On the size distribution of spots within sunspot groups

We start by considering all data on every given day, i.e. irrespective of the group’s evolutionary phase on that day. $\rm{R}$ values are then calculated for all individual group entries. Figure 2a shows the derived distributions of $\rm{R}$ from Debrecen (black histogram), Kislovodsk (green) and Pulkovo (red) data. All three distributions display some common features, e.g., a sharp primary peak at R=1 and a flatter secondary peak around $\rm{R}=0.5$. The peak at $\rm{R}$=1 comes from the single-spot groups for which $\rm{A_{big\_spot}}$= $\rm{A_{group}}$. Such groups make approximately 19% of all the listed entries in our catalogues. Since we are interested in the distribution of spots within a group, we remove these single spot groups from our sample. In addition to that, we also reject small groups, with area less than 30 $\mu$Hem, as they are likely to be associated with larger measurement uncertainties (see Muñoz-Jaramillo et al., 2015). We note, however, that a change of this threshold to different values within the range 30–80 $\mu$Hem led to the results only marginally different from those presented below, while reducing the statistics.

The final distributions, after implementing these two restrictions, are shown in Fig. 2b. Two distinct features are evident in all three distributions. First, the distributions are clearly asymmetric, being skewed towards higher ratios (skewness=0.03) and their mode values lying around $\rm{R}\approxeq$0.5. Secondly, although the peak at $\rm{R}$=1 has now disappeared, a weak secondary peak is still seen at R$\approxeq$0.97, which is pronounced in Debrecen and Pulkovo data and is significantly weaker in Kislovodsk data. We will return to this peak in Sect. 3.2.

The shape of these histograms might possibly be affected by the complex evolutionary scenarios of sunspot groups, such as fragmentation or coalescence (Schrijver et al., 1997). Furthermore, while spots emerge relatively fast, their decay times are much longer (Howard, 1992). Thus, as has been pointed out in the past, the results derived from a snapshot distribution (i.e. considering all spots independently of their evolution) can potentially be dominated by the longer decay phase of sunspot groups (Baumann & Solanki, 2005). To examine the influence of these effects on the obtained distributions, we track every group in our catalogues and re-derive the $\rm{R}$ distributions such that each group is considered only once.

We first visually inspected a subset of Debrecen images⁷⁷7Available here ftp://ftp.ngdc.noaa.gov/STP/SOLAR˙DATA/SUNSPOT˙REGIONS/Debrecen/Debrecen˙Photoheliographic˙Data/images/ from the period 1986–1999, which trace individual groups. These images are extractions of the full-disc photographs and within each of these snapshots, spots forming a group are uniquely numbered. These images allowed us to trace out 40 individual groups (chosen randomly). By following these groups from their first emergence to their disappearance, we have noticed that, during the initial growth or final decay phase, the following spots are significantly smaller than the leading spots. In other words, as has also been noticed in earlier studies, leading spots form faster than the following ones (McIntosh, 1981; Rempel & Cheung, 2014), and decay more slowly that following spots (Bumba, 1963). This leads to $\rm{R}$ values very close to (though not exactly) 1 during these periods. This is, in fact, the reason behind the weak secondary peak at R$\sim$0.97 which we see in the instantaneous distributions (Figure 2b).

Therefore, we now use the information from all the catalogues considered here to track sunspot groups over their lifetime. To minimise the errors introduced by the foreshortening corrections, only groups within central meridian distances of $\pm$65°are considered here. Following the snapshot analysis described in the previous section, we leave out single-spot ($\rm{N}$=1) groups, and groups with area less than 30 $\mu$Hem. Our aim here is to consider every group only once over its entire lifetime. Usually, such studies consider the time when a group area ($\rm{A_{group}}$) reaches its maximum value. However, in our case, the second parameter entering $\rm{R}$, the size of the biggest spot $\rm{A_{big\_spot}}$, also evolves with time and $\rm{A_{big\_spot}}$ and $\rm{A_{group}}$ do not necessarily reach their maximum at the same time. In fact, we find that only 65% of the total groups analyzed here, have the maximum of these two quantities on the same day. For 20% of all groups, $\rm{A_{big\_spot}}$ reaches its maximum after the maximum in $\rm{A_{group}}$ (termed a positive delay here) whereas 15% of the remaining groups show a negative delay. Both, the positive and the negative delays mainly happen in big complex groups with many spots, which require a separate study. For the purpose of this paper, we only consider groups with no delay, i.e. when the groups and the biggest spot areas reach their maximum on the same day.

The distribution of $\rm{R}$ derived for these groups is shown in Figure 3a. The maxima of the distributions in all three cases lie around $\rm{R}$= 0.5, and unlike the snapshot distributions, there are no secondary peaks. To further understand these distributions, we calculate the quartiles Q1, Q2 and Q3, which describe the sample 25, 50 and 75 percentiles, respectively (Ross, 2000); see Table 1. In Figure 3a, the shaded regions highlight the quartiles computed for the distribution averaged over the three datasets. The range between Q1 and Q3 represents the interquartile range (IQR=Q3-Q1), where 50% of the data points around the median value, Q2, lie. IQR is considered to be a measure of the spread of the distribution. For all the individual and the average distributions, IQR is only 0.26. Such a low IQR indicates that the data are clustered about the central tendency, i.e. the median value (Q2) of $\rm{R}$=0.6. Thus, in the majority of groups, a single large spot occupies about 48–74% of the total group area.

Some properties of sunspot groups, such as their rotation rates (Balthasar et al., 1986), growth and decay rates (Howard, 1992; Hathaway & Choudhary, 2008; Muraközy et al., 2014; Muraközy, 2020, 2021), hemispheric asymmetry (Mandal & Banerjee, 2016) etc, have been found to show a clear dependence on the group sizes. Therefore, we now investigate whether a similar behaviour also exists for $\rm{R}$. To examine this, we bin the group areas (in bins of equal widths in $\log(A_{group})$) and calculate the median value of $\rm{R}$ in each bin. Figure 3b shows the dependence of $\rm{R}$ on $\rm{A_{group}}$ for the three observatories. Barring the first few points below 100 $\mu$Hem and a single bin centred at 650 $\mu$Hem for Kislovodsk, $\rm{R}$ stays above 0.6 over the entire range of sunspot group sizes. We also observe an interesting three-phase evolution in $\rm{R}$. First, as $\rm{A_{group}}$ increases from 30 to 200 $\mu$Hem, $\rm{R}$ rises rapidly from about 0.55 to 0.65. Between about $\rm{A_{group}}$ of 200 and 600 $\mu$Hem it decreases somewhat and then eventually settles at a constant value of around 0.6 beyond $\rm{A_{group}}\geq$600 $\mu$Hem. Such a trend can be explained by the fact that most of the small groups ($\rm{A_{group}}<$50 $\mu$Hem) have a simple bipolar structure with roughly equal distribution of areas between the two constituent spots and thus $\rm{R}\approxeq$0.5. As $\rm{A_{group}}$ increases, the compact leading spot grows rapidly (McIntosh, 1981; Rempel & Cheung, 2014). This leads to an increase in $\rm{R}$ value and explains the initial rise of R from 0.55 to 0.65 below $\rm{A_{group}}\approxeq$200 $\mu$Hem. The drop beyond that to a constant value of 0.6 is unexpected and interesting. We suggest that this aspect of the data indicates a degree of scale-invariance in the flux emergence process, i.e a large active region is just a scaled up version of a small active region, with the size of all the spots increasing in proportion. In this interpretation, the slow decrease in R seen for $\rm{A_{group}}\geq$200 $\mu$Hem is a consequence of the larger bright-points in a small active region being scaled up in flux to become small spots in larger active regions.

Our findings are also of relevance for understanding giant starspots seen on highly active (rapidly rotating) sun-like stars (see, e.g., Berdyugina, 2005; Strassmeier, 2009). In Fig. 3c, we show an extrapolation of the $\rm{R}$ vs $\rm{A_{group}}$ relation up to $\rm{A_{group}}$=5000 $\mu$Hem. We perform this extrapolation by applying a linear fit to the averaged data (i.e averaged over the three individual datasets) over the range $\rm{A_{group}}$=500–1200 $\mu$Hem. The grey shaded region in Fig. 3c highlights the 1-$\sigma$ uncertainty associated with this fitting. We find that the largest spot within a huge group (of $\approx$5000 $\mu$Hem) would cover roughly 55–75% of the group area. These giant, though unresolved, spot structures have already been observed on highly active (rapidly rotating) sun-like stars and thus, our result puts constrains on the properties of such starspots. We stress, however, that this is a phenomenological projection of the solar case to other stars, and other factors might play a role in determining the size of individual spots on such stars.

Thus far we have shown that, at maximum group growth, approximately 60% of the total group area is occupied by a single big spot. We now analyse the spot distribution within the remaining 40% of the group area. For this, we consider the relationship between $\rm{N}$ and $\rm{A_{group}}$. We bin the group areas $\rm{A_{group}}$ in log scale and calculate the average values of $\rm{N}$ in each bin. The result is plotted in Figure 4. Overall, the number of individual umbrae within groups, $\rm{N}$, increases with $\rm{A_{group}}$. The rate of increase slows down with the increasing group area and the overall profile can be described with a power law function, $\rm{N}$$\propto$$\rm{A^{0.58}_{group}}$ (dashed line in Fig. 4). Since $\rm{N}$ is the number of umbrae within a given group while some spots have multiple penumbrae, the dependence of the number of spots within a group on its area would be somewhat flatter than $\rm{N}(\rm{A_{group}})$. Our result implies that, with the increasing group area, new flux emerges in the form of new spots (such that the number of spots in a group increases), while the primary biggest spot within the group keeps growing, too (which keeps the $\rm{R}$ constant).

It is well known that many sunspot parameters, such as sunspot areas, number, etc, exhibit periodic changes with the solar cycle (Hathaway, 2015). Therefore, we also analyze the behaviour of $\rm{R}$ with time. In particular, we look for changes in $\rm{R}$ from cycle to cycle, as well as during the activity maxima and minima. We first analyze the distribution of $\rm{R}$ during solar activity maxima and minima. For better statistics, we merged the data from all cycles and from all the observatories. We define cycle maxima and minima as 12-month periods centered at the times of the maxima and minima in the 13-months running mean of the sunspot area (in the merged series), respectively. Now, there are significantly more bigger spots and groups during cycle maxima than during minima, while $\rm{R}$ depends on the group area (Figure 3b). Hence, to eliminate this size dependency, we first fit a lognormal function ($\rm{R}\sim\exp{-\big{[}\rm{(\log(A_{group}}))^{2}}/{2\sigma^{2}}\big{]}$) to the data in Fig. 3b and then divide each measured $\rm{R}$ by the value this fit returns for the corresponding $\rm{A_{group}}$ to obtain the normalized $\rm{R}$ value. Figure 5a shows these normalized $\rm{R}$ distributions for cycle maxima and minima in blue and red, respectively.

These distributions are nearly identical in shape and their mean, median and mode values (listed in the legend) are very similar. We further test the hypothesis that the two samples come from the same distribution by applying the two-sided Kolmogorov-Smirnov (K–S) test which checks for the equivalence of two datasets by comparing their empirical cumulative distribution functions (ECDFs) (Berger & Zhou, 2014). The test statistics, $\rm{D}$, is a measure of the difference between the ECDFs. The null hypothesis, that both samples come from the same underlying population, is rejected if $\rm{D}>D_{crit}$. On our two $\rm{R}$ distributions, $\rm{D}$=0.06 and D_crit=0.12, with p value being 0.44. Thus, in our case the null hypothesis is true, and we therefore conclude that no substantial variation in $\rm{R}$ is observed between the minimum and maximum of a sunspot cycle.

Finally, in Figure 5b, we show the cycle-to-cycle variation of the normalized $\rm{R}$. Filled circles represent the median values (second quartile) of $\rm{R}$ over each cycle, whereas the third and first quartiles of the corresponding $\rm{R}$ distributions are shown as the upper and lower limits of the error bars. We see no significant variation in $\rm{R}$ from cycle to cycle.