Statistical Analysis of Circular-ribbon Flares

In Figure 1, the top panels show three images of the C4.2-class flare observed by AIA in 304 Å on 2015 October 16 (Zhang et al. 2016a; Dai et al. 2021). The flare was triggered by a minifilament eruption and was associated with a blowout coronal jet. In panel (a3), the flare ribbon is fitted with an ellipse, whose center is considered as the flare location (S13E42). The middle panels of Figure 1 show 304 Å images of another three cases. The bright ribbons are fitted with a circle and two ellipses, which are drawn with dashed cyan lines. The centers (S22E09, S19E32, and S13E59) of the flares are derived and pointed to by solid arrows. The bottom panels of Figure 1 show the corresponding LOS magnetograms of the three events observed by HMI. The CFs are characterized by a central positive (negative) polarity surrounded by negative (positive) polarities, which is essentially consistent with the fan-spine magnetic structure (Masson et al. 2009; Wang & Liu 2012). Most of the CFs in our sample show similar characteristics. For all 134 events, we fit the circular or quasi-circular ribbons observed in 304 or 1600 Å with a circle or an ellipse. The derived coordinates of CFs in the sixth column of Table 1 agree with those recorded in the GOES flare catalog. ⁸ In Figure 2(a), the flare centers are marked with colored dots, with cyan (red) dots signifying confined (eruptive) flares. Figure 2(b) shows latitude distribution of the CFs, where 76 (58) of which are located in the southern (northern) hemisphere. It is clear that all CFs are located in ARs like microflares, and the latitudes of CFs are between −30° and 30°. Hence, the distribution of CFs is consistent with the AR belts.

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Compared with two-ribbon flares whose ribbons show separation (Qiu et al. 2002) or elongation (Li et al. 2015; Qiu et al. 2017), the outer ribbons of CFs hardly expand. Hence, the area keeps constant. In Figure 1(a3), the apparent area of the C4.2 flare enclosed by the ellipse is ∼452 Mm². The true area (∼609 Mm²) after deprojection according to the flare longitude is defined as the total area (A_CF). In the middle panels of Figure 1, the apparent areas of the three CFs are ∼13,008, ∼985, and ∼739 Mm². The true areas are estimated to be ∼13,166, ∼1165, and ∼1423 Mm², respectively. For the 134 CFs, the areas are calculated according to the fitted circles or ellipses, which range from 123 to ∼5000 Mm² in most cases. Figure 3 shows the distribution of A_CF. Note that the areas (13,166 and 23,962 Mm²) of two events significantly exceeding 5000 Mm² are not taken into account in the histogram. We fit the distribution of A_CF with a lognormal function (Zhang et al. 2010):

$$\begin{eqnarray}&&\displaystyle \frac{1}{N}\displaystyle \frac{{dN}}{{dx}}=f(x,\mu ,\sigma )=\displaystyle \frac{1}{x\sigma \sqrt{2\pi }}{e}^{-\tfrac{{\left(\mathrm{ln}x-\mu \right)}^{2}}{2{\sigma }^{2}}},x\gt 0.\end{eqnarray} \tag{ 1 }$$

The curve fitting is performed by using mpfit.pro and the result is superposed with a red line in Figure 3, where μ = 6.51 and σ = 0.66. The minima, maxima, mean values, and median values of the flare areas are listed in Table 2.

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We notice that there is a tendency of increasing area with flare class. In Figure 4(a), the flare areas are divided into three groups of roughly the same amount. In the range of 0–462 Mm², only 13% (6/45) are M- and X-class CFs. The proportion rises to 36% (16/45) in the range of 462–915 Mm². Above 915 Mm², M- and X-class CFs account for 59% (26/44). Therefore, flares with larger magnitudes generally have larger areas. The sample of CFs in our study overlaps with that in Song & Tian (2018). They used an irregularly closed line rather than a circle or an oval to fit the flare ribbon. Although the methods are different, the derived areas of CFs are very close to each other.

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It is widely believed that the magnetic configuration of CFs consists of a null point and the associated dome-shaped fan-spine structure. Assuming that the fan surface is a semisphere (Pariat et al. 2009, 2010), the height of the null point is equivalent to the radius of circular ribbons:

$$\begin{eqnarray}&&{h}_{\mathrm{NP}}={r}_{\mathrm{CF}}=\sqrt{\displaystyle \frac{{A}_{\mathrm{CF}}}{\pi }}.\end{eqnarray} \tag{ 2 }$$

Figure 5 shows the distribution of r_CF, which lies in the range of 6–39 Mm with a mean value of ∼16 Mm (the largest two events are excluded). The estimated altitudes of null points are consistent with previous results (e.g., Sun et al. 2013; Xu et al. 2017; Hou et al. 2019; Li & Yang 2019; Yang et al. 2020). Song & Tian (2018) found that WL flares feature smaller areas than normal flares with the same flare magnitude, which is explained by the lower heights of null points where magnetic free energy is released and consequently there are larger energy fluxes in WL flares.

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We determine the flare lifetime (τ_CF) using the start and end times. The former refers to the time when the GOES 1–8 Å flux begins to increase rapidly, and the latter refers to the time when the flux declines to a nearly constant level. In Figure 6, the bottom panels show AIA 131 Å images of the M8.7-class flare occurring in AR 12242 on 2014 December 17 (Chen et al. 2019). The top panel shows the SXR light curve of the flare. The black dashed lines denote the start time (04:25:34 UT), peak time (04:51:34 UT), and end time (07:07:34 UT) of the flare. The lifetime reaches ∼162 minutes accordingly.

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If the SXR light curve has a second peak in the decay phase, which is probably due to another eruption somewhere else, the AIA 131 Å light curve of the relevant flare is used as a supplementary to determine τ_CF. In Figure 7, the bottom panels show AIA 131 Å images of the C3.0-class flare occurring in AR 11936 on 2013 December 28. The top panel shows the SXR light curve (blue line) and 131 Å light curve (orange line) of the flare. The black dashed lines denote the start time (12:40:34 UT), first peak time (12:47:34 UT), and end time (13:18:34 UT). There is only one peak in the EUV light curve, which corresponds to the first peak in SXR. The second peak (∼13:14:55 UT) in SXR is unrelated to the C3.0-class flare. Hence, the lifetime of the flare is ∼38 minutes, which is considerably shorter than the M8.7 flare.

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Figure 8 shows the distribution of τ_CF with a mean value of ∼50 minutes, which is 2 times larger than that of Lyα flares (Lu et al. 2021). Likewise, the distribution can be fitted with a lognormal function in Equation (1). The fitted curve is superposed with a red line, where μ = 3.60 and σ = 0.64. To explore the relationship between the area and lifetime of CFs, we draw a scatterplot in Figure 9. The two parameters have a positive correlation with a coefficient of ∼0.5. The timescale of conductive cooling of hot flare loops is expressed as (Cargill 1994; Zhang et al. 2019a)

$$\begin{eqnarray}&&{\tau }_{\mathrm{cc}}=4\times {10}^{-10}\displaystyle \frac{{n}_{{\rm{e}}}{L}^{2}}{{T}_{{\rm{e}}}^{5/2}},\end{eqnarray} \tag{ 3 }$$

where n_e, T_e, and L represent the electron number density, temperature, and total length of a flare loop. For CFs with fan-spine topology, A_CF ∝ L² ∝ τ_cc. That is to say, for flares with larger sizes, the cooling times become longer, which can easily interpret the linear correlation between A_CF and τ_CF.

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According to the lifetime, we divide the 134 flares into three groups of the same amount, i.e., 0–25, 25–59, and 59–205 minutes. The percentages of M- and X-class CFs in the three groups are 26% (12/46), 20% (9/45), and 63% (27/43) (see Figure 4(b)). Hence, flares with larger magnitudes tend to have longer lifetimes. In the statistical investigation of CFs (Song & Tian 2018), the end time corresponds to the time when the flux decreases to a point halfway between the peak flux and the preflare level, leading to an average lifetime of 10–20 minutes, which is systematically shorter than our results. Besides, some of the CFs may experience an EUV late phase after the main flare phase observed in "warm" emission lines (e.g., 335 Å) as a result of additional heating (Sun et al. 2013). The much-extended lifetimes in EUV 335 Å compared with SXR lifetimes are out of the scope of this study.

For the 134 CFs, the peak SXR flux in 1–8 Å ranges from ∼5.9 × 10⁻⁷ to ∼3.9 × 10⁻⁴ W m⁻². Figure 10 shows the distribution of the peak SXR flux. It is clear that the distribution could be nicely fitted with a power-law function (e.g., Dennis 1985; Lu & Hamilton 1991; Crosby et al. 1993; Christe et al. 2008) above ∼2 × 10⁻⁶ W m⁻²:

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dF}}\propto {F}^{\alpha },\end{eqnarray} \tag{ 4 }$$

where F denotes the peak flux, and α = −1.42 denotes the power-law index. The value of α is close to that of microflares at 3–6 keV and less than that of Lyα flares (Lu et al. 2021).

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As mentioned in Section 1, CFs are usually associated with remote brightenings or remote ribbons (Masson et al. 2009; Xu et al. 2017; Song et al. 2018; Chen et al. 2019). In Figure 1(a3), remote brightenings appear to the southwest of the C4.2-class flare, which is overlaid with a cyan line. The brightening has a total length (L_RB) of ∼262 Mm and an average distance (D_RB) of ∼171 Mm away from the flare center.

For the 134 CFs in this study, we searched for remote brightenings observed by AIA mainly in 304 and 1600 Å, finding that ∼57% (76/134) of CFs have associated remote brightenings, which is shown in Figure 11. Specifically, the association rates with B-, C-, M-, and X-class flares are 25%, 49%, 70%, and 88%, respectively (see Figure 12(a) and Table 3). The increasing rate with flare magnitude indicates that larger flares are more likely to produce remote brightenings. The total length of remote brightening or ribbon is denoted with L_RB, and the average distance from the flare center is denoted with D_RB. Figure 13 shows the distributions of L_RB and D_RB. It is revealed that L_RB lies in the range of 6.2–381.0 Mm, with a mean value of ∼84.1 Mm. D_RB lies in the range of 28.6–186.2 Mm, with a mean value of ∼89.1 Mm. Figure 14 shows the scatterplot between L_RB and D_RB, indicating a good correlation between the two parameters with a correlation coefficient of ∼0.65. In other words, remote brightenings further away from the main CFs are more likely to have longer extensions. A linear fitting (red dashed line) between the two parameters is performed, i.e.,

$$\begin{eqnarray}&&{D}_{\mathrm{RB}}=67.56+0.26{L}_{\mathrm{RB}}.\end{eqnarray} \tag{ 5 }$$

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To investigate the relationship between flare magnitude and association with remote brightenings, we divide L_RB and D_RB into three groups with the same numbers. The values of L_RB are divided into 0–29, 29–80, and 80–381 Mm. The values of D_RB are divided into 0–72, 72–93, and 93–186 Mm. The percentages of M- and X-class CFs in each group are plotted in Figures 4(c)–(d). The percentages increase systematically with both parameters, indicating that flares with larger magnitudes are more likely to produce longer remote brightenings or ribbons. In the 3D numerical simulations of jets confined by large-scale coronal loops (Wyper & DeVore 2016; Wyper et al. 2016), the aspect ratio of the fan-spine structure is between 1.0 and 2.7. In our study, the aspect ratio is defined as $\tfrac{{D}_{\mathrm{RB}}}{2{r}_{\mathrm{CF}}}$, which has a range of 0.8–6.7 and a mean value of ∼2.9. Hence, the results will impose constraints on numerical simulations of CFs in the future (Pontin et al. 2013).

Flare-accelerated nonthermal electrons propagating along open field lines are capable of producing type III radio bursts (Benz et al. 2005; Morosan et al. 2014; Reid & Ratcliffe 2014; Zhang et al. 2015). In Figure 15, the top panel shows the 304 Å image of an M1.4-class CF in AR 11476 on 2012 May 8. The corresponding radio dynamic spectra recorded by the e-Callisto/BLEN7M station is displayed in the bottom panel. It is obvious that type III radio bursts around 700 MHz with fast frequency drift occur during the impulsive phase, when the release rate of magnetic free energy is maximum. The existence of radio bursts is confirmed by the WIND/WAVES observation.

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For the 134 CFs, 47% (63/134) of them are accompanied with type III bursts during the impulsive phases, which is displayed in Figure 11. Note that those occurring in the preflare or decay phases are considered as unrelated events, which may result in an underestimate of the total amount. In Figure 12(b), the proportions of CFs related to type III bursts at various flare magnitudes are demonstrated, being 0%, 44%, 60%, and 38% for B-, C-, M-, and X-class CFs, respectively. The relatively lower proportion in B-class flares may be due to the limited number of samples. Hence, there is no correlation between the flare magnitude and type III bursts. In other words, the production of type III bursts depends mainly on the magnetic topology instead of flare energy (Krucker et al. 2011; Glesener et al. 2012; Duan et al. 2022).

As mentioned in Section 1, CFs are frequently associated with hot coronal jets observed in EUV wavelengths (Li & Yang 2019; Zhang & Ni 2019; Zhang 2020; Zhang et al. 2021) or cool surges observed in Hα (Wang & Liu 2012; Xu et al. 2017). In Figure 1, the C4.2-class flare is accompanied by a blowout jet (Zhang et al. 2016a; Dai et al. 2021). For the 134 events, about half of them are accompanied by jets in 304 Å, which is displayed in Figure 11. In Figure 12(c), the proportions of CFs related to jets at various flare magnitudes are demonstrated, being 25%, 52%, 53%, and 50% for B-, C-, M-, and X-class flares, respectively. Therefore, there is no preference of jet production for CFs with larger magnitudes as well (Chae et al. 1999; Shibata et al. 2007; Duan et al. 2022).

The occurrence of CFs is often associated with filament or minifilament eruptions (Wang & Liu 2012; Jiang et al. 2013; Sun et al. 2013; Yang & Zhang 2018). In this scenario, breakout-type magnetic reconnection takes place near the null point after the filament embedded in the fan dome rises up (Joshi et al. 2015; Wyper et al. 2017). The initiation of filament eruption may result from magnetic flux emergence (Li et al. 2017), rotation (Xu et al. 2017), or ideal MHD instabilities. The null-point reconnection not only accelerates nonthermal electrons to produce the circular ribbon in the chromosphere, but speeds up the filament eruption as well (Zhang et al. 2021).

Most of the homologous CFs in AR 12434 are triggered by minifilament eruptions (Zhang et al. 2016a, 2021). Figure 1(a2) shows the AIA 304 Å image of the C4.2-class flare, and the minifilament within the circular ribbon is pointed to by the arrow. For the 134 events, 38% (51/134) of them are associated with filament eruptions, which is displayed in Figure 11. Figure 12(d) shows the proportions of CFs related to minifilament eruptions: 25% (1/4), 30% (25/82), 45% (18/40), and 88% (7/8) for B-, C-, M-, and X-class flares, respectively. The increasing proportions with flare classes indicate that larger flares are more likely to be triggered by filament eruptions.

As mentioned in the previous section, CFs are sometimes triggered by filament or minifilament eruptions. The eruption may also drive a CME observed in the WL coronagraphs (Joshi et al. 2017; Liu et al. 2020; Kumar et al. 2021). In Figure 16, the left panel shows the 304 Å image of the M7.3-class CF in AR 12036 on 2014 April 18 (Joshi et al. 2015). The associated CME at a speed of ∼1203 km s⁻¹ in the LASCO/C2 FOV is displayed in the right panel. For the 134 CFs, only 28% (37/134) of them are related to CMEs, which is shown in Figure 11. It is evident that the association rate with CMEs is the lowest while the association rate with remote brightenings is the highest in our investigation. In other words, most of (∼72%) CFs are confined rather than eruptive (Wyper & DeVore 2016; Li et al. 2018; Yang & Zhang 2018). The CME association rates for B-, C-, M-, and X-class flares are 0%, 21%, 35%, and 75%, which is displayed in Figure 12(e). The increasing CME rates with flare magnitudes are consistent with previous findings, suggesting that larger flares are more likely to produce CMEs (Yashiro et al. 2005, 2006). The low percentage of CMEs is expected, since magnetic flux ropes are required to generate CMEs in most cases (Vourlidas et al. 2013). For eruptive flares, two parallel ribbons or S-shaped ribbons appear on both sides of the polarity inversion lines (Aulanier et al. 2012; Janvier et al. 2013; Savcheva et al. 2016). Therefore, the formation of a CF is unlikely in this situation.

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The distribution of the linear speed (V_CME) of the 37 CMEs is drawn in Figure 17. The values of V_CME lie in the range of 120–1203 km s⁻¹, with an average speed of ∼517 km s⁻¹, which is slightly higher than that of CMEs near solar maximum (Yashiro et al. 2004). Figure 18 shows the scatterplot between the peak flux in 1–8 Å and V_CME of the 37 eruptive flares. A positive correlation is clearly demonstrated with a correlation coefficient of ∼0.37, which is consistent with the previous finding that CMEs associated with X-class flares are remarkably faster than those associated with C-class flares (Yashiro et al. 2005).

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Out of the 37 CMEs, 30 of them are normal or partial halo CMEs, and 7 of them are full halo CMEs. Some of them are jet-like, narrow CMEs with an angular width (W_CME) less than 25°. Figures 19 and 20 show scatterplots between W_CME and the peak flux of flares and V_CME, respectively. It is clear that W_CME have positive correlations with flare peak flux and V_CME. Note that full halo CMEs (W_CME = 360°) are not included when calculating the correlation coefficients. The result in Figure 20 suggests that faster CMEs tend to be wider, which is in line with previous statistical investigation (Yashiro et al. 2004).

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According to their speeds, the 37 CMEs are divided into three groups with roughly the same numbers. Figure 21(a) shows that the percentages of M- and X-class CFs are 33% (4/12), 42% (5/12), and 85% (11/13) for 0–360, 360–583, and 583–1203 km s⁻¹, respectively. The association rates of CMEs with big flares increase with the CME speeds, which is consistent with the fact that faster CMEs are more related to flares with larger magnitudes (Yashiro et al. 2005, 2006). Figure 21(b) shows the percentages of filament eruptions for the three groups of CMEs, being 50% (6/12), 58% (7/12), and 54% (7/13). The association rates of CMEs with filament eruptions are independent of the CME speeds.

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Fast CMEs are capable of driving shock waves, which are associated with type II radio bursts (e.g., Ontiveros & Vourlidas 2009; Zucca et al. 2018; Mancuso et al. 2019). For the 37 eruptive CFs accompanied by CMEs, seven are associated with type II bursts, including two X-class (No. 3, 53) and five M-class (No. 34, 49, 77, 82, 83) flares. In Figure 22, the left panels show three CFs observed in 131 Å. The middle panels show the corresponding CMEs observed by LASCO/C2, with the apparent speeds being labeled. The related shock waves with relatively lower intensities are pointed to by the arrows. The right panels show radio dynamic spectra recorded by the e-Callisto stations, where type II radio bursts with drifting frequency are pointed to by the arrows.

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