Statistical Relation between Solar Flares and Coronal Mass Ejections with Respect to Sigmoidal Structures in Active Regions

The focus of our analysis is to determine the statistical relations among physical parameters of CMEs and solar flares simultaneously. We chose CME presence/absence, speed, and size as the CME parameters, and X-ray class, duration, and sigmoid presence/absence as solar lare parameters. Each characteristic is defined below.

3.1.1. CMEs

3.1.2. Duration of Flare Events

We define the flare rise time (t_rise), decay time (t_dec), and duration (t_dur) as follows:

$$\begin{eqnarray}&&{t}_{\mathrm{rise}}={T}_{\mathrm{peak}}-{T}_{\mathrm{st}},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{t}_{\mathrm{dec}}={T}_{\mathrm{en}}-{T}_{\mathrm{peak}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{t}_{\mathrm{dur}}={t}_{\mathrm{rise}}+{t}_{\mathrm{dec}},\end{eqnarray} \tag{ 3 }$$

where T_st, T_peak, and T_en are the flare start time, flare peak time, and flare end time, respectively. Although T_st, T_peak, and T_en are provided by the GOES event list,⁷ we re-determined them based on the method of Aschwanden & Freeland (2012), which enabled us to distinguish more clearly between long-duration events (LDEs) and impulsive flares. In the GOES event list, the flare end time is defined as time when the flux level decays to a point halfway between the maximum flux and pre-flare background level. However, this study defines the flare end time as the time when the flux level reaches approximately the pre-flare background level. The procedure is as follows.

• 1.  
  Data rebinning. The original time resolution of dt = 2 s or 3 s was rebinned to time steps of Δt = 6 s.
• 2.  
  Data smoothing. The rebinned light curve was smoothed with a boxcar average, and the width of the smoothing window measured 60 s.
• 3.  
  Detection of flare peak time. We defined the peak time, T_peak, as the local maximum, where the X-ray flux, F(T), satisfies F(T_peak − Δt) < F(T_peak), F(T_peak+Δt) < F(T_peak), and F(T_peak) > 10⁻⁵ W m⁻².
• 4.  
  Detection of flare start time. The definition of the flare start time (T_st) is equal to that of the GOES start time, which is determined by the first minute of a sequence of 4 min of steep monotonic increase.
• 5.  
  Detection of flare end time. We defined the flare end time (T_end) as the time when the soft-X-ray flux satisfies F < 3F_back, with F_back as the background flux, which is defined by the average X-ray flux 36 s before the flare start time.

Figure 1 shows an example of GOES X-class flares and our detection of the rise time (red line), decay time (blue line), and duration (red line + blue line) described above. The peak and end times from the GOES event list are represented by the vertical dotted line. The decay time of the GOES event list is quite short (∼16 min), despite visual inspection that conventionally declared this event as an LDE. In contrast, our method captured a long decay time (∼4 hr).

Zoom In Zoom Out Reset image size

It should be noted that there are 20 events which occurred at decay times of other events. Further, five events lacked data and were thus excluded from our duration analysis.

3.1.3. Sigmoids

The sigmoids can be identified as S-shaped or inverse S-shaped structures with higher intensity than the surrounding structures in soft X-ray. We identified sigmoidal structures in soft-X-ray images as follows.⁸

First, we determined the frequency distribution of the X-ray intensity in each image to obtain the XRT noise level. The XRT data were normalized using the exposure time. The left panels in Figure 2 show examples of the frequency distribution at this step. A peak exists in the range below 50 DN s^–1, which corresponds to photon noise. We determined this value by fitting the distribution with a Gaussian function for each image. The typical value of noise is 15 DN s^–1. Note that the system gain of the XRT is 57 e⁻¹/DN.

Zoom In Zoom Out Reset image size

Next, we looked for enhancement in the frequency distribution. While the distribution typically decreases as a power-law function in the range above the photon noise level, certain enhancement ranges are occasionally seen. We extracted all enhancements in the range above 100 DN s^–1, which is marked with triangles in the left column of Figure 2, as ranges where the first derivative of the frequency distribution is larger than zero. The middle column of Figure 2 shows examples of this step. The horizontal line corresponds to zero and the vertical dashed lines correspond to the lowest values of enhancement ranges detected in the analysis. Afterwards, the bright enhancement region (candidate for sigmoid detection) was drawn in an X-ray intensity image using the threshold value obtained in the analysis above. The right column of Figure 2 presents an example of enhancement regions found with this method. The colors of the contour lines correspond to those of the vertical dashed ones in the left and right columns.

At the final step of sigmoid recognition, we judged whether the bright regions are S- or J-shaped. Three authors, i.e., Y. Kawabata, Y. Iida, and T. Doi, separately carried out this step by visual inspection for all bright regions of all flare events. When two or three individuals judged that the bright region had an S- or a J-shaped structure, it was classified as a sigmoid. The flare event was judged to be accompanied by a sigmoid structure when it appeared at least in one image taken in 12 hr before the flare peak time.

We investigated the independence of four parameters through contingency tables and probabilities of intersection sets: X-ray class, duration, sigmoid (hereafter SGM) existence, and CME existence. We define large X-ray-class (hereafter LXC) events as events in which the X-ray class is more than M2.3. LDEs are defined as events in which the duration exceeds 1 hr. Both thresholds were approximately set to the median value of the 211 and 186 events, respectively.

First, we computed the occurrence probability of each event. Events without Be-thin data were excluded from the SGM analysis. Further, CME data were excluded when it was difficult to judge whether CMEs had occurred or not. An error was obtained as a Poisson noise value in the case of many events, i.e., $\sqrt{N}$, with N as event number.

Next, a contingency table of the pairs among all four parameters, e.g., LXC, LDE, SGM, and CME, was created to investigate the association of each pair of parameters. We calculated the ϕ coefficient defined by

$$\begin{eqnarray}&&\phi =\sqrt{\displaystyle \frac{{\chi }^{2}}{N}},\end{eqnarray} \tag{ 4 }$$

where χ is Pearson's chi-square and N is the number of events.

The association was also investigated from a different point of view. We focused on how large differences exist between the actual occurrence probabilities of the intersection sets and those assuming the independence of each parameter. We computed the occurrence probability of each intersection set, A ∩ B, from the individual occurrence probabilities by assuming independence. The occurrence probability of the intersection set could be obtained as a product due to the independent events in this case. The error bars of the intersection events, σ_ind, were calculated as Poisson noise for each occurrence probability. The occurrence probability of the actual intersection event was obtained from the observation result. The error value, σ_obs, was calculated as Poisson noise from the occurrence number of the actual intersection event. Finally, we compared the occurrence probability of the intersection events with independent events with that from the actual observation obtained in the analysis above, calculated by how many multiples of the error value they were separated.

In order to estimate how much SGM and CME occurrence depends on the four flare properties, i.e., (i) flare duration, (ii) rise time, (iii) decay time, and (iv) soft-X-ray flux, we conducted the Kolmogorov–Smirnov (KS) test and Anderson–Darling (AD) test for each condition. The KS and AD tests judge whether two distributions are similar or not. We derived the cumulative distribution functions (CDFs) of the four parameters (i)–(iv),

$$\begin{eqnarray}&&F(x)={\int }_{0}^{x}f(t){dt},\end{eqnarray} \tag{ 5 }$$

where F and f are CDF and frequency distribution. Both x and t are taken from parameters (i)–(iv). The KS and AD tests were conducted for the two CDFs with and without CMEs (or SGMs). The KS and AD statistics, D_KS and D_AD are calculated as

$$\begin{eqnarray}&&{D}_{\mathrm{KS}}=\sup | {F}_{1}(x)-{F}_{2}(x)| ,\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{D}_{\mathrm{AD}}=\displaystyle \frac{{mn}}{m+n}\int \displaystyle \frac{{F}_{1}(x)-{F}_{2}(x)}{G(x)(1-G(x))}{dG}(x),\end{eqnarray} \tag{ 7 }$$

where sup is the supremum, m and n are the sample number of each distribution, and G(x) = mF₁(x)−nF₂(x). The AD test gives more weight to the tails of the CDFs than the KS test. In the latter, the two populations are judged to be different distributions in the confidence interval of 100(1−α)% when the D_KS satisfies the relation

$$\begin{eqnarray}&&{D}_{\mathrm{KS}}\geqslant \sqrt{\displaystyle \frac{n+m}{{nm}}}\sqrt{-\displaystyle \frac{1}{2}\mathrm{ln}\displaystyle \frac{\alpha }{2}}.\end{eqnarray} \tag{ 8 }$$

Note that the critical value is proportional to $\sqrt{(n+m)/{nm}}$. The minimum value of α (e.g., α ∼ 0) corresponds to the probability that the two populations are in a different distribution function (i.e., ∼100% confidence), which means that CME (SGM) occurrence has some dependence on the flare parameter. In this study, we adopted α = 0.01 to judge whether the two populations are different or not. In the AD test, the critical value is also set to 99% confidence as follows (Scholz & Stephens 1987):

$$\begin{eqnarray}&&{D}_{\mathrm{AD}}\geqslant 3.752.\end{eqnarray} \tag{ 9 }$$

Figure 3 presents the frequency distribution of solar flares as a function of the soft-X-ray peak intensity, dN/dW, which includes 195 M-class flares and 16 X-class flares. The median of the X-ray class is M2.3. The power-law index of this distribution is ∼−2.0, which was derived by linearly fitting the frequency distribution. The value is consistent with the previous result (∼−1.9) derived from XRT observations (Sako et al. 2013).

Zoom In Zoom Out Reset image size

Figure 4 shows histograms of the duration, rise time, and decay time. The respective medians are 3385, 676, and 2409 s. The duration and decay time show similar distributions. More events exist in the shorter timescale (<1000 s) in the decay time histograms. Further, extremely long durations and decay times such as ∼10,000 s exist. The rise time behaves differently from the duration and decay time. It has a peak at ∼400 s.

Zoom In Zoom Out Reset image size

We categorized the 211 events into two groups: on-disk and limb events. On-disk events are located within 500'' from the disk center of the Sun and limb events describe the rest. Based on our definition, 63 on-disk events and 148 limb events occurred. The existence of SGMs and CMEs for on-disk and limb events is summarized in Table 1. If no Be-thin images were taken in 12 hr before the onset of flares, SGM judgement was impracticable. Furthermore, when it was difficult to identify which flare produced the CME, CME judgement was also impracticable.

In this section, we illustrate the dependence of SGM and CME occurrence on the duration, rise time, decay time, and X-ray flux of flares. SGM (CME) occurrence is defined as the number ratio of events in which the SGM (CME) occurred. Out of the 211 events, there were 58 events with SGMs and 62 events without. Also, there were 57 events with CMEs and 120 events without (see Table 1).

In Table 2, we give the Spearman's rank correlation coefficient ρ and Kendall's rank correlation coefficient τ of the occurrence dependences, which are shown in Figures 5–7. Bold values show the deviation from zero in the 99% confidence interval, which means that the relation of the two parameters can be represented by a monotonic function. There are significant correlations between the duration and SGM, rise time and CME, and X-ray flux and CME for all events. Significant correlation also can be seen between decay time and SGM for the on-disk events, and between rise time and CME and X-ray flux and CME for the limb events.

Zoom In Zoom Out Reset image size

Figure 5 shows the SGM and CME occurrence as functions of the respective flare properties. The length of the error bar is estimated assuming Poisson noise. Hence, a larger error bar is due to a smaller sample size (e.g., only one event exists at which the rise time ranges between 10^3.7 < t_rise < 10^3.9 s. It can be estimated whether a CME was included. Therefore, the 1σ uncertainty of the occurrence in the range equals 1.0). In order to estimate the dependence of the SGM and CME occurrence on the respective flare parameter, the histograms were fitted to a linear function using the least-squares method and considering a Poisson error scale. Only when the Spearman's rank correlation coefficient ρ and Kendall's rank correlation coefficient τ are significantly different from 0 in the 99% confidence interval, the fitted lines are shown in bold. The error value of a slope R is the 1σ uncertainty. Regarding SGM occurrence, we obtained a slope of R = 0.43 ± 0.22 for the duration, which is a positive value for a 1σ uncertainty. Regarding CME occurrence, we obtained a positive slope for two parameters: R = 0.40 ± 0.12 for the rise time, and R = 0.53 ± 0.13 for the soft-X-ray peak flux.

To compare on-disk and limb events, we derived the SGM and CME occurrence as functions of the respective flare properties. We conducted linear fitting, as above. The results are shown in Figures 6 and 7. Since the total number of events for SGMs is small (42), the occurrence error is large. For the on-disk events, the SGM occurrence as a function of the decay time possesses a steep slope (R = 0.61 ± 0.41). For the limb events, we obtained a positive slope from the CME occurrence as a function of the rise time (R = 0.45 ± 0.14), and X-ray flux (R = 0.60 ± 0.15).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We investigated the linear correlations between physical parameters by creating a scatter plot. Before presenting the result of this plot, we show Pearson's correlation coefficient C, Spearman's rank correlation coefficient ρ, and Kendall's rank correlation coefficient τ, between two parameters in Table 3. As in Table 2, bold values show the deviation from zero in the 99% confidence interval. As we can see, there are significant correlations between the duration, decay time, and rise time for all, on-disk, and limb events. Significant correlations are also seen between duration and CME width, decay time and CME width, duration and X-ray flux, and decay time and X-ray flux for all events. In the limb events, there is no correlation between duration and CME width, or decay time and CME width. Note that the results of linear fitting are shown later in this section but the linear model may be invalid when the rank correlation coefficient is not significantly different from 0 in the 99% confidence interval.

In Figures 8–10, we show the results of linear fitting, in which bold lines are used when the the values of ρ and τ are significantly different from 0 in the 99% confidence interval. The correlation coefficients C are shown for all (black), on-disk (red), and limb (blue) events. The correlation coefficient of the duration and decay time (middle panel) is extremely large, C = 0.97. However, C is relatively small (∼0.45) for the decay and rise time.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In Figure 9 the correlation coefficients are small and similar for all relations for all events, C = 0.16–0.40. In the on-disk events, however, the correlation coefficient between the duration and CME width, and decay time and CME width is relatively high, C = 0.55, 0.60. The low correlation in the limb events may come from the upper limit of the CME width (360°). There are more limb events than on-disk events. As the number of CME events with 360° width increases, the correlation coefficient decreases.

In Figure 10 the correlation coefficients for all events between the duration and X-ray class, and the decay time and X-ray class, are relatively large: C = 0.44, 0.47, respectively. In contrast, that of rise time and X-ray class is quite small, C = 0.14.

Zoom In Zoom Out Reset image size

Table 4 summarizes the occurrence probabilities. Those of LXC events and LDEs constitute roughly half. As described above, in five of 211 flare events, data are lacking in GOES. In addition, certain events occur at decay times of other events, which makes it difficult to evaluate their duration. We excluded these events from our analysis. Thus, there are 186 events in the second row in Table 4 instead of 211. The SGMs are accompanied by flares in approximately half of the events. The CME occurrence probability is 32.2%, which is slightly lower than those for LXCs, LDEs, and SGMs. No occurrence probabilities show significant variation among all, on-disk, and limb events.

The contingency tables of each two sets regarding LXC, LDE, SGM, and CME are summarized in Table 5. The fourth and seventh columns show the intersection probabilities and those of negation on both events. The fifth and sixth columns present the results for the intersection probability of negation on one event. In the SGM–CME analysis, many unaccompanied SGMs exist (39.2%), but the quantity of CMEs without SGMs is much lower (12.8%). Many LXC events exist without CMEs (25.4%), but the quantity of CMEs without LXC events is much lower (9.6%). Further, LDEs without CME are observed (26.1%), but the occurrence probability of CMEs without LDEs is much lower (8.9%). Nevertheless, no dependence on the observation location could be observed (on-disk or limb) for LXC–LDE, LXC–SGM, or LDE–SGM. The ϕ coefficient is shown in the eighth column. We employ 0.2 as the threshold value for a weak association. A ϕ coefficient larger than 0.2 is shown in bold face in Table 5. A dependence on observation location is observed. In all and limb cases, the ϕ coefficients of all pairs from LXC–LDE–CME show relatively high values. On the other hand, those from LDE–SGM–CME are larger than 0.2 but those of LXC–CME and LXC–SGM are not in the on-disk case.

Table 5.  Contingency Table of Events and Mean Square Contingency Coefficient

	A	B	Ntot	$P({\rm{A}}\cap {\rm{B}})$	$P(\overline{{\rm{A}}}\cap {\rm{B}})$	$P({\rm{A}}\cap \overline{{\rm{B}}})$	$P(\overline{{\rm{A}}}\cap \overline{{\rm{B}}})$	ϕ
All
	LXC	LDE	186	28.5 ± 3.8% (53)	19.4 ± 3.2% (36)	18.4 ± 3.1% (34)	33.9 ± 4.3% (63)	0.25
	LXC	SGM	120	21.7 ± 4.2% (26)	26.7 ± 4.7% (32)	26.7 ± 4.7% (32)	25.0 ± 4.6% (30)	0.07
	LXC	CME	177	22.6 ± 3.6% (40)	9.6 ± 2.3% (17)	25.4 ± 3.8% (45)	42.1 ± 4.9% (75)	0.31
	LDE	SGM	108	27.8 ± 5.1% (30)	21.3 ± 4.4% (23)	23.2 ± 4.6% (25)	27.8 ± 5.1% (30)	0.11
	SGM	CME	102	13.7 ± 3.7% (14)	12.8 ± 3.5% (13)	39.2 ± 6.2% (40)	34.3 ± 5.8% (35)	0.01
	CME	LDE	157	24.8 ± 4.0% (39)	26.1 ± 4.1% (41)	8.9 ± 2.4% (14)	40.1 ± 5.1% (63)	0.32
On-disk
	LXC	LDE	53	26.4 ± 7.1% (14)	28.3 ± 7.3% (15)	22.6 ± 6.5% (12)	22.6 ± 6.5% (12)	0.02
	LXC	SGM	42	28.6 ± 9.0% (12)	21.4 ± 7.1% (9)	21.4 ± 7.1% (9)	28.6 ± 8.3% (12)	0.14
	LXC	CME	51	15.7 ± 5.5% (8)	13.7 ± 5.2% (7)	27.5 ± 7.3% (14)	43.1 ± 9.2% (22)	0.13
	LDE	SGM	39	30.8 ± 8.7% (12)	18.0 ± 6.8% (7)	18.0 ± 6.8% (7)	33.3 ± 9.3% (13)	0.28
	SGM	CME	34	20.6 ± 7.8% (7)	8.8 ± 5.1% (3)	32.4 ± 9.8% (11)	38.3 ± 10.6% (13)	0.22
	CME	LDE	44	25.0 ± 7.5% (11)	36.4 ± 9.1% (16)	6.8 ± 3.9% (3)	31.8 ± 8.5% (14)	0.24
Limb
	LXC	LDE	133	29.3 ± 4.7% (39)	15.8 ± 3.5% (21)	16.5 ± 3.5% (22)	38.4 ± 5.4% (51)	0.34
	LXC	SGM	78	18.0 ± 4.8% (14)	29.5 ± 6.1% (23)	29.5 ± 6.1% (23)	23.1 ± 5.4% (18)	0.18
	LXC	CME	126	25.4 ± 4.5% (32)	7.9 ± 2.5% (10)	24.6 ± 4.4% (31)	42.1 ± 5.8% (53)	0.37
	LDE	SGM	69	26.1 ± 6.2% (18)	23.2 ± 5.8% (16)	26.1 ± 6.2% (18)	24.6 ± 6.0% (17)	0.02
	SGM	CME	68	10.3 ± 3.9% (7)	14.7 ± 4.7% (10)	41.2 ± 7.8% (28)	33.8 ± 7.1% (23)	0.12
	CME	LDE	113	24.8 ± 4.7% (27)	22.1 ± 4.4% (25)	9.7 ± 2.9% (11)	43.4 ± 6.2% (49)	0.35

Download table as:  ASCIITypeset image

The results of association analysis for the intersection sets are summarized in Table 6. The parameter P_ind represents the intersection probability assuming that two sets are independent, P_ind = P(A) × P(B); P_obs corresponds to the intersection of the events according to observations, $P(A\cap B)$ in Table 5; ΔP is the difference between P_ind and P_obs. The error value σ_Δ is obtained from the propagation of the error values assuming independent σ_ind and σ_obs, i.e., $\sqrt{{\sigma }_{\mathrm{ind}}^{2}+{\sigma }_{\mathrm{obs}}^{2}}$. For all events, LXC–CME and CME–LDE show significant probabilities (1.25 and 1.42), while LXC–SGM and SGM–CME exhibit smaller probabilities (−0.36 and −0.33) compared to other events. Although there are no significant probabilities in the on-disk events, LXC–SGM, SGM–CME, and CME–LDE show comparatively higher probabilities. For limb events, the CME–LXC and CME–LDE values show significant dependence.

We conducted KS and AD tests and estimated the dependence of the SGMs and CMEs on the flare parameters. We selected 58 events with an SGM and 62 events without (see Table 1). Also, we selected 57 events with a CME and 120 events (see Table 1).

Figure 11 shows the CDFs.

Zoom In Zoom Out Reset image size

Tables 7 and 8 show the values of the KS and AD statistics regarding SGMs and CMEs, respectively. Bold values indicate than the 99% confidence level (α < 0.01). According to Table 7, SGM occurrence is not significantly related to all parameters. However, the statistics of both KS and AD tests show relatively larger values in the decay time and duration for all and for the on-disk events, and is less related to the rise time and soft-X-ray peak flux for these events. Because the number of the sample of the limb events is larger than that of the on-disk events, the value $\sqrt{(n+m)/({nm})}$ of the limb events is lower compared to on-disk events. Therefore, the probability tends to be smaller in the limb events than the on-disk events in the KS test.

According to Table 8, in a confidence interval of 99%, the CME occurrence is related to the flare duration and decay time for all, on-disk, and limb events and to the soft-X-ray peak flux for all and the limb events. Only for all events is the rise time related to the CME occurrence in the AD test.

CME occurrences as functions of duration, rise time, and decay time show a positive slope in Figure 5, while the duration and the decay time do not show significant dependence at the 99% confidence level. Toriumi et al. (2017) also compared the histograms of the duration and decay time for eruptive and noneruptive events. In their results, the decay time histogram for eruptive events shows large values for long decay times. On the other hand their histograms of duration show little difference between eruptive and noneruptive events, whereas CME occurrence as a function of duration shows a positive slope in our study. Their conclusion that a longer decay time tends to produce CMEs is consistent with our result. However, the decay time and the duration show a similar trend in our study. This is due to the different definition of decay time. They define the decay time as the e-folding time. Therefore, their duration is more prone to being affected by the rise time than the decay time obtained using our definition. In our definition, the rise time is an order of magnitude shorter than the decay time. The duration is determined largely by the decay time from Equation (3). The relation between CME occurrence and decay time can be understood by examining the relation between decay time and flaring loop length. It is known that the decay time depends linearly on the flaring loop length (in the global thermodynamic timescale) (Reale 2002):

$$\begin{eqnarray}&&{t}_{\mathrm{dec}}\sim \displaystyle \frac{120{L}_{9}}{\sqrt{{T}_{7}}},\end{eqnarray} \tag{ 10 }$$

where L₉ and T₇ are loop length and temperature in 10⁹ cm and 10⁷ K, respectively. When a CME occurs, the reconnected field lines, i.e., flare loops, must be large because the reconnection point becomes higher and higher as the flare proceeds. Consequently, the decay time tends to be large.

In limb events, CME occurrence shows a strong dependence on X-ray flux (R = 0.60 ± 0.15), which is consistent with the results in Yashiro et al. (2006). They showed that the value of the slope is R = 0.332 for X-ray flux without on-disk events. Their is different in 1σ uncertainty from our results. We suspect that this is mainly due to different event numbers. There are 16 X-class flares in our study, and 98 in Yashiro et al. (2006). Our results indicate that CMEs are associated with approximately all events whose X-ray flux is larger than 10^−3.9 W m⁻². This is also consistent with the results in Yashiro et al. (2006).

In Figure 9, fewer correlations exist between duration, rise time, CME width, and CME speed. The correlation of CME speed and duration measures 0.19, although Toriumi et al. (2017) reported a correlation coefficient of 0.50. It should be noted that they selected events at heliographic coordinates less than 45^◦ from the disk center, which makes the CME speed uncertain. Furthermore, there are more (57) CME events in our study than in theirs (32). Therefore our results, indicating a lower correlation between duration and CME speed, should be more reliable. In addition, there is more sophisticated way of deriving the CME speed due to Lugaz et al. (2010), who studied the azimuthal property using stereoscopic observations. The application of this method to a statistical study will be a future undertaking. Regarding the CME width, it appears that the correlation coefficients are larger compared to those of the CME speed. This results from the fact that large CME sizes cause larger flare loops and longer duration and decay time, as discussed above.

In Table 5, LXC–CME and LDE–CME show a relatively high association (ϕ > 0.2) in all events and limb events, while LXC–CME does not show high association in on-disk events. The lower association between LXCs and CMEs in on-disk events might be due to their small number. This result indicates that the relation between LDEs and CMEs is stronger than that between LXCs and CMEs. This might be also affected by the detection uncertainty of each phenomenon and structure. In other words, limb events have less uncertainty for CME detection, while it is difficult to associate CMEs with flares for on-disk events. Our association analysis for intersection sets in Table 6 indicates a high dependence for LXC–CME (ΔP/σ_Δ = 1.25) and CME–LDE (ΔP/σ_Δ = 1.74) for all events. This is consistent with the results of the occurrence rate in Figure 5. However, several relations show a dependence on the position of the flare event.

The KS and AD tests indicate that in a confidence interval of 99%, the CME is related to longer duration and decay time. Further, for the limb and all events, the X-ray flux is significantly related to the CMEs. Thus, the relation between CME association and X-ray flux strengthens in limb events, which reflects that CMEs are more easily detected at the solar limb than inside the solar disk.

In Figure 5, as in the case of SGM occurrence, the positive values of the slope R = 0.43 ± 0.22 for duration. Thus, duration tends to increase when a sigmoidal structure exists. A sigmoidal structure means highly twisted magnetic field lines and is usually formed in large active regions. Hence, sigmoidal structures tend to have long field lines and can lead to long-duration flares from the discussion above. From Figures 6 and 7, SGM occurrence as a function of X-ray flux appears to exhibit a positive slope, although their dependences do not show significant correlation. A sigmoidal structure implies a large free energy in the 3D magnetic field. Naturally, a large free energy can lead to large flares. Our results, however, do not show significant evidence that a larger free energy is needed to produce larger flares for on-disk events.

From the KS and AD tests, we did not find a relation between the flare parameters and SGM existence in the 99% confidence interval. However, there were weak relations between some parameters. For all events (black line in figures), SGM existence is related to a longer decay time (probability 0.045) and for the on-disk events (red line), it is related to a longer duration and longer decay time (the probabilities are 0.037 and 0.035). In contrast, for limb events (blue line), SGM existence is not related to the duration, rise time,or decay time due to its detection uncertainty regarding active regions far away from the disk center. Moreover, according to the CDF curves in the panel of the fourth row on the left side of Figure 11, if an SGM exists, the X-ray flux becomes large only for on-disk events.

In this study, we determined SGM existence by visual inspection aided by the detection of the bright region described in Section 3.1.3. We plan to develop a fully automatic SGM detection method by applying automatic thinning processing to the sigmoidal region. Although the thinning processing of the spread area is difficult in general, one promising method is OCCULT-2, developed by Aschwanden et al. (2013).

The appearance of X-ray SGMs has been considered as an indicator for eruptive flares (Canfield et al. 1999). In their results, 50% of the non-sigmoidal region produced eruptive events, while 84% of the sigmoidal region produced these events. Our association analysis in Tables 5 and 6, however, indicates that there is less dependence of SGMs on CMEs in all and the limb events. As discussed above, we suspect that in the limb events, sigmoidal structures are difficult to identify due to projection effects. In the on-disk events, on the other hand, the CME–SGM relation shows comparatively high dependence (ϕ > 0.2) in Table 5, while it is not significant in Table 6. Our results in Table 5 show only 39% of sigmoidal structures produce CMEs in the on-disk events. These results show that the sigmoidal structure does not necessarily lead to eruptive flares. The result in Table 5 also supports this statement. Focusing on the on-disk events, while 20.6% of the flare events have both SGMs and CMEs, 32.4% of the flare events have sigmoidal structure but do not produce CMEs. Another important result is that there are extremely few flare events with CMEs and without SGMs. This result shows that the sigmoidal structure is necessary (but not sufficient) factor for the occurrence of CMEs. The methods for identifying sigmoidal structures may result in a deviation of our results from those of Canfield et al. (1999). For example, smaller sigmoidal structures, shown in the bottom panel of Figure 2, were identified in our study. This might reduce the number of non-sigmoidal but eruptive events.

Tripathi et al. (2004) studied the relation between the post-eruptive arcade (PEA) and CMEs and found that almost all PEAs as observed by SOHO/EIT were associated with CMEs. They also studied the length of the PEA at each latitude. It would be an interesting future topic to study the relation between the length of SGMs and the PEA.