arXiv subjects

Tjarda C. N. Boekholt

Publications and source records attributed to Tjarda C. N. Boekholt.

16 recordsLinked to original sources

arXiv2026

A solution to the paradox of youth in the Galactic centre

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 $f_{\mathrm{obs}} = 0.43 \pm 0.09$ 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 $f_{\mathrm{theo}} = 0.38 \pm 0.10$. 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.

astro-ph.GA↗

arXiv2025

Dynamical constraints on the S2 (S0-2) star possible companions

The center of the Galaxy harbors 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\,μ{\rm as}$. 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 $\lesssim 2\,M_{\odot}$ (otherwise it would be visible) narrows the fraction of undetectable stellar binaries to just 3.0\%.

astro-ph.GA↗

arXiv2024

Capture of field stars by dark substructures

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 $d N/d M_\bullet\sim M_\bullet^{-α}$ are expected to generate stellar systems with a luminosity function, $d N/d M_\star\sim M_\star^{-β}$, 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.

astro-ph.GA↗

arXiv2024

Isles of regularity in a sea of chaos amid the gravitational three-body problem

The three-body problem (3BP) poses a longstanding challenge in physics and celestial mechanics. Despite the impossibility of obtaining general analytical solutions, statistical theories have been developed based on the ergodic principle. This assumption is justified by chaos, which is expected to fully mix the accessible phase space of the 3BP. This study probes the presence of regular (i.e. non chaotic) trajectories within the 3BP and assesses their impact on statistical escape theories. Using numerical simulations, we establish criteria for identifying regular trajectories and analyse their impact on statistical outcomes. Our analysis reveals that regular trajectories occupy up to 32% of the phase space, and their outcomes defy the predictions of statistical escape theories. The coexistence of regular and chaotic regions at all scales is characterized by a multi-fractal behaviour. Integration errors manifest as numerical chaos, artificially enhancing the mixing of the phase space and affecting the reliability of individual simulations, yet preserving the statistical correctness of an ensemble of realizations. Our findings underscore the challenges in applying statistical escape theories to astrophysical problems, as they may bias results by excluding the outcome of regular trajectories. This is particularly important in the context of formation scenarios of gravitational wave mergers, where biased estimates of binary eccentricity can significantly impact estimates of coalescence efficiency and detectable eccentricity.

astro-ph.EP↗

arXiv2024

Individual chaotic behaviour of the S-stars in the Galactic centre

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 the two solutions diverging exponentially, defined by the separation in position space $δ_{r}$, with an average Lyapunov timescale of $\sim$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 $\rm A^{*}$ (Sgr $\rm A^{*}$), 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 $\rm A^{*}$ is considered in one of the focal points of the Keplerian orbits. Methods were considered to find a relationship between these elements and the separation in position space. Symbolic regression turns out to provide the clearest diagnostics for finding an interpretable expression for the problem. Our symbolic regression model indicates that $\left\langleδ_r\right\rangle \propto e^{2.3}$, implying that the time-averaged individual separation in position space increases rapidly with the initial eccentricity of the S-stars.

astro-ph.GA↗

arXiv2023

Gargantuan chaotic gravitational three-body systems II. Dependence on angular momentum and astrophysical scale

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.

gr-qc↗

arXiv2023

Punctuated Chaos and Indeterminism in Self-gravitating Many-body Systems

Dynamical chaos is a fundamental manifestation of gravity in astrophysical, many-body systems. The spectrum of Lyapunov exponents quantifies the associated exponential response to small perturbations. Analytical derivations of these exponents are critical for understanding the stability and predictability of observed systems. This essay presents a new model for chaos in systems with eccentric and crossing orbits. Here, exponential divergence is not a continuous process but rather the cumulative effect of an ever-increasing linear response driven by discrete events at regular intervals, i.e., punctuated chaos. We show that long-lived systems with punctuated chaos can magnify Planck length perturbations to astronomical scales within their lifetime, rendering them fundamentally indeterministic.

astro-ph.IM↗

arXiv2022

Reversible time-step adaptation for the integration of few-body systems

The time step criterion plays a crucial role in direct N-body codes. If not chosen carefully, it will cause a secular drift in the energy error. Shared, adaptive time step criteria commonly adopt the minimum pairwise time step, which suffers from discontinuities in the time evolution of the time step. This has a large impact on the functioning of time step symmetrisation algorithms. We provide new demonstrations of previous findings that a smooth and weighted average over all pairwise time steps in the N-body system, improves the level of energy conservation. Furthermore, we compare the performance of 27 different time step criteria, by considering 3 methods for weighting time steps and 9 symmetrisation methods. We present performance tests for strongly chaotic few-body systems, including unstable triples, giant planets in a resonant chain, and the current Solar System. We find that the harmonic symmetrisation methods (methods A3 and B3 in our notation) are the most robust, in the sense that the symmetrised time step remains close to the time step function. Furthermore, based on our Solar System experiment, we find that our new weighting method based on direct pairwise averaging (method W2 in our notation), is slightly preferred over the other methods.

astro-ph.IM↗

arXiv2022

On the Jacobi capture origin of binaries with applications to the Earth-Moon system and black holes in galactic nuclei

Close encounters between two bodies in a disc often result in a single orbital deflection. However, within their Jacobi volumes, where the gravitational forces between the two bodies and the central body become competitive, temporary captures with multiple close encounters become possible outcomes: a Jacobi capture. We perform 3-body simulations in order to characterise the dynamics of Jacobi captures in the plane. We find that the phase space structure resembles a Cantor-like set with a fractal dimension of about 0.4. The lifetime distribution decreases exponentially, while the distribution of the closest separation follows a power law with index 0.5. In our first application, we consider the Jacobi capture of the Moon. We demonstrate that both tidal captures and giant impacts are possible outcomes. The impact speed is well approximated by a parabolic encounter, while the impact angles follow that of a uniform beam on a circular target. Jacobi captures at larger heliocentric distances are more likely to result in tidal captures. In our second application, we find that Jacobi captures with gravitational wave dissipation can result in the formation of binary black holes in galactic nuclei. The eccentricity distribution is approximately super-thermal and includes both prograde and retrograde orientations. We conclude that dissipative Jacobi captures form an efficient channel for binary formation, which motivates further research into establishing the universality of Jacobi captures across multiple astrophysical scales.

astro-ph.EP↗

arXiv2022

A direct N-body integrator for modelling the chaotic, tidal dynamics of multi-body extrasolar systems: TIDYMESS

Tidal dissipation plays an important role in the dynamical evolution of moons, planets, stars and compact remnants. The interesting complexity originates from the interplay between the internal structure and external tidal forcing. Recent and upcoming observing missions of exoplanets and stars in the Galaxy help to provide constraints on the physics of tidal dissipation. It is timely to develop new N-body codes, which allow for experimentation with various tidal models and numerical implementations. We present the open-source N-body code TIDYMESS, which stands for ``TIdal DYnamics of Multi-body ExtraSolar Systems''. This code implements a creep deformation law for the bodies, parametrized by their fluid Love numbers and fluid relaxation times. Due to tidal and centrifugal deformations, we approximate the general shape of a body to be an ellipsoid. We calculate the associated gravitational field to quadruple order, from which we derive the gravitational accelerations and torques. The equations of motion for the orbits, spins and deformations are integrated directly using a fourth-order integration method based on a symplectic composition. We implement a novel integration method for the deformations, which allows for a time step solely dependent on the orbits, and not on the spin periods or fluid relaxation times. This feature greatly speeds up the calculations, while also improving the consistency when comparing different tidal regimes. We demonstrate the capabilities and performance of TIDYMESS, particularly in the niche regime of parameter space where orbits are chaotic and tides become non-linear.

astro-ph.IM↗

arXiv2021

Chaos in self-gravitating many-body systems: Lyapunov time dependence of $N$ and the influence of general relativity

In self-gravitating $N$-body systems, small perturbations introduced at the start, or infinitesimal errors that are produced by the numerical integrator or are due to limited precision in the computer, grow exponentially with time. For Newton's gravity, we confirm earlier results that for relatively homogeneous systems, this rate of growth per crossing time increases with $N$ up to $N \sim 30$, but that for larger systems, the growth rate has a weaker scaling with $N$. For concentrated systems, however, the rate of exponential growth continues to scale with $N$. In relativistic self-gravitating systems, the rate of growth is almost independent of $N$. This effect, however, is only noticeable when the system's mean velocity approaches the speed of light to within three orders of magnitude. The chaotic behavior of systems with more than a dozen bodies for the usually adopted approximation of only solving the pairwise interactions in the Einstein-Infeld-Hoffmann equation of motion is qualitatively different than when the interaction terms (or cross terms) are taken into account. This result provides a strong motivation for follow-up studies on the microscopic effect of general relativity on orbital chaos, and on the influence of higher-order cross-terms in the Taylor-series expansion of the Einstein-Infeld-Hoffmann equations of motion.

nlin.CD↗

arXiv2021

The relativistic Pythagorean three-body problem

We study the influence of relativity on the chaotic properties and dynamical outcomes of an unstable triple system; the Pythagorean three-body problem. To this end, we extend the Brutus N-body code to include Post-Newtonian pairwise terms up to 2.5 order, and the first order Taylor expansion to the Einstein-Infeld-Hoffmann equations of motion. The degree to which our system is relativistic depends on the scaling of the total mass (the unit size was 1 parsec). Using the Brutus method of convergence, we test for time-reversibility in the conservative regime, and demonstrate that we are able to obtain definitive solutions to the relativistic three-body problem. It is also confirmed that the minimal required numerical accuracy for a successful time-reversibility test correlates with the amplification factor of an initial perturbation. When we take into account dissipative effects through gravitational wave emission, we find that the duration of the resonance, and the amount of exponential growth of small perturbations depend on the mass scaling. For a unit mass <= 10 MSun, the system behavior is indistinguishable from the Newtonian case, and the resonance always ends in a binary and one escaping body. For a mass scaling up to 1e7 MSun, relativity gradually becomes more prominent, but the majority of the systems still dissolve. The first mergers start to appear for a mass of ~1e5 MSun, and between 1e7 MSun and 1e9 MSun all systems end prematurely in a merger. These mergers are preceded by a gravitational wave driven in-spiral. For a mass scaling >= 1e9 MSun, all systems result in a gravitational wave merger upon the first close encounter. Relativistic three-body encounters thus provide an efficient pathway for resolving the final parsec problem. The onset of mergers at the characteristic mass scale of 1e7 MSun potentially leaves an imprint in the mass function of supermassive black holes.

astro-ph.IM↗

arXiv2021

Formation of supermassive black hole seeds in nuclear star clusters via gas accretion and runaway collisions

More than two hundred supermassive black holes (SMBHs) of masses $\gtrsim 10^9\,\mathrm{M_{\odot}}$ have been discovered at $z \gtrsim 6$. One promising pathway for the formation of SMBHs is through the collapse of supermassive stars (SMSs) with masses $\sim 10^{3-5}\,\mathrm{M_{\odot}}$ into seed black holes which could grow upto few times $10^9\,\mathrm{M_{\odot}}$ SMBHs observed at $z\sim 7$. In this paper, we explore how SMSs with masses $\sim 10^{3-5}\,\mathrm{M_{\odot}}$ could be formed via gas accretion and runaway stellar collisions in high-redshift, metal-poor nuclear star clusters (NSCs) using idealised N-body simulations. We explore physically motivated accretion scenarios, e.g. Bondi-Hoyle-Lyttleton accretion and Eddington accretion, as well as simplified scenarios such as constant accretions. While gas is present, the accretion timescale remains considerably shorter than the timescale for collisions with the most massive object (MMO). However, overall the timescale for collisions between any two stars in the cluster can become comparable or shorter than the accretion timescale, hence collisions still play a crucial role in determining the final mass of the SMSs. We find that the problem is highly sensitive to the initial conditions and our assumed recipe for the accretion, due to the highly chaotic nature of the problem. The key variables that determine the mass growth mechanism are the mass of the MMO and the gas reservoir that is available for the accretion. Depending on different conditions, SMSs of masses $\sim10^{3-5} \,\mathrm{M_{\odot}}$ can form for all three accretion scenarios considered in this work.

astro-ph.GA↗

arXiv2021

Effect of mass loss due to stellar winds on the formation of supermassive black hole seeds in dense nuclear star clusters

The observations of high redshifts quasars at $z\gtrsim 6$ have revealed that supermassive black holes (SMBHs) of mass $\sim 10^9\,\mathrm{M_{\odot}}$ were already in place within the first $\sim$ Gyr after the Big Bang. Supermassive stars (SMSs) with masses $10^{3-5}\,\mathrm{M_{\odot}}$ are potential seeds for these observed SMBHs. A possible formation channel of these SMSs is the interplay of gas accretion and runaway stellar collisions inside dense nuclear star clusters (NSCs). However, mass loss due to stellar winds could be an important limitation for the formation of the SMSs and affect the final mass. In this paper, we study the effect of mass loss driven by stellar winds on the formation and evolution of SMSs in dense NSCs using idealised N-body simulations. Considering different accretion scenarios, we have studied the effect of the mass loss rates over a wide range of metallicities $Z_\ast=[.001-1]\mathrm{Z_{\odot}}$ and Eddington factors $f_{\rm Edd}=L_\ast/L_{\mathrm{Edd}}=0.5,0.7,\,\&\, 0.9$. For a high accretion rate of $10^{-4}\,\mathrm{M_{\odot}yr^{-1}}$, SMSs with masses $\gtrsim 10^3\MSun$ could be formed even in a high metallicity environment. For a lower accretion rate of $10^{-5}\,\mathrm{M_{\odot}yr^{-1}}$, SMSs of masses $\sim 10^{3-4}\,\mathrm{M_{\odot}}$ can be formed for all adopted values of $Z_\ast$ and $f_{\rm Edd}$, except for $Z_\ast=\mathrm{Z_{\odot}}$ and $f_{\rm Edd}=0.7$ or 0.9. For Eddington accretion, SMSs of masses $\sim 10^3\,\mathrm{M_{\odot}}$ can be formed in low metallicity environments with $Z_\ast\lesssim 0.01\mathrm{Z_{\odot}}$. The most massive SMSs of masses $\sim 10^5\,\mathrm{M_{\odot}}$ can be formed for Bondi-Hoyle accretion in environments with $Z_\ast \lesssim 0.5\mathrm{Z_{\odot}}$.

astro-ph.GA↗

arXiv2017

Dynamical Ejections of Stars due to an Accelerating Gas Filament

Observations of the Orion-A integral shaped filament (ISF) have shown indications of an oscillatory motion of the gas filament. This evidence is based on both the wave-like morphology of the filament as well as the kinematics of the gas and stars, where the characteristic velocities of the stars require a dynamical heating mechanism. As proposed by Stutz and Gould (2016), such a heating mechanism (the "Slingshot") may be the result of an oscillating gas filament in a gas-dominated (as opposed to stellar-mass dominated) system. Here we test this hypothesis with the first stellar-dynamical simulations in which the stars are subjected to the influence of an oscillating cylindrical potential. The accelerating, cylindrical background potential is populated with a narrow distribution of stars. By coupling the potential to N-body dynamics, we are able to measure the influence of the potential on the stellar distribution. The simulations provide evidence that the slingshot mechanism can successfully reproduce several stringent observational constraints. These include the stellar spread (both in projected position and in velocity) around the filament, the symmetry in these distributions, and a bulk motion of the stars with respect to the filament. Using simple considerations we show that star-star interactions are incapable of reproducing these spreads on their own when properly accounting for the gas potential. Thus, properly accounting for the gas potential is essential for understanding the dynamical evolution of star forming filamentary systems in the era of Gaia.

astro-ph.GA↗

arXiv2016

The Origin of Chaos in the Orbit of Comet 1P/Halley

According to Munoz-Gutierrez et al. (2015) the orbit of comet 1P/Halley is chaotic with a surprisingly small Lyapunov time scale of order its orbital period. In this work we analyse the origin of chaos in Halley's orbit and the growth of perturbations, in order to get a better understanding of this unusually short time scale. We perform N-body simulations to model Halley's orbit in the Solar System and measure the separation between neighbouring trajectories. To be able to interpret the numerical results, we use a semi-analytical map to demonstrate different growth modes, i.e. linear, oscillatory or exponential, and transitions between these modes. We find the Lyapunov time scale of Halley's orbit to be of order 300 years, which is significantly longer than previous estimates in the literature. This discrepancy could be due to the different methods used to measure the Lyapunov time scale. A surprising result is that next to Jupiter, also encounters with Venus contribute to the exponential growth in the next 3000 years. Finally, we note an interesting application of the sub-linear, oscillatory growth mode to an ensemble of bodies moving through the Solar System. Whereas in the absence of encounters with a third body the ensemble spreads out linearly in time, the accumulation of weak encounters can increase the lifetime of such systems due to the oscillatory behaviour.

astro-ph.EP↗