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

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3. Numerical results

3.1. General

Before we discuss the results for each system type in more detail, we summarize information that allows a statistical comparison of the outcomes. All cases are cold ([FORMULA]), nonrotating ([FORMULA]), and spherical ([FORMULA]) and are distinguished only by the choice of N and f.

Decay Channels.

Table 1 gives the absolute number of systems found in each decay channel (see Sect. 2.5) after our typical integration time of 300 [FORMULA]. Summing over all decay channels results in the 1,000 systems that we started with, and so the statistical significance of the numbers can be gauged directly. Of course, more decay channels are accessible for larger N. We note the following general trends:

  • The dominant decay mode is BS = one binary plus [FORMULA] singles; but TS, QpS, and BBS are not negligible when allowed.

  • For a fixed mass spectrum, larger N increases the number of higher multiplicity remnants (hierarchical triples and quadruples) in the endstates.

  • The broader [FORMULA] (EM[FORMULA]CM[FORMULA]MS, see Fig. 1), the higher the relative fraction of BS endstates compared with channels that have higher multiplicity remnants.


Table 1. Decay channel distribution.

Multiplicity Fractions. Using the above numbers, we can derive the overall fractions of remnant singles, binaries, triples, and higher-order systems. For easy comparison with observational work, we define the multiplicity fraction as the total number of each remnant type divided by the total number of remnants of all types. This definition of binary frequency BF is compatible with that of DM and others (e.g., Köhler & Leinert, 1998) who define the BF as the number of secondaries per 100 sample stars. In addition to a BF, we define in a similar fashion a single-star frequency SF, a triple frequency TF, etc. In Table 2 we give these fractions, in addition to the total number of remnants ("total") of all types.


Table 2. Multiplicity fractions.

This table is constructed from the information given in Table 1 by totaling the number of remnants of different types. Remarkably, the influence of the assumed [FORMULA] on the multiplicity fractions is much weaker than it is for the distribution of outcomes over decay channels. The multiplicity fractions are dominated by the influence of N. As a measure of the importance of higher-order remnants, consider that, if every decay followed the BS channel, we would get a BF of 0.5 (0.33, 0.25) for N = 3 (4, 5). The relative deviation from this pure-BS value increases with increasing N.

Velocity Dispersions. In Table 3, we give the one-dimensional dispersions for the endstate velocities of singles (S), binaries (B), and triples (T) as measured by the standard deviation [FORMULA] in the one-dimensional velocity distributions. The [FORMULA]'s from the three independent velocity components have been averaged together.


Table 3. One-dimensional velocity dispersion [FORMULA].

Two dominant effects are readily identified:

  • The velocity dispersions are most strongly affected by the shape of [FORMULA]. The broader [FORMULA], the more low-mass single escapers there are with relatively high velocities.

  • Smaller N-body systems give higher dispersion velocities for single escapers. Comparing [FORMULA] and [FORMULA] one finds a relative difference in [FORMULA] of 20-50%, in agreement with SD for first single-star escapers.

Dynamical Biasing. It is well known that binaries formed through dynamical capture within disintegrating small-N clusters preferentially consist of the first and second-most massive stars. Let us order the stars in our systems so that [FORMULA] is the most massive, [FORMULA] the second-most massive, and so on.

The assumptions that all decays are BS and that all binaries actually consist of [FORMULA] and [FORMULA] enables one to derive analytic binary mass-ratio distributions for a specified f (van Albada 1968a, McDonald & Clarke 1993). For [FORMULA] 3, Monaghan's statistical theory predicts deviations from this assumption of pure dynamical biasing. In Table 4, we assess the magnitude of these deviations in our own data by giving the fractions of all binary remnants built up from [FORMULA], [FORMULA], and [FORMULA]. For [FORMULA], pairings with even lower-mass components are possible, but rare. Deviations from pure dynamical biasing are in the range of 10-20%. There is a tendency for dynamical biasing to strengthen as N increases and [FORMULA] broadens.


Table 4. Dynamical biasing in binary systems.

3.2. Length of integration

For star formation, the mapping discussed in SD from pre-collapse cloud conditions to the size of the initial N-body system gives [FORMULA] yrs (for [FORMULA] and [FORMULA] A.U., see Sect. 4.3). So, beyond 1,000 [FORMULA], we compromise the applicability of our results to star formation. Narrowing the choice of our standard integration time to 300 [FORMULA] is then a compromise between good statistical characterization of the endstates and the quality (and computational cost) of the integrations.

Table 5 shows multiplicity fractions for the equal mass case after 100, 300, and 1,000 [FORMULA]. For each N, the same set of 1,000 initial conditions was used. The meanings of the various fractions is the same as in Table 2, except that here we include the systems with large energy errors in the multiplicities. As discussed in Sect. 2.4, the energy errors usually occur well after decay within high-eccentricity remnant binaries. We exclude these systems from other analyses because we are interested in energy distributions. Here, however, they must be retained in order to determine how quickly the fractions of various remnant types converge.


Table 5. Time evolution of the multiplicity fractions for equal masses.

As shown in the last column of the table, the number E of systems with bad energies increases steadily with integration time. It is clear from [FORMULA] that we need to adopt an integration time considerably less than 1,000 [FORMULA].

On the other hand, Table 5 illustrates that there are significant numbers of remnants with high multiplicity which take many hundreds of [FORMULA] to decay. Note, for instance, the TF column for [FORMULA] and the QF column for [FORMULA] and 5. Delayed decay of these systems boosts the SF and BF fractions with integration time. It is generally true, however, that the remnant fractions for the lower multiplicity types have converged to within [FORMULA] or so by 1,000 [FORMULA] and that the reservoir of higher multiplicity systems is insufficent to change these numbers by much more than this amount in subsequent times. At 300 [FORMULA], with the exception of the particularly difficult case of [FORMULA] equal masses, the fractions for remnants of lower multiplicity do not differ from those at 1,000 [FORMULA] by more than about 0.01. So, the convergence by 300 [FORMULA] is good to within a few 0.01's or better. Furthermore, by stopping at 300 [FORMULA], we keep the additional uncertainty due to rejection of the E systems at the same order or less. The uncertainties caused by a finite integration time are not too different from the intrinsic statistical errors caused by having only 1,000 system realizations, and so a larger number of integrations would not significantly improve our results.

Another measure of the completeness of our few-body system decays is the longevity of the resulting multiple remnants. Eggleton & Kiseleva (1995, hereafter EK) developed a relatively simple analytic criterion for the stability of hierarchical triple systems through fits to a large number of orbit integration results. Their criterion gives an approximate critical value for the ratio of outer to inner orbit periods below which a triple is "two-unstable", i.e., changes its hierarchical structure, usually by decay, within 102 outer orbit periods. The critical period ratio depends only on the mass ratios and orbit eccentricities, which we know from our virtual-particle analysis.

Table 6 gives the total number of hierarchical triple


Table 6. EK triple stability criterion for equal masses (energy errors [FORMULA]%).

remnants for equal mass systems at different times. In parentheses is the number of those triples that are "two-unstable" according to EK. Clearly we do not follow all these systems for 102 outer orbit periods; but we do expect that the bulk of the triples that do decay should be those satisfying the EK criterion, as borne out by the table. The interpretation is complicated by several factors. The first is that, for [FORMULA] and 5, the number of triples is replenished somewhat over time by the decay of higher-order systems. Nevertheless, there is convergence toward having mostly two-stable hierarchical triple remnants. The second complication is that, for all N, our zero angular momentum initial conditions create a relatively large number of systems with inner and outer orbit eccentricities [FORMULA]. EK only test their criterion for [FORMULA]. Moreover, in this context, "unstable" only means that the system changes its hierarchical structure; it does not necessarily imply that the system will decay. For unequal masses, similar tests give stronger results, where by 1,000 [FORMULA] there are relatively few triples left which are two-unstable by the EK criterion.

For the rest of this paper, all our experimental results are quoted for 300 [FORMULA], unless otherwise noted.

3.3. Equal masses

3.3.1. [FORMULA]

The case of [FORMULA] equal masses provides comparisons with other work and the tightest constraints on the development of analytic fits. Two sets of integrations with different modern codes are discussed in Anasova & Orlov (1994, hereafter AO) and Anasova et al. (1994, hereafter AOA).

Due to the different technical emphases of these researchers, however, there are only a few parameters available for us to compare. As shown in Table 7, our results for the average values of the final binary e and a agree about as well with AO and AOA as AO and AOA agree with each other. For AO, the precentage of systems decaying into a binary plus unbound single is determined by various "escape criteria". Our percentage decay, given at 300 [FORMULA], is based on virtual-particle analysis. In fact, by 1,000 [FORMULA], we find that more like 92.1% of the three-body systems have decayed. Of the three codes, we would expect our Mikkola & Aarseth chain regularization scheme to be best for treating the close three-body encounters responsible for smaller values of a; and this may be responsible for our lower value of [FORMULA]. Overall, the differences in Table 6 are acceptable for describing the effects of few-body dynamics on star formation.


Table 7. Comparison with other [FORMULA] results for equal masses.

The energetics of the resulting remnants are determined by the binary energy distribution [FORMULA].

Sect. 2.6 outlined arguments for [FORMULA], 7/2, or 9/2 in the analytic [FORMULA] (Eq. [8]). Rather than use some statistical measure which might build in subtle biases, we determine the overall quality of fit by eye in plots of [FORMULA], [FORMULA], and [FORMULA] and limit ourselves to selecting only one of the three discrete values of [FORMULA] for each N and f. Fig. 2 compares our experimental [FORMULA] with these power-law distributions normalized to produce the same number of binaries. We clearly favor [FORMULA].

[FIGURE] Fig. 2. Binary binding energy distribution for [FORMULA] equal masses. The dashed, solid, and dotted curves are analytic fits with [FORMULA], 7/2, and 9/2, respectively.

This choice is reinforced by comparing the computed speed distributions for remnant singles and binaries with Eqs. (12) and (13), as illustrated in Fig. 3.

[FIGURE] Fig. 3. Speed distribution of single and binary remnants for N = 3 equal masses. Curve types have the same meanings as in Fig. 2.

To avoid clutter, only the [FORMULA] curve is plotted for the binaries. For equal masses, the mapping from [FORMULA] to a is one-to-one. So, the same 7/2 choice of [FORMULA] must be used to fit [FORMULA] for self-consistency, and in fact it is also very good (see Fig. 5 below). The general properties of the [FORMULA] and [FORMULA] in our experiments have been noted in earlier three-body research, going back to Saslaw et al. (1974); but our determination of the [FORMULA] governing the underlying physics seems better, perhaps because we follow the decay process longer and do not use approximate escape criteria.

3.3.2. [FORMULA] and 5

As N increases, it is not clear a priori that Eq. (8) should work as well. Table 1 does indicate the predominance of one hard-binary formation event in disrupting the systems. We have examined figures like Fig. 2 where we include only the BS channel binaries, all binaries (BS, BBS, and TB), and all binaries plus the inner binaries of triples. Some of the BBS and TB binaries are only weakly bound and do not at all fit a power law, but their numbers are small. Ragardless of how we construct the [FORMULA] diagram, we again judge that [FORMULA] gives the best fit to the hard-binary distribution for both [FORMULA] and 5. Even more surprising is that Eq. (15) gives a remarkably good fit to the velocity data for singles, binaries, and triples, as illustrated in Fig. 4 for [FORMULA]. One exception is a noticable excess of relatively slow escaping singles which also occurs to a lesser degree for [FORMULA].

[FIGURE] Fig. 4. Speed distribution for single, binary and triple remnants for N = 5 equal masses. Only analytic fits for [FORMULA] are shown.

The goodness of the analytic fit does have a physical implication. The underlying assumption that the single escapers share equally in the escape energy must be roughly true overall. One might have expected instead to have a far greater excess of slow escapers and only one star per system carrying away most of the energy.

The binary separation distribution for all binary remnants, along with the [FORMULA] version of Eq. (9), is shown in Fig. 5 for [FORMULA], 4, and 5. Binary subcomponents of triples and quadruples are not included. As N increases, smaller binary separations are achieved primarily because m is smaller. There is more smearing away from the power law as N increases in part because one of the binaries from the BBS channel and binaries from the TB channel are usually only loosely bound.

[FIGURE] Fig. 5. Binary semi-major axis distributions for N = 3, 4 and 5 equal masses together with [FORMULA] analytic fits. The arrows along the top indicate the locations of [FORMULA] for N = 3, 4, and 5 from right to left.

Remarkably, after studying plots like Fig. 2

3.4. Realistic mass spectra

3.4.1. The [FORMULA] [FORMULA]-distribution

for unequal masses, we conclude that, for the low angular momentum ([FORMULA]) spherical cases considered in this paper, the binary energy distributions all obey essentially the same [FORMULA] power law regardless of N or f. To illustrate this point with the best possible statistics, Fig. 6 combines the data for the 6,863 binary remnants from all nine experiments listed in Table 1. Except for the loosely bound binaries contributed by the BBS and TB channels, the [FORMULA] law evidently is an excellent fit. Preliminary experiments for high angular momentum systems and for systems with extreme geometries, which are the subject of the next paper in this series, do show different power laws; and we defer until then a discussion of what factors determine the appropriate [FORMULA].

[FIGURE] Fig. 6. The distribution of binary binding energies for all N and f combined. Curve types have the same meanings as in Fig. 2.

3.4.2. Escape speeds

Table 8 summaries two important features of the escape speed distributions for single stars. Typical speeds are indicated by quoting the median speed (labelled 50%), and the nonGaussian high-velocity tail of the speed distribution is characterized by giving the speed for which 90% of the stars move more slowly in the cumulative distribution. Shown, for


Table 8. Escape speed distributions for single stars.

comparison are the same quantities obtained from the analytic distributions (12), (14), and (17) for the best-fit [FORMULA]. As a quantitive measure of the superiority of this fit, the same velocities are also tabulated for [FORMULA] and 9/2. The agreement between the data and [FORMULA] is usually within about 10% and tends to get better as N increases. With only a few exceptions, the speeds for 5/2 or 9/2 tend to be off by more like 20-50%.

Table 8, of course, manifests the same physical trends as Table 3.

Escape speeds increase as [FORMULA] becomes broader, due to the lower masses of the escapers, and decreases with increasing N because there are more objects sharing the energy released by hard-binary formation. About half the single escapers end up with speeds [FORMULA] [FORMULA], but the distribution is significantly nonGaussian. Fully 10% of the stars have speeds which are greater than the median by factors of 2.2 for equal mass cases and of 2.3-2.5 for unequal masses. The precise ratio expected for the analytic [FORMULA] fit is 2.18. About one percent of the escapers have speeds over four times the median speed. Figs. 7a and 7c show the full escape speed distributions for all single, binary, and triple remnants drawn from [FORMULA].

[FIGURE] Fig. 7a-d. Results for the clump mass spectrum CM: remnant speed (a ) and binary semi-major axis (b ) distributions for [FORMULA]; (c ) and (d ), the same for [FORMULA]. Curve types have the same meanings as in Fig. 2.

3.4.3. Binary and multiple characteristics

Figs. 7b and 7d illustrate the binary semi-major axis distributions for [FORMULA] and 5 with the CM spectrum and can be compared with Fig. 5 for the case of equal masses. In Fig. 7b, the best fit is obtained with the semi-analytic [FORMULA] "shark-fin" function, where the numerically determined [FORMULA] is used in Eq. (20). The fit is not as good for [FORMULA]. The apparent shift of the experimental results to about 20% smaller values of a is due in part to the fact that, even for equal masses, the [FORMULA] peak is not as sharp and is shifted to the left of [FORMULA]. The agreement is better for CM with [FORMULA] and for all the MS cases.

Table 9 gives the peaks and FWHM of the analytic functional fits to the semi-major axis distributions. In all cases, these are within 10-20% of the values which would be judged by eye from the histograms. Note that, as in the diagrams, the distributions are here considered to be per unit [FORMULA] not per unit da interval.


Table 9. Peak and full-width-at-half-maximum of the binary separation distributions.

For the realistic unequal mass case, the sharp power law becomes smeared out largely because of the range of possible binary mass products [FORMULA]. For all unequal mass cases, the peak of the semi-major axis distribution is [FORMULA] and the FWHM [FORMULA] with a tendency for the peak to shift to smaller a as N increases. In other words, the binaries formed by the decay of [FORMULA] to 5 systems tend to have semi-major axes about five times smaller than the original few-body system size [FORMULA], distributed with a FWHM that covers a range of about a factor 3 or 4 in a.

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Online publication: September 30, 1998