Flare Frequency Distributions Based on a Master Equation

Abstract

The Rosner & Vaiana model for flares is generalized to allow for flares that do not deplete all free energy from the system, a step that overcomes a number of objections to the original model. We obtain a probability balance equation, or master equation, describing the free energy E of an active region subject to a prescribed growth rate, Ė, and a prescribed distribution, α(E), of stochastic decay events. We argue that the solution appropriate to flares involves an energy-independent growth rate and a power-law form for α(E), which may be the result of an underlying avalanche process. The resulting model produces power-law flare frequency distributions below a high-energy rollover corresponding to the largest energy the system is likely to attain, which is set by the balance between the rate of growth and the rate of stochastic decay. There is a close correspondence between the resulting model and the avalanche model for flares.