A STUDY OF LONG-TERM SUNSPOTS AND K-INDEX GEOMETRIC CYCLES USING PROBABILISTIC MODELING

The research work done in this paper comprises the application of different well-known probability distribution models. This includes the understanding of the behavior and dynamics of 24 sunspot cycles with total data. The time-series data sets were selected from 1749 to 2014. To observe the solar activity effects on K-index activity the double cycles from 1932 to 2014 were also incorporated in the study. The comparative study is useful to observe the long-term solar-terrestrial connection. The magnetic field of the sun reverses its polarity after every 11 years of the cycle. So after every 22 years, the north pole becomes again north pole. By using the two well-known tests Kolmogorov-Smirnov (KST) and Anderson-Darling test (ADT) the probability distribution models were obtained for each sunspot cycles and compare. The significant probability models for all the sunspot cycles have been obtained. The fitted probability distribution models on selected data sets may be useful to understand the trend of solar and geomagnetic activity.


INTRODUCTION
The study of sunspots and its cycles has been unique importance in understanding the space climate, global climate and solarterrestrial relationship. The solar cycles are depending on the sunspots and on its variation that is because of polar reversibility on sun. the quiet and active durations are also strongly connected with the number of sunspots [1,15].
The other phenomena on sun like Plages, flares, solar wind and prominences are also depending on sunspots activity [14]. The astrophysics provides the information about the solar-terrestrial connection in the perspective of magnetic field of sun, electromagnetic radiation and emission of different particles. The composition of different gases on sun are H (60%), He (40%) and heavier particles (1%). The cumulative mass is 2 × 10 33 grams with 5 × 10 9 years of age. The earth is 1.5 × 10 8 Km away in the perspective of the climate of the earth has been one of the important indices to experience the variation. It categorizes from 0-9 (very quiet to extremely disturb) scale on a scale [10]. The data from any observatory represents the measured at three hours of intervals. The index depicts the third of a unit (5 − is 4 2 3 , 5o is ) [4,7]. The average of activity K-index is the index that is determined and standardized by the data of all station. A storm is labeled a major storm if it is equivalent or surpasses forty [11]. In 1932 to 1989 more than thousands of such storms were happened. A

DETAILS OF MATERIAL AND METHODS
The sun comprises numerous kinds of cycles, many features of the passing solar atmosphere and lots of assets of the quiet as well as the active sun are linked with sunspots. The number of sunspots and latitude foundation is considered almost periodic, again over an 11- year cycle [11,12]. A specific sunspot remains at persistent latitude in its lifetime; also flourishing sunspots tend to format increasingly worse altitudes, moving from the poles towards the nevertheless the main sunspot in the further hemisphere will have the reverse polarity [5].
When a large collection of sprawling spots exists then it creates a twisted magnetic field shape, a fundamentally bipolar field is existent. The polarizations will additionally be reserved in time of the forthcoming cycle of 11 years. In the north hemisphere the sunspot will prime with a magnetic polarity in south. In overall, the dipole park of the sun will variation with the intent that the north-pole of the sun will control from the geographic north-pole to the geographic south-pole. At minimum sunspot time, where the first sunspot is begins to form at higher latitudes.
It is noticed that the change of polarity occurs always. For this purpose, the sun is known to have a cycle of 22 year [3,5]. In this Manuscript, the behavior of 24 sunspots cycles have been where is depicts the positive integer [6].
The obtained models may be the most applicable distribution for the sunspot and Kindex cycles. Since the probability distribution helps to realize the potential future variation of the variables. The aim of this research work was to Test the behavior of all the sunspot cycles using probability distribution. It is obvious that 4 K-index cycles correspond to these 8 sunspot cycles in the duration [5]. The normal distribution of continuous variables is the most important distribution fitted on the selected time series. It can be demonstrated mathematically as follows.
Where, is recognizes as the mean of the sample and is the standard deviation.
Similarly, with parameters and , if the random variable Y depicts the gamma distribution, then the likelihood of Y can be indicated in the form as.
Where and are the shape and scale parameter respectively.
Moreover, it is normally have noticed that the normal distribution is shifted to gamma distribution and this gamma distribution can be further transformed to log-normal distribution [6].
For the log-normal distribution the mathematical model is expressed as, [ ] = + 2 2 ,

Equation 6 signifies the mean (E[X]) and
variance (Var (X)) of experiments respectively [6,14]. Where parameters = δ > 0 and the variable x > 0 belongs to real numbers. The lognormal distribution, has been used as a first estimate sometimes known as Landau distribution, explains the loss of energy of a heavy charged particle by ionization [6,9]. The normal distribution can be assessed the chisquare distribution and is given by The variable ≥ 0 and positive integer n is the number of degrees of freedom in the chi-square distribution [9].

Conflicts of interest/Competing interests:
The authors declare no any conflict of interest.