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We investigate the formation and evolution of a first core, protostar, andcircumstellar disc with a three-dimensional non-ideal (including both Ohmic andambipolar diffusion) radiation magnetohydrodynamics simulation. We found thatthe magnetic flux is largely removed by magnetic diffusion in the first corephase and that the plasma $\beta$ of the centre of the first core becomeslarge, $\beta>10^4$. Thus, proper treatment of first core phase is crucial ininvestigating the formation of protostar and disc. On the other hand, in anideal simulation, $\beta\sim 10$ at the centre of the first core. Thesimulations with magnetic diffusion show that the circumstellar disc forms atalmost the same time of protostar formation even with a relatively stronginitial magnetic field (the value for the initial mass-to-flux ratio of thecloud core relative to the critical value is $\mu=4$). The disc has a radius of$r \sim 1$ AU at the protostar formation epoch. We confirm that the disc isrotationally supported. We also show that the disc is massive ($Q\sim 1$) andthat gravitational instability may play an important role in the subsequentdisc evolution.

Effects of Ohmic and ambipolar diffusion on the formation and evolution of the first cores, protostars and circumstellar discs