The evolution of the stellar mass function in star clusters
aa r X i v : . [ a s t r o - ph . GA ] O c t Astronomy & Astrophysics manuscript no. KruijssenMF c (cid:13)
ESO 2018September 4, 2018
The evolution of the stellar mass function in star clusters ⋆ J. M. Diederik Kruijssen , Astronomical Institute, Utrecht University, PO Box 80000, NL-3508TA Utrecht, The Netherlands(e-mail: [email protected] ) Leiden Observatory, Leiden University, PO Box 9513, NL-2300RA Leiden, The NetherlandsReceived 20 September 2009 / Accepted 23 October 2009
ABSTRACT
Context.
The dynamical ejection of stars from star clusters affects the shape of the stellar mass function (MF) in theseclusters, because the escape probability of a star depends on its mass. This is found in N -body simulations and has beenapproximated in analytical cluster models by fitting the evolution of the MF. Both approaches are naturally restrictedto the set of boundary conditions for which the simulations were performed. Aims.
The objective of this paper is to provide and to apply a simple physical model for the evolution of the MF instar clusters for a large range of the parameter space. It should also offer a new perspective on the results from N -bodysimulations. Methods.
A simple, physically self-contained model for the evolution of the stellar MF in star clusters is derived fromthe basic principles of two-body encounters and energy considerations. It is independent of the adopted mass loss rateor initial mass function (IMF), and contains stellar evolution, stellar remnant retention, dynamical dissolution in a tidalfield, and mass segregation.
Results.
The MF evolution in star clusters depends on the disruption time, remnant retention fraction, initial-finalstellar mass relation, and IMF. Low-mass stars are preferentially ejected after t ∼
400 Myr. Before that time, massesaround 15—20% of the maximum stellar mass are lost due to their rapid two-body relaxation with the massive starsthat still exist at young ages. The degree of low-mass star depletion grows for increasing disruption times, but can bequenched when a large fraction of massive remnants is retained. The highly depleted MFs of certain Galactic globularclusters are explained by the enhanced low-mass star depletion that occurs for low remnant retention fractions. Unlessthe retention fraction is exceptionally large, dynamical evolution always decreases the mass-to-light ratio. The retentionof black holes reduces the fraction of the cluster mass in remnants because white dwarfs and neutron stars have massesthat are efficiently ejected by black holes.
Conclusions.
The modeled evolution of the MF is consistent with N -body simulations when adopting identical boundaryconditions. However, it is found that the results from N -body simulations only hold for their specific boundary conditionsand should not be generalised to all clusters. It is concluded that the model provides an efficient method do understandthe evolution of the stellar MF in star clusters under widely varying conditions. Key words. stellar dynamics – stars: kinematics – (
Galaxy: ) globular clusters: general – (
Galaxy: ) open clusters andassociations: general – galaxies: star clusters – galaxies: stellar content
1. Introduction
The evaporation of star clusters is known to change theshape of the underlying stellar mass function (H´enon1969; Chernoff & Weinberg 1990; Vesperini & Heggie 1997;Takahashi & Portegies Zwart 2000; Portegies Zwart et al.2001; Baumgardt & Makino 2003). This phenomenon hasbeen used to explain the observed MFs in globular clus-ters (Richer et al. 1991; De Marchi et al. 2007; De Marchi& Pulone 2007), which are flatter than typical initial massfunctions (IMFs, e.g. Salpeter 1955; Kroupa 2001). In ad-dition, the effect of a changing MF on cluster photome-try has been investigated (Lamers et al. 2006; Kruijssen& Lamers 2008; Anders et al. 2009). This has been shownto explain the low mass-to-light ratios of globular clusters(Kruijssen 2008; Kruijssen & Mieske 2009) and to have ⋆ The models presented in this paper will be publiclyavailable in electronic form at the CDS via anonymous ftpto http://cdsweb.u-strasbg.fr/ (130.79.128.5) and also at . Hereafter, ‘mass function’ is referred to as ‘MF’. a pronounced effect on the inferred globular cluster massfunction (Kruijssen & Portegies Zwart 2009).The existing parameterised cluster models that incor-porate a description of low-mass star depletion are re-stricted by the physically self-contained models on whichthey are based. Some studies (Lamers et al. 2006; Kruijssen& Lamers 2008) assume an increasing lower stellar masslimit to account for the evolving MF, others (Anders et al.2009) fit a changing MF slope to N -body simulations. Inboth cases, the models are accurate for a certain range ofboundary conditions, but they do not include a physicalmodel and are therefore lacking flexibility. While N -bodysimulations do include the appropriate physics, they arevery time-consuming. As a result, only a limited number ofclusters can be simulated and the applicability of the sim-ulations is thus restricted to the specific set of boundaryconditions for which they have been run.It would be desirable to obtain a simple physical modelfor the evolution of the MF, which would have a short run-time and could be used independently of N -body simula-tions. Forty years ago, a pioneering first approach to such J. M. D. Kruijssen: The evolution of the stellar mass function in star clusters a model was made by H´enon (1969), who considered thestellar mass-dependent escape rate of stars from star clus-ters. However, the applicability of his model was limiteddue to a number of assumptions that influenced the re-sults. First of all, H´enon (1969) assumed that the clustersexist in isolation and neglected the tidal field. As a conse-quence, the ejection of a star could only occur by a single,close encounter and the repeated effect of two-body relax-ation was not included. Secondly, the distribution of starswas independent of stellar mass, i.e. mass segregation wasnot included. Both mass segregation and the influence of atidal field are observed in real clusters, and can be expectedto affect the evolution of the MF.The aim of this paper is to derive a physical descriptionof the evolution of the stellar MF in star clusters, allevi-ating the assumptions that were made by H´enon (1969).This should explain the results found in N -body simula-tions and observations, while providing the required flexi-bility to explore the properties of star clusters with simple,physically self-contained models. The outline of this paperis as follows. In Sect. 2, total mass evolution of star clustersis discussed. A recipe for the evolution of the MF is derivedin Sect. 3, covering stellar evolution, the retain of stellarremnants, dynamical dissolution and mass segregation. Themodel is compared to N -body simulations in Sect. 4. InSect. 5, the model is applied to assess the evolution of theMF for different disruption times and remnant retentionfractions. The consequences for other cluster properties arealso considered. This paper is concluded with a discussionof the results and their implications.
2. The mass evolution of star clusters
The mass of star clusters decreases due to stellar evolutionand dynamical dissolution. This is expressed mathemati-cally asd M d t = (cid:18) d M d t (cid:19) ev + (cid:18) d M d t (cid:19) dis , (1)with M the cluster mass, and the subscripts ‘ev’ and ‘dis’denoting stellar evolution and dynamical dissolution. Thecontribution of stellar evolution to the mass loss is derivedfrom the decrease of the maximum stellar mass with timeand depends on the adopted stellar evolution model.The dynamical evaporation of star clusters is increas-ingly well understood. Over the past years it has becomeclear that clusters lose mass on a disruption timescale t dis that is proportional to a combination of the half-mass re-laxation time t rh and the crossing time t cr as t dis ∝ t x rh t − x cr (e.g. Baumgardt 2001; Baumgardt & Makino 2003; Gieles& Baumgardt 2008). It is found that x = 0 . c = log ( r t /r c )) or King pa-rameter ( W ) of the cluster (Baumgardt & Makino 2003).This proportionality leads to a disruption timescale thatscales with the present day mass as (Lamers et al. 2005): t dis = t M γ , (2)with M the cluster mass, t the dissolution timescale pa-rameter which sets the rapidity of dissolution and dependson the cluster environment, and γ a constant related to x .Lamers et al. (2009) find γ = 0 .
62 for W = 5 and γ = 0 .
70 for W = 7. This timescale implies a mass loss rate due todissolution that can be described with the simple relation (cid:18) d M d t (cid:19) dis = − Mt dis = − M − γ t , (3)which can be integrated for the mass evolution of the clusterdue to dynamical dissolution.The above formulation of the cluster mass evolution wasextended to include stellar remnants, photometric clusterevolution, and a simple description of the MF in the SPACE cluster models (Kruijssen & Lamers 2008). Stellar rem-nants were accounted for by assuming initial-final mass re-lations (similar to Sect. 3.1 of the present paper), while thephotometric evolution was computed by integrating stel-lar isochrones from the Padova group (Bertelli et al. 1994;Girardi et al. 2000). The description of low-mass star deple-tion followed the simple model from Lamers et al. (2006) inwhich the minimum stellar mass of the MF increases withtime.The present study provides a new description of the evo-lution of the MF which is based on fundamental principles,and does not depend on the above prescription for the to-tal mass evolution. In addition, the latest Padova models(Marigo et al. 2008) are incorporated to calculate the pho-tometric cluster evolution. These improvements update the
SPACE cluster models.
3. The evolution of the stellar mass function
To describe the evolution of the MF, the effects of stellarevolution, stellar remnant production, and dynamical dis-solution need to be included. While the focus of this paperlies with the effects of dissolution, a proper treatment ofstellar evolution is essential. This is described first, beforepresenting a model for cluster dissolution. The influence of stellar evolution on the MF is twofold. Firstof all, the maximum stellar mass decreases, because at anytime during cluster evolution the most massive stars reachthe end of their lives. Secondly, the stellar remnants thatare created upon the death of these massive stars constitutea part of the MF that can only be lost from the cluster bydynamical mechanisms.The maximum stellar mass in the cluster as a function ofits age is taken from the Padova 2008 isochrones (Marigoet al. 2008) for metallicities in the range Z = 0 . m sr are computed fromtheir progenitor stellar mass m using initial-final mass re-lations. Following Kruijssen & Lamers (2008), for whitedwarfs ( m < ⊙ ) the relation from Kalirai et al. (2008)is adopted: m wd = 0 . m + 0 .
394 M ⊙ , (4)which holds for all ages that are covered by the Padovaisochrones. For neutron stars (8 M ⊙ ≤ m <
30 M ⊙ ) therelation from Nomoto et al. (1988) is used: m ns = 0 . m − ⊙ ) + 1 .
02 M ⊙ , (5) The model presented in this paper is independent of the massloss rate and of the form of the IMF N i ( m ), but for explanatorypurposes a Kroupa (2001) IMF is adopted later on.. M. D. Kruijssen: The evolution of the stellar mass function in star clusters 3 while for black holes ( m ≥
30 M ⊙ ) a simple relation isassumed that is in acceptable agreement with theoreti-cally predicted masses of stellar mass black holes (Fryer& Kalogera 2001): m bh = 0 . m −
30 M ⊙ ) + 8 . ⊙ . (6)With these relations, the remnant MF is computed fromconservation of numbers as N sr ( m sr ) = f ret , sr ( M ) N ( m ( m sr )) d m d m sr , (7)with sr = { wd , ns , bh } denoting the appropriate remnanttype, N sr ( m sr ) representing its MF, f ret , sr ( M ) denoting thecluster mass-dependent fraction of these remnants that isretained after applying kick velocities, and N ( m ( m sr )) rep-resenting the progenitor MF.For a given velocity dispersion of remnants, the reten-tion fraction of each remnant type depends on the localescape velocity v esc , which is related to the potential ψ as v esc = √ ψ . Stellar remnants are predominantly producedin the cluster centre in the case of mass segregation, whichis reached most rapidly for massive stars (see Sect. 3.2).For a Plummer (1911, also see Eq. 9) potential this im-plies that upon remnant production v esc = p GM/r ,with G the gravitational constant and r the Plummer ra-dius. Adopting a Maxwellian distribution of velocities thatis truncated at v esc , it is straightforward to show that f ret , sr ( x ) = A " erf (cid:18) x √ (cid:19) − r π x e − x / , (8)where A is a normalisation constant to account for the trun-cation of the velocity distribution and x ≡ GM/r σ ,with σ = σ + σ , sr denoting the total velocity disper-sion of the produced remnant type, which arises from thecentral velocity dispersion in the cluster σ = GM/ r (e.g.Heggie & Hut 2003) and the velocity dispersion of the ex-erted kick σ kick . The normalisation constant then follows as A = erf √ − p /π exp ( − σ kick , sr aregiven in literature. White dwarf kicks have recently beenproposed to be of order σ kick , wd = 4 km s − (Davis et al.2008; Fregeau et al. 2009). For neutron stars σ kick , ns =100 km s − is adopted, which is a somewhat conservativeestimate with respect to theory, but it agrees reasonablywell with observed neutron star numbers in globular clus-ters and represents a compromise between single star andbinary channels (for estimates of the retention fraction anddiscussions of the ‘neutron star retention problem’ see Lyne& Lorimer 1994; Drukier 1996; Arzoumanian et al. 2002;Pfahl et al. 2002). Gravitational wave recoils are thought toexert black hole kicks of order σ kick , bh = 80 km s − (Moody& Sigurdsson 2009). This value depends on metallicity, butfor simplicity I assume a single, typical value here.The retention fractions following from Eq. 8 are shownas a function of cluster mass per unit Plummer radius inFig. 1. This quantity best reflects the retention fractionbecause x ∝ M/r in Eq. 8. Open clusters (with initialmasses M i such that typically M i /r < × M ⊙ pc − ,Larsen 2004) do not retain any neutron stars or black holes,while globular clusters ( M i /r ∼ × —10 M ⊙ pc − ,Harris 1996) retain 0.1—4% of the neutron stars and 0.3—7% of the black holes. These values are in excellent agree-ment with other studies (e.g. Pfahl et al. 2002; Moody & Fig. 1.
Retention fraction of stellar remnants as a function ofcluster mass per unit Plummer radius
M/r , for black holes(solid), neutron stars (dashed) and white dwarfs (dotted). Sigurdsson 2009), but are still lower than the large observednumber of neutron stars in a number of globular clusters(the aforementioned ‘retention problem’).
Dissolution alters the shape of the stellar MF in star clus-ters due to the effects of two-body relaxation and energyequipartition. In a pioneering paper, H´enon (1969) derivedthe escape rate of stars of different masses from an isolatedcluster. The cluster was represented by a Plummer (1911)gravitational potential: ψ ( r ) = ψ (cid:18) r r (cid:19) − / , (9)where r denotes the Plummer radius setting the concen-tration of the cluster and ψ ≡ GM/r represents the cen-tral potential, with G the gravitational constant and M thecluster mass. It was argued by H´enon (1960) that the onlyway for stars to escape such an isolated cluster is by a single,close encounter. The corresponding stellar mass-dependentescape rate was found to be (H´enon 1969):d N ( m )d t = − | E | / N ( m ) GM / Z ∞ N ( m ′ ) F (cid:16) mm ′ (cid:17) m ′ d m ′ , (10)with N ( m ) the MF, m the stellar mass, E the total energyof the cluster, and F ( µ ) a function related to the ejectionprobability for a star of mass m in a close encounter with astar of mass m ′ and a corresponding mass ratio µ ≡ m/m ′ .The expression in Eq. 10 is independent of the adoptedIMF. The function F will be referred to as the ‘H´enon func-tion’ and is shown in Fig. 2. The original expression consistsof several integrals that have to be solved numerically. InH´enon (1969), a table is given for the H´enon function, butit can also be fitted by: F ( µ ) = (cid:0) .
32 + 0 . µ . + 13 . µ . (cid:1) − . (11)This approaches the power law F ( µ ) = 0 . µ − / for µ >
1, as was derived explicitly by H´enon (1969).
J. M. D. Kruijssen: The evolution of the stellar mass function in star clusters
Fig. 2.
H´enon function F ( µ ), which is a measure for the ejectionprobability of a star of mass m in a two-body interaction withmass ratio µ ≡ m/m ′ . The dotted line shows the fit from Eq. 11. The total mass loss rate corresponding to Eq. 10 con-flicts with N -body simulations (as was already noted byWielen 1971) because only ejections by single, close en-counters are included. This restriction implies that the dis-ruption timescale t dis is proportional to the crossing time( t dis , H´enon ∝ t cr ), while N -body simulations show that itscales with a combination of the half-mass relaxation timeand the crossing time ( t dis ∝ t . t . ) due to two-bodyrelaxation, i.e. the repeated effect of soft encounters (e.g.Baumgardt & Makino 2003). Nonetheless, the escape ratefrom H´enon (1969) does accurately describe what happensif two stars interact and can therefore be used as a start-ing point for a more complete description of the evolu-tion of the MF. For that purpose, it is convenient to scaleEq. 10 to the mass loss rate found in N -body simulationsand only use the relative or ‘differential’ stellar mass de-pendence from H´enon (1969). This is allowed if the ratio t dis /t dis , H´enon = ( t rh /t cr ) . only depends on global clus-ter properties. It is straightforward to show (e.g. Spitzer1987; Heggie & Hut 2003) that indeed this is the case as t rh /t cr ∝ N/ ln Λ with ln Λ the Coulomb logarithm. Assuch, one can writed N ( m )d t = (cid:18) d M d t (cid:19) dis χ ( m ) , (12)with (d M/ d t ) dis the mass loss rate from Eq. 3 (Lamers et al.2005) and χ ( m ) the stellar mass-dependent escape rate perunit mass loss rate. The quantity χ ( m ) is completely inde-pendent of the prescription for the total mass evolution. Inorder to derive χ ( m ), I start from Eq. 10 and express χ ( m )as χ ( m ) = (13) N ( m ) R ∞ N ( m ′ ) F ( m/m ′ ) λ ( m, m ′ ) m ′ d m ′ R ∞ m ′′ N ( m ′′ ) R ∞ N ( m ′ ) F ( m ′′ /m ′ ) λ ( m ′′ , m ′ ) m ′ d m ′ d m ′′ , where λ ( m, m ′ ) represents a correction factor to accountfor additional physics (see below). The numerator reflectsthe escape rate, while the denominator is proportional tothe mass loss rate. For mathematical simplicity H´enon (1969) made thefollowing assumptions in the derivation of Eq. 10.(1) The cluster exists in isolation and the tidal field is ne-glected. Therefore, ejection can only occur by a single,close encounter and the repeated effect of soft encoun-ters (two-body relaxation) is not accounted for. Thisunderestimates the escape rate of massive stars.(2) The distribution of stars is independent of stellar mass,i.e. mass segregation is not included. Depending on thebalance between their encounter rate and their proxim-ity to the escape energy, this over- or underestimatesthe escape rate of low-mass stars from H´enon (1969).Considering the results from Baumgardt & Makino(2003), the latter seems to be the case.The remainder of this section concerns the derivation ofthe factor λ ( m, m ′ ) in Eq. 13 that corrects for the aboveassumptions.Let us assume that the distribution of stars over radiusand velocity space is initially independent of their mass.This implies that mass segregation is dynamically createdand not primordial, which is discussed in Sect. 6. For suchan initial distribution, the separation from the escape en-ergy ∆ E is independent of mass. As the cluster evolves,energy equipartition is reached between the stars and theradius, velocity and proximity to the escape energy becomea function of stellar mass. I first consider this effect on theescape rate before including the timescale on which two-body relaxation occurs for different stellar masses. Pleasenote that the formulation of Eq. 13 with λ ( m, m ′ ) appear-ing in the numerator and the denominator implies that onlythe proportionality of λ ( m, m ′ ) is important. Its exact valueis determined by constants that drop out when substitutingin Eq. 13.It is intuitive to express the dependence of the es-cape rate on the energy needed for escape as d N ( m ) / d t ∝ [∆ E ( m )] − . The energy that is required for escape ∆ E isrelated to the position and velocity of the star. For thepotential in Eq. 9 it is given by∆ E ( r, v ) = ψ ( r ) − v ψ (cid:18) r r (cid:19) − / − v , (14)with r and v the radial position and velocity of the star,and v esc ≡ p ψ ( r ) its escape velocity. If the cluster isin ‘perfect’ energy equipartition and correspondingly per-fect mass segregation, the radius and velocity become amonotonous function of stellar mass (Heggie & Hut 2003,Ch. 16). Mass segregation is strongest in the cluster centre,which for a Plummer (1911) potential can be approximatedwith a harmonic potential ψ ∝ r . For a cluster in a tidalfield the potential is truncated, and the harmonic approxi-mation serves as a crude but reasonable approximation for And because this is the only way to obtain an analyticalsolution as in Eq. 10. The energy difference ∆ E that is discussed here concernsthe energy that needs to be added to reach the escape energy.As such, it differs from the separation from the escape energy inFukushige & Heggie (2000) and Baumgardt (2001), who are con-sidering the excess energy of stars and its relation to the escapetime, resulting in the aformentioned relation t dis ∝ t . t . .. M. D. Kruijssen: The evolution of the stellar mass function in star clusters 5 the entire cluster (Heggie & Hut 2003, Ch. 16). Energyequipartition yields v ( m ) = h v i h m i m , (15)with h v i ∝ ψ the mean speed of all stars squared and h m i the mean stellar mass. For the harmonic potential, thistranslates to a similar relation for the radial position: r ( m ) = r r h m i m , (16)where r represents the typical radius of the system, inthis case the Plummer radius. This relation assumes thatthere is no particular stellar mass which dominates the massspectrum. The decrease of radial position with stellar massimplied by Eq. 16 is a direct consequence of the energy lossendured by massive stars as the system evolves towards en-ergy equipartition. Substituting Eqs. 15 and 16 into Eq. 14and dividing out the proportionality h v i ∝ ψ gives anexpression for ∆ E ( m ):∆ E ( m ) = (cid:18) h m i m (cid:19) − / − c h m i m , (17)with c = h v i / ψ denoting the ratio of the mean speedsquared to the central escape velocity squared. This con-stant mainly depends on the degree of mass segregation.Consequently, it will depend on the IMF. By comparingthe models to the N -body simulations with a mass spec-trum by Baumgardt & Makino (2003) the value is con-strained to c = 0 .
020 for a Kroupa IMF, using King (1966)potentials with King parameter W = 5—7 (see Sect. 4).For reference, an unevolved Plummer (1911) potential has h v i / ψ = 3 π/
64 = 0 . N -body simulations (pro-vided by M. Gieles, private communication) with differentIMF power law slopes α and a ratio between the maxi-mum and minimum mass of 10, the approximate relationlog c ≈ α − .
76 is found for a MF n s ∝ m − α . Fittingthe Kroupa IMF with a single power law in the mass range0.08—15 M ⊙ (as used by Baumgardt & Makino 2003) yields α = 2 .
06, resulting in c = 0 .
020 as mentioned earlier. The comparison with N -body simulations also showed thata single value of c suffices to determine the MF evolution,even though it does not remain constant over the full clusterlifetime.Because d N ( m ) / d t ∝ [∆ E ( m )] − , Eq. 17 indicates thatthe escape rate of low-mass stars is increased if a cluster isin complete energy equipartition. However, the timescale onwhich two-body relaxation occurs between different stellarmasses has not yet been considered. For a cluster startingwith a stellar mass-independent distribution of radial po-sitions and velocities, the equipartition timescale t e scalesas t e ( m, m ′ ) ∝ m − m ′− , (18) And the energy gain experienced by low-mass stars. This prescription for c implies that the condition for thestars in the cluster to be physically bound ∆ E ( m ) > α < . Nonetheless, the relation for c should be expected to exhibitsome variation for different mass ranges. for equipartition between stars of masses m and m ′ (Heggie& Hut 2003). This is a modified version of the relaxationtimescale, which shows a very similar proportionality ( t r ∝ m − ). It illustrates that two-body relaxation occurs on ashorter timescale for massive stars than for low-mass stars,increasing their escape rate d N ( m ) / d t ∝ t − .The correction factor for the escape rate λ ( m, m ′ ) thatappears in the integrals of Eq. 13 now follows from Eqs. 17and 18 as λ ( m, m ′ ) = t − ( m, m ′ )[∆ E ( m )] − = mm ′ "(cid:18) h m i m (cid:19) − / − c h m i m − . (19)It was mentioned before that the proportionalities of∆ E ( m ) and t e ( m, m ′ ) rather than their exact values sufficefor the computation of λ ( m, m ′ ) due to the renormalisationof the total mass loss rate that appears in Eq. 13: only thestellar mass-dependence is important .The influence of the tidal field is now included in twoways. First of all, the ejection of stars no longer occursby a single, close encounter but arises due to two-body re-laxation on the equipartition timescale, representing the re-peated effect of soft encounters. Secondly, the above deriva-tion of the separation from the escape energy assumes apotential which approximates tidally limited clusters. As aresult, the escape rate of massive stars is increased with re-spect to clusters in the model of H´enon (1969), which wasderived for an isolated cluster. On the other hand, the ef-fect of mass segregation is included by introducing a stellarmass-dependence for the energy needed by stars to reachthe escape velocity. Low-mass stars are closer to the tidalradius than massive stars, leading to a lower energy that isneeded for escape and an increased escape rate. It dependson the shape of the MF which mechanism dominates.The evolution of the MF of various cluster componentsis obtained from Eqs. 12, 13 and 19 by writingd log N comp ( m )d t = d log N ( m )d t , (20)where the MFs of stars, white dwarfs, neutron stars andblack holes are represented by N comp ( m ), with comp = { s , wd , ns , bh } . The overall cluster evolution is computed bycombining the results of this section with the prescriptionfor stellar evolution from Sect. 3.1.If stellar evolution is included, the resulting mass losscauses an expansion of the cluster, during which stars arelost independently of their masses. This delays the onsetof mass segregation and the stellar mass-dependent massloss that is described above. The moment of transition tostellar mass-dependent mass loss can be characterised bya certain fraction of the initial cluster mass that has beenlost by dissolution f diss ≡ M diss /M i . It is assumed that thefraction f smd of the mass loss for which the ejection ratedepends on the stellar mass grows exponentially between0 and 1 as f smd = C (cid:16) e f diss /f diss , seg − (cid:17) , (21) This form assumes that the increase of the fraction of themass loss that is stellar mass-dependent scales with the total dy-namical mass loss, which is a compromise between a step func-tion and a linear increase. J. M. D. Kruijssen: The evolution of the stellar mass function in star clusters
Fig. 3.
MF slope change ∆ α in the range m = 0 . . ⊙ versus the remaining mass fraction for a Kroupa IMF (solid),Salpeter IMF (dotted), and a power law IMF with α = 1 . m = 0 . ⊙ .The displayed relation is valid if stellar evolution is excluded. where the subscript ‘smd’ denotes ‘stellar mass-dependent’, f diss , seg ≡ M diss , seg /M i is the fraction of the initial massthat has been lost by dissolution at which mass segrega-tion is reached, and C = (e − − is a constant to nor-malise f smd = 1 at the reference value f diss = f diss , seg . For f diss > f diss , seg , per definition f smd = 1, indicating thatall mass loss is stellar mass-dependent. The timescale t seg on which mass segregation is reached and the transition tostellar mass-dependent mass loss is completed is propor-tional to the initial half-mass relaxation time ( t seg ∝ t rh , i ).It has been shown in several studies that for Roche lobe-filling clusters the disruption timescale t dis ∝ t . , i t . (Vesperini & Heggie 1997; Baumgardt & Makino 2003;Gieles & Baumgardt 2008), implying that t seg /t dis ∝ t . .The expression for t dis in Eq. 2 then leads to t seg /t dis ∝ t . M . γ i . Assuming that the cluster mass evolution isclose to linear, the first-order relation f diss , seg ∝ t seg /t dis isobtained, implying f diss , seg = c (cid:18) t t ⊙ (cid:19) . (cid:18) M i M ⊙ (cid:19) . , (22)for a King parameter of W = 5, with the dissolutiontimescale at the solar galactocentric radius t ⊙ = 21 . W = 7, the exponent of the initialcluster mass M i becomes 0.23 and t ⊙ = 10 . c represents a constantthat is fixed by comparing the model to the results of N -body simulations from Baumgardt & Makino (2003), giving c = 0 .
25 for W = 5 and c = 0 .
15 for W = 7 (see Sect. 4).The variation with King parameter arises because two-bodyrelaxation is faster for more concentrated clusters. If stellarevolution were neglected, at all ages c = 0 and f smd = 1.The modeled MF slope change ∆ α in the mass range m = 0 . . ⊙ is shown in Fig. 3 for different IMFscovering m = 0 . ⊙ . Evidently, ∆ α is a function ofthe remaining mass fraction and is insensitive to the slopeof the IMF, as long as that the ratio between the maximumand minimum mass is kept fixed and stellar evolution isexcluded. This is an interesting observation in view of the Fig. 4.
Relative ejection rate χ ( m ) /N ( m ) as a function ofstellar mass, shown for a Kroupa MF with different maximummasses. The end point of each curve (dot) marks its maximummass. The quantity χ ( m ) /N ( m ) ≡ (d log N ( m ) / d t ) / (d M/ d t )represents the ejection rate per unit mass loss rate normalisedto the number of stars at each mass (also see Eq. 13). MF evolution of globular clusters, in which m ≈ . ⊙ and stellar evolution only plays a minor role. Figure 3 showsthat the slope of the MF in globular clusters could be apossible indicator for the mass fraction that has been lostdue to dissolution, provided that the IMF does not varyand the remnant retention fractions were not substantiallydissimilar during the early evolution of different globularclusters (see Sect. 5.2 and Fig. 19).For the particular example of a Kroupa MF that is trun-cated at different maximum masses m max , the relative es-cape rate per unit mass loss rate χ ( m ) /N ( m ) (see Eqs. 12and 13) is shown in Fig. 4. This quantity is proportional tod log N ( m ) / d t and reflects the probability that a star of acertain mass is ejected. Figure 4 illustrates that the mass ofthe highest relative ejection rate is related to the maximummass of the MF. The peak occurs at intermediate masses ifthe MF is truncated at a high mass. This implies that thereis a typical mass where the stars are not too far from theescape energy and have an equipartition timescale with themassive stars that is short enough to eject them efficiently.This ‘sweet spot’ depends on the maximum mass of theMF. If the MF is truncated at an intermediate mass, thecombination of quick two-body relaxation and proximity tothe escape energy favours the ejection rate of stars at thelowest masses.The maximum stellar mass at which the transition from‘sweet spot’-depletion to low-mass star depletion happens,is determined by the proximity of the low-mass stars tothe escape energy. In Fig. 5, the mass of the peak relativeejection rate is shown as a function of the maximum stellarmass. At low truncation masses, the peak occurs at theminimum mass, indicating strong low-mass star depletion.Around m max ∼ ⊙ , the relative ejection rate at m peak ∼ . ⊙ becomes larger than its value at the lowest masses,which causes a jump in Fig. 5. For even higher values of m max , the peak relative ejection rate typically occurs at . M. D. Kruijssen: The evolution of the stellar mass function in star clusters 7 Fig. 5.
Mass of the highest relative ejection rate m peak as afunction of the maximum stellar mass of the MF m max (solidline). The dashed line represents the relation m peak = 0 . m max ,while the dotted line describes an eyeball fit for masses m max > ⊙ and includes an exponential truncation at the low-massend (see Eq. 23). m peak = 0 . m max e − M ⊙ /m max . (23)Even though its quantitative properties only hold for aKroupa MF, the variation of the relative escape rate withthe maximum mass of the MF has several implicationsfor star cluster evolution. The change of m max in Figs. 4and 5 can be interpreted as an example of what happenswhen stellar evolution removes the most massive stars inthe cluster, provided that the remnants are all ejected bytheir kick velocities. If dynamical evolution does not affectthe shape of the MF too much before m max ( t ) ∼ ⊙ , or t ∼
400 Myr, the subsequent evolution of the MF will bedominated by low-mass star depletion. If substantial dis-solution occurs earlier on, it is dominated by the ‘sweetspot’ depletion of intermediate masses. Only the retentionof massive stellar remnants will make the evolution of theMF deviate from these basic estimates, because remnantretention can provide a fixed maximum (remnant) mass ofthe MF. This is treated in more detail in Sect. 5.
4. Comparison to N -body simulations The model described in Sect. 3 can be easily verifiedby running it for the exact same boundary conditions asthe N -body simulations by Baumgardt & Makino (2003)and comparing the results. They conducted simulations ofRoche lobe-filling clusters between 8k and 128k particles,which were evolved in the Galactic tidal field at galactocen-tric radii in the range 2.833—15 kpc. The boundary condi-tions for the N -body runs of Baumgardt & Makino (2003)differ from those described in Sect. 3 by neglecting kick ve-locities and defining the Kroupa stellar IMF between 0.1and 15 M ⊙ , thereby excluding black holes. For this partic-ular comparison, the same IMF, stellar evolution prescrip- These were performed using
NBODY4 (Aarseth 1999). tion, and initial-final mass relation for stellar remnants areused in the model that is presented in this paper.In Fig. 6, the modeled evolution of the (luminous) stellarMF is compared to the N -body runs with King parameter W = 5 for a range of cluster masses and total disrup-tion times. As time progresses, the maximum stellar massdecreases due to stellar evolution and the MF is lowereddue to the dynamical dissolution of the star cluster. Theslope of the MF changes due to the preferential ejection oflow-mass stars, which have energies closer to their escapeenergies, even to the extent that it dominates over their rel-atively slow two-body relaxation. For both the models andthe N -body simulations, the MF develops a slight bend at m ∼ . ⊙ when approaching total disruption. The bendarises as an optimum between on the one hand high energiesbut slow relaxation for the lowest stellar masses, and on theother hand quick relaxation but low energies for the higheststellar masses (see the discussion at the end of Sect. 3).In all cases, the resemblance of the models and the N -body simulations is striking. The models reproduce all keyaspects of the N -body runs, such as the amount of low-mass star depletion, the changing slope at m ∼ . ⊙ for clusters close to dissolution, the survival of the Kroupabend at m = 0 . ⊙ , and the dependence of the low-massdepletion on the total lifetime of the cluster (compare thethree 32k runs). The only difference occurs at the high-massend of the MF, where the maximum stellar masses do notmatch at young ages. This is due to a minor dissimilarityof the total mass evolution (also see Lamers et al. 2005;Kruijssen & Lamers 2008). Because the maximum stellarmass only depends on the age of the cluster, this causes adifference in maximum stellar mass when showing the MFsat fixed remaining cluster mass fractions. The contrast isclearest at young ages, since there the maximum stellarmass most rapidly decreases.In the description of the model in Sect. 3, two con-stants have been determined from the N -body simulationsby Baumgardt & Makino (2003). These constants are theratio of the mean speed squared to the central escape ve-locity squared ( c , see Eq. 17) and the proportionality con-stant for the relation marking the transition to stellar mass-dependent mass loss ( c , see Eq. 22). As mentioned inSect. 3, for a Kroupa IMF and King parameter W = 5one obtains c = 0 .
020 and c = 0 .
25. To illustrate therobustness of the models, in Fig. 7 they are compared toa 64k N -body run with W = 7. For such a cluster witha higher concentration, the early mass segregation implies c = 0 .
15. Again, the model and the simulation are in ex-cellent agreement.The dependence of the MF evolution on both constantsis considered in Fig. 8. For c , the dependence of the evo-lution of the MF on its value is shown in the upper panelof Fig. 8, while for c it is shown in the bottom panel ofFig. 8. Both panels show the evolution of the MF for the64k cluster in Fig. 6 for different values of c and c .The ratio of the mean speed squared to the central es-cape velocity squared c affects the ejection probability ofthe stars with the lowest masses. Because these stars areclosest to their escape energies in a mass-segregated cluster,they are most strongly influenced by the value of c . Forhigher c , the MF gets more depleted in low-mass stars dueto their closer proximity to the escape energy, while forlower c more low-mass stars are retained as the balance J. M. D. Kruijssen: The evolution of the stellar mass function in star clusters
Fig. 6.
Comparison of the evolution of the stellar MF from the models (dashed) to the N -body runs from Baumgardt & Makino(2003, solid) for the exact same boundary conditions. The initial number of particles and the galactocentric radius are indicatedin the bottom-left corner of each panel. From top to bottom, the subsequent MFs in each panel are shown for the times at whichthe remaining cluster mass fraction equals M/M i = { , . , . , . , . , . , . , . } . between close proximity to the escape energy and slow re-laxation shifts to the latter.The proportionality constant for the transition to stellarmass-dependent dissolution c in Eq. 22 affects the MF as awhole. For lower c , the transition occurs earlier and morelow-mass stars are lost, while for higher c the onset of thedepletion is delayed and the slope of the MF remains closerto its initial value. If one were to assume a constant f diss , seg ,which is contrary to the adopted relation with cluster massin Eq. 22, this would therefore yield a stellar MF in massiveclusters that is underpopulated in low-mass stars, and a MFin low-mass clusters that is overabundant in low-mass stars.
5. Star cluster evolution
In this section, the described model is applied to computethe evolution of clusters for a variety of boundary condi-tions. The stellar content as well as integrated photometryare addressed, using the boundary conditions from Sect. 3instead of those that were adopted to compare the modelto N -body simulations in Sect. 4. The most important dif-ferences are the mass range of the IMF, the inclusion ofremnant kick velocities, and the initial-final mass relation.The model that will be referred to as the ‘standardmodel’ uses a metallicity Z = 0 .
004 (which is typical ofglobular clusters), a King parameter of W = 7 (cor-responding to γ = 0 . t = 1 Myr, and a Kroupa IMF between For W = 5, or γ = 0 .
62, the results vary only marginally.. M. D. Kruijssen: The evolution of the stellar mass function in star clusters 9
Fig. 7.
Comparison of the evolution of the stellar MF fromthe models (dashed) to the N -body run from Baumgardt &Makino (2003, solid) with W = 7 for the exact same boundaryconditions. From top to bottom, the subsequent MFs are shownfor the times at which the remaining cluster mass fraction equals M/M i = { , . , . , . , . , . , . , . } . m = 0 .
08 M ⊙ and the maximum stellar mass given bythe Padova isochrones at log t = 6 .
6, which is typically m ∼
70 M ⊙ . For the computation of the retained remnantfraction (see Eq. 8), the Plummer radius r is related to thehalf-mass radius r h as r h = 1 . r . The half-mass radius isassumed to remain constant during the cluster lifetime (e.g.Aarseth & Heggie 1998). For the relation between r h andinitial cluster mass M i the expression from Larsen (2004)is adopted: r h = 3 .
75 pc (cid:18) M i M ⊙ (cid:19) . . (24)The models that are used in this section are computed from10 yr to 1 . × yr (the maximum age of the Padovaisochrones) for initial masses between 10 M ⊙ and 10 M ⊙ ,spaced by 0.25 dex intervals. The disruption time of a cluster affects the evolution ofthe MF and of the integrated photometric properties. Toassess the influence of the disruption time on cluster evolu-tion, clusters with low and high remnant retention fractionsshould be treated separately, because the presence of mas-sive remnants also has a pronounced effect on the results(see Sect. 5.2). As shown in Fig. 1, for a given kick veloc-ity dispersion the remnant retention fraction is set by thecluster mass. This means that the division between low andhigh remnant retention fractions can be made by making acut in initial cluster mass.In Fig. 9, the impact of the disruption time on the evo-lution of the MF is shown for a cluster with initial masslog ( M i / M ⊙ ) = 4 .
5, representing the evolution for low rem-nant retention fractions. The range of the dissolutiontimescale parameter t and resulting total disruption times High remnant retention fractions will be treated in the dis-cussion of the influence of the retention fraction in Sect. 5.2.
Fig. 8.
Influence of the constants c and c on the evo-lution of the stellar MF. From top to bottom, the sub-sequent MFs in each panel are shown for the times atwhich the remaining cluster mass fraction equals M/M i = { , . , . , . , . , . , . , . } . Top panel : the values c = { . , . , . } are represented by dashed, solid anddotted lines, respectively. Bottom panel : the values c = { , . , . } are represented by dashed, solid and dotted lines,respectively. For both c and c , the second (boldfaced) valueis the one obtained from the comparison to the N -body simula-tions with W = 5 in Fig. 6. that are considered in Fig. 9 cover two orders of magnitude.As the total lifetime increases, the depletion of the low-massstellar MF close to total disruption becomes more promi-nent. Conversely, the MF of short-lived clusters is depletedaround m ∼ ⊙ . As introduced in the last paragraphsof Sect. 3, this difference is caused by the fixed timescaleon which stellar evolution decreases the maximum stellarmass, implying that the masses of the most massive starsare larger in quickly dissolving clusters than in slowly dis-solving ones. Because in short-lived clusters the massivestars are still present when the bulk of the dissolution oc-curs, their rapid two-body relaxation with intermediate-mass stars dominates over the relatively close proximity tothe escape energy of low-mass stars, yielding a depletionat intermediate masses. In long-lived clusters, this cannotoccur because the very massive stars have disappeared be-fore the mass loss by dissolution becomes important, thusresulting in the depletion of the very low-mass end of theMF. As a rule of thumb, for t <
400 Myr (which is the life-time of a 3 M ⊙ star) the depletion typically occurs around15—20% of the mass of the most massive star (see Sect. 3).In terms of the total disruption time, the transition fromintermediate-mass star depletion to low-mass star depletionoccurs around t totaldis ∼ Fig. 9.
Influence of the disruption time on the evolutionof the stellar MF for a cluster with a low remnant reten-tion fraction (log ( M i / M ⊙ ) = 4 . M/M i = { , . , . , . , . , . , . , . } . A quantifiable way to look at the evolution of the stel-lar MF in star clusters is to consider the slope of theMF n s ∝ m − α in certain mass intervals (Richer et al.1991; De Marchi et al. 2007; De Marchi & Pulone 2007;Vesperini et al. 2009). For the commonly used mass inter-vals 0 . < m/ M ⊙ < . α ) and 0 . < m/ M ⊙ < . α ),Fig. 10 shows the evolution of the slope α for the sameclusters as before. Like Fig. 9, this illustrates that for shortdisruption times the slope steepens as the cluster dissolves,while for long disruption times the slope flattens with time.The presented models and other model runs indicate that α increases with time for t totaldis < t totaldis > α shows the samebehaviour. It increases for t totaldis < . t totaldis > Fig. 10.
Influence of the disruption time on the stellar MF slope α in the range 0 . < m/ M ⊙ < . . < m/ M ⊙ < . M i / M ⊙ ) = 4 . α versus the remaining clustermass fraction. From top to bottom, for each mass range thelines represent t = { . , , } Myr, corresponding to t totaldis = { . , . , . } Gyr.
Fig. 11.
Influence of the disruption time on the
M/L V ra-tio evolution for a cluster with a low remnant retention frac-tion (log ( M i / M ⊙ ) = 4 . M/L V ratiodecrease with respect to the value expected for stellar evolu-tion ( M/L V ) stev versus the remaining cluster mass fraction. Thesolid, dashed and dotted lines represent t = { . , , } Myr,respectively, corresponding to t totaldis = { . , . , . } Gyr.
The mass-to-light (
M/L ) ratio evolution of star clustersis affected by the evolution of the MF due to the largevariations in
M/L ratio between stars of different masses.Massive stars have lower
M/L ratios than low-mass stars,implying that a cluster with a MF that is depleted in low-mass stars will have a reduced
M/L ratio (Baumgardt &Makino 2003; Kruijssen 2008; Kruijssen & Lamers 2008).As such, one would also expect a correlation between theslope of the MF and
M/L ratio.In Fig. 11, the evolution of the ratio of the V -band M/L V to the mass-to-light ratio due to stellar evolution( M/L V ) stev is shown for the same clusters as in Figs. 9 . M. D. Kruijssen: The evolution of the stellar mass function in star clusters 11 Fig. 12.
Influence of the disruption time on the combinedevolution of the MF slope α and the M/L V ratio for a clus-ter with a low remnant retention fraction (log ( M i / M ⊙ ) =4 . α versus the relative M/L V ratio decrease dueto dynamical evolution. All clusters start at the vertical line( M/L V ) / ( M/L V ) stev = 1. Solid lines denote the slope in themass range 0 . < m/ M ⊙ < .
5, dashed lines designate themass range 0 . < m/ M ⊙ < .
8, and dotted lines representthe mass range 0 . m max ( t ) < m/ M ⊙ < . m max ( t ), withfrom top to bottom t = { . , , } Myr, corresponding to t totaldis = { . , . , . } Gyr. and 10. This quantity reflects the relative
M/L V ratiochange induced by dynamical evolution with respect to evo-lutionary fading only. If the ejection rate would be indepen-dent of stellar mass, the evolution would follow a horizon-tal line at ( M/L V ) / ( M/L V ) stev = 1. However, when ac-counting for dynamical evolution, the M/L ratio is alwayssmaller than that for stellar evolution only. Somewhat sur-prisingly, this is also the case for clusters for which theslope of the MF increases (see Fig. 10). This is explainedby looking at the evolution of the entire MF in Fig. 9. Eventhough the slope at low masses increases for short disrup-tion times due to the ejection of intermediate-mass stars,the most massive stars that dominate the cluster light arestill retained. Because stars of intermediate masses are lostinstead, the
M/L ratio decreases.Because the slope of the stellar MF either increases ordecreases at masses m < ⊙ , the decrease of the M/L ratio implies a large range of MF slopes that can occurat low
M/L ratios. This is shown in Fig. 12, where therelation between α and the M/L ratio drop is presented.The slope of the stellar MF in a certain mass range doesnot necessarily reflect the
M/L ratio of the entire cluster.Considering the aforementioned rule of thumb stating thatfor total disruption times t totaldis < m max ( t ), it is useful to define the slope in a mass rangethat is related to m max ( t ). In Fig. 12, the relation betweenslope and M/L ratio is also shown for the slope in the stellarmass range 0 . m max ( t ) < m/ M ⊙ < . m max ( t ). In sucha relative mass range, the slope follows a much narrowerrelation with M/L ratio. The range between 30% and 80%of m max ( t ) was chosen to maximise this effect.For the slopes in the fixed stellar mass ranges ( α and α , see above), the relation with the M/L ratio be-
Fig. 13.
Influence of the disruption time on the V − I colour fora cluster with a low remnant retention fraction (log ( M i / M ⊙ ) =4 . V − I ) versus the remaining mass fraction. The solid, dashedand dotted lines represent t = { . , , } Myr, respectively,corresponding to t totaldis = { . , . , . } Gyr. comes better defined for long-lived clusters. It is shown inFigs. 10—12 that both the slope and the
M/L ratio de-crease for clusters with long disruption times, indicatingthat both quantities are more clearly related for globularcluster-like lifetimes.The colour of star clusters is also influenced by the evo-lution of the MF, due to the colour differences betweenstars of different masses. The V − I magnitude difference∆( V − I ) with respect to the V − I value that a clusterwould have if dynamical evolution were neglected is shownin Fig. 13. As the clusters dissolve, their colours becomeredder due to the ejection of main sequence stars. The mag-nitude difference in V − I exceeds ∆( V − I ) = 0 . ≤ . V − K colour), the difference grows to several tenthsof magnitudes. For longer total disruption times only starsof the lowest masses are ejected (see Fig. 9), which hardlycontribute to the cluster light and colour, implying that thecolours are only marginally affected. The formation of stellar remnants introduces massive bod-ies in the MF that do not end their lives due to stellar evo-lution like massive stars do. Depending on their kick veloc-ities, stellar remnants can be retained in (massive) clusters.If they are retained, they keep affecting the evolution of thestellar MF until the cluster is disrupted. Especially blackholes can have a pronounced effect on cluster evolution.The remnant retention fraction arises from the clustermass, radius and the kick velocity dispersion (see Eq. 8). Inthis section, the mass-radius relation from Eq. 24 is used.Although the results will differ for other relations, it hasbeen verified that for commonly used alternatives, thechange is only marginal and does not affect the nature ofthe conclusions. To separate the effect of remnant reten-tion from that of the disruption time, a fixed initial cluster Such as a constant radius or density.2 J. M. D. Kruijssen: The evolution of the stellar mass function in star clusters
Fig. 14.
Influence of the black hole kick velocity dispersionand disruption time on the evolution of the stellar MF foran initial cluster mass M i = 10 M ⊙ . From top to bottom,the subsequent MFs in each panel are shown for the timesat which the remaining cluster mass fraction equals M/M i = { , . , . , . , . , . , . , . } . Solid lines denote t = 1 Myr( t totaldis = 15 .
13 Gyr), while dotted lines represent t = 0 . t totaldis = 1 .
66 Gyr). mass of 10 M ⊙ is assumed while independently varying thevelocity dispersion of the remnant kick velocities and thedisruption time. The corresponding evolution of the stel-lar MF is shown in Fig. 14, for the standard model (seethe beginning of this section) with black hole kick velocitydispersions σ kick , bh = { , , } km s − , equivalent to f ret , bh = { . , . , . } for a 10 M ⊙ cluster, and fordissolution timescale parameters t = { . , } Myr, whichfor a 10 M ⊙ cluster implies t totaldis = { . , . } Gyr.Assuming an age of 12 Gyr, the present-day mass in thecase of t = 1 Myr is about M ∼ × M ⊙ , comparableto globular clusters. The remaining fraction of the initialmass is M/M i ∼ . Fig. 15.
Influence of the black hole retention fraction on thestellar MF slope α in the range 0 . < m/ M ⊙ < . . < m/ M ⊙ < . M i = 10 M ⊙ . Shown is α versus the remaining cluster massfraction. From top to bottom, for each mass range the linesrepresent σ kick , bh = { , , } km s − , corresponding to f ret , bh = { . , . , . } for a 10 M ⊙ cluster. the ejection rate of massive stars is increased with respectto high kick velocity dispersions. This arises due to thequick two-body relaxation between the massive stars andthe black holes, which will have masses larger than the mostmassive stars after a few Myr of stellar evolution. As a re-sult, the ejection rate of low-mass stars is largest in clusterscontaining only few black holes. This happens for clusterswith either long or short disruption times, but the effect islargest for long-lived clusters (the solid lines in Fig. 14). Inthese clusters the maximum stellar mass is more stronglydecreased by stellar evolution than in short-lived clusters,implying that the black hole masses are larger compared tothe most massive stars in these clusters. For long disrup-tion times, the presence of massive remnants therefore has amore pronounced effect on the ejection rate of massive starsthan for short disruption times. If these long-lived clustersretain a sufficiently large fraction of the stellar remnants,their stellar MF may even become depleted in massive stars.The top panel of Fig. 14 also shows that for a clusterwith a high remnant retention fraction, the impact of thedisruption time on the MF evolution is similar to that ofclusters with low retention fractions (see Fig. 9). However,the influence of the disruption time becomes smaller whenmore remnants are retained. This explains why Baumgardt& Makino (2003) only found a very weak dependence ofthe evolution of the MF on the disruption time (also seeFig. 6), since they neglected remnant kick velocities andretained all remnants in their simulations.Analogous to Fig. 10 in Sect. 5.1, the evolution of theMF slope in different mass ranges is shown in Fig. 15 forthe clusters with t = 1 Myr from Fig. 14. The kick For the clusters with relatively long disruption times thatare considered in this section, the variable stellar mass rangethat was introduced in Sect. 5.1 to trace the relation betweenMF slope and
M/L ratio gives an evolution of the slope that iscomparable that for the fixed mass ranges. It is omitted fromthe figures in this section to improve their clarity.. M. D. Kruijssen: The evolution of the stellar mass function in star clusters 13
Fig. 16.
Influence of the black hole retention fraction on the
M/L V ratio evolution for an initial cluster mass M i = 10 M ⊙ .Shown is the relative M/L V ratio decrease with respect to thevalue expected for stellar evolution ( M/L V ) stev versus the re-maining cluster mass fraction. The solid, dashed and dottedlines represent σ kick , bh = { , , } km s − , correspondingto f ret , bh = { . , . , . } for a 10 M ⊙ cluster. velocity dispersion has an effect on the MF that is moreuniform than the consequences of variations in the dis-ruption time, leading to very similar slope evolutions inthe two different stellar mass ranges. Independent of themass range, an increase in remnant retention fraction is re-flected by an increase of α . The model that is displayed for σ kick , bh = 40 km s − , t = 1 Myr, and M i = 10 M ⊙ (theupper dashed and solid lines in Fig. 15) marks the transi-tion between an increase or decrease of the MF slope bydynamical evolution. For an initial f ret , bh < .
25, low-massstars are preferentially ejected during most of the clusterlifetime, while for f ret , bh > .
25 mainly the massive starsescape. For shorter disruption times, the transition is lo-cated at a smaller black hole retention fraction.Because the black hole retention fraction affects theoverall slope of the stellar MF, the changes in α are matchedby corresponding changes in the M/L ratio. In Fig. 16, therelative
M/L V ratio change due to dynamical evolution isshown for same clusters as in Fig. 15. Contrary to the clus-ters with low remnant retention fractions in Sect. 5.1, the M/L ratio of the clusters in Fig. 16 does not monotonouslydecrease. Close to total disruption, the massive remnantsare the last bodies to be ejected. During that short phase ofcluster evolution, the
M/L ratio is increased by dynamicalevolution and exceeds the value it would have due to stellarevolution alone.The behaviour of
M/L ratio for different black hole kickvelocity dispersions has interesting implications for the re-lation between stellar MF slope and
M/L ratio, which isshown in Fig. 17. In combination with Fig. 12 (note thedifferent axes), it shows possible evolutionary tracks of starclusters in this plane, indicating that nearly every loca-tion may be reached. However, when limiting ourselves tolong-lived clusters, Fig. 17 illustrates that these clusters willfollow a trend of decreasing slope with decreasing
M/L ra-tio, albeit with excursions to high
M/L ratios and slightlyhigher α close to their total disruption. This explains thetrend that was found by Kruijssen & Mieske (2009), who Fig. 17.
Influence of the black hole retention fraction on thecombined evolution of the MF slope α and the M/L V ratio for aninitial cluster mass M i = 10 M ⊙ . Shown is α versus the relative M/L V ratio decrease due to dynamical evolution. All clustersstart at the vertical line ( M/L V ) / ( M/L V ) stev = 1. Solid linesdenote the slope in the mass range 0 . < m/ M ⊙ < . . < m/ M ⊙ < .
8, withfrom right to left σ kick , bh = { , , } km s − , correspondingto f ret , bh = { . , . , . } for a 10 M ⊙ cluster. considered the relation between the observed MF slopesand M/L ratios of Galactic globular clusters.The colour change due to dynamical evolution is onlyvery small for clusters with t totaldis > . V − I ) < .
03 mag). The colour change is even smallerif more massive remnants are retained, because then thestellar MF more closely resembles its initial form (see theupper panel of Fig. 14). Long-lived clusters generally ap-pear ∼ .
005 mag bluer in V − I due to dynamical evolu-tion during the last ∼ ∼ ⊙ ) stars by the blackholes. After ∼ Fig. 18.
Influence of the black hole retention fraction on thetotal remnant mass fraction. Shown is the ratio of the totalmass in stellar remnants M sr to the cluster mass M versus theremaining cluster mass fraction. The solid, dashed and dottedlines represent σ kick , bh = { , , } km s − , corresponding to f ret , bh = { . , . , . } for a 10 M ⊙ cluster.
6. Discussion and applications
The results of this paper show that the stellar MFs in starclusters differ strongly from their initial forms due to dy-namical cluster evolution. The specific kinds of these dif-ferences depend on the properties of the star clusters andtheir tidal environment, most importantly on the disruptiontime, remnant retention fraction, and IMF. A physical model for the evolution of the stellar MF ispresented in which two-body relaxation leads to a stellarmass dependence of the ejection rate. For any particularstellar mass, the ejection rate is determined by the typ-ical proximity of that mass to the escape energy and bythe timescale on which the two-body relaxation with theother stars takes place. Combined with a prescription forstellar evolution, stellar remnant production, and remnantretention using kick velocity dispersions, this provides adescription for the total evolution of the MF.
This descrip-tion is independent of the adopted total mass evolution . Themodel shows that the slope of the mass function is a possi-ble indicator for the mass fraction that has been lost due todissolution, provided that the IMF does not vary and theremnant retention fraction has been fairly similar for youngglobular clusters. For the exact same initial conditions, the model showsexcellent agreement with N -body simulations of the evolv-ing MF by Baumgardt & Makino (2003). However, an im-portant advantage of the presented model compared to the(more accurate) N -body simulations is its short runtimeand corresponding flexibility. It can be easily applied tocompute the evolution of clusters for a large range of initialconditions. The results can then be used to identify interest-ing cases for more detailed and less simplified calculationswith N -body or Monte Carlo models. Although not specifically shown in this paper (but not sur-prisingly), the differences also depend on the initial-final stellarmass relation. Any variability of the retention fraction would induce sub-stantial scatter, see Sect. 5.2 and Fig. 19.
The most important simplification of the model is ne-glecting the effect of binary encounters on the stellar massdependence of the ejection rate. To incorporate binaries, aconclusive census of the binary population in star clusterswould be required, which is not yet available. Nonetheless,it is possible to make a qualitative estimate for the effectof binaries. The encounter rate of binaries would typicallybe higher than that of individual stars, because the crosssection of binaries is larger. This would increase the relativeescape rate at the stellar mass for which the binary frac-tion peaks. This binary fraction is found to increase withprimary mass (see e.g. Kouwenhoven et al. 2009). Becausemassive stars are removed by stellar evolution, this impliesthat the binary fraction decreases with age, which is inagreement with the low binary fraction observed in glob-ular clusters ( ∼ <
50 Myr (the typical lifetime of an 8 M ⊙ star), in which case it would somewhat enhance the relativeescape rate of the most massive stars.The model is applied to investigate the influence of thedisruption time and remnant retention on the evolution ofthe MF and integrated photometric properties of star clus-ters. For total disruption times t totaldis < M/L ratio is increased by dynami-cal evolution when the cluster approaches total disruption.In all other scenarios, the
M/L ratio decreases because themost massive (luminous) stars are kept. When definingthe slope of the MF in the range 30—80% of the maxi-mum stellar mass, this gives a clear relation between theMF slope and the
M/L ratio. For slopes that are definedin fixed mass ranges, there is not necessarily a correlationbetween slope and
M/L ratio if t totaldis < V − I for total disruption times t totaldis < . The fraction of stars residing in binary or multiple systems. This process differs from a possible variability of the propor-tionality between the velocity dispersion and the cluster mass,which concerns a much shorter timescale (e.g. Boily et al. 2009).. M. D. Kruijssen: The evolution of the stellar mass function in star clusters 15 ruption time, their MF evolution will be dissimilar due totheir different remnant retention fractions and the impactof the retained remnants on the dynamical cluster evolu-tion. Alternatively, if two clusters have equal initial massesbut different total disruption times, for instance due to dif-ferences in their galactic location or environment, their MFevolution will be dissimilar due to the dynamical impact ofthe evolution of the maximum stellar mass.The larger variation of MF evolution that is found withpresented model may also be able to explain observationsof globular clusters in which the MF cannot be charac-terised by a single power law (De Marchi et al. 2000). Ifthe evolution of the MF were homologous, these featureswould likely be primordial (Baumgardt & Makino 2003),but this is not necessarily the case when using realistic re-tention fractions. Most other differences between the resultspresented in Sect. 5 and those from Baumgardt & Makino(2003) are also due to their assumption of full remnantretention. For example, their
M/L ratio evolution showsa smaller decrease than in Fig. 11. This is explained inFig. 16, where it is shown that dynamical evolution reducesthe
M/L ratio by a smaller amount if the retention fractionis larger.Studies on the fractal nature of cluster formation showthat star clusters are initially substructured (Elmegreen2000; Bonnell et al. 2003). Even though this substructureis typically erased on a crossing time, it can induce pri-mordial mass segregation in star clusters (McMillan et al.2007; Allison et al. 2009). The influence of primordial masssegregation on the evolution of the MF has recently beeninvestigated by Baumgardt et al. (2008) and Vesperiniet al. (2009). While Baumgardt et al. (2008) do not in-clude stellar evolution and concentrate on two-body relax-ation, Vesperini et al. (2009) do include stellar evolution.They show that for some degrees of primordial mass segre-gation, the mass loss by stellar evolution can induce addi-tional dynamical mass loss that strongly decreases the to-tal disruption time. For clusters that survive for a Hubbletime, the MF evolution in the case of primordial mass seg-regation is very similar to an initially unsegregated clus-ter. Vesperini et al. (2009) conclude that the evolution ofthe MF is only affected by primordial mass segregation forclusters in which the total disruption time is sufficientlydecreased by the induced mass loss. In that case, the slopeof the MF remains much closer to its initial value thanit would in clusters without primordial mass segregation.Their conclusion is consistent with the model presented inthis paper, because the evolution of the MF is determinedby the most massive stars at the time when the largest massloss occurs (see Figs. 4 and 9). This induced mass loss en-ters the model in terms of the absolute mass loss rate inEq. 3, not in the stellar mass-dependent escape rate perunit mass loss rate of Eq. 13.A change in total mass loss rate is not the only con-sequence of primordial mass segregation. Baumgardt et al.(2008) have shown that low-mass star depletion is enhancedfor clusters without stellar evolution that are primordiallymass-segregated. This occurs because energy equipartitionis reached on a shorter timescale and because of their useof a fixed ( m max = 1 . ⊙ ) maximum stellar mass. Asa result, there are no massive bodies to increase the ejec-tion rate of intermediate mass stars (see Fig. 5), implyingthat only the low-mass stars are preferentially lost. In thepresent paper, mass segregation is assumed to arise dynam- Fig. 19.
MF slope versus remaining lifetime (assuming a glob-ular cluster age of 12 Gyr). Diamonds represent the observedvalues from De Marchi et al. (2007), with typical errors asshown by the error bar in the lower right corner. The re-maining lifetimes are taken from Baumgardt et al. (2008).Dotted curves represent the model evolutionary tracks of clus-ters with log ( M i / M ⊙ ) = { , . , . , . , } from Sect. 5.2with { σ kick , wd , σ kick , ns , σ kick , bh } = { , , } km s − , corre-sponding to { f ret , wd , f ret , ns , f ret , bh } = { . , . , . } for a10 M ⊙ cluster. The solid line connects the present-day loca-tions of the modeled clusters in the diagram (crosses), while thedashed line represents the same relation for σ kick , bh = 40 km s − ( f ret , bh = 0 .
219 for a 10 M ⊙ cluster). The dash-dotted lineshows the homologous cluster evolution from Baumgardt &Makino (2003). ically, but the model could in principle be adapted to coverprimordial mass segregation by setting c = 0 and adjust-ing c to the initial velocity distribution until it is erasedby dynamical evolution (see Eq. 22), after which the valuesfrom Sect. 3 can be used. This does not necessarily yieldenhanced low-mass star depletion for clusters with a com-plete IMF (including masses m > . ⊙ ) because of thepresence of massive stars or remnants.The presented model can be applied to the MFs ofGalactic globular clusters that are observed by De Marchiet al. (2007). These MFs are more strongly depleted than isfound in the N -body simulations by Baumgardt & Makino(2003), which has been attributed to primordial mass segre-gation (Baumgardt et al. 2008). However, the observationscan also very accurately be explained with the realistic rem-nant retention fractions that are used in the present paper.This is shown in Fig. 19, where the observed MF slopesand remaining lifetimes of the globular clusters from DeMarchi et al. (2007) are compared with the globular cluster-like models from Sect. 5.2 ( t = 1 Myr). The models arein much better agreement with the data than the N -bodyruns with complete remnant retention from Baumgardt &Makino (2003). Deviations to other values of α can occurdue to variations in disruption time and remnant retention As explained in Sect 3, c represents the ratio of the meanspeed squared to the central escape velocity squared that de-pends on the degree of mass segregation (and thus on the IMF).On the other hand, c is a proportionality constant in the ex-pression for the onset of the stellar mass-dependent ejection ofstars, which depends on the concentration or King parameter.6 J. M. D. Kruijssen: The evolution of the stellar mass function in star clusters fractions, as is also shown in Fig. 19. For example, a varia-tion of the remnant kick velocity with metallicity in combi-nation with the known variation of the disruption time (see,e.g. Kruijssen & Mieske 2009; Kruijssen & Portegies Zwart2009) should be sufficient to cover the observed scatter.The above line of reasoning provides an explanation forthe the depleted MFs in Fig. 19 that is consistent with thesimulations by Vesperini et al. (2009), who showed thatthe effects of primordial mass segregation are in fact sup-pressed in long-lived clusters due to the expansion causedby stellar evolution. This increases the relaxation time andyields an evolution of the MF that is very similar to theinitially unsegregated scenario, indicating that primordialmass segregation is not a likely explanation for stronglydepleted MFs. Observations of the remnant composition ofthese globular clusters could reveal a definitive answer asto whether the depleted MFs are explained by primordialmass segregation or by dynamical evolution with a realisticremnant retention fraction.Dynamical cluster evolution does not appear to have alarge effect on the colours of old (globular) clusters. Theonly way in which the colours could be affected beyondtypical observational errors, is if globular clusters have lostsubstantial fractions of their masses during the first ∼ Gyrafter their formation. In that case, the dynamical evolutionof the stellar MF in globular clusters may have implicationsfor studies of colour bimodality (e.g. Larsen et al. 2001) orthe blue tilt (e.g. Harris et al. 2006). It could then alsopossibly explain the trend of increasing V − K colour withdecreasing M/L V ratio found by Strader et al. (2009) forglobular clusters in M31, because quickly dissolving clus-ters generally become redder and have reduced M/L ratios.More research is needed to determine the role of the chang-ing MF in the above properties of globular cluster systems.It can be concluded that the evolution of the stellar MFin star clusters is not as similar for all clusters as previouslythought. Its precise evolution is determined by cluster char-acteristics like the disruption time, the remnant retentionfraction, initial-final stellar mass relation, and the IMF. Inorder to decipher the evolution of observed star clusters,it is essential to record these characteristics and to relatethem to possible scenarios for the internal evolution of clus-ters. That way, observables like the slope of the MF, the
M/L ratio, the broadband colours, and the mass fractionin remnants can be better understood.
Acknowledgements.