Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 DOI 10.1186/s40668-016-0019-0

METHODOLOGY OpenAccess The evolution of hierarchical triple star-systems

Silvia Toonen* , Adrian Hamers and Simon 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 in triples (Tokovinin; , b;Raghavanetal.; The majority of stars are members of multiple systems. Moe and Di Stefano ) a fraction that gradually in- These include binaries, triples, and higher order hierar- creases (Duchêne and Kraus )to∼% for spectral chies. The evolution of single stars and binaries have been type B stars (Remage Evans ;Sanaetal.;Moeand studied extensively and there is general consensus over Di Stefano ). the dominant physical processes (Postnov and Yungel- The theoretical studies of triples can classically be di- son ;Toonenetal.). Many exotic systems, how- vided into three-body dynamics and stellar evolution, ever, cannot easily be explained by binary evolution, and whichbothareoftendiscussedseparately.Three-bodydy- these have often been attributed to the evolution of triples, namics is generally governed by the gravitational orbital for examples low-mass X-ray binaries (Eggleton and Ver- evolution, whereas the stellar evolution is governed by the bunt ) and blue stragglers (Perets and Fabrycky ). internal nuclear burning processes in the individual stars Ourlackofaclearunderstandingoftripleevolutionhin- and their mutual influence. ders the systematic exploration of these curious objects. Typical examples of studies that focused on the three- At the same time triples are fairly common; Our nearest body dynamics include Ford et al. (), Fabrycky and neighbour α Cen is a triple star system (Tokovinin a), Tremaine (), Naoz et al. (), Naoz and Fabrycky but more importantly ∼% of the low-mass stars are (), Liu et al. (a), and stellar evolution studies in- clude Eggleton and Kiseleva (), Iben and Tutukov *Correspondence: [email protected] (), Kuranov et al. (). Interdisciplinary studies, in Leiden Observatory, Leiden University, PO Box 9513, Leiden, The Netherlands which the mutual interaction between the dynamics and

© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 2 of 36

stellar aspects are taken into account are rare (Kratter and .. Timescales Perets ;PeretsandKratter;Hamersetal.; Fundamental timescales of stellar evolution are the dy- Shappee and Thompson ;MichaelyandPerets; namical (τdyn), thermal (τth), and nuclear timescale (τnucl). Naoz et al. ), but demonstrate the richness of the in- The dynamical timescale is the characteristic time that it teracting regime. The lack of a self consistent treatment would take for a star to collapse under its own gravitational hinders a systematic study of triple systems. This makes it attraction without the presence of internal pressure: hard to judge the importance of this interacting regime, or how many curious evolutionary products can be attributed R τ = ,() to triple evolution. Here we discuss triple evolution in a dyn Gm broader context in order to address some of these hiatus. In this paper we discuss the principle complexities of where R and m are the radius and mass of the star. It is triple evolution in a broader context (Section ). We start a measure of the timescale on which a star would expand by presenting an overview of the evolution of single stars or contract if the hydrostatic equilibrium of the star is dis- and binaries, and how to extend these to triple evolu- turbed. This can happen for example because of sudden mass loss. tion. In the second part of this paper we present heuristic A related timescale is the time required for the Sun to recipes for simulating their evolution (Section ). These radiate all its thermal energy content at its current lumi- recipes combine three-body dynamics with stellar evolu- nosity: tion and their mutual influences, such as tidal interactions and mass transfer. These descriptions are summarized in a GmR public source code TrES with which triple evolution can τ = ,() th L be studied. where L is the luminosity of the star. In other words, when 2 Background the thermal equilibrium of a star is disturbed, the star We will give a brief overview of isolated binary evolution will move to a new equilibrium on a thermal (or Kelvin- (Section .) and isolated triple evolution (Section .). We Helmholtz) timescale. discuss in particular under what circumstances triple evo- Finally, the nuclear timescale represents the time re- lution differs from binary evolution and what the conse- quired for the star to exhaust its supply of nuclear fuel at quences are of these differences. We start with a brief sum- its current luminosity: mary of single star evolution with a focus on those aspects cm that are relevant for binary and triple evolution. τ = nucl ,() nucl L

2.1 Single stellar evolution where  is the efficiency of nuclear energy production, Hydrostatic and thermal equilibrium in a star give rise to c is the speed of light, and mnucl is the amount of mass temperatures and pressures that allow for nuclear burn- available as fuel. For core hydrogen burning,  =. ing, and consequently the emission of the starlight that we and Mnucl ≈ .M. Assuming a mass-luminosity relation observe. Cycles of nuclear burning and exhaustion of fuel of L ∝ Mα, with empirically α ≈ - (e.g. Salaris and Cas- regulate the evolution of a star, and sets the various phases sisi ;Ekeretal.), it follows that massive stars live during the stellar lifetime. shorter and evolve faster than low-mass stars. The evolution of a star is predominantly determined by For the Sun, τdyn ≈  min, τth ≈  Myr, and τnucl ≈ a single parameter, namely the stellar mass (Table ). It de-  Gyr. Typically, τdyn < τth < τnucl, which allows us to quan- pends only slightly on the initial chemical composition or titatively predict the structure and evolution of stars in the amount of core overshooting.a broad terms.

.. Hertzsprung-Russell diagram Table 1 Necessary parameters to describe a single star system, a binary and a triple The Hertzsprung-Russell (HR) diagram in Figure  shows seven evolutionary tracks for stars of different Parameters Stellar Orbital masses. The longest phase of stellar evolution is known as Single star m - the main-sequence (MS), in which nuclear burning takes Binary m1, m2 a, e place of hydrogen in the stellar core. The MS occupies the Triple m1, m2, m3 imutual, ain, ein, gin, hin, aout, eout, gout, hout region in the HR-diagram between the stellar birth on the zero-age MS (ZAMS, blue circles in Figure ) and the end For stellar parameters, age and metallicity of each star can be added. The table shows that as the multiplicity of a stellar system increases from one to three, the of the MS-phase (terminal-age MS (TAMS), blue circles in problem becomes significantly more complicated. Figure ). Stars more massive than .M contract slightly Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 3 of 36

Figure 2 Evolution of stellar radius. Radius as a function of stellar Figure 1 Hertzsprung-Russell diagram. Evolutionary tracks for age for two stars with masses 4 and 6.5M at solar metallicity. Specific seven stars in the HR-diagram with masses 1, 1.5, 2.5, 4, 6.5, 10, and moments in the evolution of the stars are noted by blue circles as for 15M at solar metallicity. Specific moments in the evolution of the Figure 1. The radius evolution is calculated with SeBa (Portegies stars are noted by blue circles as explained in the text. The tracks are Zwart and Verbunt 1996; Toonen et al. 2012). The figure also shows SeBa calculated with (Portegies Zwart and Verbunt 1996, Toonen that high-mass stars evolve faster and live shorter than lower-mass et al. 2012). The dashed lines show lines of constant radii by means of stars. the Stefan-Boltzmann law. at the end of the MS when the stellar core runs out of hy- the core, the outer layers of the star expand again and the drogen. This can be seen in Figure  as the hook in the evolutionary track bends back to right in the HR-diagram. ·  tracks leading up to the TAMS. As the core of the star reaches temperatures of   K, After the TAMS, hydrogen ignites in a shell around the carbon ignites in the star (denoted by a blue circle in Fig- core. Subsequently the outer layers of the star expand ure ). rapidly. This expansion at roughly constant luminosity re- Asthecoreofthestarbecomesdepletedofhelium,he- sults in a lower effective temperature and a shift to the right lium burning continues in a shell surrounding the inert in the HR-diagram. Stars of less than M reach effective carbon-oxygen core. The star has now has reached the temperatures as low as (.K) ,K before helium ig- supergiant-phases of its life. The star ascents the asymp- nition. At this point (denoted by a blue circle in Figure ) totic giant branch (AGB) reaching its maximum size of theystarttoascendtheredgiantbranch(RGB)whichgoes about a thousand solar radii. hand in hand with a strong increase in luminosity and ra- Figure  shows the variation of the outer radius as the dius. On the right of the RGB in the HR-diagram, lies the star evolves in its lifetime. It illustrates the dramatic in- forbidden region where hydrostatic equilibrium cannot be creasesinradiusduringtheRGB-andAGB-phasesaspre- achieved. Any star in this region will rapidly move towards viously discussed. Shrinkage of star occur after helium ig- the RGB. The red giant star consists of a dense core and an nition, and to a lesser degree at the end of the MS. The extended envelope up to hundreds of solar radii. When the radial evolution is of particular interest for binaries and temperature in the core reaches K, helium core burning triples, as a star is more likely to initiate mass transfer (i.e. commences and the red giant phase has come to an end. fill its Roche lobe) when its envelope is extended e.g. on For stars less massive than M, helium ignites degener- the RGB or AGB. ately in a helium flash. For stars more massive than M, helium ignites before their effective temperature has de- .. Stellar winds creased to a few thousand Kelvin; the shift to the right in During the lifetime of a star, a major fraction of the star’s the HR-diagram is truncated when helium ignites. mass is lost by means of stellar winds (HJGLM Lamers During helium burning the stellar tracks make a loop and Cassinelli ;Owocki). The winds deposit en- in the HR-diagram, also known as the horizontal branch. riched material back into the ISM and can even collide This branch is marked in Figure  byabluecircleatitsmax- with previously ejected matter to form stellar-wind bub- imum effective temperature. The loop goes hand in hand bles and planetary nebulae. with a decrease and increase of the stellar radius. As the Stellar winds develop for almost all stars, but the mass burning front moves from the core to a shell surrounding losses increases dramatically for more evolved stars and for Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 4 of 36

Table 2 Initial stellar mass range and the corresponding more massive stars. The winds of AGB stars (see Höfner remnant type and mass  for a review) are characterized by extremely high – – – Initial mass (M) Remnant type Remnant mass (M) mass-loss rates ( - M yr ) and low terminal ve- locities (- km s–). For stars up to M,these‘su- 1-6.5 CO WD 0.5-1.1 perwinds’ remove the entire stellar envelope. AGB-winds 6.5-8 ONe WD 1.1-1.44 8-∼23 NS 1.1-2 are driven by radiation pressure onto molecules and dust 23 BH >5 grains in the cold outer atmosphere. The winds are fur- ther enhanced by the stellar pulsations that increase the Note that the given masses represent approximate values. They are dependent gas density in the extended stellar atmosphere where the on the metallicity and on stellar evolutionary processes that are not understood well, such as stellar winds and core overshooting (footnote a). dust grains form. For massive O and B-type stars, strong winds already oc- cur on the MS. These winds (e.g. Puls et al. ;Vink hand, the conditions to sustain nuclear burning are ful- ) are driven by another mechanism, i.e. radiation pres- filled, the star can evolve further as a hydrogen-poor he- sure in the continuum and absorption lines of heavy ele- lium rich star i.e. helium MS star or helium giant star. ments. The winds are characterized with high mass-loss Due to the mass loss, the initial mass ranges given in Ta- – – – rates ( - M yr ) and high velocities (several - ble  can be somewhat larger. Furthermore, if a star with – , km s ) (e.g. Kudritzki and Puls ). For stars of aheliumcoreoflessthan∼.M (e.g. Han et al. ) ∼ more than M,themass-lossrateissufficientlylarge looses its envelope as a result of mass transfer before he- that the evolution of the star is significantly affected, as lium ignition, the core contracts to form a white dwarf thetimescaleformasslossissmallerthanthenuclear made of helium instead of CO or ONe. timescale. In turn the uncertainties in our knowledge of the stellar wind mechanisms, introduces considerable un- .. Supernova explosions certainties in the evolution of massive stars. When a high-mass star reaches the end of its life and its core collapses to a NS or BH, the outer layers of the .. Stellar remnants star explode in a core-collapse supernova (SN) event. The The evolution of a star of less than ∼.M comes to an matter that is blown off the newly formed remnant, en- end as helium burning halts at the end of the AGB. Strong riches the ISM with heavy elements. Any asymmetry in the winds strip the core of the remaining envelope and this ma- SN, such as in the mass or neutrino loss (e.g. Lai ; terial forms a planetary nebula surrounding the core. The Janka ),cangiverisetoanatal-kickvk to the star. core cools and contracts to form a white dwarf (WD) con- Neutron stars are expected to receive a kick at birth of sisting of carbon and oxygen (CO). about  km s– (e.g. Cordes et al. ;LyneandLorimer Slightly more massive stars up to ∼M experience an ; Hobbs et al. ), however smaller kick velocities additional nuclear burning phase. Carbon burning leads in the range of  km s– have been deduced for neutron to the formation of a degenerate oxygen-neon (ONe) core. stars in high-mass X-ray binaries (Pfahl et al. ). Also, Stars up to ∼M follow a similar evolutionary path dis- whether or not black holes that are formed in core-collapse cussed above, but they end their lives as oxygen-neon supernova receive a kick is still under debate (e.g. Gua- whitedwarfs.Inthemassrange∼-M,theoxygen- landris et al. ; Repetto et al. ;Wongetal.; neon core reaches the Chandrasekhar mass, and collapses Repetto and Nelemans ;Zuo). to a neutron star (NS). More massive stars than ∼M go through a rapid suc- 2.2 Binary evolution cession of nuclear burning stages and subsequent fuel ex- The evolution of a binary can be described by the masses of haustion. The nuclear burning stages are sufficiently short, the stars m and m,thesemi-majoraxisa, and the eccen- that the stellar envelope hardly has time to adjust to the tricity e. A useful picture for binaries is the Roche model, hydrodynamical and thermal changes in the core. The po- which describes the effective gravitational potential of the sition of the star in the HR-diagram remains roughly un- binary. It is generally based on three assumptions: () the changed. The stellar evolution continues until a iron core binaryorbitiscircular()therotationofthestellarcompo- is formed after which nuclear burning cannot release fur- nents are synchronized with the orbit () the stellar com- ther energy. The star then collapses to form a NS or a black ponents are small compared to the distance between them. hole (BH). An overview of the initial mass ranges and the The first two assumptions are expected to hold for bina- correspondingremnantsaregiveninTable. ries that are close to mass transfer because of tidal forces When a star is part of a compact stellar system, its evo- (Section ..). Under the three assumptions given above, lution can be terminated prematurely when the star looses the stars are static in a corotating frame of reference. The its envelope in a mass-transfer phase. After the envelope is equipotential surface around a star in which material is lost, the star may form a remnant directly. If on the other gravitationally bound to that star is called the Roche lobe. Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 5 of 36

The Roche radius is defined as the radius of a sphere with While the two stars in the binary are in orbit around each the same volume of the nearly spherical Roche lobe, and is other, the stars can accrete some of the wind material of the often approximated (Eggleton )by: other star. Including wind accretion, the orbit changes as: / a˙ –m˙ m m .q wind  β β  β  RL ≈ = – + –(– ) ,() .q/ + ln( + q/) a m m m + m q. ˙ ≈ a where the star with mass m accretes at a rate of m = . . ,() ( + q) –βm˙ .NotethatEq.()reducestoEq.()forcomplete non-conservative mass transfer i.e. β =.Windaccretion where the mass ratio q = m/m. If one of the stars in the is often modelled by Bondi-Hoyle accretion (Bondi and binary overflows its Roche lobe, matter from the outer Hoyle ; Livio and Warner ). This model considers layers of the star can freely move through the first La- a spherical accretion onto a point mass that moves through grangian point L to the companion star. Binaries with ini- a uniform medium. Wind accretion is an important pro- tial periods less than several years (depending on the stellar cess known to operate in high-mass X-ray binaries (Tauris masses) will experience at least one phase of mass transfer, and van den Heuvel ;Chaty) and symbiotic stars if the stars have enough time to evolve. (Mikolajewska ; Sokoloski ). IfthestarsdonotgetclosetoRochelobeoverflow The assumptions of a fast and spherically symmetric (RLOF), the stars in a binary evolve effectively as single wind are not always valid. The former is not strictly true stars, slowly decreasing in mass and increasing in radius for all binary stars i.e. an evolved AGB-star has a wind and luminosity until the remnant stage. The binary orbit of - km s– (e.g. Höfner ), which is comparable can be affected by stellar winds, tides and angular momen- to the velocity of stars in a binary of a ≈ R.Hydro- tum losses such as gravitational wave emission and mag- dynamical simulations of such binaries suggest that the netic braking. These processes are discussed in the follow- wind of the donor star is gravitationally confined to the ing three sections. In the last three sections of this chapter Roche lobe of the donor star (Mohamed and Podsiad- we describe how RLOF affects a binary. lowski , ; de Val-Borro et al. ). The wind can be focused towards the orbital plane and in particular to- .. Stellar winds in binaries wards the companion star. This scenario (often called wind Wind mass loss affects a binary orbit through mass and an- Roche-lobe overflow (wRLOF) or gravitational focusing) gular momentum loss. Often the assumption is made that allows for an accretion efficiency of up to %, which is the wind is spherically symmetric and fast with respect to significantly higher than for Bondi-Hoyle accretion. A re- the orbit. In this approximation, the wind does not interact quirement for wRLOF to work is that the Roche lobe of with the binary orbit directly, such that the process is adi- the donor star is comparable or smaller than the radius abatic. Furthermore, the orbital eccentricity remains con- where the wind is accelerated beyond the escape veloc- stant (Huang , ). ity. wRLOF is supported by observations of detached bina- If none of the wind-matter is accreted, the wind causes ries with very efficient mass transfer (Karovska et al. ; Blind et al. ). the orbit to widen. From angular momentum conserva- Furthermore, the assumption of adiabatic mass loss is tion, it follows as: inconsistent with binaries in which the orbital timescale ˙ ˙ is longer than the mass-loss timescale. The effects of in- awind,no-acc –m = ,()stantaneous mass loss has been studied in the context a m + m   of SN explosions, and can even lead to the disruption of the binary system (see also Section ..). However, also where m and m are the masses of the stars, m˙  is the mass wind mass-loss can have a non-adiabatic effect on the bi- lost in the wind of the star with mass m (m˙  ≤ ), a is the nary orbit (e.g. Hadjidemetriou ; Rahoma et al. ; semi-major axis of the orbit, and a˙ wind,no-acc the change in the orbital separation with no wind accretion. Eq. ()can Veras et al. )ifthemass-lossrateishighandtheor- be rewritten to: bit is wide. Under the assumption that mass-loss proceeds isotropically, the wind causes the orbit to widen, as in the af m + m case for adiabatic mass loss. However, the eccentricity may = ,() ai m + m – mwind decrease or increase, and may even lead to the disrup- tion of the binary system (see e.g. Veras et al.  for a where af and ai are the semi-major axis of the orbit before detailed analysis of the effects of winds on sub-stellar bi- and after the wind mass loss, and mwind is the amount of naries in which an exoplanet orbits a host star). Toonen, matter lost in the wind (mwind ≥ ). Hollands, Gaensicke and Boekholt, in prep. show that also Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 6 of 36

(intermediate-mass) stellar binaries can be disrupted dur- only known source of GWs for GW interferometers such ing the AGB-phases when the mass loss rates are high as LIGO, VIRGO and eLISA. (–-–M yr–) for orbital separations approximately Magnetic braking extracts angular momentum from a larger then R (P ≈  yr where P is the orbital pe- rotating magnetic star by means of an ionized stellar wind riod). (Schatzman ;Huang;Skumanich). Even Lastly, anisotropic mass-loss might occur in fast-rotating when little mass is lost from the star, the wind matter can stars or systems that harbour bipolar outflows. Rotation exert a significant spin-down torque on the star. This hap- modifies the structure and evolution of a star, and as such pens when the wind matter is forced to co-rotate with the the surface properties of the star where the wind originates magnetic field. If the star is in a compact binary and forced (see Maeder and Meynet  forareview).Foranincreas- to co-rotate with the orbit due to tidal forces, angular mo- ing rate of rotation until critical rotation, the stellar winds mentum is essentially removed from the binary orbit as increasingly depart from a spherical distribution (see e.g. well (Verbunt and Zwaan ). This drain of angular mo- Georgy et al. ). Additionally, the bipolar outflows or mentum results in a contraction of the orbit. jets are associated with protostars, evolved post-AGB stars Magnetic braking plays an important role in the or- and binaries containing compact objects. Their origin is bital evolution of interacting binaries with low-mass donor most likely linked to the central object or the accretion disk stars, such as cataclysmic variables and low-mass X-ray (e.g. O’Brien ). binaries (Knigge et al. ;TaurisandvandenHeuvel The effect of anisotropic mass loss on the orbit of a bi- ). For magnetic braking to take place, the donor star is nary system is important primarily for wide binaries (e.g. expected to have a mass between .-.M, such that the Parriott and Alcock ;Verasetal.). Specifically, star has a radiative core and convective envelope to sustain Veras et al. () show that the relative contribution√ of the magnetic field. The strength of magnetic braking is still the anisotropic terms to the overall motion scale as a. under debate and several prescriptions exist (see Knigge Ifthemasslossissymmetricaboutthestellarequator,the et al. ,forareview). mass loss does not affect the orbital motion in another way than for the isotropic case. Veras et al. ()concludethat .. Tides the isotropic mass-loss approximation can be used safely The presence of a companion star introduces tidal forces to model the orbital evolution of a planet around a host star in the binary system that act on the surface of the star and until orbital separations of hundreds of AU. For a fixed to- lead to tidal deformation of the star. If the stellar rotation is tal mass of the system, the effects of anisotropic mass loss are further diminished with decreasing mass ratio (i.e. for not synchronized or aligned with the binary orbit, the tidal systems with more equal masses), such that the assump- bulges are misaligned with the line connecting the cen- tion of isotropic mass-loss is robust until even larger or- tres of mass of the two stars. This produces a tidal torque bital separations for stellar binaries. that allows for the transfer of angular momentum between the stars and the orbit. Additionally, energy is dissipated .. Angular momentum losses in the tides, which drains energy from the orbit and rota- Angular momentum loss from gravitational waves (GW) tion. Tidal interaction drives the binary to a configuration and magnetic braking act to shrink the binary orbit (e.g. of lowest energy e.g. it strives to circularize the orbit, syn- Peters ; Verbunt and Zwaan ). Ultimately this can chronize the rotation of the stars with the orbital period lead to RLOF of one or both components and drive mass and align the stellar spin with respect to the orbital spin. transfer. See Zahn ()andZahn() for recent reviews. The strength of GW emission depends strongly on the For binaries with extreme mass ratios, a stable solution semi-major axis, and to lesser degree on the eccentricity. does not exist (Darwin ;Hut). In this scenario a It affects the orbits as: star is unable to extract sufficient angular momentum from  the orbit to remain in synchronized rotation. Tidal forces  G mm(m + m)     a˙ gr =– + e + e () will cause the orbit to decay and the companion to spiral  ca( – e)/   into the envelope of the donor star. This tidal instability occurs when the angular momentum of the star J >  J , and   b  with Jb the orbital angular momentum and J = I ,whereI  Gm m (m + m )  is the moment of inertia and  the spin angular frequency. e˙ =– e     + e ,() gr  ca( – e)/  Hut () derives a general qualitative picture of tidal evolution and its effect on the orbital evolution of a binary where a˙ gr and e˙gr are the change in orbital separation and system. Hut () considers a model in which the tides as- eccentricity averaged over a full orbit (Peters ). Ac- sume their equilibrium shape, and with very small devia- cordingly, GW emission affects most strongly the com- tions in position and amplitude with respect to the equipo- pact binaries. These binaries are a very interesting and the tential surfaces of the stars. If a companion star with mass Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 7 of 36

m raises tides on a star with mass m, the change of binary dissipation mechanisms, it is generally assumed that cir- parameters due to tidal friction is: cularization and synchronization is achieved before RLOF in a binary. k R  a a˙ am q˜ q˜ TF =– ( + )  / .. Mass transfer τTF a ( – e ) Whether or not mass transfer is stable depends on the re-   /   sponse of the donor star upon mass loss, and the reaction × f e – –e f e ,() b of the Roche lobe upon the re-arrangement of mass and k R  e angular momentum within the binary (e.g. Webbink ; e˙ am q˜ q˜ TF =– ( + )  / Hjellming and Webbink ;PolsandMarinus; τTF a ( – e ) Soberman et al. ). If the donor star stays approxi-    /   mately within its Roche lobe, mass transfer is dynami- × f e – –e f e ,()  b cally stable. When this is not the case, the donor star will overflow its Roche lobe even further as mass is removed. k q˜ R   ˙ am b This leads to a runaway situation that progresses into a TF =    τTF k a ( – e ) common-envelope (CE, Paczynski ). During the CE-  phase, the envelope of the donor star engulfs both stars ×   /  f e – –e f e ,()causing them to spiral inwards until both stars merge or b the CE is expelled. Due to the mass loss, the donor star falls out of hy- where q˜ ≡ m/m,andb =π/P is the mean orbital an- drostatic and thermal equilibrium. The radius of the star gular velocity. The star with mass m has an apsidal mo- changes as the star settles to a new hydrostatic equilib- tion constant kam, gyration radius k,andspinangularfre- rium, and subsequently thermal equilibrium. The stellar quency . τTF represents the typical timescale on which response upon mass loss depends critically on the struc- significant changes in the orbit take place through tidal ture of the stellar envelope i.e. the thermal gradient and en- evolution. The parameters f (e)arepolynomialexpres- n tropy of the envelope. In response to mass loss, stars with sions given by (Hut ): a deep surface convective zone tend to expand, whereas ⎧ stars with a radiative envelope tend to shrink rapidly. ⎪          ⎪f(e )=+  e +  e +  e +  e , Therefore, giant donor stars with convective envelopes ⎪ ⎪        favour CE-evolution upon RLOF.c As giants have radii of ⎨⎪f(e )=+  e +  e +  e , several hundreds to thousands of Solar radii, the orbit at        () ⎪f(e )=+  e +  e +  e , ⎪ the onset of mass transfer is of the same order of mag- ⎪      nitude. On the other hand, donor stars on the MS with ⎪f(e )=+  e +  e , ⎩⎪ radiative envelope often lead to dynamically stable mass f (e)=+e +  e.   transfer in binaries with short orbital periods (e.g. Toonen et al. ). The degree of tidal interaction strongly increases with the ratio of the stellar radii to the semi-major axis of the orbit .. Common-envelope evolution (Eqs. (), ()and()). Therefore, tidal interaction mostly During the CE-phase, the core of the donor star and the affect the orbits of relatively close binaries, unless the ec- companion are contained within a CE. Friction between centricities are high and/or the stellar radii are large. these objects and the slow-rotating envelope is expected The tidal timescale T (Eqs. ()-()) is subject to debate to cause the objects to spiral-in. If this process does not due to quantitative uncertainties in tidal dissipation mech- release enough energy and angular momentum to drive anisms (Witte and Savonije ; Willems ;Meibom off the entire envelope, the binary coalesces. On the other and Mathieu ). Tidal dissipation causes the misalign- hand if a merger can be avoided, a close binary remains ment of the tidal bulges with the line connecting the cen- in which one or both stars have lost their envelopes. The tres of mass of the two stars. For stars (or planets) with an evolution of such a star is significantly shortened, or even outer convection zone, the dissipation is often attributedb terminated prematurely if it directly evolves to a remnant toturbulentfrictionintheconvectiveregionsofthestar star. (Goldman and Mazeh ;Zahn, ). For stars The systems that avoid a merger lose a significant with an outer radiation zone, the dominant dissipation amount of mass and angular momentum during the CE- mechanism identified so far is radiative damping of stellar phase. The orbital separation of these systems generally oscillations that are exited by the tidal field i.e. dynamical decreases by two orders of magnitude, which affects the tides (Zahn , ). Despite the uncertainties in tidal further evolution of the binary drastically. The CE-phase Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 8 of 36

plays an essential role in the formation of short-period sys- where Jb,i and Jb,f are the orbital angular momentum of the tems with compact objects, such as X-ray binaries, and pre- and post-mass transfer binary respectively. The pa- cataclysmic variables. In these systems the current orbital rameters md and ma represent the mass of the donor and separation is much smaller than the size of the progenitor accretor star, respectively, and md is the mass lost by the of the donor star, which had giant-like dimensions at the donor star. The γ -parameter describes the efficiency with onset of the CE-phase. which orbital angular momentum is used to blow away the Despite of the importance of the CE-phase and the enor- CE. mous efforts of the community, the CE-phase is not under- Valuable constraints on CE-evolution have come from stood in detail (see Ivanova et al.  for a review). The evolutionary reconstruction studies of observed samples CE-phase involves a complex mix of physical processes, of close binaries and from comparing those samples with such as energy dissipation, angular momentum transport, the results of binary population synthesis studies. The and tides, over a large range in time- and length-scales. emerging picture is that for binaries with low mass ra- A complete simulation of the CE-phase is still beyond our tios, the CE-phase leads to a shrinkage of the orbit. For reach, but great progress has been made with hydrody- the formation of compact WD-MS binaries with low-mass namical simulations in the last few years (Ricker and Taam MS companions, the orbit shrinks strongly (αλce ≈ ., ;Passyetal.b;Nandezetal.). The uncer- see Zorotovic et al. ; Toonen and Nelemans ; tainty in the CE-phase is one of the aspects of the theory Portegies Zwart ; Camacho et al. ). However, for of binary evolution that affects our understanding of the the formation of the second WD in double WDs, the or- evolutionary history of a specific binary population most bit only shrinks moderately (αλce ≈ , see Nelemans et al. (e.g. Toonen and Nelemans ;Toonenetal.). , ; van der Sluys et al. ). When binaries with Theclassicalwaytotreattheorbitalevolutiondueto approximately equal masses come in contact, mass trans- the CE-phase, is the α-formalism. This formalism consid- fer leads to a modest widening of the orbit, alike the γ - ers the energy budget of the initial and final configuration formalism (Nelemans et al. , ). The last result is (Tutukov and Yungelson ); basedonastudyofthefirstphaseofmasstransferfordou- bleWDs,inwhichthefirstWDisformed.Woodsetal. E = α(E – E ), () gr orb,i orb,f () suggested that this mass transfer episode can occur stably and non-conservatively even with donor star (early) where E is the binding energy of the envelope, E and gr orb,i on the red giant branch. Further research is needed to see if Eorb,f are the orbital energy of the pre- and post-mass trans- fer binary. The α-parameter describes the efficiency with this evolutionary path suffices to create a significant num- which orbital energy is consumed to unbind the CE. When ber of double WDs. both stars have loosely bound envelopes, such as for gi- ants, both envelopes can be lost simultaneously (hereafter .. Stable mass transfer double-CE, see Brown ;Nelemansetal.). In Whereas the duration of the CE-phase is likely of the or- der of  yr (i.e. the thermal timescale of the envelope), Eq. () Egr is then replaced by the sum of the binding en- ergy of each envelope to its host star: stable mass transfer occurs on much longer timescales. Several driving mechanisms exist for stable mass transfer

Egr, + Egr, = α(Eorb,i – Eorb,f ). () with their own characteristic mass transfer timescales. The donor star can drive Roche lobe overflow due to its nu- The binding energy of the envelope of the donor star in clear evolution or due to the thermal readjustment from Eqs. ()and()isgivenby: the mass loss. Stable mass transfer can also be driven by the contraction of the Roche lobe due to angular momentum Gmdmd,env losses in the system caused by gravitational wave radiation Egr = ,() λceR or magnetic braking. When mass transfer proceeds conservatively the change where R is the radius of the donor star, Md,env is the enve- in the orbit is regulated by the masses of the stellar com- lope mass of the donor and λce depends on the structure ponents.Forcircularorbits, of the donor (de Kool et al. ;DewiandTauris; Xu and Li ; Loveridge et al. ). The parameters λce  af md,ima,i and α are often combined in one parameter αλce. = ,() According to the alternative γ -formalism (Nelemans ai md,f ma,f et al. ), angular momentum is used to expel the enve- where the subscript i and f denote the pre- and post-mass lope of the donor star, according to: transfer values. In general, the donor star will be the more massive component in the binary and the binary orbit will Jb,i – Jb,f md = γ ,()initially shrink in response to mass transfer. After the mass Jb,i md + ma Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 9 of 36

ratio is approximately reversed, the orbit widens. In com- from the outer layers is mixed into the nuclear-burning parison with the pre-mass transfer orbit, the post-mass zone, the star is ‘rejuvenated’ (see e.g. Vanbeveren and De transfer orbit is usually wider with a factor of a few (Too- Loore ). These stars can appear significantly younger nen et al. ). than their co-eval neighbouring stars in a cluster.d Sec- If the accretor star is not capable of accreting the matter ondly, the accretor star adjusts its structure to a new equi- conservatively, mass and angular momentum are lost from librium. If the timescale of the mass transfer is shorter the system. The evolution of the system is then dictated by than the thermal timescale of the accretor, the star will how much mass and angular momentum is carried away. temporarily fall out of thermal equilibrium. The radial re- Assuming angular momentum conservation and neglect- sponse of the accretor star will depend on the structure of ing the stellar rotational angular momentum compared to the envelope (as discussed for donor stars in Section ..). the orbital angular momentum, the orbit evolves as (e.g. A star with a radiative envelope is expected to expand upon Massevitch and Yungelson ;PolsandMarinus; mass accretion, whereas a star with a convective envelope Postnov and Yungelson ): shrinks. In the former case, the accretor may swell up suf- ficiently to fill its Roche lobe, leading to the formation of a a˙ m˙ m contact binary. =– d –β d a md ma .. Supernova explosions in binaries  m If the collapsing star is part of a binary or triple, natal kick –(–β) η + d ,()  md + ma vk alters the orbit and it can even unbind the system. Under the assumption that the SN is instantaneous and the SN- where the accretor star captures a fraction β ≡ –m˙a/m˙d of shell does not impact the companion star(s), the binary or- the transferred matter, and the matter that is lost carries bit is affected by the mass loss and velocity kick (Hills ; specific angular momentum h equal to a multiple η of the Kalogera ; Tauris and Takens ; Pijloo et al. ) specific orbital angular momentum of the binary: through: J˙ J af m h ≡ = η b .()= – m˙ + m˙ m + m ai mt,i d a a d a m (v · v ) v – Different modes of angular momentum loss exist which · i i k k – –  –  ,() can lead to a relative expansion or contraction of the or- ri mt,i vc vc bit compared to the case of conservative mass transfer where ai and af are the semi-major axis of the pre-SN and (Soberman et al. ;Toonenetal.). For example, post-SN orbit, m is the mass lost by the collapsing star, the generic description of orbital evolution of Eq. ()re- mt,i is the total mass of the system pre-SN, ri is the pre-SN duces to that of conservative mass transfer (Eq. ()) for distance between the two stars, v is the pre-SN relative ˙ ˙ i β =orma = md.Also,Eq.()reducestoEq.()describ- velocity of the collapsing star relative to the companion, ing the effect of stellar winds on the binary orbit, under and the assumption of specific angular momentum loss equal to that of the donor star (h = Jd/md = ma/md · Jb/(md + ma) ≡ Gmt,i or η = ma/md). Depending on which mode of angular mo- vc () ai mentum loss is applicable, the further orbital evolution and stability of the system varies. is the orbital velocity in a circular orbit. A full derivation of Stable mass transfer influences the stellar evolution of this equation and that for the post-SN eccentricity is given the donor star and possibly that of the companion star. The in Appendix A.. Note that the equation for the post-SN donorstarisaffectedbythemassloss,whichleadstoa eccentricity of Eq. (a) in Pijloo et al. ()isincomplete. change in the radius on long timescales compared to a sit- Eq. () shows that with a negligible natal kick, a binary uation without mass loss (Hurley et al. ). Stable mass survives the supernova explosion if less than half of the transfer tends to terminate when the donor star has lost mass is lost. Furthermore, the binary is more likely to sur- most of its envelope, and contracts to form a remnant star vive if the SN occurs at apo-astron. With substantial natal or to a hydrogen-poor helium rich star. In the latter case kicks compared to the pre-SN orbital velocity, survival of the evolution of the donor star is significantly shortened, the binary depends on the magnitude ratio and angle be- and in the former it is stopped prematurely, similar to what tween the two (through vi ·vk in Eq. ()). Furthermore, the was discussed previously for the CE-phase. rangeofanglesthatleadtosurvivalislargeratperi-astron If the companion star accretes a fraction or all of the than apo-astron (Hills ). If the direction of the natal transferred mass, evolution of this star is affected as well. kick is opposite to the orbital motion of the collapsing star, Firstly, if due to accretion, the core grows and fresh fuel the binary is more likely to survive the SN explosion. Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 10 of 36

aout aout 2.3 Triple evolution wheresystemsareunstableif < |crit and qout ≡ ain ain The structures of observed triples tend to be hierarchical, m . This criterion is based on the concept of chaos and m+m i.e. the triples consist of an inner binary and a distant star the consequence of overlapping resonances. The criterion (hereafter outer star) that orbits the centre of mass of the is conservative, as the presence of chaos in some cases inner binary (Hut and Bahcall ). To define a triple star is not necessarily the same as an instability. By compari- system, no less than  parameters are required (Table ): son with numerical integration studies, it was shown that - the masses of the stars in the inner orbit m and m, Eq. () works well for a wide range of parameters (Aarseth and the mass of the outer star in the outer orbit m; and Mardling ;Aarseth). - the semi-major axis a, the eccentricity e,theargument Most observed triples have hierarchical structures, be- of pericenter g of both the inner and outer orbits. cause democratic triples tend to be unstable and short- Parameters for the inner and outer orbit are denoted lived (van den Berk et al. ). Hierarchical triples that with a subscript ‘in’ and ‘out’,respectively; are born in a stable configuration can become unstable - the mutual inclination ir between the two orbits. The as they evolve. Eq. () shows that when the ratio of the longitudes of ascending nodes h specify the semi-major axes of the outer and inner orbit decreases suf- orientation of the triple on the sky, and not the relative ficiently, the system enters the instability regime. Physi- orientation. Therefore, they do not affect the intrinsic cal mechanisms that can lead to such an event, are stellar dynamical evolution. From total angular momentum winds from the inner binary and stable mass transfer in the conservation hin – hout = π for a reference frame with inner binary (Kiseleva et al. ;IbenandTutukov; the z-axis aligned along the total angular momentum Freire et al. ; Portegies Zwart et al. ). Regard- vector (Naoz et al. ). ing wind mass losses from the inner binary exclusively, In some cases, the presence of the outer star has no sig- the fractional mass losses |m˙ |/(m + m)>|m˙ |/(m + m + nificant effect on the evolution of the inner binary, such m). Therefore, the fractional orbital increases a˙ in/ain > that the evolution of the inner and outer binary can be a˙ out/aout,followingEq.(). Perets and Kratter ()shows described separately by the processes described in Sec- that such a triple evolution dynamical instability (TEDI) tions . and .. In other cases, there is an interaction be- lead to close encounters, collisions, and exchanges be- tween the three stars that is unique to systems with mul- tween the stellar components. They find that the TEDI tiplicities of higher orders than binaries. In this way, many evolutionary channel caused by stellar winds is responsible new evolutionary pathways open up compared to binary for the majority of stellar collisions in the Galactic field. evolution. The additional processes are described in the following sections, such as the dynamical instability and .. Lidov-Kozai mechanism Lidov-Kozai cycles. Secular dynamics can play a mayor role in the evolution of triple systems. The key effect is the Lidov-Kozai mech- .. Stability of triples anism (Lidov ;Kozai), see Section . for an The long-term behaviour of triple systems has fascinated example of a triple undergoing Lidov-Kozai cycles. Due scientists for centuries. Not only stellar triples have been to a mutual torque between the inner and outer binary investigated, but also systems with planetary masses, such orbit, angular momentum is exchanged between the or- as the Earth-Moon-Sun system by none other than Isaac bits. The orbital energy is conserved, and therefore the Newton. It was soon realised that the three-body prob- semi-major axes are conserved as well (e.g. Mardling and lem does not have closed-form solutions as in the case for Aarseth ). As a consequence, the orbital inner eccen- two-body systems. Unstable systems dissolve to a lower or- tricity and mutual inclination vary periodically. The max- der systems on dynamical timescales (van den Berk et al. imum eccentricity of the inner binary is reached when the ). inclination between the two orbits is minimized. Addition- It is hard to define the boundary between stable and un- ally, the argument of pericenter may rotate periodically stable systems, as stability can occur on a range of time- (also known as precession or apsidal motion) or librate. scales. Therefore, many stability criteria exist (Mardling For a comprehensive review of the Lidov-Kozai effect, see ; Georgakarakos ), that can be divided in three Naoz (). categories: analytical, numerical integration and chaotic The Lidov-Kozai mechanism is of great importance in criteria. The commonly used criterion of Mardling and several astrophysical phenomena. For example, it can play Aarseth (): a mayor role in the eccentricity and obliquity of exoplan- ets (e.g. Holman et al. ; Veras and Ford ;Naoz aout . .i = – et al. ) including high-eccentricity migration to form π ain crit –eout hot Jupiters (e.g. Wu and Murray ; Correia et al. ; / Petrovich ), and for accretion onto black holes in the (. + qout) · ( + eout) · √ ,()context of tidal disruption events (e.g. Chen et al. ; –eout Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 11 of 36

Wegg and Nate Bode ) or mergers of (stellar and tupole term compared to the quadropole term, and is de- super-massive) black hole binaries (e.g. Blaes et al. ; fined by: Miller and Hamilton ;Antoninietal.). In par- ticular for the evolution of close binaries, the Lidov-Kozai m – m ain eout oct =  .() oscillations may play a key role (e.g. Harrington ; m + m aout –eout Mazeh and Shaham ;Kiselevaetal.;Fabrycky | |  and Tremaine ;NaozandFabrycky), e.g. for Generally, when oct ., the eccentric Lidov-Kozai black hole X-ray binaries (Ivanova et al. ), blue strag- mechanism can be of importance (Naoz et al. ; glers (Perets and Fabrycky ), and supernova type Ia Shappee and Thompson ). progenitors (Thompson ;Hamersetal.). The dynamical behaviour of a system undergoing reg- When the three-body Hamiltonian is expanded to qua- ular or eccentric Lidov-Kozai cycles can lead to extreme situations. For example, as the eccentricity of the inner drupole order in a /a , the timescale for the Lidov-Kozai in out orbit increases, the corresponding pericenter distance de- cycles is (Kinoshita and Nakai ): creases. The Lidov-Kozai mechanism is therefore linked to  a possible enhanced rate of grazing interactions, physical Pout m + m + m  / tKozai = α –eout ,()collisions, and tidal disruptions events of one of the stel- Pin m lar components (Ford et al. ;Thompson), and to the formation of eccentric semi-detached binaries (Sec- where P and P are the periods of the inner and outer in out tion ..). orbit, respectively. The dimensionless quantity α depends weakly on the mutual inclination, and on the eccentricity .. Lidov-Kozai mechanism with mass loss and argument of periastron of the inner binary, and is of Eqs. ()and() show that the relevance of the Lidov- order unity (Antognini ). The timescales are typically Kozai mechanism for a specific triple strongly depends on much longer than the periods of the inner and outer binary. the masses and mass ratios of the stellar components. If Within the quadrupole approximation, the maximum one of the components loses mass, the triple can change eccentricity e is a function of the initial mutual incli- max from one type of dynamical behaviour to another type. For nation i as (Innanen et al. ): i example, mass loss from one of the stars in the inner bi- nary, can increase |oct| significantly. As a result the triple   can transfer from a regime with regular Lidov-Kozai cycles emax = – cos (ii), ()  to a regime where the eccentric Lidov-Kozai mechanism is active. This behaviour is known as mass-loss induced in the test-particle approximation (Naoz et al. ), i.e. eccentric Kozai (MIEK) (Shappee and Thompson ; nearly circular orbits (ein =,eout = ) with one of the in- Michaely and Perets ). See also Section . for an ex- ner two bodies a massless test particle (m m, m)and ◦ ample of this evolutionary pathway. the inner argument of pericenter gin = .Inthiscase,the The inverse process (inverse-MIEK), when a triple (regular) Lidov-Kozai cycles only take place when the ini- ◦ changes state from the octupole to the quadrupole regime, tial inclination is between .-. . For larger inner ec- can also occur. Eq. () shows this is the case when mass centricities, the range of initial inclinations expands. loss in the inner binary happens to create an fairly equal For higher orders of ain/aout i.e. the octupole level of ap- mass binary, or when the semi-major axis of the outer orbit proximation, even richer dynamical behaviour is expected increases. This latter is possible when the outer star loses than for the quadrupole approximation (e.g. Ford et al. mass in a stellar wind (Section ..). ;Blaesetal.; Lithwick and Naoz ;Naoz Another example comes from Michaely and Perets et al. ; Shappee and Thompson ; Teyssandier et al. (), who studied the secular freeze-out (SEFO). In this ). The octupole term is non-zero when the outer or- scenario mass is lost from the inner binary such that the bitiseccentricorifthestarsintheinnerbinaryhaveun- Lidov-Kozai timescale increases (Eq. ()). This induces equal masses. Therefore it is often deemed the ‘eccentric aregimechangefromthequadrupoleregime,toastate Lidov-Kozai mechanism’. In this case the z-component of where secular evolution is either quenched or operates on the angular momentum of the inner binary is no longer excessively long time-scales. conserved. It allows for a flip in the inclination such that The three examples given above illustrate that the dy- the inner orbit flips from prograde to retrograde or vice namical evolution of a triple system is intertwined with the versa (hereafter ‘orbital flip’). Another consequence of the stellar evolution of its components. Thus, in order to gain a eccentric Lidov-Kozai mechanism is that the eccentricity clear picture of triple evolution, both three-body dynamics of the inner binary can be excited very close to unity. The and stellar evolution need to be taken into account simul- octupole parameter oct measure the importance of the oc- taneously. Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 12 of 36

.. Precession if an outer star is present that gives rise to Lidov-Kozai cy- Besides precession caused by the Lidov-Kozai mechanism, cles on a short timescale (Thompson ). This is impor- other sources of precession exist in stellar triples. These tant in the context of supernova type Ia and gamma-ray include general relativistic effects (Blaes et al. ): bursts. The combination of Lidov-Kozai cycles with tidal fric- a tion (hereafter LKCTF) can also lead to an enhanced for- g˙ = b .() GR c( – e) mation of close binaries (Mazeh and Shaham ;Kisel- eva et al. ). This occurs when a balance can be reached Furthermore, orbital precession can be caused by the between the eccentricity excitations of the (regular or ec- distortions of the individual stars by tides (Smeyers and centric) Lidov-Kozai mechanism and the circularisation Willems ): due to tides.f The significance of LKCTF is illustrated by Fabrycky and Tremaine (), who show that MS bina-  ries with orbital periods of .-d are produced from bi- kam   ma R g˙ e e  tides =   + + ,()naries with much longer periods up to  d. Observation- ( – e ) b   md a ally, % of close low-mass MS binaries are indeed part of and by intrinsic stellar rotation (Fabrycky and Tremaine atriplesystem(Tokovininetal.). Several studies of ): LKCTF for low-mass MS stars exist (Mazeh and Shaham ; Eggleton and Kiseleva-Eggleton ;Fabryckyand  Tremaine ; Kisseleva-Eggleton and Eggleton ; k  m + m R g˙ = am d a ,()Hamers et al. ), however, a study of the effectiveness rotate ( – e) m a b d of LKCTF for high-mass MS triples or triples with more evolved components is currently lacking. Due to the ra- where m is the mass of the distorted star that instigates d diative envelopes of high-mass stars, LKCTF is likely less the precession and m the companion star in the two-body a effective compared to the low-mass MS case. However, orbit. The distorted star has a classical apsidal motion con- evolved stars develop convective envelopes during the gi- stant k , radius R,andaspinfrequency.Theprecession am ant phases for which tidal friction is expected to be effec- rates in Eq. (), as well as Eq. ()andEq.(), are always tive. Hence, in order to understand the full significance of positive. This implies that relativistic effects, tides and stel- LKCTF for triple evolution, it is necessary to model three- lar rotation mutually stimulate precession in one direction. body dynamics and stellar evolution consistently. Note, that precession due to these processes also take place in binaries, which affects the binary orientation, but not .. Mass transfer initiated in the inner binary the evolution of the system. e In Section ..,wedescribedtheeffectofmasstrans- If the timescales for these processes become compara- fer on a circularized and synchronized binary. However, ble or smaller than the Lidov-Kozai timescales, the Lidov- as Lidov-Kozai cycles can lead effectively to RLOF in ec- Kozai cycles are suppressed. Because Lidov-Kozai cycles centric inner binaries, the simple picture of synchroniza- are driven by tidal forces between the outer and inner or- tion and circularisation before RLOF, is no longer gener- bit, the additional precession tends to destroy the reso- ally valid for triples. In an eccentric binary, there does not nance (Liu et al. a). As a result of the suppression of exist a frame in which all the material is corotating, and the cycles, the growth of the eccentricity is limited, and or- the binary potential becomes time-dependent. Studies of bital flips are limited to smaller ranges of the mutual incli- the Roche lobe for eccentric and/or asynchronous bina- nation (Naoz et al. ;Petrovich;Liuetal.a). ries, show that the Roche lobe can be substantially altered (Plavec ; Regös et al. ;Sepinskyetal.a). In .. Tides and gravitational waves an eccentric orbit, the Roche lobe of a star at periastron As mentioned earlier, the Lidov-Kozai mechanism can may be significantly smaller than that in a binary that is cir- lead to very high eccentricities that drives the stars of cularized at the same distance rp = a(–e). The Roche lobe the inner binary close together during pericenter passage. is smaller for stars that rotate super-synchronously at peri- During these passages, tides and GW emission can effec- astron compared to the classical Roche lobe (Eq. ()), and tively alter the orbit (Mazeh and Shaham ;Kiseleva larger for sub-synchronous stars. It is even possible that et al. ). Both processes are dissipative, and act to cir- the Roche lobe around the accretor star opens up. When cularize the orbit and shrink the orbital separation (Sec- mass is transferred from the donor star through L, it is not tions .. and ..). The combination of Lidov-Kozai cy- necessarily captured by the accretor star, and mass and an- cles with tides or GW emission can then lead to an en- gular momentum may be lost from the binary system. hanced rate of mergers and RLOF. For GW sources, the The modification of the Roche lobe affects the evolution merger time of a close binary can be significantly reduced, of the mass transfer phase, e.g. the duration and the mass Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 13 of 36

loss rate. Mass transfer in eccentric orbits of isolated bina- matter may be lost from the inner binary as a result of ries has been studied in recent years with SPH techniques a common-envelop phase, which we will discuss in Sec- (Layton et al. ;Regösetal.;Churchetal.; tion ... Lajoie and Sills ;vanderHelmetal.)aswellas If matter escapes the inner binary, its velocity must ex- analytical approaches (Sepinsky et al. b, , ; ceedtheescapevelocity: Davis et al. ; Dosopoulou and Kalogera a, b). These studies have shown that (initially) the mass transfer G(m + m) is episodic. The mass transfer rate peaks just after perias- vesc,in = ,() a ( + e ) tron, and its evolution during the orbit shows a Gaussian- in in like shape with a FWHM of about % of the orbital period. and analogously, to escape from the outer binary, and the The long-term evolution of eccentric binaries under- triple as a whole: going mass transfer can be quite different compared to circular binaries. The long-term evolution has been stud- ied with analytics adopting a delta-function for the mass G(m + m + m) vesc,out = .() transfer centred at periastron (Sepinsky et al. b, , aout( + eout) ; Dosopoulou and Kalogera a, b). Under these assumptions, the semi-major axis and eccentricity For stable triples, such that aout/ain  , vesc,in > vesc,out un- canincreaseaswellasdecreasedependingontheprop- less m  f (m + m). The factor f is of the order of one, erties of the binary at the onset of mass transfer. In other e.g. for circular orbits f = . In the catalogue of Tokovinin words, the secular effects of mass transfer can enhance (b) it is uncommon that m  (m + m). Out of  and compete with the orbital effects from tides. Therefore, systems, there are  systems with (m + m)

is expelled through a CE-phase. For isolated binaries the other stars (Section .). In about % of the triples in the CE-phaseanditseffectontheorbitisanunsolvedprob- Tokovinin catalogue, the outer orbit is significantly small lem, and the situation becomes even more complicated for that the outer star is expected to fill its Roche lobe at some triples. In the inner binary the friction between the stars point in its evolution (de Vries et al. ). and the material is expected to cause a spiral-in. If the outer What happens next has not been studied to great ex- star in the triple is sufficiently close to the inner binary, the tent,i.e.thelong-termevolutionofatriplesystemwith matter may interact with the outer orbit such that a sec- a mass-transferring outer star. It is an inherently compli- ond spiral-in takes place. If on the other hand, the outer cated problem where the dynamics of the orbits, the hy- star is in a wide orbit, and the CE-matter is lost in a fast drodynamics of the accretion stream and the stellar evolu- and isotropic manner, the effect on the outer orbit would tion of the donor star and its companion stars need to be be like a stellar wind (Section ..). Veras and Tout () taken into account consistently. study the effect of a CE in a binary with a planet on a wider Such a phase of mass transfer has been invoked to ex- orbit. Assuming that the CE affects the planetary orbit as plain the triple system PSR J+, consisting of a an isotropic wind, they find that planetary orbits of a few millisecond pulsar with two WD companions (Tauris and R are readily dissolved. In this scenario the CE-phase van den Heuvel ;SabachandSoker). Tauris and operates virtually as a instantaneous mass loss event, and van den Heuvel () note one of the major uncertain- therefore the maximum orbital separation for the outer or- ties in their modelling of the evolution of PSR J+ bittoremainboundisstronglydependentontheuncertain comes from the lack of understanding of the accretion onto timescale of the CE-event (Section ..). Disruption of a the inner binary system and the poorly known specific or- triple due to a CE-event may also apply to stellar triples, bital angular momentum of the ejected mass during the however, the effect is likely less dramatic, as the relative outer mass transfer phase. Sabach and Soker ()pro- mass lost in the CE-event to the total system mass is lower. poses that if the inner binary spirals-in to the envelope of Note that most hydrodynamical simulations of common- the expanding outer star, the binary can break apart from envelope evolution show that matter is predominantly lost tidal interactions. in the orbital plane of the inner binary (e.g. Ricker and To the best of our knowledge, only de Vries et al. () Taam ;Passyetal.b), however, these simulations have performed detailed simulations of mass transfer ini- are not been able to unbind the majority of the envelope. tiated by the outer star in a triple. They use the same soft- In contrast, in the recent work of Nandez et al. (), the ware framework (AMUSE,Section)asweuseforour envelope is expelled successfully due to the inclusion of code TrES.deVriesetal.()simulatethemasstrans- recombination energy in the equation of state. These sim- ferphaseinitiatedbytheouterstarfortwotriplesinthe ulations show a more spherical mass loss. Roughly % of Tokovinin catalogue, ξ Tau and HD. For both sys- theenvelopemassisejectedduringthespiral-inphasein tems, they find that the matter lost by the outer star does the orbital plane, while the rest of the mass is ejected after not form an accretion disk or circumbinary disk, but in- the spiral-in phase in a closely spherical way (priv. comm. stead the accretion stream intersects with the orbit of Jose Nandez). the inner binary. The transferred matter forms a gaseous The first scenario of friction onto the outer orbit has cloud-like structure and interacts with the inner binary, been proposed to explain the formation of two low-mass similar to a CE-phase. The majority of the matter is ejected X-ray binaries with triple components (U + from the inner binary, and the inner binary shrinks mod- (V Cyg) (Portegies Zwart et al. )andPSRJ+ erately to weakly with αλce   depending on the mutual  (Tauris and van den Heuvel )), however, it could inclination of the system. In the case of HD, this con- notberuledoutthatthedesireddecreaseintheouteror- traction leads to RLOF in the inner binary. The vast major- bital period did not happen during the SN explosion in ity of the mass lost by the donor star is funnelled through which the compact object was formed. Currently, it is un- L, and eventually ejected from the system by the inner clear if the CE-matter is dense enough at the orbit of the binary through the L Lagrangian pointg of the outer or- outer star to cause significant spiral-in. Sabach and Soker bit. As a consequence of the mass and angular momen- () suggests that if there is enough matter to bring the tum loss, the outer orbit shrinks withh η ≈ - in Eq. (). outer star closer, the CE-phase would lead to a merger in During the small number of outer periods that are mod- the inner binary. elled, the inner and outer orbits approaches contraction at the same fractional rate. Therefore the systems remain dy- .. Mass loss from the outer star namically stable. In about % of the multiples in the Tokovinin cata- Systems that are sufficiently wide that the outer star does logue of multiple star systems in the Solar neighbourhood, not fill its Roche lobe, might still be affected by mass loss the outer star is more massive than the inner two stars. from the outer star in the form of stellar winds. Soker For these systems the outer star evolves faster than the () has studied this scenario for systems where the Toonen et al. 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outer star is on the AGB, such that the wind mass loss rates which the outer star collapses in a SN event, the inner bi- are high. Assuming Bondi-Hoyle-Littleton accretion and nary is not affected, and the effect on the outer orbit can be approximated by that of an isolated binary. For a SN taking  R Racc,column ain Racc,B-H aout,()place in the inner binary, the inner binary itself is modified similar to an isolated binary (Section ..). The effect on where Racc,column is the width of the accretion column at the theouterbinarycanbeviewedasthatofanisolatedbinary binary location, and Racc,B-H the Bondi-Hoyle accretion ra- in which the inner binary is replaced by an effective star at dius, Soker ()findsthatalargefractionoftriplesthe the center of mass of the inner binary. The effective star stars in the inner binary may accrete from an accretion disk changes mass and position (as the center of mass changes) around the stars. The formation of an accretion disk de- instantaneously in the SN event. The semi-major axis of pends strongly on the orientation of the inner and outer the outer orbit is affected by the SN in the inner binary as: orbit. When the inner and outer orbits are parallel to each other, no accretion disk forms. On the other hand, when af m aim = – · – the inner orbit is orientated perpendicular to the outer or- ai mt,i rf mt,i  bit, an accretion disk forms if q .. In this case the ac-  – a(vi · vsys) vsys r – r cretion is in a steady-state and mainly towards the most a i f –  –  + i ,() massive star of the inner binary. vc vc rirf

.. Triples and planetary nebulae where mt,i is the total mass of the pre-SN triple, ri and rf Interesting to mention in the context of triple evolution are the pre-SN and post-SN distance between the star in are planetary nebulae (PNe), in particular those with non- the outer orbit and the center of mass of the inner binary, v spherical structures. The formation of these PNe is not i isthepre-SNrelativevelocityofthecenterofmassofthe v well understood, but maybe attributed to interactions be- inner binary relative to the outer star, sys is the systemic tween an AGB-star and a companion (e.g. Bond and Livio velocity the inner binary due to the SN, and ;Bond; De Marco et al. ;Zijlstra)or multiple companions (e.g. Bond et al. ;Exteretal. Gmt,i vc ≡ () ;Soker;BearandSoker). Where a binary ai companion can impose a non-spherical symmetry on the resulting PN, and even a non-axisymmetry (see e.g. Soker the orbital velocity in a circular orbit. Note that in compar- and Rappaport , for eccentric binaries), triple evo- ison with the circular velocity in binaries (Eq. ()), here lution can impose structures that are not axisymmetric, mt,i refers to m + m + m and a = aout. mirrorsymmetric, nor pointsymmetric (Bond et al. ; A full derivation of the change in semi-major axis Exter et al. ;Soker). (Eq. ()) and eccentricity (Eq. ()) of the outer orbit due Since the centers of many elliptical and bipolar PNe host to a SN in the inner orbit, are given in Appendix A..Note close binaries, the systems are expected to have undergone that the equation for the post-SN eccentricity of the outer a CE-phase. In the context of triples, PNe formation chan- orbitinPijlooetal.(), their Eq. (), is incomplete (see nels have been proposed that concern outer stars on the Appendix A.). AGB whose envelope matter just reaches or completely engulfs a tight binary system, e.g. the PN SuWt  (Bond 2.4 Quadruples and higher-order hierarchical systems et al. ;Exteretal.). Another proposed channel Although quadruples star systems are less common than involves systems with a very wide outer orbit of tens to triple systems, hierarchical quadruples still comprise about thousands of AU and in which the outer star interacts with % of F/G dwarf systems in the field (Tokovinin a, the material lost by the progenitor-star of the PN (Soker b). While for triples, there is one type of hierarchy et al. ;Soker). For a detailed review of such evo- that is stable on long-terms (compared to stellar lifetimes), lutionary channels, see Soker (). Under the assump- quadruples can be arranged in two distinct long-term tions that PN from triple evolutionary channels give rise stable configurations: the ‘+’ or ‘binary-binary’ config- to irregular PNe, Soker () and Bear and Soker () uration, and the ‘+’ or ‘triple-single’ configuration. In find that about  in - PNe might have been shaped by an the first case, two binaries orbit each others barycentre, interaction with an outer companion in a triple system. and in the latter case a hierarchical triple is orbited by a fourth body. In the sample of F/G dwarfs of Tokovinin .. Supernova explosions in triples (a, b), the ‘+’ systems comprise about / of Pijloo et al. () study the effect of a supernova explosion quadruples, and the ‘+’ about /. inatriplestarsystemunderthesameassumptionsasHills The secular dynamics of the ‘+’ systems were investi- (). The authors show that for a hierarchical triple in gated by Pejcha et al. () using N-body methods. Pejcha Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 16 of 36

et al. () find that in these systems, orbital flips and as- ics and stellar evolution need to be taken into account si- sociated high eccentricities are more likely compared to multaneously. equivalent triple systems (i.e. with the companion binary In this paper, we present a public source code TrES to viewed as a point mass). The ‘+’ configuration was stud- simulate the evolution of wide and close, interacting and ied by Hamers et al. ().i For highly hierarchical sys- non-interacting triples consistently. The code is designed tems, i.e. in which the three binaries are widely separated, for the study of coeval, dynamically stable, hierarchical, the global dynamics can be qualitatively described in terms stellar triples. The code is based on heuristic recipes that of the (initial) ratio of the Lidov-Kozai time-scales of the combine three-body dynamics with stellar evolution and two inner most binaries compared to that of the outer their mutual influences. These recipes are described here. two binaries. This was applied to the ‘+’ F/G systems The code can be used to evaluate the distinct evolution- of Tokovinin (a, b), and most (%) of these sys- ary channels of a specific population of triples or the im- tems were found to be in a regime in which the inner three portance of different physical processes in triple evolution. stars are effectively an isolated triple, i.e. the fourth body As an example, it can be used to assess the occurrence does not affect the secular dynamical evolution of the in- rate of stable and unstable mass transfer initiated in cir- ner triple. cular and eccentric inner orbits of triple systems (Toonen, We note that in the case of ‘+’ quadruples and with Hamers and Portegies Zwart in prep.). We stress that mod- a low-mass third body (in particular, a planet), the third elling though a phase of stable mass transfer in an eccentric body can affect the Lidov-Kozai cycles that would other- orbitiscurrentlynotimplementedinTrES,butweaimto wise have been induced by the fourth body. In particu- add this to the capabilities of TrES in a later version of the lar, under specific conditions the third body can ‘shield’ code. the inner binary from the Lidov-Kozai oscillations, possi- The code TrES is based on the secular approach to solve bly preventing the inner binary from shrinking due to tidal the dynamics (Section .) and stellar evolution is included dissipation, and explaining the currently observed lack of in a parametrized way through the fast stellar evolution circumbinary planets around short-period binaries (Mar- code SeBa (Section .). TrES iswrittenintheAstro- tin et al. ; Muñoz and Lai ;Hamersetal.). physics Multipurpose Software Environment, or AMUSE A similar process could apply to more massive third bod- (Portegies Zwart et al. ; Portegies Zwart ). This is ies, e.g. low-mass MS stars. a component library with a homogeneous interface struc- It is currently largely unexplored how non-secular effects ture based on Python. AMUSE can be downloaded for free such as stellar evolution, tidal evolution and mass trans- at amusecode.org and github.com/amusecode/amuse. In fer affect the evolution of hierarchical quadruple systems, the AMUSE framework new and existing code from dif- or, more generally, in higher-order multiple systems. The ferent domains (e.g. stellar dynamics, stellar evolution, hy- secular dynamics of the latter could be efficiently modelled drodynamics and radiative transfer) can be easily used and using the recent formalism of Hamers and Portegies Zwart coupled. As a result of the easy coupling, the triple code (). can be easily extended to include a detailed stellar evolu- tion code (i.e. that solve the stellar structure equations) or 3Methods a direct N-body code to solve the dynamics of triples that In the previous section, we gave an overview of the most are unstable or in the semi-secular regime (Section ..). important ingredients of the evolution of stars in single systems, binaries and triples. For example, nuclear evo- 3.1 Structure of TrES lution of a star leads to wind mass loss, that affects the The code consist of three parts: dynamics of binaries and triples, and can even lead to Step . Stellar evolution. a dynamical instability in multiple systems. Three-body Step . Stellar interaction. dynamics can give rise to oscillations in the eccentricity Step . Orbital evolution. of the inner binary system of the triple, which can lead to an amplified tidal effect and an enhanced rate of stel- At the beginning of each timestep we estimate an appro- lar mass transfer, collisions, and mergers. Additionally, a priate timestep dttrial and evolve the stars as single stars for triple system can transition from one to another dynam- this timestep (Step ). The trial timestep is estimated with: ical regime (i.e. without Lidov-Kozai cycles, regular and eccentric Lidov-Kozai cycles) due to stellar evolution, e.g. dttrial = min(dtstar, dtwind, dtR, fprevdtprev), () wind mass loss or an enhancement of tides as the stellar radius increases in time. These examples illustrate that for where dtstar, dtwind, dtR and fprevdtprev are the minimum the evolution of triple stars, stellar evolution and dynam- timesteps due to stellar evolution, stellar wind mass losses, ics are intertwined. Therefore, in order to study the evolu- stellar radius changes and the previous timestep. Each star tion of triple star systems consistently, three-body dynam- givesrisetoasinglevaluefordttrial, where the minimum is Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 17 of 36

adopted as a trial timestep in TrES.Thetimestepdtstar is are met or until the timestep is sufficiently small, dtmin = determined internally by the stellar evolution code (SeBa, – yr. If the change in the stellar parameters is too large Section .). It is the maximum attainable timestep for the (i.e. case (b) and (c)), the new trail timestep is taken to be: next iteration of this code and is mainly chosen such that the stellar masses that evolve due to winds, are not sig- ˙ nificantly affected by the timesteps. Furthermore, when a dttrial =min .dttrial, dttrial R ,() star changes its stellar type (e.g. from a horizontal branch star to an AGB star), the timestep is minimized to en- where .dttrial represents % of the previous timestep sure a smooth transition. For TrES,werequireamore ˙ and dttrial(R ) is a newly calculated timestep according to strict constraint on the wind mass losses, such that dtwind = Eq. () for which the time derivative of the radius R˙ from ˙ ˙ fwindm/mwind,wherefwind =.andmwind is the wind mass the last trial timestep is used. loss rate given by the stellar evolution code. The numeri- During mass transfer, the timestep is estimated by: cal factor fwind establishes a maximum average of % mass loss from stellar winds per timestep. Furthermore, we en- m sure that the stellar radii change by less then a percent per dt =min dt , f , f dt ,() · ˙ trial,MT trial MT m˙ MT,prev prev timestep through dtR = fRfR R/R,wherefR and fR are nu- MT merical factors. We take fR =.and

⎧ where m˙ MT is the mass loss rate from mass transfer (Sec- ˙ ⎪. for R =R/yr, ⎪ prev tion ..), and the numerical factors fMT =.and ⎪ ˙ ˙ ⎪. for R · R <(R/yr), fMT,prev = . If a star starts filling its Roche lobe in a ⎨⎪ prev timestep, dt =.dttrial.Ifdt < dttrial,MT,masstrans- f = forR˙ < R˙ and not MS, () trial trial R ⎪ prev ferisallowedtocommence. ⎪ ˙ ˙ ˙ ˙ ⎪R/Rprev for R < Rprev and MS, Step  in our procedure regards the modelling of the stel- ⎪ ⎩ ˙ ˙ ˙ ˙ lar interactions such as stable mass transfer, contact evo- Rprev/R for R > Rprev, lution and common-envelope evolution (Section .). ˙ ˙ The last step involves the simulation of the orbital evolu- where R and Rprev represent the time derivative of the ra- dius of the current and previous timestep, respectively. tion of the system by solving a system of differential equa- This limit is particularly important since the degree of tidal tions (Section .). If the evolution leads to the initiation interaction strongly depends on the stellar radius. Lastly, of RLOF during the trial timestep, both the orbit and stel- we require that dttrial < fprevdtprev,wheredtprev is the previ- lar evolution are reverted to the beginning of the timestep. ous successfully accomplished timestep and fprev anumer- If the time until RLOF is shorter than % of dtMT,thelat- ical factor with a value of . ter is taken to be the new trial timestep and mass trans- The trial timestep is accepted and the code continues to fer is allowed to commence. If not, the timestep is taken Step , only if: to be the time until RLOF that was found during the last case(a)nostarhasstartedtofillitsRochelobe, trial timestep. If during the orbital evolution the system be- case (b) stellar radii have changed by less than %, comes dynamically unstable, the simulation is terminated. case (c) stellar masses have changed by less than % within The stability criterion of Mardling and Aarseth ()is the trial timestep. used (Eq. ()). In all other cases, the trial timestep is ac- cepted, and the next iteration begins. We have tested that these percentages give accurate results with respect to the orbital evolution. Condition (b) is not applied at moments when the stellar radius changes dis- 3.2 Stellar evolution continuously, such as during the helium flash or at white Single stellar evolutionj is included through the fast stellar dwarf formation. evolution code SeBa (Portegies Zwart and Verbunt ; Conditions (b) and (c) are not applied, when a massive Nelemans et al. ;Toonenetal.;ToonenandNele- star collapses to a neutron star or black hole. Note that mans ). SeBa is a parametrized stellar evolution code when a star undergoes such a supernova explosion, the providing parameters such as radius, luminosity and core timestep is minimized through dtstar. Additionally Steps  and  are skipped and the triple is adjusted according to mass as a function of initial mass and time. SeBa is based Section ... on the stellar evolution tracks from Hurley et al. (). If conditions (a), (b) and (c) are not met, the timestep is These tracks are fitted to the results of a detailed stellar reverted and Step  is tried again with a smaller timestep evolution code (based on Eggleton , )thatsolves dttrial. This process is done iteratively until the conditions the stellar structure equations. Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 18 of 36

3.3 Orbital evolution where TrES solves the orbital evolution through a system of first-   order ordinary differential equations (ODE): I = k(m – mc)R + kmcRc ,() ⎧ ⎪˙ ˙ ˙ ˙ ˙ ⎪ain = ain,GR + ain,TF + ain,wind + ain,MT, where mc and Rc are the mass and radius of the core, ⎪ ⎪a˙ = a˙ + a˙ + a˙ + a˙ , k =.andk =.(Hurleyetal.). Thus we ap- ⎪ out out,GR out,TF out,wind out,MT ⎪˙ ˙ ˙ ˙ proximate the moment of inertia with a component for the ⎪ein = ein,b + ein,GR + ein,TF, ⎪ core and for the envelope of the star. This method works ⎪e˙ = e˙ + e˙ + e˙ , ⎪ out out,b out,GR out,TF well for evolved stars that have developed dense cores, as ⎪ ⎪g˙in = g˙in,b + g˙in,GR + g˙in,tides + g˙in,rotate, ≡ ⎪ well as for MS stars with mc M,andcompactobjects ⎨˙ ˙ ˙ ˙ ˙ gout = gout,b + gout,GR + gout,tides + gout,rotate, for which m – mc ≡ M. ˙ ˙ () ⎪hin = hin,b, The initial spin periods of the stellar components of the ⎪ ⎪θ˙ – J˙ J J θ triple are assumed to be similar to that of ZAMS stars. ⎪ = J J [ b,in( b,in + b,out ) ⎪ b,in b,out Based on observed rotational velocities of MS stars from ⎪ ˙ ⎪ + Jb,out(Jb,out + Jb,inθ)], Lang (), Hurley et al. () proposed the fit: ⎪ ˙ ˙ ˙ ˙ ⎪ = ,TF + ,I + ,wind, ⎪ ⎪˙ = ˙ + ˙ + ˙ , , M. ⎪  ,TF ,I ,wind  = yr–.() ⎩ ˙ ˙ ˙ ˙ .  = ,TF + ,I + ,wind, R . + M

As in Hamers et al. (), we make the simplifying as- where θ ≡ cos(i), J and J are the orbital angular mo- b,in b,out sumption that the stellar spin axes are aligned with the or- mentum of the inner and outer orbit, and  ,  and     bital axis of the corresponding star. For the vast majority of the spin frequency of the star with mass m , m and m ,    stellar triples the magnitude of the spin angular momenta respectively, and I the moment of inertia of the corre- are small compared to that of the orbital angular momenta. sponding star. x˙ represents the time derivative of parame- A consequence of this assumption, is that tidal friction ter x.Eq.() includes secular three-body dynamics (with from a spin-orbit misalignment is absent. The change in subscript b), general relativistic effects (GR), tidal fric- the mutual inclination in Eq. () is based on the conser- tion (TF), precession, stellar wind effects and mass trans- vation of total angular momentum (Hamers et al. ). fer (MT). The quadrupole terms of the three-body dy- The set of orbital equations of Eq. ()aresolvedby namics are based on Harrington (), and the octupole a routine based on the ODE solver routine presented in terms on Ford et al. () with the modification of Naoz Hamers et al. (). This routine uses the CVODE library, et al. (). Gravitational wave emission is included as in which is designed to integrate stiff ODEs (Cohen et al. Eqs. ()and()(Peters). Our treatment of tidal fric- ). It has been verified by comparing integrations with tion and precession is explained in Section ... Magnetic example systems presented in Ford et al. (), Blaes et al. braking is currently not included. The treatment of stel- ()andNaozetal.(), and comparing with analyt- lar winds and mass transfer is described in Sections ..- ical solutions at the quadrupole-order level of approxima- ... tion assuming J J , given by Kinoshita and Nakai The ODE solver routine uses adaptive timesteps to sim- b,out b,in (). ulate the desired timestep dttrial. Within the ODE solver, parameters that are not given in Eq. () (e.g. gyration ra- The ODE consists of a combination of prescriptions for the main physical processes for triple evolution (e.g. three- dius), are assumed to be constant during dttrial.Anexcep- tion to this is the stellar radius, mass and moment of iner- body dynamics and tides) which are described in Section . In order to set up the ODE, we have assumed that the phys- tia. Even though dttrial is chosen such that the parameters do not change significantly within this timestep (Sec- ical processes are independent of one another, such that tion .), there is a cumulative effect that can violate angu- their analytical treatments can be added linearly. Michaely ˙ and Perets () show that the dynamics of a hierarchical lar momentum conservation on longer timescales if I is not taken into account. As a non-interacting star evolves triple including mass loss and transfer can be well mod- and the mass, radius and moment of inertia change, the elled with this approach. They find that the secular ap- spin frequency of the star evolves accordingly due to con- proach shows excellent agreement with full N-body sim- servation of spin angular momentum. The change in the ulations. Additionally, we note that the ODE of Eq. ()is spin frequency is: validaslongastheincludedprocessesoccurontimescales longer than the dynamical timescale. In the next section we discuss when this criterion is violated, and describe the –I˙ ˙ = ,()alternative treatments in TrES. I I Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 19 of 36

.. The secular approach Furthermore, Luo et al. () showed recently that the ˙ The components in Eq. ()(e˙b, g˙b,andhb)thatdescribe rapid oscillations accumulate over time and alter the long- the secular three-body dynamics, including the regular term evolution of the triple systems (e.g. whether or not an Lidov-Kozai cycles, are derived using the orbital-averaging orbital flip occurs). The non-secular behaviour discussed technique (e.g. Michaely and Perets ;Naoz; by Luo et al. () occurs in systems in which the mass Luo et al. ). With this method, the masses of the of the outer star is comparable or larger than that of the components of the triple system are distributed over the inner binary, and in which the octupole term is important inner and outer orbit. The three-body dynamics is then (|oct|  .). approximated by the interaction between these ellipses. Furthermore, the Hamiltonian is expanded up to third or- 3.4 Stellar interaction der in ain/aout (i.e. the octopole term). The quadrupole .. Stellar winds in TrES level of approximation refers to the second-order expan- The mass loss rate of each star depends on the evolution- sion, which is the lowest non-trivial expansion order. The ary stage and is determined by the stellar evolution code. quadrupole level is sufficient for systems in which the oc- We make the common assumption that the wind is fast | | tupole parameter (Eq. ()) is sufficiently low, i.e. oct < and spherically symmetric with respect to the orbit, as dis- – ≈  . For example, this includes systems with m m. cussed in Section ... For the inner binary, we assume a To accurately model the dynamics of a population of sys- fraction β→ of the wind mass lost from m at a rate m˙  tems with a wide range of mass ratios, eccentricities and can be accreted by m,andβ→ for the mass flowing in orbital separations, one needs to include the octupole term the other direction. Following Eq. (), the effect on the in- as well. ner orbit is then: However, whereas the orbit-averaged equations of mo- tion are valid for strongly hierarchical systems, they be- a˙ in,wind = a˙ wind(m˙ , β→)+a˙ wind(m˙ , β→). () come inaccurate for triples with weaker hierarchies (e.g. Antonini and Perets ; Katz and Dong ;Naoz The wind mass lost from the inner binary is m˙ in =(– et al. ;Antoninietal.; Antognini et al. ; β→)m˙  +(–β→)m˙ ,ofwhichtheouterstarcanac- Bode and Wegg ;Luoetal.). For these systems, crete a fraction βin→. We do not allow the inner binary the timescale of the perturbations due to the outer star to accrete mass from the outer star. The winds widen the can become comparable to or shorter than the dynamical orbit according to: timescales of the system. This is problematic as the orbit- average treatment neglects any modulations on short or- a˙ = a˙ (m˙ , β → )+a˙ (m˙ ), () bital timescales per definition. For moderately hierarchical out,wind wind in in  wind,no-acc  systems, the outer star can significantly change the angu- see Eq. ()andEq.(). lar momentum of the inner binary between two successive The wind matter carries away an amount of angular mo- pericenter passages causing rapid oscillations in the corre- mentum which affects the spin of the star. Under the as- sponding eccentricity. The orbit-average treatment is valid sumption that the wind mass decouples from the star as a when: spherical shell: √ m a    in ˙  –ein π ,()˙ –/mwindR  m + m aout( – eout)  = .() wind I as derived by Antonini et al. (). If wind matter is accreted by a star, we assume the accre- In the non-secular regime, the inner binary can be driven to much higher eccentricities than the secular approxima- tor star spins up i.e. the stellar spin angular momentum in- tion predicts, and subsequently lead to more collisions of creases with the specific angular momentum of the wind e.g. black holes (Antonini et al. ), neutron stars (Seto matter. For example for m, the total change in the spin due ) or white dwarfs (Katz and Dong ). These are to winds is: interesting in the context of gravitational wave emission ˙  ˙ –/m,windR  and type Ia supernovae. Due to the very short timescale of ,wind = the eccentricity oscillations, and therefore rapid changes of I  the periapse distance, tidal or general relativistic effects do /β → m˙ R  +   ,wind   .() not play a role (Katz and Dong ). With the secular ap- I proach, as in TrES, the maximum inner eccentricity and therefore the number of collisions is probably underesti- .. Tides and precession mated for moderately hierarchical systems (see also Naoz Tidal friction is included in TrES as described in Eqs. ()- et al. ). (). The dominant tidal dissipation mechanism is linked Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 20 of 36

with the type of energy transport in the outer zones of the stellar models (see e.g. Ge et al. , ;Passyetal. star. We follow Hurley et al. (), and distinguish three a; Woods et al. , ). Our incomplete types: damping in stars with convective envelopes, radia- understanding of the stability of mass transfer leads to tive envelopes (i.e. dynamical tide), and degenerate stars. differences between synthetic binary populations The quantity kam/τTF of Eq. ()-()isgivenforthese (Toonen et al. ). three types of stars in their Eqs. (), ()k and (), re- The response of the Roche lobe is strongly spectively. We assume that radiative damping takes place dependent on the envelope of the donor star and the in MS stars with M >.M,inhelium-MSstarsandhori- mass ratio of the system.l Therefore the stability of zontal branch stars. Excluding compact objects, all other mass transfer is often described by a critical mass ratio stars are assumed to have convective envelopes. For the qcrit < q ≡ md/ma for different types of stars. For mass and radius of the convective part of the stellar en- unevolved stars with radiative envelopes, mass velope, we follow Hurley et al. ()(theirSection.) transfer can proceed in a stable manner for relatively and Hurley et al. () (their Eqs. ()-()), respectively, large mass ratios. We assume qcrit =,unlessthestar with the modification that MS stars to have convective en- is on the MS for which we take qcrit =.(de Mink et velopes in the mass range .-.M. Regarding the gy- al. b; Claeys et al. ). Stars with convective ration radius k, for stars with convective or radiative en- envelopes are typically unstable to mass transfer, velopes we assume k = k, for compact objects k = k (see unless the donor is considerably less massive than the Eq. ()). companion. For giants, we adopt qcrit =.+[(– – We include precession from three-body dynamics, gen- Mc/M)] ,whereMc is the core mass of the donor star eral relativistic effects (Eq. (), Blaes et al. ), tides (Hjellming and Webbink ). For naked helium (Eq. (), Smeyers and Willems ), and stellar rota- giants, low-mass MS stars (M <.M), and white tion (Eq. (), Fabrycky and Tremaine ). The latter dwarfs, we follow Hurley et al. () and adopt two equations require an expression for the apsidal mo- qcrit =., qcrit =.and qcrit =., tion constant kam (instead of kam/τTF as required for the respectively. tidal equations of Eqs. ()-()). For MS, helium-MS, and WDs, we assume kam = . (Brooker and Olle ), .. Common-envelope evolution in the inner binary for neutron stars kam = . (Brooker and Olle ), for As CE-evolution is a fast, hydrodynamic process, the ODE black holes kam =,andforotherstarskam =.(Claret solver routine is disabled during the modelling of the CE- and Gimenez ). For low-mass (m <.M)MSstars phase. If the donor star is a star without a clear distinc- that are fully or deeply convective, we take kam =. tion of the core and the envelope (i.e. MS stars, helium (Claret and Gimenez ). MS stars and remnants), we assume the phase of unstable mass transfer leads to a merger. For other types of donor .. Stability of mass transfer initiated in the inner stars,thetreatmentoftheCE-phaseconsistsofthreedif- binary ferent models that are based on combinations of the for- When one of the inner stars fills its Roche lobe, we test for malisms described in Section ...Inmodelandmodel the stability of the mass transfer: , the α-formalism (Eq. ()) and γ -formalism (Eq. ()) • Tidal instability; areusedtodeterminetheoutcomeoftheCE-phase,re- Tidal friction can lead to an instability in the binary spectively. When two giants are involved, the double-CE is system and subsequent orbital decay (see applied (Eq. ()). In the standard model, the γ -formalism Section ..). The tidal instability takes place in is applied unless the CE is triggered by a tidal instability compact binaries with extreme mass ratios. It occurs or the binary contains a remnant star. This is based on when there is insufficient angular momentum to keep modelling the evolution of double white dwarfs (Nelemans  the star in synchronization i.e. J >  Jb,withJ = I. et al., , ;Toonenetal.). The standard values When RLOF occurs due to a tidal instability, we of αλce and γ are taken to be . and . (Nelemans et al. assume that a CE develops around the inner binary. ). This will lead further orbital decay, and finally either a The companion star in the inner binary is probably not merger or ejection of the envelope. able to accrete from the overflowing material of the CE- • RLOF instability; phase, because of its relatively long thermal timescale The stability of the mass transfer depends on the compared to the short timescale on which the CE is ex- responseoftheradiusandtheRochelobetothe pected to place. Therefore, we assume that the CE occurs imposed mass loss. In the fundamental work of completely non-conservatively. Hjellming and Webbink (), theoretical stability TheeffectofaCE-phaseontheouterstarofatriple criteria are derived for polytropes. Stability criteria is poorly studied or constrained (Section ..). For sta- have been improved with the use of more realistic ble, hierarchical systems, if the CE-material is energetic Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 21 of 36

enough to escape from the inner binary, the matter is likely binaries have managed to experience a phase of fairly con- energetic enough to escape from the triple as well (Sec- servative mass transfer (e.g. φ Per Pols ), additional tion ..). We assume that the escaping CE-matter has mechanisms of angular momentum loss must play a role expanded and diluted sufficiently to avoid a second CE- during mass transfer (de Mink et al. a). phase with the outer star, as we only consider stable triples The lack of synchronisation can affect the size of the with aout/ain  . We allow the matter to escape as a fast Roche lobe significantly. For example, for a star that is ro- wind in a non-conservative way i.e. according to Eq. () tating  times faster than synchronization, the Roche with m˙  =andβin→ =,i.e. lobe is only -% of that of the classical Roche lobe (based on Sepinsky et al. a). For simplicity, we make the a˙ out,wind = a˙ wind,no-acc(m˙ in). () common assumption that any circularized system enter- ing RLOF, is and will remain synchronized during the mass There may be friction between the CE-matter and the transfer phase. outer star, if the CE-matter is primarily expelled in the or- For stable, hierarchical triples systems, the matter lost bital plane and the inner and outer orbital planes are par- by the inner binary is likely energetic enough to escape allel. Therefore we multiply Eq. ()withafactor fromthetriple(Section..), and we model this as a fast wind i.e. according to Eq. (). To incorporate the effect of |sin(i)| friction between the matter and the outer star, we multi- ffric =min , ,() |sin(icrit)| ply Eq. ()withafactor

Table 3 Initial conditions for the three triple systems We make the common assumption that the SN takes discussed in Section 4 place instantaneously. As a result, the positions of the stars Parameters Gliese 667 Eta Carinae MIEK just before and after the SN are not changed. As TrES is based on orbit-averaged techniques, we do not follow the m1 (M) 0.73 110 7 m (M) 0.69 30 6.5 position of the stars along the orbit as a function of time. 2 m3 (M)0.37306 In order to obtain the position at the moment of the SN, ain (AU) 12.6 1 10 we randomly sample the mean anomaly from a uniform aout (AU) 250 25 250 distribution. The natal kick is randomly drawn from ei- ein 0.58 0.1 0.1 eout 0.5 0.2 0.7 ther of three distributions (Paczynski ,Hansenand ◦ i ( )909060 Phinney  or Hobbs et al. ) in a random direction. mutual gin (rad) 0.1 0.1 0 Our method simply consists of two coordinate transfor- gout (rad) 0.5 0.5 π mations (thus we do not use Eqs. (), (), (), (), (), nor Eq. () directly). We convert from our standard or- bital parameters of i and a, e, g, h for the inner and outer 4Examples orbit to orbital vectors i.e. eccentricity eˆ and angular mo- In Section . and Section .,wediscussedseveralphys- mentum vector Jˆ for both orbits. After the mass of the b ical processes and how they affect the long-term evolu- dying star is reduced and the natal kick is added to it, we tion of inner binaries and triple systems. Here we illustrate convert back to the orbital elements. The reason for per- those processes by simulating the evolution of a few realis- forming two coordinate transformation, to orbital vectors tic triple star systems. For example, the evolution of Gliese and back, is that the orbital elements in the code are de-  displays the Lidov-Kozai cycles, and the evolution of fined with respect to the ‘invariable’ plane, i.e. in a frame Eta Carinae illustrates the effect of precession and stellar defined by the total angular momentum. In the case of a winds. The evolutionary pathways of the triple systems are SN, however, the total orbital angular momentum vector is simulated with the new triple code TrES, such that the ex- not generally conserved, which implies that the coordinate amples below also demonstrate the capabilities of TrES. frame changes after the SN. In contrast, the orbital vectors are defined with respect to an arbitrary inertial frame that 4.1 Gliese 667 isnotaffectedbytheSN.Thepost-SNorbitalvectorsare Gliese  is a nearby triple system in the constellation transformed to the orbital elements in the new ‘invariable’ of Scorpius. The orbital parameters of the system are de- plane, i.e. defined with respect to the new total angular scribed in Table  based on Tokovinin (). The outer momentum vector. An additional advantage of the double star is in an orbit of aout >  AU, but for simplicity, we coordinate transformation is that the pre-supernova orbit will assume aout =  AU in the following. The outer star canbecircularaswellashaveanarbitraryeccentricity. is also a planetary host-star; up to five planets have been If the post-supernova eccentricity of an orbit is larger claimed, of which two have been confirmed so far (Feroz than one, the orbit is unbound. We distinguish four situa- and Hobson ).TheorbitoftheplanetGlieseCb tions: lies just within the habitable zone, which makes this planet • Both the inner as the outer orbit remain bound, and a prime candidate in the search for liquid water and life on the system remains a triple. The simulation of the other planets (Anglada-Escudé et al. ). In the follow- evolution of the triple is continued. ing, we will neglect the dynamical effect of the presence of • When the inner orbit remains bound, and the outer planets on the evolution of the triple. orbit becomes unbound, the outer star and inner Gliese  is a prime example of a triple system undergo- binary remain as separated systems. We assume the ing Lidov-Kozai cycles. Figures  and  show the evolution outerstardoesnotdynamicallyaffecttheinnerbinary. of the inner eccentricity and mutual inclination for the first With the default options in TrES the simulation is  Myr after the birth of the system. under the assumption stopped here unless the user specifies otherwise. ◦ of eout =.,i = , gin =.andgout = .. For different • When both the inner as the outer orbit become values for the outer eccentricity, arguments of pericenters, unbound, the stars evolve further as isolated stars. As and mutual inclination the general behaviour of Figures - in the previous scenario, by default the simulation is  remains the same, but the timescale and amplitude of the stopped unless the user specifies otherwise. Lidov-Kozai cycles varies (to the point where the cycles are • The inner orbit becomes unbound, but at the moment not notable). Figures  and  show the cyclic behaviour of just after the SN the outer star remains bound the eccentricity and inclination in Gliese . When the ec- inner system. In this case, TrES cannot simulate the centricity is at its maximum, the inclination between the evolution of this system further. The evolution of these orbits is minimal. The timescale of the oscillations is a few systemsshouldbefollowedupwithanN-bodycode. . Myr, which is consistent with the order of magnitude Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 23 of 36

Figure 3 Inner eccentricity evolution. The evolution of the inner Figure 5 Inner semi-major axis evolution. The evolution of the inner semi-major axis as a function of time for the first 3 Myr of the eccentricity ein as a function of time for the first 3 Myr of the evolution of Gliese 667. The figure shows that Gliese 667 is susceptible for evolution of Gliese 667. The figure shows a decreasing semi-major axis Lidov-Kozai cycles. The initial conditions are given in Table 3. due to the combination of Lidov-Kozai cycles with tidal friction, i.e. LKCTF. Note the small scale on the y-axis, where a0 = 12.599999999.

way, LKCTF could lead to RLOF in or a merger of the inner system. Regarding the long-term evolution of this system, it is analogous to that discussed for the first  Gyr, After  Gyr (approximately the age of the Galactic thin disk, e.g. Os- walt et al. ;delPelosoanddaSilva; Salaris ), the system is still detached. The orbital separation has de- creased by only ∼ km. The stellar masses are sufficiently low, that the stars do not evolve off the MS within  Gyr, and as such do not experience a significant growth in ra- dius that could lead to RLOF. Even taking into account the low metallicity of Gliese A (Cayrel de Strobel et al. ), and the corresponding speed-up of the evolutionary timescales, a .M star is not massive enough to evolve of the MS within  Gyr. Furthermore, the stars are not Figure 4 Mutual inclination evolution. The evolution of the mutual massive enough to lose a considerable amount of matter inclination i as a function of time for the first 3 Myr of the evolution of in stellar winds, such that the triple is not affected dynam- Gliese 667. This triple shows the cyclic behaviour in inclination and ically by wind mass losses. inner eccentricity (Figure 3) related to Lidov-Kozai cycles. 4.2 Eta Carinae Eta Carinae is a binary system with two massive stars (m ∼ M and m ∼ M) in a highly eccentric or- approximation of . Myr of Eq. (). The octupole param- bit (e = .) with a period of . yr (Damineli et al. ). eter oct < ., which indicates that the eccentric Lidov- Both stars are expected to explode as supernovae at the Kozai mechanism is not of much importance here. end of their stellar lives. Eta Carinae is infamous for its For the same timescale as Figures  and ,Figure shows ‘Great Eruption’. From  to , it brightened consid- the evolution of the inner-orbital semi-major axis ain.The erably, and in  it even became the second brightest star change in the inner semi-major axis of Gliese  is neg- in the sky (de Vaucouleurs and Eggen ). The system is ligibly small, however, the figure illustrates the effect of surrounded by the Homunculus Nebulae, that was formed Lidov-Kozai cycles with tidal friction or LKCTF. When the during the Great Eruption, and heavily obscures the binary inner eccentricity is at its maximum, and the inner stars stars (Humphreys and Davidson ). The kinetic energy are at their closest approach during pericenter passage, the of the Homunculus Nebulae is large i.e. .erg (Smith inner semi-major axis decreases due to tidal forces. In this et al. ) and comes close to that of normal supernovae. Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 24 of 36

Figure 7 Semi-major axis evolution. The evolution of the inner Figure 6 Inner eccentricity evolution. The evolution of the inner eccentricity ein as a function of time for the first 20,000 yr of the eccentricity ein as a function of time for the first 20,000 yr of the evolution of triple progenitor of Eta Carinae. The figure shows the evolution of triple progenitor of Eta Carinae. The figure shows that the characteristic increase in semimajor-axes due to stellar wind mass system undergoes Lidov-Kozai cycles. The initial conditions of the loss, and the periodic decrease in inner semimajor-axis when the triple progenitor are given in Table 3. eccentricity is high due to LKCTF.

However, as both stars have survived the Great Eruption, outer orbits expand (Section ..). Initially the inner or- Eta Carinae is often referred to as a ‘supernova imposter’. bit expands faster than the outer orbit, as expected for a The cause of the Great Eruption remains unexplained. strong wind from the inner orbit (Section ..). However, Amassiveoutflow,asduringtheGreatEruption,canbe due to the combination of stellar winds with LKCTF for driven by a strong interaction between two stars (e.g. theEtaCarinaeprogenitor,itsouterorbitexpandsfaster, Harpaz and Soker ;Smith)oramergerof and the triple becomes more dynamically stable. two stars (e.g. Soker and Tylenda ). Recently, Porte- On a longer timescale, the triple moves from the ec- giesZwartandvandenHeuvel() tested the hypothe- centric Lidov-Kozai regime to the regular regime (|oct|  sis that the Eta Carinae system is formed from the merger .) as the inner binary loses matter and angular momen- of a massive inner binary of a triple system. According to tum in the stellar winds (Figure ). After  Myr, the oc- their model, the merger was triggered by the gravitational tupole parameter has decreased from oct = . initially, interaction with a massive third companion star, which is to oct =.. the current ∼M companion star in Eta Carinae. Here, The long-term evolution of the progenitor candidate we simulate the evolution of their favourite model with the of Eta Carinae shows another interesting feature in Fig- initial conditions as given by Table . Furthermore, Porte- ures  and , i.e. the Lidov-Kozai cycles are quenched. gies Zwart and van den Heuvel () assume that the ar- Here the precession due to the distortion and rotation gument of periastron does not affect the tidal evolution, of the stars dominates over the precession caused by the and therefore we arbitrarily set gin =.andgout =.. Lidov-Kozai mechanism. As a result, the amplitudes of the During the early evolution of the triple, the system ex- cycles in inner eccentricity and mutual inclination are re- periences Lidov-Kozai cycles with a timescale of a few kyr, duced. After approximately . Myr, the evolution of the see Figure .Theoctupoleparameteroct = . > ., system is completely dominated by tides, i.e. the system which indicates that the system is in the eccentric Lidov- circularizes and the inner semi-major decreases accord- Kozai regime with a timescale of order toct ∼ tKozai/oct ∼ ingly (Figure ). After circularization of the inner binary few tens of kyr. has been achieved ( Gyr), the inner semi-major axis in- The evolution of the semi-major axis shows two charac- creases again due to the stellar winds from the inner binary. teristics in Figure . Firstly, as for Gliese , the system The evolution of the Eta Carinae progenitor (Figures -) is affected by LKCTF i.e. the semi-major axis shrinks pe- illustrates that both three-body dynamics and stellar evo- riodically, due to strong tides at pericenter when the inner lution matter, and neither can be neglected. eccentricity is at its maximum. Secondly, as the primary AfteraboutMyr,theprimarystarfillsitsRochelobe star is very massive, strong winds remove large amounts of and initiates a mass transfer phase (Figure ). The inner mass while the star is still on the MS (Section ..). The semi-major axis is about . AU (R), and the mas- dynamical effect of such a fast wind is that the inner and sive primary has increased in size to R. Even though, Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 25 of 36

Figure 8 Radius and mass evolution. The evolution of the radii Figure 10 Mutual inclination evolution. The evolution of the (dashed line) and mass (dash-dotted line) as a function of time for the mutual inclination on the same timescale as Figure 9. The variation in stars in the triple progenitor of Eta Carinae. The primary star of initial the inclination decreases with time; as the stars evolve, their radii mass 110M is shown in blue. The secondary and tertiary star, both of increase, and tidal effects become stronger (Eqs. (10)-(12)). initial mass 30M,areshowninred.TheRochelobeoftheprimary and secondary are overplotted (blue solid line and red solid line respectively). The Roche lobe of the tertiary is about 2,000-3,000R. After about 3 Myr, the primary star fills its Roche lobe. tion ..). A new star is formed, that is still on the MS, with a mass of M. We assume this merger proceeds conservatively, and therefore the outer orbit is not affected, such that aout ≈  AU, eout ≈ . and P ≈ . yr. Prior to the merger, the outer semi-major axis has increased from the initial value of  AU to  AU due to the stellar winds. The resulting binary is similar to the current Eta Cari- nae system in mass and orbital period. It is not an exact match, as the evolution of this specific triple is shown for illustrative purposes, and has not been fitted to match the currently observed system. A progenitor study of Eta Cari- nae to improve the match is beyond the scope of this paper. We note that the current eccentricity of our remaining binary is low i.e. e ≈ . compared to the observed e =.. In our simulations, the post-merger eccentricity is equal to the pre-merger outer eccentricity. During the evolution of the Eta Carinae progenitor, the outer eccentricity has re- mained roughly equal to its initial value of eout =..The Figure 9 Inner eccentricity evolution. The evolution of the inner outer eccentricity is not affected strongly by stellar evo- eccentricity on a timescale of 3 Myr. The timescale of the Lidov-Kozai lution or Lidov-Kozai cycles. If we study the evolution of cycles is a few kyr, such that the lines overlap in Figure 9.Asthe an alternative progenitor similar to the favourite model of system evolves, the amplitude of the cycles reduces, until the system Portegies Zwart and van den Heuvel (), but with eout = circularizes. ., the system is dynamically unstable at birth. In order for the triple to be dynamically stable eout  . for the ◦ ◦ standard i = ,oruptoeout  . for i = .Theevo- the inner semi-major axis (and Roche lobe) of the primary lutions of the dynamically stable systems with eout  . are ∼% smaller around  Gyr, there is no RLOF yet as show similar behaviour as our initial system (Table ), and the radius of the primary star is ∼% smaller. At RLOF, the merger leads to a similar binary as in the case of our the masses of the inner stars have reduced from the initial initial system. For dynamically stable systems with higher M and M to M and M,andbothstarsare outer eccentricities, the merger time decreases strongly, still on the MS. The mass transfer phase proceeds in an and the inner system does not reach circularization before unstable manner (Section ..), and a common-envelope the merger takes place. The merger product is more mas- develops that leads to a merger of the inner stars (Sec- sive,aslessmassislostinstellarwinds.Inthesimulation Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 26 of 36

Figure 12 Radius and mass evolution. The evolution of the radii Figure 11 Inner semi-major axis evolution. The evolution of the (dashed line) and mass (dash-dotted line) as a function of time for the inner semimajor-axis on the same timescale as Figure 9 and 10.Inthe stars in a triple that transitions from a region with regular to eccentric first Myr, the evolution of the system is dominated by the Lidov-Kozai Lidov-Kozai behaviour i.e. MIEK. The initial conditions of the triple are mechanism and the inner semimajor-axis remains more or less giveninTable3, based on Shappee and Thompson (2013)and constant. In the following Myr, the system circularizes and the Michaely and Perets (2014). The primary star is shown in blue, the semimajor-axis decreases by a factor 2. After synchronisation and secondary in green and the tertiary in red. The Roche lobe of the circularisation has been reached, the inner semimajor-axis increases primary and secondary are overplotted (blue solid line and green due to the ejection of stellar winds. solid line respectively). The Roche lobe of the tertiary is 6,500-8,000R. After about 55.5 Myr, the primary star fills its Roche lobe. of (Portegies Zwart and van den Heuvel ), the outer  orbit has an eccentricity eout = . initially, but becomes oxygen-neon white dwarf of .M at  Myr. The outer ∼ highly eccentric in the merger phase due to asymmetric orbit widens to about aout  AU due to the wind mass mass loss. losses. The wind mass loss allows the triple to transition to the 4.3 MIEK-mechanism eccentric Lidov-Kozai regime at about  Myr, i.e. oct = In this section, we illustrate the dynamical effect of mass . at this time. The system is driven into extremely high loss on a triple system from the point of view of tran- eccentricities, and also the amplitude of the Lidov-Kozai sitions in dynamical regimes, e.g. the regime without cycle in inclination increases. The evolution of the system Lidov-Kozai cycles, with regular or with eccentric Lidov- asshowninFigures and  is qualitatively similar to that Kozai behaviour. Here, we focus on the transition from found by Shappee and Thompson ()basedonN-body the regular Lidov-Kozai regime to the eccentric regime, calculations and Michaely and Perets ()basedonthe i.e. where the octupole term is significant. This transition secular approach. In these studies, stellar winds are imple- has been labelled ‘mass-loss induced eccentric Kozai’ or mented ad-hoc with a constant mass loss rate for a fixed MIEK (Section ..). The canonical example of MIEK- time interval starting at a fixed time. Moreover, the system evolution is a triple with the initial conditions as given is followed for multiple Myrs after the mass loss event in ◦ by Table  (Shappee and Thompson ;Michaelyand both papers, such that the inclination rises above  ,and Perets ). To reproduce the experiment of Shappee and the inner and outer orbit become retrograde to each other. Thompson (), we simulate the evolution of this triple In our case the simulation is stopped before such a flip in with TrES including three-body dynamics and wind mass inclination develops, as RLOF is initiated in the inner bi- losses, however, without stellar evolution in radius, lumi- nary when the inner eccentricity is high. nosity, or stellar core mass etc. However, if we fully include stellar evolution, as in the Starting from the birth of the triple system, its orbit is standard version of TrES, the triple is not driven into susceptible to Lidov-Kozai cycles (Figures  and ). The the octupole regime. On the AGB, the radius of a M- timescaleofthecyclesisapproximately.Myr.Thecy- star can reach values as large as ∼,R (Figure ), cles are in the regular regime, i.e. oct =..Astime and therefore RLOF initiates before the MIEK-mechanism passes, the stars evolve. The primary star evolves off the takes place. Even if the inner binary would be an isolated MS at  Myr, and after  Myr it reaches the AGB with a binary, RLOF would occur for initial separations of a < mass of .M (Figure ). Subsequently, it quickly loses  AU. For triples, RLOF can occur for larger initial (in- a few solar masses in stellar winds, before it becomes an ner) separations, as the Lidov-Kozai cycles can drive the Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 27 of 36

For the canonical triple with ain =AUandaout =  AU, RLOF occurs at . Myr, just a few . Myr after the primary star arrives on the AGB. In that time, the radius of the primary increased by a factor ∼, and tides can no longer be neglected. The tidal forces act to circularize and synchronize the inner system, such that ein =  at RLOF. The eccentric Kozai-mechanism does not play a role at this point, i.e. oct = .. The mass of the core has not had enough time to grow to the same size as in the example without RLOF, i.e. the core mass is .M instead of .M. Stellar winds have reduced the mass of the primary star to .M.Astheprimaryhas a convective envelope and is more massive than the sec- ondary, a CE-phase develops. We envision three scenar- ios based on the different models for CE-evolution (Sec- Figure 13 Inner eccentricity evolution. The evolution of the inner tions .. and ..). First, the CE-phase leads to a merger eccentricity ein as a function of time for a triple that transitions from a of the inner binary, when the inner orbit shrinks strongly, region with regular to eccentric Lidov-Kozai behaviour i.e. MIEK. The as for the α-model of CE-evolution with αλce =.(Sec- initial conditions of the triple are given in Table 3, based on Shappee and Thompson (2013) and Michaely and Perets (2014). For this figure, tion ..). Second, the CE-phase leads to strong shrink- the stars are not allowed to evolve in TrES,exceptforwindmass age of the orbit, but not enough for the inner stars to losses. If stellar evolution is taken into account fully, RLOF initiates at merge. In this scenario, the envelope of the donor star is 55.5 Myr, before the transition to MIEK can develop. completely removed from the system, and the outer orbit widens to about  AU, under the assumption that the mass removal affects the outer orbit as a fast wind. As- suming αλce =(Eq.(), Section ..), ain ∼ . AU and oct = . after the CE-phase. In this scenario, the triples does not enter the octupole regime, and the MIEK- mechanism does not manifest. Lastly, the CE-phase does not lead to a strong shrinkage of the inner orbit, as for the γ -model of CE-evolution with γ =.(Eq.(), Sec- tion ..). The inner semimajor-axis even increases from . to . AU. In this scenario, oct = ., such that the perturbations from the octupole level become significant. In this last scenario, the triple undergoes the MIEK mech- anism, despite and because of the mass transfer phase.

5 Discussion and conclusion In this paper, we discuss the principle complexities of the evolution of hierarchical triple star systems. Hierarchical triples are fairly common and potentially long-lived, which Figure 14 Mutual inclination evolution. The evolution of the allows for their evolution to be affected by (secular) three- mutual inclination i as a function of time for the same triple as in Figure 13. The triple transitions from a region with regular to eccentric body dynamics, stellar evolution and their mutual influ- Lidov-Kozai behaviour at 56 Myr. ences. We present an overview of single star evolution and binary evolution with a focus on those aspects that are rel- evant for triple evolution. Subsequently, we describe the processes that are unique to systems with multiplicities of inner eccentricity to higher values. For wider inner bina- higher order than for binaries. ries i.e. ain >  AU, the MIEK-mechanism does not oc- In some cases, the evolution of a hierarchical triple can cur either, as the triple is dynamically unstable. This ex- be adequately described by the evolution of the inner and ample indicates that the parameter space for the MIEK- outer binary separately. In other cases, the presence of the mechanism to occur is smaller than previously thought, outer star significantly alters the evolution of the inner and so it may occur less frequently. Moreover, this example binary. Several examples of the latter are given in detail. demonstrates the importance of taking into account stellar These examples also show the richness of the regime in evolution when studying the evolution of triples. which both three-body dynamics and stellar evolution play Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 28 of 36

a role simultaneously. Moreover, the examples demon- Portegies Zwart ), which is based on Python. AMUSE strate the importance of coupling three-body dynamics including TrES can be downloaded for free at amusecode. with stellar evolution. org and github.com/amusecode/amuse. Due to the nature Additionally, we present heuristic recipes for the princi- of AMUSE, the triple code can be easily extended to include ple processes of triple evolution. These descriptions are in- a detailed stellar evolution code or a direct N-body code. corporated in a public source code TrES for simulating the Regarding the latter, this is interesting in the context of evolution of hierarchical, coeval, dynamically stable stel- triples with moderate hierarchies where the orbit-averaged lar triples. We discuss the underlying (sometimes simplify- technique breaks down (as discussed above). Furthermore, ing) assumptions of the heuristic recipes. Some recipes are it is relevant for triples that become dynamically unstable exact or adequate (e.g. gravitational wave emission, wind during and as a consequence of their evolution. For exam- mass loss or Lidov-Kozai cycles), and others are admittedly ple, Perets and Kratter ()showtriplesthatbecomedy- crude (e.g. mass transfer). The recipes are based on simple namically unstable due to their internal wind mass losses, assumptions and should be seen as a starting point for dis- are responsible for the majority of stellar collisions in the cussion and further study. When more sophisticated mod- Galactic field. Consequently, the majority of stellar colli- els become available of processes that influence triple evo- sions do not take place between two MS stars, but involve lution, these can be included in TrES,andsubsequently an evolved star of giant-dimensions. Another interesting the effect on the triple populations can be studied. For prospect is the inclusion of triples in simulations of cluster now, the accuracy levels of the heuristic recipes are suffi- evolution, where triples are often not taken into account cient to initiate the systematic exploration of triple evolu- e.g. in the initial population, through dynamical formation tion (e.g. populations, evolutionary pathways), while tak- nor a consistent treatment of the evolution of triple star ing into account three-body dynamics and stellar evolu- systems. However, dynamical encounters involving triples tion consistently. We note that simulating through a phase are common, reaching or even exceeding the encounter of stable mass transfer in an eccentric inner orbit is cur- rate involving solely single or binary stars, in particular rently beyond the scope of the project. However, appropri- in low- to moderate-density star clusters (Leigh and Sills ate methodology for eccentric mass transfer (e.g. Sepinsky ; Leigh and Geller ). Therefore, the evolution of et al. b, ; Dosopoulou and Kalogera b)has triples might not only be important for the formation and been developed that we aim to implement at a later stage. destruction of compact or exotic binaries, but also for the ThetripleevolutioncodeTrES is based on the secu- dynamical evolution of clusters in general. lar approach to solve for the dynamics of the triple sys- tem. It has been shown that this approach is in good agreement with N-body simulations of systems in which Appendix the secular approximations are valid (Naoz et al. ; A.1 Derivation of the orbital evolution of a triple Hamers et al. ;MichaelyandPerets). The advan- during a supernova explosion tage of the secular approach is that the computational time A.. Definitions and assumptions is orders of magnitudes shorter than for an N-body simula- Assume a hierarchical triple with inner binary masses m tion. The secular approach, however, is not valid when the and m,andtertiarymassm. The corresponding position evolutionary processes occur on timescales shorter than and velocity vectors of the bodies, with respect to an ar- the dynamical timescale of the system. In these cases, we bitrary inertial reference frame, are denoted with ri and either stop the simulation (e.g. during a dynamical insta- vi, respectively. Let primed quantities denote quantities bility)orsimulatetheprocessasaninstantaneousevent after the SN. It is assumed that body  collapses in a SN (such as a common-envelope phase). Lastly, the secular in such a way that the mass of body  changes instanta- approximation becomes inaccurate when the triple hier- neously from m to m = m –m and that body  receives archy is weaker (e.g. Antonini and Perets ;Katzand a kick velocity vk. Therefore the new velocity of body  Dong ; Antognini et al. ;BodeandWegg; is v = v + vk. The position of body  is assumed not to Luo et al. ). In this case, the timescale of the perturba- change, i.e. r = r. Similarly, changes are not assumed to tion from the outer star onto the inner binary during its pe- occur in the masses and instantaneous positions and ve- riastron passage, is comparable to the dynamical timescale locities of the other bodies, i.e. mi = mi, ri = ri and vi = vi of the inner binary. This can result in extremely high ec- for i ∈{, }. centricities and collisions between the stars in the inner bi- The pre-SN inner and outer relative separation vectors nary. With the secular approach, as in TrES,theseoccur- are given by rences are probably underestimated in systems with mod- erate hierarchies (see also Naoz et al. ). rin = r – r,() TrES is written in the Astrophysics Multipurpose Soft- ware Environment, or AMUSE (Portegies Zwart et al. ; rout = rcm,in – r,() Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 29 of 36

where After the SN, the inner binary specific energy is given by

mr + mr rcm,in ≡ () G(m + m – m) m + m Ein =– ain is the inner binary centre of mass position vector. Simi-   G(m + m – m) larly, the pre-SN inner and outer relative velocity vectors = vin –  rin are given by    · = vin + vk +(vin vk) vin = v – v,() G(m + m – m) G(m + m) vout = vcm,in – v,()– – rin ain where    · G m + vk +(vin vk) + . m v + m v  rin v ≡     () cm,in m + m   This relation can be rewritten as is the inner binary centre of mass velocity vector. With a m our assumptions, the corresponding post-SN quantities in = – ain m + m are given by a m (v · v ) v – × – in  – in k – k , rin = r – r = rin,()   rin m + m vc,in vc,in

rout = rcm,in – r ()

m m = rout – rin,()where vc,in istherelativeinnerorbitalspeedforacircular m + m – m m + m orbit, i.e. where we used that the change of rcm,in can be written as G(m + m) vc,in ≡ .() m m ain rcm,in = rcm,in – rin.() m + m – m m + m Eq. () is equivalent to Eq. () and to Eq. () of Pijloo The corresponding equations for the velocities are et al. (). To find the post-SN eccentricity, we use the following

vin = v – v = vin + vk,()standard relation for the specific angular momentum of the inner orbit, vout = vout + vsys,()   G(m + m)ain –e = rin × vin () where the systemic velocity vsys is given by in = r v –(r · v ).() ≡ in in in in vsys vcm,in – vcm,in

m The post-SN inner orbit specific angular momentum is =– given by m + m – m

m m G(m + m – m )a –e  · vin + – vk .()    in in m + m m ×  = rin vin A.. Orbital elements of the inner binary   ·  To derive the change of the semimajor-axis in the inner = rinvin – rin vin () orbit, we use the following standard relation between the   ·  = rin vin +(vin vk)+vk semimajor-axis, specific binding energy and the relative  position and velocity vectors, – (rin · vin)+(rin · vk) ()  G(m + m) = G(m + m)ain –ein Ein =– ain  ·  + rin (vin vk)+vk  G(m + m )     = vin – .()–(rin · vin)(rin · vk)–(rin · vk) .()  rin Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 30 of 36

Inserting the relation for the post-SN inner semimajor-axis Rewriting this relation, we find (Eq. ()), we find the following expression for the post-SN eccentricity, aout m aout m = – – aout m + m + m rout m + m + m

–e  in rout – rout +aout m + m  routrout   =  – m + m – m (vout · vsys) vsys – – ,()     v · v v vc,out vc,out × ain m ( in k) k – –  –  rin m + m vc,in vc,in where, analogously to the inner orbit, we defined ×   –ein + G(m + m)ain G(m + m + m) vc,out ≡ .() ×  ·  aout rin (vin vk)+vk  Eq. () is equivalent to Eq. ()andtoEq.()ofPijloo –(rin · vin)(rin · vk)–(rin · vk) .() et al. (). The outer orbit specific angular momentum is given by The terms in the second line of Eq. ()aremissingin  Eq.(a)ofPijlooetal.(). G(m + m + m)aout –eout  = rout × vout A.. Orbital elements of the outer binary   ·  Using the same method as above, we derive equations for = routvout –(rout vout) .() the post-SN outer binary semimajor-axis and eccentricity. The outer orbit specific energy is given by After the SN, it changes according to   G(m + m + m) G(m + m + m – m)aout –eout Eout =– aout ×  = rout vout  G(m + m + m) = v – ,()   ·  out = routvout – rout vout  rout     = r –α(rin · rout)+α r (vout + vsys) and changes according to out in – (rout · vout)–α(rin · vout) G(m + m + m – m) · ·  Eout =– +(rout vsys)–α(rin vsys) ,() aout

  G(m + m + m – m) where, for notational convenience, we define the mass ratio = vout –  rout m m α ≡ .()  G(m + m + m)  m + m – m m + m = vout –  rout Inserting the relation for the post-SN outer semimajor-axis G(m + m + m) ·  + +(vout vsys)+vsys (Eq. ()), we find the following expression for the post-SN rout outer eccentricity, G(m + m + m – m) –  r –eout out G(m + m + m) ·  aout m =– +(vout vsys)+vsys = – aout rout m + m + m G(m + m + m )     r – r (vout · vsys) vsys + +a out out – – rout out   routrout vc,out vc,out × rout m + m + m – m  – .() × m + m + m rout m + m + m m + m + m – m Toonen et al. Computational Astrophysics and Cosmology (2016)3:6 Page 31 of 36

·   i –eout + Hamers et al. (2015) derive the orbit-averaged Hamiltonian expressions G(m + m + m)aout for the ‘3+1’ as well as the ‘2+2’ configuration. j ×  ·  Note that SeBa is not used to model binary evolution in TrES. rout (rout vsys)+vsys k Note that there is an error in Eq. (42) of Hurley et al. (2002). The factor 2 5 ∝ ·   MR /a should be raised to the power 1/2, which means that kam/τTF R + –α(rin rout)+α (vout + vsys) 2 instead of kam/τTF ∝ R . l And to a lesser degree also the accretion efficiency and the +(rout · vout) corresponding angular momentum loss mode (e.g. Soberman et al. 1997; Toonen et al. 2014). × α(rin · vout)–(rout · vsys)+α(rin · vsys) Received: 29 April 2016 Accepted: 26 November 2016 – –α(rin · vout)+(rout · vsys)  – α(rin · vsys) .()References Aarseth, SJ: Formation and evolution of hierarchical systems. In: Allen, C, Scarfe, C (eds.) The Environment and Evolution of Double and Multiple The terms in the last two lines of Eq. 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