We analyze the long-term evolution of hierarchical triple systems in Newtonian gravity to second order in the quadrupolar perturbation parameter, and to sixth order in $ε= a/A$, the ratio of the semimajor axes of the inner and outer orbits. We apply the ``two-timescale'' method from applied mathematics to the Lagrange Planetary Equations for the inner and outer orbits, in which each osculating orbit element is split into an orbit averaged part that evolves on the long perturbative timescale, and an ``average-free'' part that is oscillatory in the orbital timescales. Averages over the two orbital timescales are performed using the well-known secular approximation. We also incorporate perturbative corrections to the relation between time and the orbital phases. We place no restrictions on the masses, on the relative orbit inclinations or on the eccentricities, beyond the requirement that the quadrupolar parameter and $ε$ both be small. The result is a complete set of long-timescale evolution equations for the averaged elements of the inner and outer orbits. At first order in perturbation theory, we obtain the dotriacontapole contributions explicitly at order $ε^6$. At second order in perturbation theory, i.e. quadratic in the quadrupole perturbation amplitude, we find contributions that scale as $ε^{9/2}$ (found in earlier work), $ε^{5}$, $ε^{11/2}$, and $ε^{6}$. At first perturbative order and dotriacontapole order, the two averaged semimajor axes are constant in time (and we prove that this holds to arbitrary multipole orders); but at second perturbative order, beginning at $O( ε^{5})$, they are no longer constant. Nevertheless we verify that the total averaged energy of the system is conserved, and we argue that this behavior is not incompatible with classical theorems on secular evolution of the semimajor axes.
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…
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…
We analyze the secular evolution of hierarchical triple systems to second-order in the quadrupolar perturbation induced on the inner binary by the distant third body. The Newtonian three-body equations of motion, expanded in powers of the ratio of semimajor axes $a/A$, become a pair of effective one-body Keplerian equations of motion, perturbed by a sequence of multipolar perturbations, denoted quadrupole, $O[(a/A)^3]$, octupole, $O[(a/A)^4]$, and so on. In the Lagrange planetary equations for the evolution of the instantaneous orbital elements, second-order effects arise from obtaining the fi…
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…
Max-Planck-Institut für Astrophysik（马克斯·普朗克天体物理研究所）；Monash University（莫纳什大学）；Aarhus University（奥胡斯大学）
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 bette…
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…
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…
Three-body interactions are ubiquitous in astrophysics. For instance, Kozai-Lidov oscillations in hierarchical triple systems have been studied extensively and applied to a wide range of astrophysical systems. However, mildly-hierarchical triples also play an important role, but they are less explored. In this work we consider the secular dynamics of a test particle in a mildly-hierarchical configuration. We find the limit within which the secular approximation is reliable, present resonances and chaotic regions using surface of sections, and characterize regions of phase space that allow larg…
We investigate the possibility of detecting planetary or stellar companions orbiting white dwarf binaries using the LISA gravitational radiation detector. Specifically, we consider the acceleration of the barycenter of the white dwarf binary due to the orbiting third body as well as the effect of changes in the tidal field across the binary due to the perturber's eccentric orbit. We find that the movement of the barycenter is detectable for both stellar and planetary mass objects. If circumbinary planets occur with frequencies similar to gas giant planets around isolated main sequence stars, t…
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
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…
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…
Exoplanet searches through space-based photometric time series have shown to be very efficient in recent years. However, follow-up efforts on the detected planet candidates have been demonstrated to be critical to uncover the true nature of the transiting objects. In this paper we show a detailed analysis of one of those false positives hidden as planetary signals. In this case, the candidate KOI-3886.01 showed clear evidence of a planetary nature from various techniques. Indeed, the properties of the fake planet set it among the most interesting and promising for the study of planetary evolut…
Transit timing variations in eccentric hierarchical triple exoplanetary systems. I. Perturbations on the time-scale of the orbital period of the perturber
In hierarchical triple systems, the inner binary is perturbed by a distant companion. For large mutual inclinations, the Lidov-Kozai mechanism secularly excites large eccentricity and inclination oscillations of the inner binary. The maximal eccentricity attained, $e_{\rm max}$ is simply derived and widely used. However, for mildly hierarchical systems (i.e. the companion is relatively close and massive), non-secular perturbations affect the evolution. Here we account for fast non-secular variations and find new analytic formula for $e_{\rm max}$, in terms of the system's hierarchy level, corr…
Circumbinary planets in coplanar triple-star systems: I. Minimum eccentricity variation region
Facultad de Matemática, Astronomía, Física y Computación (FAMAF), Universidad Nacional de Córdoba (UNC)（科尔多瓦国立大学数学、天文学、物理与计算学院）；Instituto de Astronomía Teórica y Experimental (IATE), CONICET-UNC（理论与实验天文研究所）；Observatorio Astronómico de Córdoba (OAC)（科尔多瓦天文台）；Univ. Grenoble Alpes, CNRS, IPAG（格勒诺布尔阿尔卑斯大学法国国家科学研究中心IPAG实验室）
