We present two approaches to determine the dynamical stability of a hierarchical triple-star system. The first is an improvement on the Mardling-Aarseth stability formula from 2001, where we introduce a dependence on inner orbital eccentricity and improve the dependence on mutual orbital inclination. The second involves a machine learning approach, where we use a multilayer perceptron (MLP) to classify triple-star systems as `stable' and `unstable'. To achieve this, we generate a large training data set of 10^6 hierarchical triples using the N-body code MSTAR. Both our approaches perform better than previous stability criteria, with the MLP model performing the best. The improved stability formula and the machine learning model have overall classification accuracies of 93 % and 95 % respectively. Our MLP model, which accurately predicts the stability of any hierarchical triple-star system within the parameter ranges studied with almost no computation required, is publicly available on Github in the form of an easy-to-use Python script.
Okinawa Institute of Science and Technology（冲绳科学技术大学院大学）；Research Center for the Early Universe, School of Science, The University of Tokyo（东京大学理学部早期宇宙研究中心）
Understanding the long-term evolution of hierarchical triple systems is challenging due to its inherent chaotic nature, and it requires computationally expensive simulations. Here we propose a convolutional neural network model to predict the stability of hierarchical triples by looking at their evolution during the first $5 \times 10^5$ inner binary orbits. We employ the regularized few-body code TSUNAMI to simulate $5\times 10^6$ hierarchical triples, from which we generate a large training and test dataset. We develop twelve different network configurations that use different combinations o…
We present evolutionary pathways for creating hot subdwarf OB (sdOB) stars from hierarchical triple configurations. We use the population synthesis code Multiple Stellar Evolution (MSE) to follow the stellar, binary, and gravitational dynamical evolution of triple-star systems. To ascertain the effect of the outer tertiary, we also consider the evolution of the inner binary with the tertiary component removed. We find we are able to create sdOB stars in single, binary and triple configurations. We also demonstrate that it is possible to form sdOBs in systems which undergo triple common envelop…
While there have been many studies examining the stability of hierarchical triple systems, the meaning of ``stability'' is somewhat vague and has been interpreted differently in previous literatures. The present paper focuses on ``Lagrange stability'', which roughly refers to the stability against the escape of a body from the system, or ``disruption'' of the triple system, in contrast to ``Lyapunov-like stability'' that is related to the chaotic nature of the system dynamics. We compute the evolution of triple systems using direct $N$-body simulations up to $10^7 P_\mathrm{out}$, which is sig…
We examine the stability of hierarchical triple systems using direct $N$-body simulations without adopting a secular perturbation approximation. We estimate their disruption timescales in addition to the mere stable/unstable criterion, with particular attention to the mutual inclination between the inner and outer orbits. First, we improve the fit to the dynamical stability criterion by \citet{Mardling1999,Mardling2001} widely adopted in the previous literature. Especially, we find that that the stability boundary is very sensitive to the mutual inclination; coplanar retrograde triples and ort…
Detailed Simulations of Massive Hierarchical Triple Star Systems - Exploring the impact of the stellar physics on the evolutionary pathways of massive hierarchical triple systems
In spite of their importance for modern astronomy, we do not fully understand how close binary stars containing at least one white dwarf form from main sequence binary stars. The discovery of EL CVn binaries, close pre-white dwarfs with A/F main sequence star companions, offers now the unique possibility to test models of close compact binary star formation. Binary evolution theories predict that these EL CVn stars descend from very close main sequence binaries with orbital periods shorter than 3 days. If this is correct, nearly all EL CVn stars should be inner binaries of hierarchical triples…
The White Dwarf Binary Pathways Survey aims at increasing the number of known detached A, F, G and K main sequence stars in close orbits with white dwarf companions (WD+AFGK binaries) to refine our understanding about compact binary evolution and the nature of Supernova Ia progenitors. These close WD+AFGK binary stars are expected to form through common envelope evolution, in which tidal forces tend to circularize the orbit. However, some of the identified WD+AFGK binary candidates show eccentric orbits, indicating that these systems are either formed through a different mechanism or perhaps t…
Young multiple systems accrete most of their final mass in the first few Myr of their lifetime, during the protostellar and protoplanetary phases. Previous studies showed that in binary systems the majority of the accreted mass falls onto the lighter star, thus evolving to mass equalisation. However, young stellar systems often comprise more than two stars, which are expected to be in hierarchical configurations. Despite its astrophysical relevance, differential accretion in hierarchical systems remains to be understood. In this work, we investigate whether the accretion trends expected in bin…
Spectroscopic orbits for three SB2s and one hierarchical triple using SALT data
Yunnan Observatories, Chinese Academy of Sciences（中国科学院云南天文台）；South African Astronomical Observatory（南非天文台）；Sternberg Astronomical Institute, Lomonosov Moscow State University（莫斯科国立大学斯滕伯格天体物理研究所）；Institute of Astronomy, Russian Academy of Sciences（俄罗斯科学院天体物理研究所）
Aims. We introduce a novel way to identify new compact hierarchical triple stars by exploiting the huge potential of Gaia DR3 and also its future data releases. We aim to increase the current number of compact hierarchical triples significantly. Methods. We utilize several eclipsing binary catalogs from different sky surveys totaling more than 1 million targets for which we search for Gaia DR3 Non-single Star orbital solutions with periods substantially longer than the eclipsing periods of the binaries. Those solutions in most cases should belong to outer orbits of tertiary stars in those syst…
Stability of hierarchical triples comprising a central massive body and a tight binary: the effect of inner and outer eccentricities on the binary breakup condition
We show how astrometric and spectroscopic errors introduced by an unresolved binary system can be combined to give estimates of the binary period and mass ratio. This can be performed analytically if we assume we see one or more full orbits over our observational baseline, or numerically for all other cases. We apply this method to Gaia DR3 data, combining the most recent astrometric and spectroscopic data. We compare inferred periods and mass ratios calculated using our method with orbital parameters measured for non-single stars in Gaia DR3 and find good agreement. Finally, we use this metho…
The three-body problem is famously chaotic, with no closed-form analytical solutions. However, hierarchical systems of three or more bodies can be stable over indefinite timescales. A system is considered hierarchical if the bodies can be divided into separate two-body orbits with distinct time- and length-scales, such that one orbit is only mildly affected by the gravitation of the other bodies. Previous work has mapped the stability of such systems at varying resolutions over a limited range of parameters, and attempts have been made to derive analytic and semi-analytic stability boundary fi…
The gas dissipation from a protoplanetary disk is one of the key processes affecting planet formation, and it is widely accepted that it happens on timescales of a few million years for disks around single stars. Over the last years, several protoplanetary disks have been discovered in multiple star systems, and despite the complex environment in which they find themselves, some of them seem to be quite old, a situation that may favor planet formation. A clear example of this is the disk around HD 98800 B, a binary in a hierarchical quadruple stellar system, which at a $\sim$10 Myr age seems t…
Under certain rather prevalent conditions (driven by dynamical orbital evolution), a hierarchical triple stellar system can be well approximated, from the standpoint of orbital parameter estimation, as two binary star systems combined. Even under this simplifying approximation, the inference of orbital elements is a challenging technical problem because of the high dimensionality of the parameter space, and the complex relationships between those parameters and the observations (astrometry and radial velocity). In this work we propose a new methodology for the study of triple hierarchical syst…
We report the discovery and complex analyses of the first two compact hierarchical triple star systems discovered with TESS in or near its southern continuous viewing zone during Year 1. Both TICs 167692429 and 220397947 were previously unknown eclipsing binaries, and the presence of a third companion star was inferred from eclipse timing variations exhibiting signatures of strong 3rd-body perturbations and, in the first system, also from eclipse depth variations. We carried out comprehensive analyses, including the simultaneous photodynamical modelling of TESS and archival ground-based WASP l…
Compact Hierarchical Triples (CHT) are systems with the tertiary star orbiting the inner binary in an orbit shorter than 1000 days. CHT with an eclipsing binary as its inner binary can help us extract a multitude of information about all three stars in the system. In this study, we use independent observational techniques to estimate the orbital, stellar, and atmospheric parameters of two triple-lined CHT: BD+442258 and KIC06525196. We find that the masses of stars in BD+442258 are $1.011\pm0.029 M_{\odot}$, $0.941\pm0.033 M_{\odot}$, and $0.907\pm0.065 M_{\odot}$ while in KIC06525196 the esti…
The isolated binary evolution model for merging neutron stars (NSs) involves processes such as mass transfer, common-envelope evolution, and natal kicks, all of which are poorly understood. Also, the predicted NS-NS merger rates are typically lower than the rates inferred from the LIGO GW170817 event. Here, we investigate merger rates of NS and black hole (BH)-NS binaries in hierarchical triple-star systems. In such systems, the tertiary can induce Lidov-Kozai (LK) oscillations in the inner binary, accelerating its coalescence, and potentially enhancing compact object merger rates. However, si…
We report a study of the eclipse timing variation (ETV) of short period ($P_1\le 6^d$) eclipsing binaries (EB) monitored during the photometric survey OGLE-IV. From the 425\,193 EBs we selected approximately 80\,000 binaries that we found suitable for further examination. Among them we identified 992 potential hierarchical triple (or multiple ) system candidates exhibiting light-travel-time effect (LTTE). Besides, we obtained the orbital parameters of these systems and carried out statistical analyses on the properties of these candidates. We found that (i) there is a significant lack of tripl…
We report the discovery of a compact triply eclipsing triple star system in the southern continuous viewing zone of the TESS space telescope. TIC 278825952 is a previously unstudied, circular eclipsing binary with a period of 4.781 days with a tertiary component in a wider, circular orbit of 235.55 days period that was found from three sets of third-body eclipses and from light travel-time effect dominated eclipse timing variations. We performed a joint photodynamical analysis of the eclipse timing variation curves, photometric data, and the spectral energy distribution, coupled with the use o…
Southern Massive Stars at High Angular Resolution (SMaSH+): Properties of hierarchical massive triples
KU Leuven（荷语鲁汶大学）；Universität zu Köln（科隆大学）；Leuven Gravity Institute, KU Leuven（荷语鲁汶大学引力研究所）；European Southern Observatory（欧洲南方天文台）；naXys Research Institute, Department of Mathematics, University of Namur（纳穆尔大学数学系naXys研究所）；Royal Observatory of Belgium（比利时皇家天文台）；Anton Pannekoek Institute for Astronomy, University of Amsterdam（阿姆斯特丹大学安东·潘内克天文研究所）；GRAPPA, University of Amsterdam（阿姆斯特丹大学GRAPPA研究中心）；Centro de Astrobiología (CAB), CSIC-INTA（西班牙国家航天技术研究所/西班牙国家科学研究委员会天体生物学中心）
We present TESS photometry of the triple star system V1031 Ori (AB)C, which exhibits short-period oscillations superimposed on an eclipsing light curve. The high-quality data were analyzed in detail and combined with the published spectroscopic measurements to obtain the fundamental stellar parameters of the program target. The masses and radii of the eclipsing pair (AB) were determined to about 0.5 \% and 0.7 \% precision, respectively. We found 23 significant frequencies in two ranges of $<$ 2 day$^{-1}$ and 10.9$-$12.7 day$^{-1}$ from the eclipse-subtracted residuals. Among them, sixteen in…
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 prese…
Unveiling the nature of G6096: a likely hierarchical triple system
Chongqing University（重庆大学）；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）；Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University（北京师范大学天文与前沿科学研究院）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学汤忠道研究院）；School of Physics and Astronomy, Shanghai Jiao University（上海交通大学物理与天文学院）；Department of Astronomy, Xiamen University（厦门大学天文学系）；School of Astronomy and Space Science, University of Chinese Academy of Sciences（中国科学院大学天文与空间科学学院）
