2 Methodology

2.1 Deviations from complete dynamical baising

SD98 found that complete dynamical biasing is violated in two ways. First, as N increases, decay channels which involve stable and metastable hierarchical multiples or which yield two binaries become more common. Even so, only one stellar pair generally has the bulk of the original cluster binding energy; and, far and away, the most common decay channels are single stars plus one binary or one multiple fragment. For example, such decays occurred for >93% of the 1000 two-step IMF N = 5 clusters considered in SD98. The second deviation from complete dynamical biasing is that resultant binaries do not always consist of the two most massive stars. Table 1, which is based on data from SD98, shows the fraction of binary decay fragments with different mass pairings for the two-step IMF from that paper. Here M₁, M₂, and M₃ are the first, second, and third-most massive stars, respectively. Pairings of lower mass components are negligible (<1%). The last row of Table 1 is explained below.

 

 

Table 1: Fractions of different binary pairings for the SD98 two-step IMF

N	M1M2	M1M3	M2M3
3	0.80	0.19	0.01
4	0.81	0.15	0.03
5	0.81	0.15	0.04
MC	0.82	0.15	0.03

2.2 Approximate Monte Carlo approach

Integrations like those of SD98 are somewhat computationally intensive, which limits ensembles of integrations to about 1000 systems. Nevertheless, properties of the results described in the above section are fairly robust. In order to consider a wide range of parameter space with much larger numbers of systems for each choice of parameters, we here adopt a simple Monte Carlo (MC) approach, similar to that of McDC93, but modified by incorporation of a two-step IMF and the deviations from complete dynamical biasing given in Table 1. We first choose a CMS $f_{\rm c}(M_{\rm tot})$ and a SMS $f_{\rm s}(M)$, including functional forms and upper and lower mass limits. An ensemble of cluster masses is then generated randomly from $f_{\rm c}$. For each $M_{\rm tot}$, stellar masses M_i are chosen from $f_{\rm s}$ until $M_{\rm tot}$ is matched within some prescribed tolerance $\tau$ according to

$$% (1 - \tau)M_{\rm tot} \le \sum_{i=1}^N M_i \le (1 + \tau)M_{\rm tot}.$$ (1)

We pick the stellar masses in two alternative ways: A) Fixed N. N stellar masses are chosen repeatedly at random from the adopted $f_{\rm s}$ until Eq. (1) is satisfied. B) Variable N. The test in Eq. (1) is made after each stellar mass is chosen. If Eq. (1) is satisfied at any point in the process, then this determines the cluster size N as well as the stellar masses. We impose an upper limit $N_{\rm max}$ on the allowed N. If (1) is not satisfied by the time $N = N_{\rm max}$, a new selection of stellar masses is begun starting with one star. Suppose we now have an ensemble of clusters (typically 100000 for a given choice of $f_{\rm c}$, $f_{\rm s}$, and N or $N_{\rm max}$). We assume, as in McDC93, that each cluster decays into one binary plus single stars. In light of the SD98 results, this is consistent with two interpretations: 1) most multiples resulting from the dominant decay channels (singles plus one multiple) do eventually decay into one hard binary plus singles. 2) Even though some multiples may be stable and not decay, the hardest binaries in the multiples are effectively included in the BF statistics, with only small errors introduced by treating the extra members of multiples as if they were single stars. For field stars, the DM91 $BF = 0.57 \pm 0.09$ does in fact include a contribution from triples and quadruples at about the same level as the stated uncertainty in BF. In young stellar populations, multiples may be more common and have a greater effect on the statistics (see Zinnecker & Mathieu 2001 and the references therein). For each cluster, we arrange the masses $\{M_i\}$ so that they decrease monotonically as i increases from 1 to N. When forming the binary, we assign probabilities to the various mass pairings according to the MC entries in Table 1. Statistics on the binary fraction as a function of mass BF(M) for different choices of primary mass can then be compiled. The resultant two-step IMF f_i(M) determined from all the stars in the clusters will be different from $f_{\rm s}$, because of the additional constraint imposed by $f_{\rm c}(M_{\rm tot})$ through Eq. (1). For N = 4 and 5, Table 2 compares the SD98 two-step IMF cluster integrations with the Monte Carlo procedure. The mass bins are the same as those used in SD98. Masses for this and all other tables are given in solar units. The MC results are based on 100000 fixed-N clusters using the same CMS and SMS as in SD98. In Table 2, BF⁺ represents BF + TF + QF where TF and QF are the triple and quadruple fractions obtained in SD98. This is the proper quantity to compare with BF(MC) because the MC approach assumes that multiples are effectively included in BF through either of the interpretations given above. The fractions in these tables are computed such that the star in the given mass range is the primary. The MC numbers have statistical errors of less than a percent, while the SD98 numbers are uncertain by a few percent.

 

 

Table 2: Comparison of SD98 binary fractions with the Monte Carlo (MC) approach

N	mass bin	BF(SD98)	BF+(SD98)	BF(MC)
4	0.1-0.2	0.00	0.00	0.00
	0.2-0.5	0.15	0.19	0.19
	0.5-1.2	0.52	0.68	0.64
	>1.2	0.73	0.93	0.89
5	0.1-0.2	0.01	0.01	0.00
	0.2-0.5	0.15	0.25	0.21
	0.5-1.2	0.35	0.60	0.55
	>1.2	0.56	0.91	0.86

The BF(MC)'s in Table 2 tend to be systematically low by about 0.04 to 0.05. This is due in part to the fact that the triples and quadruples lumped into BF⁺(SD98) can have more than one star in the same mass bin. Effectively, the most massive star in the multiple is counted as a binary primary in BF⁺. For comparison with BF(MC), some stars from the same mass bin in the same multiple should be counted as singles but are not in the BF⁺ accounting. Also, when the multiples decay, the resulting binaries will not always contain the most massive star. Both these effects tend to inflate BF⁺(SD98) relative to BF(MC), but the differences are typically not much larger than the statistical uncertainties in the SD98 numbers. Table 2 demonstrates that the MC approach does a reasonably good job of reproducing SD98 binary statistics, but the IMF resulting from the CMS and SMS chosen in SD98 is not realistic. Armed with the MC method, we can now consider binary statistics for other, more realistic mass distributions and for other ways of choosing cluster sizes without resorting to N-body calculations. One uncertainty is that we extrapolate SD98 results from $N \le 5$ to N > 5. N-body calculations of cluster decay for large ensembles of systems have not yet been done with N > 5. We plan to remedy this deficiency in the near future. Meanwhile, Table 1 is an improvement over McDC93; and, anyhow, we find that our results are more affected by the two-step process than by modifications to dynamical biasing.

2.3 Choices for ${\mathsfsl f}_{\mathsf c}{\mathsfsl {(M}}_{\mathsf {tot}}{\mathsfsl )}$ and ${\mathsfsl f}_{\mathsf s}{\mathsfsl {(M)}}$

Unfortunately, the physical processes which might underly the CMS and SMS are still poorly known. Because our goal in this paper is to illustrate the differences in binary characteristics between one-step and two-step approaches, we deliberately select our CMS and SMS so that, after two-step mass selection, they yield an acceptable stellar IMF. Our CMS is

$$% f_{\rm c}(M_{\rm tot}) \sim {M_{\rm tot}}^{-\alpha_{\rm c}}$$ (2)

for $0.5~M_{\odot} \le M_{\rm tot} \le 10.0~M_{\odot}$ and zero otherwise. Our standard choice for $\alpha_{\rm c}$ is 1.5. For the SMS, we use a lognormal distribution (see Scalo 1986)

$$f_{\rm s}(M) \sim M^{-1}{\exp}\{-[\log_{10}(M/M_{\odot}) - \mu]^2/2\sigma^2\}$$ (3)

for $M_{\rm min} \le M \le 10.0~M_{\odot}$ and zero otherwise with $\mu = {\rm log}_{10}0.20$ and $\sigma = \log_{10}0.35$. The functions $f_{\rm c}$ and $f_{\rm s}$ give fractions of systems or stars per unit mass interval. Stars form from clouds with limited efficiency, however, and so the mass range of the CMS is lower than that of cloud cores. Mot98 note, for instance, that the pre-stellar clumps within the cores they observe represent only about 5% to 40% of the core gas mass, consistent with other estimates of star formation efficiency between 10% and 20% (Lada et al. 1991). So the cluster mass $M_{\rm tot}$ represents some typical fraction of a molecular cloud core mass. Parameters $\mu$ and $\sigma$ in the lognormal $f_{\rm s}$ were varied until f_i from an $N_{\rm max} = 10$ cluster ensemble agreed to within about 20% with the Kroupa et al. (1993, hereafter KTG93) piecewise power-law IMF over the range $0.3~M_{\odot} < M < 3~M_{\odot}$ for $M_{\rm min} = 0.08~M_{\odot}$. The KTG93 IMF is consistent with modern IMF's for stellar masses (Kroupa 1998) and is simple to characterize. It is a continuously decreasing mass spectrum with a power-law index of 1.3 for $0.08~M_{\odot} < M < 0.5~M_{\odot}$, 2.2 for $0.5~M_{\odot} < M < 1.0~M_{\odot}$, and 2.7 for $M > 1.0~M_{\odot}$. Figure 1 compares the KTG93 IMF, $f_{\rm s}$, and f_i for all stars in 100000 $N_{\rm max} = 10$ clusters created by the two-step IMF process with $\tau = 0.1$. There is an excess of high-mass stars in f_i relative to $f_{\rm s}$, because, for large cluster masses, random mass selections from $f_{\rm s}$ tend to be rejected until they include at least one high-mass star.

Figure 1: The SMS $f_{\rm s}$ (dotted) and resultant two-step IMF fi (thick solid curve) for the $N_{\rm max} = 10$ ensemble of clusters generated by Eq. (1) to (3) with $\tau = 0.1$ and $M_{\rm min} = 0.08~M_{\odot}$. The thin solid curve is the KTG93 IMF normalized to the same total number of stars. All histograms are fractions of stars in each logarithmic mass interval of 0.05 dex

Although Scalo (1998) has recently disavowed the Miller & Scalo (1979) lognormal IMF, others have argued for its plausibility (Adams & Fatuzzo 1996). In Eq. (3), we use a lognormal for the SMS; but, after two-step mass selection, the two-step IMF is not lognormal. As shown in Fig. 1, it rolls over to a lognormal at low masses but has a power-law shape from about 0.6 to $6~M_{\odot}$. The CMS in (2) with $\alpha_{\rm c} = 1.5$ has a power-law index similar to that of the molecular cloud mass spectrum (Elmegreen & Falgarone 1996). Klessen & Burkert (2000) might argue that (2) should be lognormal as well. We will offer some discussion of how parameter and function choices affect results in Sects. 3.5 and 3.6.