Context. Radiative shocks play a dominant role in star formation. The accretion shocks on first and second Larson cores involve radiative processes and are thus characteristic of radiative shocks. Aims. In this study, we explore the formation of the first Larson core and characterize the radiative and dynamical properties of the accretion shock, using both analytical and numerical approaches. Methods. We developed both numerical radiation-hydrodynamics calculations and a semi-analytical model that characterize radiative shocks in various physical conditions, for radiating or barotropic fluids. Then, we performed 1D spherical collapse calculations of the first Larson core, using a grey approximation for the opacity of the material. We considered three different models for radiative transfer: the barotropic approximation, the flux limited diffusion approximation, and the more complete M1 model. We investigate the characteristic properties of the collapse and of the first core formation. Comparison between the numerical results and our semi-analytical model for radiative shocks shows that the latter reproduces the core properties obtained with the numerical calculations quite well. Results. The accretion shock on the first Larson core is found to be supercritical; i.e., the post and pre-shock temperatures are equal, implying that all the accretion shock energy on the core is radiated away. The shock properties are described well by the semi-analytical model. The flux-limited diffusion approximation is found to agree quite well with the results based on the M1 model of radiative transfer, and is thus appropriate for studying the star formation process and allows a tractable and relatively correct treatment of radiative transfer in multidimensional radiation-hydrodynamics calculations. In contrast, the barotropic approximation does not correctly describe the thermal properties of the gas during the collapse.
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