Are coronae of magnetically active stars heated by flares?: III.: Analytical distribution of superposed flares

We study the hypothesis that observed X-ray/extreme-ultraviolet emission from coronae of magnetically active stars is entirely (or to a large part) due to the superposition of flares, using an analytic approach to determine the amplitude distribution of flares in light curves. The flare-heating hypothesis is motivated by time series that show continuous variability suggesting the presence of a large number of superposed flares with similar rise and decay timescales. We rigorously relate the amplitude distribution of stellar flares to the observed histograms of binned counts and photon waiting times, under the assumption that the flares occur at random and have similar shapes. Our main results are as follows: (1) The characteristic function (Fourier transform of the probability density) of the expected counts in time bins Deltat is phi(F)(s, Deltat) = exp (-T-1 integral(-infinity)(infinity) dt {1 - 0(a)[sXi(t,Deltat)]}), where T is the mean flaring interval, phi(a)(s) is the characteristic function of the flare amplitudes, and Xi(t, Deltat) is the flare shape convolved with the observational time bin. (2) The probability of finding n counts in time bins Deltat is P-c(n) = (2pi)(-1) integral(0)(2pi) ds e(-ins)phi(F)(s, Deltat). (3) The probability density of photon waiting times x is P-delta(x) = partial derivative(x)(2)phi(F)(i,x)/[r], with [r] = partial derivative(x)phi(F)(i,x)\x=0 the mean count rate. An additive independent background is readily included. Applying these results to Extreme Ultraviolet Explorer/Deep Survey instrument observations of the flaring star AD Leo, we find that the flare amplitude distribution can be represented by a truncated power law with a power-law index of 2.3 +/- 0.1. Our analytical results agree with existing Monte Carlo results of Kashyap et al. and Gudel et al. The method is applicable to a wide range of further stochastically bursting astrophysical sources such as cataclysmic variables, gamma-ray burst substructures, X-ray binaries, and spatially resolved observations of solar flares.