Fluctuating Sunspot Numbers Exhibit A Non-Markovian Damped Stochastic Process

Abstract

The rise and fall in the number of sunspots have served as a lynchpin in many investigations on solar dynamics. Arising from magnetic disturbances in the sun, variations in sunspot numbers have helped define a solar cycle of around eleven years which to date is yet to be fully understood. We model the fluctuation of sunspot numbers as a modulated Brownian motion characterized by a memory parameter {\mu} and a decay parameter \b{eta}. By matching the theoretical and empirical mean square deviation of the sunspot numbers, the values of {\mu} and \b{eta} are determined for each solar cycle. This allows us to obtain an exact form of a probability density function (PDF) which closely matches the dataset for sunspots. This novel PDF for sunspot numbers exhibit a memory behavior from which some insights could be obtained. In particular, the values of {\mu} indicate that consecutive sunspot numbers are negatively correlated for large times. The values of \b{eta}, on the other hand, when viewed as a time series from one solar cycle to another, indicate a positive trend towards increasing values which could possibly suggest a diminishing solar activity.