Andrey G. Tlatov1 and Alexei A. Pevtsov2
1. Kislovodsk Solar Station of Pulkovo Observatory, PO Box 145, Gagarina Str., 100, Kislovodsk, 357700 Russian Federation; firstname.lastname@example.org
2. National Solar Observatory, Sunspot, NM 88349, U.S.A.; email@example.com
Sunspots are the hallmark of solar activity and are the longest studied activity phenomena on the Sun. Sunspots vary by size from tiny 4.5 millionths of the solar hemisphere (MSH) in area (which corresponds to a circular feature of about 4200 km in diameter) to behemoth “Great Sunspot of 1947” with area exceeding 6100 MSH. For comparison, during the last Venus transit, the area of solar disk blocked by the Venusian disk was about 470 MSH. Recent advances in high resolution observations and numerical modeling have brought a better understanding of the fine structure of individual sunspots, their emergence, and evolution. Still, knowledge of statistical properties of sunspots is important in order to understanding their physical characteristics as a distinct “family” of solar features, which in turn is invaluable in order to model this solar phenomenon.
Sunspots are magnetic features. At the outer boundary of the spot, the outside gas pressure is balanced by the sum of magnetic and gas pressure inside the spot. Based on this simple model, one would expect a relationship between the size of sunspot and the strength of its magnetic field. Indeed, early studies[1,2] showed that there is a correlation between magnetic flux (maximum field strength) and area of sunspots (As) albeit with a significant scatter in both parameters. This scatter is significantly reduced when one uses observations from space-borne instruments.
Figure 1 shows a scatter plot of total magnetic flux as a function of area of pores and sunspots observed by HMI/SDO line-of-sight magnetograms during June 2010 – September 2012. Because pores and small sunspots in this plot partially overlap, horizontal bars mark approximate range in their areas. Also marked are ranges for regular sunspots and transition between small and regular sunspots. The total magnetic flux is a sum of the flux within an individual sunspot or pore as determined by an intensity threshold to define the outer penumbral or umbral boundary.
It is interesting to note that the slope is different for areas smaller than about 20 MSH and for those larger than 100 MSH. Linear fits to the area-flux dependence in Figure 1 (white solid lines) show a clear change between small sunspots, transition and regular sunspots (compare inclination of three linear segments fitted to the data in Figure 1). If one fit a continuous function to the data in Figure 1, As × sin(log As) works reasonably well. The curve corresponding to this function overlaps with the three linear segments shown in Figure 1, and thus, is not shown in the Figure.
Geometric properties of “small” and “regular” sunspots form two distinct groups. Figure 2 shows a scatter plot of area of sunspot umbra (Au) as a function of total area of sunspot (As). For regular spots (As ≥100 MSH), the umbral area changes linearly with the size of sunspot with a relatively small scatter. The distribution of ratio of umbral to sunspot areas for this group is very narrow, and it peaks at about Au/As=0.156±0.034. For small sunspots (20 MSH < As <100 MSH), the distribution of Au/As ratio is very broad with the average of 0.359±0.119. For transition sunspots, Au/As=0.231±0.099.
This slight but clear difference in mean properties of sunspots suggests that sunspots of different size may represent different evolutionary stages of this class of objects, or that they are the result of different physical processes.
Bimodal distribution of sunspots by their area and magnetic flux (maximum field strength) was recently noted by several researchers. Thus, for example,  have shown that the distribution of sunspot areas can be represented by a composite of two log-normal distributions, corresponding to “small” and “large” sunspots. They interpreted the presence of these two components as an indication of two spatially separated dynamos, with smaller sunspots forming at more shallow depths, and larger sunspots forming deeper in the solar convection zone. More recently,  showed that the distribution of sunspot areas can be fitted by a combination of Weibull and lognormal functions, and also interpreted this as evidence of two separate mechanisms: one directly connected to the global component of the dynamo, and the other with the small-scale component of the dynamo.
 Nicholson S.B.: 1933, “The Area of a Sun-Spot and the Intensity of Its Magnetic Field”, Pub. of Astron. Soc. of the Pacific, 45, 51.
 Houtgast, J., van Sluiters, A.: 1948, “Statistical investigations concerning the magnetic fields of sunspots“, Bull. Astron. Inst. Netherlands, 10, 325
 Tlatov, A.G. and Pevtsov, A.A.: 2014, “Bimodal Distribution of Magnetic Fields and Areas of Sunspots”, Solar Physics, 289, 1143-1152, DOI: 10.1007/s11207-013-0382-9
 Nagovitsyn, Yu. A., Pevtsov, A.A., and Livingston, W.C.: 2012, “On a Possible Explanation of the Long-Term Decrease in Sunspot Field Strengths”, Astrophysical Journal Letters, 758, L20, DOI: 10.1088/2041-8205/758/1/L20
 Muñoz-Jaramillo, A., et al.: 2015, “Area and Flux Distributions of Active Regions, Sunspot Groups, and Sunspots: A Multi-Database Study”, Astrophys. J, in press