A method for following fragmentation simulations further in time using smoothed particle hydrodynamics (SPH) is presented. In a normal SPH simulation of the collapse and fragmentation of a molecular cloud, high-density regions of gas that form protostars are represented by many particles with small separations. These high-density regions require small time steps, limiting the time for which the simulation can be followed. Thus, the end result of the fragmentation can never be de nitively ascertained, and comparisons between cloud fragmentation calculations and the observed characteristics of stellar systems cannot be made.In this paper, each high-density region is replaced by a single, non-gaseous particle, with appropriate boundary conditions, which contains all the mass in the region and accretes any infalling mass. This enables the evolution of the cloud and the resulting protostars to be followed for many orbits or until most of the original cloud mass has been accreted.The Boss & Bodenheimer standard isothermal test case for the fragmentation of an interstellar cloud is used as an example for the technique. It is found that the binary protostellar system that forms initially does not merge, but instead forms a multiple system. The collapse is followed to 4 initial cloud free-fall times when approximately 80 per cent of the original mass of the cloud has been accreted by the protostars, or surrounds them in discs, and the remainder of the material has been expelled out to the radius of the initial cloud by the binary.

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We present a new model to explain stellar mass distributions in different stellar environments. In our model, the protostar phase is terminated, when the protostellar core embedded in a molecular clump experiences a collision with another star or protostellar clump, which ejects the protostellar core from its parent clump. Such dynamical interactions are necessarily important, if stars preferentially form in dense clusters. We show that, in a simple model, the initial mass function approaches a simple, asymptotic form, which strongly resembles observed mass functions. The model has important consequences for star formation in different environments. We also discuss the implications of the model for our understanding of pre-main-sequence stellar evolution. COLLISIONS A N D THE INITIAL MASS FUNCTION (IMF)Modern observations of star-forming regions show that stars preferentially form in very dense, but ultimately unbound clusters (for references see Podsiadlowski and Price 1992). This suggests that collisions between young stellar objects (YSOs) are important. We have studied two types of collisions: collisions between two protostellar cores (i.e., protostars embedded in dense envelopes from which they accrete) and collisions between a protostellar core and a star (i.e., a YSO which is no longer embedded in a massive envelope). The cross section for collisions between protostellar cores is mainly determined by the geometric size of their envelopes. The most likely outcome, if the collision is supersonic, is that the protostellar cores are separated from their envelopes, thus terminating the protostar accretion phase. A collision between a protostellar core and a star will also separate a protostar from its envelope, if the kick velocity imparted to the protostar (relative to its envelope) by a star passing through the core exceeds the central escape velocity. We estimate that, for characteristic core properties, the cross section for the latter collisions is of order 10 % of the cross section for direct core-core collisions. In the simulations we present below, direct core-core collisions dominate initially, until the number of stars significantly exceeds the number of protostellar cores.To illustrate how such collisions lead to a mass spectrum, we adopt a very simple model for the star-forming process. We assume that stars form at a constant rate (per unit gas mass), a, and that protostars accrete mass at a constant rate, M, until they are ejected (the individual rates for core-core collisions and core-star collisions are /? and (3Z [Z ~ 0