Protostellar discs formed from rigidly rotating cores

We use three-dimensional smoothed particle hydrodynamic simulations to investigate the collapse of low-mass pre-stellar cores and the formation and early evolution of protostellar discs. The initial conditions are slightly supercritical Bonnor-Ebert spheres in rigid rotation. The core mass and initial radius are held fixed at M(O) = 6.1 M(circle dot) and R(O) = 17 000 au, and the only parameter that we vary is the initial angular speed Omega(O). Protostellar discs forming from cores with Omega(O) < 1.35 x 10(-13) s(-1) have radii between 100 and 300 au and are quite centrally concentrated; due to heating by gas infall on to the disc and accretion on to the central object, they are also quite warm, <(T)over bar> > 100 K, and therefore stable against gravitational fragmentation. In contrast, more rapidly rotating cores form discs which are less concentrated and cooler, and have radii between 400 and 1000 au; as a consequence they are prone to gravitational fragmentation and the formation of multiple systems. We derive a criterion that predicts whether a rigidly rotating core having given M(O), R(O) and Omega(O) will produce a protostellar disc which fragments whilst material is still infalling from the core envelope. We then apply this criterion to core samples for which M(O), R(O) and Omega(O) have been estimated observationally. We conclude that the observed cores are stable against fragmentation at this stage, due to their low angular speeds and the heat delivered at the accretion shock where the infalling material hits the disc.