Evolutionary Relationship between Sunspot Groups and Soft X-Ray Flares over Solar Cycles 21–25

Studying the interaction between solar flares and sunspot groups (SGs) is crucial for understanding and predicting solar activity. We examined the distribution, correlation, and flaring rates in the northern and southern hemispheres to reveal the relationship between different classes of soft X-ray (SXR) flares and different magnetic classifications of SGs. We discovered a significant north–south asymmetry in SXR flares and SG distribution over Solar Cycles (SC) 21–25. In the rising phase of SC24, the northern hemisphere’s activity is significantly excessive. In the declining phase of SC24, the southern hemisphere’s activity becomes significantly excessive. The total numbers of various SXR flares and SGs vary between the northern and southern hemispheres over the solar cycle. B-class flares are negatively correlated with all SGs at maximum but positively correlated at minimum. C-class flares correlate best with α and β SGs. M-class flares correlate best with β γ δ and β SGs. X-class flares correlate highest with β γ δ SGs. The flaring rate of each flare class is lowest for α SGs and highest for β γ δ SGs. The flaring rates are higher in the southern hemisphere than in the northern hemisphere. Our results demonstrate that solar flares originate from different sources of solar active regions; the high-energy flares tend to be caused by more complex magnetic fields.


Introduction
Solar flares are large-scale and extremely intense energy release processes occurring in the solar atmosphere, which are directly related to the safety of the space environment of the Sun and Earth (Fletcher et al. 2011;Emslie et al. 2012;Benz 2016).Many studies have shown that solar flares are closely related to sunspots.Therefore, studying the evolutionary relationship between solar flares and sunspots is crucial.
When solar flares erupt, there is a sudden increase in electromagnetic radiation at almost all wavelengths from radio to γ-rays (Fletcher et al. 2011;Shibata & Magara 2011).The behavior of flares with different energies is different, with many more small-energy flares than large-energy flares (Gao & Xu 2016;Gao & Zhong 2016;Xiong et al. 2021).Solar flares are divided into five classes based on the peak intensity of soft X-ray (SXR) flux between 0.1 and 0.8 nm observed by the Geostationary Orbiting Environmental Satellites (GOES) satellites.The flare classes from weak to strong are A (<10 −7 W m −2 ), B (10 −7 to 10 −6 W m −2 ), C (10 −6 to 10 −5 W m −2 ), M (10 −5 to 10 −4 W m −2 ), and X (10 −4 W m −2 ).The first four levels are divided linearly into subclasses from 1 to 9 based on intensity, while the strongest X-class has no upper limit.
Sunspots are the main indicators of solar activity intensity, usually appearing in sunspot groups (SGs).The temporal and spatial behaviors of various types of SGs are different (Kilcik et al. 2011(Kilcik et al. , 2014;;Nagovitsyn et al. 2012;Gao 2019).There are three main classification methods for SGs: the Zurich classification (Cortie 1901), the McIntosh classification (McIntosh 1990), and the Mount Wilson classification (Hale et al. 1919;Künzel 1960).In this study, we will classify SGs using the Mount Wilson Classification.The Mount Wilson Classification classifies SGs into α, β, γ, δ, βγ, βδ, γδ, and βγδ on the morphology and magnetic polarity of the SGs.α are unipolar SGs; β are bipolar SGs with a regular distribution of positive and negative polarities; γ are SGs with a very irregular distribution of positive and negative polarities; βγ are complex bipolar SGs with no obvious boundary between opposite polarities; δ are bipolar SGs where there is an umbra of opposite polarities sharing the same penumbra; βδ, γδ, and βγδ all contain the structure of δ SGs.In this scheme, the most complex SGs are βγδ SGs.
Solar flares are closely correlated with SG evolution.Relevant studies show that the intensity of solar flare eruptions is related to the size of SGs, the number of sunspots in SGs, and the complexity of the magnetic field in SGs.The more complex the structures and magnetic field polarities of SGs are, the more likely large solar flares will erupt (Greatrix 1963;Zirin & Liggett 1987;Kilcik et al. 2011;Norquist 2011;McCloskey et al. 2016;Eren & Kilcik 2017).Sammis et al. (2000) found that βγδ SGs with an area larger than 1000 millisolar hemispheres (MSH) have a 40% probability of producing X1 or larger flares.Furthermore, X4 or larger flares occur within the βγδ SGs exceeding more than 1000 MSH in area.Ternullo et al. (2006) found that SGs with more sunspots (over 15), larger area (greater than 1000 MSH), and more complex magnetic field structures (e.g., β, βγ, and βγδ) are more prone to producing M-class and X-class flares.Lee et al. (2012) found that as the area of SGs increases, the occurrence rate and probability of solar flares increase significantly, especially for large solar flares.Eren & Kilcik (2017) investigated the relationship between the number of SXR flares and SGs, finding that the primary source of SXR flares is complex and large SGs.Oloketuyi et al. (2023) found that from 1996 July to 2018 December 100 out of 144 X-class flares are associated with βγδ SGs.Out of 1703 M-class flares, 696, 468, and 457 are associated with βγδ, βγ, and β SGs, respectively.This suggests that X-class flares mainly originate from βγδ SGs and M-class flares primarily originate from βγδ, βγ, and β SGs.
The evolutionary relationship between different classes of solar flares and magnetic classifications of SGs is different.Many researches have shown that X-class flares mainly originate from SGs containing δ structures (Mayfield & Lawrence 1985;Shi & Wang 1994;Yang et al. 2017;Gao 2019).Guo et al. (2014)  The above studies reveal the intrinsic mechanism connecting solar flares of different classes and SGs of different magnetic types, providing essential clues for predicting solar flare activity corresponding to the evolution of various SG types.
It is well known that solar activity exhibits significant asymmetry between the northern and southern hemispheres (Roy 1977;Norton et al. 2014;Deng et al. 2016;Schüssler & Cameron 2018;Zhang et al. 2022).To further reveal the differences in the evolution of SXR flares and SGs, we analyzed their distribution, correlation, and flaring rates in the two hemispheres from 1982 January to 2022 December.In the rest of this paper, we introduce data sources and methods in Section 2. Section 3 shows the evolutionary relationship between the different classes of flares and the different magnetic classifications of SGs in the northern and southern hemispheres.Section 4 presents conclusions and discussions.

Data
In this work, we used the data of SXR The SXR data are observed by GOES satellites and provided by the National Oceanic and Atmospheric Administration (NOAA),5,6 containing information such as start time, end time, NOAA active region, latitude, longitude, and X-ray class.Since the number of A-class flares is very small, this study analyzed B-class, C-class, M-class, and X-class flares.Figure 1 shows the time series of the monthly number of B-class, C-class, M-class, and X-class flares for the northern and southern hemispheres.
The SG data are provided by the Royal Greenwich Observatory (RGO) and the US Air Force (USAF)/NOAA,7 containing information such as occurrence time, magnetic group type, Zurich/McIntosh group type, area, number of spots, latitude, and longitude.Since the number of γ, δ, and γδ SGs is very small, this study analyzed α, β, βγ, βδ, and βγδ SGs. Figure 1 shows the time series of the monthly number of α, β, βγ, βδ, and βγδ SGs for the northern and southern hemispheres.

The Normalized North-South Asymmetry Index
The normalized north-south (N-S) asymmetry index can effectively describe the relative activity intensity of solar activity indicators between the northern and southern hemispheres (Newton & Milsom 1955;Swinson et al. 1986;Carbonell et al. 1993;Li et al. 2002;Knaack et al. 2004;Bankoti et al. 2011).The index is expressed as follows: where N and S represent the number of SXR flares (or SGs) in the northern and southern hemispheres, respectively.When A is less than 0, it indicates that southern hemisphere activity dominates.When A is greater than 0, it indicates that northern hemisphere activity dominates.

The North-South Asymmetry of the Random Distribution
To examine the statistical significance of A, we used the N-S asymmetry of the random distribution (Letfus & Ruzickova-Topolova 1980;Joshi 1995;Joshi & Chandra 2019).Its calculation formula is as follows: , the N-S asymmetry can be divided into three categories: insignificant, significant, and highly significant.

The Pearson Correlation Coefficient
In statistics, one way to quantify the relationship between two time series is to use the Pearson correlation coefficient.The formula of the Pearson correlation coefficient between two time series x and y is as follows (Sedgwick 2012): where x is the mean of data x, ȳ is the mean of data y, and n is the data length.

The Probable Error
The probable error (PE) of the correlation coefficient provides an estimate of the variability, which helps determine the statistical significance and reliability of the correlation value.The calculation formula for the PE is as follows (Student 1908;Odell 1926): When the value of r is equal to or greater than 6 times the PE, it indicates that the correlation analysis is statistically significant and reliable.

Distribution of Hemispheres with the Solar Cycle
To study the distribution of SXR flares and SGs in the northern and southern hemispheres, we analyzed their N-S asymmetry (as shown in Tables 1 and 2), cumulative distribution (as shown in Figures 2 and 3), and evolution of total number (as shown in Figure 4).
In Tables 1 and 2, we present the number of B-class, C-class, M-class, and X-class flares and the number of α, β, βδ, βγ, and βγδ SGs in the northern and southern hemispheres in different solar cycles and calculate the N-S asymmetry index (A) of the data.In most cases, A is less than 0, indicating that the southern hemisphere dominates solar activity.In order to evaluate the statistical significance of A, we calculate the N-S asymmetry of the random distribution (!ASY) of SXR flares and SGs.The results show that the N-S asymmetry of SXR flares and SGs in different cycles is significant or highly significant (except for a few marked with "-").This suggests that the difference in the number of SXR flares and SGs between the northern and southern hemispheres is real.
Compared to other solar cycles, SC24 exhibits different behaviors.From Tables 1 and 2, we can find that in other cycles the southern hemisphere activity is basically dominant.However, in SC24, northern hemisphere activity dominates in B-class flares, α SGs, and β SGs.As shown in Figure 2, during the rising phase of SC24, the northern hemisphere's activity is significantly excessive across different classes of solar flares.The southern hemisphere's activity becomes significantly excessive during the declining phase of SC24, except for B-class flares, which are consistently excessive in the northern hemisphere.In Figure 3, different types of SGs in SC24 also show the same behavior.However, SXR flares and SGs do not show significant excess activity in the northern hemisphere during the rising phases of SC22, SC23, and SC25.
Figure 4 shows that the total numbers of various SXR flares and SGs in the two hemispheres exhibit different evolutionary trends over the solar cycle.From SC22 to SC24, for SXR flares, the total number of B-class flares in the northern and southern hemispheres first increases and then decreases, while C-class, M-class, and X-class flares decrease continuously.For SGs, the total numbers of α and β SGs in the northern and southern hemispheres decrease continuously; the total number of βδ SGs in the northern hemisphere decreases constantly, while the total number in the southern hemisphere first decreases and then increases; and the total numbers of βγ and βγδ SGs in the northern and southern hemispheres first increase and then decrease.

The Correlation
To study the relationship between SXR flares and SGs in the northern and southern hemispheres, we calculated the correlation coefficients and PE between them from 1982 January to 2022 December (as shown in Table 3).
Table 3 shows that the correlation between various flares and SGs is different.flares are negatively correlated with SGs, and the correlation is very low.C-class flares are most correlated with β (0.75 and 0.76, respectively) and α (0.66 and 0.69, respectively) SGs and least correlated with βδ (0.38 and 0.44, respectively) SGs, and the correlation coefficient in the southern hemisphere is slightly higher than that in the northern hemisphere.M-class flares are most correlated with βγδ SGs (0.62 and 0.58, respectively), followed by β (0.52 and 0.60, respectively) and α (0.49 and 0.54, respectively) SGs.X-class flares are most correlated with βγδ SGs (0.51 and 0.47, respectively) but less correlated with other SGs.
In Figure 5, the monthly data of B-class flares and SGs are all normalized, and the thick red and blue lines represent the 13 month smoothed data.As shown in Figure 5, B-class flares are obviously negatively correlated with various SGs during the maximum of each cycle.However, B-class flares are clearly positively correlated with SGs during the minimum of each cycle, e.g., during the declining phase of SC22 and the ascending phase of SC23, the declining phase of SC23, and the ascending phase of SC24.This behavior is most significant in α, β, and βγ SGs, and more significant in the northern hemisphere than in the southern hemisphere.

The Flaring Rate
To study the associations between SXR flares and SGs in the northern and southern hemispheres, we calculated the number of SXR flares in active regions of different magnetic types (as shown in Tables 4 and 5) and the flaring rates (as shown in Table 6).
Table 4 shows the number of different classes of SXR flares associated with SGs of different magnetic types in the northern hemisphere.In the northern hemisphere, 12,309 SXR flares associated with active regions are observed, of which C-class flares are the most frequently observed (8559) and X-class flares are the least frequently observed (120).Among the 12,309 flares, the occurrence frequency is the highest for those associated with β SGs (6549) and the lowest for those associated with βδ SGs (453).For B-class and C-class flares, the occurrence frequency is the highest for those associated with β SGs, followed by βγ SGs.Among the 2224 B-class flares, 1449 (65.15%) are associated with β SGs, and 329 (14.79%) are associated with βγ SGs.Among the 8559 C-class flares, 4510 (52.69%) are associated with β SGs, and 2252 (21.68%) are associated with βγ SGs.For M-class and X-class flares, the occurrence frequency is the highest for those associated with β SGs and βγδ SGs and the lowest for those associated with α SGs.Among the 1406 M-class flares, 559 (39.76%) are associated with β SGs, 446 (31.72%) are associated with βγδ SGs, and only 71 (5.05%) are associated with α SGs.Among the 120 X-class flares, 49 (40.83%) are associated with βγδ SGs, 31 (25.83%) are associated with β SGs, and only 1 (0.83%) is associated with α SGs.Similar results are also found in the southern hemisphere in Table 5.
Table 6 lists the flaring rates of the northern and southern hemispheres.These values are obtained by dividing the count of each SXR flare class by the corresponding number of active regions (Oloketuyi et al. 2023).Table 6 shows that the flaring rates of various SXR flares increase continuously as the magnetic field complexity of SGs increases.For example, for B-class flares, the flaring rate increases from 0.022 in α SGs to 0.24 in βγδ SGs in the northern hemisphere and from 0.032 in α SGs to 0.319 in βγδ SGs in the southern hemisphere.For the northern and southern hemispheres, the flaring rates are highest Note.Those marked with "-" in the "Dominant Hemisphere" column are insignificant, and the rest are significant or highly significant.
for C-class flares associated with βγδ SGs (1.976 and respectively) and lowest for X-class flares associated with α SGs (0.00009 and 0.0006, respectively).For a given type of SGs, the flaring rates are highest for C-class flares, indicating that C-class flares are the most common and frequent.
Compared with the northern hemisphere, the flaring rates in the southern hemisphere are higher (except for C-class flares associated with βγ SGs, M-class flares associated with α and βγδ SGs, and X-class flares associated with βδ SGs).

Conclusions and Discussions
The interaction between different classes of SXR flares and different magnetic classifications of SGs is important for forecasting solar flare activity.North-south asymmetry is a fundamental feature of the solar cycle.We analyzed their distribution, correlation, and flaring rates in the northern and southern hemispheres from SC21 to SC25.The results are as follows: 1. Different types of SXR flares and SGs exhibit significant N-S asymmetry.The distribution of SXR flares and SGs varied with the solar cycle.2. SC24 exhibits distinguishable behaviors.The northern hemisphere is significantly excessive during the rising phase of SC24, while the southern hemisphere is excessive during the declining phase.highest for βγδ SGs.The flaring rates are higher in the southern hemisphere than in the northern hemisphere.
The N-S asymmetry of the Sun is one of the most important characteristics of solar activity (Vizoso & Ballester 1990;  Carbonell et al. 1993;Oliver & Ballester 1994;Bankoti et al. 2011).Researchers found that the N-S asymmetry of the Sun varies with the solar cycle, and some periods are more dominant in one hemisphere (Waldmeier 1971;Goel & Choudhuri 2009;Li et al. 2009;Bankoti et al. 2010       dominated by the northern hemisphere in SC24, and βδ SGs dominated by the northern hemisphere in SC23.In the ascending phase of SC25, M-class flares, βδ SGs, and βγδ SGs are dominated by the northern hemisphere, and the rest are dominated by the southern hemisphere.The variation differs among various solar flares, SGs, and sunspots (Lefèvre & Clette 2011;Kilcik et al. 2014;Gao & Zhong 2016;Kilcik et al. 2020).Lefèvre & Clette (2011) and Kilcik et al. (2014) showed that the number of the smallest sunspots decreased by more than half from SC22 to SC23, while the number of larger sunspots showed no significant change.Gao (2019) found that the ratios of simple, mediumsized, large, and decaying SGs from SC22 to SC23 are 1.38, 1.06, 0.80, and 0.74, respectively.Gao & Zhong (2016) found that, compared with SC22, the peak value of C-class flares in SC23 only shows a slight decrease, while the frequency of M-class and X-class flares decreases by more than half.Kilcik et al. (2020) found that the number of C-class flares increases by about 16% from SC23 to SC24, while the number of M-class and X-class flares decreases by about 32%.We find that the variation regularities differ between categories.Furthermore, the same categories exhibit hemispheric differences in total number variations.
Compared with SC22 and SC23, the number of various SXR flares and SGs in SC24 decreased, indicating that SC24 is a weak cycle.In addition, the cumulative numbers of SXR flares and SGs in SC24 behave differently from those in SC22 and SC23.According to previous studies, SC24 has a weaker magnetic field (Salabert et al. 2015;Tripathy et al. 2015), has a lower solar activity (Gao et al. 2014;Prasad et al. 2022), and is at the lowest point of the Centennial Gleissberg Cycle (Feynman & Ruzmaikin 2011;Li et al. 2011).Chowdhury et al. (2013) and El-Borie et al. (2019) found that during the rising phase of SC24, the number and area of sunspots are significantly excessive in the northern hemisphere.Joshi & Chandra (2019) pointed out that the cumulative behavior of SC24 is different from that of SC22 and SC23 but similar to that of SC21 (Temmer et al. 2001;Joshi et al. 2015).
The eruption of solar flares is closely related to the evolution of SGs, and the correlation between different classes of flares and different magnetic classifications of SGs is different.We find that B-class flares are significantly negatively correlated with SGs at the cycle maximum but positively correlated at the cycle minimum.C-class flares have the highest correlation coefficients with β and α SGs, M-class flares have the highest correlation coefficients with βγδ and β SGs, and X-class flares have the highest correlation coefficients with βγδ SGs.Oloketuyi et al. (2019Oloketuyi et al. ( , 2023) also found that during SC23 and SC24 B-class flares are negatively correlated with SGs, while other flares are positively correlated and synchronized.Oloketuyi et al. (2019) pointed out that B-class flares are in phase with SGs from the descending phase of SC23 to the rising phase of SC24.When the number of SGs in the two solar cycles rises or falls to about 100, B-class flares begin to deviate from the SGs.Our results support that solar flares originate from different sources of solar active regions, with high-energy flares (such as M-class and X-class flares) tending to originate from SGs with stronger magnetic fields and more complex structures (such as βγδ SGs).
The associations between various flares and SGs are different.Gao (2019) investigated the associations of X-class flares with various SGs in SC22 and SC23, finding that the fewest are associated with α SGs (1.5% and 3.7%, respectively) and the most are associated with βγδ SGs (35.4% and 44.9%, respectively).In our work, in both hemispheres, M-class flares are most associated with β (39.76% and 42.62%, respectively) and βγδ SGs (31.72% and 28.39%, respectively) but least associated with α SGs (5.05% and 3.29%, respectively).X-class flares are most associated with βγδ SGs (40.83% and 48.89%, respectively), but least associated with α SGs (0.83% and 4.44%, respectively).Our results further indicate that in both hemispheres the higherenergy X-class flares mainly originate from βγδ SGs with the most complex magnetic fields.
The flaring rate of each flare class is lowest for α SGs and highest for βγδ SGs.Compared to the northern hemisphere, the southern hemisphere has higher flaring rates.This means that in the southern hemisphere, not only are the numbers of SGs and flares higher, but also the flaring rates are higher.
It is worth noting that the frequency distribution of solar flares generally follows a decreasing power law with increasing flare size (Hudson 1991;Ryan et al. 2012).However, in Figure 1 and Table 1, the number of B-class flares is less than that of C-class flares.One possible reason is that when the background radiation is high, GOES satellites have difficulty detecting B-class or smaller flare events, especially during solar maximum periods (Wagner 1988;Bornmann 1990;Aschwanden 1994;Aschwanden & Freeland 2012;Ryan et al. 2012).Therefore, more sensitive detectors or correction methods are needed to accurately capture small flare events.Xiaojuan Zhang (张小娟) https://orcid.org/0000-0002-8412-6289 statistically studied the relationship between solar flares and SGs of different magnetic classifications from 1983 April to 2011 December.They found that 83.34% of X-class flares, 62.35% of M-class flares, 43.18% of C-class flares, and 25.47% of B-class flares occur in βγδ SGs.Oloketuyi et al. (2019) investigated the distribution of different classes of solar flares and the number of SGs from 1996 July to 2016 December.They found that the number of B-class flares is negatively correlated with the number of SGs, while the numbers of C-class, M-class, and X-class flares are positively correlated with the number of SGs.Oloketuyi et al. (2023) pointed out that C-class flares are consistent with the temporal evolution behavior of β SGs, M-class and X-class flares are consistent with βγ and βγδ SGs, while B-class flares are not consistent with any SGs.

Figure 1 .
Figure 1.The monthly number of SXR flares and SGs from 1982 January to 2022 December.The black line represents the northern hemisphere, and the red line represents the southern hemisphere.
3. B-class flares are negatively correlated with SGs at maximum but positively correlated at minimum.C-class flares are best correlated with α and β SGs.M-class flares are best correlated with βγδ and β SGs.X-class flares are best correlated with βγδ SGs. 4. The flaring rate of each flare class is lowest for α SGs and

Figure 3 .
Figure 3.The cumulative distribution of α, β, βδ, βγ, and βγδ SGs in the northern and southern hemispheres under different solar cycles.

Figure 4 .
Figure 4.The evolution of the total number between various SXR flares and SGs in the northern and southern hemispheres under different cycles, where the left column shows SXR flares and the right column shows SGs.

Figure 5 .
Figure 5.The monthly time series of B-class flares and various SGs in the northern (left) and southern (right) hemispheres.

Table 1
Statistics on the Number of B-class, C-class, M-class, and X-class Flares in the Northern and Southern Hemispheres under Different Cycles

Table 2
Statistics on the Number of α, β, βδ, βγ, and βγδ SGs in the Northern and Southern Hemispheres under Different Cycles Figure 2. The cumulative distribution of B-class, C-class, M-class, and X-class flares in the northern and southern hemispheres under different solar cycles.

Table 3
The Correlation Coefficients between SXR Flares and SGs in the Northern and Southern Hemispheres The values in parentheses are the PE.

Table 4
The Number of SXR Flares Associated with SGs in the Northern Hemisphere

Table 5
The Number of SXR Flares Associated with SGs in the Southern Hemisphere