Dust-Dust Collisional Charging and Lightning in Protoplanetary Discs

We study the role of dust-dust collisional charging in protoplanetary discs. We show that dust-dust collisional charging becomes an important process in determining the charge state of dust and gas, if there is dust enhancement and/or dust is fluffy, so that dust surface area per disc volume is locally increased. We solve the charge equilibrium equations for various disc environments and dust number density $\eta$, using general purpose graphic processors (GPGPU) and {\sc cuda} programming language. We found that as dust number density $\eta$ increases, the charge distribution experience four phases. In one of these phases the electrostatic field $E$ caused by dust motion increases as $E \propto \eta^4$. As a result, macroscopic electric discharge takes place, for example at $\eta = 70$ (in units of minimum-mass solar nebula (MMSN) values, considering two groups of fluffy dust with radii $10^{-2}\unit{cm}$, $10^{2}\unit{cm}$). We present a model that describes the charge exchange processes in the discs as an electric circuit. We derive analytical formulae of critical dust number density for lightning, as functions of dust parameters. We estimate the total energy, intensity and event ratio of such discharges (`lightning'). We discuss the possibility of observing lightning and sprite discharges in protoplanetary discs by Astronomically Low Frequency ({\em ALF}) waves, {\em IR} images, {\em UV} lines, and high energy gamma rays. We also discuss the effects of lightning on chondrule heating, planetesimal growth and magnetorotational instability of the disc.