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Statistical properties of visual binaries as tracers of the formation of young stellar clusters

In this review, we summarize the observed statistical properties of visual binaries in various young stellar clusters. The binary frequency, orbital period distribution and mass ratio distribution are considered for populations of both low-mass and high-mass stars. These properties are then compared to numerical models of the dynamical evolution of stellar clusters and to the predictions of binary formation mechanisms.


Introduction
The current paradigm for star formation is that most stars form in stellar clusters rather than in Taurus-like star-forming regions. Since binary systems are more frequent than single stars, the study of multiple systems in clusters is expected to provide us with some insight on the stellar formation process itself. For instance, it has been suggested that the gravitational collapse of molecular clouds leads to a binary fraction close to 100 %, which would decrease over time to about 60 %, the value observed on the Main Sequence (Ghez et al. 1993). The physical processes leading to such an evolution and the timescale over which they act still need to be asserted. Alternatively, the impact of local physical conditions in the molecular cloud, such as the magnetic field strength or gas temperature, on the output of the star formation process has still to be ascertain and might lead to different binary frequencies in various environments (Durisen & Sterzik 1994).
Observing stellar clusters with a wide range of physical parameters allows to perform a direct comparison between binary properties and model predictions. Clusters with ages ranging from one to several hundred million years are known within 500 pc from the Sun with some hosting many massive stars while other 2 G. Duchêne, J. Bouvier, J. Eislöffel & T. Simon containing only low-mass stars. The stellar density can vary over more than an order of magnitude from cluster to cluster, as well as the physical conditions of the parent molecular cloud. In this contribution, we review the observed properties of visual binaries in clusters and compare them with cluster properties and with model predictions regarding the formation and evolution of stellar clusters.

Visual binary frequency
The advent of several high-angular imaging techniques makes the binary frequency a relatively straightforward property to derive in stellar populations. For instance, HST imaging (Prosser et al. 1994), speckle interferometry (Leinert et al. 1993, Patience et al. 1998) and adaptive optics (Bouvier et al. 1997) have been used, mostly in the near-infrared in order to facilitate the detection of relatively faint and cold low-mass companions. The low-mass populations of extremely young clusters (the Trapezium cluster, IC 348) as well as of more evolved ones (α Per, the Pleiades, the Hyades, Praesepe) have been studied, which span an age range from ∼1 to several 100 Myr. The range of semi-major axes probed by these surveys usually extends from a few 10 to ∼2000 AU. The Hyades cluster, which is the closest to the Sun, is the only one for which visual binaries could be resolved down to about 5 AU.
Most surveys miss the faintest companions so that a statistical completeness correction has to be applied in order to estimate the true binary frequency from the observed one. This correction can be large, especially at small separations for adaptive optics surveys due to the separation-dependent detection limit inherent to this technique. The correction factor also depends upon assumptions regarding the distribution of mass ratios and/or orbital periods of the binary population, which obviously increases the statistical uncertainties on the derived binary frequency.
With these limitations, all studied clusters are found to exhibit a visual binary frequency consistent with that of field G-dwarfs (Duquennoy & Mayor 1991), to within 10 % or less (e.g. Prosser et al. 1994, Bouvier et al. 1997. This result does not seem to depend upon the age of the cluster between ∼1 Myr and 600 Myr. One possible exception might be the Hyades cluster which appears to host more tight (5-50 AU) binaries than the field population does, though statistical uncertainties could account for this slight excess (Patience et al. 1998). It hence appears that the fraction of binaries in very young clusters, at an age of about a million years, is already similar to that measured for field dwarfs and that it does not significantly evolve as the clusters dynamically relax on the ZAMS.

Orbital period distribution
While the overall binary frequency integrated over a range of semi-major axes already provides clues to the properties of binary populations in clusters, one would ideally wish to compare the distribution of their orbital periods. One is thus led to convert the observed angular separations into orbital periods assum-Visual binaries in young stellar clusters 3 Figure 1.
The distribution of orbital periods for Pleiades G and Ktype binaries. Spectroscopic binaries are taken from Mermilliod et al. (1992) and visual binaries from Bouvier et al. (1997). The solid curve is the gaussian fit to the field G dwarf binary distribution (Duquennoy & Mayor 1991). Both the shape of the distribution and the fraction of binaries in each orbital period bin appear similar in Pleiades and field binaries. ing an average system mass and applying a statistical correction factor to take into account projection effects.
Over a limited range of orbital periods, the distribution of semi-major axes in cluster binaries again appears to be consistent with that of field G-dwarf binaries (Duquennoy & Mayor, 1991). The comparison between cluster and field binaries can be further investigated by considering not only visual binaries but spectroscopic binaries as well. This has been done for ZAMS clusters, where the low-mass stars have already been searched for short-period companions. Within statistical uncertainties, the results reinforce the similarity between cluster and field binaries, as shown for the Pleiades in Figure 1.
In order to improve the statistics, we combined the results obtained for several ZAMS clusters. This yields the orbital period distribution shown in Figure 2 for visual binaries. While the distribution of period in cluster binaries agrees with that of field binaries over the orbital range 10 4.4 -10 6.4 days, a paucity of wide (sep ≥ 300 AU) cluster binaries appears relative to the field at the 2.5 σ level. Such a deficit cannot be related to a detection bias, since the distant companions of wide binaries are the most easily detected. The uncompleteness correction which was applied here affects only the first orbital bin of the distribution.
This difference between field and cluster binaries is reinforced by the evidence that the orbital period distribution peaks at a smaller value in clusters than in the field (Patience et al. 2000). The mode of the distribution is 5 AU for cluster binaries and 40 AU for field binaries, the former being estimated under the assumption that all clusters, including the Hyades, can be combined into a single sample. Furthermore, Scally et al. (1999) have shown that there are no very wide binaries (sep ≥ 2000 AU) in the Trapezium cluster.
Hence, even though clusters at various evolutionary stages appear to host a roughly similar fraction of low-mass binary systems, which itself is similar to Figure 2.
The orbital period distribution for solar-type binary systems in three main sequence clusters (α Per, the Pleiades and Praesepe). To ensure identical detections limits for all clusters, only results from adaptive optics surveys are considered here. Completeness corrections mostly affect the first period bin, and are similar for all clusters. Error bars represent 1 σ Poisson uncertainties. The dashed curve is the gaussian fit to the field G-dwarf binary population (Duquennoy & Mayor 1991).
the field binary fraction, the distribution of orbital periods appears to exhibit small but significant differences between clusters and field binary populations.

Mass ratio distribution
The mass of the primaries and secondaries of binary systems on the ZAMS can be reliably estimated from their near-infrared flux using main sequence massluminosity relationships. In order to increase the statistical significance of the derived mass ratio distribution, the combined results for three clusters observed with similar set-ups are shown in Figure 3. The observed distribution appears to be essentially flat over the range q = M b /M a = 0.1-1. This agrees with the q-distribution derived by Duquennoy & Mayor (1991) for low-mass field binaries if the limited sensitivity of our observations, which prevents us from detecting tight systems with low mass ratios, is taken into account.
Contrary to main sequence binaries, young low-mass systems frequently exhibit significant continuum excesses at near-infrared wavelengths and this may affect the derivation of mass ratios from near-IR flux ratios. However, it is usually assumed on a statistical basis that both stars in a young binary system exhibit similar excesses, and that the overall flux ratio distribution can be safely transformed into a mass ratio distribution. Baraffe et al. (1998)'s models were used to get the mass-luminosity relationship relevant for such young stars. Both in the Taurus star-forming region (Leinert et al. 1993) and in the Trapezium cluster (Prosser et al. 1994), the observed q-distribution is consistent with being flat down to q = 0.1, and therefore not different from the ZAMS and field estimates.
Current results thus suggest that the mass ratio distribution is similar for all low-mass binary populations studied so far, independent of their evolutionary Visual binaries in young stellar clusters 5 Figure 3.
The mass ratio distribution derived for various populations of low-mass binaries. Shaded histogram: field G-dwarf distribution from Duquennoy & Mayor (1991) corrected for the detection limits of current adaptive optics binary surveys in ZAMS clusters. This correction removes some of the low-q binaries, and thus flattens the intrinsic distribution. Solid histogram: adaptive optics observations of ZAMS cluster members (α Per, Pleiades, and Praesepe), without applying any completeness correction; Dashed histogram: T Tauri stars in Taurus from Leinert et al. (1993); Dotted histogram: low-mass members of the Trapezium cluster from Prosser et al. (1994). All distributions are undistinguishable within statistical error bars. status and environmental conditions (field or clusters). No peak is found in the distribution towards q ≈ 1 nor towards low q. Furthermore, the ZAMS clusters mass ratio distribution does not seem to vary with separation for visual binaries (Patience et al. 2000).

Massive binaries
Studies conducted so far have mostly focussed on the low-mass binary population of clusters. More recently, however, the investigation of massive binaries has started, with the additional difficulty of a huge contrast between the bright primary and the low-mass companion which is a major limitation of current massive binary surveys.
The recent HST study of thirteen OB stars in the Trapezium cluster by Preibisch et al. (1999) already revealed eight visual companions with separations ranging from 15 to 450 AU. The binary frequency of high-mass stars in this cluster thus appears significantly higher than that of their low-mass counterparts (Prosser et al. 1994), and the difference is further enhanced when completeness corrections are taken into account. Similarly, an adaptive optics survey of about fifty OB stars in the distant cluster NGC 6611 revealed eight companions in the separation range 200-2000 AU (Duchêne 2000), indicative of a high binary fraction compared to G-type field dwarfs if the orbital period distribution is the same for high-mass and low-mass binaries. A study of young, mostly isolated, Herbig AeBe stars conducted by Bouvier & Corporon (2000) yields an uncor- Mass ratio distributions for young B stars located in the Trapezium cluster (Preibisch et al. 1999), in NGC 6611 (Duchêne 2000) and isolated Herbig Be stars (Bouvier & Corporon 2000). Low q are much more common in high-mass binaries than in solar-type binaries. However, there are hints that the mass ratio distribution moight be different for cluster and isolated young massive stars.
rected binary frequency which is significantly higher than that of lower mass Main Sequence stars for both Be and Ae PMS stars.
The mass-ratio distribution of massive binaries can be estimated from the flux ratios measured by imaging surveys. The q-distribution derived for massive binaries with B0-B7 primaries is shown in Figure 4. The sample includes members of the Trapezium cluster and NGC 6611 as well as isolated Herbig Be stars, with a semi-major axis ranging from 100 to 2000 AU. The mass ratio distribution is strongly peaked towards small values: only about 25 % of the binaries have q > 0.5. Patience et al. (2000) obtained a similar trend for A and F stars in various ZAMS clusters. This distribution is dramatically different from that of low-mass binaries, indicating that the primary's mass is a relevant parameter when investigating the formation and evolution of binary systems.

Interpretation
The observed properties of young binary systems hold clues to their formation mechanism and their early evolution. In the following, we consider both classes of models, binary formation and dynamical evolution of clusters, and compare their predictions with observations.

Dynamical evolution of stellar clusters
The fact that all clusters studied so far exhibit a similar fraction of visual binaries regardless of their age rules out a long-term decrease of the binary frequency past the first million years after gravitational collapse. In addition, since similarlyaged loose associations and clusters display different binary frequencies, it is tempting to conclude that environmental effects play a key role in fixing the binary content of a star forming region.
Visual binaries in young stellar clusters 7 Figure 5.
Comparison of the observed orbital period distribution for low-mass stars in ZAMS clusters to the end state of Kroupa et al. (1999)'s simulations of the dynamical evolution of a Trapezium-like cluster. Models represent respectively a virial equilibrium (model A1), a rapidly expanding (model B1) and a collapsing cluster (model C1). Most simulations disrupt wide binaries, in good agreement with the observations, to the exception of model B1 which only disrupts those binaries with orbital period larger than about 10 7.5 days.
N -body simulations of the dynamical evolution of stellar clusters were ran by Kroupa et al. (1999) and compared with observations of the Trapezium cluster. They found that close encounters between cluster members could efficiently disrupt binaries in less than a million years while the number of binaries would remain constant afterwards. This results from the fact that the number of encounters falls off dramatically after one relaxation time. If the initial low-mass binary fraction is of the order of 100 % at the end of the protostellar collapse, such a scenario would naturally explain the high binary frequency observed in T Tauri associations: in these regions, the stellar density is several orders of magnitudes too low to generate numerous enough close-by interactions so as to impact on the initial binary content (Clarke & Pringle 1991). In contrast, frequent gravitational encounters in dense clusters would rapidly erode the initially high binary fraction and lead to a binary frequency similar to that of the field after a million years of evolution or so.
The observed properties of binary populations described in Section 2 further support the hypothesis that dynamical evolution of clusters may largely drive the binary properties. In dynamical models, wide binaries are preferentially disrupted, because their binding energy is small compared to the average kinetic energy of the cluster (see the review by P. Kroupa in this volume). This prediction agrees with the observed distribution of orbital periods for binaries in ZAMS clusters which exhibits a paucity of wide systems, as shown in Figure 5. The higher binary fraction found for massive stars is also in qualitative agreement with this model since the potential well of massive primaries is deeper than that of solar-mass stars, making it more difficult to disrupt more massive binaries. The "softening boundary" which delineates hard and soft systems, the latter being prone to disruption, is thus moved towards wider separations.
A number of issues remain however to be addressed before accepting this paradigm. For instance, dynamical models predict that low-q systems are preferentially disrupted, introducing a non-negligible modification in the shape of this distribution. This constrasts with the derivation of similar mass ratio distributions for low-mass binaries in clusters and associations. Quantitatively, Kroupa (1995) estimated that 70 % of the initial binaries must have a mass ratio q < 0.4 in order to account for the observed distribution of field binaries. This is unlike the q-distribution observed for T Tauri associations. Another puzzling result is the fact that all clusters seem to have very similar binary frequency and orbital period distributions. Hence, binary properties appear to be weakly dependent on small differences in initial conditions. Dyamical models have focussed on reproducing the binary population of the Trapezium cluster and it remains to be seen whether the same results would be obtained for slightly less dense clusters such as IC 348.
According to these models, wide binaries found in the solar neighbourhood cannot have formed in clusters, since they would have been rapidly disrupted. One thus has to assume that they were born in loose associations that have subsequently dissipated in the field. However, the field binary population does not appear to be a simple combination of cluster and association members: a weighted mixture of these 2 populations that reproduces the observed fraction of tight and wide binaries also yields a large excess of systems in the 5-50 AU separation range compared to field binaries. The issue of the efficiency of star formation in clusters as opposed to associations is still open.

Binary formation models
If the dynamical evolution of clusters is responsible for the observed binary properties, then the formation process itself must yield a large initial binary fraction. Furthermore, the masses of the components must be taken at random from the initial mass function, as suggested by Kroupa et al. (1995). We note that the observations of high-mass binaries are qualitatively consistent with random pairing, with a limited number of q > 0.5 pairs, though this result is still preliminary due to the small sample size. We will now discuss whether binary formation mechanisms do predict such properties for initial binaries, prior to cluster's evolution.
The formation of a stellar cluster probably results from the fragmentation of the parent molecular cloud into a large number of individual cores. In this environment, wide binary systems can form either by capture or through the fragmentation of a small collapsing core. Models considering capture in small-N (N < 10) subclusters predict that the two most massive members are usually associated in a binary system, while the other stars are ejected (McDonald & Clarke 1993). This leads to a mass ratio distribution which either peaks towards unity or is roughly flat, depending on the value of N . However, taking into account the dissipative effect of protostellar disks during close encounters leads to a secondary mass distribution that mimics the IMF (McDonald & Clarke 1995). Furthermore, the model predicts a higher binary frequency for more massive stars. Similar simulations made by Sterzik & Durisen (1998) additionally predict that the mass ratio distribution is more heavily weighted towards low-q values in higher mass systems. All these predictions are consistent with the observations, indicating that stars in clusters may form in compact subclusters. However, such a scenario does not end up with a 100 % initial binary frequency, since some single stars are ejected from each subcluster.
Capture during the relaxation of small-N subgroups appears to be the model that best fits current observations, and especially the random pairing of secondary masses. The alternative model of scale-free fragmentation (Clarke 1996) predicts that binary properties do not depend on the primary mass, which conflicts with the observations. Binary properties could also result, at least partly, from the accretion of residual material onto two gravitationally bound seeds (Bate & Bonnell 1997, Bate 2000. However, these models predict a larger number of q ∼ 1 binaries for tighter systems as well as for more massive binaries, which is not verified in the observational trends. Finally, binaries formed through disk fragmentation should preferentially have small separations and low mass ratios, a prediction which contrasts with the apparent lack of a dependence of the mass ratio distribution upon separation.

Summary
The advent of high-angular resolution imaging techniques have offered the opportunity to systematically survey binaries in young clusters, thus complementing spectroscopic studies. We reviewed some statistical properties of visual binaries such as their frequency, orbital period distribution and mass ratio distribution; both low-mass and high-mass stars were considered. Observational results can be summarized as follows: the low-mass binaries in cluster have similar properties as Main Sequence (field) binaries, except for a significant deficit of systems with a separation larger than 300 AU. High-mass stars host many more companions than do solar-type stars, and most massive binaries have a small mass ratio. These observational constraints are consistent with the results of numerical simulations describing the early dynamical evolution of stellar clusters provided that the initial binary frequency is of the order of 100 % for low-mass stars. Diskassisted capture in small subclusters is the formation model that best approaches the initial conditions required to account for the observations. In both classes of models, however, several issues still need to be investigated and additional simulations over a wider range of physical conditions have to be run before these scenarios can be considered as offering an acceptable new paradigm for the star and binary formation process. In parallel, future observations of young embedded clusters will help to derive the properties of protobinaries at an even earlier stage of evolution than discussed in this review, which will undoubtly lead to new clues to their formation.