Insights into the first and second hydrostatic core stages from numerical simulations

Abstract: The theory of how low mass stars form from the collapse of a dense molecular cloud core has been well-established for decades. Thanks to significant progress in computing and numerical modelling, more physical models have been developed and a wider parameter space explored to understand the early stages of star formation more fully. In this review, I describe the expected physical properties of the first and second core stages and how the inclusion of different physics affects those predicted characteristics. I provide an overview of chemical models and synthetic observations, looking towards the positive identification of the first core in nature, which remains elusive. However, there are a few likely candidate first cores, which are listed, and I briefly discuss the recent progress in characterising the youngest protostellar sources. Chemistry will be instrumental in the firm identification of the first core so we require robust theoretical predictions of the chemical evolution of protostellar cores, especially of the first and second core outflows. Looking ahead, simulations can shed light on how the protostellar collapse phase shapes the evolution of the protostellar disc. Simulations of dust evolution during protostellar core collapse show there is significant enhancement in grain size and abundance towards the centre of the core. Chemical models show that the warm, dense conditions of the first core drive chemical evolution. There is a wide scope for further study of the role that the first and second core stages play in determining the structure and composition of the protostellar disc and envelope and, of course, the eventual influence on the formation of planets.