1 Introduction

Sympathetic events which mean that a pair of solar events appear almost simultaneously in different locations on the solar disk have been detected and studied previously. Firstly, it is found in solar flare events which are called sympathetic flares (i.e., flare-flare interactions) (Mullan 1976; Bumba and Klvana 1993; Changxi et al. 2000; Moon et al. 2002; Titov et al. 2010). Sympathetic events are found in other solar events, such as sympathetic filaments (i.e., filament-filament interaction) (Wang et al. 2016; Jiang et al. 2011; Li et al. 2017), coronal mass ejections (CMEs) (CME-CME homologous) (Moon et al. 2003; Cheng et al. 2005), sympathetic CME-Pseudostreamers (Lynch and Edmondson 2013), sympathetic coronal bright points (Zhang and Ji 2013), sympathetic filament-coronal dimmings connection (Jiang et al. 2011), and sympathetic active regions (Pearce and Harrison 1990). Sequential sympathetic active regions led to the flare occurrence (Pearce and Harrison 1990; Wang et al. 2001; Pucci et al. 2012).

Sympathetic flares may be associated with other solar events such as CME (Jiang et al. 2009; Wang et al. 2001; Joshi et al. 2016), dimming (Wang et al. 2001), or filaments (Yang et al. 2012; Shen et al. 2012; Joshi et al. 2016).

The sympathetic flares term was studied by Fritzova-Svestkova et al. (1976), which means that a pair of solar flares appear almost simultaneously in different locations on the solar disk. They found sympathetic flares have a short time interval equal to 20 minutes. Abramov-Maximov et al. (2007) presented three sympathetic pairs for flares with an interval time delay of 10–12 min. at wavelength 1.76 cm. Biesecker and Thompson (2000) reported that there was no increase in the number of flares due to the EIT wave due to a comparison of the rate of flaring in the interval before an EIT wave to the rate of flaring while the wave traverses the solar disk.

The distance between paired flares was studied by Fritzova-Svestkova et al. (1976), who found pairs of active regions are closer than 30° to each other between both flares. While Abramov-Maximov et al. (2007) noted large distances for one day, more than 90° in latitude.

The flares of sympathy had three ribbons. One ribbon happened in a single polarity area, whereas two ribbons emerged in an active region (Changxi et al. 2000). Sympathetic flares were caused by the interaction of a sheared lower loop in the active region and a higher loop connecting the two regions.

Toeroek et al. (2011) suggest a 3-dimensional magnetohydrodynamic simulation for sympathetic eruptions, along with two possible magnetic trigger mechanisms. They consider a configuration with two coronal flux ropes within a pseudo-streamer and one rope next to it. The eruption of the flux rope next to the streamer starts a chain reaction of eruptions. The rope’s expansion causes two successive reconnection events, each of which causes the eruption of a flux rope by removing a sufficient amount of overlying flux. Joshi et al. (2016) observed chain reconnections in sympathetic eruptions.

Sympathetic flares were studied from observations from Burst and Transient Source Experiment (BATSE), Geostationary Operational Environmental Satellite (GOES), the Solar and heliospheric observatory SOHO/Extreme, Ultraviolet Imaging Telescope (EIT) (Biesecker and Thompson 2000), Solar Dynamics Observatory and STEREO (Titov et al. 2012), and the Solar Magnetic Field Telescope (SMFT) at the Huairou Solar Observing Station of Beijing Astronomical Observatory (Changxi et al. 2000).

In this study, we will study all X-ray sympathetic flares detected by GOES during the period 1975–2017, to identify the spatial and temporal relationship between both pairs of solar flares. We will develop methods for studying the new characteristics of sympathetic flares that have not been studied before.

2 Approach and algorithm

2.1 Matching criteria

Sympathetic flares are a pair of successive flares that occur nearly simultaneously in different active regions that have a certain physical connection, at least with statistical significance.

The solar flare has three times records, start- (\(T_{s}\)), max- or peak (\(T_{m}\)), and end- (\(T_{e}\)) times. These times are available on any data recorded by ground or space telescopes such as GOES, which is used in our study.

Previous studies reported that the sympathetic flares may have an interval time equal to 20 min., while we restricted this interval. The selection of sympathetic flares can be considered under the following condition:

$$ T_{e} (i-1)\leq T_{s} (i)\geq T_{s}(i-1) $$

Or we can write this condition as

$$ T_{e} (i)\leq T_{s} (i-1)\geq T_{e} (i) $$
(1)

Where \(T_{s}\) and \(T_{{e}}\) are the start and end times of the flare. But \(T_{s}\) is the start time. While \(I\) denote the number of records or flare index. It is a secondary flare (second flare). But \(i-1\) denotes the primary flare (first flare).

2.2 Sympathetic flare phasing

The secondary flares may occur during the rising phase (during start and peak times) of the primary flare, or during the decay phase (during peak and end times). By following the condition, we can find these phasing categories.

Rising Phase:

$$ T_{s} (i-1)\leq T_{s} (i)\geq T_{m}(i-1) $$

Recovering phase:

$$ T_{m} (i-1)\leq T_{s} (i)\geq T_{e} (i-1) $$
(2)

2.3 Estimation of distance

By assuming that solar flares occur on the solar surface, which is considered a sphere. The angular distance of their great circle determines the distance between the sympathetic flares, which are pointed by latitude (\(\varphi \)) and longitude (\(\lambda \)) (in spherical coordinates). The spherical triangle is defined by three points: a pair of flares and a solar pole, as shown in Fig. 1. We can estimate the angular distance (\(D\)) between the primary flare \(F_{1}\) and the secondary flare \(F_{2}\) using the cosine-cosine formula.

$$\begin{aligned} & \cos(D) = \cos(90-\varphi _{1}) \cos(90-\varphi _{2}) \\ &\quad +\sin(90-\varphi _{1}) \sin(90-\varphi _{2}) \cos(\lambda _{2} - \lambda _{1} ) \\ & D = \cos^{-1}\bigl[\sin(\varphi _{1}) \sin(\varphi _{2}) + \cos(\varphi _{1}) \cos(\varphi _{2}) \cos(\Delta \lambda )\bigr] \end{aligned}$$
(3)

Where

$$ \Delta \lambda =\lambda _{2} - \lambda _{1} $$
(4)

Where \(\varphi _{1}\) and \(\varphi _{2}\) are the latitudes of the primary and secondary flares, respectively. \(\lambda _{1}\) and \(\lambda _{1}\) are the longitudes of the primary and secondary flares, respectively.

Fig. 1
figure 1

The spherical triangle of the sympathetic flares (primary flare \(F_{1}\), secondary flare \(F_{2}\), and solar pole \(P\))

Full size image

Figure 1 shows an illustration of the solar surface as a sphere. \(C\) is the center of the Sun. \(P\) is the solar north pole. The spherical triangle is denoted by \(P\) and \(F_{1}\) and \(F_{2}\) of the sympathetic flares.

2.4 Polar inclination

We need to estimate the inclination of the great circle line on the solar equator. Because it helps us to understand the association of this sympathetic solar flare alignment line (denoted by a great circle) with the main polar magnetic field lines.

Figure 1 shows the spherical triangle of three points located in the photosphere. \(\varphi _{1}\) and \(\lambda _{1}\) are the solar latitude and longitude of the primary flare. \(\varphi _{2}\) and \(\lambda _{2}\) are the solar latitude and longitude of the secondary flare. \(P\) is the solar pole.

By applying a cotangent four-part formula to the spherical triangle \(F_{1}\), \(F_{2,}\) and \(P\). We can write

$$\begin{aligned} &\cos (90-\varphi _{1}) \times \cos \Delta \lambda \\ & = \cot (90-\varphi _{2}) \times \sin (90-\varphi _{1}) - \cot(Q) \times \sin(\Delta \lambda ) \\ &\sin(\varphi _{1}) \times \cos \Delta \lambda \\ & = \tan(\varphi _{2}) \times \cos(\varphi _{1}) - \cot(Q) \times \sin(\Delta \lambda ) \\ &\cot(Q) \times \sin(\Delta \lambda ) \\ & = \tan(\varphi _{2}) \times \cos(\varphi _{1}) - \sin(\varphi _{1}) \times \cos(\Delta \lambda ) \\ &\cot(Q) = \bigl[\cos(\varphi _{1}) \times \tan(\varphi _{2}) \\ &\quad - \sin(\varphi _{1})\mbox{second} \times \cos(\Delta \lambda )\bigr] / \sin(\Delta \lambda ) \\ &Q = \tan^{-1}\bigl[\sin(\Delta \lambda ) \div \bigl[\cos(\varphi _{1}) \times \tan(\varphi _{2}) \\ &\quad -\sin(\varphi _{1}) \times \cos(\Delta \lambda )\bigr]\bigr] \end{aligned}$$
(5)

Where \(Q\) is the clockwise direction from the north pole.

2.5 Timing ratio

The appearance of sympathetic flare events means that there is a physical connection between both flares (Changxi et al. 2000). So, it makes sense to study when the second solar flare appears during the life (duration) of the primary solar flare.

The timing ratio \(R\) is the ratio obtained by the following formula:

$$ R = T / D $$
(6)

Where \(T \) is the rise time between the start times of the primary and secondary solar flares. It is determined by the interval between the start and peak times of the solar flare. While \(D\) is the duration of the primary flare. It is determined by the time between the solar flares’ start and end times.

3 Data source

We used the soft X-ray solar flares obtained by GOES satellites during the period 1975–2017 (solar cycles 21 to ∼24). We have 77604 recorded solar flare events during this period. It is the URL:

https://www.ngdc.noaa.gov/stp/space-weather/solar-data/solar-features/solar-flares/x-rays/goes/xrs/

The soft X-rays (SXR) brightening is routinely measured with NOAA’s Geostationary Operational Environmental Satellite (GOES; Donnelly et al. 1977) network, with the X-ray Sensor (XRS) onboard. XRS measures spatially unresolved light curves in two wavelength bands: 1–8°A. and 0.5–4°A. The flux levels in GOES XRS are used to report flares and classify their size according to their brightness in the 1–8°A band.

We compared the sympathetic flare distribution with the yearly mean total sunspot number obtained from SIDC (Solar Influences Data Analysis Center) [http://www.sidc.be/silso/DATA/SN_y_tot_V2.0.txt].

4 Results and discussions

By using our criteria mentioned above for matching between sympathetic flares, we found 2204 sympathetic solar flares during the solar period 1975–2017 (i.e., solar cycles 21, 22, 2,3, and 24) out of all 77604 solar flare events. We examined the characteristics of these selected events in the following subsections.

4.1 Occurrence and indexing

As shown in Table 1 and Fig. 3, the majority of sympathetic flares (1728 events, 78.6%) belong to the same class. We indexed sympathetic flares by putting an association with two classifications of both flares, such as “BC”. The first character, such as “B-”, is a classification of the primary solar flare. The second character “-C” is a classification of the secondary flare.

Table 1 The number of sympathetic flares that occurred between 1975 – 2017
Full size table

As we found in Table 1, we can classify the sympathetic flare into two classes:

  1. 1)

    Twins: such as BB, CC, MM, and XX. It means that the second flare is a homogeneous copy of the primary flare. This type represents most of them.

  2. 2)

    Non-twins: it is associated with a heterogeneous pair of solar flares, such as XC.

We found that sympathetic flares have a nearby homogeneity class. I.e., B class will not be associated with X classes yet. The primary flare is followed by another flare that has the same class dominantly (such as BB, CC, MM, XX), or it may be associated with nearby classes (such as CM, BC). There is no far power pair (such as BX or XB class events). So, we can say that the second solar flare is just a homogeneous copy of the primary flare.

Sympathetic flares are occurring in the decay phase of the solar cycle dominantly, as shown in Fig. 2. This means that the energy can be stored in an active region for a longer time.

Fig. 2
figure 2

The annual occurrence of sympathetic flares. Left Y-axis represents the number of sympathetic flares, while right Y-axis represents the sunspot number SSN

Full size image
Fig. 3
figure 3

Histogram of the classification of sympathetic flares

Full size image

During strong solar cycles (21 and 22), we have an enormous number of sympathetic flares compared to the next weaker solar cycles (23 and 24), which have fewer solar flare events. This indicates that the number of sympathetic flares increases with strong solar cycles while it decreases with weaker solar cycles. We show that the trend of the occurrence of sympathetic flares depends on solar cycle activity.

We expect the reason for this occurrence number is that solar flares are related to solar magnetic field strength and solar polarity. The solar flares are correlated to the sunspot number.

The number of sympathetic solar flares increases with the increase in solar activity and decreases as solar activity decreases.

Solar polarity was positive in solar cycles 21 and 23, but it reversed in solar cycles 22 and 24 to be negative (Youssef et al. 2012). In our study, we found only 2 reversible magnetic polarities during the studied period. The correlation between solar flares and magnetic polarity may lead to the occurrence of sympathetic flares in the following solar cycles 25 and 26.

From this result, we are expecting that the future occurrences of sympathetic flares during the current solar cycle 25 will be a few.

During solar cycle 23, there are few sympathetic flares. However, during the fall of its solar cycle in 2005, we observed a large number of occurrences.

4.2 The sunspot association

Most solar flares occur in the active region associated with the sunspots. We counted the sympathetic flares that occurred with the sunspot or without it. In addition, we split the countable into two groups. The first type occurred during the rising phase of the primary solar flare, and the second occurred during the decay phase of the primary flare. Table 2 shows these counts.

Table 2 The count of sympathetic flare events during solar cycles 21–24
Full size table

As shown in the result location in Table 2, the sympathetic flares occur without sunspots. Most sympathetic flares occur with sunspots. Sympathetic flares occurred during the recovering phase of the primary flare for all solar cycles except solar cycle 23. During solar cycle 23 only, the sympathetic flares occurred during the rising phase of the primary flare. This case of occurrence during the rising phase (cycle 23) cannot be relied upon because the number of solar flares cycled during cycle 23 was insignificant.

During the weaker solar cycle 24, we show the most sympathetic flares occurred without sunspots.

Finally, most sympathetic flare events are associated with sunspots. Most of those events are associated with different sunspots. This is because the sympathetic flares depend on the solar magnetic field. Therefore, it often needs active regions.

This means probably that multiple events occur more frequently during the maximum solar activity, which is very natural, as more energy is accumulated in active regions and any flares are more possible to trigger a second event. A sympathetic flare, as it is called at times (Simnett 1974).

4.3 Timing ratio

By using equation (6), the time ratio of the secondary flare start time during the duration of the primary flare has been plotted as shown in Fig. 4.

Fig. 4
figure 4

The time series of the annual mean value of the timing ratio of sympathetic flares compared to SSN during the period 1975–2017

Full size image

It is noticeable that the relationship between the timing ratio and the strength of the solar cycle appears to be negative. The timing ratio is the lowest value possible during the active sun, while it reaches its maximum value during the quiet sun. Also, the stronger the solar cycle, the lower the maximum value of the timing ratio at the quiet Sun’s epoch, while the timing ratio’s value reaches a higher value during the weaker cycles.

Previously, Fritzova-Svestkova et al. (1976) found sympathetic flares have a short interval time that was equal to 20 minutes. Abramov-Maximov et al. (2007) found the interval time delay is 10–12 minutes of three sympathetic pairs of flares at wavelength 1.76 cm.

4.4 Sympathetic distance

By using equation (3), we estimated the distance between both flares of sympathetic flares. We found clearly that the relationship between distance and SSN is a positive relationship with strong coherence. During the epoch of the active sun, the distance between the two solar flares is as far as possible, while that distance becomes shorter during the epoch of the quiet sun. Figure 5 shows us this result. Both time series are approximate.

Fig. 5
figure 5

The time series of the annual mean value of the angular distance of sympathetic flares as compared to SSN during the period 1975–2017

Full size image

Our result is different from that concluded by Pearce and Harrison (1990). Because they examined the physical heliographic separations angle of sympathetic flares. They found active region pairs separated by smaller than 35° which shows a significant departure from random coincident flare activity. Fritzova-Svestkova et al. (1976) found pairs of active regions are closer than 30° to each other between both flares. While Abramov-Maximov et al. (2007) noted large distances are more than 90° in latitude.

We split sympathetic flares into two groups according to their timing ratio. In the case of the ratio \(R\) smaller than 0.5, i.e., the secondary flare occurred during the rising phase of the primary flare, it is called a “Rising sympathetic flare”. While in the case of an \(R\) greater than 0.5, i.e., the secondary flare occurred during the decay phase of the primary flare, it is called a “failing sympathetic flare”. The histogram is plotted as shown in Fig. 6. We found that sympathetic solar flares have a small angular distance in the case of \(R<0.5\). But they have a wide angular distance in the case of \(R>0.5\). We concluded that the rising sympathetic flares are mainly about polar alignment. While declining, sympathetic flares probably have equatorial alignment.

Fig. 6
figure 6

A histogram of the angular distance between the sympathetic flares. The left plot is for rising sympathetic flares, while the right plot is for failing sympathetic flares

Full size image

4.5 Inclination and wing diagram

The sympathetic flares are represented by a line that passes through both flares and creates a great circle. This line has an inclination angle from the polar direction (or longitudinal direction). We estimated this inclination using equation (5). It is plotted as shown in Fig. 7. We found that most sympathetic flare events have an equatorial alignment (latitudinal direction). While we see polar alignment (longitudinal alignment) in a few sympathetic flares. This result confirms that the alignment of the pair flares of the sympathetic flare is dependent on the strength of the solar polarity.

Fig. 7
figure 7

A histogram of the polar inclination of the great circle line path from primary to follower sympathetic flares

Full size image

We notice that the majority of the secondary flares are occurring at higher latitudes and longitudes than the primary flare.

The counts of the sympathetic flares according to their classification are determined. It found a greater inclination of the sympathetic flare with an associated X-class of primary flare than there are not associated with X-class primary flares.

We examined the variation of the inclination of the sympathetic flares with solar activity progression. It has a positive relationship, as shown in Fig. 8. We repeated this plot for all the data as shown in Fig. 9. It becomes the wing shape of birds. These birds are moving toward the Y axis, and their wings are down. We have one bird per solar cycle. The Wing Diagram proves that the inclination of the sympathetic flares has a higher value at the beginning of the solar cycle than at the end of it. This inclination value decreases with the development of solar activity during each solar cycle. It becomes a lower value at the end of the solar cycle. Its linear variation has a slope equal to about −20°.

Fig. 8
figure 8

The time series of the annual mean value of the inclination of the sympathetic flares compared to SSN during the period 1975–2017

Full size image
Fig. 9
figure 9

Wing Diagram. The body of the bird is above the main inclination and the remaining events lower the main body of the bird. Red points represent the sympathetic flares. As we can see, most of the sympathetic flare events are crowded at the top in the form of a line with a negative curvature, which is represented by a blue line, and this crowded area is similar to the body of the bird, which is flying in the direction of the left. While the rest of the events are distributed below that line with separation, which resembles a bird’s wing. Negative curvature indicates that the inclination decreases during the evolution of the solar activity cycle, as it begins with a higher inclination and decreases at the end of the cycle

Full size image

4.6 Waiting time

The interval time between both sympathetic flares has been estimated. We found that the dominant interval time between both sympathetic flares is between 0–20 minutes. The interval time may reach 2 hours. The interval time is small in the quiet sun (0–10 minutes). The probability of increasing interval time values as the solar cycle progresses (0–200 minutes). It reaches the maximum probability range of the interval time during an active sun. This result is shown clearly in Fig. 10.

Fig. 10
figure 10

A time series of the interval values of sympathetic flares

Full size image

5 Conclusions

We have studied the sympathetic flares detected by the GOES satellites during the solar period 1975–2017 (i.e., solar cycles 21–24). We found 2204 sympathetic solar flares out of all 77604.

The frequency of the occurrence of sympathetic flares depends on the solar cycle activity. The number of sympathetic solar flares increases with the increase in solar activity and decreases as solar activity decreases.

Sympathetic flares have a near or homogeneous GOES class. So, we can say that the second solar flare is just a homogeneous copy of the primary solar flare of the sympathetic flare. There are two types:

  1. 1)

    Twins: such as BB, CC, MM, and XX. It means that the secondary flare is a homogeneous copy of the primary flare. This type represents most of them (∼78.6%).

  2. 2)

    Non-twins: It is an associated heterogeneous pair of solar flares, such as XC.

Sympathetic flares occur in the decay phase of the solar cycle dominantly. This means that the energy can be stored in an active region for a longer time.

Also, we examined the time ratio from the interval between the two start times of the primary and secondary flares by the primary flare duration time. This timing ratio has a negative relationship with the solar cycle progress. The timing ratio has the lowest value possible during the active sun, while it reaches its maximum value during the quiet sun. Also, the stronger the solar cycle, the lower the maximum value of the timing ratio at the quiet Sun’s epoch, while the timing ratio’s value

The distance between paired flares of sympathetic flares has been examined. The relationship between distance with SSN is positive. During the epoch of the active sun, the distance is as far as possible, while that distance becomes shorter during the epoch of the quiet sun.

Additionally, we estimated sympathetic flares’ inclinations. Most sympathetic flare events have an equatorial alignment (latitudinal direction). While we see polar alignment (longitudinal direction) in a few sympathetic flares. This result confirms that the alignment of the pair flares of the sympathetic flare is dependent on the strength of the solar polarity.

The greater inclination of sympathetic flares is for association with X-class primary flares more than those not associated with X-class primary flares.

The annual mean value of the inclination of the sympathetic flares has a positive relationship with solar activity progress. While it becomes a Wings shape for all events. It is called the”Wings Diagram”. It has a slope of about −20° each cycle.

The dominant interval time between both sympathetic flares is between 0–20 minutes. The interval time may reach 2 hours. In addition, the interval time is small in the quiet sun (0–10 minutes). The probability of increasing interval time values as the solar cycle progresses (0-200 minutes). It reaches the maximum probability range of the interval time during an active sun.