objects first interact non-chaotically with their orbits tracing out a well-defined pattern with a clear degree of symmetry. A schematic illustration of this primary orbital interaction assuming perfect symmetry is shown in Fig. 11. Although the interaction in this example promptly ends as a BS[23] endstate, the outcomes do vary depending on the exact ICs. In a few cases, we even observed that the system after this primary interaction entered a quasi-stable state which it stayed in for 5–10 cycles. We note here that the more stable a three-body state is, the less likely it is to form (e.g. Hut 1983a), which implies that fully stable three-body states are not assembled through our binary–single experiments (Newtonian three-body states cannot be stable at plus infinity if they are not stable at minus infinity). Of course, the inclusion of GR in the EOM immediately changes the situation, as the implied GW emission destabilizes the system within a finite time. Further studies on the possible assembly of quasi-stable three-body states with and without GR are beyond this paper, but still of great theoretical interest (e.g. Šuvakov & Dmitrašinović 2013). Figure 11. View largeDownload slide Schematic illustration of the orbital trajectories associated with an interaction leading to a BS[23] endstate found near f ≈ 1.3, b ≈ 1.25. As seen, the objects are in this interaction clearly moving in a non-chaotic way on orbits that trace out an elegant symmetrical pattern. We find that most of the interactions near the considered values for f, b actually go through this orbital evolution, but with different outcomes to follow depending on the exact ICs. In this illustration, the colours red, green, and blue refer to objects 1, 2, and 3, respectively, where the arrows denote the direction of motion. The considered BS[23] outcome corresponds to the case where the binary enters from the left and moves towards the right. Further discussions are found in Section 5.1. Figure 11. View largeDownload slide Schematic illustration of the orbital trajectories associated with an interaction leading to a BS[23] endstate found near f ≈ 1.3, b ≈ 1.25. As seen, the objects are in this interaction clearly moving in a non-chaotic way on orbits that trace out an elegant symmetrical pattern. We find that most of the interactions near the considered values for f, b actually go through this orbital evolution, but with different outcomes to follow depending on the exact ICs. In this illustration, the colours red, green, and blue refer to objects 1, 2, and 3, respectively, where the arrows denote the direction of motion. The considered BS[23] outcome corresponds to the case where the binary enters from the left and moves towards the right. Further discussions are found in Section 5.1. Although the state illustrated in Fig. 11 is not stable, it certainly classifies as a special interaction due to its symmetry and interesting orbital topology. However, the classification of such interactions is not entirely clear and to our knowledge still unexplored. A systematic search through the f, b space could reveal if this is a member of a larger family of peculiar interactions, or just a single interesting incident. 5.2 Zoom Region 2 The bottom plot shows a zoom in on a transition zone between RIs (lower left) and DIs (upper right). As seen, the only pattern that is clearly visible consists of nested rings with a spacing that decreases as one moves closer to the DI boundary from the RI side. Focusing on the BS[13] endstate bands, we find that the corresponding IMS binary configuration is such that when the single returns from its first excursion, the resultant IMS binary–single interaction leads to a prompt BS[13] endstate. This happens in each band, with the only difference being how many times the IMS binary rotates before the single returns. As a result, there is a band for each integer n, where n is the number of times the IMS rotates before the single returns (by visual expectation, we find that the band in the lower left corner corresponds to n = 10). As E1BS approaches zero as one moves towards the DI region, the number n will correspondingly approach infinity. As a result, the spacing between the blue bands becomes smaller and smaller, approaching zero near the boundary. This explains why substructures with infinitely small spacings exist and where they are located. The precess of such structures implies that the true richness of microtopological structures cannot be studied with a finite grid approach as the one we have presented here. 6 DOUBLE GW MERGERS All GW inspirals form during the binary–single interaction while the three objects are still bound to each other in an IMS (Samsing et al. 2014). Ignoring dynamical kicks caused by asymmetric GW emission (e.g. Baker et al. 2006), this interestingly implies that the BH formed from the GW inspiral merger and the remaining single are still bound to each other after the three-body interaction. A compelling question is how long time it takes for this binary to merge through GW emission, or equivalently, what the time span is from the first to the second GW merger. If the time span is short enough, it might be possible for LIGO to observe both GW mergers, a scenario that undoubtfully would be extremely exiting. In this section, we explore for which combinations of f and b the time span from first to second GW merger, denoted by Δt12, takes its shortest value. A more detailed study of this double GW merger scenario is reserved for a separate paper, and the following discussions are therefore kept short. An illustration of our proposed double GW merger channel is shown in Fig. 12, and results presented below. Figure 12. View largeDownload slide Illustration of a binary–single interaction that results in the formation of two GW mergers. As seen, the interaction first results in the formation of a BBH that promptly undergoes a GW inspiral (first GW merger). After this, the resultant BH formed through the first GW inspiral then subsequently undergoes a GW merger with the remaining bound single (second GW merger). If the time span between the first and the second GW merger is relative short, then it might be possible to observe both GW mergers with a detector such as LIGO. In Section 6.1, we explore which values for f, b that lead to the shortest time span. Figure 12. View largeDownload slide Illustration of a binary–single interaction that results in the formation of two GW mergers. As seen, the interaction first results in the formation of a BBH that promptly undergoes a GW inspiral (first GW merger). After this, the resultant BH formed through the first GW inspiral then subsequently undergoes a GW merger with the remaining bound single (second GW merger). If the time span between the first and the second GW merger is relative short, then it might be possible to observe both GW mergers with a detector such as LIGO. In Section 6.1, we explore which values for f, b that lead to the shortest time span. 6.1 Time from first to second GW merger The distribution of log (Δt12/T0), where T0 is the orbital time of the initial target binary, as a function of f, b for γ = 0, is shown in Fig. 13. For this, we have included all GW inspirals and BH collisions that form with a bound companion, where Δt12 is estimated using the equations from Peters (1964). To keep this primary study simple and clear, we have not incorporated effects related to relativistic mass losses or asymmetric GW emission; corrections that in general would lead to an increase in Δt12. Figure 13. View largeDownload slide Distribution of log (Δt12/T0), where Δt12 is the time span from the first to the second GW merger, and T0 is the orbital time of the initial target binary, derived for a0 = 10−4 au, γ = 0. As described in Section 6.1, the horizontal band for which Δt12 takes its minimum value corresponds to where the total angular momentum of the three-body system equals zero. Figure 13. View largeDownload slide Distribution of log (Δt12/T0), where Δt12 is the time span from the first to the second GW merger, and T0 is the orbital time of the initial target binary, derived for a0 = 10−4 au, γ = 0. As described in Section 6.1, the horizontal band for which Δt12 takes its minimum value corresponds to where the total angular momentum of the three-body system equals zero. As seen in the figure, the minimum value for Δt12 is highly sensitive to b, but not f. The reason is that only b relates to the total angular momentum, which in turn directly relates to the corresponding GW inspiral time (Peters 1964). This further explains why the value for b that minimizes Δt12 is simply the value for which the total three-body angular momentum equals zero. As derived in Section 2.2, the angular momentum of the single and the binary cancels out for b ≈ −0.87, which clearly is where Δt12 takes its minimum as confirmed by the simulation results shown in the figure. Close to this value of b, the three objects can, as a result of the vanishing angular momentum, promptly merge without any significant use of GW emission. This leads to values of Δt12 that are of the order of the orbital time of the initial binary, instead of its initial GW inspiral time. The angular momentum cancellation is less efficient in out-of-plane interactions, which explains why the co-planar interaction case is the relevant channel to consider for generating double GW mergers. 7 CONCLUSIONS In this paper, we explored how the outcomes of binary–single BH interactions distribute as a function of the ICs. For this, we studied the corresponding graphical representation of the distribution (Fig. 2) – a representation we loosely refer to as the topology in analogy with Hut (1983a). Our study shows for the first time how this binary–single topology changes when GR effects at the 2.5 PN order, which accounts for dynamical GW emission, and BH finite sizes are included in the N-body EOM. Such corrections give rise to a non-negligible distribution of BH collisions and GW inspirals that is directly observable by LIGO (e.g. Samsing et al. 2014; Samsing & Ramirez-Ruiz 2017). Using zoom-in simulations, we resolved for the first time the emergence of microtopological structures residing inside RI regions (Fig. 10). By visually inspecting orbits near these structures, we found that many of the interactions (in the RI region centred around f ≈ 1.3, b ≈ 1.25) start out in a similar dynamical way characterized by a particular ordered and symmetrical orbital evolution (Fig. 11). To our knowledge, the classification and assembly of such symmetrical orbital evolutions are still unexplored, and a systematic search for these throughout the f, b space could therefore be of potential interest. Finally, we pointed out that the inclusion of GW emission in the N-body EOM naturally leads to interactions resulting in two GW mergers. In this regard, we described how the co-planar case can lead to scenarios where the time span from the first to the second GW merger is short enough for LIGO to observe both events. This double GW merger channel could have profound astrophysical consequences, and will therefore be explored in greater detail in an upcoming paper. From an astrophysical perspective, our most relevant result is that the distribution of outcomes of binary–single interactions is highly dependent on the ICs. While this certainly applies to small variations to the ICs, due to the chaotic nature of the three-body problem, we surprisingly find that also large-scale variations to the ICs give rise to corresponding large-scale differences in the outcome distribution. For example, in the co-planar case (γ = 0), the distribution is clearly different between prograde (b > 0) and retrogade (b < 0) interactions (Fig. 2). This also applies to how the relative number of BH mergers distributes with the ICs (Fig. 6). Clear differences are also seen when the orbital-plane angle γ is varied (Fig. 7). This means that if properties of these distributions, such as the eccentricity distribution at merger, could be observed by LIGO, then one would be able to roughly deduce the underlying distribution of ICs, which could give important hints about the astrophysical origin. As an example, in AGN environments (e.g. McKernan et al. 2012; Bellovary et al. 2016; McKernan et al. 2017), the interactions will not only be close to co-planer, but they will also likely be unequally distributed between prograde or retrograde interactions, depending on the formation history of the interacting BHs and their interaction with the disc. Our idealized study indicates that the outcome distribution is clearly different for these cases. A natural extension to our presented work could therefore be to derive observables related to BH collisions, GW inspirals, and high eccentricity BH mergers, for different scenarios including retrograde interactions, prograde interactions, in-plane interactions, out-of-plane interactions, and for varying mass hierarchy. Although our paper presents details to a level that is not directly observable, we do note that similar studies on how the distribution of BH mergers and related observables changes with the ICs are likely to play a key role in distinguishing different astrophysical GW merger channels from each other. An exercise that will become highly relevant when the number of observed GW mergers starts to grow. ACKNOWLEDGEMENTS Support for this work was provided by NASA through Einstein Postdoctoral Fellowship grant number PF4-150127 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. REFERENCES Abbott B. P. et al.  , 2016a, Physical Review Letters , 116, 061102 https://doi.org/10.1103/PhysRevLett.116.061102 CrossRef Search ADS   Abbott B. P. et al.  , 2016b, Physical Review Letters , 116, 241103 https://doi.org/10.1103/PhysRevLett.116.241103 CrossRef Search ADS   Abbott B. P. et al.  , 2016c, ApJ , 818, L22 https://doi.org/10.3847/2041-8205/818/2/L22 CrossRef Search ADS   Abbott B. P. et al.  , 2016d, ApJ , 833, L1 https://doi.org/10.3847/2041-8205/833/1/L1 CrossRef Search ADS   Abbott B. P. et al.  , 2017, Phys. Rev. Lett. , 118, 221101 https://doi.org/10.1103/PhysRevLett.118.221101 CrossRef Search ADS PubMed  Antognini J. M. O., Thompson T. A., 2016, MNRAS , 456, 4219 https://doi.org/10.1093/mnras/stv2938 CrossRef Search ADS   Antonini F., Rasio F. A., 2016, ApJ , 831, 187 https://doi.org/10.3847/0004-637X/831/2/187 CrossRef Search ADS   Askar A., Szkudlarek M., Gondek-Rosińska D., Giersz M., Bulik T., 2017, MNRAS , 464, L36 https://doi.org/10.1093/mnrasl/slw177 CrossRef Search ADS   Baker J. G., Centrella J., Choi D.-I., Koppitz M., van Metre J. R., Miller M. C., 2006, ApJ , 653, L93 https://doi.org/10.1086/510448 CrossRef Search ADS   Bartos I., Kocsis B., Haiman Z., Márka S., 2017, ApJ , 835, 165 https://doi.org/10.3847/1538-4357/835/2/165 CrossRef Search ADS   Bellovary J. M., Mac Low M.-M., McKernan B., Ford K. E. S., 2016, ApJ , 819, L17 https://doi.org/10.3847/2041-8205/819/2/L17 CrossRef Search ADS   Blanchet L., 2014, Living Reviews in Relativity , 17, 2 CrossRef Search ADS PubMed  Fregeau J. M., Cheung P., Portegies Zwart S. F., Rasio F. A., 2004, MNRAS , 352, 1 https://doi.org/10.1111/j.1365-2966.2004.07914.x CrossRef Search ADS   Goodman J., Tan J. C., 2004, ApJ , 608, 108 https://doi.org/10.1086/386360 CrossRef Search ADS   Gültekin K., Miller M. C., Hamilton D. P., 2006, ApJ , 640, 156 https://doi.org/10.1086/499917 CrossRef Search ADS   Heggie D. C., 1975, MNRAS , 173, 729 https://doi.org/10.1093/mnras/173.3.729 CrossRef Search ADS   Horn B., Lyra W., Mac Low M.-M., Sándor Z., 2012, ApJ , 750, 34 https://doi.org/10.1088/0004-637X/750/1/34 CrossRef Search ADS   Hut P., 1983a, AJ , 88, 1549 https://doi.org/10.1086/113445 CrossRef Search ADS   Hut P., 1983b, ApJ , 268, 342 https://doi.org/10.1086/160957 CrossRef Search ADS   Hut P., Bahcall J. N., 1983, ApJ , 268, 319 https://doi.org/10.1086/160956 CrossRef Search ADS   Just A., Yurin D., Makukov M., Berczik P., Omarov C., Spurzem R., Vilkoviskij E. Y., 2012, ApJ , 758, 51 https://doi.org/10.1088/0004-637X/758/1/51 CrossRef Search ADS   Kocsis B., Tremaine S., 2011, MNRAS , 412, 187 https://doi.org/10.1111/j.1365-2966.2010.17897.x CrossRef Search ADS   Kocsis B., Tremaine S., 2015, MNRAS , 448, 3265 https://doi.org/10.1093/mnras/stv057 CrossRef Search ADS   Kocsis B., Yunes N., Loeb A., 2011, Phys. Rev. D , 84, 024032 https://doi.org/10.1103/PhysRevD.84.024032 CrossRef Search ADS   Levin Y., 2007, MNRAS , 374, 515 https://doi.org/10.1111/j.1365-2966.2006.11155.x CrossRef Search ADS   McKernan B., Ford K. E. S., Lyra W., Perets H. B., Winter L. M., Yaqoob T., 2011, MNRAS , 417, L103 https://doi.org/10.1111/j.1745-3933.2011.01132.x CrossRef Search ADS   McKernan B., Ford K. E. S., Lyra W., Perets H. B., 2012, MNRAS , 425, 460 https://doi.org/10.1111/j.1365-2966.2012.21486.x CrossRef Search ADS   McKernan B. et al.  , 2017, preprint (arXiv:1702.07818) Ostriker J. P., 1983, ApJ , 273, 99 https://doi.org/10.1086/161351 CrossRef Search ADS   Peters P., 1964, Phys. Rev. , 136, B1224 https://doi.org/10.1103/PhysRev.136.B1224 CrossRef Search ADS   Rodriguez C. L., Chatterjee S., Rasio F. A., 2016, Phys. Rev. D , 93, 084029 https://doi.org/10.1103/PhysRevD.93.084029 CrossRef Search ADS   Roupas Z., Kocsis B., Tremaine S., 2017, ApJ , 842, 90 https://doi.org/10.3847/1538-4357/aa7141 CrossRef Search ADS   Samsing J., Ramirez-Ruiz E., 2017, ApJ , 840, L14 https://doi.org/10.3847/2041-8213/aa6f0b CrossRef Search ADS   Samsing J., MacLeod M., Ramirez-Ruiz E., 2014, ApJ , 784, 71 https://doi.org/10.1088/0004-637X/784/1/71 CrossRef Search ADS   Samsing J., MacLeod M., Ramirez-Ruiz E., 2017, ApJ , 846, 36 CrossRef Search ADS   Samsing J., MacLeod M., Ramirez-Ruiz E., 2018, ApJ , 853, 140 Stone N. C., Metzger B. D., Haiman Z., 2017, MNRAS , 464, 946 https://doi.org/10.1093/mnras/stw2260 CrossRef Search ADS   Šuvakov M., Dmitrašinović V., 2013, Phys. Rev. Lett. , 110, 114301 https://doi.org/10.1103/PhysRevLett.110.114301 CrossRef Search ADS PubMed  Syer D., Clarke C. J., Rees M. J., 1991, MNRAS , 250, 505 https://doi.org/10.1093/mnras/250.3.505 CrossRef Search ADS   Wang L. et al.  , 2016, MNRAS , 458, 1450 https://doi.org/10.1093/mnras/stw274 CrossRef Search ADS   © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society Oxford University Press

# Topology of black hole binary–single interactions

, Volume 476 (2) – May 1, 2018
13 pages

/lp/ou_press/topology-of-black-hole-binary-single-interactions-1SLFuIKLnJ
Publisher
The Royal Astronomical Society
ISSN
0035-8711
eISSN
1365-2966
D.O.I.
10.1093/mnras/sty197
Publisher site
See Article on Publisher Site

### Abstract

Abstract We present a study on how the outcomes of binary–single interactions involving three black holes (BHs) distribute as a function of the initial conditions; a distribution we refer to as the topology. Using a N-body code that includes BH finite sizes and gravitational wave (GW) emission in the equation of motion (EOM), we perform more than a million binary–single interactions to explore the topology of both the Newtonian limit and the limit at which general relativistic (GR) effects start to become important. From these interactions, we are able to describe exactly under which conditions BH collisions and eccentric GW capture mergers form, as well as how GR in general modifies the Newtonian topology. This study is performed on both large- and microtopological scales. We further describe how the inclusion of GW emission in the EOM naturally leads to scenarios where the binary–single system undergoes two successive GW mergers. gravitation, gravitational waves, stars: kinematics and dynamics 1 INTRODUCTION Dynamical interactions of black holes (BHs) in dense stellar environments are likely to play an important role for the assembly of binary black hole (BBH) mergers observable by gravitational wave (GW) interferometers such as LIGO (e.g. Abbott et al. 2016c; Antonini & Rasio 2016; Rodriguez, Chatterjee & Rasio 2016). Major effort is especially being devoted to understand the formation and evolution of BHs in globular cluster (GC) systems (e.g. Samsing, MacLeod & Ramirez-Ruiz 2014; Rodriguez, Chatterjee & Rasio 2016; Wang et al. 2016; Askar et al. 2017; Samsing & Ramirez-Ruiz 2017; Samsing, MacLeod & Ramirez-Ruiz 2018), where the derived BBH merger rate interestingly seems to be comparable with that inferred from the current sample of LIGO detections (e.g. Abbott et al. 2016a,b,d, 2017). Related few-body studies are therefore often focused on deriving cross-sections and rates assuming the orbital and angular momentum initial conditions (ICs) follow an isotropic distribution at infinity, which would be the case for most dense stellar systems (e.g. Hut & Bahcall 1983; Gültekin, Miller & Hamilton 2006; Samsing, MacLeod & Ramirez-Ruiz 2014; Antognini & Thompson 2016; Samsing, MacLeod & Ramirez-Ruiz 2017; Samsing & Ramirez-Ruiz 2017). However, not all few-body interactions take place under isotropic conditions. Dynamical interactions of BHs in active galactic nuclei (AGNs) discs provide one example (e.g. Bellovary et al. 2016; Bartos et al. 2017; McKernan et al. 2017; Stone, Metzger & Haiman 2017), where gas torques force objects to move and interact in approximately the same orbital plane as the disc itself (e.g. Ostriker 1983; Syer, Clarke & Rees 1991; Goodman & Tan 2004; Levin 2007; Kocsis, Yunes & Loeb 2011; McKernan et al. 2011; Just et al. 2012). Although the number of BHs in AGN discs is currently unknown, it has interestingly been pointed out that if BHs do exist in such environments, then they are likely to pile up and undergo frequent interactions in migration-traps similar to those found in planetary discs (e.g. Horn et al. 2012; McKernan et al. 2012, 2017; Bellovary et al. 2016). If the BHs have different masses, then the corresponding difference in migration time can also lead to interactions throughout the disc (e.g. McKernan et al. 2017). Disc-like systems can also form as a result of vector resonant relaxation (e.g. Kocsis & Tremaine 2011, 2015; Roupas, Kocsis & Tremaine 2017). Motivated by these recent ideas, we explore in this paper the dynamics and outcomes of co-planar binary–single interactions. For this, we perform a detailed study of the binary–single phase-space, a representation we broadly refer to as the binary–single topology in analogy with Hut (1983a). Our work should partly be considered as a continuation of that by Hut (1983a), where the topology of co-planar binary–single interactions in the Newtonian point-particle limit was studied. In our work, we expand on that study by including general relativity (GR) corrections in the N-body equation of motion (EOM) as well as BH finite sizes. These corrections break the scale-free nature of the Newtonian point-particle binary–single problem (e.g. Hut 1983a; Hut & Bahcall 1983), leading to several new effects including the formation of GW captures and BH collisions (Samsing et al. 2014). Parts of our presented study describe highly idealized situations, however, we see this as a necessary first step, as the three-body problem with GR effects is largely unexplored for anisotropic distributions; a proper understanding of the BBH outcome distribution and how it relates to the binary–single ICs is crucial for connecting observations of GW events to their astrophysical origin. We further introduce a new unexplored dynamical pathway for generating two successive GW merger events originating from the same binary–single interaction. Here, the first GW merger happens during the interaction while the three BHs are still in a bound state (e.g. Samsing et al. 2014), where the second GW merger happens between the BH formed in the first merger and the remaining bound BH. Interestingly, we find the time span from the first to the second GW merger in co-planar interactions to occasionally be short enough for LIGO to observe both GW mergers. This suggests that AGN discs might be a likely birthplace for such double GW merger events. The paper is organized as follows. In Section 2, we introduce the binary–single interaction channel, together with the ICs and outcomes that are relevant for this study. After this follows in Section 3, a detailed study of the binary–single topology for both the classical and the relativistic limit. In Section 4, we explore in greater detail how the inclusion of GW emission in the N-body EOM affects the resultant topology and range of outcomes. Results from zoom-in simulations performed to study microtopological structures are given in Section 5. Finally, in Section 6, we present our proposed channel for generating double GW merger events through binary–single interactions. Conclusions are given in Section 7. 2 BH BINARY–SINGLE INTERACTIONS In this work, we study interactions between a circular binary and an incoming single, and how the corresponding outcomes distribute as a function of the ICs. Throughout the paper, we assume the three objects to be identical BHs each with a mass m = 10 M⊙, and corresponding Schwarzschild radius Rs. All interactions are performed with our N-body code presented in Samsing et al. (2017), which includes GR corrections using the post-Newtonian (PN) formalism (e.g. Blanchet 2014). To approach the problem in a systematic way, we have for this paper chosen to sorely focus on how the loss of energy and angular momentum through GW emission affects the outcome distributions. Our implementation of GR effects is therefore at the 2.5 PN level, as the lower order terms conserve energy (Blanchet 2014). With the insight provided in this primary study, we will in upcoming work systematically study how each term in the PN expansion affects the results. In this section, we first summarize a few dynamical properties of binary–single interactions with GW emission and finite sizes included in the EOM. We then list the ICs relevant for our considered binary–single study together with the range of possible outcomes. The last part contains a description of what we refer to as the topology. We refer the reader to Hut (1983a) for further details. 2.1 Three-body energetics and interactions The range of possible outcomes is broadly determined by the orbital velocity of the initial binary, vorb, relative to the velocity of the incoming single at infinity, v∞. In the equal mass case, one finds that the initial orbital binary–single energy is negative for v < 1 and positive for v > 1, where $$v \equiv v_{\infty }/(v_{\rm orb}\sqrt{3})$$, as further described in Hut & Bahcall (1983). These two limits are usually referred to as the hard-binary (HB) limit and the soft-binary (SB) limit, respectively (e.g. Heggie 1975). We describe a few characteristics of these two limits below. We will argue that only the HB limit is relevant for our study. 2.1.1 SB limit In the SB limit, the binary–single interaction is always prompt (Hut 1983b), any dynamical corrections to the N-body EOM will therefore rarely have a chance to affect the standard Newtonian distribution of outcomes. As a result, the inclusion of GW emission, as well as collisions and tidal interactions, does not play a significant role in this limit (e.g. Samsing et al. 2014). The topology of the SB limit was studied by Hut (1983a), and will therefore not be discussed further in this paper. 2.1.2 HB limit In the HB limit, the binary–single system either undergoes a direct interaction (DI) or a resonant interaction (RI), depending on the exact ICs (e.g. Samsing et al. 2014). For equal mass interactions, these two interaction types appear statistically in near equal numbers. However, the RI channel has a much higher probability than the DI channel of entering a state where GR effects and finite sizes can affect the three-body dynamics, as described in the following. A DI is characterized by a duration time that is similar to the orbital time of the initial binary, which makes it somewhat similar to an SB limit interaction. The kinematics of a DI is often such that the incoming single promptly pairs up with one of the binary members through a sling-shot interaction that ejects the remaining member. The short nature of this interaction type implies that GR effects and collisions rarely have a chance to affect the outcome. An RI lasts instead from a few to several thousand of initial orbital times (e.g. Hut & Bahcall 1983). The inclusion of GR effects and finite sizes in the EOM therefore have a relative high chance of affecting the dynamics during this type of interaction. How this in particular leads to an increase in the number of BH mergers can be understood as follows. As envisioned by Samsing et al. (2014), an RI can be described as a series of intermediate states (IMS), each of which is characterized by an IMS binary with a bound single. Due to pair-wise semichaotic exchanges of energy and angular momentum during the resonating evolution, each IMS binary has a finite probability for being formed with a high eccentricity, even when the initial binary is circular. If the eccentricity is high enough, GR will affect the EOM of the IMS binary (e.g. Peters 1964), which most notably leads to an inspiral driven by GW emission that is followed by a prompt merger (Gültekin et al. 2006; Samsing et al. 2014). For even higher eccentricities, the IMS binary members will instead undergo a prompt collision (e.g. Samsing et al. 2017, 2018). 2.2 Initial conditions The IC parameters that are relevant for our presented binary–single study are listed and described in the following. An accompanying illustration is shown in Fig. 1. Figure 1. View largeDownload slide ICs and notations for binary–single scatterings. The black dots represent the three interacting BHs, where the indices ‘1’ and ‘2’ denote the initial target binary members, and ‘3’ the incoming single. The impact parameter b and the relative velocity v are both defined at infinity. The angle f denotes the orbital phase of the binary at the time the distance between the binary COM and the incoming single is exactly 20 × a0, where a0 is the initial binary SMA. The angle γ denotes the angle of the incoming single at infinity relative to the initial binary orbital plane. In this work, we study how the outcome of a binary–single interaction depends on f, b for fixed values of v, γ. The ICs are further described in Section 2.2. Figure 1. View largeDownload slide ICs and notations for binary–single scatterings. The black dots represent the three interacting BHs, where the indices ‘1’ and ‘2’ denote the initial target binary members, and ‘3’ the incoming single. The impact parameter b and the relative velocity v are both defined at infinity. The angle f denotes the orbital phase of the binary at the time the distance between the binary COM and the incoming single is exactly 20 × a0, where a0 is the initial binary SMA. The angle γ denotes the angle of the incoming single at infinity relative to the initial binary orbital plane. In this work, we study how the outcome of a binary–single interaction depends on f, b for fixed values of v, γ. The ICs are further described in Section 2.2. Semimajor axis (a0): The semimajor axis of the initial target binary is denoted by a0. As described later, both the classical and the relativistic limit can be studied by simply varying the value of a0. Velocity (v): We denote the rescaled relative velocity between the initial binary COM and the incoming single at infinity by v, as described in Section 2.1. In this work, we set v = 0.01 for all our scatterings, however, none of our conclusions depend on the exact value of v, as long as v ≪ 1. We note that Hut (1983a) only considered interactions for which v ≥ 0.5, our study is therefore the first to explore the ‘ultra’ HB limit. Binary phase (f): The orbital phase of the binary when the single is exactly 20 × a0 away from the binary COM is denoted by f. The value of f is defined relative to an axis that is parallel to the velocity vector of the incoming single at infinity. Impact parameter (b): We define the rescaled impact parameter at infinity by b ≡ b∞/bc, where b∞ is the physical value and bc = a0/v. The value of b can be positive or negative, where a positive (negative) b here corresponds to that the angular momentum vectors of the binary and the incoming single, respectively, have the same (opposite) sign. The initial three-body angular momentum in the co-planar case will therefore equal zero for $$b = -\sqrt{3/4} \approx - 0.87$$ (e.g. Hut 1983a). We further note that the pericentre distance of the single w.r.t. the binary COM, rp, relates to b as rp/a0 = 3b2/8. Orbital plane angle (γ): The angle between the orbital planes of the initial binary and the incoming single is denoted by γ. In this notation, a co-planar interaction will have γ = 0. 2.3 Outcomes and kinematical properties The set of binary–single outcomes and kinematical properties we keep track of in this work is listed and described below. Further details on defining and identifying outcomes in N-body simulations can be found in, e.g. Fregeau et al. (2004) and Samsing et al. (2014, 2017). 2.3.1 Outcomes Binary-single (BS[ij]): The most common endstate is a binary with an unbound single. We denote this endstate by BS[ij], where ‘BS’ is short for ‘binary-single’, and [ij] refers to the object pair in the final binary. Throughout the paper, we denote the two initial binary members by ‘1’ and ‘2’, and the incoming single by ‘3’. GW inspiral (GW): A close passage between any two of the three interacting objects occasionally leads to a GW capture, where the two binary members undergo an inspiral through the emission of GWs (e.g. Gültekin et al. 2006; Samsing et al. 2014). We generally refer to this endstate by a ‘GW inspiral’, but will in our figures simply use ‘GW’ to keep the notation short. As described in Samsing et al. (2018), the probability for a three-body interaction to form a GW inspiral is approximately PGW ≈ 0.04 × [(m/10 M⊙)/(a0/10−2 au)]5/7. For m ∼ 10 M⊙, GR is therefore expected to significantly alter the outcome distribution for a0 ≲ 10−4 au. Collision (Coll): If two objects pass each other at a distance smaller than the sum of their radii without undergoing a GW inspiral first, then the outcome is denoted a ‘collision’, or in short by ‘Coll’. In our case, the collisional distance is simply 2Rs. Following Samsing et al. (2018), the probability for a three-body interaction to undergo a collision is approximately PColl ≈ 0.002 × [(m/10 M⊙)/(a0/10−2 au)]. Note here that $$P_{\rm GW}/P_{\rm Coll} \propto a_{0}^{2/7}$$, i.e. the relative importance of GR increases with a0. Very long interaction (VLint): A finite fraction of the interactions can in theory last infinitely long time. This happens generally when the single in an IMS is sent out on a nearly unbound orbit. In this work, we denote an interaction that did not finish within 2500 × Torb by ‘VLint’, where Torb is the orbital time of the initial target binary – with the only exception of the results shown in Fig. 2, where the limit is set to 1000 × Torb. Figure 2. View largeDownload slide Topological map of binary–single endstates. The figure shows the distribution of endstates derived from equal mass BH binary–single interactions, as a function of the binary phase f and impact parameter b, for BH mass m = 10 M⊙, initial SMA a0 = 10−2 au, relative velocity v = 0.01, and orbital plane angle γ = 0.0. We generally refer to this distribution as the binary–single endstate topology, as described in Section 2.4. The large-scale patterns and structural regions seen in the figure clearly illustrate that the outcome of a binary–single interaction is not completely random, although the three-body problem is known to be chaotic. In this work, we use topological mappings, similar to the one shown here, as a tool to gain further insight into the binary–single problem with finite size effects and GR corrections included in the three-body EOM. Figure 2. View largeDownload slide Topological map of binary–single endstates. The figure shows the distribution of endstates derived from equal mass BH binary–single interactions, as a function of the binary phase f and impact parameter b, for BH mass m = 10 M⊙, initial SMA a0 = 10−2 au, relative velocity v = 0.01, and orbital plane angle γ = 0.0. We generally refer to this distribution as the binary–single endstate topology, as described in Section 2.4. The large-scale patterns and structural regions seen in the figure clearly illustrate that the outcome of a binary–single interaction is not completely random, although the three-body problem is known to be chaotic. In this work, we use topological mappings, similar to the one shown here, as a tool to gain further insight into the binary–single problem with finite size effects and GR corrections included in the three-body EOM. 2.3.2 Kinematical properties Number of IMS (NIMS): For each interaction, we count the number of times the binary–single system splits into an IMS. The total number from initial interaction to final outcome is what we denote NIMS. Minimum distance (Rmin): For all interactions ending as a BS[ij] endstate, we save the minimum distance any two of the three objects undergo during the interaction. This distance is denoted by Rmin. Binary–single energy (EBS): At the end of each interaction, we calculate the orbital energy of the single and the COM of the remaining pair. We denote this energy by EBS. In this notation, a BS[ij] endstate will have EBS > 0, where collisions and GW inspirals generally have EBS < 0, as they preferentially form during the interaction. 2.4 Topological mappings The graphical representation of the distribution of binary–single outcomes as a function of the ICs is what we loosely refer to as the topology, in analogy with Hut (1983a). As an example, Fig. 2 shows the distribution of binary–single endstates as a function of f, b, for a0 = 10−2 au and γ = 0, derived on a 1000 × 1000 grid. In our terminology, we refer to this distribution as the endstate topology, or the endstate-map. Building on the work of Hut (1983a), we focus in this study on topological maps derived from varying f and b, while keeping v and γ fixed. 3 TOPOLOGY OF BINARY–SINGLE INTERACTIONS In this section, we explore the binary–single topology on large scales. For this, we first study the classical limit, which represents the idealized case where the objects are all point-like particles interacting through Newtonian gravity only. After this follows, a study of the relativistic limit, in which GR corrections and finite size effects start to affect the classical topology. In the last part, we briefly explore how the topological maps change for varying γ. For all scatterings performed in this section, we keep finite sizes and GW emission in the EOM for consistency; the classical and the relativistic limits are therefore studied by simply varying a0. We note that f conveniently happens to approximately be the phase of the binary as the incoming single enters the binary. For example, at f ≈ 0, b ≈ 0, we find that all three objects are approximately on a line as the single enters. In our following descriptions, we will therefore simply refer to f instead of the exact binary phase as the single enters the binary. Furthermore, we often refer to just a single value of f, however, due to the rotational symmetry of the binary, results are similar for any phase f + nπ, where n is an integer. 3.1 Classical limit If the three interacting objects are point-like and only Newtonian gravity is included in the EOM, the system is said to be scale-free, as the topology and the relative number of each endstate do not depend on the SMA a0 and mass m, as long as v remains constant (Hut 1983a; Hut & Bahcall 1983). Although such an idealized system does not exist in nature, one approaches this classical scale-free limit as a0 increases relative to the radius of the interacting objects. In this section, we study the topology for a0 = 1 au, which provides a reasonable representation of the classical limit. Results are shown in Fig. 3, and discussed below. Figure 3. View largeDownload slide Topological maps derived from binary–single interactions with a0 = 1 au, v = 0.01, and γ = 0.0. Top left: endstates. Top right: total orbital energy between the endstate binary COM and the remaining single, log (EBS). Bottom left: number of times the binary–single system splits into an IMS before reaching an endstate, NIMS. Bottom right: minimum distance any two of the three objects undergo throughout an interaction, Rmin, in units of the initial SMA a0, log (Rmin/a0). These maps are used to study the classical limit, as described in Section 3.1. Further descriptions of outcomes, endstates, and ICs are found in Section 2. Figure 3. View largeDownload slide Topological maps derived from binary–single interactions with a0 = 1 au, v = 0.01, and γ = 0.0. Top left: endstates. Top right: total orbital energy between the endstate binary COM and the remaining single, log (EBS). Bottom left: number of times the binary–single system splits into an IMS before reaching an endstate, NIMS. Bottom right: minimum distance any two of the three objects undergo throughout an interaction, Rmin, in units of the initial SMA a0, log (Rmin/a0). These maps are used to study the classical limit, as described in Section 3.1. Further descriptions of outcomes, endstates, and ICs are found in Section 2. 3.1.1 Endstate topology The endstate-map (Fig. 3, upper left) shows a clear large-scale topological pattern. We find this to emerge from the distribution of two different small-scale topological pattern types, which we in the following refer to as the homogeneous-type and the random-type. The homogeneous-type is characterized by that neighbouring points share the same endstate, whereas the random-type is characterized by that neighbouring points have a semirandom distribution of endstates. As seen on the figure, the random-type forms one fully connected large-scale structure, which divides the homogeneous-type into a few disconnected islands. This is a feature of the ultra HB limit for which v ≪ 1, as the random-type structure starts to separate into isolated parts as v approaches 1 (e.g. Hut 1983a). Comparing the NIMS-map (Fig. 3, lower left) with the endstate-map shows that the random-type generally originates from RIs and the homogeneous-type from DIs. This is a consequence of the chaotic nature of RIs, which makes the outcomes very sensitive to the ICs. Neighbouring points on the f, b grid in RI regions are therefore likely to have a semirandom distribution of endstates. In contrast, DIs are always prompt. Small variations to the ICs will therefore not have a chance to undergo any significant growth during this type of interaction. Neighbouring points on the f, b grid in DI regions therefore often have similar endstates. The EBS-map (Fig. 3, upper right) relates closely to the other maps, as it is the orbital energy of the single w.r.t. the COM of the first binary that forms (endstate binary or IMS binary), denoted here by E1BS, that determines if the interaction proceeds as a DI or an RI. By definition, E1BS > 0 leads to a DI, in which case EBS = E1BS, whereas E1BS < 0 leads to an RI. As the distribution of E1BS seems to vary smoothly across the f, b space, we conclude that the boarders separating DIs from RIs – or equivalently the coastline of the DI islands – are simply the path for which E1BS = 0. As E1BS approaches zero along these boarders, the associated binary–single interaction time correspondingly approaches infinity, which explains why the ‘VLint’ endstate generally appears between DI and RI regions. As seen, there is a clear asymmetry in the large-scale topology along the b-axis, despite corresponding symmetry in the positional configuration of the three objects. This asymmetry results from the way the binary rotates relative to the incoming single, where b > 0 and b < 0 correspond to prograde and retrograde motion, respectively. For v ≪ 1, the incoming single almost has the same velocity as the binary members as it passes the binary. In the case of prograde orbits, this leads to situations where the three objects almost appear stationary w.r.t. each other. As a result, for certain combinations of f, b the incoming single does not directly encounter any of the binary members, which give rise to the large wave-like BS[12] structures seen for b > 0. In contrast, negative values of b always result in encounters between the single and a binary member due to the retrograde motion. A comparison with the high velocity results shown in Hut (1983a) confirms that the relative orbital motion indeed is the origin of the b-axis asymmetry. For example, the results in Hut (1983a) clearly show how the topology along the b-axis becomes increasingly symmetric as v increases, or in other words, as the velocity of the binary members become less important. 3.1.2 Kinematical examples Having described a few topological relations in the sections above, we now turn to a more detailed kinematical analysis. We note that an analytical solution for describing the dynamics of the SB limit is possible (e.g. Hut 1983b), where a statistical approach has been shown to provide valuable insight in the HB limit (e.g. Samsing et al. 2014). We divide our analysis below into the study of the topology around two orbital configurations: Configuration A (f ≈ 0) and Configuration B (f ≈ π/2). All results presented below are found by visually inspecting a representative set of three-body interactions. Configuration A: We first consider the change in topology for varying b at f ≈ 0. For this configuration, we refer to the binary member that is closest to the incoming single as it enters by ‘1’, the remaining member by ‘2’, and thereby the incoming single by ‘3’ (this notation matches the endstate labels on our figures for f ≈ 0 + π). This configurative labelling is also used in Fig. 4. Figure 4. View largeDownload slide Schematic illustrations showing the dynamics of two examples each representing an exchange interaction through a DI, where the left (E1) leads to BS[13], and the right (E2) to BS[23]. The grey large dots show the three objects at their initial positions, whereas the large black dots show the final configuration. The grey dotted circle illustrates the initial orbit of the two binary members, ‘1’ and ‘2’, before the incoming object ‘3’ enters the system. A simplified version of the orbital paths from initial to final state is shown with thin black lines. These two examples are further described in Section 3.1.2. Figure 4. View largeDownload slide Schematic illustrations showing the dynamics of two examples each representing an exchange interaction through a DI, where the left (E1) leads to BS[13], and the right (E2) to BS[23]. The grey large dots show the three objects at their initial positions, whereas the large black dots show the final configuration. The grey dotted circle illustrates the initial orbit of the two binary members, ‘1’ and ‘2’, before the incoming object ‘3’ enters the system. A simplified version of the orbital paths from initial to final state is shown with thin black lines. These two examples are further described in Section 3.1.2. Starting at b ≈ 0, we see that the topology changes rather abruptly from a BS[23] DI region to a BS[13] DI region. The kinematics leading to these two neighbouring regions is as follows. For slightly negative values of b, object 3 first encounters 1 from below which leads to an upwards deflection of 3 and a downwards deflection of 1. Along its new orbit 1 then interacts with 2, which sends 1 up towards 3 while 2 is being ejected downwards. As a result, [13] escapes as a binary. This interaction is illustrated to the left (labelled ‘E1’) in Fig. 4. For slightly positive b, object 3 instead encounters 1 from above, which gives rise to an interaction where the two objects almost perfectly replace each other, which leads to 1 flying to the left and 3 moving upwards. Object 1 then interacts with 2 from above, which sends 1 downwards and 2 upwards where it pairs up with 3. As a result, [23] escapes as a binary. This interaction is illustrated to the right (labelled ‘E2’) in Fig. 4. At b ≈ −1.5, we see a region that is somewhat similar to the one at b ≈ 0, but with the two BS[23] and BS[13] regions switched. For the region at b ≲ −1.5, we find that 3 first interacts with 2 from below after which the binary has rotated ≈π/2. This interaction sends 2 downwards and 3 up towards 1. 1 and 3 then undergo a near parabolic hook-interaction, which sends 3 back towards 2 by ejecting 1 further upwards. As a result, [23] escapes as a binary. For the region at b ≳ −1.5, the first interaction between 3 and 2 is simply so strong that 2 is immediately kicked out by 3 which then takes its place in the binary. As a result, [13] ends up as a binary. Configuration B: We now consider the change in topology for varying b near f ≈ π/2. For this configuration, we refer to the binary member that moves towards the incoming single by ‘1’, the remaining member by ‘2’, and thereby the single by ‘3’. This configuration is similar to that shown in Fig. 4, but for the binary rotated clockwise by π/2. We first notice that for any value of b ≈ 0 ± 0.5, the endstate will always be of the same type, in contrast to Configuration A considered above. The kinematics near b ≈ 0 are as follows. As object 3 enters the binary, it first accelerates 1 in the upwards direction along its orbital tangent. Object 3 then interacts with 2 from below, which results in 2 being ejected downwards and 3 being sent up towards 1. As a result, [13] escapes as a binary. This basically happens for all b for which 3 enters the binary on an orbit that goes in between 1 and 2. Across larger variations of b, significant differences in the topology appear. For example, interactions near b ≈ 1.5 all evolve as an RI, whereas interactions near b ≈ −1.5 instead evolve as a DI. We find the kinematical differences to be as follows. At b ≈ −1.5, object 3 no longer moves in between 1 and 2, but instead passes the binary from below. As it passes, 3 and 1 first interact, pulling 1 out of its circular motion along an orbit pointing towards the initial motion of 3. Object 3 then interacts with 2 which sends 3 back towards 1 by ejecting 2 in the opposite direction. As a result, [13] escapes as a binary. At b ≈ 1.5, the kinematics is similar to that of b ≈ 0, however, instead of 3 ejecting 2 into an unbound orbit, 2 remains bound after which the interaction then proceeds as an RI. 3.1.3 Close encounter topology The Rmin-map (Fig. 3, lower right) shows a new structural pattern that is characterized by connected wavy bands. These bands map the closest distance any two of the three objects undergo before the system splits into its first binary–single state (BS[ij] or IMS). For example, the band that intersects f ≈ 0, b ≈ 0 and rises until f ≈ π/2, b ≈ 1.0 originates from the first close encounter between objects 1 and 3, where ‘1’ here refers to the binary member that is closest to the incoming single ‘3’ at f ≈ 0, b ≈ 0. Since the trajectory of 3 is not perturbed significantly before it first encounters 1, the value of b for which 1 and 3 undergo a close encounter is therefore approximately that of the shadow image of 1 along an axis parallel to b. The first part of the considered band therefore takes the shape of a sinusoidal curve, which explains why it first rises and then peaks at f ≈ π/2. However, when it goes back towards b ≈ 0 near f ≈ π the sinusoidal curve is strongly perturbed, because object 2 comes in between 1 and 3, which leads to deflections of 3 before it encounters 1. For velocities v > 1, object 3 is hardly deflected, and the bands will therefore trace out a near perfect sinusoidal form. The Rmin-map also shows a population of low Rmin values inside the RI regions. This population originates from IMS binaries that are formed with high eccentricity and corresponding small pericentre distance. 3.2 Relativistic limit We now study the topology of the relativistic limit, where finite sizes and GR corrections start to play a role. To this end, we perform binary–single scatterings with the initial SMA set to a0 = 10−4 au, as this is the scale for which GR corrections are expected to significantly change the outcome distributions (see Section 2.3.1). This unrealistic low value of a0 has only been chosen for illustrative purposes; however, the following results do apply for any a0 – in fact, it is the wide binaries that dominate the rate of relativistic outcomes, due to their larger interaction cross-section (Samsing et al. 2014). Results are shown in Fig. 5, and discussed below. Figure 5. View largeDownload slide Topological maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstates. Bottom plot: number of times the binary–single system splits into an IMS before reaching an endstate, NIMS. These maps are used to study the relativistic limit, as described in Section 3.2. Figure 5. View largeDownload slide Topological maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstates. Bottom plot: number of times the binary–single system splits into an IMS before reaching an endstate, NIMS. These maps are used to study the relativistic limit, as described in Section 3.2. 3.2.1 BH collisions Looking at the endstate-map (Fig. 5, top plot) and the NIMS-map (Fig. 5, bottom plot), we see that BH collisions form through both DIs and RIs. These two collision populations are described in the following. The DI-collisions trace the wavy Rmin bands shown in Fig. 3, with only small differences due to the modified dynamics resulting from including GW emission in the EOM. The bands broaden as the value of b increases. This is due to the corresponding decrease in the relative velocity between the binary member in question and the incoming single, which leads to an increase in the gravitational focusing, and thereby to the seen broadening in b. As the value of b decreases, the relative velocity increases, which makes the gravitational focusing less efficient. The orbit of the incoming single therefore has to be relatively fine-tuned to collide with a binary member, which explains the thinner bands at negative b. The RI-collisions originate from the population of IMS binaries that are formed with a pericentre distance that is smaller than the collisional distance 2Rs. Despite the apparent randomness of the RI regions, the collision endstates do not seem to be completely randomly distributed in these regions. For example, the relative number of collisions clearly decreases for b ≳ 2. This partially results from the corresponding increase in the total angular momentum, which naturally leads to fewer low angular momentum orbits. Further insight into the collision endstates distribution is provided in Fig. 6, which shows what pair that collides. Besides bands originating from collisions with the incoming single (Coll[13], Coll[23]), we interestingly see a band passing through f ≈ 0.7, b ≈ −0.5 that is formed by collisions of the binary members themselves (Coll[12]). We find that the corresponding dynamics is such that the incoming single, during its first passage, simply perturbs the two binary members into a prompt collisional orbit. Although this seems like a fine-tuned problem, the corresponding bands appear relative broad, which indicates a non-negligible formation probability. Figure 6. View largeDownload slide Topological map of collision and GW inspiral endstates derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. This map provides information about what object pair that collides or undergoes a GW inspiral. As seen, the red colours show GW inspirals, where the grey colours show collisions. This map is discussed in Section 3.2. Figure 6. View largeDownload slide Topological map of collision and GW inspiral endstates derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. This map provides information about what object pair that collides or undergoes a GW inspiral. As seen, the red colours show GW inspirals, where the grey colours show collisions. This map is discussed in Section 3.2. 3.2.2 GW inspirals A clear population of GW inspirals is seen near the boarders of the wavy collision bands. The origin of this inspiral population is not surprising, as the collision boarders are characterized by near collisional orbits. When GW emission is included in the EOM, these orbits therefore naturally result in strong GW emission, which then leads to the observed populations of GW inspirals. Just as the population of collisions in the wavy bands forms through DIs, so do the GW inspirals forming along the bands, as seen in the bottom plot in Fig. 5. We note that there are a few collisional bands located in the DI regions with no GW inspirals forming along their edges. We find here that the dynamics leading to the prompt collision at the same time ejects the third BH into an unbound orbit. The endstates populating these bands are therefore characterized by a collision with an unbound companion. This explains why no GW inspirals form along their edges, as the binary–single system splits up into a BS[i,j] state before the GW inspiral is complete. The two binary objects will of course still undergo the GW inspiral, but it will happen without a bound companion. Moving away from the wave-like bands, we see how GW inspirals clearly form throughout the RI regions. We note here that collisions and GW inspirals do not have to follow the same distribution, as the formation of a GW inspiral is not related to a particular value of the IMS binary pericentre distance unlike a collision. Instead, the formation of a GW inspiral relates to how much energy it can lose through GW emission during the orbital time it is isolated from the bound single (Samsing et al. 2014). As a result, GW inspirals can form from IMS binaries with both small and large pericentre distances, depending on the corresponding isolation time, and can therefore appear in regions where collisions are relative rare. For example, in the upper plot in Fig. 5, there are many GW inspirals forming for b > 2.5 but only relative few collisions. A more detailed picture is provided in Fig. 6, which also shows what pair that undergoes the GW inspiral. 3.3 Out-of-plane interactions For completeness, we here study the change in topology for varying orbital-plane angle γ. Endstate-maps for γ = π/4 and γ = π/2 are shown in Fig. 7, and briefly discussed below. Figure 7. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au and varying γ. Top plot: endstate topology for γ = π/4. Bottom plot: endstate topology for γ = π/2. The corresponding endstate topology for γ = 0 is shown in the top of Fig. 5. Figure 7. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au and varying γ. Top plot: endstate topology for γ = π/4. Bottom plot: endstate topology for γ = π/2. The corresponding endstate topology for γ = 0 is shown in the top of Fig. 5. The most notable difference between the co-planar case (γ = 0) and our presented out-of-plane cases (γ = π/4, π/2) is a breakup of the connected wave-like collision and GW inspiral bands seen for γ = 0. To understand this change, we note that the incoming single can only undergo a prompt collision or a prompt GW inspiral near regions where its orbital plane intersects with the binary orbit. When γ = 0, the incoming single intersects for sinusoidal variations in f, b, however, when γ > 0, the orbital plane of the single is only able to intersect the binary orbit at maximum two points, namely at {−π/2, +π/2}. This correspondingly results in isolated regions in the topological map instead of connected bands. As with the co-planar case, the DI collision regions for γ > 0 are surrounded by GW inspirals that form through a prompt capture. However, a large population of both DI and RI collisions and GW inspirals also form beyond these DI regions. In fact, we find that the total number of collisions and GW inspirals is not strongly dependent on γ, although the distribution of orbital parameters at merger might depend on γ due to the corresponding change of angular momentum. We save such a study for future work. 4 THE ROLE OF GW EMISSION We now perform a slightly more controlled study on how the inclusion of GW emission in the EOM affects the topology on different scales. For this, we explore two different topological regions, with and without GW emission included in the EOM. For all interactions, we set a0 = 10−4 au and γ = 0 to have GW emission effects stand out clearly. Results are discussed below. 4.1 Large-scale effects We start by analysing how the inclusion of GW emission affects the large-scale topology. Results are shown in Fig. 8. Figure 8. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstate-map derived without the inclusion of GW emission in the N-body EOM. Bottom plot: endstate-map derived with the inclusion of GW emission in the N-body EOM. Results are discussed in Section 4.1. Figure 8. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstate-map derived without the inclusion of GW emission in the N-body EOM. Bottom plot: endstate-map derived with the inclusion of GW emission in the N-body EOM. Results are discussed in Section 4.1. When GW emission is not included in the EOM (top plot), we see that the collision regions simply appear superimposed on the classical endstate-map, with a distribution that follows the Rmin distribution as expected. Other than occupying part of the map, the inclusion of collisions does not affect the large-scale topological structure, as the EOM is unchanged. When GW emission is included (bottom plot), we see that this not only leads to the formation of GW inspirals, but also a slight modification of the topology. For example, at f ≈ 0.75, b ≈ 0.75, where the collision band intersects the RI region in the classical map, the inclusion of GW emission clearly smoothes out the transition region, by turning a part of the BS[1,3] endstates formed in the DI region into a semirandom mix of outcomes. Generally, one finds that the largest topological changes indeed are found near regions where collision endstates, DIs, and RIs intersect. This results from two factors: the GW emission is strong (the objects are close to colliding) and the orbital energy E1BS is near zero (the boarder between a DI and an RI is characterized by E1BS = 0). This combination makes it possible for the GW emission to change a classical DI into an RI, by emitting the orbital energy out of the system that otherwise would have ejected the single into an unbound orbit. In fact, in Samsing et al. (2017), it was briefly discussed if this way of ‘capturing’ the single into an RI through GW emission would be able to change the relative frequency of endstates. We have here shown and argued that this indeed happens, but only in a very small part of the available phase-space. The effect is therefore negligible from a statistical perspective. 4.2 Small-scale effects We now study how the topology changes on small scales as GW emission is included in the EOM. Results are shown in Fig. 9. Figure 9. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstate-map derived without the inclusion of GW emission in the N-body EOM. Bottom plot: endstate-map derived with the inclusion of GW emission in the N-body EOM. Results are discussed in Section 4.2. Figure 9. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3, but with a0 = 10−4 au. Top plot: endstate-map derived without the inclusion of GW emission in the N-body EOM. Bottom plot: endstate-map derived with the inclusion of GW emission in the N-body EOM. Results are discussed in Section 4.2. In the top plot, where GW emission is not included, we first notice that the considered RI region has a surprisingly rich and complex microtopology. This will be studied in greater detail in Section 5. As in the large-scale example, we see here how the collision regions simply appear as sharp shadows superimposed on top of the classical map. We now turn to the bottom plot where GW emission is included. As seen, the inclusion drastically changes the topology close to the classical collision bands. Not only is a clear population of GW inspirals forming along the bands, but the GW emission further results in band-like structures that alter between collisions and GW inspirals. To understand the kinematics leading to this alternation, we visually inspected a set of binary–single interactions for f ≈ π/2 and varying b. For this configuration, we here denote the binary member moving away from the incoming single by ‘2’, the remaining member by ‘1’, and thereby the incoming single by ‘3’. We find the kinematics to be as follows. First, the large collisional band at the bottom (b ≈ 1.15) is formed by 2 directly colliding with 3 as it enters the binary, as also described in Section 3.1.3. Above this is an inspiral band (b ≈ 1.3), which we find consists of interactions where 3 instead of directly colliding with 2, slightly grazes it, which then leads to a prompt GW inspiral between the two. The subsequent collision band (b ≈ 1.4) is formed by 3 first being captured into a relative wide low angular momentum orbit by 2, after which 3 and 2 collide at their first pericentre passage along their new orbit. The following narrow BS[23] band (b ≈ 1.42) arises from a rather fine-tuned dynamical configuration where 1 becomes kinematically unbound just before 2 and 3 undergo a collision or a GW inspiral. This band is therefore sensitive to the tidal threshold used for labelling a BS[ij] endstate, and is therefore not a fundamental feature. The next few bands form in a similar way to the previous collision band, only with 2 and 3 changing between undergoing a collision or a GW inspiral. To conclude, these topological changes all relate to the capture of object 3 into a bound orbit partly through GW emission. As the capture happens at low angular momentum, and angular momentum further is radiated away during the capture, the system evolves in a state that is likely to result in either a prompt collision or a GW inspiral. 5 TOPOLOGICAL MICROSTRUCTURES We now explore the topology of the microstructures that surprisingly seem to appear within RI regions. For this, we perform scatterings for a0 = 1 au, and γ = 0, with collisions and GW emission included in the EOM. However, we do note that our considered small-scale topology is likely to change slightly if additional PN terms are added to the EOM, we therefore limit ourselves to general conclusions. Results from two selected zoom-in regions, referred to as ‘Zoom Region 1’ and ‘Zoom Region 2’, respectively, are shown in Fig. 10, and discussed below. Figure 10. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3 with a0 = 1 au and γ = 0. Top plot: zoom in on the centre of one of the larger RI regions. Bottom plot: zoom in on the boundary between one of the larger RI regions and its neighbouring DI region. As seen in both plots, the considered regions do not just consist of a semirandom distribution of endstates as it appears to on larger scales. Instead, the regions have in fact a rich variety of microtopological structures. The origin of these structures is discussed in Section 5. Figure 10. View largeDownload slide Topological endstate-maps derived in a similar way to the ones shown in Fig. 3 with a0 = 1 au and γ = 0. Top plot: zoom in on the centre of one of the larger RI regions. Bottom plot: zoom in on the boundary between one of the larger RI regions and its neighbouring DI region. As seen in both plots, the considered regions do not just consist of a semirandom distribution of endstates as it appears to on larger scales. Instead, the regions have in fact a rich variety of microtopological structures. The origin of these structures is discussed in Section 5. 5.1 Zoom Region 1 The top plot shows a zoom in on one of the larger RI regions. As seen, the region does not only consist of semirandomly distributed endstates, as it appears to on large scales, but in fact hosts clear microtopological structures of various kinds. To gain insight into the dynamics of this region, we visually investigated a representative set of binary–single interactions. From this exercise, we found that most of the interactions near the centre (f ≈ 1.3, b ≈ 1.2) start out in a similar way, where the objects first interact non-chaotically with their orbits tracing out a well-defined pattern with a clear degree of symmetry. A schematic illustration of this primary orbital interaction assuming perfect symmetry is shown in Fig. 11. Although the interaction in this example promptly ends as a BS[23] endstate, the outcomes do vary depending on the exact ICs. In a few cases, we even observed that the system after this primary interaction entered a quasi-stable state which it stayed in for 5–10 cycles. We note here that the more stable a three-body state is, the less likely it is to form (e.g. Hut 1983a), which implies that fully stable three-body states are not assembled through our binary–single experiments (Newtonian three-body states cannot be stable at plus infinity if they are not stable at minus infinity). Of course, the inclusion of GR in the EOM immediately changes the situation, as the implied GW emission destabilizes the system within a finite time. Further studies on the possible assembly of quasi-stable three-body states with and without GR are beyond this paper, but still of great theoretical interest (e.g. Šuvakov & Dmitrašinović 2013). Figure 11. View largeDownload slide Schematic illustration of the orbital trajectories associated with an interaction leading to a BS[23] endstate found near f ≈ 1.3, b ≈ 1.25. As seen, the objects are in this interaction clearly moving in a non-chaotic way on orbits that trace out an elegant symmetrical pattern. We find that most of the interactions near the considered values for f, b actually go through this orbital evolution, but with different outcomes to follow depending on the exact ICs. In this illustration, the colours red, green, and blue refer to objects 1, 2, and 3, respectively, where the arrows denote the direction of motion. The considered BS[23] outcome corresponds to the case where the binary enters from the left and moves towards the right. Further discussions are found in Section 5.1. Figure 11. View largeDownload slide Schematic illustration of the orbital trajectories associated with an interaction leading to a BS[23] endstate found near f ≈ 1.3, b ≈ 1.25. As seen, the objects are in this interaction clearly moving in a non-chaotic way on orbits that trace out an elegant symmetrical pattern. We find that most of the interactions near the considered values for f, b actually go through this orbital evolution, but with different outcomes to follow depending on the exact ICs. In this illustration, the colours red, green, and blue refer to objects 1, 2, and 3, respectively, where the arrows denote the direction of motion. The considered BS[23] outcome corresponds to the case where the binary enters from the left and moves towards the right. Further discussions are found in Section 5.1. Although the state illustrated in Fig. 11 is not stable, it certainly classifies as a special interaction due to its symmetry and interesting orbital topology. However, the classification of such interactions is not entirely clear and to our knowledge still unexplored. A systematic search through the f, b space could reveal if this is a member of a larger family of peculiar interactions, or just a single interesting incident. 5.2 Zoom Region 2 The bottom plot shows a zoom in on a transition zone between RIs (lower left) and DIs (upper right). As seen, the only pattern that is clearly visible consists of nested rings with a spacing that decreases as one moves closer to the DI boundary from the RI side. Focusing on the BS[13] endstate bands, we find that the corresponding IMS binary configuration is such that when the single returns from its first excursion, the resultant IMS binary–single interaction leads to a prompt BS[13] endstate. This happens in each band, with the only difference being how many times the IMS binary rotates before the single returns. As a result, there is a band for each integer n, where n is the number of times the IMS rotates before the single returns (by visual expectation, we find that the band in the lower left corner corresponds to n = 10). As E1BS approaches zero as one moves towards the DI region, the number n will correspondingly approach infinity. As a result, the spacing between the blue bands becomes smaller and smaller, approaching zero near the boundary. This explains why substructures with infinitely small spacings exist and where they are located. The precess of such structures implies that the true richness of microtopological structures cannot be studied with a finite grid approach as the one we have presented here. 6 DOUBLE GW MERGERS All GW inspirals form during the binary–single interaction while the three objects are still bound to each other in an IMS (Samsing et al. 2014). Ignoring dynamical kicks caused by asymmetric GW emission (e.g. Baker et al. 2006), this interestingly implies that the BH formed from the GW inspiral merger and the remaining single are still bound to each other after the three-body interaction. A compelling question is how long time it takes for this binary to merge through GW emission, or equivalently, what the time span is from the first to the second GW merger. If the time span is short enough, it might be possible for LIGO to observe both GW mergers, a scenario that undoubtfully would be extremely exiting. In this section, we explore for which combinations of f and b the time span from first to second GW merger, denoted by Δt12, takes its shortest value. A more detailed study of this double GW merger scenario is reserved for a separate paper, and the following discussions are therefore kept short. An illustration of our proposed double GW merger channel is shown in Fig. 12, and results presented below. Figure 12. View largeDownload slide Illustration of a binary–single interaction that results in the formation of two GW mergers. As seen, the interaction first results in the formation of a BBH that promptly undergoes a GW inspiral (first GW merger). After this, the resultant BH formed through the first GW inspiral then subsequently undergoes a GW merger with the remaining bound single (second GW merger). If the time span between the first and the second GW merger is relative short, then it might be possible to observe both GW mergers with a detector such as LIGO. In Section 6.1, we explore which values for f, b that lead to the shortest time span. Figure 12. View largeDownload slide Illustration of a binary–single interaction that results in the formation of two GW mergers. As seen, the interaction first results in the formation of a BBH that promptly undergoes a GW inspiral (first GW merger). After this, the resultant BH formed through the first GW inspiral then subsequently undergoes a GW merger with the remaining bound single (second GW merger). If the time span between the first and the second GW merger is relative short, then it might be possible to observe both GW mergers with a detector such as LIGO. In Section 6.1, we explore which values for f, b that lead to the shortest time span. 6.1 Time from first to second GW merger The distribution of log (Δt12/T0), where T0 is the orbital time of the initial target binary, as a function of f, b for γ = 0, is shown in Fig. 13. For this, we have included all GW inspirals and BH collisions that form with a bound companion, where Δt12 is estimated using the equations from Peters (1964). To keep this primary study simple and clear, we have not incorporated effects related to relativistic mass losses or asymmetric GW emission; corrections that in general would lead to an increase in Δt12. Figure 13. View largeDownload slide Distribution of log (Δt12/T0), where Δt12 is the time span from the first to the second GW merger, and T0 is the orbital time of the initial target binary, derived for a0 = 10−4 au, γ = 0. As described in Section 6.1, the horizontal band for which Δt12 takes its minimum value corresponds to where the total angular momentum of the three-body system equals zero. Figure 13. View largeDownload slide Distribution of log (Δt12/T0), where Δt12 is the time span from the first to the second GW merger, and T0 is the orbital time of the initial target binary, derived for a0 = 10−4 au, γ = 0. As described in Section 6.1, the horizontal band for which Δt12 takes its minimum value corresponds to where the total angular momentum of the three-body system equals zero. As seen in the figure, the minimum value for Δt12 is highly sensitive to b, but not f. The reason is that only b relates to the total angular momentum, which in turn directly relates to the corresponding GW inspiral time (Peters 1964). This further explains why the value for b that minimizes Δt12 is simply the value for which the total three-body angular momentum equals zero. As derived in Section 2.2, the angular momentum of the single and the binary cancels out for b ≈ −0.87, which clearly is where Δt12 takes its minimum as confirmed by the simulation results shown in the figure. Close to this value of b, the three objects can, as a result of the vanishing angular momentum, promptly merge without any significant use of GW emission. This leads to values of Δt12 that are of the order of the orbital time of the initial binary, instead of its initial GW inspiral time. The angular momentum cancellation is less efficient in out-of-plane interactions, which explains why the co-planar interaction case is the relevant channel to consider for generating double GW mergers. 7 CONCLUSIONS In this paper, we explored how the outcomes of binary–single BH interactions distribute as a function of the ICs. For this, we studied the corresponding graphical representation of the distribution (Fig. 2) – a representation we loosely refer to as the topology in analogy with Hut (1983a). Our study shows for the first time how this binary–single topology changes when GR effects at the 2.5 PN order, which accounts for dynamical GW emission, and BH finite sizes are included in the N-body EOM. Such corrections give rise to a non-negligible distribution of BH collisions and GW inspirals that is directly observable by LIGO (e.g. Samsing et al. 2014; Samsing & Ramirez-Ruiz 2017). Using zoom-in simulations, we resolved for the first time the emergence of microtopological structures residing inside RI regions (Fig. 10). By visually inspecting orbits near these structures, we found that many of the interactions (in the RI region centred around f ≈ 1.3, b ≈ 1.25) start out in a similar dynamical way characterized by a particular ordered and symmetrical orbital evolution (Fig. 11). To our knowledge, the classification and assembly of such symmetrical orbital evolutions are still unexplored, and a systematic search for these throughout the f, b space could therefore be of potential interest. Finally, we pointed out that the inclusion of GW emission in the N-body EOM naturally leads to interactions resulting in two GW mergers. In this regard, we described how the co-planar case can lead to scenarios where the time span from the first to the second GW merger is short enough for LIGO to observe both events. This double GW merger channel could have profound astrophysical consequences, and will therefore be explored in greater detail in an upcoming paper. From an astrophysical perspective, our most relevant result is that the distribution of outcomes of binary–single interactions is highly dependent on the ICs. While this certainly applies to small variations to the ICs, due to the chaotic nature of the three-body problem, we surprisingly find that also large-scale variations to the ICs give rise to corresponding large-scale differences in the outcome distribution. For example, in the co-planar case (γ = 0), the distribution is clearly different between prograde (b > 0) and retrogade (b < 0) interactions (Fig. 2). This also applies to how the relative number of BH mergers distributes with the ICs (Fig. 6). Clear differences are also seen when the orbital-plane angle γ is varied (Fig. 7). This means that if properties of these distributions, such as the eccentricity distribution at merger, could be observed by LIGO, then one would be able to roughly deduce the underlying distribution of ICs, which could give important hints about the astrophysical origin. As an example, in AGN environments (e.g. McKernan et al. 2012; Bellovary et al. 2016; McKernan et al. 2017), the interactions will not only be close to co-planer, but they will also likely be unequally distributed between prograde or retrograde interactions, depending on the formation history of the interacting BHs and their interaction with the disc. Our idealized study indicates that the outcome distribution is clearly different for these cases. A natural extension to our presented work could therefore be to derive observables related to BH collisions, GW inspirals, and high eccentricity BH mergers, for different scenarios including retrograde interactions, prograde interactions, in-plane interactions, out-of-plane interactions, and for varying mass hierarchy. Although our paper presents details to a level that is not directly observable, we do note that similar studies on how the distribution of BH mergers and related observables changes with the ICs are likely to play a key role in distinguishing different astrophysical GW merger channels from each other. An exercise that will become highly relevant when the number of observed GW mergers starts to grow. ACKNOWLEDGEMENTS Support for this work was provided by NASA through Einstein Postdoctoral Fellowship grant number PF4-150127 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. REFERENCES Abbott B. P. et al.  , 2016a, Physical Review Letters , 116, 061102 https://doi.org/10.1103/PhysRevLett.116.061102 CrossRef Search ADS   Abbott B. P. et al.  , 2016b, Physical Review Letters , 116, 241103 https://doi.org/10.1103/PhysRevLett.116.241103 CrossRef Search ADS   Abbott B. P. et al.  , 2016c, ApJ , 818, L22 https://doi.org/10.3847/2041-8205/818/2/L22 CrossRef Search ADS   Abbott B. P. et al.  , 2016d, ApJ , 833, L1 https://doi.org/10.3847/2041-8205/833/1/L1 CrossRef Search ADS   Abbott B. P. et al.  , 2017, Phys. Rev. Lett. , 118, 221101 https://doi.org/10.1103/PhysRevLett.118.221101 CrossRef Search ADS PubMed  Antognini J. M. O., Thompson T. A., 2016, MNRAS , 456, 4219 https://doi.org/10.1093/mnras/stv2938 CrossRef Search ADS   Antonini F., Rasio F. A., 2016, ApJ , 831, 187 https://doi.org/10.3847/0004-637X/831/2/187 CrossRef Search ADS   Askar A., Szkudlarek M., Gondek-Rosińska D., Giersz M., Bulik T., 2017, MNRAS , 464, L36 https://doi.org/10.1093/mnrasl/slw177 CrossRef Search ADS   Baker J. G., Centrella J., Choi D.-I., Koppitz M., van Metre J. R., Miller M. C., 2006, ApJ , 653, L93 https://doi.org/10.1086/510448 CrossRef Search ADS   Bartos I., Kocsis B., Haiman Z., Márka S., 2017, ApJ , 835, 165 https://doi.org/10.3847/1538-4357/835/2/165 CrossRef Search ADS   Bellovary J. M., Mac Low M.-M., McKernan B., Ford K. E. S., 2016, ApJ , 819, L17 https://doi.org/10.3847/2041-8205/819/2/L17 CrossRef Search ADS   Blanchet L., 2014, Living Reviews in Relativity , 17, 2 CrossRef Search ADS PubMed  Fregeau J. M., Cheung P., Portegies Zwart S. F., Rasio F. A., 2004, MNRAS , 352, 1 https://doi.org/10.1111/j.1365-2966.2004.07914.x CrossRef Search ADS   Goodman J., Tan J. C., 2004, ApJ , 608, 108 https://doi.org/10.1086/386360 CrossRef Search ADS   Gültekin K., Miller M. C., Hamilton D. P., 2006, ApJ , 640, 156 https://doi.org/10.1086/499917 CrossRef Search ADS   Heggie D. C., 1975, MNRAS , 173, 729 https://doi.org/10.1093/mnras/173.3.729 CrossRef Search ADS   Horn B., Lyra W., Mac Low M.-M., Sándor Z., 2012, ApJ , 750, 34 https://doi.org/10.1088/0004-637X/750/1/34 CrossRef Search ADS   Hut P., 1983a, AJ , 88, 1549 https://doi.org/10.1086/113445 CrossRef Search ADS   Hut P., 1983b, ApJ , 268, 342 https://doi.org/10.1086/160957 CrossRef Search ADS   Hut P., Bahcall J. N., 1983, ApJ , 268, 319 https://doi.org/10.1086/160956 CrossRef Search ADS   Just A., Yurin D., Makukov M., Berczik P., Omarov C., Spurzem R., Vilkoviskij E. Y., 2012, ApJ , 758, 51 https://doi.org/10.1088/0004-637X/758/1/51 CrossRef Search ADS   Kocsis B., Tremaine S., 2011, MNRAS , 412, 187 https://doi.org/10.1111/j.1365-2966.2010.17897.x CrossRef Search ADS   Kocsis B., Tremaine S., 2015, MNRAS , 448, 3265 https://doi.org/10.1093/mnras/stv057 CrossRef Search ADS   Kocsis B., Yunes N., Loeb A., 2011, Phys. Rev. D , 84, 024032 https://doi.org/10.1103/PhysRevD.84.024032 CrossRef Search ADS   Levin Y., 2007, MNRAS , 374, 515 https://doi.org/10.1111/j.1365-2966.2006.11155.x CrossRef Search ADS   McKernan B., Ford K. E. S., Lyra W., Perets H. B., Winter L. M., Yaqoob T., 2011, MNRAS , 417, L103 https://doi.org/10.1111/j.1745-3933.2011.01132.x CrossRef Search ADS   McKernan B., Ford K. E. S., Lyra W., Perets H. B., 2012, MNRAS , 425, 460 https://doi.org/10.1111/j.1365-2966.2012.21486.x CrossRef Search ADS   McKernan B. et al.  , 2017, preprint (arXiv:1702.07818) Ostriker J. P., 1983, ApJ , 273, 99 https://doi.org/10.1086/161351 CrossRef Search ADS   Peters P., 1964, Phys. Rev. , 136, B1224 https://doi.org/10.1103/PhysRev.136.B1224 CrossRef Search ADS   Rodriguez C. L., Chatterjee S., Rasio F. A., 2016, Phys. Rev. D , 93, 084029 https://doi.org/10.1103/PhysRevD.93.084029 CrossRef Search ADS   Roupas Z., Kocsis B., Tremaine S., 2017, ApJ , 842, 90 https://doi.org/10.3847/1538-4357/aa7141 CrossRef Search ADS   Samsing J., Ramirez-Ruiz E., 2017, ApJ , 840, L14 https://doi.org/10.3847/2041-8213/aa6f0b CrossRef Search ADS   Samsing J., MacLeod M., Ramirez-Ruiz E., 2014, ApJ , 784, 71 https://doi.org/10.1088/0004-637X/784/1/71 CrossRef Search ADS   Samsing J., MacLeod M., Ramirez-Ruiz E., 2017, ApJ , 846, 36 CrossRef Search ADS   Samsing J., MacLeod M., Ramirez-Ruiz E., 2018, ApJ , 853, 140 Stone N. C., Metzger B. D., Haiman Z., 2017, MNRAS , 464, 946 https://doi.org/10.1093/mnras/stw2260 CrossRef Search ADS   Šuvakov M., Dmitrašinović V., 2013, Phys. Rev. Lett. , 110, 114301 https://doi.org/10.1103/PhysRevLett.110.114301 CrossRef Search ADS PubMed  Syer D., Clarke C. J., Rees M. J., 1991, MNRAS , 250, 505 https://doi.org/10.1093/mnras/250.3.505 CrossRef Search ADS   Wang L. et al.  , 2016, MNRAS , 458, 1450 https://doi.org/10.1093/mnras/stw274 CrossRef Search ADS   © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society

### Journal

Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations

Abstract access only

18 million full-text articles

Print

20 pages / month

PDF Discount

20% off