## 2. Methodology## 2.1. Numerical approachOur 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 ## 2.2. Initial conditionsFor a given
In these units, "Hénon's virial radius" (Hénon 1972) For and equal masses, is equal to the true virial radius, defined exactly in that limit as the harmonic mean of the particle separations. 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" and the "crossing time" can be defined by ## 2.3. Mass spectraLet be the probability of choosing a star
with mass
In dimensionless integration units, this mass spectrum is
effectively tapered at the upper and lower mass cutoffs due to the
= 1 renormalization (see Fig. 1). We
distinguish distributions of dimensionless masses
For each , we then use Eq. (4) to
determine the individual stellar masses subject
to the constraint that the choices sum to to
within 5%. We refer to the combined dimensional
mass spectrum as , and its nondimensional
counterpart as . Fig. 1 intercompares
## 2.4. Numerical integrationsFor each type of few-body system, the particle orbits of all 1,000 realizations are integrated for 300 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 . 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 ; but we find it preferrable to integrate all systems for the same number of , 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 . 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 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 . After 300 , the errors are checked. Any system with a relative energy error % is reintegrated for the same initial conditions with a tolerance parameter of . For the worst case in this paper ( 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 ( to 30) of the systems have relative energy errors 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 after 300 . ## 2.5. Hierarchical virtual-particle analysisA 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 There is an implicit assumption in this analysis that, after 300 , 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 singles TS: triple plus singles Q Q BBS: two binaries plus singles TB: triple plus binary U: undecayed nonhierarchical system E: relative energy error % 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 approachAnalytic 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 resulting from the decay of nonhierarchical bound triples should have the form where is the fraction of binaries in the
interval . In various contexts, Eq. (6) is
referred to as "Heggie's Law" (e.g., Valtonen & Mikkola 1991). The
dependence of this formula can be understood
crudely as follows. A factor 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 reflects the
energy-dependence of (the cross-section times
the relative velocity) for encounters that give rise to binary
energies of . The binaries are produced mostly
by close encounters for which one expects the cross-section to vary
roughly as , where while the encounter speed 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- We expect 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 , binary formation in the BS mode is the only true decay channel, and so (8) should be directly applicable. For 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 , the semi-major axis distribution expected from Eq. (8) is in dimensionless units, where is the
fraction of BS binaries with For (4, 5), (1/8, 2/25). The maximum of (9) is at , and the average value is given by . For the BS case of , the escape speeds of the
single star and of the binary
are completely determined from 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 and binaries as and where and are the fractions of singles and binaries in the intervals and , respectively. For 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 is the average number of single escapers per single-emiting decay. By analogy to the momentum argument () that applies exactly only for the three-body case, we can generalize (14) to all remnant types as where The subscript There is an important difference in the final kinetic energy
distribution for the case of very unequal masses with any where and is now the ensemble average mass for
remnants of multiplicity (see Eq. [7] of Saslaw et al. 1974). We now generalize in (9) to the case of a mass spectrum. Replace in (9) by , the dimensionless product of the masses in a binary remnant, and assume that (9) applies with the same for every possible value of . Then, the full separation distribution can be constructed by folding together (9) and the distribution of binary mass products, where gives the fraction of binary remnants in . We get where is the fraction of binaries in
© European Southern Observatory (ESO) 1998 Online publication: September 30, 1998 |