ALBERT

Description: We have coupled a fast, parametrized star cluster evolution code to a Markov Chain Monte Carlo code to determine the distribution of probable initial conditions of observed star clusters, that may serve as a starting point for future N -body calculations. In this paper, we validate our method by applying it to a set of star clusters which have been studied in detail numerically with N -body simulations and Monte Carlo methods: the Galactic globular clusters M4, 47 Tucanae, NGC 6397, M22, Centauri, Palomar 14 and Palomar 4, the Galactic open cluster M67, and the M31 globular cluster G1. For each cluster, we derive a distribution of initial conditions that, after evolution up to the cluster's current age, evolves to the currently observed conditions. We find that there is a connection between the morphology of the distribution of initial conditions and the dynamical age of a cluster and that a degeneracy in the initial half-mass radius towards small radii is present for clusters that have undergone a core collapse during their evolution. We find that the results of our method are in agreement with N -body and Monte Carlo studies for the majority of clusters. We conclude that our method is able to find reliable posteriors for the determined initial mass and half-mass radius for observed star clusters, and thus forms an suitable starting point for modelling an observed cluster's evolution.

Description: According to Muñoz-Gutiérrez et al. 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 yr, 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 yr. 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.

Description: Approximately 0.2 ± 0.2 of white dwarfs (WDs) show signs of pollution by metals, which is likely due to the accretion of tidally disrupted planetary material. Models invoking planet–planet interactions after WD formation generally cannot explain pollution at cooling times of several Gyr. We consider a scenario in which a planet is perturbed by Lidov–Kozai oscillations induced by a binary companion and exacerbated by stellar mass-loss, explaining pollution at long cooling times. Our computed accretion rates are consistent with observations assuming planetary masses between ~0.01 and 1 M Mars , although non-gravitational effects may already be important for masses 0.3 M Mars . The fraction of polluted WDs in our simulations, ~0.05, is consistent with observations of WDs with intermediate cooling times between ~0.1 and 1 Gyr. For cooling times 0.1 Gyr and 1 Gyr, our scenario cannot explain the high observed pollution fractions of up to 0.7. Nevertheless, our results motivate searches for companions around polluted WDs.

Description: We discuss the events that led to the giant eruption of Eta Carinae, and find that the mid-nineteenth century (in 1838–1843) giant mass-loss outburst has the characteristics of being produced by the merger event of a massive close binary, triggered by the gravitational interaction with a massive third companion star, which is the current binary companion in the Eta Carinae system. We come to this conclusion by a combination of theoretical arguments supported by computer simulations using the Astrophysical Multipurpose Software Environment. According to this model the ~90 M  present primary star of the highly eccentric Eta Carinae binary system is the product of this merger, and its ~30 M  companion originally was the third star in the system. In our model, the Homunculus nebula was produced by an extremely enhanced stellar wind, energized by tidal energy dissipation prior to the merger, which enormously boosted the radiation-driven wind mass-loss. The current orbital plane is then aligned with the equatorial plane of the Homunculus, and the symmetric lobes are roughly aligned with the argument of periastron of the current Eta Carina binary. The merger itself then occurred in 1838, which resulted in a massive asymmetric outflow in the equatorial plane of the Homunculus. The 1843 outburst can in our model be attributed to the subsequent encounter when the companion star (once the outermost star in the triple system) plunges through the bloated envelope of the merger product, once when it passed periastron again. We predict that the system has an excess space velocity of order 50 km s –1 in the equatorial plane of the Homunculus. Our triple model gives a viable explanation for the high runaway velocities typically observed in LBVs.

Description: We study the secular gravitational dynamics of quadruple systems consisting of a hierarchical triple system orbited by a fourth body. These systems can be decomposed into three binary systems with increasing semimajor axes, binaries A, B and C. The Hamiltonian of the system is expanded in ratios of the three binary separations, and orbit averaged. Subsequently, we numerically solve the equations of motion. We study highly hierarchical systems that are well described by the lowest order terms in the Hamiltonian. We find that the qualitative behaviour is determined by the ratio $\mathcal {R}_0$ of the initial Kozai–Lidov (KL) time-scales of the binary pairs AB and BC. If $\mathcal {R}_0\ll 1$ , binaries AB remain coplanar if this is initially the case, and KL eccentricity oscillations in binary B are efficiently quenched. If $\mathcal {R}_0\gg 1$ , binaries AB become inclined, even if initially coplanar. However, there are no induced KL eccentricity oscillations in binary A. Lastly, if $\mathcal {R}_0\sim 1$ , complex KL eccentricity oscillations can occur in binary A that are coupled with the KL eccentricity oscillations in B. Even if binaries A and B are initially coplanar, the induced inclination can result in very high eccentricity oscillations in binary A. These extreme eccentricities could have significant implications for strong interactions such as tidal interactions, gravitational wave dissipation, and collisions and mergers of stars and compact objects. As an example, we apply our results to a planet+moon system orbiting a central star, which in turn is orbited by a distant and inclined stellar companion or planet, and to observed stellar quadruples.

Description: We specify the range to which perturbations penetrate a planetesimal system. Such perturbations can originate from massive planets or from encounters with other stars. The latter can have an origin in the star cluster in which the planetary system was born, or from random encounters once the planetary system has escaped its parental cluster. The probability of a random encounter, either in a star cluster or in the Galactic field depends on the local stellar density, the velocity dispersion and the time spend in that environment. By adopting order of magnitude estimates, we argue that the majority of planetary systems born in open clusters will have a Parking zone , in which planetesimals are affected by encounters in their parental star cluster but remain unperturbed after the star has left the cluster. Objects found in this range of semimajor axis and eccentricity preserve the memory of the encounter that last affected their orbits, and they can therefore be used to reconstruct this encounter. Planetary systems born in a denser environment, such as in a globular cluster are unlikely to have a Parking zone. We further argue that some planetary systems may have a Frozen zone , in which orbits are not affected either by the more inner massive planets or by external influences. Objects discovered in this zone will have preserved information about their formation in their orbital parameters.

Description: We specify the range to which perturbations penetrate a planetesimal system. Such perturbations can originate from massive planets or from encounters with other stars. The latter can have an origin in the star cluster in which the planetary system was born, or from random encounters once the planetary system has escaped its parental cluster. The probability of a random encounter, either in a star cluster or in the Galactic field depends on the local stellar density, the velocity dispersion and the time spend in that environment. By adopting order of magnitude estimates, we argue that the majority of planetary systems born in open clusters will have a Parking zone , in which planetesimals are affected by encounters in their parental star cluster but remain unperturbed after the star has left the cluster. Objects found in this range of semimajor axis and eccentricity preserve the memory of the encounter that last affected their orbits, and they can therefore be used to reconstruct this encounter. Planetary systems born in a denser environment, such as in a globular cluster are unlikely to have a Parking zone. We further argue that some planetary systems may have a Frozen zone , in which orbits are not affected either by the more inner massive planets or by external influences. Objects discovered in this zone will have preserved information about their formation in their orbital parameters.

Description: We compare the observed size distribution of circumstellar discs in the Orion Trapezium cluster with the results of N -body simulations in which we incorporated an heuristic prescription for the evolution of these discs. In our simulations, the sizes of stellar discs are affected by close encounters with other stars (with discs). We find that the observed distribution of disc sizes in the Orion Trapezium cluster is excellently reproduced by truncation due to dynamical encounters alone. The observed distribution appears to be a sensitive measure of the past dynamical history of the cluster, and therewith on the conditions of the cluster at birth. The best comparison between the observed disc-size distribution and the simulated distribution is realized with a cluster of N  = 2500 ± 500 stars with a half-mass radius of about 0.5 pc in virial equilibrium (with a virial ratio of Q  = 0.5, or somewhat colder Q ~= 0.3), and with a density structure according to a fractal dimension of F ~= 1.6. Simulations with these parameters reproduce the observed distribution of circumstellar discs in about 0.2–0.5 Myr. We conclude that the distribution of disk sizes in the Orion Trapezium cluster is the result of dynamical interactions in the early evolution of the cluster.

Description: Transiting circumbinary planets are more easily detected around short-period than long-period binaries, but none have yet been observed by Kepler orbiting binaries with periods shorter than seven days. In triple systems, secular Kozai–Lidov cycles and tidal friction (KLCTF) have been shown to reduce the inner orbital period from ~10 4 to a few days. Indeed, the majority of short-period binaries are observed to possess a third stellar companion. Using secular evolution analysis and population synthesis, we show that KLCTF makes it unlikely for circumbinary transiting planets to exist around short-period binaries. We find the following outcomes. (1) Sufficiently massive planets in tight and/or coplanar orbits around the inner binary can quench the KL evolution because they induce precession in the inner binary. The KLCTF process does not take place, preventing the formation of a short-period binary. (2) Secular evolution is not quenched and it drives the planetary orbit into a high eccentricity, giving rise to an unstable configuration, in which the planet is most likely ejected from the system. (3) Secular evolution is not quenched but the planet survives the KLCTF evolution. Its orbit is likely to be much wider than the currently observed inner binary orbit, and is likely to be eccentric and inclined with respect to the inner binary. These outcomes lead to two main conclusions: (1) it is unlikely to find a massive planet on a tight and coplanar orbit around a short-period binary, and (2) the properties of circumbinary planets in short-period binaries are constrained by secular evolution.

Description: We developed a Keplerian-based Hamiltonian splitting for solving the gravitational N -body problem. This splitting allows us to approximate the solution of a general N -body problem by a composition of multiple, independently evolved two-body problems. While the Hamiltonian splitting is exact, we show that the composition of independent two-body problems results in a non-symplectic non-time-symmetric first-order map. A time-symmetric second-order map is then constructed by composing this basic first-order map with its self-adjoint. The resulting method is precise for each individual two-body solution and produces quick and accurate results for near-Keplerian N -body systems, like planetary systems or a cluster of stars that orbit a supermassive black hole. The method is also suitable for integration of N -body systems with intrinsic hierarchies, like a star cluster with primordial binaries. The superposition of Kepler solutions for each pair of particles makes the method excellently suited for parallel computing; we achieve 64 per cent efficiency for only eight particles per core, but close to perfect scaling for 16 384 particles on a 128 core distributed-memory computer. We present several implementations in sakura , one of which is publicly available via the amuse framework.