Star Cluster Evolution, Dynamical Age Estimation and the Kinematical Signature of Star Formation

Abstract

We use N-body integration to follow the evolution of clusters of 200 binary systems with different initial half mass radii $R_{0.5}$. We also simulate single-star clusters. All clusters evolve according to the same $n(t)$ curve, where $n(t)$ is the number density of stars in the central 2~pc sphere at time $t$. $n(t)$ and the lifetime are independent of (i) the inital proportion of binaries and (ii) the initial $R_{0.5}$. Mass segregation measures the dynamical age of the cluster. The proportion of binaries in the central cluster region is a sensitive indicator of the initial cluster concentration. If most stars form in binaries in a typical embedded cluster which is located at the edge of a giant molecular cloud, then we estimate that at most about 10~per cent of all pre-main sequence stars achieve near escape velocities from the molecular cloud. The large ejection velocities resulting from close encounters between binary systems imply a `halo' distribution of young stars over large areas surrounding star forming sites which is expected to have a significantly reduced binary proportion and a significantly increased proportion of stars with depleted circumstellar disks. We compare the time dependent model single star and system luminosity function in the central cluster region with the observational Hyades and Pleiades luminosity functions and find no evidence for different dynamical properties of stellar systems at birth in the Hyades, Pleiades and Galactic field stellar samples. The observed proportion of binary stars in the very young Trapezium Cluster is consistent with the early dynamical evolution of a cluster with a very high initial stellar number density.

Full text

Page 1

arXiv:astro-ph/9507017v2 13 Nov 1995
Star Cluster Evolution, Dynamical Age Estimation
and the Kinematical Signature of Star Formation
K3
Pavel Kroupa
Astronomisches Rechen-Institut
M¨ onchhofstraße 12-14, D-69120 Heidelberg, Germany
e-mail: S48@ix.urz.uni-heidelberg.de
MNRAS, in press
1

Page 2

Abstract. We distribute 400 stars in Nbin= 200 binary systems in clusters with initial half mass radii
0.077 ≤ R0.5 ≤ 2.53pc and follow the subsequent evolution of the stellar systems by direct N-body
integration. The stellar masses are initially paired at random from the KTG(1.3) initial stellar mass
function. The initial period distribution is flat ranging from 103to 107.5days, but we also perform
simulations with a realistic distribution of periods which rises with increasing P > 3 days and which is
consistent with pre-main sequence observational constraints. For comparison we simulate the evolution
of single star clusters. After an initial relaxation phase, all clusters evolve according to the same
n(t) ∝ exp(−t/τe) curve, where n(t) is the number density of stars in the central 2 pc sphere at time
t and τe ≈ 230 Myrs. All clusters have the same lifetime τ. n(t) and τ are thus independent of (i)
the inital proportion of binaries and (ii) the initial R0.5. Mass segregation measures the dynamical age
of the cluster: the mean stellar mass inside the central region increases approximately linearly with
age. The proportion of binaries in the central cluster region is a sensitive indicator of the initial cluster
concentration: it declines within approximately the first 10–20 initial relaxation times and rises only
slowly with age, but for initial R0.5< 0.8 pc, it is always significantly larger than the binary proportion
outside the central region. If most stars form in binaries in embedded clusters that are dynamically
equivalent to a cluster specified initially by (Nbin,R0.5) = (200,0.85pc), which is located at the edge of a
1.5×105M⊙molecular cloud with a diameter of 40 pc, then we estimate that at most about 10 per cent
of all pre-main sequence stars achieve near escape velocities from the molecular cloud. The large ejection
velocities resulting from close encounters between binary systems imply a distribution of young stars
over large areas surrounding star forming sites. This ‘halo’ population of a molecular cloud complex is
expected to have a significantly reduced binary proportion (about 15 per cent or less) and a significantly
increased proportion of stars with depleted or completely removed circumstellar disks. In this scenario,
the distributed population is expected to have a similar proportion of binaries as the Galactic field
(about 50 per cent). If a distributed population shows orbital parameter distributions not affected by
stimulated evolution (e.g. as in Taurus–Auriga) then it probably originated in a star-formation mode
in which the binaries form in relative isolation rather than in embedded clusters. The Hyades Cluster
luminosity function suggests an advanced dynamical age. The Pleiades luminosity function data suggest
a distance modulus m − M = 6, rather than 5.5. The total proportion of binaries in the central region
of the Hyades and Pleiades Clusters are probably 0.6–0.7. Any observational luminosity function of
a Galactic cluster must be corrected for unresolved binaries when studying the stellar mass function.
Applying our parametrisation for open cluster evolution we estimate the birth masses of both clusters.
We find no evidence for different dynamical properties of stellar systems at birth in the Hyades, Pleiades
and Galactic field stellar samples. Parametrising the depletion of low mass stars in the central cluster
region by the ratio, ζ(t), of the stellar luminosity function at the ‘H2–convection peak’ (MV≈ 12) and
‘H−plateau’ (MV≈ 7), we find good agreement with the Pleiades and Hyades ζ(t) values. The observed
proportion of binary stars in the very young Trapezium Cluster is consistent with the early dynamical
evolution of a cluster with a very high initial stellar number density.
Keywords: stars: low mass, formation, luminosity function – binary stars – Galactic clusters and
associations: dynamical evolution, individual: Hyades, Pleiades, Trapezium
2

Page 3

1 INTRODUCTION
We refer to the stellar mass function, the proportion of stellar systems (singles, binaries, triples, etc.) and
the distribution of their orbital parameters as the dynamical properties of stellar systems. Studying Galactic
clusters is important because they represent fossils of discrete star formation events. From them we can
hope to obtain information on the dependence of the dynamical properties of stellar systems on the birth
conditions.
The majority of stars in the Galactic disk may be born in embedded clusters rather than in isolation.
This conclusion is drawn by Lada & Lada (1991) after studying the distribution of young stellar objects in the
L1630 molecular cloud of the Orion complex. In it they find no significant distributed population of young
stars but four embedded clusters containing at least about 627 objects. Strom, Strom & Merrill (1993), on
the other hand, find a distributed population of about 1500 young stars in the L1641 molecular cloud in
Orion, as well as seven clusters, in which 10–50 young stellar objects have been detected, and one partially
embedded cluster, in which 150 young stellar objects have been detected. The ages (< 1 Myrs) of the stellar
objects in the detected clusters appear to be younger than in the distributed population (5–7 Myrs). This is
consistent with the latter having originated in aggregates which are now dissolved. Nevertheless, it is evident
that the observational evidence for a predominant clustered star formation mode remains suggestive rather
than conclusive. However, if it is assumed that star formation nearly always produces a binary star (as
suggested by observations of pre-main sequence stars) then clustered star formation must be the dominant
mode, rather than distributed star formation (Kroupa 1995a, hereafter K1).
This paper is the third in a series of three papers K1, K2 and K3. In K1 we study the evolution of a
binary star population in stellar aggregates and use the results for inverse dynamical population synthesis
on the observed dynamical properties of stellar systems in the Galactic field to deduce that a dominant
clustered mode of star formation may exist. In K1 we suggest that most Galactic field stars may have been
born in aggregates that are dynamically equivalent to the ‘dominant mode cluster’, which is defined by
(Nbin,R0.5) = (200,0.85pc), where Nbinis the initial number of binaries and R0.5 is the initial half mass
radius. We also derive an initial distribution of periods. In K2 (Kroupa 1995b) we study in detail the
dynamical properties of Galactic field stellar systems if they form in the dominant mode cluster. In this
paper (K3) we make use of the N-body simulations of K1 and K2 to study the overall evolution of the stellar
clusters.
The realistic KTG(1.3) stellar mass function (Section 2) is adopted, and all stars are initially in ag-
gregates, or clusters, of binary systems. These have a period and mass ratio distribution consistent with
pre-main sequence data. Our initial very large proportion of primordial binaries is also consistent with
observational evidence that a large proportion of binaries may reside in the central region of at least one
Galactic cluster (the Praesepe Cluster, Kroupa & Tout 1992). As a control experiment we simulate single-
star clusters. We are interested to learn if and how the initial conditions of our simulations are observable
in real Galactic clusters. By studying the rate at which the clusters dissolve and the evolution of mass and
binary star segregation the evolutionary history of a Galactic cluster can probably be traced. The stellar
luminosity function in the central regions of clusters evolves as a result of mass segregation, evaporation
of stars and increase of the proportion of binary systems. We also investigate if the distribution of stellar
systems in space and in velocity after dissolution of the aggregates can be used to obtain clues about the
initial dynamical configuration.
We concentrate on readily observable properties of stellar clusters and do not mean to provide a compre-
hensive treatment of their evolution which has been extensively studied elsewhere: Mathieu (1985) describes
the internal kinematics and the structure of Galactic clusters, Wielen (1985) discusses their dynamics, and
Aarseth (1988a,b) reviews the dynamical evolution of open clusters and discusses core collapse, respectively,
based on the extensive and detailed simulations of Terlevich (1987). The ejection of stars from open clusters
containing binaries is studied in the context of OB runnaway stars by Leonard & Duncan (1990). The
hypothesis that blue stragglers may result from collisions and merging of finite sized stars in clusters is
discussed by Leonard & Linnell (1992). Hut (1985) emphasises that binaries can be considered a dynamical
energy source in a cluster similar to nuclear reactions in a star. A comprehensive review of the role of bina-
ries in Globular cluster dynamics is provided by Hut et al. (1992). Heggie & Aarseth (1992) and McMillan
& Hut (1994) consider the evolution of globular clusters consisting of equal-mass stars and containing an
3

Page 4

initial proportion of binaries of up to 20 per cent. These authors, and the model of the binary star evolution
in a globular cluster devised by Hut, McMillan & Romani (1992), provide many valuable insights into the
processes that govern the interaction of the cluster with the population of binary systems.
In Section 2 we briefly summarise the assumptions, definitions and our method. The evolution of four
binary star clusters and two single star clusters is discussed in Section 3 and compared with the evolution of
the dominant mode cluster in Section 4, where we also elaborate on the concept of ‘dynamical equivalence’
introduced in K1. The kinematical signature of star formation is addressed in Section 5. In Section 6
we apply our simulations to three clusters and discuss the stellar luminosity function. We conclude with
Section 7.
2. ASSUMPTIONS, DEFINITIONS AND METHOD
A detailed description of our assumptions and method can be found in section 3 of K1, which we briefly
summarise here.
We distribute Nbin = 200 binary systems according to a Plummer density law with half mass radii
R0.5 = 0.077,0.25,0.77,2.53 pc. R0.5 = 0.08 pc corresponds to an initially highly concentrated cluster
similar to the Trapezium Cluster, whereas R0.5= 2.53 pc corresponds to a loose cluster which approximates
distributed star formation (e.g. Taurus–Ariga). We also distribute Nsing= 400 single stars in clusters with
R0.5= 0.077,0.25 pc. These are of academic interest only and serve as a comparison with the realistic binary
star clusters. Virial equilibrium is assumed. For the N-body simulation of the dynamical evolution of each
cluster we use the program nbody5 written by Aarseth (1994). We model a standard Galactic tidal field (see
K1).
Stellar masses in the range 0.1M⊙≤ m ≤ 1.1M⊙are obtained from the initial mass function which is
conveniently approximated by ξ(m) ∝ m−αi, α1= 1.3 for 0.08M⊙≤ m < 0.5M⊙, α2= 2.2 for 0.5M⊙≤
m < 1.0M⊙, α3 = 2.7 for 1.0M⊙ ≤ m and ξ(m)dm is the number of stars with masses in the range
m to m + dm (Kroupa et al. 1993). We refer to ξ(m)dm as the KTG(α1) mass function. Our adopted
stellar mass range allows us to ignore post-main sequence stellar evolution which simplifies the computation
and allows focusing on purely stellar dynamical evolution. If we have a population of stars with masses
0.1M⊙≤ m < A, then for A = 10(50)M⊙, about 8 per cent of these are more massive than 1.1M⊙, and
contribute 35 (39) per cent to the total mass. Stars more massive than about 5 (10)M⊙ will affect the
dynamical evolution of the clusters within the first 7 × 107(2 × 107) yrs, but will be insignificant during
most of the lifetime of the clusters studied here. The mean stellar mass in our models is 0.32M⊙and the
mass of each cluster is 128M⊙, which is near the peak of the mass function of Galactic clusters (Battinelli,
Brandimarti & Capuzzo-Dolcetta 1994).
To build the initial binary stars we combine the stellar masses at random and distribute orbital periods,
P (in days), from a flat distribution, fP(log10P) = [log10(Pmax) − log10(Pmin)]−1(equation 3 in K1), with
log10Pmin= 3, log10Pmax= 7.5 and Pmin≤ P ≤ Pmax. The initial mass-ratio and period distribution are
consistent with pre-main sequence binary star data (K1). The initial eccentricity distribution is assumed to
be dynamically relaxed, fe(e) = 2e (equation 4 in K1), but is not critical here.
For each binary and single-star cluster we perform Nrun= 5 and 3 simulations, respectively. All results
quoted here are averages of Nrunsimulations.
In addition to the simulations of the above six clusters we perform Nrun= 20 simulations of the dominant
mode cluster studied by K2 which has R0.5= 0.85 pc initially. The Nbin= 200 binary systems per simulation
have a mass-ratio distribution and a birth eccentricity distribution as above, but a birth period distribution
given by equation 8 in K2. The birth orbital parameters of the short period (log10P < 2 − 3) binary
systems are assumed to eigenevolve during the proto-stellar accretion phase on a timescale of approximately
105yrs (see section 2 in K2). The resulting distribution of orbital elements is the initial (t = 0) distribution
for the N-body simulation, and is consistent with the mass-ratio and eccentricity distributions observed for
short-period G dwarfs, i.e. a bias towards equal mass components and a bell shaped eccentricity distribution,
respectively. The minimum orbital period is about 3 days, and in our model about 3 per cent of all systems are
merged binaries at the start of the N-body simulation. The initial period distribution can be approximated by
fP,in(log10P) = 3.50log10P /[100+(log10P)2] (equation 11 in K1, see also fig. 7 in K2). Stimulated evolution
(i.e. the changes in orbital parameters due to interactions with other systems) does not significantly change
the orbital parameters of binaries with log10P < 3.
4

Page 5

Physical parameters for each cluster discussed here are listed in table 1 of K1, and in Table 1 we repeat
some of these.
Table 1. Initial stellar clusters
R0.5 NbinNsing ftot
pc
n
stars stars
pc−3pc−3
12
12
11
5
12
12
10
log10(nc)σ
km
sec−1
1.7
0.9
0.5
0.3
1.7
0.9
0.5
Tcr
Myrs
Trelax
Myrst = 0
0.08 200
0.25 200
0.77 200
2.53 200
0.08 0
0.25 0
0.85 200
0
0
0
0
400
400
0
1
1
1
1
0
0
1
5.6
4.1
2.7
1.1
5.6
4.1
2.5
0.094
0.54
3.0
17
0.094
0.54
3.5
0.30
1.8
9.5
56
0.30
1.8
11
ncis the central number density; σ the average velocity dispersion; Tcrand Trelaxare, respectively, the
crossing and relaxation times (for further details see section 3 in K1).
We refer to a single star as a single star system and define the overall proportion of binary systems
at time t to be ftot(t) = Nbin(t)/(Nbin(t) + Nsing(t)) (equation 2 in K1). This definition extends to stellar
systems in any sub-domain, e.g. if orbits only in a specific period range are accessible (cf. with equation 7
in K1) or if ftot(t) is evaluated in a particular volume of space or mass range.
The simulations of the dynamical evolution of the above seven clusters allow an intercomparison of the
evolution of overall properties of the stellar systems such as stellar number density, mean stellar mass and
binary star segregation, and the distribution of centre-of-mass kinetic energies (in the local rest frame). To
quantify the evolution of the clusters we measure the number density, n(t), mean stellar mass, m(t), and
the overall proportion of binaries, ftot(t), within a standard spherical volume, V , with radius R centered on
the number density maximum of the cluster. For example, if we take R = 2 pc then we refer to V as the
central 2 pc sphere. ftot(t) is defined above, and in this context Nsing(t) and Nbin(t) are the number of single
and binary systems in V , respectively. Similarly, m(t) = Mstars(t)/(Nsing(t) + 2Nbin(t)), where Mstars(t) is
the total stellar mass in V , and n(t) = (Nsing(t) + 2Nbin(t))/V . The three quantities n(t), m(t) and ftot(t)
are readily accessible to an observer (t =cluster age). If an instrumental flux limit and/or a resolution limit
limits the observations then we can in principle apply the same bias to our model values n(t), m(t) and
proportion of binaries. This will be necessary in a case-by-case treatment of individual Galactic clusters.
For conciseness we write Nbin= Nbin(0), Nsing= Nsing(0), and R0.5= R0.5(0) in the knowledge that
the half mass radius of a cluster evolves. We do not evaluate R0.5(t > 0) because this is non trivial requiring
exact knowledge of escapers. In Section 3.1 we show that all clusters with 400 stars disintegrate completely
after 800 Myrs, and from hereon we refer to the final distributions of, say binary star binding energies, as
the distributions evaluated at t = 1 Gyr that result after cluster disintegration.
3. CLUSTER EVOLUTION
In this section we concentrate on the evolution of the number density, n(t), mean stellar mass, m(t), and
proportion of binaries, ftot(t), in the four stellar clusters with Nbin= 200 and R0.5= 0.077,0.25,0.77,2.53pc,
and in the two single-star clusters with Nsing= 400 and R0.5= 0.077,0.25 pc.
3.1 Lifetime and birthrate
From Fig. 1 we infer that there is no significant difference in the evolution of n(t) between the various
clusters. After the first two relaxation times, n(t) for the R0.5= 2.53 pc cluster joins the other evolutionary
curves and the clusters depopulate at the same rate, irrespective of the presence of primordial binaries (some
of which are hard - see Fig. 5 below) and of the initial cluster concentration. A reasonable approximation of
the depopulation of the central 2 pc sphere is
5