arXiv:1109.2896v1 [astro-ph.GA] 13 Sep 2011 o.Nt .Ato.Soc. Astron. R. Not. Mon. igeadbnr otnso aatcfil populations field galactic of contents binary stellar and the single Constructing synthesis: population Dynamical cetd??.Rcie ??;i rgnlfr ????? form original in ?????; Received ????. Accepted

onadClge -al mrsatouibn.e(MM) [email protected] Universities e-mail: the Cologne; at and Bonn and cent Astronomy for (IMPRS) per 50 ⋆ 1992; and Throughout 2010). al. of stars al. et Mayor et Raghavan consists single late- 1992; 1991; Marcy M) cent & Mayor stellar Fischer & to (Duquennoy per field G binaries 50 (spectral-types Galactic roughly population (MW) type Milky-Way’s The INTRODUCTION 1 c 2 1 ihe Marks Michael readrIsiuefrAtooy nvriyo on Au H¨ugel Bonn, dem of Auf f¨ur Radioastronomie, University Max-Planck-Institut Astronomy, for Institute Argelander ebro h nentoa a lnkRsac School Research Planck Max International the of Member 00RAS 2010 1 , 000 2 ,⋆ –4(00 rne 5My21 M L (MN 2018 May 25 Printed (2010) 1–14 , n ae Kroupa Pavel and aayi agrtelwrtemnmmcutrmass, cluster minimum the the in lower proportion the binary larger the is that galaxy found a is It embedd to Synthesis). power-law rise Population gives a This from functions up. added tribution selected then are are function which mass clu st m clusters single enters star particular of star population contributions in a the individual different has distributions dissolution The which characteristics. cluster star achieved these On binary with is stars. the situation binary evolve and equilibrium single an to until used Myr d is few recently Oh) A as events. & form formation calcu Kroupa star stars is discrete all population in properties binary that variant and supposition single well-constrained stellar observationally late-type field’s galactic The ABSTRACT eeal omdcmatsa lses fallt-yebinaries, late-type all Of M clusters. indep format star star with compact discrete formed agree individual generally the IGBDFs that theoretical suggesting mentioned distributions, afore the cluster, iayfeunyi litclglxe slwrta nsia n ndwarf in and explicitly are spiral galaxies in words: these the than Key in lower in distributions predic mass-ratio is systems, work and galaxies binary This period stars. elliptical in The single in born being up frequency are ends binary M-dwarfs majority point all the is field model it Galactic present particular In the clusters. in mass-segregated populatio binary in M-dwarf formed and stars G-dwarf the of Comparison clusters. omto ae F,teseprteebde trcutrmas cluster star embedded the index steeper by the SFR, rate, formation r oeldusing IGBD eccentricity modelled and mass-ratio- are period-, particular, In galaxy. lsesadascain:gnrl–slrnihorod–metho – neighbourhood solar – general associations: and clusters ie yosrain.For observations. by tified ecl . 300 β n h agrtetpclsz ffrigsa lsesi h cons the in clusters star forming of size typical the larger the and ) M ⊙ aay tla otn iais eea trcutr:gener clusters: star – general binaries: – content stellar Galaxy: lses hl 0prcn falsnl tr eebr in born were stars single all of cent per 50 while clusters, IBF)rsmln aatcfil’ tla otn (Dynamical content stellar field’s galactic a resembling (IGBDFs) M 9 -32 on Germany Bonn, D-53121 69, ecl of 1 e ¨gl7,511Bn,Germany Bonn, 53121 H¨ugel 71, dem f , min r h 5 = ouain,Cre ta.(05 n o ihpoe mo- proper high for 28 find that (2005) stars tion al. et Carney populations, homogeneous a provides data DM91 by set. the performed it data survey i.e. since yrs) team, compared, (13 single long-term being a existing is only Galactic data the canonical which is the here- to as (1991, population G-dwarfs Mayor field & for Duquennoy results the DM91) use after will we work this ≈ 0 M nasetocpcsre fhl,tikadti disc thin and thick halo, of survey spectroscopic a In . 1 ⊙ − F=3 SFR= , A 0 T . E c h afms aiso nembedded an of radius half-mass the pc, 3 tl l v2.2) file style X ± e eto l ea-or([Fe/H] metal-poor all of cent per 3 M ⊙ nertdglci edbnr dis- binary field galactic integrated yr − 1 M and ecl , min β vlpdto (Marks, tool eveloped 0prcn tmfrom stem cent per MW 50 the in events ion sfrteMlyWay Milky the for Fs r n iaisfrom binaries and ars ucin(described function s dotta although that out ed ds:numerical h oe h star the lower the , hc r jus- are which 2 = niae htthe that indicates n netyobserved endently iaiswt in- with binaries h aatcfield galactic the aatcfil of field galactic dsa cluster star ed M sta today’s that ts predicted. ik Way’s Milky ecl ae nthe on lated tr o a for sters & -galaxies. l–open – al xueof ixture 10 idered 6 4 M − 1) ⊙ 2 Michael Marks and Pavel Kroupa centre-of-mass (cm-)systems1 are binaries. A similar number ple but powerfull argument that the vast majority of stars (26 ± 3 per cent) is found for the more metal-rich stars in must form in binaries: The lack of a significant single-star their sample ([Fe/H]> −1). These values compare well with population in dynamically not evolved star forming regions a binary-fraction of 22 per cent identified by DM91 over the means that stars cannot form in higher-order multiple sys- same period range (1.9

c 2010 RAS, MNRAS 000, 1–14 Dynamical population synthesis 3

ries (such as between period and eccentricity, Kroupa 1995b) where ξecl(Mecl) is the embedded cluster mass function as seen in observations. For a summary of these processes (ECMF). For simplicity it is assumed that all star clus- the reader is referred to Paper I. ters have formed with comparable half-mass radii, rh. The integration ranges from a minimum-cluster-mass, Mecl,min to some maximum-cluster-mass, Mecl,max ≡ Mecl,max(SFR). 2.1 Binary distributions in star clusters The maximum-cluster-mass that can form in a galaxy which has a given star formation rate (SFR) is determined by In order to construct the field population by adding up the (Weidner et al. 2004), single stars and binaries in individual clusters, the evolution 0.75 of binary populations has to be understood in terms of the Mecl,max(SFR) × SFR = 84793 −1 . (6) initial properties of their host cluster (Mecl,rh). M⊙  M⊙ yr  In Paper I an efficient method is provided to analytically Star clusters are distributed according to a power-law describe the first 5 Myr of the evolution of orbital-parameter ECMF with index β, BDFs in N-body computations of star clusters. The com- × −β putations start with 100 per cent binaries distributed ac- ξecl(Mecl)= kecl Mecl , (7) cording to an initially rising period BDF derived in Kroupa which is normalized such that the sum of the masses of all (1995b), which is consistent with constraints for pre-main clusters equals the total mass of the freshly formed star clus- sequence and Class I protostellar binary populations (see ter system, Paper I). Their method quantifies a stellar dynamical op- erator (Kroupa 2002, 2008), ΩMecl,rh (t), in dependence of Mecl,max(SFR) dyn M = M ξ (M ) dM . (8) ≡ 3 SCS ecl ecl ecl ecl the initial cluster mass density, ρecl 3Mecl/8πrh, where ZMecl,min Mecl is the total mass in stars that formed in the embed- The total mass, MSCS, needed to find the normalization con- ded cluster and rh is its initial half-mass radius. This op- erator transforms an initial (t = 0), perhaps invariant (see stant kecl is determined from the SFR and the formation time-scale, δt, of the star cluster system, Paper I), orbital-parameter BDF, Din, into an evolved one, DMecl,rh (t), after some time t of stimulated evolution, MSCS = SFR × δt . (9) DMecl,rh Mecl,rh × D (t) = Ωdyn (t) in . (1) Weidner et al. (2004) found that about every δt = 10 Myr of ongoing star formation an ECMF is fully populated. Mecl,rh In the formulation of Paper I Ωdyn acts in particular on The binary population will enter a galactic field with the initial BDF for binding-energies, Φ , i.e. log10 Eb,in characteristics set at time tfreeze. This is reached after suf-

Mecl,rh Mecl,rh ficient time for cluster internal stimulated evolution when Φ (t) = Ω (t) × Φlog E ,in , (2) log10 Eb dyn 10 b only hard binaries are left (Paper I), or if a cluster suddenly but extraction of other BDFs (period, semi-major axis, expands rapidly, e.g. as a result of residual-gas expulsion, mass-ratio, eccentricity) is also possible with their model, which inhibits further stimulated evolution. This state is al- given the interrelation between the orbital-parameters via ready reached after a few Myr or even in less than 1 Myr for Kepler’s laws (Sec. 2.4 below). dense configurations (Kroupa 1995a; Duchˆene et al. 1999; Let Ncms = Ns + Nb be the number of systems, i.e. Fregeau et al. 2009; Parker et al. 2009, Paper I). For our the sum of all single stars and binaries with primary- purpose, we choose the freeze-in time to coincide with the ≈ star mass near m1 in a stellar population. Then a BDF, time of the occurence of the first supernovae, tfreeze 3 Mecl,rh Myr. If the cluster is still embedded, supernovae are ex- Φx (m1) (x = log10 E,a,e,q,...), is defined as the dis- tribution of binary-fractions, fb = Nb/Ncms, as a function of pected to drive out the residual-gas of the embedded cluster x, rapidly, leading to cluster expansion and destruction of the majority of clusters in a star cluster system (≈90 per cent, Mecl,rh dfb(x) 1 dNb(x) e.g. Lada & Lada 2003). The exact time star clusters are al- Φx (m1)= = , (3) dx Ncms dx lowed to evolve their population is not that important since where dNb(x) is the number of binaries in the interval the time-scale on which binary dissolution occurs is short. [x,x + dx]. The total binary fraction equals the area below In particular, the difference in the binary fraction between the BDF, 3 and 5 Myr in the N-body models used in Paper I is of the ∞ order of a few per cent only. Mecl,rh fb(m1)= Φx (m1) dx . (4) Z−∞ 2.3 Normalization Since the stellar-dynamical operator (eq. 2) transforms be- 2.2 Integrated orbital-parameter distributions tween BDFs independently of the number of systems in a The galactic field’s binary population is the sum over the star cluster, one has to take care to preserve the definition populations in all star clusters that ever formed in a galaxy, and normalization for individual clusters (eqs. 3 and 4) also having evolved for at least a time-span tfreeze after which the for the integrated population. Consider therefore as an ex- binary orbital-parameter properties become frozen-in, ample two clusters consisting of Ncms = 10 and 100 systems, with fb = 40 and 60 per cent, respectively. Evaluating eq. (5) Mecl,max Drh Mecl,rh D for the two clusters would result in fb = 50 per cent for the GF = Ωdyn (tfreeze) in ξecl(Mecl) dMecl , (5) ZMecl,min combined (or integrated) population, but the true resulting

c 2010 RAS, MNRAS 000, 1–14 4 Michael Marks and Pavel Kroupa population has fb = 64/110 = 58 per cent. Thus, the num- Table 1. Adopted mass-ranges for single-stars or primary- ber of systems making up the population has to be taken components of binaries with different spectral-type (SpT). into account. Define a BDF for a galaxy analogous to eq. (3), SpT F G K M m∗/M⊙ 1.04-1.4 0.8-1.04 0.45-0.8 0.08-0.45 GF GF GF dfb (x) 1 dNb (x) Φx (m1)= = GF , (10) dx Ncms dx constructed by removing an appropriate amount of binaries GF GF where dNb (x) and Ncms are the number of binaries in the with a given binding-energy from the library according to interval of size dx and the number of cm-systems in a whole the number-ratio of binaries in the resulting and initial en- galaxy, respectively. The goal is thus to calculate the num- ergy IGBDFs. The remaining binaries are used to construct ber of binaries per x-interval and number of systems (sin- the P −,e−, q− and a−IGBDF. In order to extract sub- gle+binary) in a galaxy from the respective numbers in star distributions, such as for a special spectral-type or period- clusters separately. range, only those remaining binaries are used which fullfill For an initial cluster mass, Mecl, the total number of the additional criteria. In the forthcoming sections, mass- freshly hatched stars in that particular cluster is calculated ranges for single-stars or primary-component masses for from binary-stars, respectively, are adopted, as shown in Tab. 1. Mecl N∗(M )= , (11) ecl m where m ≈ 0.4M⊙ is the average mass of the canonical stel- 3 RESULTS lar IMF (Kroupa 2001). The total number of binaries is re- According to eq. (15), the properties of an IGBDF depend lated to N∗(M ) via ecl on the minimum embedded cluster mass and the SFR (i.e. Nb(Mecl)= N∗(Mecl) − Ncms(Mecl) . (12) the maximum cluster mass, eq. 6) of the considered galaxy, the index, β, of the ECMF (eq. 7), and the average half- Inserting this in fb = Nb/Ncms and rearranging gives the mass radius, rh, of the star clusters. The influence of these number of cm-systems in that particular cluster, parameters on the integrated binary properties by means of N∗(Mecl) the global binary fraction will be investigated. A discussion Ncms(Mecl)= . (13) 1+ fb of the Galactic field binary population of the Milky-Way

Mecl,rh (MW) and what can be learned about the initial star cluster Since Φx is known (eq. 2) and therefore fb (eq. 4), system of the MW from the observed binary population ends Ncms and Nb can be calculated. The number of binaries per this section. interval dx becomes, dNb(Mecl) Mecl,rh = Ncms(Mecl) × Φ . (14) 3.1 The importance of low-mass clusters for the dx x Galactic field binary population Therefore, eq. (10) becomes, From which type of cluster do most Galactic field binaries Mecl,max(SFR) dNb(Mecl) ξecl(Mecl) dMecl GF Mecl,min dx originate? Dissolving low-mass embedded clusters will each Φx (m1)= .(15) RMecl,max(SFR) contribute only a small number of systems to a Galactic field Ncms(Mecl) ξecl(Mecl) dMecl Mecl,min population but a large fraction of binaries will be among R Eq. (15) is referred to as the integrated galactic-field binary them due to inefficient stimulated evolution (Paper I). For distribution function (IGBDF). It can be calculated for sin- high-mass clusters the situation is the other way round. gle stars and binaries with primary-mass in an interval ∆m However, the ECMF (eq. 7) is steep, i.e. there are many around m1, e.g. 0.8 − 1.04M⊙, or for all late-type stellar more low-mass than high-mass clusters. Thus, this question is not simple to answer qualitatively. systems m1/M⊙ ∈ [0.08, 2] (Tab. 1). Therefore the number of systems, binaries and single stars which are added to the Galactic field by all clusters of 2.4 P-, e-, q- and a-IGBDF a given mass Mecl is calculated by evaluating GF In order to extract IGBDFs for different orbital-parameters dNcms/b/s = Ncms/b/s(Mecl) ξecl(Mecl) . (16) (period, P , eccentricity, e, mass-ratio, q, and semi-major dMecl axis, a) from the known energy IGBDF, the procedure is The result is depicted in the left panel of Fig. 1 for the MW basically as described in sec. 4.2 of Paper I for orbital- star cluster system with β = 2.0, rh = 0.2 pc, Mecl,min = parameter distributions in single clusters. The idea is to −1 5M⊙ and SFR=3M⊙ yr (see Sec. 3.3 below). It shows compile a large library of binaries (N = 107 binaries for lib that the steepness of the ECMF dominates such that most the present purpose) whose properties are selected accord- systems in the Galactic field stem from low-mass clusters. ing to the recipe in Kroupa (1995b, see Paper I). Here, the For the same star cluster system parameters, the right panel library consists only of binaries with primary masses up to of Fig. 1 plots the normalized cumulative number, 2M⊙ in order to resemble a galactic field and it contains Mecl GF the values for m1,m2,Eb,P,e,q and a. The total number of 1 dN N GF,cum(M )= cms/b/s dM (17) initial binaries in the integrated population, calculated ac- cms/b/s ecl GF ecl Ncms/b/s ZMecl,min dMecl cording to Sec. 2.3, is scaled so as to match the library size GF of Nlib binaries. Following this, the final distributions are where Ncms/b/s is the total number of systems, binaries or

c 2010 RAS, MNRAS 000, 1–14 Dynamical population synthesis 5

1010 systems 1 systems single single binary binary 109 0.8

108 ) ecl

ecl 0.6 (M 107 / dM GF,cum cms/b/s

GF cms/b/s 0.4

6 N dN 10

0.2 105

104 0 100 101 102 103 104 105 106 100 101 102 103 104 105 106 cluster mass Mecl / Msun cluster mass Mecl / Msun

Figure 1. Left panel: Number of all singles (dotted), binaries (dash-dotted) and all systems (single+binary, solid) per cluster mass in the Galactic field (eq. 16), respectively, that form within one star cluster system formation time-scale δt (eq. 9). Lines are drawn for the −1 MW star cluster system (β = 2.0, rh = 0.2 pc, Mecl,min = 5M⊙ and SFR=3M⊙ yr ). Low-mass clusters contribute most systems to the Galactic field population owing to the steepness of the ECMF. The intersection between the single-stars and binary-stars line is the cluster mass at which fb = 50 per cent for the used parameters. Right panel: Cumulative number of singles, binaries and all systems as a function of Mecl normalized to the respective total number (eq. 17) for the same star cluster system parameters as in the left panel. 4 Low-mass clusters (Mecl . 300M⊙) are the dominant binary contributors while high-mass clusters (Mecl & 10 M⊙) donate most single stars. singles in the Galactic field, respectively, i.e. eq. (16) in- fore on average populations with larger binary-fractions en- tegrated from Mecl,min to Mecl,max. It demonstrates that ter the field and fb increases. about 50 per cent of all systems in the Galactic field formed ... the larger the minimum cluster mass, the fewer bina- 3 in clusters with initial masses Mecl . 2 × 10 M⊙, while ries exist in the field. Increasing Mecl,min means cutting out 50 per cent of all Galactic field binaries originate from clus- of low-density clusters which would contribute populations ters with Mecl . 300M⊙ only. For single stars the situation with large binary proportions. The remaining higher-mass is inverted. Roughly 50 per cent of all single stars come from clusters shed dynamically more evolved populations into a 4 star clusters with masses Mecl & 10 M⊙. galactic field. Fig. 2 distincly shows that the parameter space is degen- erate. E.g., the increase in fb due to a steeper ECMF can 3.2 Parameter study & degeneracy be compensated for by introducing smaller cluster sizes (left panel) or assuming a larger Mecl,min (right panel). Similarly, Fig. 2 depicts the global binary fraction (color-coded), i.e. a larger typical rh can be counteracted with a higher SFR including all late-type binaries (m1 6 2M⊙), of a galax- (middle panel), and so on. Therefore constraints on at least ies’ stellar population in dependence of the four parameters some of the four parameters from independent sources are (Mecl,min, SFR,β,rh) in eq. (15). We learn the following from needed to reduce the allowed solution space, when compar- these diagrams: If all other parameters are fixed... ing model predictions with observed distributions. Those are available for the MW. ... the higher the SFR, the smaller the field-binary popu- lation. A higher SFR yields a larger Mecl (eq. 6). A higher- mass cluster is generally denser than a lower-mass cluster 3.3 The Milky-Way and will therefore contribute a population with a lower bi- nary fraction (Paper I). Therefore fb drops with increasing In order to calculate a synthetic stellar population for the SFR. MW’s Galactic field, we need to estimate the parameters ... the larger rh, the larger the galactic field binary frac- entering the IGBDF for our Galaxy. tion. Larger typical half-mass radii imply lower densities, The current global SFR of the MW using different −1 resulting in dynamically less evolved binary populations be- methods is determined to lie between ≈ 1 and 5M⊙ yr fore dissolution (higher fb, Paper I). Therefore fb increases (Smith et al. 1978; Diehl et al. 2006; Misiriotis et al. with increasing rh. 2006; Calzetti et al. 2009; Murray & Rahman 2010; ... the steeper the ECMF (larger index β), the larger the Robitaille & Whitney 2010). These values are compatible field-binary population. A steeper ECMF amounts to a lesser with the assumption that the total stellar mass in the disk 10 proportion of high-mass, i.e. higher-density clusters. There- and bulge of the MW (5 × 10 M⊙, Binney & Tremaine

c 2010 RAS, MNRAS 000, 1–14 6 Michael Marks and Pavel Kroupa

-1 SFR=1 M yr , M =5M fb [%] -1 sun ecl,min sun ECMF slope β=2, M =5 M fb [%] r =0.2 pc, SFR=1 M yr fb [%] 1 100 ecl,min sun 4 h sun 106 100 10 100 0.9 70% 80% 90% 30% 40% 30% 4 50% / pc 0.8 80 10 80 80 h -1 3 0.7 60% 10sun

yr 2 60% 60 10 60 0.6 60

/ M 40%

sun MW 70% 0.5 0 10 2 40 80% 40 10 40 0.4 50% ecl,min 50% -2

10 M 0.3 20 SFR / M 90% 20 20 40% MW 60% 0.2 1 10-4 10 half-mass radius r MW 0.1 0 0 0 1.8 1.9 2 2.1 2.2 2.3 2.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.8 1.9 2 2.1 2.2 2.3 2.4 β β ECMF slope half-mass radius rh / pc ECMF slope

Figure 2. Study of the influence of the parameters in eq. (15) determining the integrated stellar population. In each panel the color- coding shows the global binary-fraction, fb, i.e. including all late-type binaries (m1 6 2M⊙), according to the bar on the right of each panel (in per cent). The solid overlaid lines indicate curves of constant binary fraction. The big solid white dot in each panel marks the position of the MW (Sec. 3.3). The binary population increases with decreasing SFR, with increasing cluster half-mass radii, rh, with steepening of the ECMF (increasing index β) and with decreasing minimum cluster mass, Mecl,min.

2008) has assembled continously during the last 13.7 Gyr 0.3 −1 −1 initial with a SFR of 3.6M⊙ yr . Assuming SFR=3M⊙ yr rh / pc = 0.8 × 5 r / pc = 0.6 yields Mecl,max = 1.9 10 M⊙ (eq. 6) for the most rh / pc = 0.4 massive cluster having formed in the MW disk, compa- 0.25 rh / pc = 0.2 rh / pc = 0.1 rable to the most massive open cluster in Piskunov et al. h b

(2007)’s sample of 236 open clusters within 1 kpc from E

5 10 0.2 the Sun (1.1 × 10 M⊙ for Sco OB5). Taking the min- imum cluster mass similar to that of a Taurus-Auriga log ∆ like group (Kroupa & Bouvier 2003, Mecl,min = 5M⊙) 0.15 ) / and an ECMF index β = 2 similar to the observed b slopes for Galactic embedded clusters (Lada & Lada E 10 2003; de la Fuente Marcos & de la Fuente Marcos 2004; 0.1 (log

Gieles et al. 2006), we can evaluate the properties of the b f Galactic field binary population by numerically integrating eq. (15) for different embedded cluster half-mass radii. 0.05

3.3.1 Energy distribution 0 -4 -2 0 2 4 6 2 -2 Fig. 3 shows the energy IGBDF for the above parameters, log10Eb / Msun km s for different cluster half-mass radii and for primaries of all masses. It is found that the binary-fraction in the Galactic Figure 3. Global energy IGBDFs for different typical initial clus- field (the area under the distributions) is smaller if the typ- ter half-mass radii in the MW. The smaller the clusters are, the ical cluster radii are smaller. Smaller average radii lead to larger are their densities and the more efficient is the binary higher initial densities in star clusters which results in more destruction initially (Sec. 3.2). Therefore the binary fraction is efficient binary dissolution and thus a lower binary-fraction smaller the lower the initial value of rh is, ranging from fb = 0.34 (Sec. 3.2). For rh = 0.1 pc the overall binary-fraction is for rh = 0.1 pc to fb = 0.71 for rh = 0.8 pc. The area under each fb = 0.34 and for rh = 0.8 pc, fb = 0.71. distribution equals the total model binary-fraction in the Galactic field.

3.3.2 Period distribution size of rh = 0.1 pc, where fb(G) = 0.58. A similar study by For the adopted initial conditions in star clusters (Paper I) Raghavan et al. (2010) investigating the multiplicity prop- and the above values for the SFR, β and Mecl,min in the erties of 454 solar-type stars selected from the Hipparcos MW, the period IGBDFs in Fig. 4 for different values of rh catalogue is in agreement with the DM91 data and thus demonstrate that the initially rising period distribution in with the compact formation of star clusters. star clusters translates into a bell-shaped form in the Galac- Fischer & Marcy (1992) studied the M-dwarf binary tic field, at least if the star clusters from which the Galactic population within 20 pc from the Sun, showing that the pe- field population originates, were rather compact. Compari- riod distribution of M-dwarfs can also be described by a log- son with the corrected observed Galactic field period BDF normal distribution. Comparison with the IGBDF models for G-dwarfs (DM91) suggests that clusters formed quite (Fig. 4, right panel) favors their formation in slightly more compact (left panel). The G-dwarf field binary fraction of extended clusters. For rh = 0.3 pc the binary-frequency ≈ 57 per cent is best reproduced with a typical initial cluster of 41 per cent compares well with the observational result

c 2010 RAS, MNRAS 000, 1–14 Dynamical population synthesis 7

0.25 Duquennoy & Mayor (1991, corrected) Fischer & Marcy (1992) 0.2 Raghavan et al. (2010) Bergfors et al. (2010)

initial 0.2

P (Kroupa 1995b) P

10 0.15 10 initial (Kroupa 1995b)

log log 0.15 ∆ rh=0.8pc ∆ rh=0.6pc 0.1 r =0.4pc P) / h P) / 0.1 10 10

rh=0.8pc (log (log

b b r =0.6pc f 0.05 f h 0.05 rh=0.4pc

rh=0.2pc rh=0.2pc rh=0.1pc rh=0.1pc 0 0 0 2 4 6 8 10 0 2 4 6 8 10 log10(P / d) log10(P / d)

Figure 4. Period IGBDFs for the MW model (Sec 3.3) with different typical initial cluster radii for G-dwarf (left) and M-dwarf (right) binaries (histograms). Comparison with the G-dwarf period BDF by DM91 (circles) and Raghavan et al. (2010, triangles) suggests that clusters should have formed very compact. The period IGBDF with a half-mass radius of rh = 0.1 pc matches the observed binary- frequency best. The observed M-dwarf binary-frequency (Fischer & Marcy (1992, robust) and Bergfors et al. (2010)) favours formation in rh = 0.3 pc sized clusters. This difference in rh between the G-dwarf and M-dwarf solution is discussed in Sec. 4. The line-types are as in Fig. 3.

(fb = 0.42). Recently, Bergfors et al. (2010) surveyed 124 M- mation models, as above for the period IGBDFs. Addition type stars for binary separations a . 200AU for their multi- of the mass ratio IGBDFs for the different spectral types plicity properties. The observed semi-major axis distribution (Tab. 1 and Fig. 8 below) results in the flat distribution for has been translated into periods using Kepler’s laws and are the complete mass ratio IGBDF (left panel in Fig. 5). The incorporated in Fig. 4 (right panel). Their data are com- binary fraction of 35 per cent in the observations of 106 G- to patible with the Fischer & Marcy (1992) data. For a sub- M-type systems in the analysis by Reid & Gizis (1997) are sample of the model binary population with a . 200 AU, as best compatible with the rh = 0.1 pc model where fb = 0.34. in the observations, the rh = 0.2 pc-model binary-frequency In contrast to the declining q-distribution for G-dwarfs is fb = 0.31 and agrees with the observed binary-fraction found by DM91, note that the results of Raghavan et al. of ≈ 32 per cent. Bergfors et al. (2010) note, however, (2010) suggest a mass-ratio distribution for binaries with a that there might be some overabundance of systems with solar-type primary which is flat between q ≈ 0.2 and 0.9, P < 20 d, corresponding to a . 80 AU for an average system while finding a similar period distribution (see discussion in mass of ≈ 0.5M⊙, due to the sample selection and that cor- Sec. 4). Fischer & Marcy (1992) and Bergfors et al. (2010) rections might be necessary for non-physical (optical) pairs. extracted also a mass ratio distribution for their respective samples of M-dwarfs whose rising shapes are well reproduced by the models (right panel). As expected from the period IGBDFs (Fig. 4), the same r = 0.3 and 0.2 pc models, re- 3.3.3 Mass-ratio distribution h spectively, are consistent with the observations, being again The result that star formation in rather compact structures larger than for G-dwarfs. is favoured is independently confirmed by considering the distribution of mass-ratios. Fig. 5 depicts the mass ratio IGBDFs for the complete field population (left, IGBDF for all binaries with primary masses m1 6 2M⊙), for G-dwarf binaries only (middle) and for M-dwarfs (right). It is appar- 3.3.4 Eccentricity distribution ent that the complete distribution is flat between roughly q = 0.2 and 0.9 while the G-type binary sub-distribution is The eccentricity IGBDF for G-dwarf binaries is in agree- decreasing and the M-dwarf IGBDF is increasing with in- ment with the observational data (Fig. 6). The distribu- creasing q. We emphasize, that all three mass ratio IGBDFs tion of eccentricities is bell-shaped for log10 Pcirc = 1.06 < result from initially sampling the two birth components of a log10 P < 3, where Pcirc is the circularization period identi- binary randomly from the same underlying stellar IMF (e.g. fied by DM91 below which the orbits are circular (e ≈ 0). Kroupa 1995a). The decreasing trend is also evident for F- Circularisation occurs through pre-main sequence eigenevo- and K-type primaries (Fig. 8 below) and is seen in the obser- lution (Kroupa 1995b, Paper I). The eccentricity IGBDF vations of G-dwarfs by DM91 (middle panel of Fig. 5). The follows the thermal distribution for log10 P > 3 because it observational data compare very well with the compact for- is invariant to stimulated evolution.

c 2010 RAS, MNRAS 000, 1–14 8 Michael Marks and Pavel Kroupa

3 2.5 1.6 Reid & Gizis (1997) initial Duquennoy & Mayor (1991, corrected) Fischer & Marcy (1992) Bergfors et al. (2010) initial 1.4 2.5 initial 2 0.8 pc 1.2 0.6 pc 0.8 pc 0.4 pc 2 0.6 pc

q 1 q 0.4 pc q 1.5 ∆ 0.2 pc ∆ ∆ 0.8 pc 0.6 pc 1.5 0.8 0.1 pc 0.2 pc 0.4 pc (q) / (q) / (q) / b b b

f f f 1 0.6 0.2 pc 1 0.1 pc 0.1 pc 0.4 0.5 0.5 0.2

0 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 mass-ratio q=m2/m1 mass-ratio q=m2/m1 mass-ratio q=m2/m1

Figure 5. Mass-ratio IGBDFs for all late-type binaries (m1 6 2M⊙, left panel), G-dwarf (middle panel) and M-dwarf binaries (right panel), for the MW model (Sec. 3.3) with different half-mass radii (histograms). The line-types are as in Fig. 3. While the distribution between q = 0.2 and 0.9 is flat for the complete distribution it declines for the G-dwarfs and increases for M-type binaries with increasing q. The peak at mass ratio’s in the model close to unity (an effect of pre-main sequence eigenevolution) is also evident in the observational data for all primary-masses combined (Reid & Gizis 1997, left) and M-dwarf binaries (Fischer & Marcy 1992; Bergfors et al. 2010, right), while it is less pronounced, if at all, for the G-dwarf observations by DM91 (middle). Note that agreement is obtained for the similar rh as for the period IGBDFs (Fig. 4).

4.5 2.5 Duquennoy & Mayor (1991) Duquennoy & Mayor (1991, corrected) P =3 4 circ 10 10 e e

∆ 3.5 ∆ 2

3 1.5 2.5

2 1 1.5

1 0.5 fraction of all binaries / fraction of all binaries / 0.5

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 eccentricity e eccentricity e

Figure 6. Eccentricity IGBDFs for G-dwarf binaries in comparison with the observations by DM91 in the indicated period ranges. Pcirc = 11.6 d is the circularization period derived from the observational data. Note that in both panels the data is (exceptionally) normalized to the total number of binaries instead of systems, in order to be able to compare to the observations. This is also why IGBDF models with different rh (histograms, different line-types) can hardly be distinguished. Left panel: The eccentricity BDF is bell-shaped for orbital periods below 103 d due to pre-main sequence eigenevolution as in the observational data. Right panel: For P > 103 d the e-distribution follows the thermal distribution (fb(e) = 2e, solid line) for the IGBDF model as well as in the observations. The thermal eccentricity distribution is invariant of stimulated evolution.

3.3.5 Single and binary population in dependence of nary components at birth and after eigenevolution (Sec. 2) spectral type about ≈ 90 per cent of all initial binaries carry a com- panion of spectral type M. Thus, everytime a binary dis- The fraction of binaries in the field is a function of the spec- solves, in 9 out of 10 cases at least one M-dwarf will end tral type (i.e. mass) of the primary. The later the primary as a single star (two if a M-dwarf binary dissolves). Each spectral type the lower is the resulting binary fraction in M-dwarf contributes to Ncms(M), and therefore fb(M) = the field (Fig. 7). Especially M-dwarfs have a lower binary- Nb(M)/Ncms(M) shrinks. fraction than binaries with a F-, G- or K-type primary. The reason for this is twofold. Secondly, binaries with a lower binding-energy, Eb, are First, due to the shape of the stellar IMF M-type stars more prone to dissolution (Paper I) and the binding-energy are most numerous so that upon random pairing of the bi- is proportional to the mass of the primary, Eb ∝ m1. Since,

c 2010 RAS, MNRAS 000, 1–14 Dynamical population synthesis 9

to 55 − 66 per cent and is in excellent agreement with the 1 estimate by Lada (2006), that about 2/3 of all primary stars β=2.4 are single. β=2.0 0.8 (ST)

b 4 DISCUSSION & MODEL PREDICTIONS 0.6 4.1 Formation in compact star clusters Comparison of the observational data with the model has 0.4 suggested that MW star clusters typically formed quite com- pact (rh ≈ 0.1 − 0.3 pc, Sec. 3.3). Such small radii compare DM91 with the observational lower end of the sizes of dense cores Raghavan et al. binary fraction f in giant molecular clouds. However, the sizes of these dense 0.2 Mayor et al. Fischer & Marcy cores range up to 2 pc and the spatial extends of embedded Bergfors et al. Delfosse et al. clusters are typically comparable (e.g. Lada & Lada 2003). Kroupa et al. But there is evidence that the forming stars within the em- 0 F G K M bedded cluster start dynamically cold (Walsh et al. 2004; Spectral Type Peretto et al. 2006; Lada et al. 2008) and that protostellar objects are more confined than more evolved young stellar objects (Teixeira et al. 2006; Muench et al. 2007). Thus, an Figure 7. Comparison of the binary fractions among systems embedded cluster will collapse to a smaller configuration. of one spectral type (SpT) for IGBDF models with different rh (filled and open circles for an ECMF slope β = 2 and 2.4, re- In that sense the here derived half-mass radii may be inter- spectively, in each column from top to bottom: 0.8, 0.6, 0.4, 0.2 preted as the sizes of clusters when they reach their peak and 0.1 pc). The initial binary fraction for all spectral types is stellar density and stimulated evolution is most efficient. fb = 1 (dotted horizontal line). M-dwarf binaries have the low- Additionaly the IGBDF model implicitely assumed that est binary-fraction when they enter the Galactic field. The flatter a typical half-mass radius for all clusters exists (Sec. 2.2). the ECMF (smaller β), the lower is the binary-fraction (Sec. 3.2). If a mass-radius relation for embedded clusters better de- Observed binary fractions agree with the β = 2, rh = 0.1 − 0.2 pc scribes reality (e.g. Harris & Pudritz 1994, for virialised gas IGBDF model best (data taken from DM91; Mayor et al. 1992; cores in giant molecular clouds), the typical rh for MW clus- Fischer & Marcy 1992; Kroupa et al. 1993; Delfosse et al. 2004; ters can be seen as an average value for all clusters of any Raghavan et al. 2010; Bergfors et al. 2010). mass and size. Note that if such a mass-radius relation ex- ists, upon averaging the inferred best rh will be closer to the true value for low-mass clusters, which are most im- by construction, all binaries intially follow the same initial portant for the Galactic field binary population (Sec. 3.1). period distribution (Kroupa 1995b, Fig. 4), the M-dwarf Higher mass clusters will then occupy a somewhat larger ra- (low m1) energy IGBDF is shifted to slightly lower energies dius range. However, a trend of radius with luminosity (or compared to F-, G- and K-type binaries. Thus, it is gener- mass) in young star clusters has been shown to be shallow ally easier to dissolve M-dwarf binaries than binaries with a for the galaxy merger NGC 3256 (Zepf et al. 1999), for stel- primary of an earlier spectral type. Furthermore, low-mass lar clusters in 18 spiral galaxies (Larsen 2004) and for young binaries are more frequent, again owing to the shape of the clusters in M 51 (Scheepmaker et al. 2007). stellar IMF. In the model, ≈ 57 per cent of all binaries have a The typical rh is arrived at assuming β = 2 down to M-dwarf primary initially. Dynamical encounters including 5M⊙. The ECMF might however flatten (β → 0) below M-dwarfs will therefore occur often in the clusters. In turn, ≈ 50 − 100M⊙ (Lada & Lada 2003), implying that fewer each disruption of a M-dwarf binary will reduce Nb(M) by low-mass clusters are present compared to the numbers used one, increase Ncms(M) by one and, in turn, reduce fb(M). in the model. According to a computation with a broken All observational data in Fig. 7 again compare best with power-law ECMF, where β = 0 for Mecl 6 100M⊙ and star formation in compact star clusters (rh = 0.1−0.2 pc and β = 2 otherwise was chosen, typical half-mass radii nedd to β = 2). Although all stars are locked up in binaries initially, be larger by ≈ 0.1 pc in order to agree with the observations. more than half of all systems in the Galactic field end up A small difference in the solutions for rh (0.1 vs. 0.3 pc) as single stars due to stimulated evolution in star clusters in the solutions for G- and M-dwarf binaries might be evi- before they dissolve (Fig. 2). M-dwarfs in the IGBDF model dent (Fig. 4). Since the estimated cluster size is also a mea- constitute about ≈ 80 per cent of the single star population sure of how dense the region is in which the respective sub- in the Galactic field. Only ≈ 13, 3 and 2 per cent of all single population has formed, the possible difference might simply stars in the IGBDF model have spectral type K, G and F, indicate their formation in different locations of the same respectively (the remaining 2 per cent are of spectral type cluster. A primordial mass-segregated cluster, where the G- A, for a Galactic field population which consists of systems dwarfs would be more centrally confined to a region of higher with m1 6 2 M⊙, Sec. 2.4). density while M-dwarfs form out to larger radii, would nat- For rh ≈ 0.1 − 0.2 pc and β = 2 the single star fraction, urally account for the apparently different ranges of allowed fs(SpT) = 1 − fb(SpT), in the model becomes fs(M) ≈ cluster radii. 0.75 − 0.85, fs(K) ≈ 0.36 − 0.48, fs(G) ≈ 0.31 − 0.42 and The uncertainty in the inferred typical cluster size of fs(F) ≈ 0.28 − 0.39. The total single star fraction amounts 0.2 pc does not appear to be very large, given the estimates

c 2010 RAS, MNRAS 000, 1–14 10 Michael Marks and Pavel Kroupa

for rh by comparison of the model with independent observa- dictions. The combined IGBDFs lie typically between the tions (Sec. 3.3). Even if a possible error in the ECMF index, M-dwarf distribution and the earlier types. β, of up to 0.5 is considered (Larsen 2009) the uncertainty Note that it is not possible to show a common initial in rh is of order 0.1 − 0.3 pc only (Fig. 2, left panel). mass ratio distribution since it depends, in contrast to the To calculate the composition of the Galactic field stellar other IGBDFs, on the considered spectral type (see also population a (constant, average) global SFR was adopted. Kouwenhoven et al. 2009). This is a result of the random A declining SFR history might instead be better suitable to selection of birth binary component masses and the upper describe the evolution of the MW disc (Boissier & Prantzos limit for the mass of the primary star when a spectral type- 1999; Naab & Ostriker 2006; Sch¨onrich & Binney 2009). If limited sample is investigated2. While at least the shape early on the SFR has been higher than average, clusters of the initial mass ratio distributions is similar for F-, G- more massive than allowed for the adopted SFR (eq. 6) and K-binaries it is completeley different for M-dwarfs (com- would have been able to form, which would have contributed pare, e.g., the middle with the right panel in Fig. 5 for the a larger number of single stars. Later, when the SFR sank initial distributions of G- and M-dwarfs, respectively). The below the average SFR the highest mass clusters wouldn’t IGBDFs decline with increasing mass ratio for F-, G- and be able to form any more and more binaries would en- K-type binaries, while the mass ratio IGBDF for M-dwarfs ter the field originating in the low-mass clusters. These ef- is increasing with increasing q. We explicitly note that the fects might eventually compensate each other, but will de- same observational data for M-dwarfs, for which typically a pend on the actual history, which can’t be tested without flat q-distribution is inferred, is consistent with the increas- a modification of the used code. Also, the global history ing trend in the models. The mass ratio distribution is flat might be non-representative for the solar neighbourhood only if all primary-masses are combined to construct a mass (Boissier & Prantzos 1999). However, even if the SFR has ratio IGBDF. been much higher in the past only and settled to the present As opposed to DM91, Raghavan et al. (2010) find a value, the inferred typical rh wouldn’t change strongly. Re- mass ratio distribution for binaries with a solar-type primary ducing or enhancing the SFR by up to two orders of mag- that is flat between q ≈ 0.2 and 0.9, while finding a simi- nitude and at the same time retaining the observed binary- lar period distribution as DM91. This can not be expected fraction requires, respectively, a rh smaller by ≈ 0.1 pc or within the framework of the IGBDF model. Since the range larger by ≈ 0.2 − 0.3 pc only (as evident from Fig. 2, middle of considered primary masses in Raghavan et al. (2010)’s panel). and DM91’s study are comparable, their mass ratio distribu- In this sense, the typical cluster size seems to be well tions should be in agreement if the period distributions are, constrained by the IGBDF model. and vice versa. A flat distribution from the models is only obtained if the complete late-type binary population is con- sidered (Fig. 5, left panel, and Fig. 8). Although this appar- ent inconsistency might be due to model assumptions (see 4.2 MW orbital-parameter BDFs Sec. 4.4), the presented results are otherwise very successfull Since the IGBDF models for the MW (Sec. 3.3) with typical in describing independent observational data for the MW, star cluster sizes of rh ≈ 0.2 pc agree well with independent adjusting only one free parameter (the typical half-mass ra- observational data, these models are used to predict the pe- dius rh, Sec. 3.3). It is noted that DM91 monitored radial riod, semi-major axis, mass-ratio and eccentricity IGBDFs velocities for their sample over a period of 13 years find- for the solar neighbourhood. ing accurate orbital solutions, while Raghavan et al. (2010) Fig. 8 depicts the resulting distributions for binaries compile data from various sources covering many tech- of different spectral type and the combined distributions niques. However, in order to understand these issues better (m1 6 2M⊙; no additional cuts, e.g. in period, are applied). it will be necessary to study the consistency between the For the period, mass-ratio and eccentricity IGBDFs the dis- Raghavan et al. (2010) and the Reid & Gizis (1997) data - tributions for F-, G-, and K-type binaries are very similar how can both, the (essentially) G-dwarf and the all-primary and therefore probably hard to distinguish by observations. combined mass ratio distributions be flat at the same time? However, the mass-ratio IGBDF might hold the ability to test the model prediction for different spectral types at low (q ≈ 0.1 − 0.2) and high (q ≈ 0.9 − 1) mass-ratios since dif- 4.3 Dependence on galaxy morphology ferences are more pronounced there. In particular, there is a larger gap between the G- and K-binaries around q ≈ 0.15 The binary properties of galaxies within the IGBDF model and between the F- and later type binaries at q ≈ 0.05 which are strongly dependent on the SFR (Sec. 3.2). Since SFRs might be visible in observations, too. in galaxies are observed to cover a large range from ≈ −1 −1 The M-dwarf binary IGBDFs for all quantities are dis- 0 M⊙ yr for giant ellipticals to & 1000 M⊙ yr for ultra tinct from the corresponding distributions for earlier spec- luminous infrared galaxies (ULIRGS, Grebel 2011), binary- tral types since breaking-up of binaries having at least one frequencies and binary-properties are expected to vary be- M-type component happens frequently before their birth tween galaxies of different morphology. This might have cos- clusters dissolve (Sec. 3.3.5). Thus, by the time the M-dwarfs mological implications, such as for the SN type Ia rates in emerge from the clusters their majority are single stars con- galaxies. sistent with observations (Lada 2006, Sec. 3.3.5), despite be- ing born as binaries. Differences in observed distributions for M-dwarf binary populations and those of later types should be apparent and are thus suited to test the IGBDF pre- 2 By definition the secondary mass can only be lower.

c 2010 RAS, MNRAS 000, 1–14 Dynamical population synthesis 11

0.2 0.3 initial initial F F G G K 0.25 K M M all 0.15 all a P

10 0.2 10 log log ∆ ∆ 0.1 0.15 P) / a) / 10 10 (log (log 0.1 b b f f 0.05

0.05

0 0 -2 0 2 4 6 8 10 -2 -1 0 1 2 3 4 period log10(P / d) semi-major axis log10(a / AU)

1.8 1.6 F initial G F 1.6 K G M 1.4 K all M 1.4 all 1.2 1.2 1 q e

∆ 1 eigenevolution peaks ∆ 0.8

(q) / 0.8 (e) / b f b f 0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 mass-ratio q=m2/m1 eccentricity e

Figure 8. Predictions for the period (log10 P ), semi-major-axis (a), mass-ratio (q) and eccentricity (e) IGBDFs for different spectral types and the combined distributions in the solar neighbourhood for the MW models (Sec. 3.3) which best fit the observational data (rh ≈ 0.2 pc). The log10 P -, a- and e-IGBDFs for F-, G- and K-binaries appear very similar, but can probably be distinguished at low and high q-values in the mass-ratio IGBDF. The peaks seen for q = 0.9 − 1 in the mass ratio distributions are due to pre-main sequence eigenevolution (Sec. 2). All IGBDFs for M-dwarfs and the all-primary mass IGBDFs are distinct from those for F- to K-binaries. Note that a common initial distribution for the q-IGBDFs can not be given since they are different for different spectral types (see discussion in Sec. 4).

4.3.1 Elliptical galaxies inferred. It is noted that a non-shallow mass-radius relation (Sec. 4.1), which is not considered in the present models, Ellipticals (Es) are nowadays more or less free of cold gas and might affect the results for such high SFR. If, additionally, are pressure- or random-stellar-motion-supported with low during star bursts low-mass clusters are not able to form, or no star formation activity. Even the suspected precursors i.e. Mecl,min is larger, this would further lower the binary- of giant ellipticals, quasars at high redshift (z ≈ 6, tuniverse . fraction in Es (Fig. 2, right panel). Thus, if E galaxies formed 1 Gyr), reveal supersolar indicating a starburst rapidly they ought to have low binary fractions. that quickly enriched the material with metals (Fan et al. 2001). This suggests that Es had a large SFR initially until their gas reservoir was depleted. One of the highest-redshift −1 4.3.2 Spiral galaxies quasars known has a SFR of ≈ 1000M⊙ yr as derived 3 −1 for ULIRGS (Fan 2006). For SFR= 10 M⊙ yr , from the In terms of the SFR, spiral galaxies are intermediate objects middle panel of Fig. 2, a binary fraction of the order ≈ between Es and dwarf galaxies. SFRs in spirals like the MW −1 30−40 per cent for a typical cluster size of rh = 0.2 pc can be lie between 0.1 and 10M⊙ yr (e.g. Lee et al. 2009, 2011).

c 2010 RAS, MNRAS 000, 1–14 12 Michael Marks and Pavel Kroupa

initial 1.6 initial 0.2 -4 -1 -4 -1 SFR=10 Msun yr (dIrr) SFR=10 Msun yr (dIrr) -1 1.4 -1 SFR=1 Msun yr (S) SFR=1 Msun yr (S) 4 -1 4 -1 SFR=10 Msun yr (E) SFR=10 Msun yr (E) 1.2

P 0.15 10 1 log q ∆ ∆

P)/ 0.8 0.1 (q)/ b 10 f

0.6 (log b f

0.05 0.4

0.2

0 0 -2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 period log (P / d) 10 mass-ratio q=m2/m1

Figure 9. Predictions for the period (log10 P ) and mass-ratio (q) IGBDFs in dependence of galaxy morphology. Model parameters are as those for the MW (Sec. 3.3, rh = 0.2 pc) except that SFRs are used according to the galaxy type (Sec. 4.3). Elliptical (E) galaxies have the lowest binary-fraction (the area below the distribution) and the period IGBDF peaks at shorter periods than for spirals (S) and dwarf irregulars (dIrr). The shapes of the mass ratio IGBDFs for the different galaxies resemble each other. The pre-main sequence eigenevolution peak in the mass ratio IGBDF is visible, as in Fig. 8.

−1 For the same cluster size and a SFR of 1M⊙ yr , from 4.4 Model Limitations Fig. 2 a global binary frequency of ≈ 40 − 50 per cent is The validity of the results and predictions outlined in this expected, similar to what is seen in the solar neighbourhood analysis depend on the accuracy of the assumptions entering (Sec. 3.3). the IGBDF model. While the model formulated in Sec. 2.2 appears robust, the results obtained by performing the in- tegration in eq. (15) depend on the properties of the birth and initial binary population (see Sec. 2, Kroupa 1995a,b, 4.3.3 Dwarf irregular galaxies Paper I) and the resulting analytical description of the evo- lution of binary properties in the N-body models of Paper Dwarf irregular (dIrr) galaxies have very low SFRs rang- I, which use this initial binary population. The two ma- −5 −1 ing down to 10 M⊙ yr (Lee et al. 2009, 2011). For jor assumptions are that (i) the birth binary population is rh ≈ 0.2 pc dIrrs would be expected to exhibit a sig- formed via random-pairing of the two binary components nificantly larger binary fraction, which is of the order of from the canonical stellar IMF and that (ii) the birth pop- 70 − 80 per cent. ulation evolves into the initial population via pre-main se- quence eigenevolution, which was specifically parameterised for G-dwarf binaries (Kroupa 1995b). If, e.g., instead of the declining mass ratio distribution found in DM91 the flat mass ratio distribution in Raghavan et al. (2010) for solar- 4.3.4 IGBDFs for galaxy types type stars were correct, this would possibly imply a birth The above results hold if cluster formation in the MW is rep- pairing method for binaries which is different from random- pairing and, in turn, would require alteration of the binary resentative for other galaxy types (Sec. 3.3, rh = 0.2 pc, but with different SFRs). Then, the expected period and mass birth population (see also Kouwenhoven et al. 2009). Alter- ratio IGBDF for ellipticals, spirals and dIrrs for a population natively, the eigenevolution model might need adjustments, e.g. through primary star dependent eigenevolution param- of binaries with m1 6 2M⊙ (no cuts in period or primary- mass) are expected to be as in Fig. 9. The period IGBDF eters λ = λ(m1) and χ = χ(m1), so that the initial binary of Es peaks at shorter periods than the corresponding dis- population may be mildly different for other spectral types. tributions for S and dIrr galaxies. It is due to the initial These limitations should always be kept in mind, but the star burst in Es (high SFR), namely that more dynamically overall results of the IGBDF model would not be affected. evolved binary populations from high-mass clusters, which are not present in the lower SFR spirals and dIrrs (eq. 6), contribute to the galactic field of Es (Sec. 3.2). The mass 5 SUMMARY & OUTLOOK ratio IGBDFs for all morphological types look alike, a dif- ference only appearing through the varying binary-fraction Following observational evidence which implies all stars less between the galaxies (the area below the distributions). massive than about 2M⊙ to form as binaries with compo-

c 2010 RAS, MNRAS 000, 1–14 Dynamical population synthesis 13 nent masses picked randomly from the stellar IMF in dis- (high SFR) is predicted to be significantly smaller than in crete star formation events (i.e. embedded star clusters) spiral (intermediate SFR) and dwarf galaxies (low SFR) and and allowing for pre-main sequence eigenevolution (Sec. 2) their period- and mass-ratio IGBDFs are calculated. and stimulated evolution, the concept of integrated galactic- The IGBDF model will be extended to allow calcula- field binary distribution functions (IGBDFs) is introduced. tion of binary populations in galaxies which had a strongly Adding up the stellar populations that ever formed in star varying star formation history, i.e. a time-dependence clusters which are selected from an ECMF yields the statisti- will be incorporated, via SFR(t). It will be possible to cal binary properties of a galactic field population (Dynam- synthesize binary properties in individual nearby (dwarf) ical Population Synthesis). This approach is similar to the galaxies, and the case of the MW may be revisited. A full IGIMF theory which adds up the stellar IMFs in individual synthetic galaxy might be constructed and ”observed” in clusters to calculate the galaxy-wide IMF (Sec. 1). the computer to mimic real data in order to provide more The IGBDFs depend on the minimum cluster mass, sophisticated means of comparison with observations. It Mecl,min, the galaxy-wide SFR, the steepness of the ECMF will also allow to test their influence on observationally and the typical, or average value for cluster half-mass radii, derived properties, especially in populations where binaries rh, in a star cluster system. The galactic field binary-fraction cannot be resolved, such as velocity dispersions (used to increases with decreasing Mecl,min and SFR, with increas- calculate dynamical masses). ing index β of the ECMF (eq. 7) and with increasing rh (Fig. 2). It is found that low-mass (i.e. low-density) clusters Acknowledgments contribute most binaries to the field since stimulated evolu- MM was supported for this research through a stipend from tion is least effective in them and low-mass clusters are most the International Max Planck Research School (IMPRS) numerous due to the steep ECMF. High-mass (high-density) for Astronomy and Astrophysics at the Universities of clusters donate most single stars since binary destruction is Bonn and Cologne. We thank K. M. Menten for useful effective and the number of stars forming in a high-mass suggestions. We thank D. Raghavan for discussions and cluster is larger than in a low-mass cluster. help with their data. 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