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

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4. Applications

In 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 fractions

Going 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[FORMULA] [FORMULA] 0.2[FORMULA]

MK:  M & K-type stars, 0.2[FORMULA] [FORMULA] 0.5[FORMULA]

KG:  Solar-type stars, 0.5[FORMULA] [FORMULA] 1.2[FORMULA]

F+:  F-type stars & earlier, 1.2[FORMULA] [FORMULA] 10[FORMULA]

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 systems whose most-massive star (primary) is in the mass bin, and we emphasize that the multiplicity fractions are only for systems with primaries in the mass bin by adding a subscript p.


Table 10. Multiplicity fractions for remnant systems with primaries in four mass bins.

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 N without dissipation, it is very difficult to achieve a reasonable distribution of BFp's along the main sequence. Also, the lowest-mass stars (or brown dwarfs) allowed by the fragmentation IMF tend to be entirely single and to dominate the population of remnants escaping with moderate to high speeds. In real SFR's, however, one expects there to be a spectrum of cloud masses undergoing collapse. Then the total few-body system mass that results is constrained by the clump mass, and lower-mass stars can sometimes end up as the most-massive and hence binary-producing stars in their system. The CM systems are in fact constructed in this way and, as a result, exhibit a more realisitically uniform distribution of SF's and BFp's among the mass bins. It is still the case that the lowest-mass objects are almost entirely singles, but now there are also nonnegligible BFp's even for the MK bin. There is a distinct tendency for both f's to produce rather large fractions of higher multiplicity systems for F+ with [FORMULA] and 5.

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 [FORMULA]% (Leinert et al. 1997) to [FORMULA]% (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 [FORMULA]% (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 N's. However, the general trends of the BFp distributions in our CM simulations are in agreement with observed trends. Because [FORMULA] with a CM spectrum comes closest to observed multiplicity values, we concentrate the subsequent analysis on this case.

4.2. Mass ratios

Observations also suggest trends in the distribution of binary mass ratio [FORMULA] as a function of primary mass. For G-dwarf primaries (DM), the number of secondaries increases as q decreases and peaks near [FORMULA] 0.2-0.4. Very low-mass companions ([FORMULA] 0.0-0.1) seem to be rare. On the other hand, FM show that the q-distribution for M-dwarf primaries is relatively flat and possibly even decreases as q decreases; but their results are restricted to the range [FORMULA] 0.4-1.0 to avoid a selection-loss bias for the lowest-mass secondaries. Fig. 8 shows the q-distributions from the N = 4 CM spectrum for our four primary mass bins. Our MK and KG mass bins exhibit q-distributions with characteristics similar to those reported by FM and DM, respectively.

[FIGURE] Fig. 8. The distribution of binary mass ratios for N = 4 and the CM spectrum. Curve types refer to the primary mass bins defined in the text.

This match of the mass-ratio distributions is a natural consequence of binary formation in decaying, small-N clusters. In 1968, van Albada already described the general tendencies. Using a modern IMF and assuming pure dynamical biasing of the formed binary, McDonald & Clarke (1993, 1995) found they were able to fit the observed distributions of BFp and q, provided they included effects of dissipative encounters. More recently, Valtonen (1997, 1998) found reasonable agreement in the q-distributions of wide binaries by applying Monaghan's (1976a, 1977) statistical theory for three-body encounters. In particular, he showed that the Tokovinin (1997) multiple-star catelog exhibits trends with primary mass similar to those in our Fig. 8. Our work strengthens these results, because we are deriving the binary properties directly with a high-precision numerical integrator. We match the observed trends in BFp and q without appeal to the dissipative effects of gas by using the two-level IMF of the CM spectrum.

4.3. Speed and binary semi-major axis distributions

Results 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 [FORMULA] 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 [FORMULA][FORMULA] appears to be consistent with the stellar mass fraction within typical fragmenting cloud cores. Typical separations of the "seeds" in fragmentation calculations are [FORMULA] 100 A.U.; SD used [FORMULA] A.U. These choices of [FORMULA] and [FORMULA] imply [FORMULA] km/s.

In a given SFR, it is plausible that the [FORMULA] and [FORMULA] 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 N-values. It is beyond the scope of this paper to define such relations or distributions. Instead, for illustrative purposes, we assume there is a relation of the form [FORMULA] and consider three limiting cases which allow easy scaling and a direct application of results from Sect. 3:

  1. [FORMULA] constant. We choose the SD value [FORMULA] 125 A.U. For this case, [FORMULA] constant and [FORMULA].

  2. [FORMULA] 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., [FORMULA] constant. A relation of this type could result if fragmentation during isothermal collapse occurred below some threshold ratio of thermal to gravitational energy.

  3. [FORMULA] constant. Using the SD choices, [FORMULA] ergs. For this case, [FORMULA].

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 [FORMULA] and the CM spectrum. The full lines correspond to the [FORMULA] 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 [FORMULA]. So, the [FORMULA] curve in Fig. 9a results simply from multiplying the dimensionless speeds (as in Figs. 7a and 7c) by the constant [FORMULA] 4.6 km/s. This distribution is bracketed by the [FORMULA] (dotted) and [FORMULA] (dashed) cases, which give broader and narrower distributions, respectively. In Fig. 9b, it is the [FORMULA] case for which the dimensionless a's need only be multiplied by the constant scale factor [FORMULA]. The [FORMULA] and [FORMULA] cases are broadened by the [FORMULA]-dependence of the [FORMULA] scale factor. Figs. 9c and d show the speed and semi-major axis distributions with [FORMULA] for each of the mass bins defined in Sect. 4.1.

[FIGURE] Fig. 9a-d. Scaled results for the CM spectrum and N = 4: a single remnant speeds for [FORMULA] = 0, 1 and 2 in km/sec b binary semi-major axis distributions for [FORMULA] = 0, 1 and 2 in A.U. c remnant speeds for [FORMULA] = 1, curve types refer to mass bins d binary semi-major axis distributions for [FORMULA] = 1, curve types refer to mass bins.

In Table 11, we compile results for the following observable quantities by mass bin for the particular case [FORMULA]: 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 [FORMULA] (in dec's).


Table 11. One-dimensional velocity dispersions for single stars and binaries in km/s, plus means in A.U. and standard deviations in dec's of the binary semi-major axis distributions for [FORMULA] and [FORMULA] km/s. Stars indicate poor binary statistics for that mass bin; no entry means a complete absence of binaries.

A number of important conclusions can be drawn from Table 11:

  1. For [FORMULA], typical one-dimensional velocity dispersions for single stars are in the range 3-4 km/s, largely independent of [FORMULA], N, and the mass bin.

  2. For [FORMULA], 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).

  3. 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.

  4. The typical mean semi-major axis is [FORMULA] 30 A.U. with a shift toward larger separations ([FORMULA] 100 A.U.) for the earlier-type stars and toward smaller separations ([FORMULA] 10 A.U.) for the latest types.

  5. 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 non Gaussian and has a power-law high-velocity tail. Conclusions 1, 2, and 4 above are sensitive to the choice of [FORMULA]; conclusions 3 and 5 are not.

4.4. Implications for star formation

If 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 [FORMULA] or [FORMULA] 104 yrs, and will be preserved over a much longer time, provided that no further dynamic interactions with other components of the SFR takes place. In this sense, our results should be more relevant to loose T Associations like Taurus than to densely clustered SFR's like Orion. However, molecular clouds exhibit a broad spectrum of length scales and core masses. The few-body systems considered here might be subcomponents of larger clusters or groups, like those considered by Kroupa (1995, 1998), and could have a significant influence on their dynamical evolution.

Independent of any scaling assumptions, we have shown that the typical binary semi-major axis for remnants of few-body decay is [FORMULA] 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 [FORMULA] example given in the preceeding section, the difference in dispersion velocities for single stars (3-4 km/s) and binaries ([FORMULA] 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 107 yrs could fill a sphere with a radius [FORMULA] 50 pc, a volume not easily surveyed by conventional observational methods for discovering young stars. If [FORMULA], even single G, F, and A-type stars can obtain roughly the same large dispersion speeds as those of single M and K-type stars, a result which surprised even us. The recent ROSAT-discoveries of broad spatial distributions of young stars (see Sect. 1.3) could, in part, be a manifestation of large dispersion velocities caused by few-body decays. Although we have not included substellar mass objects in our IMF, it is clear from our results for the M mass bin that these would all be ejected as single bodies. A spectacular HST/NICMOS observation shows a very cool object near a young binary in Taurus. This could be an escaping planet or brown dwarf which has been dynamically ejected in the manner considered here (Terebey et al. 1998).

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 N-body decay has played a significant or dominant role in shaping these properties of binary stars.

4.5. Strengths and limitations

We 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 N's, [FORMULA]'s, [FORMULA]'s, [FORMULA], etc. In this paper, we have considered only rather unrealistic zero angular momentum systems; but we plan to include effects of extreme fragment geometries and initial velocity fields in our next paper. A more fundamental limitiation is that we omit all effects of continued gas accretion by the few-body system (McDonald & Clarke 1995, Bonnell et al. 1997), a difficiency that will be difficult to remedy with our current orbit integrator. We also ignore the possibility of star/star collisions or tidal interactions and have not attempted to keep track of closest approaches. For most of our resulting binaries, with separations of 10's A.U., this is probably not a concern, because low-mass stars typically have radii only a few times larger than the Sun's. Of course, if the stars have disks, star/disk and disk/disk collisions will be important (McDonald & Clarke 1995). For these reasons, our pure point-mass treatment is not directly applicable to the origin of close binaries ([FORMULA] A.U.).

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 [FORMULA], except that the BFp's and q-distributions show more realistic trends with mass when a two-step IMF is used. As realistic conditions and outcomes of cloud collapse become better defined, our mappings to final stellar remnant distributions, as well as the tools to produce additional such mappings, will be available.

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Online publication: September 30, 1998