## 4. ApplicationsIn Sect. 3, we used only dimensionless variables. Here, we first consider several aspects of the results in physical units which depend only on the physical stellar masses, namely the multiplicity fractions as a function of mass bin and the mass ratios of the remnant binaries. Then, we apply an additional physical scaling to derive speed and binary separation distributions in physical units. Although we make some comparisons with observations, the analyses in this section are only illustrative of the possible effects of few-body decay on star formation. Definitive results require input not yet available concerning initial cloud core conditions and reliable statistical mappings from these conditions to few-body outcomes of cloud collapse and fragmentation. ## 4.1. Multiplicity fractionsGoing back to the original physical masses, we divide the stars into four mass bins which give roughly equal numbers of stars for the CM spectrum. The labels are meant to reflect the approximate spectral types that stars of these masses would have on the main sequence: M: Late M-type stars, 0.1 0.2 MK: M & K-type stars, 0.2 0.5 KG: Solar-type stars, 0.5 1.2 F+: F-type stars & earlier, 1.2 10 We do not bin more finely in order to have good statistics.
Table 10 summaries the multiplicity fractions within each group.
The "total" here refers to the total number of remnant
The results for the MS mass spectrum exhibit features described by
McDonald & Clarke (1993, see also van Albada 1968a, Valtonen 1997,
1998). Because of dynamical biasing, when simply selecting stars from
a realistic IMF for small An increasing binary fraction with primary mass is an established observational fact, at least for main-sequence stars. For M-dwarf primaries the reported binary frequencies are in the range of % (Leinert et al. 1997) to % (FM). FM find a ratio of about 4:1 for the appearance of binary to higher-order systems. The G-dwarf binary frequency is higher, estimated to be % (DM). For masses in the F+ range, observations suggest even higher binary frequencies, but at lower statistical significance (see, e.g., Abt et al. 1991, McAlister et al. 1993, Petr et al. 1998). We do not expect that the statistical outcome of our simulations
will match the observed quantities in every respect. We know, for
instance, that our initial conditions in this paper (zero angular
momentum, spherical, no remnant gas) are probably not realistic
outcomes of fragmentation. It is also likely that cloud fragmentation
will lead to a distribution of ## 4.2. Mass ratiosObservations also suggest trends in the distribution of binary mass
ratio as a function of primary mass. For
G-dwarf primaries (DM), the number of secondaries increases as
This match of the mass-ratio distributions is a natural consequence
of binary formation in decaying, small- ## 4.3. Speed and binary semi-major axis distributionsResults presented so far are applicable to any gravitationally bound, point-mass system evolving under mutual gravitational forces. Two characteristic scales are necessary to determine a complete set of physical units in this case. These can be a characterisitc length and time, or an energy and a mass, or any other two independent quantities in Eqs. (1) to (3). As discussed in the preceeding section, binary fractions and mass ratio distributions are fully described by scaling back to its dimensional value. Speed and binary semi-major axis distributions can then be given in dimensional units only when an additional scaling is applied, but there are many ways to do this. It is at this level where we are forced to introduce further assumptions about the conditions that are realistic and typical for early stages of low-mass star formation. In SD, we gave arguments for the choice of typical total system masses and sizes based on outcomes of fragmentation calculations. A total system mass appears to be consistent with the stellar mass fraction within typical fragmenting cloud cores. Typical separations of the "seeds" in fragmentation calculations are 100 A.U.; SD used A.U. These choices of and imply km/s. In a given SFR, it is plausible that the and
resulting from fragmenting collapses are
related due to correlations in pre-collapse cloud properties. It is
also possible that there might be some dispersion in these parameters,
as well as a variety of -
constant. We choose the SD value 125 A.U. For this case, constant and . -
constant. Again, we use the SD value of 3.3 km/s. This scaling law results in a constant specific energy for each few-body system, i.e., constant. A relation of this type could result if fragmentation during isothermal collapse occurred below some threshold ratio of thermal to gravitational energy. -
constant. Using the SD choices, ergs. For this case, .
In Figs. 9a and b, we compare the speed and the binary
semi-major axis distributions in physical units for these three
scaling relations using and the CM spectrum. The
full lines correspond to the scaling law.
Fig. 9a displays the speed distribution for single stars.
According to the nondimensionalization resulting from (1) and (2), the
velocity unit in Sect. 3 corresponds to .
So, the curve in Fig. 9a results simply
from multiplying the dimensionless speeds (as in Figs. 7a and 7c)
by the constant 4.6 km/s. This distribution is
bracketed by the (dotted) and
(dashed) cases, which give broader and
narrower distributions, respectively. In Fig. 9b, it is the
case for which the dimensionless
In Table 11, we compile results for the following observable quantities by mass bin for the particular case : a) one-dimensional velocity dispersions for single stars (in km/s), b) one-dimensional velocity dispersions for primaries in binaries (in km/s), c) the mean binary semi-major axis (in A.U.), d) the standard deviation of the binary semi-major axis distribution expressed as a distribution in (in dec's).
A number of important conclusions can be drawn from Table 11: -
For , typical one-dimensional velocity dispersions for single stars are in the range 3-4 km/s, largely independent of , *N*, and the mass bin. -
For , the velocity dispersions vary with stellar mass from 2-3 km/s (for F-type stars) up to 5-6 km/s (for M-type stars). -
The COM velocity dispersions for binaries are lower than those of single stars but are not negligible. They are typically a factor 3 to 6 times smaller than those for the single stars. -
The typical mean semi-major axis is 30 A.U. with a shift toward larger separations ( 100 A.U.) for the earlier-type stars and toward smaller separations ( 10 A.U.) for the latest types. -
The standard deviation of the semi-major axis distribution in a mass bin is typically a factor of 2.
It is important to remember that, although we quote velocity
dispersions here, the velocity distribution is distinctly ## 4.4. Implications for star formationIf multiple fragmentation during cloud collapse, followed by
dynamic decay of the few-body system, is a frequent mode of star
formation, then specific signatures will be imprinted on the
endproducts. This imprinting should mostly occur soon after
fragmentation, typically within tens of or
10 Independent of any scaling assumptions, we have shown that the typical binary semi-major axis for remnants of few-body decay is 5 times smaller than the original few-body system size. This step bridges the length scale from typical cloud collapse outcomes (100's A.U.) to typical binary separations (10's A.U.), a reduction in scale which is otherwise not well understood. Our application of physical scalings which are consistent with
available observational and theoretical constraints yields interesting
results for remnant velocities, binary frequencies, and binary mass
ratios. For the example given in the
preceeding section, the difference in dispersion velocities for single
stars (3-4 km/s) and binaries (
1 km/s) would cause spatial segregation of these remnants over
time. A careful analysis of the BF in a SFR should therefore take into
account the possibility that single stars may be dispersed over a few
times larger radial extent around the birth site than multiple stars
of the same age. Single TTS as old 10 By adopting the reasonable assumption that there is a cloud mass
spectrum which constrains the choice of fragment masses, leading to
the two-step IMF of the CM spectrum, we have shown that observed
trends in binary fractions and mass ratio distributions can be matched
even by simple gas-free few-body decays. This strengthens the classic
notion (see Sects. 1.1 and 1.3) that ## 4.5. Strengths and limitationsWe have characterized the statistical outcome of few-body decay,
with emphasis on multiplicity fractions, speed distributions, and
binary properties. Here, the decay is viewed as one step in the
overall process of star formation. We cannot present definitive
results because critical input is lacking, namely, the distribution of
cloud collapse outcomes, the Despite these limitations, we have, in some respects, been more
successful than expected. For the zero angular momentum case, we now
have precise results which can be readily adopted by other
researchers, like McDonald & Clarke (1993) or Valtonen (1997,
1998), who study implications of few-body decay without recourse to
orbit integration. Statistical selections of mass configurations can
now be mapped to multiplicity fractions which include first-order
corrections to pure-BS and pure dynamical biasing assumptions. We have
also derived and tested precise analytic approximations for the speed
distributions of all remnants and for the semi-major axis distribution
of binaries. Many aspects of these results have proven to be fairly
insensitive to the precise choice of , except
that the BF © European Southern Observatory (ESO) 1998 Online publication: September 30, 1998 |