BHP universality and gaussianity in sunspot numbers fluctuations

arXiv:0802.2880v4 [physics.data-an] 15 Mar 2008 BHP universality and gaussianity in sunspot numbers fluctuations R. Gon¸calves a,∗A. A. Pinto b aFaculdade de Engenharia R. Dr. Roberto Frias, 4200 - 465, Porto, Portugal bUniversidade do Minho Campus de Gualtar, 4710 - 057 Braga, Portugal Abstract We analyze the famous Wolf’s sunspot numbers. We discovered that the distribu- tion of the sunspot number fluctuations is a mixture of the BHP distribution with the Gaussian distribution. Key words: Wolf’s sunspot numbers, solar physics, statistical mechanics, self-organized criticality. PACS: 96.60.qd, 96.60.-j, 64.60.-i ∗Universidade do Porto, R. Dr. Roberto Frias, 4200-465 Porto Portugal, tel/Fax: +351 225081707/1446 Email address: rjasg@fe.up.pt (R. Gon¸calves). 1 Introduction The sunspots are relatively dark areas on the surface of the sun caused by strong concentration of magnetic flux. Lu et al. [7] stated that the physical picture that arises, associated to sunspots, is that solar flares are avalanches of many reconnection events analogous to avalanches of sands in the models of Bak et al. [4]. The relation between small-scale processes and the statis- tics of global flare properties, that follows from the self-organized magnetic field configuration, provides a way to learn about the physics of unobserv- able small-scale reconnection processes. On the other hand, Bramwell et al. [2] showed a relation between self-organized criticality and the universal BHP distribution, named after the work of Bramwell, Holdsworth and Pinton [1]. We make a bridge between the two literatures, showing that the distribution of the sunspot numbers fluctuations is close to a mixture of the BHP with a Gaussian distribution. 2 A BHP and a gaussian mixture fit of the sunspot numbers fluc- tuations The average duration of the sunspot cycle is 133 months, but cycles as short as 9 years and as long as 14 years have been observed by Rabin et al. [8]. Following Bramwell et al. [3] and J´anosi et al. [6], we define the sunspot numbers mean period wµ(t) by wµ(t) = 1 T T−1 X j=0 w(t + j ∗133) (1) 2 and the sunspot numbers standard deviation period wσ(t) by wσ(t) = sPT−1 j=0 w(t + j ∗133)2 T −wµ(t)2 , (2) where T = 23 is the number of observed cycles. The sunspot numbers fluctu- ations wf(t) is given by wf(t) = w(t) −wµ(t) wσ(t) . (3) -6 -4 -2 0 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 w_f ln(p(w_f)) Sunspots BHP Gaussian fit Fig. 1. Histogram of the sunspot numbers fluctuations with a Gaussian and BHP fits, in a semi-log plot. In Figure 1, we show a fit of the BHP pdf to the histogram of the sunspot num- bers fluctuations. For sunspot numbers fluctuations away of the mode value, the BHP pdf is close to the histogram. In the range -0.9 to -0.3, close to the mode value of the histogram, we notice a higher concentration of the sunspot numbers fluctuations compared with the BHP pdf. Hence, we add a Gaussian fit to describe the concentration of the sunspot numbers fluctuations in this range. We conclude that the histogram of the sunspot numbers fluctuations can be well approximated by a mixture of a Gaussian pdf with the BHP pdf. 3 Acknowledgements We would like to thank Nico Stollenwerk for showing us the relevance of the BHP distribution. References [1] Bramwell, S.T., Holdsworth, P.C.W., & Pinton, J.F. (1998) Universality of rare fluctuations in turbulence and critical phenomena, Nature 396, 552–554. [2] Bramwell, S.T., Christensen, K., Fortin, J.Y., Holdsworth, P.C.W., Jensen, H.J., Lise, S., Lopez, J.M., Nicodemi, M., Pinton, J.F. & Sellitto, M. (2000) Universal Fluctuations in Correlated Systems, Phys. Rev. Lett. 84, 3744–3747. [3] Bramwell, S.T., Fennell, T., Holdsworth, P.C.W. & Portelli, B. (2002) Universal Fluctuations of the Danube Water Level: a Link with Turbulence, Criticality and Company Growth, Europhysics Letters 57, 310. [4] Bak, P., Tang, C., & Wiesenfeld, K. (1988) Self-organized criticality, Phys. Rev. Lett.A 38, 364–374. [5] Faria, E., Melo, W. & Pinto, A. (2006) Global hyperbolicity of renormalization for Cr unimodal mappings, Annals of Mathematics, 164 , p. 731–824. [6] J´anosi, I.M., & Gallas, J.A.C. (1999) Growth of companies and water-level fluctuations of the River Danube, Physica A 271, 448–457. [7] Lu, E. & Hamilton, R. (1991) Avalanches and the distribution of solar flares, The astrophysical journal, 380, L89–L92. [8] Rabin, D., Wilson, R. & Moore, R. (1986) Bimodality of the Solar Cycle, Geophysical Research Letters, 13(4), 352354. 4

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