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Astron. Astrophys. 339, 95-112 (1998)

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1. Introduction

Although many basic principles of few-body dynamics were established by the mid-1970's, there are two major reasons for revisiting the problem. First, in the context of point-mass dynamics, the complete decay of nonhierarchical few-body systems with [FORMULA] has not been studied in a comprehensive and definitive way using modern integration schemes. Second, in the context of star formation, it is clear that molecular cloud core collapse leads to multiple fragmentation under a wide variety of conditions. Understanding the relationship between molecular cloud conditions and the final multiplicity distribution of stars produced by a star forming region (SFR) requires that we know how post-collapse protostellar fragment clusters evolve. These two considerations provide the impetus for our work. Because this is the first paper in a series on this topic, we discuss the motivations in some detail.

1.1. Classic work

The decay of few-body systems is a classic problem in numerical astrophysics. One question addressed early by van Albada (1968a, b) is the formation of double stars through the disintegration of small groups. Already in this work, van Albada presented the hypothesis that small star clusters with diameters [FORMULA] 100-1000 A.U. which result from fragmentation during star formation could account for the observed properties of binary stars through their subsequent decay. Several authors focused on the three-body problem (e.g., Anosova 1969, Szebehely 1972) and developed various "escape criteria" to determine when one star becomes unbound. Standish (1972) followed the dynamical evolution of triple stars and determined distributions of several quantities of astrophysical interest like times of disintegration, velocities and masses of escapers, and the semi-major axes and eccentricities of the remnant binaries. Significant insight into the decay process was gained by theoretical analyses of three-body encounters (Heggie 1975) and by a statistical theory for three-body disruption (Monaghan 1976a, b). Quantitative comparisons with numerical results of the same vintage yielded satisfactionary agreement with regard to escape velocities and final binary energies and eccentricities (Saslaw et al. 1974, Valtonen 1976). The importance of the initial system total energy was recognized early, but the system angular momentum can also be important (e.g., Standish 1972, Monaghan 1976b, Mikkola & Valtonen 1986). In order to understand the production and abundance of triple stars through few-body decay, Harrington (1974, 1975) also performed a limited number of numerical experiments for [FORMULA] and 5.

The general behavior of decaying few-body systems, as characterized by the end of the 1970's, includes the following:

  • The characterisitic system lifetime is on the order of tens of crossing times.

  • Encounters between stars redistribute energy in such a way that some stars escape and leave behind a more tightly bound subsystem, usually a binary.

  • The remnant binary generally consists of the two most massive stars of the initial system, an effect called "dynamical biasing". The escaping single stars are generally the least massive ones.

  • Tightly-bound binaries are rare. Spectroscopic binaries are difficult to explain by few-body decay.

  • Stability of higher-order systems is generally achieved by hierarchical organization.

  • Outcomes of numerical experiments are roughly consistent with the analytic work of Heggie and Monaghan.

The three-body problem has been revisited in the 1980's and 90's with modern computer codes, but with more of an emphasis on scattering problems than system decay (see reviews by Anosova & Kirsanov 1991 and Valtonen & Mikkola 1991; see also McMillan & Hut 1996 and earlier papers in the same series). Sophisticated orbit integration techniques permit precise treatment of frequent close encounters during the evolution of triple systems. A statistical approach is generally used to investigate the fate of systems which span a range of the initial phase space. There has been considerable modern work on the stability of planar and three-dimensional three-body systems (e.g., Black 1982, Anosova & Orlov 1994, Anosova et al. 1994, Kiseleva et al. 1994a, b, Eggleton & Kiseleva 1995). Some more specific initial configurations have been considered in great detail, such as the Pythagorean three-body problem (Aarseth et al. 1994a, b). However, modern orbit integrators have not yet been used for a thorough characterization of the complete decay for small-N clusters, including escape speed distributions and internal structures of all remnant subsystems.

1.2. Fragmentation calculations

As van Albada foresaw, it is now widely accepted, through results of hydrodynamics calculations, that fragmentation during molecular cloud core collapse is the initial step in the production of binary and multiple stars, at least for systems with large separations (see reviews by Boss 1988, Bodenheimer et al. 1993, Bodenheimer 1995, Burkert et al. 1998). For a wide range of conditions, isothermal cloud collapse is likely to produce more than two fragments. Although an important caveat has been given recently regarding numerical resolution in fragmentation calculations (Truelove et al. 1997, 1998), it is still clear that multiple fragmentation, with fragment numbers of two to five or more, is a likely outcome of protostellar cloud collapse (Bate & Burkert 1997, Burkert et al. 1997). Resulting nonhierarchical fragment configurations are quite varied and include, for instance, rings (Monaghan & Lattanzio 1991), thin strings or filaments (Bonnell et al. 1992, Boss 1993, Burkert & Bodenheimer 1993, Monaghan 1994), and cold thin disks (Boss 1996).

For reasons of computational cost, most collapse calculations are terminated once dense fragments containing only a few percent of the cloud mass have formed. Modern computational techniques have so far permitted only very few well-resolved calculations to be carried through to almost complete accretion of the cloud mass onto the fragments (e.g., Burkert & Bodenheimer 1996, Burkert et al. 1998). Burkert et al. (1998) describe the fragment evolution as "chaotic", involving successive stages of fragment formation and merger. This results in a multiple system at the end of the cloud accretion phase with essentially unpredictable and hence, to some degree, randomized orbital parameters. These collapse calculations demonstrate the formation of nonhierarchical multiple systems, and they provide the necessary step of mapping from interstellar cloud conditions on the 0.01 pc scale to fragment systems with typical scales of order 100 A.U. The parameter space of possible initial cloud states is still too vast for a proper computational characterization of collapse endstates, but it is clear that a wide range of N and of spatial and orbital configurations is possible. Once accretion ends, nonhierarchical stellar multiples will decay. It is this second stage of star formation that is the focus of our work - the mapping of young few-body systems through dynamical decay into stable and long-lived stellar remnants, such as single stars, binaries, and hierarchical multiples. As we will show, this second mapping involves another, fairly well-defined decrease in system scale.

1.3. Observational motivation

Significant information has now accumulated on the multiplicity fractions and binary orbit separations for low-mass stars in the solar neighborhood (Duquennoy & Mayor 1991, Fischer & Marcy 1992, hereafter DM and FM, respectively) and for young stellar objects and pre-main sequence stars in nearby SFR's (as reviewed by Mathieu 1994). The DM binary period distribution for solar-type stars is extremely broad. Recent observations suggest that binary frequencies (BF's) vary significantly among SFR's (Ghez et al. 1992, Leinert et al. 1992, Prosser et al. 1994, Padgett et al. 1997, Petr et al. 1998) and that the width of the separation distribution may be much narrower in individual SFR's than the DM distribution (Brandner & Köhler 1998). The DM distribution may thus be a blend of many narrower separation distributions contributed by different SFR's. There is some indication that the stellar density in SFR's anticorrelates with the observed BF (e.g., Reipurth & Zinnecker 1993, Bouvier et al. 1997, Petr et al. 1998). Two plausible mechanisms have so far been put forward for these differences: the disruption of primordial soft binaries in star clusters (Kroupa 1995) and an intrinsically lower parameter space available for (wide) binary formation in a higher temperature ambient cloud (Durisen & Sterzik 1994).

Both mappings of scale in van Albada's two-step scenario (cloud [FORMULA] fragment system [FORMULA] stellar remnants) need to be characterized in order to develop a theoretical explanation for observed binary and multiple star properties. An attempt to clarify the second process was made by McDonald & Clarke (1993) by using van Albada's analytic statistical approach. They assumed that all clusters produced exactly one binary consisting of the first and second-most massive stars of the original system, rather than perform direct orbit integrations. In a subsequent study, they found that they could match the observed binary mass-ratio distribution and binary fractions when system stars were chosen randomly from an IMF, but only if dissipative encounters were included (McDonald & Clarke 1995). Recently, Valtonen (1997, 1998) found fairly good agreement with mass-ratio distributions for wide binaries and multiple stars using Monaghan's (1976a, 1977) statistical theory of three-body disruption.

The kinematics of young stars may also be affected by the dynamics of few-body fragment clusters. It is generally believed that young, low-mass stars diffuse away from their SFR with velocity dispersions [FORMULA] to 3 km/s (e.g., Jones & Herbig 1979, Hartmann et al. 1991), comparable to the external dispersion velocities of their parental cloud cores. We were originally motivated to consider the possibility of ejecting T Tauri stars (TTS) from their SFR's at higher speeds via few-body dynamics (Sterzik & Durisen 1995, hereafter SD) by the observation that young stars identified in the ROSAT All-Sky Survey (RASS) and confirmed by follow-up optical studies are widely distributed in space around all nearby SFR's examined (Sterzik et al. 1995, Neuhäuser et al. 1995, Alcala et al. 1995, Magazzu et al. 1997, Neuhäuser et al. 1997, Alcala et al. 1997, Covino et al. 1997; for a recent review see Neuhäuser 1997). There are also a number of young stars in the immediate solar neighborhood, far from any site of recent star formation (Sterzik & Schmitt 1997, Soderblom et al. 1998) and a few cases where known kinematics suggest a relatively high-speed ejection (e.g., Neuhäuser et al. 1998). SD concluded that, while such "RATTS" (run-away T Tauri stars) were a likely consequence of few-body decay after multiple fragmentation, dynamic ejection could not explain the large number of "halo" TTS. It now seems likely that the widely distributed young RASS-selected stars represent a Gould Belt population (Sterzik et al. 1998, Guillout et al. 1998). Nevertheless, if multiply fragmenting collapse is a common mode of star formation, then few-body decay dynamics will necessarily have an effect on the kinematics of the young stars and stellar remnants that result.

Formation of stars within larger groups of hundreds to thousands of stars seems to be common (Lada et al. 1991). This can be thought of as a higher level in a hierarchy of star formation and does not preclude the formation of few-body systems through the collapse and fragmentation of subcomponents of larger groups. So, our work is complementary to that of Kroupa (1995, 1998), who has been studying the dynamics of young aggregates with larger numbers of stars ([FORMULA]). Kroupa computes the kinematics and binary properties resulting from the dissolution of such aggregates for various stellar densities and velocities. The inclusion of primordial binaries enhances the proportion of high-velocity escapers. If few-body decay also occurs, it will influence the results through kinetic energy input by the decay remnants and through the presence of "primordial" multiple remnants.

1.4. Goals

Unfortunately, a clean separation between the phase of cloud gas accretion after fragment formation and the stellar dynamic evolution of the fragment system is probably not really possible. Significant accretion onto the fragments may continue for [FORMULA] yrs, which is also an important time interval for orbital evolution. Orbit integrations with continuing accretion (e.g., Bonnell et al. 1997) are desirable but are difficult to do accurately and realisitically for more than a few realizations of possible systems. The disks likely to be present around the stars during the early stellar dynamic phase will also affect the outcome (Clark & Pringle 1991, 1993, McDonald & Clarke 1995) through disk/disk and star/disk interactions, but again it is difficult to study more than a few representations of realistic systems. The philosophy of our own work is to begin with an idealized separation of the two stages and avail ourselves of powerful modern techniques for the computation of pure few-body dynamics. A particular emphasis is identifying features of the endstate distributions which are relatively robust and independent of physical assumptions. From this point of view, it makes sense to begin with a thorough investigation of initially gas-free few-body systems and work systematically toward the inclusion of more physical effects and greater complexity in the calculations.

In SD, we developed a way to generate initial conditions for few-body orbit integrations that mimicked various plausible fragment distributions (velocity fields and geometries) and system sizes seen in cloud collapse calculations. SD then studied only the first single stars detected to be escaping from such systems ("first escapers"). We now set the more ambitious goal of determining the full dynamic decay of few-body systems, starting with relatively well-understood cases and moving systematically toward more realistic conditions. Our aim is to characterize statistically the decay channels, remnant escape speeds, and the internal characteristics of the bound binary and multiple stellar remnants. Sect. 2 of the present paper explains the way in which we specify initial conditions and how we integrate and analyze gas-free systems. Results are presented in Sect. 3 for cold, spherical, low angular momentum [FORMULA], 4, and 5 systems with three different initial stellar mass spectra. Later papers in this series will consider the effects of distorted geometries, systematic motions, and other physics. Sect. 4 discusses applications to star formation, while Sect. 5 summarizes our main conclusions.

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998