Abstract
The images of sunspots in 16 active regions taken at the University College of the Cayman Islands (UCCI) Observatory on Grand Cayman during June–November 2015 were used to determine their fractal dimensions using the perimeter–area method for the umbral and the penumbral region. Scale-free fractal dimensions of \(2.09 \pm0.42\) and \(1.72 \pm0.4\) were found, respectively. This value was higher than the value determined by Chumak and Chumak (Astron. Astrophys. Trans. 10, 329, 1996), who used a similar method, but only for the penumbral region of their sample set. The umbral and penumbral fractal dimensions for the specific sunspots are positively correlated with \(r = 0.58\). Furthermore, a similar time-series analysis was performed on eight images of AR 12403, from 21 August 2015 to 28 August 2015 taken from the Debrecen Photoheliographic Data (DPD). The correlation is \(r = 0.623\) between the umbral and penumbral fractal dimensions in the time series, indicating that the complexity in morphology indicated by the fractal dimension between the umbra and penumbra followed each other in time as well.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Imagej.nih.gov. “ImageJ.” accessed 14 November 2016. https://imagej.nih.gov/ij/index.html .
swpc.noaa.gov. “Sunspots/Solar Cycle|NOAA/NWS Space Weather Prediction Center.” accessed 14 November 2016. http://www.swpc.noaa.gov/phenomena/sunspotssolar-cycle .
References
Baranyi, T., Győri, L., Ludmány, A.: 2016, On-line tools for solar data compiled at the Debrecen Observatory and their extensions with the Greenwich sunspot data. Solar Phys. 291, 3081. DOI .
Bershadskii, A.G.: 1990, Large-scale fractal structure in laboratory turbulence, astrophysics, and the ocean. Sov. Phys. Usp. 33, 1073. DOI .
Boffetta, G., Carbone, V., Giuliani, P., Veltri, P., Vulpiani, A.: 1999, Power laws in solar flares: self-organized criticality or turbulence? Phys. Rev. Lett. 83, 4662. DOI .
Chumak, O.: 2005, Self-similar and self-affine structures in the observational data on solar activity. Astron. Astrophys. Trans. 24, 93. DOI .
Chumak, O.V., Chumak, Z.N.: 1996, Sunspots. The model of “elastic sceletons”. Estimation of sunspot umbra fractal dimension. Astron. Astrophys. Trans. 10, 329.
Chumak, O.V., Zhang, H.Q.: 2003, Size-flux relation in solar active regions. Chin. J. Astron. Astrophys. 3, 175. DOI .
Conlon, P.A., Gallagher, P.T., McAteer, R.T.J., Ireland, J., Young, C.A., Kestener, P., Hewett, R.J., Maguire, K.: 2008, Multifractal properties of evolving active regions. Solar Phys. 248, 297. DOI .
De Toma, G., Chapman, G.A., Preminger, D.G., Cookson, A.M.: 2013, Analysis of sunspot area over two Solar Cycles. Astrophys. J. 770, 89. DOI .
Deng, L.: 2016, Multi-fractal property and long-range correlation of chaotic time series. In: 2016 3rd International Conference on Information Science and Control Engineering (ICISCE), IEEE Press, New York, 1361. DOI .
Deng, L.H., Qu, Z.Q., Yan, X.L., Wang, K.R.: 2013, Phase analysis of sunspot group numbers on both solar hemispheres. Res. Astron. Astrophys. 13, 104. DOI .
Deng, L.H., Li, B., Xiang, Y.Y., Dun, G.T.: 2014, On mid-term periodicities of high-latitude solar activity. Adv. Space Res. 54, 125. DOI .
Deng, L.H., Li, B., Xiang, Y.Y., Dun, G.T.: 2016a, Comparison of chaotic and fractal properties of polar faculae with sunspot activity. Astron. J. 151, 2. DOI .
Deng, L.H., Xiang, Y.Y., Qu, Z.N., An, J.M.: 2016b, Systematic regularity of hemispheric sunspot areas over the past 140 years. Astron. J. 151, 70. DOI .
Feng, S., Xu, Z., Wang, F., Deng, H., Yang, Y., Ji, K.: 2014, Automated detection of low-contrast solar features using the phase-congruency algorithm. Solar Phys. 289, 3985. DOI .
Fletcher, L., Dennis, B.R., Hudson, H.S., Krucker, S., Phillips, K., Veronig, A., Gallagher, P.: 2011, An observational overview of solar flares. Space Sci. Rev. 159, 19. DOI .
Gayathri, R., Selvaraj, R.S.: 2010, Predictability of solar activity using fractal analysis. J. Indian Geophys. Union 14, 89.
Georgoulis, M.K.: 2005, Turbulence in the solar atmosphere: manifestations and diagnostics via solar image processing. Solar Phys. 228, 5. DOI .
Georgoulis, M.K.: 2012, Are solar active regions with major flares more fractal, multifractal, or turbulent than others? Solar Phys. 276, 161. DOI .
Greenkorn, R.A.: 2009, Analysis of sunspot activity cycles. Solar Phys. 255, 301. DOI .
Issa, M.A., Issa, M.A., Islam, M.S., Chudnovsky, A.: 2003, Fractal dimension—a measure of fracture roughness and toughness of concrete. Eng. Fract. Mech. 70, 125. DOI .
Jaeggli, S., Lin, H., Uitenbroek, H.: 2012, On molecular hydrogen formation and the magnetohydrostatic equilibrium of sunspots. Astrophys. J. 745, 133. DOI .
Lawrence, J.K., Cadavid, A.C., Ruzmaikin, A.A.: 1995, Turbulent and chaotic dynamics underlying solar magnetic variability. Astrophys. J. 455, 366. DOI .
Losa, G.A.: 2012, Fractals and their contribution to biology and medicine. Medicographia 34, 364.
Mandelbrot, B.B.: 1983, The Fractal Geometry of Nature, Macmillan & Co., London.
Mazzi, B., Vassilicos, J.C.: 2004, Fractal-generated turbulence. J. Fluid Mech. 502, 65. DOI .
Morse, D., Lawton, J.H., Dodson, M., Williamson, M.H.: 1985, Fractal dimension of vegetation and the distribution of anthropod body lengths. Nature 314, 73. DOI .
Panas, E.: 2001, Estimating fractal dimension using stable distributions and exploring long memory through ARFIMA models in Athens Stock Exchange. Appl. Financ. Econ. 11, 395. DOI .
Preminger, D.G., Walton, S.R.: 2006, Modeling solar spectral irradiance and total magnetic flux using sunspot areas. Solar Phys. 235, 387. DOI .
Price, C.P., Prichard, D., Hogenson, E.A.: 1992, Do the sunspot numbers form a chaotic set. J. Geophys. Res. 97, 113. DOI .
Qin, Z.: 1994, A fractal study on sunspot relative number. Chin. Astron. Astrophys. 18, 313. DOI .
Qin, Z.: 1996, A nonlinear prediction of the smoothed monthly sunspot numbers. Astron. Astrophys. 310, 646.
Rempel, M.: 2011, Subsurface magnetic field and flow structure of simulated sunspots. Astrophys. J. 740, 15. DOI .
Scherrer, P.H., Schou, J., Bush, R.I., Kosovichev, A.G., Bogart, R.S., Hoeksema, J.T., et al.: 2012, Solar Phys. 275, 207. DOI .
Shenshi, G., Zhiqian, W., Jitai, C.: 1999, The fractal research and predicating on the times series of sunspot relative number. Appl. Math. Mech. 20, 84. DOI .
Solanki, S.K.: 2003, Sunspots: an overview. Astron. Astrophys. Rev. 11, 153. DOI .
Uritsky, V.M., Davila, J.M.: 2012, Multiscale dynamics of solar magnetic structures. Astrophys. J. 748, 60. DOI .
Vertyagina, Y., Kozlovskiy, A.: 2013, Study of solar activity from the position of multifractal analysis. New Astron. 23, 36. DOI .
Watari, S.: 1995, Fractal dimensions of solar activity. Solar Phys. 158, 365.
Weitz, D.A., Huang, J.S., Lin, M.Y., Sung, J.: 1985, Limits of the fractal dimension for irreversible kinetic aggregation of gold colloids. Phys. Rev. Lett. 54, 1416. DOI .
Yan, X.L., Qu, Z.Q.: 2007, Rapid rotation of a sunspot associated with flares. Astron. Astrophys. 468, 1083. DOI .
Yan, X.L., Qu, Z.Q., Xu, C.L., Xue, Z.K., Kong, D.F.: 2009, The causality between the rapid rotation of a sunspot and an X3.4 flare. Res. Astron. Astrophys. 9, 596. DOI .
Zelenyi, L.M., Milovanov, A.V.: 1991, Fractal properties of sunspots. Sov. Astron. Lett. 17, 425.
Acknowledgements
The authors would like to thank the anonymous referee for very valuable comments that improved the manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Disclosure of Potential Conflicts of Interest
The authors declare that they have no conflicts of interest.
Rights and permissions
About this article
Cite this article
Rajkumar, B., Haque, S. & Hrudey, W. Fractal Dimensions of Umbral and Penumbral Regions of Sunspots. Sol Phys 292, 170 (2017). https://doi.org/10.1007/s11207-017-1184-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11207-017-1184-2