The Physics of Star Cluster Formation and Evolution
Martin G. H. Krause, Stella S. R. Offner, Corinne Charbonnel, Mark Gieles, Ralf S. Klessen, Enrique Vazquez-Semadeni, Javier Ballesteros-Paredes, Philipp Girichidis, J. M. Diederik Kruijssen, Jacob L. Ward, Hans Zinnecker
NNoname manuscript No. (will be inserted by the editor)
The Physics of Star Cluster Formation and Evolution
Martin G. H. Krause · Stella S. R. Offner · Corinne Charbonnel · Mark Gieles · Ralf S. Klessen · Enrique V´azquez-Semadeni · Javier Ballesteros-Paredes Philipp Girichidis · J. M. Diederik Kruijssen · Jacob L. Ward · Hans Zinnecker
Received: 31 Jan 2020 / Accepted: dateMartin G. H. KrauseCentre for Astrophysics Research, School of Physics, Astronomy and Mathematics, Universityof Hertfordshire, College Lane, Hatfield, Hertfordshire AL10 9AB, UKE-mail: [email protected] S. R. OffnerDepartment of Astronomy, The University of Texas, Austin TX, 78712, U.S.A.Corinne CharbonnelDepartment of Astronomy, University of Geneva, Chemin de Pegase 51, 1290 Versoix, Switzer-land; IRAP, CNRS & Univ. of Toulouse, 14, av.E.Belin, 31400 Toulouse, FranceMark GielesInstitut de Ci`encies del Cosmos (ICCUB-IEEC), Universitat de Barcelona, Mart´ı i Franqu`es1, 08028 Barcelona, Spain; ICREA, Pg. Lluis Companys 23, 08010 Barcelona, SpainRalf S. KlessenUniversit¨at Heidelberg, Zentrum f¨ur Astronomie, Institut f¨ur Theoretische Astrophysik,Albert-Ueberle-Str. 2, 69120 Heidelberg, GermanyEnrique V´azquez-SemadeniInstituto de Radioastronom´ıa y Astrof´ısica, Universidad Nacional Aut´onoma de M´ex´ıco, Cam-pus Morelia, Apdo. Postal 3-72, Morelia 58089, M´exicoJavier Ballesteros-ParedesInstituto de Radioastronom´ıa y Astrof´ısica, Universidad Nacional Aut´onoma de M´ex´ıco, Cam-pus Morelia, Apdo. Postal 3-72, Morelia 58089, M´exicoPhilipp GirichidisLeibniz-Institut f¨ur Astrophysik (AIP), An der Sternwarte 16, 14482 Potsdam, GermanyJ. M. Diederik KruijssenAstronomisches Rechen-Institut, Zentrum f¨ur Astronomie der Universit¨at Heidelberg,M¨onchhofstraße 12-14, 69120 Heidelberg, GermanyJacob L. WardAstronomisches Rechen-Institut, Zentrum f¨ur Astronomie der Universit¨at Heidelberg,M¨onchhofstraße 12-14, 69120 Heidelberg, GermanyHans ZinneckerNucleo de Astroquimica y Astrofisica, Universidad Autonoma de Chile, Avda Pedro de Valdivia425, Providencia, Santiago de Chile, Chile a r X i v : . [ a s t r o - ph . GA ] M a y Martin G. H. Krause et al.
Abstract
Star clusters form in dense, hierarchically collapsing gas clouds. Bulkkinetic energy is transformed to turbulence with stars forming from cores fed byfilaments. In the most compact regions, stellar feedback is least effective in remov-ing the gas and stars may form very efficiently. These are also the regions where,in high-mass clusters, ejecta from some kind of high-mass stars are effectively cap-tured during the formation phase of some of the low mass stars and effectivelychanneled into the latter to form multiple populations. Star formation epochs instar clusters are generally set by gas flows that determine the abundance of gas inthe cluster. We argue that there is likely only one star formation epoch after whichclusters remain essentially clear of gas by cluster winds. Collisional dynamics isimportant in this phase leading to core collapse, expansion and eventual disper-sion of every cluster. We review recent developments in the field with a focus ontheoretical work.
Keywords galaxies: star clusters: general · ISM: kinematics and dynamics · openclusters and associations: general · stars: formation Star clusters have caught human attention since ancient times, as evidenced forexample by depictions of the Pleiades on cave walls and the Nebra Disk (Rap-pengl¨uck, 2001; Mozel, 2003). They continue to be a fascinating topic today, thanksto new and puzzling observations challenging the theoretical models.Spitzer has traced the dense gas in a number of nearby young clusters (Fig 1)and shown its connection to young stellar objects (e.g., Gutermuth et al., 2011).Thanks to GAIA (Gaia Collaboration et al., 2018), we now know the kinematics ofmany clusters on a star-by-star basis (e.g., Ward & Kruijssen, 2018; Karnath et al.,2019; Kuhn et al., 2019). Chemistry is traced by spectroscopic and photometricsurveys (Bastian & Lardo, 2018; Gratton et al., 2019); cluster winds have beendetected in spaceborne X-ray observations (Kavanagh et al., 2011) and youngsuper star clusters show evidence of MASER emission (Gorski et al., 2019).These observations place strong constraints on theoretical modelling. The latterhas been typically attempted from different angles with a view on explaining aparticular subset of observations. A simulation that includes gas dynamics, stellardynamics and chemistry to sufficient accuracy and from cloud collapse to clusterdispersal remains beyond reach for the foreseeable future. Approaches that focuson each aspect separately, or combine some aspects making some approximationstherefore have to form the basis of our understanding of stellar clusters.This review aims to provide an overview of the different theoretical approaches,puts them in context with each other, and aims to paint a comprehensive and co-herent picture of the physics of star cluster formation and evolution. We are notaware of past projects with such an ambition, but previous reviews that havesignificant overlap with the present one include Mac Low & Klessen (2004); Zin- necker & Yorke (2007); Portegies Zwart et al. (2010); Gratton et al. (2012); Renzini(2013); Kruijssen (2014); Krumholz (2014); Longmore et al. (2014); Charbonnel(2016); Klessen & Glover (2016); Bastian & Lardo (2018); Gratton et al. (2019) andKrumholz et al. (2019). After defining star clusters in §
2, we first review the onsetof star formation in molecular clouds ( §
3) and the formation of stars in clusters he Physics of Star Cluster Formation and Evolution 3 ( § § §
6) and nucleosynthesis ( §
7) evolve independently. Each section includes, how-ever, links to the other fields. In particular, the chemistry in the predominantlyold, multiple population clusters, discussed in §
7, refers back to the formationepoch, where all the different processes are coupled. We conclude with a summaryand outlook in § We adopt the ontological definition that a star cluster is a gravitationally boundgroup of stars inside a closed tidal surface if this volume is1. not dark matter-dominated and2. contains at least 12 stars.The first condition distinguishes star clusters from galaxies. The second one frommultiple star systems. This definition essentially follows Krumholz et al. (2019,though we do not distinguish here between different overdensities required in dif-ferent environments). Groups of stars that are not gravitationally bound are calledassociations (Blaauw, 1964; Gieles & Portegies Zwart, 2011; Gouliermis, 2018, andAdamo et al. 2020, in prep.). For the Milky Way, bound star clusters have beensubdivided into open clusters in the disc and globular clusters associated withthe bulge and halo. Open clusters are generally young ( (cid:46) (cid:46) M (cid:12) ) while globular clusters are generally old ( > (cid:38) M (cid:12) ), quite typically survivors from the early Universe, representing therelics of star formation at high redshift. In fact, the oldest globular clusters inthe Milky Way have a likely age >
13 Gyr and provide an important constraintfor the age of the Universe (Krauss & Chaboyer, 2003; O’Malley et al., 2017).The distinction between open and globular clusters happens to correspond closelyto a fundamental distinction in photometric properties and chemical abundancepatterns: Open clusters are mostly single population clusters with a single mainsequence in the colour-magnitude diagram, while almost all globular clusters havemultiple main sequences and strong star to star variations in light-element abun-dances, i.e., multiple stellar populations. A more useful classification of star clustersis therefore between single and multiple population clusters (Carretta et al., 2010;Bastian & Lardo, 2018).
Star clusters form from molecular clouds, which are the densest regions in theinterstellar medium, and consist mostly of molecular hydrogen and several other molecules, which are used as tracers for observing these regions and their sub-structure. Molecular clouds range in mass from ∼ to ∼ M (cid:12) , and haveextremely complex hierarchical (or fractal) morphologies (Elmegreen & Falgarone,1996), with the densest regions embedded in larger, lower-density ones, and so on(e.g., Blitz & Williams, 1999). It has been suggested that the internal structure Martin G. H. Krause et al.
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Fig. 1
Three-color Spitzer images (3.6 (blue), 5.8 (green), and 24 µ m (red)) of young, nearby( d < and dynamics of molecular clouds is instrumental in determining the early struc-ture and kinematics of star clusters (e.g., Klessen et al., 2000; Klessen & Burkert,2000, 2001; Offner et al., 2009; Kruijssen et al., 2012; Girichidis et al., 2012b;V´azquez-Semadeni et al., 2017, hereafter VS17).3.1 The Gravoturbulent (GT) scenarioMolecular clouds are known to have internal supersonic non-thermal motions (Wil-son et al., 1970), which follow a relation between the observed linewidth and thespatial scale (Larson’s relation Larson, 1981; Hennebelle & Falgarone, 2012), al-though with substantial scatter (Ballesteros-Paredes et al., 2011; Miville-Deschˆenes et al., 2017). These supersonic motions were originally interpreted as large-scaleradial motions, likely to originate from global collapse (Liszt et al., 1974; Goldre-ich & Kwan, 1974). However, this interpretation was soon rejected because, asit was argued, it would lead to excessively large star formation rates (SFRs) andshould produce systematic velocity differences (i.e., red or blue line shifts) between he Physics of Star Cluster Formation and Evolution 5 emission lines produced by HII regions located at the centers of the clouds andabsorption lines produced at the outskirts of the clouds. Since such shifts were notobserved, the supersonic motions were then interpreted as small-scale supersonicturbulence that produces a turbulent pressure capable of supporting the cloudsagainst their self-gravity (Zuckerman & Palmer, 1974; Zuckerman & Evans, 1974).The requirement for the motions to be confined to small scales was necessary inorder to avoid the generation of the unobserved line shifts and to produce anisotropic pressure that could support the clouds.Since then, the prevailing paradigm for molecular clouds is that they are sup-ported by some agent against their self-gravity, typically turbulence and/or mag-netic fields. These are invoked in part to explain the observed star-formation effi-ciencies per free-fall time, (cid:15) ff – the fraction of gas mass converted to stars over a freefall time – of ∼
1% for most giant molecular clouds (GMCs) (Zuckerman & Evans,1974; Krumholz et al., 2019). Since turbulence is known to dissipate rapidly, typi-cally in a crossing time, it was first proposed that the motions consisted of Alfv´enicturbulence, because Alfv´en waves were thought to be less dissipative than shocks(e.g., Shu et al., 1987). However, subsequent numerical simulations of MHD turbu-lence showed that it dissipates as rapidly as hydrodynamic turbulence (Mac Lowet al., 1998; Stone et al., 1998; Padoan & Nordlund, 1999), implying that constantdriving of the turbulence must be present to maintain it. In this gravoturbulent (GT) scenario (e.g., Klessen et al., 2000; V´azquez-Semadeni et al., 2003; MacLow & Klessen, 2004), the clouds are supported globally by the pressure of the(continuously driven) supersonic, small-scale, isotropic turbulence, while locally,shocks are produced that in turn generate density fluctuations (sheets, filaments,and clumps), which may locally become Jeans unstable and collapse. Moreover,if magnetised turbulence provides support such that clouds are neither dispersingnor globally collapsing, it is assumed that the clouds are in approximate virialequilibrium between turbulence and self-gravity (Krumholz et al., 2006; Gold-baum et al., 2011). This assumption is consistent with observations (e.g., Larson,1981; Heyer et al., 2009). In particular, the Larson (1981) linewidth-size relationobserved in molecular clouds is interpreted as the manifestation of the energyspectrum, E ( k ) ∝ k − , corresponding to strongly compressible, highly supersonicturbulence.3.2 The Global Hierarchical Collapse (GHC) scenarioOn the other hand, there is evidence that the process of formation of the molecularclouds is important for their subsequent dynamical evolution. The clouds seem toform by accreting tenuous ( n ∼
10 cm − ) atomic gas, which often appears gravi-tationally bound to the molecular gas it surrounds (Fukui et al., 2009). Moreover,molecular clouds exhibit a hierarchical structure, so that their internal dynamicsare governed by very similar processes. On smaller scales, star-forming cores ac-crete material from the scale of their parent clumps (i.e., the cores are said to be clump-fed ; Liu et al., 2015; Yuan et al., 2018), and longitudinal, multi-parsec scaleflows are routinely observed along filamentary clouds, which feed the main cores(or hubs ) within the filaments (e.g., Myers, 2009b; Schneider et al., 2010; Kirket al., 2013; Peretto et al., 2014; Wyrowski et al., 2016; Hacar et al., 2017; Chenet al., 2019b). Additionally, numerical simulations of the formation and evolution Martin G. H. Krause et al. of cold, dense atomic clouds from large-scale compressions in the warm, diffusegas also suggest that the clouds engage into global, hierarchical collapse (GHC;V´azquez-Semadeni et al., 2019) soon after they reach their thermal Jeans mass(V´azquez-Semadeni et al., 2007, 2009; Heitsch et al., 2008). In what follows, wefocus on this scenario, as it provides a direct link between the processes occurringin the gas during the collapse and the structural properties of the resulting stellarcluster(s).
The clouds are expected to rapidly reach and exceed their thermal Jeans massbecause the Jeans mass in the dense, cold gas is ∼ times smaller than in thediffuse, warm gas (G´omez & V´azquez-Semadeni, 2014) and simulations indicatethat the clouds actively accrete from their diffuse environment (Ballesteros-Paredeset al., 1999; Hartmann et al., 2001; V´azquez-Semadeni et al., 2006; Heitsch & Hart-mann, 2008; Banerjee et al., 2009; Heiner et al., 2015; Wareing et al., 2019). Thisaccretion implies that the clouds generally grow in mass, allowing them to becomemagnetically supercritical (i.e., unsupported by the magnetic field), gravitationallyunstable, and molecular at roughly the same column density ( ∼ cm − ) forsolar-neighbourhood pressures and metallicities (Hartmann et al., 2001; Heitschet al., 2009; V´azquez-Semadeni et al., 2011; Heiner et al., 2015).Simulations of the self-consistent formation and evolution of clouds by converg-ing streams of diffuse gas (e.g., Heitsch et al., 2005, 2006; Audit & Hennebelle,2005, 2010; V´azquez-Semadeni et al., 2006, 2007; Hennebelle et al., 2008; Banerjeeet al., 2009) show that the very formation process of the cloud causes the genera-tion of moderately supersonic (with respect to the sound speed in the cold, densegas) turbulence by the combined action of various instabilities, such as the non-linear thin-shell instability (Vishniac, 1994), thermal instability (Field, 1965) andKelvin-Helmholtz instability (see Heitsch et al., 2006, and Klessen & Hennebelle,2010a, for further discussions). Similar effects have also been shown for shellsof expanding bubbles (Krause et al., 2013a). The turbulence generates nonlineardensity fluctuations in which the free-fall time τ ff = (cid:112) π/ (32 Gρ ) is significantlyshorter than the average in the cloud.The energy in the turbulent motions generated by the instabilities, which areonly moderately supersonic with respect to the cold gas, and subsonic with respectto the warm gas, quickly becomes overwhelmed by the gravitational energy ofthe whole cloud (actually, a cloud complex), which then begins to undergo globalgravitational contraction. In the GHC scenario, thus, the apparent near-virial stateof molecular clouds and their substructures is not due to turbulent support, butrather to the infall motions driven by the self-gravity (Ballesteros-Paredes et al.,2011). It should be noted, however, that the infall is highly chaotic and so atruly random (turbulent) component is in fact maintained by the collapse (Klessen& Hennebelle, 2010b; Robertson & Goldreich, 2012; Murray & Chang, 2015; Li, he Physics of Star Cluster Formation and Evolution 7 et al., 2012; Bovino et al., 2013; Federrath et al., 2014; Schober et al., 2015; Klessen,2019).The presence of turbulent density fluctuations with nonlinear amplitudes, to-gether with the generally amorphous and flattened or filamentary shape of theclouds, has the important implication that realistic collapse is far from homol-ogous (uniform spherical configurations, all material in the sphere reaching thecenter at the same time). It is well known that already in non-uniform sphericalconfigurations (“cores”) with centrally-peaked radial density profiles, the central,densest parts terminate their collapse (i.e., reach protostellar densities) earlierthan the outer parts, and then the rest of the material, which was initially atlower densities, continues to accrete onto the previously collapsed material (e.g.,Larson, 1969; Penston, 1969; Shu, 1977; Hunter, 1977; Whitworth & Summers,1985; Foster & Chevalier, 1993; Mohammadpour & Stahler, 2013; Keto et al.,2015; Naranjo-Romero et al., 2015). In a turbulent system, the nonlinear densityfluctuations have free-fall times significantly shorter than that of the whole cloud,and so they can collapse faster, as soon as they become locally gravitationallyunstable (compare V´azquez-Semadeni et al., 2019).Under this regime, the cloud evolves towards containing a large number ofthermal Jeans masses, in agreement with the observation that molecular cloudstypically have masses M c upwards of 10 M (cid:12) (e.g., Mac Low & Klessen, 2004, andreferences therein). Thus, the cloud becomes a system of collapses within collapses ,with an ever-larger hierarchy of collapsing scales, each one accreting from the nextlarger scale (V´azquez-Semadeni et al., 2019). This is a mass cascade, in some sensessimilar to the turbulent energy cascade (Field et al., 2008). This is also essentiallyHoyle’s fragmentation (Hoyle, 1953), except with nonlinear density fluctuationsand non-spherical geometry of the clumps (V´azquez-Semadeni et al., 2019). Also,it can be considered as an extension of the competitive accretion scenario (Bonnellet al., 2001; Bonnell & Bate, 2006), with the accretion extending to cloud scales( ∼
10 parsecs or more), and with the added ingredient that a whole hierarchy ofchaotic, gravitational contraction flows is present.
At sufficiently advanced stages of a cloud’s evolution, when its mass M c is muchlarger than the Jeans mass, it must behave essentially as a pressureless collapse,because precisely the meaning of M c (cid:29) M J is that the gravitational energy over-whelms the internal energy of the cloud. But it is known that pressureless collapseamplifies anisotropies, so that a triaxial ellipsoid contracts first along its shortestdimension to form a sheet, and then an elliptical sheet contracts again along itsshortest dimension to form a filament (Lin et al., 1965). Therefore, it is expectedthat multi-Jeans mass molecular clouds should evolve to develop filaments, whichare actually akin to “rivers” funnelling the mass from large to small scales (G´omez& V´azquez-Semadeni, 2014). This is consistent with the observation that densemolecular cloud cores appear as “hubs” at the intersection of filaments (e.g., My- ers, 2009a), with the filaments feeding material to the hubs (e.g., Schneider et al.,2010; Sugitani et al., 2011; Kirk et al., 2013; Peretto et al., 2014; Chen et al.,2019b).Since the majority ( ∼ Martin G. H. Krause et al. star formation is initiated already in the flows feeding the hubs. This mecha-nism was referred to as “conveyor belt cluster formation” by Longmore et al.(2014) and in Krumholz et al. (2019), in opposition to “monolithic cluster forma-tion”, in which the gas first collapses and subsequently forms stars in a centrally-concentrated cluster. The conveyor-belt mechanism is also observed in simulationsof self-consistent cloud formation and evolution, in which the filaments form spon-taneously by anisotropic gravitational contraction (G´omez & V´azquez-Semadeni,2014; V´azquez-Semadeni et al., 2019).
Another expected consequence of the global collapse and continued accretion ontothe star-forming hubs observed in the simulations is an acceleration of the starformation before massive stars form. This increase in the star formation rate (SFR)is routinely observed in simulations of cloud evolution (e.g., V´azquez-Semadeniet al., 2010, 2017; Hartmann et al., 2012; Col´ın et al., 2013; Lee et al., 2015;Li et al., 2018), and predicted by models of clouds dominated by gravity (e.g.,Zamora-Avil´es et al., 2012; Zamora-Avil´es & V´azquez-Semadeni, 2014; Caldwell& Chang, 2018). Observational evidence of the acceleration is provided by, forexample, a) the age histograms of young embedded clusters, which systematicallyshow a maximum at either the smallest ages, or at a certain, relatively recent age,together with a tail of older stars, of ages up to several Myr (e.g., Ballesteros-Paredes et al., 1999; Palla & Stahler, 2000, 2002; Huff & Stahler, 2006; Da Rioet al., 2010); b) a superlinear ( ∼ t ) temporal dependence of the total number ofstars formed at time t in several young clusters (Caldwell & Chang, 2018).In the GHC scenario, the increase of the SFR during the early stages of star-forming regions is due to the growth in mass, density, and size of the regions dueto accretion from their parent structures. The increase in density implies an in-crease of the SFR because a larger fraction of the mass is at densities high enoughthat their free-fall time is much shorter than that of the mean density of the par-ent structure (Zamora-Avil´es & V´azquez-Semadeni, 2014). But, additionally, thelarger mass of the more evolved regions provides a larger mass reservoir, allowingfor the formation of more massive stars. So, the star-forming regions evolve towardsforming more massive stars, meaning that the formation of massive stars is delayedwith respect to that of the first low mass stars, by several Myr in moderate-massregions, according to the simulations (V´azquez-Semadeni et al., 2009, 2017, topleft panel of Fig. 2). Note that low-mass stars always form, but the maximum massof the stars that can form is capped by the instantaneous mass of the hub wherethey form, and increases with time as long as the hub’s mass increases. Even-tually, however, the stellar feedback begins to erode the hub in the simulations,decreasing its density and mass, and also eroding the filamentary accretion flow,decreasing the maximum stellar mass that can form. A model for the developmentof the high-mass slope of the IMF based on the same principle, of the mass of the most massive star being bounded by the mass of the hub in which it forms,has been developed by Oey (2011). This delayed formation of massive stars alsoimplies that the age range of the massive stars is smaller (and they are younger)than that of the low-mass stars, which begin to form since the onset of the starformation activity in the region. Equivalently, the mass range of the younger stars he Physics of Star Cluster Formation and Evolution 9 Fig. 2
Top left:
Normalised cumulative stellar mass histograms of star forming regions ina numerical simulation of GHC (V´azquez-Semadeni et al., 2017) at various times. As timeproceeds, a larger fraction of the stars are seen to be massive, until feedback begins to disruptthe cloud.
Top right:
Mass versus age of the members of a stellar group in the same simulation,at time t = 22 . Bottom left:
Mass versus distance from center of mass of the group members at t = 23 . ≈ . Bottom right:
Groups constituting thecluster in the simulation at time t = 30 . is larger and extends to higher masses than that of the older stars (top right panelof Fig. 2). Star formation occurring in the filaments generally involves lower-mass cores, be-cause they are themselves part of the flow falling onto the main hubs, which are themain accreting centers. That is, the stars formed in the filaments do so in secondarygravitational potential wells, while the hubs are the primary wells. Therefore, themore extended secondary star formation in the filaments generally produces lower- mass stars, and thus tends to produce a primordial mass segregation in the cluster(bottom left panel of Fig. 2), independent of any N -body processes that may occurafterwards ( § stars tends to be larger than that of those nearer the hub. In the simulation fromFig. 2 a median-age gradient of ∼ − is found (Getman et al., 2018),consistent with the gradient observed in the clusters in the MYStIX (Feigelsonet al., 2013) and SFiNCs (Getman et al., 2017) star-forming region catalogs.The hierarchical and filamentary structure of the collapse flow is imprinted onthe structure of the cluster itself, which therefore adopts a self-similar, fractal-likespatial distribution, and retains traces of a filamentary morphology, as seen in thebottom right panel of Fig. 2, which shows the groups constituting the cluster in thesimulation at time t = 30 . n -body interactions (see below). There-fore, these “primordial” structural features are expected to be more prominent inyounger clusters. The formation and evolution of star clusters is a multi-scale process that dependsin detail on the formation of the constituent individual stars. In turn, forming starsinfluence the accretion and dynamics of their neighbours through stellar feedback,including winds, radiation and supernovae.Stars accreting from a common gas clump or protostellar core may competewith one another for fuel, while accretion disk properties depend on the localionising flux, which is set by the distribution of nearby massive stars. Ionisingradiation, winds and jets from protostars interact with their own accretion streams,as well as those of other stars. Finally, there is the puzzling observation of multiplepopulations in clusters ( § els that depend on feedback and the larger protostellar environment. This divisionis mainly for convenience, since in practice, accretion is determined by a variety ofnonlinear processes that span a broad range of times and physical scales. Finally,we discuss models for how and why accretion ultimately ceases, which is criti-cal for understanding the accretion histories of individual stars as well as global he Physics of Star Cluster Formation and Evolution 11 properties such as the star formation efficiency, star formation rate and lifetime ofmolecular clouds.4.1 Observational Signatures of Accretion: Protostellar Luminosities, Outflowsand Spectral LinesProtostellar outflows are a direct byproduct of the accretion process (see ChapterProcesses for more details). If a fixed fraction of accreting material is flung out-wards in an outflow, then in principle by measuring the outflow mass flux it ispossible to reverse engineer the accretion rate and history. Observations of pro-tostellar outflows suggest protostellar accretion rates of 10 − − − M (cid:12) yr − ,where younger or more massive protostars have higher inferred accretion rates(Bally, 2016).Outflow morphology also gives important insights in the accretion process.Outflows and jets (highly-collimated flows, usually observed in optical emission),frequently exhibit regularly spaced clumps along the outflow axis, “bullets” (Bally,2016; Zhang et al., 2016). The spacing of the bullets indicates that accretion isvariable on timescales of hundreds to thousands of years (Bachiller et al., 1991;Lee et al., 2009; Arce et al., 2013). In the most extreme events, the accretion rate,and hence the source luminosity, rises by several orders of magnitude over a periodof years in a brief “episodic” accretion burst (Audard et al., 2014).Outflows also exhibit precession or changes in direction, providing a windowinto the angular momentum of accreting material and the impact of binarity onthe accretion process (Shepherd et al., 2000; Hirano et al., 2010; Lee et al., 2017a).A number of outflows appear to have two components: a highly-collimated compo-nent, likely launched close to the protostar, and a wider-angle, slower componentthat likely arises from the accretion disk (Hirano et al., 2010; Arce et al., 2013).A variety of uncertainties underpin the connection between outflows and ac-cretion (Dunham et al., 2014b). Much of the outflowing material is entrainedcore material (Offner & Chaban, 2017), so the outflow is not a direct measure ofaccreting gas. The typical outflow dynamical time, as measured by the outflowextent and gas velocity, t dyn ∼ L out / (2 v out ) ∼ yr, is shorter than the expectedprotostellar lifetime, and thus provides only a narrow window into the total ac-cretion history (Bally, 2016). This is probably related to accretion physics ( § − (e.g., Kirk et al., 2010; Foster et al., 2015), a protostar would requireof the order of 10 years to move out of a filament or core of 0.1 pc thickness.Protostellar luminosities provide another constraint on accretion (see Chapterprocesses). At early times and for low masses, i.e., before the intrinsic luminosityof the protostar becomes significant, the luminosity is directly proportional to theaccretion rate. By assuming reasonable properties for the protostellar mass andradii, it is possible to set limits on the accretion rate. However, protostellar evolu- tion remains uncertain in part because it is itself sensitive to the accretion history(Palla & Stahler, 1991; Baraffe et al., 2009; Hosokawa et al., 2011). Observationsof clusters of protostars show orders of magnitude scatter, such that on averagethe luminosity is weakly dependent, at best, on the protostellar class (Dunhamet al., 2014a; Fischer et al., 2017). The difficulty of mapping classes to evolutionary stage further confuses accretion trends over time (Robitaille et al., 2006; Dunhamet al., 2010; Offner et al., 2012).Time-domain studies of protostellar luminosities are more informative and sup-port the highly variable nature of accretion suggested by outflow observations.Changes in luminosity are observed on timescales of days to decades spanningchanges from as little as a few percent to several orders of magnitude in bright-ness (Rebull et al., 2014; Audard et al., 2014). Low magnitude, shorter timescalevariations, which are quite common, are likely caused by stellar activity or disk oc-cultations, while more extreme and rarer luminosity changes can only be explainedby accretion fluctuations (Hillenbrand & Findeisen, 2015).Early observations of protostars noted that they were on average about 10times dimmer than simple accretion models and timescale arguments would sug-gest (Kenyon et al., 1990; Kenyon & Hartmann, 1995; Evans et al., 2009). Thisbecame known as the “protostellar luminosity problem”. A variety of theoreticalsolutions have been proposed that resolve this problem, including episodic andslow accretion ( § (cid:39) × yr old, indicate that accretion declines steeply atlate times (Hartmann et al., 2016). However, the accretion rate depends on bothage and mass, which are difficult to disentangle due to measurement and modeluncertainties. Observations of Balmer continuum, photometry and emission linessuggest ˙ M ∝ M α ∗ , where α = 1 . − . M ∝ t β , where β = − . − − . §
3) efficiently proceedsfrom ∼ . m = 46 . c s G = 7 . × − (cid:18) T
10 K (cid:19) / M (cid:12) yr − , (1)where c s is the sound speed. If the gas is isothermal and centrally condensed theinfall solution is self-similar and can be written (Shu, 1977):˙ m = 0 . c s G = 1 . × − (cid:18) T
10 K (cid:19) / M (cid:12) yr − . (2) Figure 3a shows the isothermal sphere accretion rate and a variety of other an-alytical predictions as defined below. In these models, the accretion is by naturetime-invariant and independent of stellar mass.These idealised solutions, however, gloss over a great deal of important physics.Cores are observed to be magnetised and turbulent (Crutcher, 2012; Kirk et al., he Physics of Star Cluster Formation and Evolution 13 a. Analytic Models for Protostellar Accretion b. MHD Simulation of Accreting Protostars
Fig. 3
Stellar accretion rates as a function of time. Left (a): different analytic model pre-dictions for protostellar accretion. Right (b): accretion of individual protostars in an MHDsimulation of protostars forming in a turbulent giant molecular cloud (figure adapted from Liet al. (2018) and reproduced with permission). Turbulence, dynamics and protostellar outflowstogether significantly modulate the accretion histories. c eff = c s (1 + 2 α + β ) / , where α = P B /P th and β = P turb /P th are the ratio ofmagnetic and turbulent pressure to thermal pressure, respectively (Stahler et al.,1980). However, both turbulence and magnetic fields are intrinsically anisotropic,which suggests this approach over-simplifies their true impact on the accretionrate and over-estimates their contributions to pressure support. Also, these mod-els implicitly assume that star formation can be represented by discrete collapsingregions and thus, arguably, are applicable only for isolated, low-mass star forma-tion (cf. § M ∝ M / M / f , where M is theinstantaneous stellar mass and M f is the final mass of the star at the end of accre-tion. This naturally implies that high-mass stars have higher accretion rates thanlow-mass stars, form faster and that their accretion increases in time. Hydrody-namic calculations of high-mass star formation, which adopt high-column density,high-mass cores as initial conditions exhibit these trends (Krumholz et al., 2012; Rosen et al., 2016).Hydrodynamic simulations of forming star clusters paint a very dynamicalpicture, particularly in clusters with high-stellar densities. In the “competitive ac-cretion” model protostars begin as small seeds formed by local collapse, whichcompete with one another for the available gas (Bonnell et al., 2001). Birth lo-4 Martin G. H. Krause et al.
Stellar accretion rates as a function of time. Left (a): different analytic model pre-dictions for protostellar accretion. Right (b): accretion of individual protostars in an MHDsimulation of protostars forming in a turbulent giant molecular cloud (figure adapted from Liet al. (2018) and reproduced with permission). Turbulence, dynamics and protostellar outflowstogether significantly modulate the accretion histories. c eff = c s (1 + 2 α + β ) / , where α = P B /P th and β = P turb /P th are the ratio ofmagnetic and turbulent pressure to thermal pressure, respectively (Stahler et al.,1980). However, both turbulence and magnetic fields are intrinsically anisotropic,which suggests this approach over-simplifies their true impact on the accretionrate and over-estimates their contributions to pressure support. Also, these mod-els implicitly assume that star formation can be represented by discrete collapsingregions and thus, arguably, are applicable only for isolated, low-mass star forma-tion (cf. § M ∝ M / M / f , where M is theinstantaneous stellar mass and M f is the final mass of the star at the end of accre-tion. This naturally implies that high-mass stars have higher accretion rates thanlow-mass stars, form faster and that their accretion increases in time. Hydrody-namic calculations of high-mass star formation, which adopt high-column density,high-mass cores as initial conditions exhibit these trends (Krumholz et al., 2012; Rosen et al., 2016).Hydrodynamic simulations of forming star clusters paint a very dynamicalpicture, particularly in clusters with high-stellar densities. In the “competitive ac-cretion” model protostars begin as small seeds formed by local collapse, whichcompete with one another for the available gas (Bonnell et al., 2001). Birth lo-4 Martin G. H. Krause et al. cation and dynamical interactions determine the protostellar locations within thegravitational potential well and thus their rate of gas accretion. More massive starsnaturally form in the center of the cluster and are best positioned to rapidly accretegas (Fig. 2, Bonnell et al., 2001; Bonnell & Bate, 2006). In this scenario accretioncontinues until the gas runs out, which occurs on ∼ a global free-fall time. As aresult, all stars have the same formation time, which is set by the cluster environ-ment. Analytically, this corresponds to accretion rates of ˙ M ∝ M / for stars ingas-dominated potentials (Bonnell et al., 2001). At late times when the stellar massexceeds the gas mass, accretion limits to ˙ M ∝ M (compare Ballesteros-Paredeset al., 2015; Kuznetsova et al., 2017, 2018). Numerical simulations following theformation of massive stars from a 250 pc scale interstellar medium region suggestthat massive stars form over a longer time period via converging, filamentary gasflows (Padoan et al., 2019). Their cores are less massive than the final stellar massat any given time, i.e., massive stars do not form from progenitor massive turbulentcores. Inflow rather than competition drives the accretion behavior.Stellar accretion is highly variable but does not increase with stellar mass aspredicted by both the turbulent core and competitive accretion models. All core-regulated models fall somewhere in the continuum between constant accretionrate and constant accretion time, between the highly dynamical and isolated starformation paradigms.4.3 Disk-Regulated Accretion ModelsObservations suggest that mass does not pass smoothly from the outer envelopeto the protostar but instead accretes in a more variable process as mediated byan accretion disk. Thus, the semi-analytic models outlined above only describethe time-average accretion behavior. Variability in models that do not explicitlyinclude disk physics arises purely from variation in the environment and evolutionof the host gas reservoir (Padoan et al., 2014; Li et al., 2018; Padoan et al., 2019).The formation of a disk is a direct consequence of angular momentum in thestar formation process. In the absence of angular momentum or in the limit ofperfectly efficient angular momentum transport accretion disks would not exist.However, observations tell us disks are common (Tychoniec et al., 2018; Andrewset al., 2018). They act as a repository for high-angular momentum gas and ef-fectively sort low-angular momentum material, which moves inwards towards theprotostar, and high-angular momentum material, which moves outwards. The twodominant processes for angular momentum transport in disks, viscous torques dueto turbulence (Balbus & Hawley, 1994) and gravitational instability (GI) (Toomre,1964; Laughlin & Bodenheimer, 1994), both produce variability in the accretionflow.Viscous torques require the activation of the magnetorotational instability(MRI), which depends on the local ionization fraction (Balbus & Hawley, 1994). If the gas is not sufficiently ionized then the magnetic field is poorly coupled,reducing the efficacy or shutting off the MRI entirely (Blaes & Balbus, 1994).Thus, MRI-regulated accretion disks may undergo periods with little or no accre-tion during which material builds up in the disk, followed by periods when thegas is thermally ionized initiating a burst of accretion (Zhu et al., 2009b). During he Physics of Star Cluster Formation and Evolution 15 these bursts accretion may be elevated by several orders of magnitude, similar toobserved FU Ori bursts (Audard et al., 2014).The requisite ionization for the MRI may be provided by the parent star andits environment including FUV radiation, x-rays, and cosmic rays (Umebayashi &Nakano, 1981; Semenov et al., 2004; Glassgold et al., 2007; Perez-Becker & Chiang,2011). Consequently, disk surface layers are generally strongly ionized such thataccretion continues in a layered fashion, where gas accretes in the surface layerswhile the disk mid-plane remains predominantly neutral and is an MRI “deadzone” (Gammie, 1996). High periods of accretion may in turn increase the x-rayand cosmic-ray ionization towards the disk mid-plane, prompting accretion deeperin the disk and boosting the magnitude of the accretion burst (Offner et al., 2019).Gravitational torques, which are the dominant transport mechanism in theouter disk, may also prompt large accretion variations (Kratter & Lodato, 2016).If mass builds up in the inner disk, the disk may undergo GI and form small clumps.If these clumps migrate inwards they produce burst events as they accrete onto thestar (Vorobyov & Basu, 2005, 2006). Mild GI, in the form of spiral arms, to severeGI, which causes catastrophic disk fragmentation, produce accretion variabilityfrom factors of a few to orders of magnitude (Audard et al., 2014).Dynamical interactions between stars or close binary companions can also pro-duce accretion variability (Adams & Lin, 1993). Close passage gravitationally per-turbs the disk, prompting instability and elevated accretion (Bonnell, 1994). Fi-nally, variation of the angular momentum of the infalling gas on larger scales mayalso create luminosity variations, either through direct accretion (Padoan et al.,2014) or by affecting disk properties (Lee et al., 2017b).The frequency and magnitude of disk-mediated accretion bursts depend bothon disk microphysics and the larger disk environment (Kratter et al., 2010). Cur-rent observations show a heterogeneous distribution of large, small, smooth andstructured disks. Likely both, GI and MRI, play a role in disk evolution (Armitageet al., 2001; Zhu et al., 2009a). The corresponding scatter in protostellar luminosi-ties provides one solution for the protostellar luminosity problem (Kenyon et al.,1990; Offner & McKee, 2011; Dunham & Vorobyov, 2012; Padoan et al., 2014).4.4 Feedback-Regulated Accretion ModelsStellar feedback, in the form of protostellar outflows, winds and radiation, alsoshapes the accretion process, either by reducing the mass reservoir available foraccretion (as found in observational and numerical work, see e.g. Dale et al., 2015;Ginsburg et al., 2016) or by dispersing bound gas and halting accretion alto-gether. The earliest semi-analytic model for feedback-regulated accretion weighedthe competition between accretion and outflow feedback (Norman & Silk, 1980).Feedback-regulated models are often formulated more generally in terms of a distri-bution of stopping times or probabilities, which has the advantage that the model can be agnostic about the particular mechanism halting accretion. For example,several more recent models assume accretion durations follow the probability dis-tribution, f ( t ) = 1 /τ e − t/τ , where τ is the mean accretion time (of the order of10 yr). Such models can reproduce the stellar IMF and match the observed pro-tostellar luminosity distributions without appealing to overly long accretion times or significant periods of episodic accretion (Basu & Jones, 2004; Myers, 2009b,2012).A variety of hydrodynamic simulations of accreting protostars including proto-stellar outflows have been carried out, which demonstrate that outflows can indeedefficiently expel 30-60% of the dense core material and reduce overall star forma-tion efficiencies by ∼
30% (Hansen et al., 2012; Machida & Hosokawa, 2013; Offner& Arce, 2014; Federrath, 2015; Offner & Chaban, 2017; Tanaka et al., 2017). Sim-ulations of isolated dense cores including protostellar outflows find that the mainphase of accretion continues for 0.3-0.5 Myr, depending on the degree of turbu-lence and magnetic field strength (Machida & Hosokawa, 2013; Offner & Arce,2014; Offner & Chaban, 2017). The accretion rate of a protostar accreting withina turbulent, magnetised dense core can be described in terms of the current pro-tostellar mass, m , and its final mass, m f : ˙ m = m (cid:16) mm f (cid:17) / m / f (cid:20) − (cid:16) mm f (cid:17) / (cid:21) , where m ∝ Σ / c is a constant coefficient related to the surface density of thecore, Σ c , and both m and m f are in solar masses (Offner & Chaban, 2017). Thisis effectively the predicted turbulent core model accretion rate (McKee & Tan,2003), tapered by a multiplicative factor. While the final masses are influenced bythe core magnetic field and turbulence, the accretion history can be analyticallydescribed independently of the gas physical properties. Simulations of the impactof outflows on accretion within forming star clusters find wide variation in theaccretion histories as shown for example in Figure 3b with some accretion ratessteadily declining over time to ˙ m = 10 − M (cid:12) yr − and others declining and thenrising again due to protostellar dynamics (Li et al., 2018).Feedback, turbulence and gravitational interactions may all play importantroles in setting the accretion histories of individual stars. These same processes alsodrive the global evolution of the molecular cloud, gas dispersal ( §
5) and star clusterdynamics ( § After the initial star formation process, clusters become exposed, i.e., no densegas is found in clusters from this stage onward. The process of a cluster becomingexposed may be driven by collective stellar feedback and may influence the dy-namics of the stars. Later, star cluster winds can convey feedback energy to larger scales. Cooling flows have been discussed in the context of secondary star forma-tion episodes, although age spreads in clusters are small, such that secondary starformation is likely restricted to associations.Star clusters have a closed tidal surface and usually contain a focal point, theminimum of the gravitational potential. It is therefore generally expected that a he Physics of Star Cluster Formation and Evolution 17 global pattern for the gas dynamics will form, which may in principle be inflow,outflow, or hydrostatic equilibrium. Contrary to galaxies, (even approximate) hy-drostatic equilibrium is probably not relevant for star clusters.5.1 Impossibility of hydrostatic equilibriumTo see this, we present a simple argument and show that starting from a situationclose to hydrostatic equilibrium, stellar feedback would alter the gas propertiesquickly. Either cooling would take over leading to inflow, or heating, leading tooutflow. Let us start with the hydrostatic equilibrium condition :d Φ d r = − ρ d p d r (3)Approximating gradients by the absolute change out to the half-mass radius, wecan write eq. (3) as : G ( M/ /r h = p/ρ = k B T / ( µm p ). Radiative cooling willreduce the gas pressure. To maintain hydrostatic equilibrium, the cooling time t c = k B T / ( nΛ ) therefore must at least exceed the crossing time t x = 2 r h /σ (Krause et al., 2019). Using also the definition σ = GM/ ( ηr h ) (4)with η = 7 . n < G / µm p M / η / Λr / = 65 cm − (cid:18) Λ − erg cm s − (cid:19) − (cid:18) M M (cid:12) (cid:19) / (cid:18) r h (cid:19) − / . (5)For the relevant densities, Λ is of the order of 10 − erg cm s − (Bialy & Sternberg,2019), which we have used for the scaling in eq. (5).The immediate effect of stellar feedback is to add mass and energy to theintracluster gas. Hydrostatic equilibrium may only be maintained, if the energyinput matches the energy loss via gas cooling. The particle density in the clusterincreases at a rate (e.g., Mathews & Brighenti, 2003):˙ n = αM/ πr / − Myr − (cid:16) α − s − (cid:17) (cid:18) M M (cid:12) (cid:19) (cid:18) r h (cid:19) − . (6)Here, we have scaled the mass loss factor α = ˙ M/M to 10 − s − , a value thatwould be expected for a very young ( ≈ Myr) stellar population (Leitherer et al.,1999; Gaibler et al., 2005; Krause et al., 2013b).Therefore, within a short timescale compared to the timescale of a cluster’sevolution, stellar feedback would increase the gas density beyond the cooling limit.
Hydrostatic equilibrium could then only be maintained, if the energy input was Φ : gravitational potential, r : radius, ρ : gas density, p : pressure M : cluster mass, µ : mean molecular weight k B : Boltzmann constant, T: temperature, n: particle density, Λ : cooling function. r h : half-mass radius, σ : line-of sight velocity dispersion.8 Martin G. H. Krause et al. spatially fine-tuned and arranged to increase in time as required for the increasingcooling rates. Since cooling rates are determined by atomic physics and energyinput by stellar physics, this will not be the case.The late time evolution of α can be approximated as α = 4 . × − s − ( t/
13 Gyr) − . (Mathews & Brighenti, 2003). Therefore, if at late times the clusterwas for some reason in a state of hydrostatic equilibrium, it would take longerfor the stellar feedback to increase the gas density beyond the stability limit.However, the relevant timescales also grow, such that the cluster would alwaysbecome unstable on a timescale that is shorter than its age. The analysis dependsonly weakly on cluster radius. For smaller masses, the argument becomes stronger.Therefore, we can conclude that stellar feedback generally inhibits hydrostaticequilibrium in star clusters at all times.5.2 Initial gas clearanceHow much gas is left over from the star formation process and how violently this isremoved from the cluster has wide-ranging implications for star formation. Fromabundances and ages of clusters and associations, Lada & Lada (2003) concludedthat stars generally form in dense clusters and get dispersed due to violent gasexpulsion and the associated change in gravitational potential ( infant mortality ).Subsequent work has superseded this initial picture, showing that the statisticsdepend crucially on the surface density threshold for the definition of star clusters(Bressert et al., 2010), as well as the initial gas density at which stars are forming(Kruijssen, 2012). Many recent studies show that star formation proceeds at avariety of densities and spatial scales (e.g., Bastian et al., 2007; Sun et al., 2018;Rodr´ıguez et al., 2019) and detailed analysis of OB associations shows that theydid not evolve from significantly smaller structures (Wright et al., 2014; Ward &Kruijssen, 2018; Ward et al., 2019). A good example is the Gaia study of the closestOB association, Sco-Cen OB2 (Wright & Mamajek, 2018), for which alternativeformation scenarios based on multi-wavelength observations have been suggested(Krause et al., 2018). While the formation of bound clusters and their dispersalmay be less common than once thought (compare also Kruijssen, 2012; Krumholzet al., 2019), it is still interesting to ask what fate the gas experiences and whatroles it can play in any given cluster.
We use the term gas expulsion to refer to a special kind of gas removal, where asignificant mass of gas ( (cid:38)
50% of the total mass) is removed quickly (compared tothe crossing time for stars, i.e. impulsively) from the cluster such that some or allstars are left unbound and escape (Hills, 1980). The only situation where this canhappen is at the end of the initial formation of the star cluster from the primordialgas cloud, hence the frequent use of the term primordial gas expulsion.
Assuming that the gas retains the same spatial profile as the young stellarcluster, the effect of primordial gas expulsion has been studied extensively in pure N -body simulations, where the stars are represented by a large number of gravi-tationally interacting bodies and the gas by a smooth potential that is varied intime (e.g., Portegies Zwart et al., 2010; Banerjee & Kroupa, 2017, for reviews). he Physics of Star Cluster Formation and Evolution 19 Fig. 4
Two time frames of a numerical simulation of the evolution of a molecular cloudwith stellar feedback. Upper panels are maps of the column density. Lower panels are thecorresponding x − profiles of the gravitational potential at the y position of each sink particle.As the expansion of the H ii region proceeds, the gravitational potential is flipped-up, andthus, the stars are pulled out toward the edges. Figure from Zamora-Avil´es et al. (2019). Baumgardt & Kroupa (2007) show in a large parameter study that most clustersare completely destroyed or lose a substantial number of stars. Those that survivehave expanded by a typical factor of 3-4. More recent N-body simulations vary,e.g., the kinematic state at gas expulsion or the level of substructure (Smith et al.,2013; Farias et al., 2015) and find that cluster dispersal becomes more difficult inmore realistic scenarios (Farias et al., 2018).Hydrodynamic simulations by Geen et al. (2018) and Zamora-Avil´es et al.(2019, Fig. 4) have shown that the dispersal of the parental molecular cloud couldhave a “gravitational feedback” effect on the newborn stellar cluster: feedback fromthe newborn massive stars expels the gas from the collapse centre. Since neitherthe parental clouds, nor the formed shells are distributed symmetrically aroundthe H ii region, net forces can even accelerate the stars towards the edges of thecavity and may produce a “Hubble flow-like” ( v ∝ r ) expansion. Several candidates for stellar groups undergoing expansion or dispersal relatedto gas expulsion have been found with
Gaia
Data Release 2, for clusters with masses up to 10 M (cid:12) : Kuhn et al. (2019) study the kinematics of 28 young stellargroups with typically 100 stars with proper motion measurements each. For 75%of their objects, they find a positive offset of the generally Gaussian distributionsof the cluster-centric radial velocities, i.e. an expansion of the system. Some oftheir groups are likely unbound and may have formed as associations (comparealso Bravi et al., 2018; Wright et al., 2019), while some could have undergonean expansion phase and are settling in virial equilibrium. There is an interestingvariety in expansion states also for compact systems. Kuhn et al. (2019) findthe partially embedded Orion Nebula cluster (ONC) ( r h = 0 . v out = 0 . ± .
20 km s − ), but the likewise partially embeddedcluster Cep B ( r h = 1 . § ≈
10 Myr), exposed, massive ( (cid:38) M (cid:12) ) star clusters also appearfrequently to have velocity dispersions above the expectation for virial equilibrium,given the mass expected for the observed luminosity and age (e.g., Bastian &Goodwin, 2006; Goodwin & Bastian, 2006; Gieles et al., 2010a; Portegies Zwartet al., 2010). This has been discussed as evidence for dissolution after gas expulsion(Goodwin & Bastian, 2006). However, N -body simulations show that many of theseclusters would have re-virialised by the time of observation (Baumgardt & Kroupa,2007; Gieles et al., 2010a; Portegies Zwart et al., 2010). An interpretation in termsof a large contribution from binaries to the velocity dispersion (compare,e.g., Leighet al., 2015; Oh et al., 2015) seems more plausible (Gieles et al., 2010a; Cottaaret al., 2012; H´enault-Brunet et al., 2012). It can be shown that there exist a critical compactness
M/r h above which gas ex-pulsion with associated dispersal of stars can no longer work in a star cluster evenif the gas dominates the gravitational potential at the time when massive star feed-back becomes effective: while the gravitational binding energy E b is proportionalto (1 − (cid:15) SF ) M /r h , the cumulative feedback energy by winds and supernovae atany given cluster age is only linear in the mass: E f ∝ (cid:15) SF M . Therefore, gravitymust eventually win.If we demand that for successful gas expulsion to happen, the provided feedbackenergy must exceed a critical energy proportional to the binding energy, i.e., E f > a − E b , (7)with a constant a − that will depend on the details of the feedback physics, thenwe can derive a critical star formation efficiency, defined here as the ratio of stellarmass in the cluster to its total mass during the embedded phase, for gas expulsion to succeed: (cid:15) > (cid:15) crit ( C ) = aC (cid:32) −
12 + (cid:114)
14 + 1 aC , (cid:33) (8) (cid:15) SF : stellar mass M over total mass (stars + gas) of an embedded clusterhe Physics of Star Cluster Formation and Evolution 21 Fig. 5
Gas dynamical constraint on the star formation efficiency for successful gas expulsion.If the star formation efficiency, i.e. the ratio between stellar and total mass in the embeddedcluster, is less than the given value, the given type of feedback will not be able to expel thegas on the crossing timescale of the cluster. N-body simulations find an upper limit of thestar formation efficiency of 50%, if gas expulsion is to significantly affect the stars (non-shadedregion). This results in an upper limit on the compactness of a star cluster, C , above whichgas expulsion cannot lead to significant expansion, loss of stars or dispersal. Curves for threeassumptions on the type of feedback responsible for the gas expulsion are shown: Stellar windsat metallicity [Fe / H] = − . where we have defined the compactness index as C = M/r h M (cid:12) pc − = (cid:16) σ . − (cid:17) (9)and used eq. (4) in the final equality above.The function (cid:15) crit ( C ) tends towards zero for small C ( σ ) and towards one forvery high cluster compactness. Krause et al. (2016) have shown that a thin-shellsuperbubble model reproduces this equation (Fig. 5).Thin-shell superbubble models (Krause et al., 2012, 2016) compute the kine-matics of the supershell assuming some prescription for the energy input andspherical symmetry. They can, however, take 3D effects into account by evalu-ating a criterion for the shell’s acceleration. The shell will be destroyed by theRayleigh-Taylor instability as soon as modes comparable to the size of the shellbecome unstable. The hot, pressurised bubble interior then escapes through holesin the shell, and the dense shell gas falls back. The more stars there are comparedto the amount of gas, the stronger the feedback, and the easier to push out thegas without making the shell unstable.Successful gas expulsion therefore requires the star formation efficiency to beabove a certain limit. There is, however, also an upper limit ( ≈ Krause et al. (2016) show that the critical compactness index is C ≈ σ =7 . − , also compare Fig. 5).The thin shell models effectively correspond to the assumption of maximumefficiency for stellar feedback: the hot gas is assumed to be always more central thanthe cold gas, thus maximising the outward push on the cold gas. The sphericalshell prevents hot gas from escaping, thus all of it can be used to act on the cold gas. Finally, Krause et al. (2012, 2016) assume 80% of the released feedbackenergy to be radiated away, thus 20% to be available for gas dynamics. This islikely a generous assumption, given the high efficiency of mixing and associatedradiative losses seen in recent 3D superbubble simulations with time-dependentdriving (Krause et al., 2013a; Vasiliev et al., 2017; Gentry et al., 2019).Stellar winds become less efficient at low metallicities Z , their energy outputscaling with Z . (Maeder & Meynet, 2012). If stellar winds (augmented by pho-toionisation and radiation pressure effects further away from the massive stars,compare below) dominate feedback in young clusters, rather than supernovae, forwhich there is some evidence from the timescales observed for massive clustersto become exposed (Hollyhead et al., 2015; Sokal et al., 2016; Kruijssen et al.,2019; Chevance et al., 2019), the critical compactness index becomes smaller atlow metallicities, C = 0 . / H] = − .
5. It is also possible to increase it byextreme assumptions on stellar feedback. If the most massive stars in a clusterexploded as hypernovae, all releasing ten times the conventional supernova energyoutput of 10 erg (e.g., Mazzali et al., 2014; L¨u et al., 2018), this would increasethe critical compactness index to C ≈
30. A more comprehensive analytic treat-ment by Matzner & Jumper (2015) that takes into account accretion and variousfeedback processes separately find the threshold at 3 km s − ( C = 0 . Crocker et al. (2018) consider the effect of the radiation pressure taking into ac-count re-radiated infrared radiation due to the presence of dust. They argue thatindirect radiation pressure on dust would first expand the gas gently, and thatdirect radiation pressure would later, but still before the first supernova, expel thegas on the dynamical timescale. For favourable assumptions, they find a maximumstellar surface density of 10 M (cid:12) pc − at which up to (cid:15) SF = 50% of the mass in acluster can be stars without them forcing the remaining gas mass out of the cluster.Taking a typical cluster radius of 1 pc converts this result to a compactness index C = 0 .
3. Hence, radiation pressure is expected to be somewhat less effective atexpelling gas than stellar winds, but of comparable order of magnitude (comparealso Reissl et al., 2018).Rahner et al. (2017) use a self-gravitating thin-shell model to predict gas re-moval in clusters with 0 . < C < . < C < . concentrated clouds with their steepest gas density power-law index of -2, whereit can reach 50%. As they consider only gas removal on any timescale, and not thespecific condition for gas expulsion on the dynamical timescale, their critical starformation efficiencies are somewhat lower than the ones of Krause et al. (2012,2016) despite them including radiation pressure into their calculations. he Physics of Star Cluster Formation and Evolution 23 Multi-dimensional simulations of star cluster formation that take into account theactual formation of the stars and follow feedback from individual massive starstypically find a reduced effect of feedback compared to the much more idealisedworks above. Dale et al. (2015) study the evolution of a turbulent molecular cloudwith photoionisation and conservatively implemented stellar wind feedback us-ing smooth particle hydrodynamics. They report a variety of conditions for starformation, including tenuous and very dense regions, with the overall number ofexpelled stars remaining low. Gavagnin et al. (2017) conducted a similar study us-ing adaptive mesh refinement hydrodynamics together with photoionisation fromindividual stars in an initially subvirial cloud. They report runs with different feed-back strength. The fraction of unbound stars depends only weakly on the feedbackstrength, and ejections are mainly due to gravitational star-star interactions. Sur-prisingly, their star cluster without feedback disperses at the end of the simulation,whereas the cluster with the strongest feedback forms a subvirial system, despite80% of the gas being ejected. This is, because the feedback efficiently slows theoverall collapse, such that the stellar density remains lower, and less dynamicalinteractions between stars take place.The accuracy with which the strength of feedback is predicted by these modelsmay be subject to further improvement. The different feedback processes (accre-tion radiation, protostellar jets and outflows, photoionisation, radiation pressure,stellar winds and supernovae) require very different computational methods. Thesimulations discussed above all include photoionisation. Dale et al. (2015) exclu-sively use the momentum from stellar winds. This is an underestimate, becausethe energy in the winds will not be entirely radiated away, but produce someadditional momentum. That the simulations generally underestimate feedback isunderlined by the fact that many runs do not terminate star formation (e.g. Dale,2017) within an observationally required time frame of 3 Myr (Chevance et al.,2019). This is particularly relevant, given that the timescale of gas loss strongly af-fects any expansion or dispersal (Smith et al., 2013). The virial state is expected tohave a strong influence on the fraction of bound stars (Farias et al., 2015). Hence,simulations with subvirial clouds, only, (Gavagnin et al., 2017) cannot provide thefull picture.More recently, Li et al. (2019) simulated star cluster formation from turbulentclouds in different kinematic states with a moving mesh hydrodynamics code.In each run, they form a variety of stellar structures, hierarchically merging intobigger ones. They apply feedback via mass and momentum deposition around eachstar, which is varied within a factor of 20. The latter range reflects the still existinguncertainty on the feedback strength. Gas expulsion with associated dispersal ofstars seems to occur in some of their simulations with the highest level of feedback.Most of their simulations do, however, not show a strong unbinding of stars due tobound structures being generally subvirial prior to the gas expulsion treatment.
Analytical and semi-analytical models combined with N-body simulations tendto overestimate the effects of stellar feedback and predict strong gas expulsioneffects with cluster expansion and dispersal for a virial velocity dispersion σ <4 Martin G. H. Krause et al.
Analytical and semi-analytical models combined with N-body simulations tendto overestimate the effects of stellar feedback and predict strong gas expulsioneffects with cluster expansion and dispersal for a virial velocity dispersion σ <4 Martin G. H. Krause et al. − − . They firmly exclude strong effects of gas expulsion on the clusterstars for σ > − unless one assumes non-standard mechanisms. Multi-dimensional simulations tend to underestimate feedback and usually see little ef-fects of gas expulsion and a small amount of unbound stars. However, tuning upthe feedback strength, such effects have also been reported (Zamora-Avil´es et al.,2019; Li et al., 2019). Gaia stellar kinematics observations suggest that gas ex-pulsion may be responsible for expansion and possibly dispersal of some stellargroups, while compact clusters appear unaffected, implying slow gas outflow orexhaustion of gas turned into newly formed stars. All reported cases where expan-sion or dispersal may take place have σ < − , consistent with the theoreticalconstraints.These results are in good agreement with direct observations of gas and stars inyoung massive clusters and progenitor clouds: there are no clouds compact enough,so that a young massive cluster could form with the same structure as that cloud(Longmore et al., 2014; Walker et al., 2015, 2016). The implication is that starformation has to proceed as the cloud collapses (compare § q ( r ) and Q ( r ), re- spectively.3. Top-hat flat source profile, i.e. , q ( r ) = ˙ M/V , Q ( r ) = ˙ E/V for r < R , and Q ( r ) = q ( r ) = 0, otherwise.4. Gravity is negligible. ˙ M : total mass loss rate; ˙ E total energy release rate; V = 4 πr / Assumption 2 above restricts the theory effectively to massive star clusters (andgalaxies, of course). For the wind phase, this is because of the strong dependenceof the stellar wind strength on the stellar mass. For example, Krause et al. (2013a)show in 3D hydrodynamics simulations that for a group that harbours star of 25,30 and 60 M (cid:12) (typical for a 1000 M (cid:12) cluster using the initial mass function fromKroupa et al., 2013), the 60 M (cid:12) star completely dominates the gas dynamics aslong as it exists. Interacting stellar winds and supernovae will heat the clusterto typically, 10 K, which corresponds to a sound speed of ≈
500 km s − . For atypical cluster diameter of, say, 10 pc, the dynamical timescale is then 20,000 yr. Ifwe require one supernova per dynamical timescale, we need roughly 1500 massivestars, which we expect for a cluster with ≈ M (cid:12) . Steady-state winds in clustersare therefore frequently referred to as super star cluster winds. Cant´o et al. (2000)show using 3D hydrodynamics simulations that in a cluster with 30 massive starswith similar properties, the 1D case with smooth source functions is approximatelyrecovered.Given these conditions, the 1D hydrodynamics equations can be solved ana-lytically (see also Zhang et al., 2014). Pressure, density and outward velocity aregiven by pρu = p ∗ ˙ M / ˙ E / R − ρ ∗ ˙ M / ˙ E − / R − u ∗ ˙ M − / ˙ E / , (10)where the functions containing the radial dependencies are given by : u ∗ = 2 M M + γ − (11) ρ ∗ = r a ∗ πu ∗ (12) p ∗ = 2 ρ ∗ γ (cid:16) M + γ − (cid:17) (13)with r ∗ = r/R , a = 1 (-2) for r ∗ < r ∗ > M : r ∗ = (cid:16) γ − M − γ +1 (cid:17) γ +12+10 γ (cid:16) γ + M − γ (cid:17) − γ +15 γ +1 r ∗ < (cid:16) γ − M − γ +1 (cid:17) γ +14 γ − M γ − r ∗ > . (cid:113) ˙ E/ ˙ M . As an example we give here parameters for the Arches cluster, one of the mostmassive, young (2-3 Myr, Lohr et al., 2018) star clusters in the Milky Way. Clarket al. (2019) estimate (cid:38)
50 stars with masses (cid:38) M (cid:12) . Using the initial massfunction from Kroupa et al. (2013), this translates to a total mass of 5 × M (cid:12) γ : adiabatic index, 5 / Fig. 6
Steady-state wind solution by Chevalier & Clegg (1985). Shown are the dimensionlessquantities given in eqs. (11-13). Section 5.3 for details. (consistent with the kinematic measurement, Clarkson et al., 2012), and a totalnumber of massive stars > M (cid:12) of ≈ M ≈ − M (cid:12) yr − and˙ E ≈ × erg s − (similar estimates can be found in Stevens & Hartwell, 2003).Within the cluster this yields particle densities and temperatures of n = 300 cm − (cid:18) ˙ E × erg s − (cid:19) − / (cid:18) ˙ M − M (cid:12) yr − (cid:19) / (cid:18) ˙ R . (cid:19) − (15) T = 9 × K (cid:18) ˙ E × erg s − (cid:19) (cid:18) ˙ M − M (cid:12) yr − (cid:19) − (16)Arches and similar young, massive and compact clusters are therefore expectedto be faint diffuse X-ray emitters (e.g., Cant´o et al., 2000; A˜norve-Zeferino et al.,2009), which has been confirmed by X-ray observations for the Arches cluster,Westerlund 1 and possibly also the Quintuplet cluster (Yusef-Zadeh et al., 2002;Wang et al., 2006; Kavanagh et al., 2011). Generally the temperature is some-what lower than predicted by the Chevalier & Clegg (1985) model (Stevens &Hartwell, 2003). This may be related to unaccounted for effects of non-equilibriumionisation (Ji et al., 2006) or limitations of our understanding of mass loading andthermalisation efficiency of the winds. Also, the spatial distribution of the mas-sive stars plays a role. X-ray emission is also expected from the interaction of thecluster wind with the surrounding gas. The superbubble is particularly bright insoft, ≈ stars ( > M (cid:12) ) are present in the cluster. They find that clusters are always ina steady outflow regime similar to the Chevalier-Clegg model, unless the energyinput is significantly overestimated (factor (cid:38)
20) by current population synthesismodels or the wind loads a significant amount of gas that was leftover from thestar formation event. If those conditions applied, part of the massive star ejecta he Physics of Star Cluster Formation and Evolution 27 would cool, be compressed by the remaining hot gas into UV-shielding filamentsand form stars in an extended or second star formation episode (Palouˇs et al.,2014). The total cold gas dropout from the wind can reach 1-6% of the totalstellar mass of a cluster (W¨unsch et al., 2017). The population of stars formedwould be (moderately due to the mass loading) enriched in He-burning productsejected from Wolf-Rayet stars and supernovae (W¨unsch et al., 2011). The latter twopredictions disfavour this mechanism as explanation for the frequently observedchemically distinct populations in globular clusters (compare § t ) = 7 × − M − (cid:12) yr − (cid:18) t Gyr (cid:19) − . . (17)For a 10 M (cid:12) cluster at an age of 100 Myr, this yields about 15 events per Myr.This is near the limit where one might consider the energy injection continuousand apply cluster wind models. D’Ercole et al. (2008) show in a 1D hydrodynamicsimulation with individual SN Ia that even one event can turn the cluster into anoutflow state. A SN Ia rate comparable to the one in the field would leave starclusters in a continuous outflow state.SN Ia occur in binary systems (e.g., Diehl et al., 2014), which may be morefrequent in massive star clusters (Leigh et al., 2015). Direct searches for type IaSNe in massive star clusters have, however, so far only produced upper limits(Washabaugh & Bregman, 2013). Dynamical effects should lead to a high netdestruction rate for binaries in clusters, so that, at least at late times, they mayhave actually fewer SN Ia than the field (Cheng et al., 2018; Belloni et al., 2019).5.4 Cooling flows?After the end of the type II supernova phase ( ≈ −
40 Myr after star formation)and before SN Ia start to occur ( ≈
100 Myr, e.g., Liu & Stancliffe, 2018) a starcluster has little internal energy production and may in principle have a coolingflow. D’Ercole et al. (2008) show that mass loss and energy injection are dominatedby AGB stars: α = ˙ M/M = 3 × − s − (18)˙ E/M = αv / erg s − M (cid:12)− (19)The cooling flow is robust for these parameters as they also show that more thanten times higher energy input would be required to turn the cooling flow into awind.D’Ercole et al. (2010, 2012, 2016) and others have argued that the gas massflowing to the centre may initiate secondary star formation. Challenges in explain- ing the chemically distinct multiple populations in globular clusters (compare § of the remaining intermediate-mass stars would keep the gas photo-dissociatedand too warm for star formation. The accumulating gas would only form starswhen the UV luminosity has declined enough to allow the formation of molecularhydrogen. However, gas cooling can also take place very efficiently in atomic gasvia C + (Glover & Clark, 2012). Also, it is unclear if type Ia SNe would be delayedsufficiently for the model to work (Lyman et al., 2018).Dense gas or late star formation as postulated in the above cooling flow modelsis generally not observed in star clusters (e.g., Cabrera-Ziri et al., 2015; Longmore,2015; Bastian & Lardo, 2018). This calls into question our understanding of thegas dynamics in star clusters with ages between the type II and type Ia supernovaphases. One possibility is that the cooling flow gas accretes on to the dark rem-nants, i.e., the stellar mass black holes and neutron stars (Krause et al., 2013b;Roupas & Kazanas, 2019). The energy released in jets, winds and radiation couldthen drive a cluster wind. D’Ercole et al. (2008) derive a critical energy input of6 × erg s − for their 10 M (cid:12) cluster. Even the typical luminosity of one X-raybinary (few 10 erg s − , Jord´an et al., 2004) would be sufficient to accomplishthis. Pulsar winds can keep star clusters in an outflow state (Naiman et al., 2020). After the gas cloud a star cluster formed from has been partially transformed intostars and dispersed, its fate is governed by gravity (i.e. collisional dynamics andtidal perturbations) and mass loss of the stars due to stellar evolution. Here wediscuss the various physical processes separately, but it is important to keep inmind that most processes act simultaneously and an important area of research isunderstanding the interplay between them, which is often non-linear.6.1 Stellar evolutionStellar evolution leads to a decrease of the total cluster mass, at a rate that is slowcompared to the orbital frequencies of the stars, such that the cluster can approx-imately maintain its virial equilibrium. The removal of mass leads to a reductionof the binding energy and an increase of the cluster radius. If the stellar mass losshappens throughout the cluster with no preferred location then the cluster radiusis inversely proportional to the mass (Hills, 1980). For mass segregated clustersmost stellar mass loss occurs in the cluster centre where the binding energy islarger, resulting in a faster expansion.6.2 Tidal shocksDuring the first 0.1-1 Gyr, cluster dissolution is likely dominated by tidal ‘shocks’, i.e. impulsive tidal perturbations from Galactic substructure, such as transientspiral arms and molecular gas clouds (e.g. Gieles et al., 2006; Elmegreen & Hunter,2010; Kruijssen et al., 2011). These perturbations boost the energy of stars in thecluster, some of which will exceed the escape energy and will therefore becomeunbound (Spitzer, 1958). The rate of shock-driven mass loss scales inversely with he Physics of Star Cluster Formation and Evolution 29 the mass volume density of the cluster, and is proportional to the surface density ofthe individual clouds and the ISM density in the host galaxy disc (Spitzer, 1958).When integrated over the lifetime of a cluster, this mass loss mechanism coulddominate the total mass loss budget (Elmegreen, 2010; Kruijssen, 2015), even inenvironments of relatively low gas density such as the solar neighbourhood (e.g.Spitzer, 1958; Gieles et al., 2006; Lamers & Gieles, 2006), where a single encounterwith a GMC ( (cid:38) M (cid:12) ) can completely disrupt a modest open cluster ( ∼ M (cid:12) ,Wielen, 1985; Terlevich, 1987). By scaling N -body models of individual encounters,it was found that in gas-rich environments like galaxy discs, GMCs dominatethe disruption of clusters (Gieles et al., 2006; Webb et al., 2019), decimating theinitial globular cluster population to the survivors that remain at the present day(Elmegreen, 2010; Kruijssen, 2015).Tidal shocks do not only drive considerable mass loss, they also dominate thestructural evolution of stellar clusters: after an initial phase of expansion due to theescape of unbound stars (Webb et al., 2019), the remaining cluster of bound starsmay shrink due to energy conservation (centrally concentrated clusters shrink,while low-concentration clusters expand, see Gieles & Renaud, 2016). When ig-noring other effects, a density increase makes tidal shocks self-limiting (Gnedin& Ostriker, 1999). However, a higher density makes two-body relaxation moreimportant which tends to reduce the cluster density, thereby counteracting theshock-induced density increase. Under the assumption that statistical equilibriumis reached, eventually the ratio of the shock dissolution time-scale and the relax-ation time-scale will become constant, resulting in a shallow mass-radius relation( r h ∝ M / , Gieles & Renaud, 2016). The normalisation of the predicted mass-radius relation depends on the environment, such that clusters are smaller athigher ISM densities (which is likely already the case at formation, see Choksi &Kruijssen 2019), slowing down their shock-driven disruption. However, even forcorrespondingly more compact clusters, Kruijssen (2015) predict that the totalshock-driven mass loss dominates over relaxation-driven mass loss when consider-ing the dynamical evolution of globular clusters over a Hubble time.To obtain a complete understanding of the interplay between shocks and re-laxation, a comprehensive parameter study of N -body simulations including bothprocesses is required, which again highlights that this is an important area for fu-ture research. A complementary approach to controlled N -body experiments wouldbe to use direct N -body simulations with realistic particles numbers ( N (cid:38) )and evolve them in the time-dependent tidal field extracted from models of galaxyformation at the epoch of GC formation.6.3 Two-body relaxationThe importance of collisional dynamics in the evolution of star clusters dependson the evolutionary stage of the cluster and the timescale that is considered. Givensufficient time, all clusters will dissolve due to collisional effects, even the clusters that are not in a Galactic tidal field (Baumgardt et al., 2002).Globular clusters are the archetypical collisional systems, that survived the ini-tial phase of tidal shock-driven disruption, meaning that orbital energy diffusionvia gravitational interactions – so-called two-body relaxation, or collisional dynam-ics – plays an important role in their evolution. This is because the velocities of stars are relatively low ( ∼
10 km/s) and stellar densities are high ( ∼ − pc − ),making two-body encounters frequent and long-lasting. Another way of sayingthis is that the relaxation timescale is short (few Gyr) compared to their ages(10-12 Gyr). Two-body relaxation is also relevant during the formation phase ofclusters, contrary to some propositions made in the literature (e.g. Fall & Zhang,2001; Krumholz et al., 2014). To explain this, we start by painting a broad-brushpicture of the classical theory of relaxation that was developed for (old) globularclusters.The consequences of two-body relaxation are reasonably well understood forthe idealised case of a single-mass cluster, which is often regarded to be a rea-sonable approximation for globular clusters, because their stars are confined to anarrow range of masses. A single-mass cluster, with stars initially in hydrostaticequilibrium, without primordial binaries, develops a radial energy flow as a resultof energy diffusion such that energy flows through the half-mass radius ( r h ) at arate ∼ | E | /τ rh (ignoring constants of order unity). Here E ∼ − GM /r h is the totalenergy of the cluster, with G the gravitational constant and M the total clustermass. The timescale τ rh is the half-mass relaxation time, which we shall definebelow (equation 22). This energy flow originates from the core, where stars losekinetic energy to the stars outside the core via two body interactions. As a result,the velocity dispersion of the stars in the core reduces – i.e. they ‘cool’ – and thecore radius contracts, while the stars outside the core heat up. The stars in thecore now experience a higher binding energy and due to the virial theorem, thestars now move faster. This somewhat paradoxical result is a direct consequenceof the negative heat capacity of self-gravitating systems. The time evolution of thecluster structure can be solved in various ways. Lynden-Bell & Eggleton (1980)used a set of equations that are strikingly similar to the stellar structure equations(so-called gaseous models, or continuum models) and found that this process ofcore contraction continues until the core has an infinite density and zero mass:the core has collapsed. In reality this mathematical endpoint is never reached, anda binary star forms when the number of stars in the core has reduced to ∼ E bin ∝ − | E ext | τ rh . (20)Here E ext is the external energy of the cluster, which is the total energy excludingthe negative energy locked up in binaries (i.e. E ext ∼ − GM /r h ). Because of energyconservation, the external energy must increase at a rate˙ E ext = − ˙ E bin . (21)The insight that there must exists a balance between the rate of energy produc-tion in the core and the energy flow through the cluster came from Michel H´enon(H´enon, 1975) and is a fundamental building block in the theory of cluster evo-lution. It allowed him to derive two models for the post-collapse evolution: in he Physics of Star Cluster Formation and Evolution 31 M [ M (cid:12) ] − − l og ρ h [ M (cid:12) / p c ] kpc kpc kpc kpc kpc NGC2419 ω CenM54
GC ‘isochrones’ (Gieles et al. 2011)Milky Way GCs < > G a l a c t o ce n t r i c d i s t a n ce [ k p c ] Fig. 7
The half-mass density of Milky Way globular clusters as a function of their mass ( x -axis) and Galactocentric radius (colour coding). ‘Isochrones’ from the cluster evolution modelof Gieles et al. (2011) are shown. This diagram is the equivalent of the Hertzsprung-Russelldiagram of stars. Collisional dynamics is not yet important for objects that are well below themodel lines. the absence of a Galactic tidal field, the energy increase leads to an expansion ofthe cluster at an approximately constant mass (H´enon, 1965), while tidally limitedclusters lose mass over the tidal boundary (sometimes referred to as ‘evaporation’)at a constant rate, while maintaining a constant density (H´enon, 1961). These twosolutions describe the two extreme ends of the life cycle of tidally limited starclusters with high initial density. By smoothly ‘stitching’ the two models the re-laxation driven evolution of star clusters can be described from the moment theyemerge from a gas-rich environment (i.e. once tidal shocks no longer dominate theinstantaneous disruption rate) to their eventual dissolution (Gieles et al., 2011).Thanks to its analytic nature, this simple model for the relaxation driven evolutionof star clusters readily provides expressions for M ( t ) and ρ h ( t ) at different Galacto-centric radii, which can be used to construct evolutionary ‘tracks’ and ‘isochrones’of globular cluster radius (or density) and mass as a function of location in theGalaxy, which – despite their first order nature – provide a satisfactory match withthe observed mass-density distribution of globular clusters (see Fig. 7), supportingH´enon’s suggestion that collisional dynamics is important for almost all globularclusters and shapes these relations.6.4 Two-body relaxation in young clustersWe now turn to the relevance of relaxation for young clusters. To get an idea for the physical time it takes for relaxation to become important, we write theexpression for τ rh from Spitzer & Hart (1971) as τ rh (cid:39)
18 Myr ψ − M M (cid:12) (cid:18) ρ h M (cid:12) / pc (cid:19) − / , (22) where ρ h = 3 M/ (8 πr ) is the average mass density within r h . To derive this,we assumed an average mass of stars of 0 . M (cid:12) and assumed that the slightdependence of the Coulomb logarithm on the number of stars can be neglected(ln Λ = 8). The term ψ ≥ r h , and often the assumption of single-mass clusters is made (i.e. ψ = 1). It takesabout 16 initial τ rh for a single-mass cluster to reach core collapse (e.g. Cohn,1980), which even for the relatively low-mass and high-density scaling adopted inequation (22) corresponds to a relatively long time of ∼
300 Myr, i.e. well after starformation ceased, suggesting that relaxation plays no role in the early evolutionof clusters.However, there are several important differences between young clusters andtheir older counterparts that are important for this discussion and make colli-sional dynamics important at young ages. Most importantly, young clusters havea well populated (initial) stellar mass function between ∼ . M (cid:12) and ∼ M (cid:12) .The presence of the high-mass stars significantly speeds up the relaxation process,because the energy transfer from high-mass stars to low-mass stars is more effi-cient than between stars of the same mass . With numerical N -body experiments,Portegies Zwart & McMillan (2002) find that core collapse happens after 0 . τ rh (with τ rh defined with ψ = 1) , i.e. 2 orders of magnitude faster than for single-mass clusters (Gieles et al., 2010b), which corresponds to ∼ τ rh of the first sub-clumps to form to ∼ yr, i.e. well within the timescale of clus-ter formation itself. In addition, the massive stars may form in the centre of thesub-clumps/cluster (see Section 6.5), resulting in a cluster forming in a core col-lapsed state, setting off binary activity immediately. Finally, several models of theformation of star clusters (Longmore et al., 2014; V´azquez-Semadeni et al., 2017;Gieles et al., 2018; Krumholz & McKee, 2019) and massive stars (Padoan et al.,2019) suggest that gas inflow from larger scale is important (compare § by reducing τ rh as the cluster gains mass (Moeckel & Clarke, 2011). The single-mass approximation, therefore, also breaks down for globular clusters with astellar-mass black hole population (Breen & Heggie, 2013; Giersz et al., 2019; Kremer et al.,2019; Wang, 2020; Antonini & Gieles, 2020).he Physics of Star Cluster Formation and Evolution 334 Martin G. H. Krause et al.
300 Myr, i.e. well after starformation ceased, suggesting that relaxation plays no role in the early evolutionof clusters.However, there are several important differences between young clusters andtheir older counterparts that are important for this discussion and make colli-sional dynamics important at young ages. Most importantly, young clusters havea well populated (initial) stellar mass function between ∼ . M (cid:12) and ∼ M (cid:12) .The presence of the high-mass stars significantly speeds up the relaxation process,because the energy transfer from high-mass stars to low-mass stars is more effi-cient than between stars of the same mass . With numerical N -body experiments,Portegies Zwart & McMillan (2002) find that core collapse happens after 0 . τ rh (with τ rh defined with ψ = 1) , i.e. 2 orders of magnitude faster than for single-mass clusters (Gieles et al., 2010b), which corresponds to ∼ τ rh of the first sub-clumps to form to ∼ yr, i.e. well within the timescale of clus-ter formation itself. In addition, the massive stars may form in the centre of thesub-clumps/cluster (see Section 6.5), resulting in a cluster forming in a core col-lapsed state, setting off binary activity immediately. Finally, several models of theformation of star clusters (Longmore et al., 2014; V´azquez-Semadeni et al., 2017;Gieles et al., 2018; Krumholz & McKee, 2019) and massive stars (Padoan et al.,2019) suggest that gas inflow from larger scale is important (compare § by reducing τ rh as the cluster gains mass (Moeckel & Clarke, 2011). The single-mass approximation, therefore, also breaks down for globular clusters with astellar-mass black hole population (Breen & Heggie, 2013; Giersz et al., 2019; Kremer et al.,2019; Wang, 2020; Antonini & Gieles, 2020).he Physics of Star Cluster Formation and Evolution 334 Martin G. H. Krause et al. (Poveda et al., 1967, see also Gavagnin et al. 2017), possibly explaining the ori-gin of the O-stars that are found with high velocities ( (cid:38)
30 km s − ), far fromstar forming regions (Blaauw, 1961). In addition, a large fraction if not all mas-sive stars are expected to form in binaries and higher order multiples (e.g., Sanaet al., 2012), which have larger gravitational cross section than single stars makingbinary-binary interactions an additional channel for ejecting massive stars from anongoing cluster formation site (Leonard & Duncan, 1990). The removal of massivestars reduces the mechanical and radiation feedback from massive stars on thecluster and the more distributed feedback in the low(er) density ISM has conse-quences for galaxy formation (Ceverino & Klypin, 2009). Finally, the high centraldensity of massive stars in the centre affect the ionisation level (and thus accretionrate) and survival of discs around smaller mass starsIn conclusion, collisional dynamics is likely important from the very beginningof cluster evolution and it may have played a role in the origin of the multiple popu-lations in GCs (see § For a long time, nucleosynthesis (or, in other words, internal chemical evolution)has been ignored in star cluster modelling, based on both theoretical and observa-tional arguments. Galaxies have a deep potential well and are hence expected toretain even some of the ejecta that massive stars shed at high velocity. This hasrecently been confirmed by measurements of Doppler kinematics of the radioactivedecay line of unstable Al, which traces high-mass star ejecta (Kretschmer et al.,2013; Krause et al., 2015; Rodgers-Lee et al., 2019). The high observed velocitiessuggest that a large fraction of the ejecta is blowing away from their birth places athigh speeds. The scale height of the order of kpc is in agreement with expectationsfrom fountain-flow super-bubbling disc models where ejecta diffuse into the hothalo and return in part on a Gyr timescale (Pleintinger et al., 2019; Rodgers-Leeet al., 2019).The need for a sufficiently deep potential well to retain the gas despite theenergetic feedback from the massive stars and eventually recycle it internally tomake new stars is supported by the fact that open clusters present no (withinmeasurement uncertainties) spread in Fe-peak, α , and s-process elements (here-after heavy metals). These specific species actually vary only in the most massiveglobular clusters (hereafter GCs), with the extreme case being Omega Cen whichis thought to be the remnant of a dwarf galaxy nucleus (Butler et al., 1978; Zin-necker et al., 1988; Marino et al., 2011). Such rare objects (recently called TypeII GCs; see e.g. Milone et al. 2017 and Marino et al. 2018) possibly make the link between star clusters (open clusters and Type I GCs) and chemically evolveddwarf galaxies.Interestingly, large variations in carbon, nitrogen, oxygen, sodium, magnesium,and aluminium (C, N, O, Na, Mg, Al, hereafter light elements) were discoveredin GCs already in the 1970’s (among bright red giants, Osborn 1971; see Kraft he Physics of Star Cluster Formation and Evolution 35 Color variations and/or separations in the CMD are used to infer He abundancevariations among MSPs (typically between 0.003 and 0.19 in mass fraction, with Heenrichment increasing with the present-day mass; Norris e.g. 2004; Piotto et al. e.g.2005; King et al. e.g. 2012; Sbordone et al. e.g. 2011; Milone e.g. 2015; Nardielloet al. e.g. 2015; Milone et al. e.g. 2018; Lagioia et al. e.g. 2019). Importantly, the photometric approach revealed the presence of MSPs similar to GC ones inextragalactic massive star clusters with ages down to ∼ ∼ In the ‘AGB model’ (e.g. Ventura et al., 2001; D’Ercole et al., 2012; Ventura et al.,2013; D’Antona et al., 2016) it is assumed that a second generation of stars formsfrom material that is polluted by AGB winds from a first generation. The modelstarts from the point that AGB winds are slow enough so that they may be un-able to escape the potential well of a massive star cluster. It is conjectured thatafter the type II supernova phase, AGB winds would be the only energy sourcefor the intracluster gas, which would be insufficient to overcome radiative losses.Consequently, a cooling flow forms, and stars would form in the centre of thecluster. This scenario has been criticised in multiple ways: AGB nucleosynthe-sis builds an O–Na correlation instead of the observed anti-correlation (Forestini& Charbonnel, 1997), and it releases He-burning products, thus predicting totalC+N+O variations that are only observed in a few GCs (e.g. Decressin et al.,2009; Yong et al., 2015). The AGB stars need to be massive enough to undergohot-bottom burning to reach the required temperatures ( ∼ . M (cid:12) , Ventura et al.,2001), which implies that not enough mass is available to pollute (in some cases) (cid:38)
80% of the stars. This is commonly referred to as the mass budget problem (e.g.Prantzos & Charbonnel, 2006; Schaerer & Charbonnel, 2011; Gieles & Charbon-nel, 2019). Some of the ideas that have been put forward to overcome the massbudget problem, such as the loss of (cid:38)
90% of the first generation stars over thetidal boundary (D’Ercole et al., 2008), do not address the GC mass dependence of the fraction of polluted stars, i.e. the requirement of more polluted materialper unit of GC mass in more massive GCs. We refer to this as the specific massbudget problem. Whether star clusters have a cooling flow phase at all is ques-tionable ( § he Physics of Star Cluster Formation and Evolution 37 clusters ( § § § (cid:46)
50% on the amount of massthat GCs could have lost (Larsen et al., 2012). This is referred to as the externalmass budget problem. Recent models combining the variety of dynamical massloss mechanisms discussed in Section 6 predict that globular clusters were only2–4 times more massive at birth (Kruijssen, 2015; Reina-Campos et al., 2018).
Ordinary massive stars ( ∼ − M (cid:12) ) have the correct central temperature tocreate the O-Na anti-correlation, but they are not convective and the material doestherefore not reach the surface. To overcome this, Decressin et al. (2007) assumefast-rotating massive stars (FRMS) which undergo rotationally induced mixingand possibly lose a large amount of material through a mechanical wind via anoutflowing (decretion) disk. Krause et al. (2013b) laid out a detailed scenario,showing that 2P stars may form in the decretion discs from material spreadingthrough the disc from the surface of the star, and accreting pristine gas duringa somewhat extended embedded phase of ≈
10 Myr. The embedded phase wouldbe longer in massive star clusters, because stellar feedback would not be strongenough to remove the gas. However, the FRMS scenario also faces the mass budgetproblem (Prantzos & Charbonnel, 2006; Krause et al., 2013b). In addition, FRMSreach high-enough temperatures to activate the MgAl chain only when the Hefraction has increased significantly, predicting a larger He spread in GCs witha Mg–Al anti-correlation than observed (e.g. Chantereau et al., 2016; Nardielloet al., 2015; Milone et al., 2015b).
Finally, supermassive stars (SMS) have also been suggested as polluters. SMSmodels with masses between ∼ × M (cid:12) and ∼ × M (cid:12) reach the requiredcentral temperature to activate the MgAl chain ( ∼ −
78 MK) already at the verybeginning of the evolution on the main sequence, when the He abundance is closeto pristine (Denissenkov & Hartwick, 2014; Prantzos et al., 2017). Consequently,in this early evolutionary phase the H-burning products of SMSs show remarkableagreement with the various observed abundance anti-correlations and Mg isotopicratios (Denissenkov & Hartwick, 2014). A formation scenario has been proposedby Gieles et al. (2018). Similar to the picture described in detail in §
3, a proto-GC would form at the confluence of gas filaments in gas-rich environments. Atthe highest density peak, inflow motions are converted to turbulence, and, wheregravity dominates locally, the gas fragments into proto-stars. Inflow over at leastthese two hierarchies leads to cancellation of angular momentum by the timethe gas reaches the proto-stars. Accretion of this low-angular momentum gas then reduces the specific angular momentum (i.e. the angular momentum per unit mass)and the stellar density increases as ρ ∝ M (Bonnell et al., 1998). An SMS wouldthen form by runaway collisions. The SMS is assumed to be fully convective andthe nucleosynthesis yields are efficiently brought to the surface, streaming off viathe usual radiatively driven wind. The wind is initially fast (cid:38) − , but brakes down quickly by interaction with dense gas in the still embedded cluster.The ejecta would then mix into the star-forming dense gas that is accreting on theproto-stars in the cluster or collapse locally to form stars independently. If SMSs form via stellar collisions, then it may be possible to keep the Heabundance low and also produce an order of magnitude more polluted materialthereby addressing the mass budget problem (Gieles et al., 2018). This model alsoprovides a pathway to solve the specific mass budget problem because the dynam-ical models predict a super-linear scaling between the amount of polluted materialreleased via the SMS wind and GC mass. As of today, SMS thus appear to be themost appealing candidate, provided that these stars really exist in nature and arefully convective (cf. Haemmerl´e et al., 2018) so they can release the material atthe very beginning of their evolution to avoid overproduction of He.It may come as a surprise that SMS would be difficult to find observationally.Martins et al. (2020) predicted the detectability of cool SMS in proto-GCs at highredshift through deep imaging with JWST NIRCAM camera. One problem at lowredshift however is that clusters that would be massive enough are not found inthe Milky Way. R136 in the Large Magellanic cloud is, at 50 kpc distance, theclosest example of a young massive cluster that may just be massive and compactenough to show some signs of massive star formation via collisions. Indeed somevery massive stars ( > M (cid:12) ) have been observed (Crowther et al., 2010). SinceSMS are expected to occur in embedded star clusters, absorption would likely bean issue.An interesting alternative might be MASER emission. GHz MASERs are re-liable tracers of massive star formation (Ellingsen et al., 2018; Billington et al.,2019). SMS might hence be expected to show particularly bright MASER emis-sion. Active Galactic Nuclei are an interesting analogue, as they also come withmolecular tori of significant optical depth and strong central UV source. StrongMASER emission is frequently seen in heavily-absorbed AGN (Castangia et al.,2019). A good example is the archetypical nearby AGN in NGC 1068, where theouter accretion disc is spatially and kinematically resolved, and the mass of thecentral object has thus been measured (Murayama & Taniguchi, 1997; Gallimoreet al., 2001).Similar to the case of AGN, forming super star clusters have also been foundto be associated with 22 GHz H O MASER emission (Gorski et al., 2019). Inparticular, Gorski et al. (2019) find a nuclear kilomaser in NGC 253, also associatedwith a forming super star cluster. The spectrum has more than one componentand a total width of ≈
170 km s − . Whether this relates to SMSs in their centres,and if rotation curves of any SMS disc can be obtained, remains to be seen.Many other models have been proposed in the literature (de Mink et al., 2009;Bastian et al., 2013; Elmegreen, 2017; Kim & Lee, 2018; Sz´ecsi & W¨unsch, 2019).A recent example is the model by Zinnecker at al. (2020, submitted). The model suggests that first, only convective, still accreting high-mass stars form (Hosokawa& Omukai, 2009), with slow, heavily cooling winds (Vink, 2018) producing a chem-ically anomalous population of predominantly low-mass stars. In this model, anSMS is not needed, yet it also solves the mass budget problem and the He over-production problem. More research into such models is required. he Physics of Star Cluster Formation and Evolution 39 It is well established that a complex cycle of matter and energy takes place withingalaxies. The non-linear flow patterns in the multi-scale multi-phase interstellarmedium are the central engine of galaxy evolution, they determine where and atwhat rate stellar clusters and associations form in our Milky Way. They build up inregions of the interstellar medium that become unstable under their own weight.These are the star forming clumps and cores in the deep interior of molecularclouds.Altogether, star cluster formation can be seen as a three-phase process. First,supersonic turbulence creates a highly transient and inhomogeneous molecularcloud structure that is characterised by large density contrasts. Some of the high-density fluctuations are transient. Others exceed the critical mass for gravitationalcontraction, they begin to collapse and eventually decouple from the complex cloudenvironment. Second, the collapse of these unstable cores leads to the formationof individual stars in clusters and associations. In this phase, a nascent protostargrows in mass via accretion from the infalling envelope until the available gasreservoir is exhausted or stellar feedback effects become important and removethe parental cocoon. Third, stellar feedback becomes so efficient that all the re-maining gas is cleared. This reveals the young cluster to optical telescopes, and itssubsequent secular evolution is then largely dominated by gravitational processesrather than by the complex competition between gravity and many other physicalagents.We begin our discussion with a definition of what we actually mean whentalking about star clusters in Section 2. Then we present evidence from analyticstudies and numerical models that indicate that the proto-cluster gas is heavilyinfluenced by the initial conditions and the dynamical properties of the parentalcloud in Section 3.As gas contracts to form stars, the density increases by more than 25 orders ofmagnitude and the temperature rises by a factor of a million. The process comesto an end when nuclear burning processes set in and provide stability: a star isborn. We discuss the different models and suggestions to describe this evolutionaryphase in Section 4.Star formation is controlled by the intricate interplay between the self-gravityof the ISM and various opposing agents, such as supersonic turbulence, magneticfields, radiation and gas pressure. The evolution is modified by the thermodynamicresponse of the gas, which is determined by the balance between heating andcooling, which in turn depends on the chemical composition of the material and thedetailed interaction of gas and dust with the interstellar radiation field. Altogether,stellar feedback provides enough energy and momentum to remove the parentalgas from the cluster. It may also be responsible for clusters being in a globaloutflow state for the rest of their life. The various physical agents contributing tothis process are discussed in Section 5.Once gas is removed, the subsequent dynamical evolution of a star cluster becomes relatively simple. It is solely governed by the mutual gravitational at-traction of the stars in the cluster, modified only by tidal forces exerted from thelarger-scale galactic environment, which are weak except near the galactic centeror when clusters pass nearby overdensities (such as giant molecular clouds, GMCs,spiral arms, or other clusters), and by mass loss due to the internal evolution of the constituent stars. Large self-gravitating systems such as star clusters exhibitcomplex dynamical behavior which we discuss in Section 6.The chemical composition of stars can provide important constraints on theorigin of stellar clusters and help us to distinguish between different physical sce-narios. We therefore introduce in Section 7 key aspects of stellar nucleosynthesisand discuss their relation to cluster formation and evolution. Specifically, we spec-ulate about the physical reasons for the observed O-Na anti-correlation observedin globular clusters.These different perspectives emphasise the interdependence of the differentprocesses: How long gas remains in a state of turbulence before accreting ontoa star ( § §
4) is crucial to understand how, and what kind of massivestars can pollute the gas out of which the low-mass stars form in massive starclusters and why this is not happening in lower-mass clusters ( § § § §
7) and the dynamics of the stars ( § Acknowledgements
This review emerged from a workshop at the International Space Sci-ence Institute in Bern, Switzerland in May 2019. The authors thank the staff of ISSI for theirgenerous hospitality and creating the inspiring atmosphere that initiated this project. Wethank the referee for a constructive report which has improved the manuscript.
Conflict of interest
J.M.D.K. gratefully acknowledges funding from the German Research Founda-tion (DFG) in the form of an Emmy Noether Research Group (grant numberKR4801/1-1) and a DFG Sachbeihilfe Grant (grant number KR4801/2-1), fromthe European Research Council (ERC) under the European Union’s Horizon 2020 he Physics of Star Cluster Formation and Evolution 41 research and innovation programme via the ERC Starting Grant MUSTANG(grant agreement number 714907), and from Sonderforschungsbereich SFB 881“The Milky Way System” (subproject B2) of the DFG. S.S.R.O. acknowledgesfunding from NSF Career grant AST-1650486. P.G. acknowledges funding fromthe European Research Council under ERC-CoG grant CRAGSMAN-646955.R.S.K. acknowledges support from the Deutsche Forschungsgemeinschaft viathe SFB 881 The Milky Way System (subprojects B1, B2, and B8) as well as fund-ing from the Heidelberg Cluster of Excellence STRUCTURES in the frameworkof Germanys Excellence Strategy (grant EXC-2181/1 - 390900948).JBP acknowledges UNAM-DGAPA-PAPIIT support through grant numberIN-111-219.MG acknowledges support from the ERC (CLUSTERS, StG-335936).E.V.-S. acknowledges financial support from CONACYT grant 255295.
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