Tjarda C. N. Boekholt

Context. Through observations of gravitational waves, mergers of black holes (BHs) have been shown to be a ubiquitous phenomenon in the Universe. However, uncertainties remain in our theoretical understanding of the evolution of the BHs prior to their merger. Black hole progenitors are seemingly born primarily in triples or higher-order multiples, and the presence of a tertiary object can perturb a black hole binary (BHB) enough to precipitate a merger. Aims. We aim to provide a detailed overview of BH mergers in wide triples by evolving systems from stellar birth to BH merger, using non-orbit-averaged methods for both the dynamics and BH spins from the point of BHB formation, and including triples that become dynamically unstable due to stellar evolution. This allows us to obtain a more complete picture of BH mergers in triples, shedding light on the initial orbital properties required to produce a merger in a triple, as well as the final properties of the merging BHBs. Methods. We used population synthesis coupled with orbit-averaged descriptions of triple dynamics to evolve a population of triples with initially wide orbits and massive progenitors. Wide orbits were chosen to avoid stellar interaction prior to BH formation. Systems that remain bound and form an inner BHB were evolved using a direct n-body code with post-Newtonian terms up to an order of 2.5. We also simulated the precession of the BH spin vectors by coupling the n-body solver with the differential equations for the spins. Results. Forming a wide, bound BHB with a tertiary companion requires the system to survive pre-BH stellar evolution and at least two supernovae in the inner binary. Consequently, only a small fraction of the massive triple population are born with the parameters required to achieve this configuration. For the dynamically stable triples with inner BHBs, only the systems with inclinations close to 90° can excite the eccentricity in the inner binary to values high enough for gravitational waves (GWs) to dissipate sufficient energy to merge the system with an estimated merger rate density of ∼5 Gpc ⁻³ yr ⁻¹ . Mergers also occur in triples that become dynamically unstable at a rate of ∼1.4 Gpc ⁻³ yr ⁻¹ . At the point of entering the 10 Hz gravitational wave frequency band, the merging inner binaries exhibit eccentricities between 10 ⁻⁴ and 10 ⁻² . Finally, we find that the final effective spin of a BHB that merges through this channel can display a wide range of values between −1 and 1, with a slight tendency towards χ _eff ≈ 0. This is a result of the strong three-body dynamics experienced by the merging triples before the inner binary begins to shrink due to GW emission. The inner angular momentum can explore the full phase space before the binary rapidly shrinks and decouples from the tertiary, effectively freezing out the effective spin to its value at the time of the highest inner eccentricity.

The supermassive black hole at the Galactic centre, Sgr A*, is surrounded by an S cluster of young stars. These stars are predominantly B-type stars, but their origin remains uncertain. Unlike massive stars in the Galactic field, which exhibit a binary fraction of about 70%, those in the S cluster show a significantly lower fraction. It is key to understand this discrepancy for constraining their formation. We show that the binary deficit arises naturally from dynamical interactions between stellar binaries and Sgr A* in an in situ formation scenario. Using K-amplitude data, we derived an observational binary fraction of f _obs = 0.43 ± 0.09 for the S-cluster stars. Our dynamical modelling predicts that interactions with Sgr A* cause 18% of the binaries to merge, 20% to be disrupted, and 62% to survive, yielding a theoretical binary fraction of f _theo = 0.38 ± 0.10. The predicted binary fraction with distance from Sgr A* is also consistent with independent observational constraints. These results show that the decline in binarity towards the Galactic centre is driven by stronger tidal forces, and they support an in situ origin of the S cluster in the same event that produced the Fermi bubbles.

The centre of the Galaxy harbours a supermassive black hole, Sgr A*, which is surrounded by a massive star cluster known as the S-cluster. The most extensively studied star in this cluster is the B-type main-sequence S2 star (also known as S0-2). These types of stars are commonly found in binary systems in the Galactic field, but observations do not seem to detect a companion to S2. This absence may be attributed to observational biases or to a dynamically hostile environment caused by phenomena such as tidal disruption or mergers. Using a N -body code with first-order post-Newtonian corrections, we investigate whether S2 can host a stellar or planetary companion. We perform 10 ⁵ simulations adopting uniform distributions for the orbital elements of the companion. Our results show that companions may exist for orbital periods shorter than 100 days, eccentricities below 0.8, and across the full range of mutual inclinations. The number of surviving companions increases with shorter orbital periods, lower eccentricities, and nearly coplanar orbits. We also find that the disruption mechanisms include mergers driven by Lidov–Kozai cycles and breakups that occur when the companion surpasses the Hill radius of its orbit. Finally, we find that the presence of a companion would alter S2’s astrometric signal by no more than 5 μas. Current radial-velocity detection limits constrain viable stellar binary configurations to approximately 4.4% of the simulated cases. Including astrometric limits reduces to 4.3%. Imposing an additional constraint that any companion must have a mass ≲2 M _⊙ (otherwise it would be visible) narrows the fraction of undetectable stellar binaries to just 3.0%.

Within hierarchical triple stellar systems, there exists a tidal process unique to them, known as tertiary tides. In this process, the tidal deformation of a tertiary in a hierarchical triple drains energy from the inner binary, causing the inner binary’s orbit to shrink. Previous work has uncovered the rate at which tertiary tides drain energy from inner binaries, as a function of orbital and tidal parameters, for hierarchical triples in which the orbits are all circular and coplanar. However, not all hierarchical triples have orbits which are circular and coplanar, which requires an understanding of what happens when this condition is relaxed. In this paper, we study how eccentricities affect tertiary tides, and their influence on the subsequent dynamical evolution of the host hierarchical triple. We find that eccentricities in the outer orbit undergo tidal circularisation as quickly as binary tidal synchronisation, and are therefore trivial, but that eccentricities in the inner binary completely change the behaviour of tertiary tides, draining energy from the outer orbit as well as the inner orbit. As with the circular orbit case, tertiary tides become significant when the tertiary is large enough to come close to filling its Roche Lobe, and dominate tidal evolution when interactions between the inner binary pair are weak. Empirical equations that approximate this behaviour are provided for ease of implementing this process in other stellar evolution codes, and the implications of these results are discussed.

Recently, we estimated that about 5 per cent of unstable supermassive black hole triple systems are fundamentally unpredictable. These gargantuan chaotic systems are able to exponentially magnify Planck length perturbations to astronomical scales within their dynamical lifetime. These results were obtained in the zero angular momentum limit, which we naively expected to be the most chaotic. Here, we generalize to weakly and non-hierarchical triple systems with a range of non-zero angular momenta by systematically varying the initial virial ratio. We find the surprising result that increasing the angular momentum enhances their ability to amplify small perturbations, which we tracked up to 90 orders of magnitude using the arbitrary-precision N-body code brutus. This result is partially explained by the fact that triples with higher angular momenta have longer lifetimes, allowing for a prolonged exponential growth. However, we also find that for a fixed lifetime, triples with higher angular momenta can amplify perturbations to larger values, indicating that the Lyapunov exponent is also a function of angular momentum. These empirical results provide targets for three-body theories which invoke chaos drivers such as resonance overlap and punctuated chaos. Comparing the zero angular momentum case to our new ensembles with higher angular momenta, we conclude that the percentage of unpredictable supermassive black hole triples increases up to about 30 per cent. A further increase up to about 50 per cent is reached when considering triples on smaller astrophysical scales. Fundamental unpredictability is thus a generic feature of chaotic, self-gravitating triple populations.

We use analytical and N-body methods to study the capture of field stars by gravitating substructures moving across a galactic environment. The majority of stars captured by a substructure move on temporarily-bound orbits that are lost to galactic tides after a few orbital revolutions. In numerical experiments where a substructure model is immersed into a sea of field particles on a circular orbit, we find a population of particles that remain bound to the substructure potential for indefinitely-long times. This population is absent from substructure models initially placed outside the galaxy on an eccentric orbit. We show that gravitational capture is most efficient in dwarf spheroidal galaxies (dSphs) on account of their low velocity dispersions and high stellar phase-space densities. In these galaxies ‘dark’ sub-subhaloes which do not experience in-situ star formation may capture field stars and become visible as stellar overdensities with unusual properties: (i) they would have a large size for their luminosity, (ii) contain stellar populations indistinguishable from the host galaxy, and (iii) exhibit dark matter (DM)-dominated mass-to-light ratios. We discuss the nature of several ‘anomalous’ stellar systems reported as star clusters in the Fornax and Eridanus II dSphs which exhibit some of these characteristics. DM sub-subhaloes with a mass function |${\rm d}N/{\rm d}M_\bullet \sim M_\bullet ^{-\alpha }$| are expected to generate stellar systems with a luminosity function, |${\rm d}N/{\rm d}M_\star \sim M_\star ^{-\beta }$|⁠, where β = (2α + 1)/3 = 1.6 for α = 1.9. Detecting and characterizing these objects in dSphs would provide unprecedented constraints on the particle mass and cross section of a large range of DM particle candidates.

We investigate the thermodynamics of close encounters between stellar mass black holes (BHs) in the gaseous discs of active galactic nuclei (AGNs), during which binary black holes (BBHs) may form. We consider a suite of 2D viscous hydrodynamical simulations within a shearing box prescription using the Eulerian grid code athena++. We study formation scenarios where the fluid is either an isothermal gas or an adiabatic mixture of gas and radiation in local thermal equilibrium. We include the effects of viscous and shock heating, as well as optically thick cooling. We co-evolve the embedded BHs with the gas, keeping track of the energetic dissipation and torquing of the BBH by gas and inertial forces. We find that compared to the isothermal case, the minidiscs formed around each BH are significantly hotter and more diffuse, though BBH formation is still efficient. We observe massive blast waves arising from collisions between the radiative minidiscs during both the initial close encounter and subsequent periapsis periods for successfully bound BBHs. These ‘disc novae’ have a profound effect, depleting the BBH Hill sphere of gas and injecting energy into the surrounding medium. In analysing the thermal emission from these events, we observe periodic peaks in local luminosity associated with close encounters/periapses, with emission peaking in the optical/near-infrared (IR). In the AGN outskirts, these outbursts can reach 4 per cent of the AGN luminosity in the IR band, with flares rising over 0.5–1 yr. Collisions in different disc regions, or when treated in 3D with magnetism, may produce more prominent flares.

We investigate close encounters by stellar mass black holes (BHs) in the gaseous discs of active galactic nuclei (AGN) as a potential formation channel of binary black holes (BBHs). We perform a series of 2D isothermal viscous hydrodynamical simulations within a shearing box prescription using the Eulerian grid code Athena++. We co-evolve the embedded BHs with the gas keeping track of the energetic dissipation and torquing of the BBH by gas gravitation and inertial forces. To probe the dependence of capture on the initial conditions, we discuss a suite of 345 simulations spanning BBH impact parameter (b) and local AGN disc density (ρ₀). We identify a clear region in b − ρ₀ space where gas assisted BBH capture is efficient. We find that the presence of gas leads to strong energetic dissipation during close encounters between unbound BHs, forming stably bound eccentric BBHs. We find that the gas dissipation during close encounters increases for systems with increased disc density and deeper periapsis passages r_p, fitting a power law such that |$\Delta E \propto \rho _0^{\alpha }r_{\mathrm{p}}^{\beta }$| where {α, β} = {1.01 ± 0.04, −0.43 ± 0.03}. Alternatively, the gas dissipation is approximately ΔE = 4.3M_dv_Hv_p, where M_d is the mass of a single BH minidisc just prior to the encounter when the binary separation is 2r_H (two binary Hill radii), v_H and v_p are the relative BH velocities at 2r_H and at the first closest approach, respectively. We derive a prescription for capture which can be used in semi-analytical models of AGN. We do not find the dissipative dynamics observed in these systems to be in agreement with the simple gas dynamical friction models often used in the literature.

Located at the core of the Galactic centre, the S-star cluster serves as a remarkable illustration of chaos in dynamical systems. The long-term chaotic behaviour of this system can be studied with gravitational N-body simulations. By applying a small perturbation to the initial position of star S5, we can compare the evolution of this system to its unperturbed evolution. This results in two solutions that diverge exponentially, defined by the separation in position space δ_r, with an average Lyapunov timescale of ∼420 yr, corresponding to the largest positive Lyapunov exponent. Even though the general trend of the chaotic evolution is governed in part by the supermassive black hole Sagittarius A^∗ (Sgr A^∗), individual differences between the stars can be noted in the behaviour of their phase-space curves. We present an analysis of the individual behaviour of the stars in this Newtonian chaotic dynamical system. The individuality of their behaviour is evident from offsets in the position space separation curves of the S-stars and the black hole. We propose that the offsets originate from the initial orbital elements of the S-stars, where Sgr A^∗ is considered in one of the focal points of the Keplerian orbits. Methods were considered to find a relation between these elements and the separation in position space. Symbolic regression provides the clearest diagnostics for finding an interpretable expression for the problem. Our symbolic regression model indicates that ⟨δ_r⟩ ∝ e^2.3, implying that the time-averaged individual separation in position space increases rapidly with the initial eccentricity of the S-stars.

Recently we estimated that about 5 percent of supermassive black hole triple systems are fundamentally unpredictable. These gargantuan chaotic systems are able to exponentially magnify Planck length perturbations to astronomical scales within their interaction timescale. These results were obtained in the zero angular momentum limit, which we naively expected to be the most chaotic. Here, we generalise to triple systems with arbitrary angular momenta by systematically varying the initial virial ratio. We find the surprising result that increasing the angular momentum enhances the chaotic properties of triples. This is not only explained by the longer life times, allowing for a prolonged exponential growth, but also the maximum Lyapunov exponent itself increases. For the ensemble of initially virialised triple systems, we conclude that the percentage of unpredictable supermassive black hole triples increases to about 30 percent. A further increase up to about 50 percent is reached when considering triples on smaller astrophysical scales. Fundamental unpredictability is thus a generic feature of chaotic, self-gravitating triple populations.