ner_tags: O O O O O B-Model O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O
tokens: Larson core. We focus on niMHD calculations; iMHD calculations are discussed in Sect 5 . Fig. 8 Visualisation of the disk structures. Top row : panels a) and b) show side and top views, respectively, of the disk density (orange), with the velocity vectors superimposed for . Panels c) and d) are the same side and top views, but showing the plasma parameter (colour map), over which we plot the magnetic field direction for the side view c) and the velocity vectors for the top view d) . Bottom row : same as for the top row , but for the simulation. Each row has a different spatial scale. 4.1. Aligned case Conservation of angular momentum during the collapse yields a very high specific angular momentum close to the protostar, which eventually leads to the formation of rotationally dominated structures with a strong toroidal velocity component and in some cases to genuine disks with Keplerian velocity profiles. To define a “disk” in the simulations, we used the criteria defined in Joos et al. (2013) . Structures in which the magnetic pressure either dominates or is at least not negligible, however, may satisfy these criteria while being very different from the flat rotationally supported structures characteristic of Keplerian disks. It is important to clearly identify and distinguish these two types of structures, since the existence of flat disks around Class-0 objects remains a subject of heated debate, and this issue is addressed below. Following Joos et al. (2013) , a piece of fluid belongs to the disk if it fulfils all the following constraints (using ): it is close to hydrostatic equilibrium:; it has strong rotational support: ; its rotational support is stronger than the thermal support: ; it has a high density: . In our fiducial aligned case with , we do observe the formation of a disk according to the above criteria that strongly resembles a Keplerian disk. Its characteristic properties are given in Table 3 and the results of the simulations are shown in Fig. 8 (top row). The disk is well defined in density, beginning just at the edge of the core and exhibiting sharp edges at its periphery. The breaking of symmetry observed in the top view (Fig. 8 b) is due to the evolution over several orbital periods at the core radius, with small numerical errors leading to a bar-like instability that evolves into two spiral arms. The disk then grows to a radius of au and is flat. The velocity profile in the disk is displayed Fig. 9 and is very close to a Keplerian one, with an estimated mass for the central object of compared to (Table 3 ) (see Appendix C for an important remark about assuming a Keplerian profile to estimate the mass of the central object). Table 3 Properties of the disks that are formed in the non-ideal MHD simulations. The evolution of the disk for our two values of magnetisation is displayed in Fig. 10 (solid lines for , dashed lines for ). In niMHD, the pinching of the field lines the midplane is greatly reduced and the radial