arXiv subjects

S. Portegies Zwart

Publications and source records attributed to S. Portegies Zwart.

At least 19 recordsLinked to original sources

arXiv2026

The exponential growth of infinitesimal perturbations in the long-term evolution of simulated galaxies

Self-gravitating systems of $N$ particles are chaotic. We study how chaotic the Galaxy is, and what the consequences are. We therefore simulated the dynamical evolution of a galaxy-scale distribution of point masses in order to measure the degree of chaos in such a system. These calculations were performed using the softened gravitational $N$-body tree-code Bonsai, with up to 40 million equal-mass particles. Smaller simulations were performed to establish the scaling of the Lyapunov time $t_L$ with $N$. We established the relations between the degree of chaos, the number of particles, and the softening length in the gravitational force calculation of large-scale $N$-body simulations. The moment in which the bar forms appears to be insensitive to infinitesimal perturbations to the initial realisation. In contrast, the bar strength and its further evolution sensitively depend on these perturbations. Interestingly enough, the maximum in the run-to-run variation in the bar strength is at about the maximum bar strength, and it drops until the bar buckles. The galaxies we simulated are highly chaotic, but the softening in the simulations suppresses chaos. Still, our models show considerable variations in the macroscopic behaviour caused by infinitesimal perturbations to the initial conditions. Real galaxies, however, are expected to be orders of magnitude more chaotic than our simulations, and we are unable to quantify the consequences of this. Smooth galactic potentials for studying individual stellar orbits should be handled with caution on timescales longer than the Lyapunov time. When we extrapolate our results to the number of stars in the Galaxy without planets and other minor bodies, we conclude that Milky Way-size galaxies are chaotic on a timescale of less than a million years.

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arXiv2026

Spin-disc misalignment drives periodic accretion and pulse profile asymmetry in X-ray pulsars

We investigate magnetospheric accretion in disc-fed X-ray pulsars assuming that the neutron star spin axis is not perpendicular to the disc plane. We focus on X-ray pulsars, where a geometrically thin disc is truncated far from the stellar surface the channelled part of the accretion flow is expected to be guided by the large-scale dipolar magnetic field after coupling to it near the disc-magnetosphere boundary. Using numerical simulations of plasma motion from the inner disc edge to the neutron star surface, we show that a finite inclination between the disc normal and the stellar spin axis leads to periodic modulation of the mass accretion rate onto the magnetic poles even for a steady mass supply through the disc. This purely geometrical effect arises because stellar rotation changes the orientation of the magnetosphere relative to the disc, producing phase-dependent mass loading of magnetic field lines. The amplitude and shape of the modulation are determined by the system geometry and by the ratio of the stellar spin period to the flow time through the magnetosphere. The resulting variability affects the structure and luminosity of emitting regions near the neutron star surface and leads to asymmetric X-ray pulse profiles. Even without intrinsic asymmetries of the emission regions, this mechanism breaks the time-reversal symmetry expected for stationary accretion and naturally contributes to the observed asymmetry of pulse profiles and phase-resolved spectral features. The effect may also be relevant for ULX pulsars, where intrinsic accretion rate modulation can help preserve strong pulsations in the presence of geometric beaming.

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arXiv2025

A Systematic North-South asymmetry in the Steady-state Climate of rapidly-rotating Oblique Water Worlds

Planetary obliquity (axial tilt) plays an important role in regulating the climate evolution and habitability of water-covered planets. Despite the suspicion of large obliquities in several exoplanetary systems, this phenomenon remains hard to observe directly. We aimed to study the effect of mass, obliquity, and rotation on the steady state climate of water-covered planets. We simulated the climate evolution of such planets with varying obliquities, rotational speed, and mass using a general circulation model (GCM) of intermediate complexity, assuming aqua-planet configurations. High obliquity supports an asymmetry between the equilibrium climatological conditions in the northern and southern hemispheres. The polar temperature ratio deviates further from unity with increasing obliquity and rotation rate. Cloud coverage patterns also shift with obliquity, displaying distinct latitudinal bands and increased cloudiness in the warmer hemisphere. The climate of habitable-zone aqua-planets turns out to be sensitive to changes in obliquity and rotation rate, but are independent of planet mass. Our results highlight the importance of considering these factors when assessing the surface conditions of exoplanets. As a consequence, surface condition asymmetries in water-world exo-planets can be used to infer the planet's obliquity and rotation rate.

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arXiv2024

Magnetospheric Flows in X-ray Pulsars I: Instability at super-Eddington regime of accretion

Within the magnetospheric radius, the geometry of accretion flow in X-ray pulsars is shaped by a strong magnetic field of a neutron star. Starting at the magnetospheric radius, accretion flow follows field lines and reaches the stellar surface in small regions located close to the magnetic poles of a star. At low mass accretion rates, the dynamic of the flow is determined by gravitational attraction and rotation of the magnetosphere due to the centrifugal force. At the luminosity range close to the Eddington limit and above it, the flow is additionally affected by the radiative force. We construct a model simulating accretion flow dynamics over the magnetosphere, assuming that the flow strictly follows field lines and is affected by gravity, radiative and centrifugal forces only. The magnetic field of a NS is taken to be dominated by the dipole component of arbitrary inclination with respect to the accretion disc plane. We show that accretion flow becomes unstable at high mass accretion rates and tends to fluctuate quasi-periodically with a typical period comparable to the free-fall time from the inner disc radius. The inclination of a magnetic dipole with respect to the disc plane and strong anisotropy of X-ray radiation stabilise the mass accretion rate at the poles of a star, but the surface density of material covering the magnetosphere fluctuates even in this case.

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arXiv2022

A Monte Carlo study of early gas expulsion and evolution of star clusters: new simulations with the MOCCA code in the AMUSE framework

We introduce a new prescription for the evolution of globular clusters (GCs) during the initial embedded gas phase into a Monte Carlo method. With a simplified version of the Monte Carlo MOCCA code embedded in the AMUSE framework, we study the survival of GCs after the removal of primordial gas. We first test our code and show that our results for the evolution of mass and Lagrangian radii are in good agreement with those obtained with N-body simulations. The Monte Carlo code enables a more rapid exploration of the evolution of systems with a larger number of stars than N-body simulations. We have carried out a new survey of simulations to explore the evolution of globular clusters with up to $N = 500000$ stars for a range of different star formation efficiencies and half-mass radii. Our study shows the range of initial conditions leading to the clusters' dissolution and those for which the clusters can survive this early evolutionary phase.

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arXiv2022

Stellar triples on the edge; Comprehensive overview of the evolution of destabilised triples leading to stellar and binary exotica

Hierarchical triple stars are ideal laboratories for studying the interplay between orbital dynamics and stellar evolution. Both stellar wind mass loss and three-body dynamics cooperate to destabilise triples, which can lead to a variety of astrophysical exotica. So far our understanding of their evolution was mainly built upon results from extensive binary-single scattering experiments. Starting from generic initial conditions, we evolve an extensive set of hierarchical triples using a combination of the triple evolution code TRES and an N-body code. We find that the majority of triples preserve their hierarchy throughout their evolution, which is in contradiction with the commonly adopted picture that unstable triples always experience a chaotic, democratic resonant interaction. The duration of the unstable phase is much longer than expected, so that stellar evolution cannot be neglected. Typically an unstable triple dissolve into a single star and a binary; sometimes democratically (the initial hierarchy is lost and the lightest body usually escapes), but also in a hierarchical way (the tertiary is ejected in a slingshot, independent of its mass). Collisions are common, and mostly involve the two original inner binary components still on the main-sequence. This contradicts the idea that collisions with a giant during democratic encounters dominate. Together with collisions in stable triples, we find that triple evolution is the dominant mechanism for stellar collisions in the Milky Way. Furthermore, our simulations produce runaway and walk-away stars with speeds up to several tens km/s, with a maximum of a few 100km/s. We suggest that destabilised triples can alleviate the tension behind the origin of the observed run-away stars. Lastly, we present a promising indicator to make general predictions for the fate of a specific triple, based on the initial inclination of the system.

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arXiv2022

Survey of Orion Disks with ALMA (SODA) I: Cloud-level demographics of 873 protoplanetary disks

(Abridged) Surveys of protoplanetary disks in nearby star-forming regions (SFRs) have provided important information on their demographics, but due to their sample sizes, they cannot be used to study how disk properties vary with the environment. We conduct a survey of the unresolved millimeter continuum emission of 873 protoplanetary disks identified by Spitzer in the L1641 and L1647 regions of the Orion A cloud. This is the largest such survey yet, allowing us to identify even weak trends in the median disk mass as a function of position in the cloud and cluster membership. The sample detection rates and median masses are also compared to those of nearby (<300 pc) SFRs. The sample was observed with ALMA at 225 GHz, with a median rms of 0.08 mJy/beam (1.5 M$_{\oplus}$). The data were reduced and imaged using an innovative parallel data processing approach. We detect 58% (502/873) of the observed disks. This includes 20 disks with dust masses >100 M$_{\oplus}$, and two objects associated with extended dust emission. We infer a median disk dust mass in the full sample of $2.2^{+0.2}_{-0.2}$ M$_{\oplus}$. In L1641 and L1647 median dust masses are $2.1^{+0.2}_{-0.2}$ M$_{\oplus}$ and $2.6^{+0.4}_{-0.5}$ M$_{\oplus}$, respectively. The disk mass distribution of the full sample is similar to that of nearby low-mass SFRs at similar ages of 1-3 Myr. We find only weak trends in disk (dust) masses with galactic longitude and between the YSO clusters identified in the sample, with median masses varying by $\lesssim$ 50%. Age differences may explain these median disk mass variations. Apart from this, disk masses are essentially constant at scales of ~100 pc. This also suggests that the majority of disks, even in different SFRs, are formed with similar initial masses and evolve at similar rates, assuming no external irradiation, with disk mass loss rates of $\sim 10^{-8}$ M$_{\odot}$/yr.

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arXiv2020

APEX-SEPIA660 Early Science: Gas at densities above $10^7$ cm$^{-3}$ towards OMC-1

Context. The star formation rates and stellar densities found in young massive clusters suggest that these stellar systems originate from gas at densities n(H$_2$) $>10^7$ cm$^{-3}$. Until today, however, the physical characterization of this ultra high density material remains largely unconstrained in observations. Aims. We investigated the density properties of the star-forming gas in the OMC-1 region located in the vicinity of the Orion Nebula Cluster (ONC). Methods. We mapped the molecular emission at 652 GHz in OMC-1 as part of the APEX-SEPIA660 Early Science. Results. We detect bright and extended N$_2$H$^+$ (J=7-6) line emission along the entire OMC-1 region. Comparisons with previous ALMA data of the (J=1-0) transition and radiative transfer models indicate that the line intensities observed in this N$_2$H$^+$ (7-6) line are produced by large mass reservoirs of gas at densities n(H$_2$) $>10^7$ cm$^{-3}$. Conclusions. The first detection of this N$_2$H$^+$ (7-6) line at parsec-scales demonstrates the extreme density conditions of the star-forming gas in young massive clusters such as the ONC. Our results highlight the unique combination of sensitivity and mapping capabilities of the new SEPIA660 receiver for the study of the ISM properties at high frequencies.

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arXiv2020

The evolution of stellar triples: The most common evolutionary pathways

Many stars do not live alone, but instead have one or more stellar companions. Observations show that these binaries, triples and higher-order multiples are common. Whereas the evolution of single stars and binaries have been studied extensively, the same is not true for the evolution of stellar triples. To fill this gap in our general understanding of stellar lives, we aim to systematically explore the long-term evolution of triples and to map out the most common evolutionary pathways that triples go through. We quantitatively study how triples evolve, which processes are most relevant, and how this differs from binary evolution. We simulate the evolution of several large populations of triples with a population synthesis approach. We make use of the triple evolution code TRES to simulate the evolution of each triple in a consistent way; including three-body dynamics (based on the secular approach), stellar evolution and their mutual influences. We simulate the evolution of the system up until mass transfer starts, the system becomes dynamically unstable, or a Hubble time has passed. We find that stellar interactions are common in triples. Compared to a binary population, we find that the fraction of systems that can undergo mass transfer is about 2 to 3 times larger in triples. Moreover, whereas in binaries the orbits typically reach circularisation before Roche-lobe overflow, this is not true anymore in triples. In our simulations, about 40% of systems retain an eccentric orbit. Additionally, we discuss various channels of triple evolution in detail such as those where the secondary or the tertiary is the first star to initiate a mass transfer event.

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arXiv2019

Planets In Young Massive Clusters: On the survivability of planets in young massive clusters and its implication of planet orbital architectures in globular clusters

As of August 2019, among the more than 4000 confirmed exoplanets, only one has been detected in a globular cluster (GC) M4. The scarce of exoplanet detections motivates us to employ direct $N$-body simulations to investigate the dynamical stability of planets in young massive clusters (YMCs), which are potentially the progenitors of GCs. In an $N=128{\rm k}$ cluster of virial radius 1.7 pc (comparable to Westerlund-1), our simulations show that most wide-orbit planets ($a\geq 20$~au) will be ejected within a timescale of 10 Myr. Interestingly, more than $70\%$ of planets with $a<5$~au survive in the 100 Myr simulations. Ignoring planet-planet scattering and tidal damping, the survivability at $t$ Myr as a function of initial semi-major axis $a_0$ in au in such a YMC can be described as $f_{\rm surv}(a_0, t)=-0.33 \log_{10}(a_0) \left(1 - e^{-0.0482t} \right) + 1$. Upon ejection, about $28.8\%$ of free-floating planets (FFPs) have sufficient speeds to escape from the host cluster at a crossing timescale. The other FFPs will remain bound to the cluster potential, but the subsequent dynamical evolution of the stellar system can result in the delayed ejection of FFPs from the host cluster. Although a full investigation of planet population in GCs requires extending the simulations to multi-Gyr, our results suggest that wide-orbit planets and free-floating planets are unlikely to be found in GCs.

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arXiv2019

Galactic Tide and Local Stellar Perturbations on the Oort Cloud: Creation of Interstellar Comets

Comets in the Oort cloud evolve under the influence of internal and external perturbations, such as giant planets, stellar passages, and the galactic tidal field. We aim to study the dynamical evolution of the comets in the Oort cloud, accounting for external perturbations (passing stars and the galactic tide). We first construct an analytical model of stellar encounters. We find that individual perturbations do not modify the dynamics of the comets in the cloud unless very close (< 0.5pc) encounters occur. Using proper motions, parallaxes, and radial velocities from Gaia DR2, we construct an astrometric catalogue of 14,659 stars that are within 50pc from the Sun. For all these stars we calculate the time and the closest distance to the Sun. We find that the cumulative effect of relatively distant ($\leq1$ pc) passing stars can perturb the comets in the Oort cloud. Finally, we study the dynamical evolution of the comets in the Oort cloud under the influence of multiple stellar encounters within 2.5pc from the Sun and the galactic tidal field over $\pm10$Myr. We considered two models for the Oort cloud, compact (a $\leq$0.25 pc) and extended (a$ \leq0.5$ pc). We find that the cumulative effect of stellar encounters is the major perturber of the Oort cloud for a compact configuration while for the extended, the galactic tide is the major perturber. In both cases, the effect of passing stars and the galactic tide raises the semi-major axis of $\sim1.1$\% of the comets at the edge of the cloud up to interstellar regions ($a >0.5$pc). This leads to the creation of transitional interstellar comets, which might become interstellar objects due to external perturbations. This raises the question about the existence of a cloud of objects in the interstellar space which might overlap with our Oort cloud if we consider that other planetary systems face similar processes for the ejection of comets.

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arXiv2018

Fast spectrophotometry of WD 1145+017

WD 1145+017 is currently the only white dwarf known to exhibit periodic transits of planetary debris as well as absorption lines from circumstellar gas. We present the first simultaneous fast optical spectrophotometry and broad-band photometry of the system, obtained with the Gran Telescopio Canarias (GTC) and the Liverpool Telescope (LT), respectively. The observations spanned $5.5$ h, somewhat longer than the $4.5$-h orbital period of the debris. Dividing the GTC spectrophotometry into five wavelength bands reveals no significant colour differences, confirming grey transits in the optical. We argue that absorption by an optically thick structure is a plausible alternative explanation for the achromatic nature of the transits that can allow the presence of small-sized ($\simμ$m) particles. The longest ($87$ min) and deepest ($50$ per cent attenuation) transit recorded in our data exhibits a complex structure around minimum light that can be well modelled by multiple overlapping dust clouds. The strongest circumstellar absorption line, Fe II $λ$5169, significantly weakens during this transit, with its equivalent width reducing from a mean out-of-transit value of $2$ Å to $1$ Å in-transit, supporting spatial correlation between the circumstellar gas and dust. Finally, we made use of the Gaia Data Release 2 and archival photometry to determine the white dwarf parameters. Adopting a helium-dominated atmosphere containing traces of hydrogen and metals, and a reddening $E(B-V)=0.01$ we find $T_\mathrm{eff}=15\,020 \pm 520$ K, $\log g=8.07\pm0.07$, corresponding to $M_\mathrm{WD}=0.63\pm0.05\ \mbox{$\mathrm{M}_{\odot}$}$ and a cooling age of $224\pm30$ Myr.

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arXiv2018

Gone with the wind: the impact of wind mass transfer on the orbital evolution of AGB binary systems

In low-mass binary systems, mass transfer is likely to occur via a slow and dense stellar wind when one of the stars is in the AGB phase. Observations show that many binaries that have undergone AGB mass transfer have orbital periods of 1-10 yr, at odds with the predictions of binary population synthesis models. We investigate the mass-accretion efficiency and angular-momentum loss via wind mass transfer in AGB binary systems. We use these quantities to predict the evolution of the orbit. We perform 3D hydrodynamical simulations of the stellar wind lost by an AGB star using the AMUSE framework. We approximate the thermal evolution of the gas by imposing a simple effective cooling balance and we vary the orbital separation and the velocity of the stellar wind. We find that for wind velocities $v_{\infty}$ larger than the relative orbital velocity of the system $v_\mathrm{orb}$ the flow is described by the Bondi-Hoyle-Lyttleton approximation and the angular-momentum loss is modest, leading to an expansion of the orbit. For low wind velocities an accretion disk is formed around the companion and the accretion efficiency as well as the angular-momentum loss are enhanced, implying that the orbit will shrink. We find that the transfer of angular momentum from the orbit to the outflowing gas occurs within a few orbital separations from the center of mass of the binary. Our results suggest that the orbital evolution of AGB binaries can be predicted as a function of the ratio $v_{\infty}/v_\mathrm{orb}$. Our results can provide insight into the puzzling orbital periods of post-AGB binaries and suggest that the number of stars entering into the common-envelope phase will increase. The latter can have significant implications for the expected formation rates of the end products of low-mass binary evolution, such as cataclysmic binaries, type Ia supernova and double white-dwarf mergers. [ABRIDGED]

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arXiv2017

Changes in orientation and shape of protoplanetary discs moving through an ambient medium

Misalignments between the orbital planes of planets and the equatorial planes of their host stars have been observed in our solar system, in transiting exoplanets, and in the orbital planes of debris discs. We present a mechanism that causes such a spin-orbit misalignment for a protoplanetary disc due to its movement through an ambient medium. Our physical explanation of the mechanism is based on the theoretical solutions to the Stark problem. We test this idea by performing self-consistent hydrodynamical simulations and simplified gravitational $N$-body simulations. The $N$-body model reduces the mechanism to the relevant physical processes. The hydrodynamical simulations show the mechanism in its full extent, including gas-dynamical and viscous processes in the disc which are not included in the theoretical framework. We find that a protoplanetary disc embedded in a flow changes its orientation as its angular momentum vector tends to align parallel to the relative velocity vector. Due to the force exerted by the flow, orbits in the disc become eccentric, which produces a net torque and consequentially changes the orbital inclination. The tilting of the disc causes it to contract. Apart from becoming lopsided, the gaseous disc also forms a spiral arm even if the inclination does not change substantially. The process is most effective at high velocities and observational signatures are therefore mostly expected in massive star-forming regions and around winds or supernova ejecta. Our $N$-body model indicates that the interaction with supernova ejecta is a viable explanation for the observed spin-orbit misalignment in our solar system.

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arXiv2017

Disc truncation in embedded star clusters: Dynamical encounters versus face-on accretion

Observations indicate that the dispersal of protoplanetary discs in star clusters occurs on time scales of about 5 Myr. Several processes are thought to be responsible for this disc dispersal. Here we compare two of these processes: dynamical encounters and interaction with the interstellar medium, which includes face-on accretion and ram pressure stripping. We perform simulations of embedded star clusters with parameterisations for both processes to determine the environment in which either of these processes is dominant. We find that face-on accretion, including ram pressure stripping, is the dominant disc truncation process if the fraction of the total cluster mass in stars is $\lesssim 30\,\%$ regardless of the cluster mass and radius. Dynamical encounters require stellar densities $\gtrsim 10^4$ pc$^{-3}$ combined with a mass fraction in stars of $\approx 90\,\%$ to become the dominant process. Our results show that during the embedded phase of the cluster, the truncation of the discs is dominated by face-on accretion and dynamical encounters become dominant when the intra-cluster gas has been expelled. As a result of face-on accretion the protoplanetary discs become compact and their surface density increases. In contrast, dynamical encounters lead to discs that are less massive and remain larger.

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arXiv2017

Characterising face-on accretion onto and the subsequent contraction of protoplanetary discs

Observations indicate that stars generally lose their protoplanetary discs on a timescale of about 5 Myr. Which mechanisms are responsible for the disc dissipation is still debated. Here we investigate the movement through an ambient medium as a possible cause of disc dispersal. The ram pressure exerted by the flow can truncate the disc and the accretion of material with no azimuthal angular momentum leads to further disc contraction. We derive a theoretical model from accretion disc theory that describes the evolution of the disc radius, mass, and surface density profile as a function of the density and velocity of the ambient medium. We test our model by performing hydrodynamical simulations of a protoplanetary disc embedded in a flow with different velocities and densities. We find that our model gives an adequate description of the evolution of the disc radius and accretion rate onto the disc. The total disc mass in the simulations follows the theoretically expected trend, except at the lowest density where our simulated discs lose mass owing to continuous stripping. This stripping may be a numerical rather than a physical effect. Some quantitative differences exist between the model predictions and the simulations. These are at least partly caused by numerical viscous effects in the disc and depend on the resolution of the simulation. Our model can be used as a conservative estimate for the process of face-on accretion onto protoplanetary discs, as long as viscous processes in the disc can be neglected. The model predicts that in dense gaseous environments, discs can shrink substantially in size and can, in theory, sweep up an amount of gas of the order of their initial mass. This process could be relevant for planet formation in dense environments.

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arXiv2016

The evolution of hierarchical triple star-systems

Field stars are frequently formed in pairs, and many of these binaries are part of triples or even higher-order systems. Even though, the principles of single stellar evolution and binary evolution, have been accepted for a long time, the long-term evolution of stellar triples is poorly understood. The presence of a third star in an orbit around a binary system can significantly alter the evolution of those stars and the binary system. The rich dynamical behavior in three-body systems can give rise to Lidov-Kozai cycles, in which the eccentricity of the inner orbit and the inclination between the inner and outer orbit vary periodically. In turn, this can lead to an enhancement of tidal effects (tidal friction), gravitational-wave emission and stellar interactions such as mass transfer and collisions. The lack of a self-consistent treatment of triple evolution, including both three-body dynamics as well as stellar evolution, hinders the systematic study and general understanding of the long-term evolution of triple systems. In this paper, we aim to address some of these hiatus, by discussing the dominant physical processes of hierarchical triple evolution, and presenting heuristic recipes for these processes. To improve our understanding on hierarchical stellar triples, these descriptions are implemented in a public source code TrES which combines three-body dynamics (based on the secular approach) with stellar evolution and their mutual influences. Note that modeling through a phase of stable mass transfer in an eccentric orbit is currently not implemented in TrES , but can be implemented with the appropriate methodology at a later stage.

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arXiv2016

The rate of stellar encounters along a migrating orbit of the Sun

The frequency of Galactic stellar encounters the Solar system experienced depends on the local density and velocity dispersion along the orbit of the Sun in the Milky Way galaxy. We aim at determining the effect of the radial migration of the solar orbit on the rate of stellar encounters. As a first step we integrate the orbit of the Sun backwards in time in an analytical potential of the Milky Way. We use the present-day phase-space coordinates of the Sun, according to the measured uncertainties. The resulting orbits are inserted in an N-body simulation of the Galaxy, where the stellar velocity dispersion is calculated at each position along the orbit of the Sun. We compute the rate of Galactic stellar encounters by employing three different solar orbits ---migrating from the inner disk, without any substantial migration, and migrating from the outer disk. We find that the rate for encounters within $4\times10^5$ AU from the Sun is approximately 21, 39 and 63 Myr$^{-1}$, respectively. The stronger encounters establish the outer limit of the so-called parking zone, which is the region in the plane of the orbital eccentricities and semi-major axes where the planetesimals of the Solar system have been perturbed only by interactions with stars belonging to the Sun's birth cluster. We estimate the outer edge of the parking zone at semi-major axes of 250--1300 AU (the outward and inward migrating orbits reaching the smallest and largest values, respectively), which is one order of magnitude smaller than the determination made by Portegies Zwart & Jílková (2015). We further discuss the effect of stellar encounters on the stability of the hypothetical Planet 9.

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