Numerical simulations of star and disc formation

Magnetized, cold, dense molecular cloud cores provide the birth environment for stars, discs, and planets. The multi-scale scenario of low-mass star formation occurs via the formation of two quasi-hydrostatic cores. Furthermore, the conservation of angular momentum can lead to the formation of a disc around the second core (i.e.~the forming protostar). During these early stages of star formation, magnetically driven outflows and jets can be launched from the first and second cores, respectively. Star, disc, and outflow formation involve complex physical processes, which require a robust, self-consistent numerical treatment. In this thesis, we use numerical simulations to probe the gravitational collapse scenario that involves the transition of an isolated molecular cloud core to a hydrostatic core with a surrounding disc. We use the \emph{PLUTO} code to perform radiation (magneto-)hydrodynamic (MHD) collapse simulations, using one- and two-dimensional (2D) grids. We include the effects of self-gravity and a grey flux-limited diffusion approximation for the radiative transfer. Additionally, we use for the gas equation of state density- and temperature-dependent thermodynamic quantities to account for the dissociation, ionisation, and molecular vibrations and rotations. Our spherically symmetric simulations span seven orders of magnitude in spatial scale. We survey a wide range of initial low- to high-mass (0.5 -- 100~$M_{\odot}$) molecular cloud cores, yielding the largest parameter scan so far. Our results highlight the dependence of the first and second hydrostatic core properties on the initial cloud core properties. These simulations indicate that in the high-mass regime, the first hydrostatic cores do not have enough time to form due to large accretion rates. We further expand our studies to three different sets of 2D simulations using axial and midplane symmetry. First, we perform 2D simulations for non-rotating molecular cloud cores with masses of 1~$M_{\odot}$, 5~$M_{\odot}$, 10~$M_{\odot}$, and 20~$M_{\odot}$. For each of these cases, we use an unprecedented resolution to model the evolution of the second core for $\geq$~100 years after its formation. For the first time, we demonstrate that convection is generated in the outer layers of the second core. This supports the intriguing possibility that dynamo-driven magnetic fields may be generated during the earliest phases of star formation. Following which, for the 1~$M_{\odot}$ case, we analyse the effects of solid-body rotation on the properties of the hydrostatic cores and disc formation. In this model, the first hydrostatic core evolves into a more oblate, pseudo-disc like structure and a sub-au disc starts forming after the formation of the second core. Finally, we explore the effects of ideal and non-ideal (including Ohmic resistivity) MHD during the collapse of rotating molecular cloud cores. We investigate the dependence of molecular outflows and disc formation on the initial cloud core mass, rotation, resistivity, and magnetic field strength. We find the presence of magnetically driven outflows launched from both first and second cores in the resistive models. We also reveal ongoing disc formation in some of our resistive simulations. In conclusion, we use detailed thermodynamical modelling to quantify the properties of the hydrostatic cores, outflows, and discs for collapse scenarios with a wide range of initial cloud core properties. The models presented herein will serve as the foundation for follow-up studies that link these theoretical insights with observational signatures