Conditions for the nuclear battery effect in the protoplanetary disc mid-plane. The top left plot shows the pebble column density for three values of gas depletion fg and the stopping column of positrons. The mid-plane layer transitions from optically thick to optically thin to positron stopping at a gas mass of 10% of the nominal value in the asteroid belt. The top right plot shows the breakdown value of the electric field. The bottom left plot shows the scale-height of the mid-plane layer (dotted line) together with the stopping length of the positrons at the breakdown value of the electric field. The mid-plane layer is clearly much thicker than the optimal thickness for transferring the positron energy to the electric field, but substructures of 1% of the mid-plane thickness are more in agreement with the optimal width (dot-dashed line). The bottom right plot shows the mid-plane density of gas, the Roche density and the conditions for chondrule formation. The lack of isotope fractionation and stabilisation of melted silicate can be achieved at solids-to-gas ratios of 10 (at 1 AU) to 100 (at 2.5 AU), but the stricter density conditions imposed by the observed high abundance of Na (Alexander et al. 2008) can only be achieved interior of 1 AU where the Roche density is high.
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