Probability Distribution Functions of Sunspot Magnetic Flux

Abstract

We have investigated the probability distributions of sunspot area and magnetic flux by using the data from Royal Greenwich Observatory and USAF/NOAA. We have constructed a sample of 2995 regions with maximum-development areas $\ge$ 500 MSH (millionths of solar hemisphere), covering 146.7 years (1874--2020). The data were fitted by a power-law distribution and four two-parameter distributions (tapered power-law, gamma, lognormal, and Weibull distributions). The power-law model was unfavorable compared to the four models in terms of AIC, and was not acceptable by the classical Kolmogorov-Smirnov test. The lognormal and Weibull distributions were excluded because their behavior extended to smaller regions ($S \ll 500$ MSH) do not connect to the previously published results. Therefore, our choices were tapered power-law and gamma distributions. The power-law portion of the tapered power-law and gamma distributions was found to have a power exponent of 1.35--1.9. Due to the exponential fall-off of these distributions, the expected frequencies of large sunspots are low. The largest sunspot group observed had an area of 6132 MSH, and the frequency of sunspots larger than $10^4$ MSH was estimated to be every 3 -- 8 $\times 10^4$ years. We also have estimated the distributions of the Sun-as-a-star total sunspot areas. The largest total area covered by sunspots in the record was 1.67 % of the visible disk, and can be up to 2.7 % by artificially increasing the lifetimes of large sunspots in an area evolution model. These values are still smaller than those found on active Sun-like stars.