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

TL;DR

This paper develops a stochastic model for solar active region energy, deriving observable flare distributions and numerically analyzing flare statistics with power-law and exponential behaviors.

Contribution

It extends previous models by providing a generalized stochastic framework and an efficient numerical method for analyzing flare energy distributions and waiting times.

Findings

Flare frequency-energy distribution follows a power-law below a high-energy rollover.

Waiting-time distribution is approximately exponential despite energy-dependent flare rates.

Numerical solutions show flare statistics consistent with observed solar flare behaviors.

Abstract

A stochastic model for the energy of a flaring solar active region is presented, generalising and extending the approach of Wheatland & Glukhov (1998). 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 downwards 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 flare-like 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.