Size Distributions of Solar Proton Events and Their Associated Soft X-Ray Flares: Application of the Maximum Likelihood Estimator

We use the maximum likelihood estimator to determine the slope (alpha) of the power-law size distribution of the peak proton fluxes of a subsampled set of 106 similar to 25 MeV solar energetic proton (SEP) events from 1997 to 2016 associated with western hemisphere soft X-ray (SXR) flares: alpha = 1.28 +/- 0.03. For the peak SXR fluxes of a subsample of 110 SEP-associated flares, we find alpha = 1.51 +/- 0.05. In addition, we obtained a slope of 1.61 +/- 0.05 for the peak SXR fluxes of a sample of 128 >= M1 SXR flares from 1996 to 2005 that were associated with coronal mass ejections (CMEs) with speeds >= 1000 km s(-1). The slopes of both of these SXR peak-flux distributions (similar to 1.5 and similar to 1.6) are closer to that for proton events (similar to 1.3) than to the a value of similar to 2.1 (2.09 +/- 0.08) determined for a subsample of 177 western hemisphere >= M1 SXR flares considered from 1996 to 2005. These results are consistent with those of a previous study, based on a less reliable method (for small samples), in which it was argued that the flatter size distribution generally found for SEP events versus those for flare electromagnetic emissions was due to the fact that SEP flares are an energetic subset of all flares, characterized by their similar to 100% association with fast/wide CMEs that drive coronal/interplanetary shock waves. Shock formation in the corona requires CMEs with speeds greater than or similar to 400 km s(-1), a threshold effect that further distinguishes SEP flares from the general population of all flares.