Fast methods for tracking grain coagulation and ionization. I. Analytic
  derivation
  Dust grains play a major role in many astrophysical contexts. They affect 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 numerically implement by
itself, it is difficult to couple it with multidimensional hydrodynamics
numerical simulations because of its high computational cost. We propose here a
simple method for tracking 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 processes other than coagulation, it is
mathematically exact, fast, inexpensive, and can be used to evaluate the effect
of grain coagulation in most astrophysical contexts. It is applicable to all
coagulation kernels in which local physical conditions and grain properties can
be separated. We also describe another method for calculating the average
electric charge of grains and the density of ions and electrons in environments
that are 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 that are required to model nonideal
magnetohydrodynamics.
