Fast methods to track grain coagulation and ionization. I. Analytic derivation

Dust grains play a major role in many astrophysical contexts. They influence the chemical, magnetic, dynamical and optical properties of their environment, from galaxies down to the interstellar medium, star-forming regions, and protoplanetary disks. Their coagulation leads to shifts in their size distribution and ultimately to the formation of planets. However, although the coagulation process is reasonably uncomplicated to implement numerically by itself, it is difficult to couple it with multi-dimensional hydrodynamics numerical simulations because of its high computational cost. We propose here a simple method to track the coagulation of grains at far lower cost. Given an initial grain size distribution, the state of the distribution at time t is solely determined by the value of a single variable integrated along the trajectory, independently of the specific path taken by the grains. Although this method cannot account for other processes than coagulation, it is mathematically exact, fast, inexpensive, and can be used to evaluate the impact of grain coagulation in most astrophysical contexts. Although other processes modifying the size-distribution, as fragmentation cannot be coupled to this method, it is applicable to all coagulation kernels in which local physical conditions and grain properties can be separated. We also describe another method to calculate the average electric charge of grains and the density of ions and electrons in environments shielded from radiation fields, given the density and temperature of the gas, the cosmic-ray ionization rate and the average mass of the ions. The equations we provide are fast to integrate numerically, and can be used in multidimensional numerical simulations to self-consistently calculate, on-the-fly, the local resistivities needed to model non-ideal magnetohydrodynamics.