Wavelet Analysis of Solar Flare Hard X-Rays

We apply a multiresolution analysis to hard X-ray (HXR) time profiles f(t) of solar flares. This method is based on a wavelet transform (with triangle-shaped wavelets), which yields a dynamic decomposition of the power at different timescales T, the scalogram P(T, t). For stationary processes, time-averaged power coefficients, the scalegram S(T), can be calculated. We develop an algorithm to transform these (multiresolution) scalegrams S(T) into a standard distribution function of physical timescales, N(T). We analyze 647 solar flares observed with the Compton Gamma Ray Observatory (CGRO), recorded at energies ≥25 keV with a time resolution of 64 ms over 4 minutes in each flare. The main findings of our wavelet analysis are:

1. In strong flares, the shortest detected timescales are found in the range T_min ≈ 0.1-0.7 s. These minimum timescales are found to correlate with the flare loop size r (measured from Yohkoh images in 46 flares), according to the relation T_min(r) ≈ 0.5(r/10⁹ cm) s. Moreover, these minimum timescales are subject to a cutoff, T_min(n_e) ≳ T_Defl(n_e), which corresponds to the electron collisional deflection time at the loss-cone site of the flare loops (inferred from energy-dependent time delays in CGRO data).

2. In smoothly varying flares, the shortest detected timescales are found in the range T_min ≈ 0.5-5 s. Because these smoothly varying flares exhibit also large trap delays, the lack of detected fine structure is likely to be caused by the convolution with trapping times.

3. In weak flares, the shortest detected timescales cover a large range, T_min ≈ 0.5-50 s, mostly affected by Poisson noise.

4. The scalegrams S(T) show a power-law behavior with slopes of β_max ≈ 1.5-3.2 (for strong flares) over the timescale range of [T_min, T_peak]. Dominant peaks in the timescale distribution N(T) are found in the range T_peak ≈ 0.5-10² s, often coinciding with the upper cutoff of N(T). These observational results indicate that the fastest significant HXR time structures detected with wavelets (in strong flares) are related to physical parameters of propagation and collision processes. If the minimum timescale T_min is associated with an Alfvénic crossing time through elementary acceleration cells, we obtain sizes of r_acc ≈ 75-750 km, which have a scale-invariant ratio r_acc/r ≈ 0.03 to flare loops and are consistent with cell sizes inferred from the frequency bandwidth of decimetric millisecond spikes.