The Energetics of a Flaring Solar Active Region and Observed Flare Statistics

Abstract

A stochastic model for the energy of a flaring solar active region is presented, generalizing and extending the approach of Wheatland and Glukhov. The probability distribution for the free energy of an active region is described by the solution to a master equation involving deterministic energy input and random jump transitions downward in energy (solar flares). It is shown how two observable distributions, the flare frequency-energy distribution and the flare waiting-time distribution, may be derived from the steady state solution to the master equation, for given choices for the energy input and for the rates of flare transitions. An efficient method of numerical solution of the steady state master equation is presented. Solutions appropriate for flaring, involving a constant rate of energy input and power-law distributed jump transition rates, are numerically investigated. The flarelike solutions exhibit power-law flare frequency-energy distributions below a high-energy rollover, set by the largest energy the active region is likely to have. The solutions also exhibit approximately exponential (i.e., Poisson) waiting-time distributions, despite the rate of flaring depending on the free energy of the system.