? III . Analytical Distribution of Superimposed Flares
نویسندگان
چکیده
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 super-position 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 superimposed flares with similar rise and decay time scales. 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: (1) The characteristic function (Fourier transform of the probability density) of the expected counts in time bins ∆t is φ F (s, ∆t) = exp{−T −1 ∞ −∞ dt [1 − φ a (s Ξ(t, ∆))]}, where T is the mean flaring interval, φ a (s) is the characteristic function of the flare amplitudes, and Ξ(t, ∆t) is the flare shape con-volved with the observational time bin. (2) The probability of finding n counts in time bins ∆t is P c (n) = (2π) −1 2π 0 ds e −ins φ F (s, ∆t). (3) The probability density of photon waiting times x is P δ (x) = ∂ 2 x φ F (i, x)/r with r = ∂ x φ F (i, x)| x=0 the mean count rate. An additive independent background is readily included. Applying these results to EUVE/DS 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. (2002) and Güdel et al. (2003). 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.
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