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Context. Observations suggest that low-mass stars condense out of dense, relatively isolated, molecular cloud cores, with each core spawning a small-N cluster of stars.\ud Aims. Our aim is to identify the physical processes shaping the collapse and fragmentation of a 5.4 core, and to understand how these processes influence the mass distribution, kinematics, and binary statistics of the resulting stars.\ud Methods. We perform SPH simulations of the collapse and fragmentation of cores having different initial levels of turbulence ( = 0.05, 0.10, 0.25). We use a new treatment of the energy equation that captures (i) excitation of the rotational and vibrational degrees of freedom of H2, dissociation of H2, ionisation of H and He; and (ii) the transport of cooling radiation against opacity due to both dust and gas (including the effects of dust sublimation, molecules, and H- ions). We also perform comparison simulations using a standard barotropic equation of state.\ud Results. We find that – when compared with the barotropic equation of state – our more realistic treatment of the energy equation results in more protostellar objects being formed, and a higher proportion of brown dwarfs; the multiplicity frequency is essentially unchanged, but the multiple systems tend to have shorter periods (by a factor ~3), higher eccentricities, and higher mass ratios. The reason for this is that small fragments are able to cool more effectively with the new treatment, as compared with the barotropic equation of state. We also note that in our simulations the process of fragmentation is often bimodal, in the following sense. The first protostar to form is usually, at the end, the most massive, i.e. the primary. However, frequently a disc-like structure subsequently forms round this primary, and then, once it has accumulated sufficient mass, quickly fragments to produce several secondaries.\ud Conclusions. We believe that this delayed fragmentation of a disc-like structure is likely to be an important source of very low-mass stars in nature (both low-mass hydrogen-burning stars and brown dwarf stars); hence it may be fundamental to understanding the way in which the statistical properties of stars change – continuously but monotonically – with decreasing mass. However, in our simulations the individual cores probably produce too many stars, and hence too many single stars. We list the physical and numerical features that still need to be included in our simulations to make them more realistic; in particular, radiative and mechanical feedback, non-ideal magneto-hydrodynamic effects, and a more sophisticated implementation of sink particles.

We explore, by means of a large ensemble of SPH simulations, how the level of turbulence affects the collapse and fragmentation of a star-forming core. All our simulated cores have the same mass ( $5.4 ~M_\odot$), the same initial density profile (chosen to fit observations of L1544), and the same barotropic equation of state, but we vary (a) the initial level of turbulence (as measured by the ratio of turbulent to gravitational energy, $\alpha_{\rm turb} \equiv U_{\rm turb}/\vert\Omega\vert = 0,\,0.01,\,0.025,\, 0.05,\,0.10\;{\rm and}\;0.25$ ) and (b), for fixed $\alpha_{\rm turb}$, the details of the initial turbulent velocity field (so as to obtain good statistics).\ud \ud A low level of turbulence ( $\alpha_{\rm turb} \sim 0.05$) suffices to produce multiple systems, and as $\alpha_{\rm turb}$ is increased, the number of objects formed and the companion frequency both increase. The mass function is bimodal, with a flat low-mass segment representing single objects ejected from the core before they can accrete much, and a Gaussian high-mass segment representing objects which because they remain in the core grow by accretion and tend to pair up in multiple systems.\ud \ud The binary statistics reported for field G-dwarfs by Duquennoy & Mayor (1991, A&A, 248, 485) are only reproduced with $\alpha_{\rm turb} \sim 0.05$. For much lower values of $\alpha_{\rm turb}$ ( $\la $0.025), insufficient binaries are formed. For higher values of $\alpha_{\rm turb}$ ( $\ga $0.10), there is a significant sub-population of binaries with small semi-major axis and large mass-ratio (i.e. close binaries with components of comparable mass). This sub-population is not present in Duquennoy & Mayor's sample, although there is some evidence for it in the pre-Main Sequence population of Taurus analyzed by White & Ghez (2001, ApJ, 556, 265). It arises because with larger $\alpha_{\rm turb}$, more low-mass objects are formed, and so there is more scope for the binaries remaining in the core to be hardened by ejecting these low-mass objects. Hard binaries thus formed then tend to grow towards comparable mass by competitive accretion of material with relatively high specific angular momentum.