MNRAS 000,1–34 (0000) Preprint February 19, 2021 Compiled using MNRAS LATEX style file v3.0 Multiple Stellar Evolution: a population synthesis algorithm to model the stellar, binary, and dynamical evolution of multiple-star systems Adrian S. Hamers1?, Antti Rantala1, Patrick Neunteufel1, Holly Preece1, and Pavan Vynatheya1 1Max-Planck-Institut f¨urAstrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany Accepted 2021 January 29. Received 2021 January 29; in original form 2020 November 9 ABSTRACT In recent years, observations have shown that multiple-star systems such as hierarchical triple and quadruple-star systems are common, especially among massive stars. They are potential sources of interesting astrophysical phenomena such as compact object mergers, leading to supernovae, and gravitational wave events. However, many uncertainties remain in their often complex evolution. Here, we present the population synthesis code Multiple Stellar Evolution (MSE), designed to rapidly model the stellar, binary, and dynamical evolution of multiple-star systems. MSE includes a number of new features not present in previous population synthe- sis codes: (1) an arbitrary number of stars, as long as the initial system is hierarchical, (2) dynamic switching between secular and direct N-body integration for efficient computation of the gravitational dynamics, (3) treatment of mass transfer in eccentric orbits, which occurs commonly in multiple-star systems, (4) a simple treatment of tidal, common-envelope, and mass transfer evolution in which the accretor is a binary instead of a single star, (5) taking into account planets within the stellar system, and (6) including gravitational perturbations from passing field stars. MSE, written primarily in the C++ language, will be made publicly available and has few prerequisites; a convenient Python interface is provided. We give a de- tailed description of MSE and illustrate how to use the code in practice. We demonstrate its operation in a number of examples. Key words: binaries: general – stars: kinematics and dynamics – methods: statistical – grav- itation – planets and satellites: dynamical evolution and stability – stars: evolution 1 INTRODUCTION history is Mizar and Alcor (e.g., Mamajek et al. 2010). Currently, even two systems with seven stars are known: AR Cassiopeiae, and 1.1 The importance of multiple-star systems Nu Scorpii (Tokovinin 1997, 2018). Multiple-star systems, stellar systems containing three or more From a more statistical viewpoint, observations of F and G stars, are common. For example, the closest stellar system to the dwarfs within 67 pc of the Sun (Tokovinin 2014b,a) show that Sun, α Centauri, is a hierarchical triple system (Innes 1917). The ≈ 10% of stellar systems with Solar-like components are triple Algol system, well known for the Algol paradox (i.e., the less mas- systems, and ≈ 1% are quadruple systems. In a similar study of sive star in the binary is more evolved than its higher-mass compan- Solar-type stars within 25 pc of the Sun (Raghavan et al. 2010), arXiv:2011.04513v3 [astro-ph.SR] 18 Feb 2021 ion, which can be explained by mass transfer; e.g., Pustylnik 1998), the triple fraction was found to be (8 ± 1)%, and the fraction of is also a triple system (Meltzer 1957). The binary star Eta Carinae, higher-multiplicity systems (3 ± 1)%. For systems with more mas- known for its high mass (∼ 90 M primary and ∼ 30 M secondary, sive (primary) stars, the multiplicity fraction is significantly higher Damineli et al. 1997; Davidson & Humphreys 1997) and its giant (Ducheneˆ & Kraus 2013). Further, Moe & Di Stefano(2017) find mass eruption event in the nineteenth century (e.g., Humphreys & that, among O-type stars, the multiplicity fraction is (35 ± 3)% for Davidson 1999), may originally have been a triple system (Porte- triple stars, and (38±11)% for quadruple stars, showing that, among gies Zwart & van den Heuvel 2016; Sun et al. 2020). More exoti- massive stellar systems in the field, triples and quadruples signifi- cally, through centuries of observations in which more companions cantly outnumber both single and binary stars. have been discovered, Castor, one of the brightest stars in the night Multiple-star systems are usually arranged in a hierarchical sky, is known to harbour six stars in a ‘(2+2)+2’ configuration (e.g., configuration, since they would otherwise be short lived. The sim- Heintz 1988). A similar sextuple system with a long observational plest hierarchical configuration occurs in triple systems in which two stars are orbited by a more distant, tertiary star. If the inner and ? E-mail: [email protected] outer orbits in such a configuration are initially mutually highly in- © 0000 The Authors 2 Hamers, Rantala, Neunteufel, Preece, Vynatheya clined, then the gravitational torque of the outer orbit can induce nen et al. 2012; van Haaften et al. 2013). In particular, BSE (Hurley high-amplitude eccentricity oscillations in the inner binary, known et al. 2002, hereafter HTP02), based on the rapid evolution algo- as Lidov-Kozai (LK) or von Zeipel-Lidov-Kozai (ZLK) oscilla- rithm SSE (Hurley et al. 2000, hereafter HPT00), has been an in- tions (von Zeipel 1910; Lidov 1962; Kozai 1962; see Naoz 2016; dustry standard for nearly two decades. Also, BSE, and the SSE an- Shevchenko 2017; Ito & Ohtsuka 2019 for reviews). alytic stellar evolution tracks on which it is based, have formed the ZLK oscillations have important implications for a large vari- basis for many other codes such as SeBa (Portegies Zwart & Ver- ety of three-body systems, not limited to triple-star systems. Gen- bunt 1996; Nelemans et al. 2001a,b; Toonen et al. 2012; Toonen & erally, the high eccentricities that can be reached during these oscil- Nelemans 2013), StarTrack (Belczynski et al. 2002b,a, 2008), bi- lations can drive strong interactions such as efficient tidal dissipa- nary c (Izzard et al. 2004, 2006, 2009; Claeys et al. 2014; de Mink tion, or even collisions. For example, they have been considered as et al. 2014; Abate et al. 2015), MOBSE (Giacobbo et al. 2018; Gi- a possible pathway to produce short-period binaries (e.g., Mazeh & acobbo & Mapelli 2018), COMPAS (Stevenson et al. 2017; Barrett Shaham 1979; Kiseleva et al. 1998; Eggleton & Kiseleva-Eggleton et al. 2018; Vigna-Gomez´ et al. 2018; Broekgaarden et al. 2019; 2001; Eggleton & Kisseleva-Eggleton 2006; Fabrycky & Tremaine Stevenson et al. 2019; Neijssel et al. 2019), and COSMIC (Breivik 2007; Naoz & Fabrycky 2014) and hot Jupiters (e.g., Wu & Mur- et al. 2020). Other codes based on other methods (i.e., not using an- ray 2003; Fabrycky & Tremaine 2007; Naoz et al. 2012; Petrovich alytic fits to stellar evolution tracks) include the Brussels code PNS 2015a; Anderson et al. 2016; Petrovich & Tremaine 2016), enhanc- (De Donder & Vanbeveren 2004; Mennekens et al. 2010; Vanbev- ing mergers of compact objects (e.g., Blaes et al. 2002; Thompson eren et al. 2012), BPAS (Eldridge & Stanway 2009; Eldridge 2012; 2011; Hamers et al. 2013; Antonini et al. 2017; Silsbee & Tremaine Eldridge & Stanway 2016; Eldridge et al. 2017; Stanway & El- 2017; Liu & Lai 2017, 2018; Hamers et al. 2018; Hoang et al. 2018; dridge 2018), SEVN (Spera et al. 2015; Spera & Mapelli 2017; Randall & Xianyu 2018a,b; Toonen et al. 2018; Fragione & Loeb Spera et al. 2019), ComBinE(Kruckow et al. 2018), dart board 2019), affecting the evolution of protoplanetary or accretion disks (Andrews et al. 2018), and METISSE (a single stellar evolution in binaries (e.g., Martin et al. 2014; Fu et al. 2015; Zanazzi & Lai code based on interpolation; Agrawal et al. 2020). The compara- 2017; Lubow & Ogilvie 2017; Zanazzi & Lai 2018; Franchini et al. tive study of Toonen et al.(2014) found that in the context of low 2019; Martin & Franchini 2019), triggering white dwarf pollution and intermediate-mass binaries, discrepancies between simulation by planets (e.g., Hamers & Portegies Zwart 2016b; Petrovich & results of binary c, PNS, SeBa, and StarTrack are typically small Munoz˜ 2017), and producing blue straggler stars (e.g., Perets & and can be explained by different physical assumptions. Fabrycky 2009; Antonini et al. 2016; Stephan et al. 2016; Fragione More recently, population synthesis codes have been devel- & Antonini 2019). Also, ZLK cycles can couple with stellar evolu- oped that can model the evolution of triple stars (as long as the sys- tion in triples, giving rise to strong interactions during or after the tem is hierarchical) taking into account both stellar/binary evolu- main sequence (MS; e.g., Hamers et al. 2013; Shappee & Thomp- tion, and gravitational dynamics1. The code triple c (Hamers et al. son 2013; Michaely & Perets 2014; Toonen et al. 2016; Stephan 2013) combined binary evolution in binary c to model the inner et al. 2016; Antonini et al. 2017; Toonen et al. 2018; Stephan et al. binary system with secular dynamics using the orbit-averaged and 2019; Hamers & Thompson 2019a,b). expanded equations of motion (e.g., Harrington 1968; Ford et al. ZLK oscillations can become more complicated when the 2000; Blaes et al. 2002; Naoz et al. 2013a). It included tidal evo- triple system is marginally hierarchical, such that higher-order ex- lution and common envelope (CE) evolution, both processes which pansion terms become important (e.g., Lithwick & Naoz 2011; can be enhanced due to high-amplitude ZLK eccentricity oscilla- Katz et al. 2011; Li et al. 2014; Hamers & Portegies Zwart 2016a; tions. Tres (Toonen et al. 2016), implemented within the AMUSE Will 2017). Similarly, the dynamics become more complex when framework (Pelupessy et al. 2013; Portegies Zwart et al. 2009), uses more bodies are added to the system (while maintaining a hierar- similar routines as triple c to model the secular dynamics, whereas chical configuration with widely separated orbits). In hierarchical SeBa is used to model stellar and binary evolution. quadruples, which occur in either the ‘2+2’ (two binaries orbiting each other’s center of mass) or ‘3+1’ (triple orbited by a fourth body) configurations, secular evolution can be more efficient com- 1.3 Limitations pared to triples (Pejcha et al.