Toonen et al. RESEARCH The evolution of hierarchical triple star-systems S. Toonen*, A. Hamers and S. Portegies Zwart Abstract 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 behaviour 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 modelling 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. Keywords: binaries (including multiple): close; stars: evolution 1 Introduction The theoretical studies of triples can classically be di- The majority of stars are members of multiple sys- vided into three-body dynamics and stellar evolution, tems. These include binaries, triples, and higher order which both are often discussed separately. Three-body hierarchies. The evolution of single stars and binaries dynamics is generally governed by the gravitational have been studied extensively and there is general con- orbital evolution, whereas the stellar evolution is gov- sensus over the dominant physical processes (Postnov erned by the internal nuclear burning processes in the and Yungelson, 2014; Toonen et al., 2014). Many ex- individual stars and their mutual influence. otic systems, however, cannot easily be explained by Typical examples of studies that focused on the binary evolution, and these have often been attributed three-body dynamics include Fabrycky and Tremaine to the evolution of triples, for example low-mass X-ray (2007); Ford et al.(2000); Liu et al.(2015a); Naoz and binaries (Eggleton and Verbunt, 1986) and blue strag- Fabrycky(2014); Naoz et al.(2013), and stellar evo- glers (Perets and Fabrycky, 2009). Our lack of a clear lution studies include Eggleton and Kiseleva(1996); understanding of triple evolution hinders the system- Iben and Tutukov(1999); Kuranov et al.(2001). In- arXiv:1612.06172v1 [astro-ph.SR] 19 Dec 2016 atic exploration of these curious objects. At the same terdisciplinary studies, in which the mutual interaction time triples are fairly common; Our nearest neighbour between the dynamics and stellar aspects are taken α Cen is a triple star system (Tokovinin, 2014a), but into account are rare (Hamers et al., 2013; Kratter more importantly ∼ 10% of the low-mass stars are in and Perets, 2012; Michaely and Perets, 2014; Naoz triples (Moe and Di Stefano, 2016; Raghavan et al., et al., 2016; Perets and Kratter, 2012; Shappee and 2010; Tokovinin, 2008, 2014b) a fraction that gradu- Thompson, 2013), but demonstrate the richness of the ally increases (Duch^eneand Kraus, 2013) to ∼ 50% interacting regime. The lack of a self consistent treat- for spectral type B stars (Moe and Di Stefano, 2016; ment hinders a systematic study of triple systems. This Remage Evans, 2011; Sana et al., 2014). makes it hard to judge the importance of this interact- ing regime, or how many curious evolutionary products * Correspondence: [email protected] can be attributed to triple evolution. Here we discuss Leiden Observatory, Leiden University, PO Box 9513 Leiden, The Netherlands triple evolution in a broader context in order to ad- Full list of author information is available at the end of the article dress some of these hiatus. Toonen et al. Page 2 of 40 Table 1 Necessary parameters to describe a single star system, a binary and a triple. For stellar parameters, age and metallicity 2.1.1 Timescales of each star can be added. The table shows that as the Fundamental timescales of stellar evolution are the dy- multiplicity of a stellar system increases from one to three, the problem becomes significantly more complicated. namical (τdyn), thermal (τth), and nuclear timescale (τnucl). The dynamical timescale is the characteristic Parameters Stellar Orbital Single star m - time that it would take for a star to collapse under its Binary m1, m2 a, e own gravitational attraction without the presence of Triple m1, m2, m3 imutual, ain, ein, gin, hin internal pressure: aout, eout, gout, hout r R3 τ = ; (1) In this paper we discuss the principle complexities dyn Gm of triple evolution in a broader context (Sect.2). We start by presenting an overview of the evolution of sin- where R and m are the radius and mass of the star. gle stars and binaries, and how to extend these to It is a measure of the timescale on which a star would triple evolution. In the second part of this paper we expand or contract if the hydrostatic equilibrium of present heuristic recipes for simulating their evolution the star is disturbed. This can happen for example (Sect.3). These recipes combine three-body dynam- because of sudden mass loss. ics with stellar evolution and their mutual influences, A related timescale is the time required for the Sun such as tidal interactions and mass transfer. These to radiate all its thermal energy content at its current descriptions are summarized in a public source code luminosity: TrES with which triple evolution can be studied. Gm2R τ = ; (2) 2 Background th L We will give a brief overview of isolated binary evolu- tion (Sect. 2.2) and isolated triple evolution (Sect. 2.3). where L is the luminosity of the star. In other words, We discuss in particular under what circumstances when the thermal equilibrium of a star is disturbed, triple evolution differs from binary evolution and what the star will move to a new equilibrium on a thermal the consequences are of these differences. We start with (or Kelvin{Helmholtz) timescale a brief summary of single star evolution with a focus Finally, the nuclear timescale represents the time re- on those aspects that are relevant for binary and triple quired for the star to exhaust its supply of nuclear fuel evolution. at its current luminosity: 2.1 Single stellar evolution c2m τ = nucl ; (3) Hydrostatic and thermal equilibrium in a star give rise nucl L to temperatures and pressures that allow for nuclear burning, and consequently the emission of the starlight where is the efficiency of nuclear energy production, that we observe. Cycles of nuclear burning and exhaus- c is the speed of light, and mnucl is the amount of tion of fuel regulate the evolution of a star, and sets mass available as fuel. For core hydrogen burning, = the various phases during the stellar lifetime. 0:007 and Mnucl ≈ 0:1M. Assuming a mass-luminosity The evolution of a star is predominantly determined relation of L / M α, with empirically α ≈ 3 − 4 (e.g. by a single parameter, namely the stellar mass (Tbl.1). Eker et al., 2015; Salaris and Cassisi, 2005), it follows It depends only slightly on the initial chemical compo- that massive stars live shorter and evolve faster than sition or the amount of core overshooting[1]. low-mass stars. For the Sun, τdyn ≈ 30 min, τth ≈ 30 Myr, and [1] Overshooting refers to a chemically mixed region τnucl ≈ 10 Gyr. Typically, τdyn < τth < τnucl, which beyond the boundary of the convective core (e.g. allows us to quantitatively predict the structure and Maeder and Meynet, 1991; Massevitch et al., 1979; evolution of stars in broad terms. Stothers, 1963), as predicted by basic stellar evolution- ary theory, i.e. the Schwarzschild criterion. A possible 2.1.2 Hertzsprung-Russell diagram mechanism is convection carrying material beyond the The Hertzsprung-Russell (HR) diagram in Fig.1 boundary due to residual velocity. For the effects of shows seven evolutionary tracks for stars of different overshooting on stellar evolution, see e.g. Bressan et al. masses. The longest phase of stellar evolution is known (2015). as the main-sequence (MS), in which nuclear burning Toonen et al. Page 3 of 40 Figure 1 Hertzsprung-Russell diagram Evolutionary tracks for to a few thousand Kelvin; the shift to the right in the seven stars in the HR-diagram with masses 1, 1.5, 2.5, 4, 6.5, HR-diagram is truncated when helium ignites. 10, and 15M at solar metallicity. Specific moments in the During helium burning the stellar tracks make a evolution of the stars are noted by blue circles as explained in the text. The tracks are calculated with SeBa (Portegies Zwart loop in the HR-diagram, also known as the horizon- and Verbunt, 1996; Toonen et al., 2012). The dashed lines tal branch. This branch is marked in Fig.1 by a blue show lines of constant radii by means of the circle at its maximum effective temperature. The loop Stefan{Boltzmann law.