tokens: different dust sizes: Ohmic diffusivity (solid line), Hall diffusivity (dotted for positive and dash-dotted for negative), and ambipolar diffusivity (dashed line) vs. radius. The plots show at time kyr, when central temperature is K and central density cm. The asterix denotes that the mean dust cross section is minimal in model S5, and the mean dust radius is not a unique characteristic of dust properties. 4.1. Sign reversal of the Hall effect Owing to the total charge neutrality condition where is the charge of species, there are only three independent parameters that can be defined as , , and (see also Sect. 2 in Xu Bai 2016 ). In Fig. 3 we show the dimensionless parameters , , and . These parameters show how smaller dust size leads to larger inequality and the transition to the dust-ion regime and even dust-dust regimes, i.e. when main charge carriers are ions and (model S3), or and (model S5). Fig. 3 Various properties of charged species and magnetic diffusivities radial profiles shown for models S1 (black), S2 (red), S3 (blue), S4 (green), and S5 (lilac). The colours refer to the same models in all following figures. Plots show, from left to right , the density of ion vs. radius, the relation of electron density to the ion density, and the relative abundances of charged dust to the ion density ( bottom ). The Hall effect term, given in Eq. ( 2 ), becomes negative when the Hall conductivity is negative, . In case of single-charged dust and local charge neutrality, Eq. ( 4 ) can be rewritten as where we reduce the dust coupling parameter to because it is insensitive to the sign of the charge, i.e. . In our simulations, we observe that the coupling parameters everywhere, and and for radii AU (see also Fig. 4 ). Thus, we simplify the equation for further and derive the threshold value for , When , the Hall effect changes sign to the negative, which may lead to the interesting MHD effects during the collapse. The Hall effect becomes negative when the number of negative charge carriers, weighted over coupling parameter , dominates the number of -weighted positive carriers. This is likely to happen when both the negatively charged dust is abundant and the magnetic fields are strong enough for the strong coupling of ions (). Fig. 4 Top : relative number densities of positively and negatively charged dust, and , compared to the threshold value in Eq. ( 9 ). Middle : coupling parameter for electrons, ions, and dust (Eqs. ( 3 )–( 5 )). Bottom : magnetic diffusivities depending on the magnetic field strength with colours showing . From the previous equations we learn that both the amount of negatively charged dust and also the quality of coupling to the magnetic fields are important. It is now a natural question to ask what happens if we change the cloud magnetisation. We consider cases with . The various magnetisations are shown with colours in Fig. 4 for models S2 and S5. We observe that the radial position of the sign reversal in Hall effect shifts
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