Abstract
Many natural processes exhibit a power-law behavior. The power-law exponent is linked to the underlying physical process, and therefore its precise value is of interest. With respect to the energy content of nanoflares, for example, a power-law exponent steeper than 2 is believed to be a necessary condition for solving the enigmatic coronal heating problem. Studying power-law distributions over several orders of magnitudes requires sufficient data and appropriate methodology. In this article we demonstrate the shortcomings of some popular methods in solar physics that are applied to data of typical sample sizes. We use synthetic data to study the effect of the sample size on the performance of different estimation methods. We show that vast amounts of data are needed to obtain a reliable result with graphical methods (where the power-law exponent is estimated by a linear fit on a log-transformed histogram of the data). We revisit published results on power laws for the angular width of solar coronal mass ejections and the radiative losses of nanoflares. We demonstrate the benefits of the maximum likelihood estimator and advocate its use.
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Notes
Our dataset expresses energy in data numbers (DN). Berghmans, Clette, and Moses (1998) reported a factor of \(2\times10^{20}~\mbox{erg}/\mbox{DN}\) to convert the flare energies to the physical units used in the histograms. However, we suspect a typographical error in this conversion factor. We needed to rescale our histogram by a factor of \(2\times10^{24}~\mbox{erg}/\mbox{DN}\) to reproduce the results of Berghmans, Clette, and Moses and to obtain reasonable energies for solar flares.
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Acknowledgements
The authors are grateful to V. Delouille and the PROBA2/SWAP team for valuable input. We also thank the anonymous referee for insightful comments that helped us improve this article. This research was co-funded by a Supplementary Researchers Grant offered by the Belgian Science Policy Office (BELSPO) in the framework of the Scientific Exploitation of PROBA2, the Inter-University Attraction Poles Programme initiated by BELSPO (IAP P7/08 CHARM), and the European Union’s Seventh Framework Programme for Research, Technological Development and Demonstration under Grant Agreements No. 284461 (Project eHeroes, www.eheroes.eu ) and No. 269299 (Project SOLSPANET, www.solspanet.eu ). These results were also obtained in the framework of the projects GOA/2015-014 (KU Leuven), G.0729.11 (FWO-Vlaanderen) and C 90347 (ESA Prodex). E. D’Huys and D.B. Seaton additionally acknowledge support from BELSPO through the ESA-PRODEX program, grant No. 4000103240. This paper uses data from the CACTus CME catalog, generated and maintained by the SIDC at the Royal Observatory of Belgium ( www.sidc.be/cactus ).
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D’Huys, E., Berghmans, D., Seaton, D.B. et al. The Effect of Limited Sample Sizes on the Accuracy of the Estimated Scaling Parameter for Power-Law-Distributed Solar Data. Sol Phys 291, 1561–1576 (2016). https://doi.org/10.1007/s11207-016-0910-5
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DOI: https://doi.org/10.1007/s11207-016-0910-5