Two-size approximation: a simple way of treating the evolution of grain size distribution in galaxies

Full calculations of the evolution of grain size distribution in galaxies are in general computationally heavy. In this paper, we propose a simple model of dust enrichment in a galaxy with a simplified treatment of grain size distribution by imposing a two-size approximation; that is, the whole grain population is represented by small grains (radius a < 0.03 mu m) and large grains (a > 0.03 mu m). We include in the model dust supply from stellar ejecta, destruction in supernova shocks, dust growth by accretion, grain growth by coagulation and grain disruption by shattering, and consider how these processes work on the small and large grains. We show that this simple framework reproduces the main features of full calculations of grain size distributions as follows. Dust enrichment starts with a supply of large grains from stars. At a metallicity level, referred to as the critical metallicity of accretion, the abundance of small grains formed by shattering becomes large enough to increase grain abundance rapidly by accretion. Associated with this epoch, the mass ratio of small grains to large grains reaches a maximum. Afterwards, this ratio converges to the value determined by the balance between shattering and coagulation, and the dust-to-metal ratio is determined by the balance between accretion and shock destruction. Using a Monte Carlo simulation, we demonstrate that the simplicity of our model has an advantage in predicting statistical properties. We also show some applications for predicting observational dust properties such as extinction curves.