Physica A 387 (2008) 1077–1087 www.elsevier.com/locate/physa From solar flare time series to fractional dynamics Krzysztof Burnecki a,∗ , Joseph Klafter b , Marcin Magdziarz a , Aleksander Weron a a Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland b School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel Received 31 July 2007; received in revised form 2 October 2007 Available online 13 October 2007 Abstract We demonstrate that continuous-time FARIMA processes with α-stable noise provide a new stochastic tool for studying the solar flare phenomenon in the framework of fractional Langevin equation. Simple computer tests to check the origins of α-stability and self-similarity are implemented for empirical time series describing the energy of solar flares. Based on observed physical time series we solve the challenging problem of how to detect long-range dependence from real data and how to model it via fractional dynamics (Langevin or Fokker–Planck). We employ here codifference as a proper measure for long-range dependence. It is applicable to empirical data from the distribution lacking the second moment. c  2007 Elsevier B.V. All rights reserved. Keywords: Solar flares; Long-range dependence; Stable noise; Self-similarity; Fractional dynamics 1. Introduction Solar flares are the most energetic and violent events occurring in the solar atmosphere [1]. Observations of solar flare phenomena in X-rays became possible in the 1960s with the availability of space-borne instrumentation. Since 1974 broadband soft X-ray emission of the Sun has been measured almost continuously by meteorological satellites operated by the National Oceanic and Atmospheric Administration (NOAA) such as the Synchronous Meteorological Satellite (SMS) and the Geostationary Operational Environment Satellite (GOES) [2]. Understanding the long-term solar variability and predicting the solar activity is an actual problem for solar physics [1]. It is important to predict the time and strength of such events since such disturbances can pose serious threats to man-made spacecrafts, can disrupt electronic communication channels and influence Earth temperature fluctuations [3–5]. Ever since the pioneering work by Mandelbrot [6], and Montroll and Scher [7], L´ evy-stable processes have enjoyed great popularity as flexible modeling tools in economics and natural sciences. The importance of L´ evy-stable distributions or processes in physics, astronomy and related areas has long been widespread [8–18]. Consequently, L´ evy-stable type anomalous diffusion has been treated on a similar footing as Brownian diffusion [19–30]. ∗ Corresponding author. Tel.: +48 713203530; fax: +48 713202654. E-mail addresses: krzysztof.burnecki@pwr.wroc.pl (K. Burnecki), klafter@post.tau.ac.il (J. Klafter), marcin.magdziarz@pwr.wroc.pl (M. Magdziarz), aleksander.weron@pwr.wroc.pl (A. Weron). 0378-4371/$ - see front matter c  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.10.024