Context. Planets form in protoplanetary discs. Their masses, distribution, and orbits sensitively depend on the structure of the protoplanetary discs. However, what sets the initial structure of the discs in terms of mass, radius and accretion rate is still unknown. Aims. It is therefore of great importance to understand exactly how protoplanetary discs form and what determines their physical properties. We aim to quantify the role of the initial dense core magnetisation, rotation, turbulence, and misalignment between rotation and magnetic field axis as well as the role of the accretion scheme onto the central object. Methods. We performed non-ideal magnetohydrodynamics numerical simulations using the adaptive mesh refinement code Ramses of a collapsing, one solar mass molecular core to study the disc formation and early, up to 100 kyr, evolution. We paid particular attention to the impact of numerical resolution and accretion scheme. Results. We found that the mass of the central object is almost independent of the numerical parameters such as the resolution and the accretion scheme onto the sink particle. The disc mass and to a lower extent its size, however heavily depend on the accretion scheme, which we found is itself resolution dependent. This implies that the accretion onto the star and through the disc are largely decoupled. For a relatively large domain of initial conditions (except at low magnetisation), we found that the properties of the disc do not change too significantly. In particular both the level of initial rotation and turbulence do not influence the disc properties provide the core is sufficiently magnetised. After a short relaxation phase, the disc settles in a stationary state. It then slowly grows in size but not in mass. The disc itself is weakly magnetised but its immediate surrounding on the contrary is highly magnetised. Conclusions. Our results show that the disc properties directly depend on the inner boundary condition, i.e. the accretion scheme onto the central object. This suggests that the disc mass is eventually controlled by a small-scale accretion process, possibly the star-disc interaction. Because of ambipolar diffusion and its significant resistivity, the disc diversity remains limited and except for low magnetisation, their properties are weakly sensitive to initial conditions such as rotation and turbulence.
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