## 3. Numerical results## 3.1. GeneralBefore 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 (), nonrotating
(), and spherical () and
are distinguished only by the choice of
Table 1 gives the absolute number of systems found in each
decay channel (see Sect. 2.5) after our typical integration time
of 300 . 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 -
The dominant decay mode is BS = one binary plus singles; but TS, Q _{p}S, 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 (EMCMMS, see Fig. 1), the higher the relative fraction of BS endstates compared with channels that have higher multiplicity remnants.
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 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
Two dominant effects are readily identified: -
The velocity dispersions are most strongly affected by the shape of . The broader , the more low-mass single escapers there are with relatively high velocities. -
Smaller *N*-body systems give higher dispersion velocities for single escapers. Comparing and one finds a relative difference in of 20-50%, in agreement with SD for first single-star escapers.
The assumptions that all decays are BS and that all binaries
actually consist of and
enables one to derive analytic binary
mass-ratio distributions for a specified
## 3.2. Length of integrationFor star formation, the mapping discussed in SD from pre-collapse
cloud conditions to the size of the initial Table 5 shows multiplicity fractions for the equal mass case
after 100, 300, and 1,000 . For each
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 that we need to adopt an integration time considerably less than 1,000 . On the other hand, Table 5 illustrates that there are significant numbers of remnants with high multiplicity which take many hundreds of to decay. Note, for instance, the TF column for and the QF column for 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 or so by 1,000 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 , with the exception of the particularly difficult case of equal masses, the fractions for remnants of lower multiplicity do not differ from those at 1,000 by more than about 0.01. So, the convergence by 300 is good to within a few 0.01's or better. Furthermore, by stopping at 300 , 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 10 Table 6 gives the total number of hierarchical triple
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 10 For the rest of this paper, all our experimental results are quoted for 300 , unless otherwise noted. ## 3.3. Equal masses## 3.3.1.The case of 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
The energetics of the resulting remnants are determined by the binary energy distribution . Sect. 2.6 outlined arguments for , 7/2,
or 9/2 in the analytic (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
, , and
and limit ourselves to selecting only one of
the three discrete values of for each
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.
To avoid clutter, only the curve is plotted
for the binaries. For equal masses, the mapping from
to ## 3.3.2. and 5As
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 version of Eq. (9), is shown in
Fig. 5 for , 4, and 5. Binary subcomponents
of triples and quadruples are not included. As
Remarkably, after studying plots like Fig. 2 ## 3.4. Realistic mass spectra## 3.4.1. The -distributionfor unequal masses, we conclude that, for the low angular momentum
() spherical cases considered in this paper, the
binary energy distributions all obey essentially the same
power law regardless of
## 3.4.2. Escape speedsTable 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
comparison are the same quantities obtained from the analytic
distributions (12), (14), and (17) for the best-fit
. As a quantitive measure of the superiority of
this fit, the same velocities are also tabulated for
and 9/2. The agreement between the data and
is usually within about 10% and tends to get
better as Table 8, of course, manifests the same physical trends as Table 3. Escape speeds increase as becomes broader,
due to the lower masses of the escapers, and decreases with increasing
## 3.4.3. Binary and multiple characteristicsFigs. 7b and 7d illustrate the binary semi-major axis
distributions for 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
"shark-fin" function, where the numerically
determined is used in Eq. (20). The fit
is not as good for . The apparent shift of the
experimental results to about 20% smaller values of 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 not per unit
For the realistic unequal mass case, the sharp power law becomes
smeared out largely because of the range of possible binary mass
products . For all unequal mass cases, the peak
of the semi-major axis distribution is and the
FWHM with a tendency for the peak to shift to
smaller © European Southern Observatory (ESO) 1998 Online publication: September 30, 1998 |