1 Introduction

1.1 Two-step initial mass function

One of the fundamental goals of star formation theory is to determine the relationship between molecular cloud conditions and the mass and multiplicity distributions of the resulting stars. To address aspects of this problem, Sterzik & Durisen (1995, 1998, hereafter SD95 and SD98, respectively) used the Mikkola & Aarseth (1990, 1993) chain regularizaton code to compute the dynamic decay of nonhierarchical N-body clusters with N = 3, 4, and 5. It was already known from classic work of the 1970's and earlier that, in the absence of dissipation, the decay of pure point-mass few-body systems is dominated, within a few tens of system crossing times, by the formation of a hard binary which ejects most if not all the other stars (e.g., van Albada 1968a,b; Standish 1972; Saslaw et al. 1974; Harrington 1974, 1975; Heggie 1975; Monaghan 1976a,b; Valtonen 1976). In SD95 and SD98, the few-body systems were explicitly interpreted as products of molecular cloud core collapse and fragmentation. The decay products were characterized statistically by considering large numbers of cluster calculations for each choice of N with the goal of determining distribution functions of fragment properties, such as binary fractions and mass-ratio distributions. For statistical studies of few-body systems, one must adopt a procedure for choosing component stellar masses. One common approach (e.g., McDonald & Clarke 1993, hereafter McDC93) is to pick masses at random from an assumed stellar mass spectrum (SMS). In this case, the SMS is also the overall IMF for the stars. SD98 also introduced a two-step process where they first chose a cluster total mass from a cluster mass spectrum (CMS) and then required that the sum of the stellar masses, chosen randomly from a different stellar mass spectrum (SMS), should equal the chosen cluster mass. In this approach, because random mass selections from the SMS which do not match the chosen cluster mass are discarded, the final IMF of the selected stars differs from the SMS through the imposition of the total cluster mass constraint. In this paper, we refer to the distribution of stellar masses which results from a large ensemble of such two-step choices as a "two-step'' IMF. SD98 found that the distributions of some binary properties agree better with observations when stellar masses are chosen in this way. The distribution which improves the most is the overall binary fraction BF as a function of stellar mass M. The reason for the improvement is straightforward. Few-body cluster decay most often leads to production of one hard binary from the two most massive stars, an effect referred to as "dynamical biasing''. Assuming this always occurs, i.e., "complete'' dynamical biasing, McDC93 demonstrated that, when stars are chosen at random from a realisitic IMF in a one-step process, solar-mass stars almost always end up being in binaries for small N. On the other hand, observations show that the BF for nearby solar-type stars in the field is only $0.57 \pm 0.09$ (Duquennoy & Mayor 1991, hereafter DM91). A BF this low can be attained for $N \ge 10$ or so, but then the mass ratio distribution does not agree well with observations. Moreover, for $N \ge 10$, the BF for M stars becomes too low. Observations of field M stars suggest BF in the range $0.26 \pm 0.09$ (Leinert et al. 1997, hereafter L97) to $0.42 \pm 0.09$ (Fischer & Marcy 1992, hereafter FM92). With a two-step IMF, when total cluster masses are chosen from a moderately steep and broad CMS, solar-mass stars are often not the most massive stars in their clusters. By the same token, for the same N, M stars more often are the most massive stars. The overall result is a more uniform distribution of BF with mass from a single value of N.

1.2 Motivations and goals

A two-step IMF might be a reasonable representation of star cluster formation because the masses of stars in a small cluster must be constrained to some extent by the total cloud mass from which the cluster forms. For many years, Larson (1992) has advocated that processes on different physical scales imprint themselves on the IMF; and Elmegreen (1997) has also proposed a related two-process IMF where the low and high-mass parts of the IMF are shaped by different physical constraints. Observations by Motte et al. (1998, hereafter Mot98) of $\rho$ Ophiuchus indicate that pre-collapse cloud cores can be already "fragmented'' into few-body systems of stellar mass clumps. These clumps do appear to have the same mass spectrum as the IMF of young stars in the same region (Luhman & Rieke 1999; Testi & Sargent 1998). The observations seem consistent with the "prompt fragmentation'' scenario (Clarke & Pringle 1991, 1993) which is the starting point of the McDC93 analyses. Of course, there are other ways to explain the observed BF(M). For instance, McDonald & Clarke (1995, hereafter McDC95) showed that cluster evolutions which include the dissipative effects of collisions involving massive circumstellar disks give a flatter BF(M). Cluster evolution will also deviate from point-mass dynamics when cluster fragments accrete competitively in a gas-rich environment (e.g., Zinnecker 1982; Burkert & Bodenheimer 1996; Bonnell et al. 1997; 1998, Smith et al. 1997; Bate 1999). One could also appeal to slow dissolution of clusters through loss of residual gas. All these effects would drive results away from pure dynamical biasing. In light of the many uncertainties, we do not claim that a two-step IMF is the final or only answer. Our objective here is to illustrate, by using reasonable choices of the CMS and SMS and by making some simple statistical calculations based on the results of SD98, that a two-step IMF does improve binary characteristics and therefore needs to be kept in mind as a possibly important component of the star formation process, at least in a statistical sense. We do not attempt to explain the stellar IMF itself but choose our parameters so that the IMF resulting from our Monte Carlo approach is reasonable. Section 2 describes our methodology and offers some motivation for our choices of mass spectra and other parameters. Our main results are given in Sect. 3 and discussed briefly in Sect. 4. Section 5 summarizes our conclusions.