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

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2. Methodology

2.1. Numerical approach

Our basic approach is to compute the orbital evolution of a large number of realizations (1,000) for various particular types of initial few-body system. The different types of system are distinguished by the total number of particles N, the assumed mass spectrum of the constituent stars, the shape of the system (spherical, prolate, or oblate), the magnitude of the particle random motions, and the magnitude and direction of any systematic rotation. Although, procedurally, our initial few-body system realizations are in physical units, the units are transformed for integration to a dimensionless form which highlights the generality of the results. We characterize the complete decay of the few-body system by orbit integrations which are long enough so that the system has broken into a subset of spatially well-separated, long-lived remnants which are not bound to each other, which themselves are internally bound, which suffer no further close interactions with other remnants, and which are far enough apart that the gravitational interactions between them are negligible. We find that integrations over 300 initial crossing times [FORMULA] suffice to converge the multiplicity fractions of remnants to within a few hundreths by intercomparing results after 100, 300, and 1,000 [FORMULA] for a few cases. Of course, some of the remnants are hierarchical multiples which are metastable and will decay further on longer time scales; and some systems (usually only a few per 1,000) remain nonhierarchical and undecayed even after 1,000 [FORMULA]. For applications to star formation, a few long-lived metastable configurations and rare singular outcomes are not a serious concern.

2.2. Initial conditions

For a given N, we specify the masses, geometry, and velocities of the systems by a Monte Carlo realization of 1,000 systems from a set of assumed initial distributions.

Masses. N masses [FORMULA], [FORMULA] to N, with total mass [FORMULA], are first chosen randomly from a prescribed mass spectrum (see 2.3 below).

Geometry. The positions of the masses are then chosen randomly within an ellipsoidal volume with semi-axes [FORMULA]. For most cases in this paper, we consider "spherical" systems in the statistical sense that [FORMULA] = [FORMULA] = [FORMULA]. Effects of severe distortions in the fragment distributions, as suggested by some collapse calculations, will be explored in the next paper of this series. At this point, the total gravitational energy W of the system is computed.

Initial Velocities. Once positioned, the particles can be given randomly chosen velocity components and/or a systematic uniform rotation about a specified axis through the center of mass (COM), and the coordinates and velocities are transformed into the COM frame. Velocities are now rescaled so that the total random kinetic energy [FORMULA] and the total rotational kinetic energy [FORMULA] satisfy prescribed values of [FORMULA] and [FORMULA]. If desired (see Sect. 4.3), the lengths and velocities may be further rescaled so that some target system size or total energy [FORMULA] or energy per unit mass [FORMULA] is achieved, while preserving [FORMULA] and [FORMULA]. However, this has no effect on the integrations, which are done in dimensionless units. For most integrations in this paper, [FORMULA], i.e., the systems are "cold" and have zero net angular momentum.

Dimensionless Units. The system is transformed to dimensionless code units by setting (Heggie & Mathieu 1986)


In these units, "Hénon's virial radius" (Hénon 1972)


For [FORMULA] and equal masses, [FORMULA] is equal to the true virial radius, defined exactly in that limit as the harmonic mean of the particle separations. [FORMULA] is a good measure of the physical size that the few-body system would have if it relaxed into virial equilibrium, and so it provides a natural reference scale for the integrations. Similarly, the "virial speed" [FORMULA] and the "crossing time" [FORMULA] can be defined by


2.3. Mass spectra

Let [FORMULA] be the probability of choosing a star with mass M in the mass interval dM. Three choices of f are used in this paper.

Equal Masses (EM). This case is considered for two reasons. Some precise and even analytic results are available in this limit for [FORMULA]. Also, it is necessary to define the base state for equal masses in order to determine which aspects of the endstate distributions are sensitive to [FORMULA] and which are not.

Miller-Scalo Power Law (MS). While the mass function for the fragment systems that result from cloud collapse is still not known, there is evidence (DM, FM) that the secondary stars in low-mass binaries are chosen randomly from the overall stellar initial mass function (IMF). So one set of calculations is done with a power-law f that represents a crudely realistic IMF for low-mass stars. The exponent is chosen to match the slope of the Miller-Scalo (1979) IMF near solar-mass stars,



In dimensionless integration units, this mass spectrum is effectively tapered at the upper and lower mass cutoffs due to the [FORMULA] = 1 renormalization (see Fig. 1). We distinguish distributions of dimensionless masses m from their dimensional counterparts by an asterisk, in this case [FORMULA].

[FIGURE] Fig. 1. The MS (solid) and CM (dashed) mass spectra for our [FORMULA] systems histogrammed in even logarithmic intervals. Heavy curves are the [FORMULA] in physical units; thin curves are the dimensionless [FORMULA]. Shown for comparison is part of the Kroupa et al. 1991 IMF (dotted curve).

Clump Mass Spectrum (CM). Molecular cloud complexes are know to be clumpy on a variety of scales with a possibly fractal structure that gives a power-law clump mass spectrum with an index of about -1.5 to -2.0 (e.g., Elmegreen & Falgarone 1996). Thus the cloud cores that collapse and fragment may themselves have a power-law mass distribution. To mimic this case, we first choose an [FORMULA] from


For each [FORMULA], we then use Eq. (4) to determine the individual stellar masses [FORMULA] subject to the constraint that the choices sum to [FORMULA] to within [FORMULA] 5%. We refer to the combined dimensional mass spectrum as [FORMULA], and its nondimensional counterpart as [FORMULA]. Fig. 1 intercompares f and [FORMULA] for both the MS and CM mass spectra of our [FORMULA] systems. The dotted curve shows the Kroupa et al. (1991) IMF normalized to produce 5,000 stars in the mass range of Eq. (4). Our [FORMULA] prescription results in a similar IMF.

2.4. Numerical integrations

For each type of few-body system, the particle orbits of all 1,000 realizations are integrated for 300 [FORMULA] using the Mikkola & Aarseth (1990, 1993) chain regularization method. By guaranteeing large spatial separations of the decay remnants, these long integrations permit an unambiguous and automatic identification of the remnants themselves, a nontrivial exercise for [FORMULA]. By minimizing residual gravitiational interactions between remnants, it also allows us to determine accurate escape speeds (the limit of infinite remnant separations), without large, ill-defined corrections. Most systems actually decay within dozens of [FORMULA]; but we find it preferrable to integrate all systems for the same number of [FORMULA], rather than apply decay criteria in a time-dependent way during integration.

Unfortunately, the length of the integrations does create some difficulties. Despite the accuracy of the integration scheme, large energy errors can accumulate in systems where a high-eccentricity binary forms because it can make many thousands of close periapse passages over 300 [FORMULA]. The errors usually become significant only well after the complete decay of the system has occurred. We are not interested in mapping initial and final points in phase space with high precision, but we do wish to determine binary semi-major axis distributions and the escape speeds of remnants. These distributions are affected by the distribution of final energies, so we keep the relative error in [FORMULA] below 0.1% by the following procedure. For each type of few-body system, the initial 1,000 integrations are performed with a tolerance parameter (see Mikkola & Aarseth 1993) of [FORMULA]. After 300 [FORMULA], the [FORMULA] errors are checked. Any system with a relative energy error [FORMULA]% is reintegrated for the same initial conditions with a tolerance parameter of [FORMULA]. For the worst case in this paper ([FORMULA] equal masses), no more than a few dozen reintegrations per 1,000 systems are necessary. The final energy errors are again checked. Usually, after this step, no more than a few percent ([FORMULA] to 30) of the systems have relative energy errors [FORMULA] 0.1%. These systems are flagged and omitted from any further analysis which might be sensitive to energy. The vast majority of the integrations are of high quality, as demonstrated by the fact that the median relative energy error for 1,000 system integrations is typically [FORMULA] after 300 [FORMULA].

2.5. Hierarchical virtual-particle analysis

A statistical characterization of complete few-body system decay requires the unambiguous identification of remnant subsystems at the endpoint of the integrations. For this purpose, we developed an analysis technique involving successive passes through the endstate data.

We start with the final positions and velocities of the N particles. For each pairing of the stars, we decide whether the pair is a bound binary subsystem. To be so-classified, the pair must satisfy the following two criteria: i.) In the COM of the pair considered as an isolated system, the two-body orbital energy must be negative. ii.) No other star is present within a spherical volume centered on either member of the pair with a radius equal to the major axis of the pair's relative orbit. In all cases, at this first level of the hierarchical treatment, we find that binary pairs are uniquely identified. These binaries are then treated as virtual particles for the next level of analysis. The virtual particle is given the COM velocity and position for the pair and a mass equal to the sum of the masses. Binary orbit characteristics are computed at this stage in dimensionless units and stored. If [FORMULA] binaries are identified in the first pass, there are now [FORMULA] "particles", real stars plus binaries, in the next level of analysis. The procedure for the next pass is the same; all pairs of particles, whether representing binaries or real singles, are again checked according to the same two criteria (i) and (ii). At this level, it is possible to discover new subsystems where a binary virtual particle is found to be bound to another binary (a "binary quadruple") or to a single star. This idenitifies hierarchical triples and hierarchical pairs of binaries. The orbit characteristics of these systems are also computed and stored. These two levels of hierarchical structure analysis suffice for [FORMULA]; but, for [FORMULA] and 5, one more level is required to identify "planetary quadruples", where a hierarchical triple has a fourth component orbiting at a large distance.

There is an implicit assumption in this analysis that, after 300 [FORMULA], the substructure of the remnants will be hierarchical. If no virtual particles are found, it is assumed that the system has remained undecayed. At the end of the analysis, a self-consistency check is made to ensure that the virtual particles identified have positive kinetic energy in the system COM and that this positive kinetic energy is larger in magnitude than the absolute error in the total system energy.

Altogether the following categories of decay products are possible:

BS: binary plus [FORMULA] singles

TS: triple plus [FORMULA] singles

QpS: planetary quadruple plus [FORMULA] singles

QbS: binary quadruple plus [FORMULA] singles

BBS: two binaries plus [FORMULA] singles

TB: triple plus binary

U: undecayed nonhierarchical system

E: relative energy error [FORMULA]%

Here, single, binary, triple, and quadruple refer to independent bound remnants, which are unbound from other remnants. Triples and quadruples are understood to be hierarchical. We do not attempt to distinquish between hierarchical and nonhierarchical bound quintuples. One can think of these different categories as the "decay channels" for the few-body systems.

2.6. Analytic approach

Analytic predictions for escape speed and binary separation distributions can be made from the pioneering work of Heggie (1975). Heggie showed by approximate analytic arguments that, in the limit of low system angular momentum, the distribution of final binary energies [FORMULA] resulting from the decay of nonhierarchical bound triples should have the form


where [FORMULA] is the fraction of binaries in the interval [FORMULA]. In various contexts, Eq. (6) is referred to as "Heggie's Law" (e.g., Valtonen & Mikkola 1991). The [FORMULA] dependence of this formula can be understood crudely as follows. A factor [FORMULA] arises simply from expressing the binary relative motion phase space in terms of orbital elements and integrating out the eccentricity and angular coordinate terms. The other factor of [FORMULA] reflects the energy-dependence of [FORMULA] (the cross-section times the relative velocity) for encounters that give rise to binary energies of [FORMULA]. The binaries are produced mostly by close encounters for which one expects the cross-section to vary roughly as [FORMULA], where a is the semi-major axis of the resulting binary relative orbit given by


while the encounter speed v at large distances is just [FORMULA] and independent of [FORMULA].

In the same 1975 paper, Heggie also considered the creation rate of hard binaries from soft binaries through close two-body encounters between one component of the binary and a third star in a large-N cluster. The creation rate in the interval [FORMULA] is then proportional to [FORMULA] due to a different energy-dependence of [FORMULA]. One might expect some binary formation events to be better described in this way, especially for [FORMULA]. The binary creation and destruction rates in Heggie's 1975 paper are only approximate (e.g., Heggie & Sweatman 1991). Monaghan (1976a, b) and Nash & Monaghan (1978) took the different point of view that, statistically, the binaries formed from three-body systems would be uniformly distributed through allowed phase space by the quasi-ergodic nature of the close interactions. A simplified version of this argument, where one ignores angular momentum conservation and assumes three-dimensional systems, yields a [FORMULA] distribution based on the phase-space volume for binary orbits. Given these uncertainties, we, like other researchers in similar contexts (Mikkola & Valtonen 1986), consider a "generalized Heggie's law"


We expect [FORMULA] to be in the range 5/2 to 9/2.

In the remainder of this section, we use Eq. (8) to generate analytic escape speed and binary separation distributions. Our approximate arguments result from efforts to obtain reasonable fits to our experimental results. For [FORMULA], binary formation in the BS mode is the only true decay channel, and so (8) should be directly applicable. For [FORMULA] and 5, the dominant decay channels are BS and TS, where a single close binary (either a free remnant or the smaller component of the hierarchical triple) releases the bulk of the energy that unbinds the system, and so again we expect (8) to apply.

For equal masses [FORMULA], the semi-major axis distribution [FORMULA] expected from Eq. (8) is


in dimensionless units, where [FORMULA] is the fraction of BS binaries with a in the interval da and where


For [FORMULA] (4, 5), [FORMULA] (1/8, 2/25). The maximum of (9) is at [FORMULA], and the average value [FORMULA] is given by [FORMULA].

For the BS case of [FORMULA], the escape speeds of the single star [FORMULA] and of the binary [FORMULA] are completely determined from a by momentum and energy conservation as


One consequence of momentum conservation is that the single star has exactly two-thirds of the final kinetic energy that unbinds the sytem. Using Eq. (9), we can then write the escape speed distributions for singles [FORMULA] and binaries [FORMULA] as




where [FORMULA] and [FORMULA] are the fractions of singles and binaries in the intervals [FORMULA] and [FORMULA], respectively.

For [FORMULA] and 5, there is more freedom in dividing momentum and energy among the remnants. Suppose that, overall, the energy available to the singles is shared by them equally; but otherwise the relative kinetic energy share of the singles is still, on average, about two-thirds. Then,


where [FORMULA] is the average number of single escapers per single-emiting decay. By analogy to the momentum argument ([FORMULA]) that applies exactly only for the three-body case, we can generalize (14) to all remnant types as




The subscript µ here refers to the multiplicity of the remnant (single, binary, triple, or quadruple), [FORMULA] is the mass for remnant type µ, [FORMULA] is the average number of remnants of multiplicity µ per decay that produces a remnant of that type, and superscript e refers to equal masses.

There is an important difference in the final kinetic energy distribution for the case of very unequal masses with any N, because the single escapers tend to be much less massive than the binaries and multiples. As a result, the single escapers tend to carry off almost all the kinetic energy that unbinds the system. So, for unequal masses,




and [FORMULA] is now the ensemble average mass for remnants of multiplicity µ. To make the dependence on basic system parameters more evident, we note that, when we do not introduce [FORMULA] in Eq. (8), the only change in (17) and (18) is that


(see Eq. [7] of Saslaw et al. 1974).

We now generalize [FORMULA] in (9) to the case of a mass spectrum. Replace [FORMULA] in (9) by [FORMULA], the dimensionless product of the masses in a binary remnant, and assume that (9) applies with the same [FORMULA] for every possible value of [FORMULA]. Then, the full separation distribution can be constructed by folding together (9) and the distribution [FORMULA] of binary mass products, where [FORMULA] gives the fraction of binary remnants in [FORMULA]. We get


where [FORMULA] is the fraction of binaries in da.

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998