A&A 372, 105-116 (2001)
DOI: 10.1051/0004-6361:20010477
R. S. Klessen^{1,2,3} - P. Kroupa^{4}
1 - UCO/Lick Observatory, University of California, 499 Kerr Hall,
Santa Cruz, CA 95064, USA
2 - Otto Hahn Fellow,
Max-Planck-Institut für Astronomie, Königstuhl 17, 69917
Heidelberg, Germany
3 - Sterrewacht Leiden, Postbus 9513, 2300-RA Leiden,
The Netherlands
4 -
Institut für Theoretische Physik und Astrophysik, Universität
Kiel, 24098 Kiel, Germany
Received 11 July 2000 / Accepted 26 March 2001
Abstract
Applying the mean surface density of companions, ,
to the
dynamical evolution of star clusters is an interesting approach to
quantifying structural changes in a cluster. It has the advantage
that the entire density structure, ranging from the closest binary
separations, over the core-halo structure through to the density
distribution in moving groups that originate from clusters, can be
analysed coherently as one function of the stellar separation r. This contribution assesses the evolution of
for clusters
with different initial densities and binary populations. The changes
in the binary, cluster and halo branches as the clusters evolve are
documented using direct N-body calculations, and are correlated with
the cluster core and half-mass radius. The location of breaks in the
slope of
and the possible occurrence of a binary gap can
be used to infer dynamical cluster properties.
Key words: stars: binaries: general - open clusters and associations: general - stars: formation - stellar dynamics
These results prompted the reanalysis of the Taurus-Auriga data by other authors (Simon 1997; Bate et al. 1998; Gladwin et al. 1999), as well as the subsequent investigation of additional star-forming regions. Mean surface densities of companions as a function of angular separation have been derived for Orion (Simon 1997; Bate et al. 1998; Nakajima et al. 1998), for the -Ophiuchus cloud (Simon 1997; Bate et al. 1998; Nakajima et al. 1998; Gladwin et al. 1999), and for the star-forming regions in Chamaeleon, Vela and Lupus (Nakajima et al. 1998). Common to all studies is that the companion surface density is best described as a double power-law, with slopes of -2 in the binary branch and slopes between -0.9 and -0.1 in the large-scale clustering regime. However, the length scales where the break of the distribution is found to vary considerably, from AU in the Trapezium cluster in Orion, over AU in Ophiuchus and AU in Taurus, to AU for the Orion OB region. This fact raises considerable doubts about the interpretation of the break location as being determined by the Jeans condition in the cloud. This would imply quite different Jeans masses which in turn should lead to deviations of the initial mass function, which have not been observed.
A thorough theoretical evaluation of the mean surface density of companions, , and a discussion of viable interpretations can be found in Bate et al. (1998). Altogether the following picture emerges: at small separations, traces the separations of binary stars and higher-order multiple stellar systems. The slope -2 results from the frequency distribution of binary separations being roughly uniform in logarithm (Duquennoy & Mayor 1991, for main sequence stars). The break occurs at the "crowding'' limit, i.e. at separations where wide binaries blend into the "background'' density of the cluster. At larger separations, simply reflects the large-scale spatial structure of the stellar cluster. Bate et al. (1998) pointed out that can be strongly affected by boundary effects and that a non-integer power-law slope in the cluster branch does not necessarily imply fractal structure. They showed that a simple core-halo structure, as is typical for evolved stellar clusters, will result in a non-integer slope for separations larger than the core radius. They also speculated about possible effects of dynamical cluster evolution on the properties of .
It is the aim of the present paper to investigate, for the first time, evolutionary effects on as derived from realistic N-body computations. We use models studied by Kroupa (1995a,b,c, 1998, hereinafter K1-K4) for a comparison with a "standard'' dynamical analysis, where the binary population is analysed separately from the bulk cluster properties. For comparison, the mean surface density of companions for models of protostellar clusters that form and evolve through turbulent molecular cloud fragmentation is discussed in Klessen & Burkert (2000, 2001).
The structure of the paper is as follows. In the next section (Sect. 2) we mathematically define the mean surface density of companions, , and briefly discuss its limitations. In Sect. 3 we describe the star cluster models and their properties. In Sect. 4, we investigate the influence of cluster evolution on and in particular discuss wide-binary depletion. The effects of averaging and projection are analysed in Sect. 5, and a possible observational bias is discussed in Sect. 6. Section 7 discusses features in and their relation to cluster morphology. Finally, our results are summarised in Sect. 8.
Model | Model | Model | Model | Model | Model | ||
n | 5 | 5 | 5 | 5 | 3 | 3 | |
f | 1 | 1 | 1 | 1 | 0 | 0 | |
(pc) | 0.08 | 0.25 | 0.8 | 2.5 | 0.08 | 0.25 | |
(yr) | 0.094 | 0.54 | 3.0 | 17 | 0.094 | 0.54 | |
log | (stars/pc^{3}) | 5.6 | 4.1 | 2.7 | 1.1 | 5.6 | 4.1 |
log | (stars/pc^{2}) | 4.3 | 3.3 | 2.3 | 1.2 | 4.3 | 3.3 |
(yr) | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |
(yr) | 1.19 | 6.67 | 37.5 | 211 | 1.19 | 6.68 | |
(yr) | 11.9 | 66.7 | -- | -- | 11.9 | 66.8 | |
(yr) | 297 | 300 | 300 | 296 | 297 | 300 |
(1) |
Stellar surveys have finite area and boundary effects may occur. For stars closer than a distance to the boundary all annuli with extend beyond the limits of the survey and companion stars may be missed. This has little effect when considering small separations r as only a small fraction of all stars is affected, however, when r becomes close to the survey size the missing companions result in a steep decline of . Several methods have been proposed to correct for that effect (Bate et al. 1998, and references therein), where either the accepted range of r is reduced to separations much smaller than the survey size, or additional assumptions about the background density of stars are made. Neither approach is completely satisfactory.
In the present study we do not attempt to adopt any of the correction methods for the following reasons. First, in the numerical simulations all information about the system is accessible. There are no observational constraints and we always consider all stars in the cluster. The survey area can be arbitrary large and is chosen such that it includes the complete cluster. Second, although the considered clusters are subject to the tidal field of the Galaxy we do not include Galactic-field stars in consideration. Hence, there is no confusion limit, where contamination with foreground or background stars becomes important.
A range of star-cluster models are constructed, and their dynamical evolution is calculated using NBODY5 (Aarseth 1999), which includes a standard Galactic tidal field (Terlevich 1987). The cluster models have been discussed extensively in K1-K4, so that only a short outline is provided here.
The stellar systems initially follow a Plummer density distribution (Aarseth et al. 1974) with half-mass radius , and the average stellar mass is independent of the radial distance, r, from the cluster centre.
Stellar masses are distributed according to the solar-neighbourhood IMF (Kroupa et al. 1993) with . Larger masses are omitted so as to avoid complications arising from stellar evolution. Binaries are created by pairing the stars randomly, giving a birth binary proportion , where and are the number of single-star and binary systems, respectively. The initial mean system mass is , with being the average stellar mass. This results in an approximately flat mass-ratio distribution at birth (Fig. 12 in K2). Periods and eccentricities are distributed following K1. The initial periods range from 10^{3} to 10^{7.5} days, and the eccentricity distribution is thermal, i.e. the relative number of binaries increases linearly with eccentricity being consistent with observational constraints.
The parameters are listed in Table 1. Four clusters with f=1 are constructed spanning a wide range of central densities, from log to 5.6 [stars/pc^{3}]. Each model contains N=400 stars and has a mass . The initial tidal radius in all cases is pc. All stars are kept in the calculation to facilitate binary-star analysis, but those with experience unphysical accelerations in the linearised local tidal field and rotating coordinate system (Terlevich 1987), so that the density distribution of stellar systems at large radii does not reflect the true distribution in the moving group. Five different renditions are calculated for each model to increase the statistical significance of our results. In addition, two clusters with f=0 are constructed for comparison with the binary-rich cases. The evolution of these models is calculated for three cluster realisations each. The computations cover years, but we consider only a sub-set of all possible snap-shots in the current analysis.
Figure 1: The mean surface density of companions, , as a function of separation r for star clusters to with an initial binary fraction of 100% at different states of the dynamical evolution: initially (t=0.0, left column), at and (2. and 3. column) and at (right column). The corresponding time in units of years is indicated in the upper right corner of each plot. is obtained as an average over n=5 different cluster realizations for each model as a projection into the xy-plane. The error bars indicate Poisson errors. | |
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In this section, we discuss the influence of the dynamical cluster evolution on the resulting mean surface density of companions . For each star cluster, is calculated as an average over the set of n individual model realisations, and we restrict ourselves to discussing the projection into the xy-plane. The influence of averaging and projection is discussed in Sect. 5.
Figure 2: The mean surface density of companions, , as a function of separation r for star clusters and . These models are equivalent to and , respectively, but initially contain no binary stars. The figure is analogous to Fig. 1, except that n=3 different realizations of each model are used in the averaging process. | |
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At any time and for all models, the mean surface density of companions exhibits, at low separations r, a well defined power-law behaviour . It implies an approximate uniform distribution of binary separations in (Bate et al. 1998). For larger r, the binary branch blends into the plateau of constant companion density corresponding to the core of the star cluster. This first break of the distribution occurs at separations r_{1}where the number of chance projections of cluster members becomes equal to the number of binaries in that separation bin. At larger rchance projections completely dominate , and wide binaries, if in fact present, can no longer be identified as such. The location of the first break therefore depends on the binary fraction and on the central density of the cluster. On larger scales the stellar clusters follow a Plummer radial density profile with half-mass radius . Therefore, a second break occurs at r_{2} and declines sharply for separations (see Sect. 7).
As the dynamical evolution progresses, the clusters expand and the density declines. Hence, the projected mean surface density of companions decreases as well. While many of the binaries with separations comparable to the mean distance of stellar systems in the cluster core become disrupted, some new binaries may form by capture. Usually these are higher-order multiples (K2) with separations close to the first break or smaller. As the cluster expands, the binary branch becomes less affected by crowding and the first break in shifts to greater separations. This behaviour is clearly visible in Fig. 1. For all models the core plateau in "decreases in height'' and "moves'' to larger separations as time progresses. At late stages of the evolution the entire binary branch is uncovered and a few long-period orbits appear through capture. This is also documented in Figs. 3 and 4 in K4.
The clusters develop core-halo structures through energy equipartition. Low-mass stars gain kinetic energy through encounters with more massive stars. The low-mass stars move away from the cluster centre, forming the halo, whereas the more massive stars sink towards the centre. The trajectories of halo stars that trespass beyond the tidal radius of the cluster are dominated by the Galactic tidal field, and most become unbound. Hence, the clusters expand until they fill their tidal radii. When this stage is reached (roughly after years), the different clusters evolve identically (see also Fig. 1 in K3), mostly loosing stars through their first and second Lagrange points (Terlevich 1987; Portegies Zwart et al. 2001). As a result, extends beyond the second break to increasingly larger separations with an increasingly shallower slope, which we identify as the halo branch in . Near the tidal radius , a third break occurs, as stars with become unbound. The trajectories of stars belonging to these unbound moving groups are not followed with sufficient resolution (see Sect. 3). Also, these stars are likely to be severely contaminated by field stars in the Galaxy, and we refrain from a further discussion of this outermost branch in .
As can be seen from Fig. 1, knowledge of the initial global properties of the system is effectively erased through the dynamical evolution. The cluster and halo branches in look quite indistinguishable in the final frames. The situation changes, however, when considering the binary branch, as discussed in Sect. 4.2. The f=1 versus f=0experiments demonstrate that there is no significant difference in bulk cluster evolution between clusters containing a large primordial binary proportion and no binaries (K3). This is also evident by studying ; concentrating only on the cluster and halo branches, the upper two final panels in Figs. 1 and 2 are indistinguishable.
Figure 3: The function for models to . The depicted times, averaging and projection are analogous to Fig. 1. The plots concentrate on the properties of the binary branch, where the horizontal dashed lines indicate its initial slope. The depletion of the distribution at late stages in the interval pc pc is the result of wide-binary disruption. The rising part of at separations greater than the first break corresponds to the cluster core, and the following decline for is the contribution from the cluster halo. To demonstrate the effects of dynamical evolution, the dotted lines for t>0.0 indicate the initial distribution. | |
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However, as the clusters evolve, binaries are not only created but also destroyed. Due to their smaller binding energies, wide binaries are more vulnerable to dynamical processes than close ones. If the initial binary fraction is high, then the destruction processes dominate over binary formation, and cluster evolution leads to a depletion of wide binaries. As a result, steepens on the large-separation side of the binary branch. In extreme cases, some annuli r of may become completely depopulated, and consequently a gap between the binary and the cluster branches opens up, as is noticeable in Fig. 1.
The efficiency of binary disruption depends strongly on the initial density of the cluster. For high stellar densities, the typical impact parameters of stellar encounters are small. Hence, there is a relatively high frequency of encounters for which the energy exchange exceeds the binding energy of typical binary systems, which subsequently dissolve. In our suite of models, the effect of binary depletion is largest in , which has the highest central density , and decreases with increasing half-mass radius as becomes smaller.
This is demonstrated in Fig. 3. Unlike the previous figures it shows a reduced range of separations, concentrating on the binary branch, and it plots to make it easier to determine the power-law slope and deviations from it. At t=0.0, for all models is constant in the binary branch. This reflects the initial distribution of the binary separations, which is uniform in the logarithm within the range pc to pc. Values pc come from the projection of the 3-dimensional distribution into the xy-plane. For model the complete binary branch is visible and fully segregated from the cluster branch (which corresponds to the rising part of the plot). This is because the half-mass radius of the cluster is large enough that the projected mean separation between cluster members exceeds the separation of the widest binary system. As both branches are clearly separated initially, dynamical evolution does not alter the binary distribution significantly. There is little sign of wide binary depletion, even at .
Figure 4: Mean surface density of companions, , of cluster at time for the projection into the xy-plane. The figure illustrates the effect of the averaging process. The small plots show for the five different realizations to of model , whereas the large plot gives the resulting averaged function. | |
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In the other models, the typical separations in the cluster core are smaller than pc, and binary and cluster branches overlap in the beginning. This is most significant for model , where the initial central density is highest. Consequently a large number of wide binaries are disrupted during the dynamical evolution of the system, and drops considerably below its initial value (indicated by the dashed line) in the range pc pc. Because the size of this gap depends on the age and the initial central concentration of the cluster, analysing the signatures in could be used to constrain the initial state of observed stellar clusters. This fact, namely that the binary population retains a memory of its past dynamical environment, is also used in K1 to infer the typical structures in which most Galactic-field stars form, by studying the shape of the binary period distribution ("inverse dynamical population synthesis'').
Figure 5: Illustration of the effect of projection. Mean surface density of companions for cluster realization at time for three different projections. The upper panel shows the stellar distribution within pc of the cluster centre and the lower panels show the resulting . The global features of are independent of projection. | |
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The invariance to changes in projection is demonstrated in Fig. 5, which shows cluster again at . The function is essentially independent of the projection, and slight differences occur only where there are smaller numbers of stars. Analogue to the variations between different model realisations discussed above, this is the case at very large and very small r, and at separations where wide-binary depletion occurs. At large separations, the tidal field breaks the symmetry. For small r and at the binary gap, it depends on the projection which separation bin stays populated and which may become empty. However, besides these details the overall structure of is projection invariant.
Figure 6: Mean surface density of companions, , of cluster for different observational minimum mass limits . First row of panels: all stars in the cluster are considered (analogue to top row in Fig. 1). Second row: only stars with contribute to . Third row: . Fourth row: , and lowest row: . Times and averaging procedure are equivalent to Figs. 1 to 3. There are 400 stars in each cluster realization in the mass range to following the Kroupa et al. (1993) IMF. The fraction of cluster stars considered in each panel is 100%, 51%, 17%, 6%, and 1%, respectively. | |
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With increasing , the total number of detected stars decreases, and as a result is reduced. Also the shape of changes. This effect is small for low detection thresholds ( ), as the overall star distribution in the cluster is still well sampled. However, it becomes significant at large cluster distances when only the brightest stars can be detected. The binary branch becomes severely under-sampled, and wide gaps open up. For the binary branch disappears in all models.
The inferred cluster core radius also depends quite sensitively on the completeness of the stellar sample. As the clusters evolve dynamically, high-mass stars sink towards the cluster centre due to mass segregation, whereas low-mass stars move outwards, building up the extended halo (Fig. 2 in K3). Sub-populations of higher-mass stars therefore exhibit smaller core radii as time progresses relative to the low-mass population.
This trend is clearly seen in Fig. 6, where the
second break moves to smaller separations as
increases.
At very late stages of the dynamical evolution and for very large
cut-off masses, the core radius may become too small, so that the
second break is no longer noticeable. For example, the function
follows an almost perfect r^{-2}-power-law for
at
,
exhibiting a smooth
transition from the binary regime to the halo regime without any sign
of the cluster core, which is present when taking all stars into
account. When considering only stars with
,
then the
signature of the cluster core disappears at all times. This bias needs
be taken into account when interpreting observational data on star
clusters.
Model | time | t yr) | r_{1} (pc) | r_{2} (pc) | (pc) | (pc) | (pc) |
Model | 0.0 | 0.0014 | 0.079 | 0.03 | 0.08 | 8.0 | |
1.19 | 0.0018 | 0.16 | 0.04 | 0.1 | 8.0 | ||
11.9 | 0.0035 | 0.32 | 0.08 | 0.4 | 8.0 | ||
300 | 0.011 | 1.0 | 0.5 | 2.1 | 6.0 | ||
Model | 0.0 | 0.005 | 0.20 | 0.08 | 0.25 | 8.0 | |
6.67 | 0.004 | 0.32 | 0.09 | 0.4 | 8.0 | ||
66.7 | 0.01 | 0.63 | 0.25 | 1.2 | 7.5 | ||
300 | 0.022 | 1.6 | 0.7 | 2.6 | 6.2 | ||
Model | 0.0 | 0.016 | 0.79 | 0.32 | 0.8 | 8.0 | |
37.5 | 0.016 | 1.26 | 0.26 | 1.0 | 8.0 | ||
300 | 0.035 | 2.0 | 0.75 | 2.6 | 6.7 | ||
Model | 0.0 | 0.032 | 2.0 | 1.1 | 2.5 | 8.0 | |
211 | 0.05 | 1.9 | 1.0 | 2.5 | 7.2 | ||
296 | 0.05 | 2.0 | 0.83 | 2.5 | 6.7 | ||
Model | 0.0 | -- | 0.12 | 0.03 | 0.08 | 8.0 | |
1.19 | -- | 0.11 | 0.03 | 0.1 | 8.0 | ||
11.9 | 0.0028 | 0.34 | 0.05 | 0.3 | 8.0 | ||
297 | 0.022 | 1.6 | 0.28 | 2.0 | 6.4 | ||
Model | 0.0 | -- | 0.31 | 0.07 | 0.25 | 8.0 | |
6.67 | -- | 0.44 | 0.07 | 0.3 | 8.0 | ||
66.7 | 0.0025 | 1.1 | 0.14 | 0.9 | 7.8 | ||
300 | 0.022 | 2.0 | 0.63 | 2.2 | 6.9 |
As has been elucidated above, the mean surface density of companions shows distinct branches, the extend of which appear to couple with the dynamical state of the cluster. In this section we consider this in more detail.
Simple bulk cluster properties that can be used to describe the
dynamical state of a cluster are the core radius, ,
the
half-mass radius, ,
and the tidal radius, .
The
core radius is approximated by calculating the density-weighted radius
(Heggie & Aarseth 1992),
(2) |
We define two breaks in , r_{1} and r_{2}, by fitting power laws to the three distinct branches of (the binary branch, the flat central plateau, and the cluster halo out to the tidal radius ) and determining the separation at the intersection of the fits. These values are listed in Table 2, and plots of vs. r_{1} and vs. r_{2} are presented in Fig. 7.
Figure 7: Correlation between the first break radius r_{1} and cluster core radius (left panel), and the radius r_{2} of the second break and the half mass radius (right panel). | |
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The figure shows that the quantities are well
correlated. Specifically, we find (the uncertainties are mean absolute
deviations)
The above scaling relations apply for the specific low-mass cluster models that we investigate. These models do not suffer significant core collapse, which is partly given by the relatively fast evaporation time (0.5-1 Gyr, K3), and the ubiquitous binary stars which oppose core collapse. More massive clusters are likely to show different correlations, notably between and , and between r_{1} and r_{2}, since core collapse leads to the contraction of but an expansion of (Giersz & Spurzem 2000). Analysis of more massive clusters using is a future goal, and it will be interesting to see if correlations 4 and 5 remain valid.
Our study confirms that different projections of the same data do not change significantly during the evolution of initially spherical clusters in the Galactic tidal field. Also, different numerical renditions of the same models lead to indistinguishable results. Hence, they can be combined to improve the statistical significance of the ensemble average .
To allow for a proper assessment of stellar cluster properties using , it is important to consider as complete a census of cluster stars as possible. Observational bias (i.e. non-detection of faint stars) may complicate the interpretation of and limit its applicability for inferring the initial stage of the cluster under study (see also Bate et al. 1998). Our tests show, however, that remains a useful quantity even when stars with are not detected. We also find that mass segregation is evident in through the location of the second break in dependence of the mass-range used to construct .
Future analysis of numerical models of rich clusters, for which mass segregation and possibly core collapse play important roles in the late phases of the dynamical evolution, will be performed to deepen the issues raised in this pilot study.
Acknowledgements
We thank Sverre Aarseth for distributing NBODY5 freely. RSK acknowledges support by a Otto-Hahn-Stipendium from the Max-Planck-Gesellschaft and partial support through a NASA astrophysics theory program at the joint Center for Star Formation Studies at NASA-Ames Research Center, UC Berkeley, and UC Santa Cruz. PK acknowledges support from DFG grant KR1635.