The sunspot as an autonomous dynamical system: A model for the growth and decay phases of sunspots

This paper presents a model for describing the sunspot as an autonomous dynamical system both in its growth and decay phases. The model consists of a two-dimensional system of ordinary differential equations (ODE) of first order with respect to time. The two time-dependent functions are the area of the sunspot on the photosphere, A(t), and the mean spatial value of the magnetic field strength inside the sunspot, B(t). The model reproduces both the sunspot growth and decay phases. Emphasis is placed on the three main decay laws supported by the observational data, namely the linear, the parabolic and the exponential. In particular, the model reproduces each one of the three decay laws for different times of the decay phase. We also calculate an upper limit for the area of sunspots. The experimental log-normal distribution of the maximum sunspot areas is satisfactorily derived. By setting the initial area of sunspots equal to the area of the finest structure observed in the quiet Sun, namely the granules, a method for calculating the mean dimensions of granules is deduced. This is achieved by implementing a fitting method, based on the q-norm, between the theoretical and the experimental distributions of the maximum areas. We show that the method of the absolute deviations minimization (q=1) performs the largest sensitivity in regard to the alternative fitting methods based on other q-norms. Finally, we consider a non-integrable extension of the model which exhibits chaotic behavior.