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.

💡 Research Summary

The paper investigates how collisional charging between dust grains influences the electrical state of protoplanetary disks (PPDs) and under what conditions this process can trigger macroscopic electric discharges (“lightning”). The authors begin by reformulating the charge‑balance equations as an electric‑circuit model: gas electrons and ions correspond to currents, dust‑dust charge exchange to voltage sources, and the resulting electric field to the circuit’s potential difference. This representation makes the highly non‑linear problem tractable and allows independent variation of key parameters such as dust number density (η), grain radius (a), and porosity (φ).

Using CUDA‑based general‑purpose GPU (GPGPU) calculations, the authors solve the circuit equations for a wide range of η (expressed in units of the Minimum‑Mass Solar Nebula, MMSN) and for two representative fluffy‑dust populations with radii 10⁻² cm and 10² cm. They identify four distinct regimes as η increases:

1. Ion‑dominated regime (η≲10) – gas‑dust charging controls the charge balance; the electric field is weak.
2. Mixed regime (10≲η≲30) – dust‑gas charge exchange becomes significant; dust grains start to accumulate net charge.
3. Dust‑dust dominated regime (30≲η≲70) – collisional charging between grains dominates; the electric field grows non‑linearly as (E\propto η^{4}).
4. Lightning regime (η≳70) – the field exceeds the breakdown threshold (≈10⁴ V m⁻¹) and a macroscopic discharge occurs.

An analytical expression for the critical dust density η_c is derived from the condition (E=E_{\rm crit}). The result, \