Probability Distribution Functions of Sunspot Magnetic Flux

TL;DR

This study analyzes the probability distributions of sunspot areas and magnetic flux using historical data, finding that tapered power-law and gamma distributions best fit the data, with implications for understanding large sunspot occurrence rates.

Contribution

It identifies the most suitable statistical models for sunspot area distributions and estimates the frequency of large sunspots over historical and stellar contexts.

Findings

Power-law model was less suitable than tapered power-law and gamma distributions.

Large sunspots (>10^4 MSH) are extremely rare, occurring every 30,000 to 80,000 years.

Maximum observed sunspot area was 6132 MSH, with potential maximum coverage up to 2.7%.

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.