Quantitative Analysis of Dynamics of Solar Activity Cycles in the Perspective of Data Distribution and Tail Analysis

Abstract

Sun is the main source of energy for the earth and other planets. Its activity in one or the other way influences the terrestrial climate. Particularly, the solar activity manifested in the form of sunspots is found to be more influential on the earth’s climate and its magnetosphere. Links of the variability in terrestrial climate, sunspot cycles and associated magnetic cycles have been the concern of many recent studies. The role of the sun and its activities to understand the space weather and the earth’s environment interaction has been the unique importance in all eras. In this dissertation, we have fitted some adequate probability distributions and stochastic modeling on solar activity (particularly sunspots and solar flares) cycles and terrestrialmagnetic (K-index) activity data comparatively. The 24 cycles (1749-2014) of sunspots including 24th cycle that is in progress, last 4 cycles (20, 21, 22 and 23) of solar flares (1966-2008) and terrestrial K-index activity data (1932-2014) are used in the research work. We have compared the solar activity cycles and K-index activity cycles (associated with solar activity cycle) in the perspective of probability distributions. Comparing both the data we have distributed the time series (1932-2014) among 22-year cycle (2 solar cycles) of each. This kind of distribution is based on the period of one magnetic cycle of sun in which polarity is changed after each 11 years. The magnetosphere’s and magnetic field’s variation of earth can be detected and analyzed by the change in K-index data on which earth climate is depends. The geomagnetic activity is the one of the best recorded sign on earth of solar activity variations. It is basically showing a relationship between space weather and earth's climate. Results obtained in this dissertation show the quasi-regular (persistent) dynamics of solar activity and K-index activity cycles along with the total time series data from the perspective of fractal dimension. Long-range dependence for each activity cycle is also calculated in terms of Hurst exponent. Theoretical instrument is developed between solar and K-index activity cycles to understand their long term relationship. Stochastic modeling is also fitted on the solar activity and K-index cyclic as well as on the total time series data. The result shows the heavy tail for the sunspots and K-index activity time series data used in this dissertation. The stochastic model FARIMA (Fractional Auto Regressive Integrated Moving Average) is applied on the cycles along with their total time series data as used time series are long ringing dependence (LRD). FARIMA has a capability to use on short and long term conditions. Fractional differencing parameter and heavy tails parameter are calculated to understand the strength and peak of each cycle. The parameters of FARIMA model are obtained by MLE (Maximum Likelihood Estimator). Goodness-of-fit (AIC, BIC and HIC) are used to select the best fitted model among FARIMA (0, d, 0), (1, d, 0), (0, d, 1) and (1, d, 1). The log - likelihood is also estimated for further verification of significant model. Any time series that have heavy tail, fitting FARIMA modeling for them can be more useful to understand their expected behavior in future. The underlying physics of solar activity and K-index activity cycles is modeled by FARIMA (p, d, q) in this dissertation. Finally, we have analyzed and verified that the sunspots and K-index activities are followed Markov process. Transition matrices for both the activities are estimated to understand their physical behavior in 4 different selected states. Stationarity for stochastic matrices is observed in this dissertation to understand similar physical behavior in the used activity data. 2-dimensional correlation between stochastic matrices of sunspots and K-index activity cycles are calculated to understand how much relationship strong between them. In this connection 2-dimensional correlation is also obtained between sunspots and ENSO data to observe the sunspots effects on the earth’s climate. Bayesian posterior and prior are also observed in the estimated stochastic matrices as Bayesian approach is more adequate to understand the complex in the models. By the results obtained we can say that all the activities used in this dissertation are correlated and predictable. We can use probabilistic and stochastic approach to model them. The topic is wide that we could not cover by single dissertation, the same can be done with other solar, geomagnetic and global indices that we did not use in this research work to understand the space weather and earth climate interaction more intensely.