arXiv subjects

Tjarda Boekholt

Publications and source records attributed to Tjarda Boekholt.

14 recordsLinked to original sources

arXiv2025

Tertiary Tides with Eccentric Orbits

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.

astro-ph.SR↗

arXiv2024

Gas Assisted Binary Black Hole Formation in AGN Discs

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 local AGN disc density ($ρ_0$) and impact parameter ($b$) space. We identify a clear region in $b - ρ_0$ 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 $ΔE \propto ρ_0^αr_p^β$ where $\{α,β\} = \{1.01\pm0.04,-0.43\pm0.03\}$. Alternatively, the gas dissipation is approximately $ΔE = 4.3 M_\text{d} v_\text{H} v_p$, where $M_\text{d} $ is the mass of a single BH minidisc just prior to the encounter when the binary separation is $2r_\text{H}$ (two binary Hill radii), $v_\text{H}$ and $v_p$ are the relative BH velocities at $2r_\text{H}$ 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.

astro-ph.GA↗

arXiv2024

Disc Novae: Thermodynamics of Gas Assisted Binary Black Hole Formation in AGN Discs

We investigate the thermodynamics of close encounters between stellar mass black holes (BHs) in the gaseous discs of active galactic nuclei (AGN), 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-IR. In the AGN outskirts, these outbursts can reach 4% of the AGN luminosity in the IR band, with flares rising over 0.5-1year. Collisions in different disc regions, or when treated in 3D with magnetism, may produce more prominent flares.

astro-ph.HE↗

arXiv2023

Black Hole Binaries in AGN Accretion Discs II: Gas Effects on Black Hole Satellite Scatterings

The black hole (BH) binaries in active galactic nuclei (AGN) are expected to form mainly through scattering encounters in the ambient gaseous medium. Recent simulations, including our own, have confirmed this formation pathway is highly efficient. We perform 3D smoothed particle hydrodynamics (SPH) simulations of BH scattering encounters in AGN disks. Using a range of impact parameters, we probe the necessary conditions for binary capture and how different orbital trajectories affect the dissipative effects from the gas. We identify a single range of impact parameters, typically of width $\sim0.86-1.59$ binary Hill radii depending on AGN disk density, that reliably leads to binary formation. The periapsis of the first encounter is the primary variable that determines the outcome of the initial scattering. We find an associated power-law between the energy dissipated and the periapsis depth to be $ΔE\propto r^{-b}$ with $b=0.42\pm0.16$, where deeper encounters dissipate more energy. Excluding accretion physics does not significantly alter these results. We identify the region of parameter space in initial energy vs impact parameter where a scattering leads to binary formation. Based on our findings, we provide a ready-to-use analytic criterion that utilises these two pre-encounter parameters to determine the outcome of an encounter, with a reliability rate of >90\%. As the criterion is based directly on our simulations, it provides a reliable and highly physically motivated criterion for predicting binary scattering outcomes which can be used in population studies of BH binaries and mergers around AGN.

astro-ph.HE↗

arXiv2023

Punctuated chaos and the unpredictability of the Galactic center S-star orbital evolution

We investigate the chaotic behavior of the S-star cluster in the Galactic center using precise $N$-body calculations, free from round-off or discretization errors. Our findings reveal that chaos among the Galactic center S-stars arises from close encounters, particularly among pairs and near the massive central body. These encounters induce perturbations, causing sudden changes in the orbital energies of the interacting stars. Consequently, neighboring solutions experience roughly exponential growth in separation. We propose a theory of "punctuated chaos" that describes the S-star cluster's chaotic behavior. This phenomenon results from nearly linear growth in the separation between neighboring orbits after repeated finite perturbations. Each participating star's orbit experiences discrete, abrupt changes in energy due to the perturbations. The cumulative effect of these events is further amplified by the steady drift in orbital phase. In the Galactic center, perturbations originate from coincidental encounters occurring within a distance of $\aplt 100$\,au between at least two stars (in some cases, three stars). Our model satisfactorily explains the observed exponential growth in the 27 S-star cluster. We determine that the S-star system has a Lyapunov time scale of approximately 462 +/-74 years. For the coming millennium, chaos in the S-star cluster will be driven mainly by a few of the closest orbiting stars: S2, S5, S6, S8, S9, S14, S18, S31, S21, S24, S27, S29, and S38.

astro-ph.GA↗

arXiv2023

Black Hole Binary Formation in AGN Discs: From Isolation to Merger

Motivated by the increasing number of detections of merging black holes by LIGO-VIRGO-KAGRA, black hole (BH) binary mergers in the discs of active galactic nuclei (AGN) is investigated as a possible merger channel. In this pathway, BH encounters in the gas disc form mutually bound black hole binary systems through interaction with the gas in the disc and subsequently inspiral through gravitational torques induced by the local gas. To determine the feasibility of this merger pathway, we present the first 3D global hydrodynamic simulations of the formation and evolution of a stellar-mass BH binaries AGN discs with three different AGN disc masses and five different initial radial separations. These 15 simulations show binary capture of prograde and retrograde binaries can be successful in a range of disc densities including cases well below that of a standard radiatively efficient alpha disc, identifying that the majority of these captured binaries are then subsequently hardened by the surrounding gas. The eccentricity evolution depends strongly on the orbital rotation where prograde binaries are governed by gravitational torques form their circumbinary mini-disc, with eccentricities being damped, while for retrograde binaries the eccentricities are excited to > 0.9 by accretion torques. In two cases, retrograde binaries ultimately undergo a close periapsis passage which results in a merger via gravitational waves after only a few thousand binary orbits. Thus, the merger timescale can be far shorter than the AGN disc lifetime. These simulations support an efficient AGN disc merger pathway for BHs.

astro-ph.GA↗

arXiv2022

The paradox of infinitesimal granularity: Chaos and the reversibility of time in Newton's theory of gravity

The fundamental laws of physics are time-symmetric, but our macroscopic experience contradicts this. The time reversibility paradox is partly a consequence of the unpredictability of Newton's equations of motion. We measure the dependence of the fraction of irreversible, gravitational N-body systems on numerical precision and find that it scales as a power law. The stochastic wave packet reduction postulate then introduces fundamental uncertainties in the Cartesian phase space coordinates that propagate through classical three-body dynamics to macroscopic scales within the triple's lifetime. The spontaneous collapse of the wave function then drives the global chaotic behavior of the Universe through the superposition of triple systems (and probably multi-body systems). The paradox of infinitesimal granularity then arises from the superposition principle, which states that any multi-body system is composed of an ensemble of three-body problems.

nlin.CD↗

arXiv2021

Radio astronomy and Space science in Azores: enhancing the Atlantic VLBI infrastructure cluster

Radio astronomy and Space Infrastructures in the Azores have a great scientific and industrial interest because they benefit from a unique geographical location in the middle of the North Atlantic allowing a vast improvement in the sky coverage. This fact obviously has a very high added value for: i) the establishment of space tracking and communications networks for the emergent global small satellite fleets ii) it is invaluable to connect the radio astronomy infrastructure networks in Africa, Europe and America continents using Very Large Baseline Interferometry (VLBI) techniques, iii) it allows excellent potential for monitoring space debris and Near Earth Objects (NEOs). There is in S. Miguel island a 32-metre SATCOM antenna that could be integrated in advanced VLBI networks and be capable of additional Deep Space Network ground support. This paper explores the space science opportunities offered by the upgrade of the S. Miguel 32-metre SATCOM antenna into a world-class infrastructure for radio astronomy and space exploration: it would enable a Deep Space Network mode and would constitute a key space facility for data production, promoting local digital infrastructure investments and the testing of cutting-edge information technologies. Its Atlantic location also enables improvements in angular resolution, provides many baseline in East-West and North-South directions connecting the emergent VLBI stations in America to Europe and Africa VLBI arrays therefore contributing for greater array imaging capabilities especially for sources or well studied fields close to or below the celestial equator, where ESO facilities, ALMA, SKA and its precursors do or will operate and observe in the coming decades.

astro-ph.IM↗

arXiv2020

Gargantuan chaotic gravitational three-body systems and their irreversibility to the Planck length

Chaos is present in most stellar dynamical systems and manifests itself through the exponential growth of small perturbations. Exponential divergence drives time irreversibility and increases the entropy in the system. A numerical consequence is that integrations of the N-body problem unavoidably magnify truncation and rounding errors to macroscopic scales. Hitherto, a quantitative relation between chaos in stellar dynamical systems and the level of irreversibility remained undetermined. In this work we study chaotic three-body systems in free fall initially using the accurate and precise N-body code Brutus, which goes beyond standard double-precision arithmetic. We demonstrate that the fraction of irreversible solutions decreases as a power law with numerical accuracy. This can be derived from the distribution of amplification factors of small initial perturbations. Applying this result to systems consisting of three massive black holes with zero total angular momentum, we conclude that up to five percent of such triples would require an accuracy of smaller than the Planck length in order to produce a time-reversible solution, thus rendering them fundamentally unpredictable.

astro-ph.IM↗

arXiv2019

Newton vs the machine: solving the chaotic three-body problem using deep neural networks

Since its formulation by Sir Isaac Newton, the problem of solving the equations of motion for three bodies under their own gravitational force has remained practically unsolved. Currently, the solution for a given initialization can only be found by performing laborious iterative calculations that have unpredictable and potentially infinite computational cost, due to the system's chaotic nature. We show that an ensemble of solutions obtained using an arbitrarily precise numerical integrator can be used to train a deep artificial neural network (ANN) that, over a bounded time interval, provides accurate solutions at fixed computational cost and up to 100 million times faster than a state-of-the-art solver. Our results provide evidence that, for computationally challenging regions of phase-space, a trained ANN can replace existing numerical solvers, enabling fast and scalable simulations of many-body systems to shed light on outstanding phenomena such as the formation of black-hole binary systems or the origin of the core collapse in dense star clusters.

astro-ph.GA↗

arXiv2018

Towards completing Planetary Systems: The role of minor bodies on life growth and survival

The search for extrasolar planets in the past decades has shown that planets abound in the Solar neighborhood. While we are still missing an Earth twin, the forthcoming space missions and ground-based instrumentation are already driven to achieve this goal. But, in order to fully understand the conditions for life appearing in the Solar System, we still miss some pieces of the planetary system jigsaw puzzle, namely a deeper understanding of the minor bodies. Trojans, moons, and comets are tracers of the formation and evolution processes of planetary systems. These missing pieces are also critical to understand the emergence and evolution of life over millions of years. With the large crop of planetary systems discovered so far and yet to be detected with the forthcoming missions, the hunt for minor bodies in extrasolar systems is a natural continuation of our search for real Solar System- and, in particular, Earth- analogs. This white paper is focused on detection of these minor components and their relevance in the emergence, evolution and survival of life.

astro-ph.EP↗

arXiv2018

Numerical verification of the microscopic time reversibility of Newton's equations of motion: Fighting exponential divergence

Numerical solutions to Newton's equations of motion for chaotic self gravitating systems of more than 2 bodies are often regarded to be irreversible. This is due to the exponential growth of errors introduced by the integration scheme and the numerical round-off in the least significant figure. This secular growth of error is sometimes attributed to the increase in entropy of the system even though Newton's equations of motion are strictly time reversible. We demonstrate that when numerical errors are reduced to below the physical perturbation and its exponential growth during integration the microscopic reversibility is retrieved. Time reversibility itself is not a guarantee for a definitive solution to the chaotic N-body problem. However, time reversible algorithms may be used to find initial conditions for which perturbed trajectories converge rather than diverge. The ability to calculate such a converging pair of solutions is a striking illustration which shows that it is possible to compute a definitive solution to a highly unstable problem. This works as follows: If you (i) use a code which is capable of producing a definitive solution (and which will therefore handle converging pairs of solutions correctly), (ii) use it to study the statistical result of some other problem, and then (iii) find that some other code produces a solution S with statistical properties which are indistinguishable from those of the definitive solution, then solution S may be deemed veracious.

astro-ph.IM↗

arXiv2014

On the Reliability of N-body Simulations

The general consensus in the N-body community is that statistical results of an ensemble of collisional N-body simulations are accurate, even though individual simulations are not. A way to test this hypothesis is to make a direct comparison of an ensemble of solutions obtained by conventional methods with an ensemble of true solutions. In order to make this possible, we wrote an N-body code called Brutus, that uses arbitrary-precision arithmetic. In combination with the Bulirsch--Stoer method, Brutus is able to obtain converged solutions, which are true up to a specified number of digits. We perform simulations of democratic 3-body systems, where after a sequence of resonances and ejections, a final configuration is reached consisting of a permanent binary and an escaping star. We do this with conventional double-precision methods, and with Brutus; both have the same set of initial conditions and initial realisations. The ensemble of solutions from the conventional simulations is compared directly to that of the converged simulations, both as an ensemble and on an individual basis to determine the distribution of the errors. We find that on average at least half of the conventional simulations diverge from the converged solution, such that the two solutions are microscopically incomparable. For the solutions which have not diverged significantly, we observe that if the integrator has a bias in energy and angular momentum, this propagates to a bias in the statistical properties of the binaries. In the case when the conventional solution has diverged onto an entirely different trajectory in phase-space, we find that the errors are centred around zero and symmetric; the error due to divergence is unbiased, as long as the time-step parameter, eta <= 2^(-5) and when simulations which violate energy conservation by more than 10% are excluded.

astro-ph.IM↗

arXiv2014

On the minimal accuracy required for simulating self-gravitating systems by means of direct N-body methods

The conservation of energy, linear momentum and angular momentum are important drivers for our physical understanding of the evolution of the Universe. These quantities are also conserved in Newton's laws of motion under gravity \citep{Newton:1687}. Numerical integration of the associated equations of motion is extremely challenging, in particular due to the steady growth of numerical errors (by round-off and discrete time-stepping, \cite{1981PAZh....7..752B,1993ApJ...415..715G,1993ApJ...402L..85H,1994LNP...430..131M}) and the exponential divergence \citep{1964ApJ...140..250M,2009MNRAS.392.1051U} between two nearby solution. As a result, numerical solutions to the general N-body problem are intrinsically questionable \citep{2003gmbp.book.....H,1994JAM....61..226L}. Using brute force integrations to arbitrary numerical precision we demonstrate empirically that ensembles of different realizations of resonant 3-body interactions produce statistically indistinguishable results. Although individual solutions using common integration methods are notoriously unreliable, we conjecture that an ensemble of approximate 3-body solutions accurately represents an ensemble of true solutions, so long as the energy during integration is conserved to better than 1/10. We therefore provide an independent confirmation that previous work on self-gravitating systems can actually be trusted, irrespective of the intrinsic chaotic nature of the N-body problem.

astro-ph.IM↗