Estimating the Properties of Hard XRay Solar Flares by Constraining Model Parameters
Abstract
We wish to better constrain the properties of solar flares by exploring how parameterized models of solar flares interact with uncertainty estimation methods. We compare four different methods of calculating uncertainty estimates in fitting parameterized models to Ramaty High Energy Solar Spectroscopic Imager Xray spectra, considering only statistical sources of error. Three of the four methods are based on estimating the scalesize of the minimum in a hypersurface formed by the weighted sum of the squares of the differences between the model fit and the data as a function of the fit parameters, and are implemented as commonly practiced. The fourth method is also based on the difference between the data and the model, but instead uses Bayesian data analysis and Markov chain Monte Carlo (MCMC) techniques to calculate an uncertainty estimate. Two flare spectra are modeled: one from the Geostationary Operational Environmental Satellite X1.3 class flare of 2005 January 19, and the other from the X4.8 flare of 2002 July 23. We find that the four methods give approximately the same uncertainty estimates for the 2005 January 19 spectral fit parameters, but lead to very different uncertainty estimates for the 2002 July 23 spectral fit. This is because each method implements different analyses of the hypersurface, yielding methoddependent results that can differ greatly depending on the shape of the hypersurface. The hypersurface arising from the 2005 January 19 analysis is consistent with a normal distribution; therefore, the assumptions behind the three nonBayesian uncertainty estimation methods are satisfied and similar estimates are found. The 2002 July 23 analysis shows that the hypersurface is not consistent with a normal distribution, indicating that the assumptions behind the three nonBayesian uncertainty estimation methods are not satisfied, leading to differing estimates of the uncertainty. We find that the shape of the hypersurface is crucial in understanding the output from each uncertainty estimation technique, and that a crucial factor determining the shape of hypersurface is the location of the lowenergy cutoff relative to energies where the thermal emission dominates. The Bayesian/MCMC approach also allows us to provide detailed information on probable values of the lowenergy cutoff, E_{c} , a crucial parameter in defining the energy content of the flareaccelerated electrons. We show that for the 2002 July 23 flare data, there is a 95% probability that E_{c} lies below approximately 40 keV, and a 68% probability that it lies in the range 736 keV. Further, the lowenergy cutoff is more likely to be in the range 2535 keV than in any other 10 keV wide energy range. The lowenergy cutoff for the 2005 January 19 flare is more tightly constrained to 107 ± 4 keV with 68% probability. Using the Bayesian/MCMC approach, we also estimate for the first time probability density functions for the total number of flareaccelerated electrons and the energy they carry for each flare studied. For the 2002 July 23 event, these probability density functions are asymmetric with long tails orders of magnitude higher than the most probable value, caused by the poorly constrained value of the lowenergy cutoff. The most probable electron power is estimated at 10^{28.1} erg s^{1}, with a 68% credible interval estimated at 10^{28.1}10^{29.0} erg s^{1}, and a 95% credible interval estimated at 10^{28.0}10^{30.2} erg s^{1}. For the 2005 January 19 flare spectrum, the probability density functions for the total number of flareaccelerated electrons and their energy are much more symmetric and narrow: the most probable electron power is estimated at 10^{27.66 ± 0.01} erg s^{1} (68% credible intervals). However, in this case the uncertainty due to systematic sources of error is estimated to dominate the uncertainty due to statistical sources of error.
 Publication:

The Astrophysical Journal
 Pub Date:
 June 2013
 DOI:
 10.1088/0004637X/769/2/89
 arXiv:
 arXiv:1304.8117
 Bibcode:
 2013ApJ...769...89I
 Keywords:

 methods: data analysis;
 methods: statistical;
 Sun: flares;
 Sun: Xrays;
 gamma rays;
 Astrophysics  Solar and Stellar Astrophysics
 EPrint:
 doi:10.1088/0004637X/769/2/89