Magnetic Flux of Active Regions Determining the Eruptive Character of Large Solar Flares

The SDO satellite has already provided a rich database since its launch in 2010 February. Its observation period lasts about 10 yr and almost spans the entirety of solar cycle 24. Thus, it provides a good opportunity to carry out a statistical analysis of the flare-CME mechanism based on SDO observations. First, we examined the database RibbonDB presented by Kazachenko et al. (2017) and selected all 302 flare events larger than M1.0 that occurred within 45° from the central meridian, from 2010 June until 2016 April. To extend the time period, we checked for the GOES soft X-ray (SXR) flare catalog to search for flare events from 2016 May to 2019 June and found 20 flares of GOES class M1.0 and greater. A total of 322 flare events are involved in our database over a 9 year period. Second, for each flare, its CME association was determined by checking the CME catalog⁷ (Gopalswamy et al. 2009) of the Solar and Heliospheric Observatory (SOHO)/Large Angle and Spectrometric Coronagraph (LASCO). We regarded a flare as eruptive if the CME onset time was within 90 minutes of the flare start time and the position angle of the CME agreed with the quadrant on the Sun where the flare occurred. Moreover, we also inspected the observations of the twin Solar Terrestrial Relations Observatory (STEREO; Howard et al. 2008; Kaiser et al. 2008) to check from a different viewing angle if there is an associated CME. For the events where it was difficult to determine the classification, e.g., there are two flares within a short time or the CME is too weak to be identified, we checked the EUV observations from the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) on board the SDO and identified the coronal EUV wave manually. If a global coronal EUV wave was associated with the flare, the flare was classified as eruptive. Out of these 322 flares, 170 (∼53%) events were eruptive (155 M- and 15 X-) and 152 (∼47%) were confined (146 M- and 6 X-).

We investigated the relations between the AR parameters (unsigned AR magnetic flux ${{\rm{\Phi }}}_{\mathrm{AR}}$ and AR area) and the eruptive character of large solar flares. The AR parameters in the RibbonDB catalog (Kazachenko et al. 2017) are calculated based on the full-disk Helioseismic and Magnetic Imager (HMI; Scherrer et al. 2012) vector magnetogram data series (hmi.B_720s) before the flare onset time. To avoid noisy magnetic fields, only pixels that host a normal component of the magnetic field $| {B}_{n}| \gt 100$ G are considered. For the flare events that are not included in the RibbonDB catalog, we use the available vector magnetograms (hmi.sharp_cea_720s) from Space-Weather HMI AR Patches (Bobra et al. 2014) before the flare onset. The magnetograms were re-mapped using a cylindrical equal area projection with a pixel size of ∼05 and presented as (B_r, ${B}_{\theta }$, B${}_{\phi }$) in heliocentric spherical coordinates corresponding to (B_z, −B_y, B_x) in heliographic coordinates (Sun 2013). Similarly, to calculate the AR magnetic flux and AR area, we consider all pixels of $| {B}_{r}| \gt 100$ G. Moreover, the RibbonDB catalog (Kazachenko et al. 2017) also includes the parameters of flare ribbons such as the flare ribbon reconnection flux ${{\rm{\Phi }}}_{\mathrm{ribbon}}$, the cumulative flare ribbon area S_ribbon, the ratio of the AR magnetic flux involved in the flare reconnection R_flux (${{\rm{\Phi }}}_{\mathrm{ribbon}}$/${{\rm{\Phi }}}_{\mathrm{AR}}$), and the area ratio R_S (S_ribbon/S_AR). In this work, we also use these parameters of 302 flare events larger than M1.0 and investigate their distributions and correlations in eruptive and confined flares.

The role of the background coronal fields in confined and eruptive flares was estimated by calculating the decay index n above the ARs. In order to extrapolate the 3D magnetic field in the entire AR volume, we use the Fourier transformation method (Alissandrakis 1981) to extrapolate the potential field. The method yields the local potential field with a resolution around 0.72 Mm, the same as the resolution of the boundary condition. The boundary condition is the normal component of the photospheric magnetic field from Space-Weather HMI AR Patches (Bobra et al. 2014) observed prior to the flare start. From the extrapolated field, the mean value of the horizontal magnetic field, $\langle {B}_{\mathrm{hor}}\rangle$, as a function of height is obtained along the main polarity inversion line (PIL) and an average decay index, $\langle n\rangle$, is then derived. Here, the main PIL was identified as a zero Gauss contour in the bottom vertical magnetic field (B_r) image from the extrapolated potential fields (Bokenkamp 2007; Vasantharaju et al. 2018). To analyze the structure and dynamics of typical flare examples, we used the E(UV) observations from the AIA, with a resolution of ∼06 per pixel and a cadence of 12(24) s. Five channels of AIA 1600, 304, 171, 94, and 131 Å were mainly applied to analyze the appearances of the flares. The full-disk line-of-sight (LOS) magnetic field data from the HMI are also used to present the ARs producing typical flare examples.

Figure 1(a) shows a scatter plot of the flare peak X-ray flux versus ${{\rm{\Phi }}}_{\mathrm{AR}}$. Blue (red) circles are the eruptive (confined) flares. When ${{\rm{\Phi }}}_{\mathrm{AR}}$ is small enough, the flares tend to be eruptive (Area A in Figure 1(a)). About 92% (36 of 39) of events occurring in ARs with ${{\rm{\Phi }}}_{\mathrm{AR}}$ smaller than $3.0\times {10}^{22}$ Mx are eruptive. An overwhelming majority of flares that are hosted by ARs with a large magnetic flux do not generate CMEs (Area C in Figure 1(a)). The proportion of confined flares of GOES class $\geqslant$ M1 is ∼93% (26 of 28), corresponding to the AR with ${{\rm{\Phi }}}_{\mathrm{AR}}$ larger than $1.0\times {10}^{23}$ Mx. We examined two special eruptive events (M4.0-class flare on 2014 October 24 in Figure 10 and X1.8-class event on 2014 December 20) in Area C of Figure 1(a) and found that they either were located at the edge of the AR or caused a sympathetic eruption of a nearby quiescent filament. If the AR has a moderate magnetic flux (larger than $3.0\times {10}^{22}$ Mx and smaller than $1.0\times {10}^{23}$ Mx), the likelihood of eruptive and confined events appears to be almost equal (132 eruptive flares and 126 confined events of 258 in Area B of Figure 1(a)). The scatter plot of the flare peak X-ray flux versus total AR area shows a similar trend (Figure 1(b)). All flares in ARs with an area smaller than $5.0\times {10}^{19}\,{\mathrm{cm}}^{2}$ are eruptive (Area A in Figure 1(b)) and all flares in ARs larger than $3.0\times {10}^{20}\,{\mathrm{cm}}^{2}$ are confined (Area C in Figure 1(b)).

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Figures 1(c)–(d) display the histograms for confined and eruptive events. There are significant differences in distributions of AR magnetic flux and AR area between the confined and eruptive cases. The confined events have larger AR magnetic flux and AR area. The averages of the log values of ${{\rm{\Phi }}}_{\mathrm{AR}}$ (indicated by vertical dotted lines) are $6.3\times {10}^{22}$ Mx and $4.4\times {10}^{22}$ Mx for confined and eruptive cases, respectively. The log-mean values of AR area for the confined and eruptive events are $1.5\times {10}^{20}\,{\mathrm{cm}}^{2}$ and $1.2\times {10}^{20}\,{\mathrm{cm}}^{2}$, respectively. Based on the the number distributions of AR magnetic flux and AR area between the confined and eruptive cases, we display the relations of the proportions of eruptive flares P_E (P_E = N_E/(N_E + N_C), N_E, and N_C are numbers of eruptive and confined events, respectively) with ${{\rm{\Phi }}}_{\mathrm{AR}}$ and AR area in Figures 1(e)–(f). It can be seen that P_E decreases with ${{\rm{\Phi }}}_{\mathrm{AR}}$. The proportion P_E has a strong anticorrelation with ${{\rm{\Phi }}}_{\mathrm{AR}}$ at the Spearman rank-order correlation coefficient r_s of –0.97. The Spearman rank correlation provides a measure of the monotonic relationship between two variables. The linear fitting to the scatter plot provides the relation of

$$\begin{eqnarray}&&{P}_{E}=(-0.75\pm 0.06)\mathrm{log}| {{\rm{\Phi }}}_{\mathrm{AR}}| +(17.53\pm 1.27),\end{eqnarray} \tag{ 1 }$$

where ${{\rm{\Phi }}}_{\mathrm{AR}}$ is in units of [Mx].

Similarly, the proportion P_E shows a strong anticorrelation with AR area (r_s = −0.95), and provides the relation of

$$\begin{eqnarray}&&{P}_{E}=(-0.76\,\pm \,0.09)\mathrm{log}{S}_{\mathrm{AR}}+(15.70\pm 1.86),\end{eqnarray} \tag{ 2 }$$

where S_AR is in units of [cm²].

We investigate the role of the background coronal fields by calculating the decay index n above the ARs. Figure 2 shows four examples including one eruptive flare in Area A of Figure 1(a) and three confined flares in Area C of Figure 1(a). The black asterisks denote the $\langle {B}_{\mathrm{hor}}\rangle$ versus h profiles and blue diamonds are the $\langle n\rangle$ versus h profiles. The error bars mark the corresponding standard deviation. The critical height for torus instability h_crit corresponds to the height where $\langle n\rangle$ reaches a value of 1.5 (Kliem & Török 2006; Fan & Gibson 2007). Clearly, the h_crit value of AR 11305 (∼17 Mm) with a small magnetic flux is lower than those of three other ARs (36–60 Mm) with larger magnetic fluxes, which means that the constraining field above AR 11305 producing an eruptive flare decays more rapidly than other ARs with confined flares, and therefore a perturbation in the lower corona may cause the CME seed to erupt out more easily (Wang & Zhang 2007; Liu et al. 2018).

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Following the procedure described above, we estimated the critical decay index heights h_crit for 82 events (including all the events in Areas A and C of Figure 1(a) and 15 flares in Area B of Figure 1(a)). Figure 3(a) shows the scatter plot of h_crit versus ${{\rm{\Phi }}}_{\mathrm{AR}}$. It can be seen that h_crit increases with ${{\rm{\Phi }}}_{\mathrm{AR}}$. This indicates that ARs with a larger magnetic flux tend to have a stronger constraining field. The critical decay index height has a strong correlation with AR magnetic flux at the Spearman rank-order correlation coefficient r_s of 0.86. The linear fitting to the scatter plot provides the relation

$$\begin{eqnarray}&&{h}_{\mathrm{crit}}=(38.31\pm 2.37)\mathrm{log}| {{\rm{\Phi }}}_{\mathrm{AR}}| +(-834.53\pm 53.92),\end{eqnarray} \tag{ 3 }$$

where h_crit and ${{\rm{\Phi }}}_{\mathrm{AR}}$ are in units of [Mm] and [Mx], respectively.

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Using this equation, an ${{\rm{\Phi }}}_{\mathrm{AR}}$ value of $3.0\times {10}^{22}$ Mx yields a h_crit of ∼27 Mm (left vertical and bottom horizontal lines in Figure 3(a)) and $1.0\times {10}^{23}$ Mx corresponds to a h_crit of about 47 Mm (right vertical and top horizontal lines in Figure 3(a)). In Figure 3(b), we plot the flare peak X-ray flux versus h_crit. All flares $\geqslant$ M1 (28 events) with a h_crit value smaller than 27 Mm are eruptive (Area A in Figure 3(b)), and about 95% (20 of 21) of events with h_crit larger than 47 Mm are confined (Area C in Figure 3(b)). The results of Figures 1 and 3 suggest that stronger strapping fields over the ARs with a larger magnetic flux play the main role in confining the eruption.

Figure 4 shows scatter plots of flare ribbon reconnection flux and cumulative flare ribbon area versus flare peak X-ray flux. We find that flare reconnection flux ${{\rm{\Phi }}}_{\mathrm{ribbon}}$ correlates with flare peak X-ray flux F_SXR at a moderate rank-order correlation coefficient r_s of 0.51 for all the flares (Figure 4(a)). By fitting the data, we obtained their empirical relationship

$$\begin{eqnarray}&&\mathrm{log}| {{\rm{\Phi }}}_{\mathrm{ribbon}}| =(0.51\pm 0.04)\mathrm{log}{F}_{\mathrm{SXR}}+(24.02\pm 0.17),\end{eqnarray} \tag{ 4 }$$

where ${{\rm{\Phi }}}_{\mathrm{ribbon}}$ and F_SXR are in units of [Mx] and [W m⁻²], respectively.

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The rank-order correlation coefficient r_s for the subset of eruptive flares (r_s = 0.58) is larger than r_s for the confined flares (r_s = 0.42). The corresponding fitting parameters for the subsets of confined and eruptive flares show no significant differences.

The ribbon area and flare peak X-ray flux (Figure 4(b)) also show a moderate correlation with a rank-order correlation coefficient r_s of 0.58 and their relation is

$$\begin{eqnarray}&&\mathrm{log}{S}_{\mathrm{ribbon}}=(0.49\pm 0.03)\mathrm{log}{F}_{\mathrm{SXR}}+(21.12\pm 0.14),\end{eqnarray} \tag{ 5 }$$

where S_ribbon and F_SXR are in units of [cm²] and [W m⁻²], respectively.

Similarly, there are no significant differences in the fitting parameters when considering confined and eruptive flares separately.

In Figures 5(a)–(b), we display the histograms of flare reconnection flux ratio R_flux (${{\rm{\Phi }}}_{\mathrm{ribbon}}$/${{\rm{\Phi }}}_{\mathrm{AR}}$) and ribbon area ratio R_S (S_ribbon/S_AR) for confined and eruptive events. It can be seen that the distributions of both R_flux and R_S show evident differences, with R_flux and R_S for confined events smaller than those for eruptive flares. R_flux ranges between 1% and 41% for eruptive flares and ranges between 1% and 21% for confined events. The proportion of eruptive flares reaches ∼89% (39 of 44), corresponding to a flux ratio R_flux higher than 15%. The log averages of flux ratio R_flux are 6.3% and 9.5% for confined and eruptive events, respectively. Similarly, the confined flares have a smaller area ratio R_S (1%–18%) than eruptive events (1%–30%). The log-mean values of area ratio R_S are 4.0% and 6.1% for confined and eruptive cases, respectively.

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We investigate the dynamic evolution of confined flares originating from ARs with a large magnetic flux ($\geqslant 1.0\times {10}^{23}$ Mx), including 26 events from 5 different ARs (ARs 11339, 11520, 11967, 12192, and 12242). After examining the AIA observations of these confined flares, we find that they have common characteristics: slipping reconnection, strong shear, and a stable filament. Here, two confined events from ARs 11520 and 12242 are taken as examples to analyze the flare dynamics and magnetic topological structures in detail.

On 2012 July 10, a confined M1.7 flare occurred in the sigmoidal region of AR 11520 with ${{\rm{\Phi }}}_{\mathrm{AR}}$ of $1.24\times {10}^{23}$ Mx. The flare was initiated at 04:58 UT and the GOES SXR flux peaked at 05:14 UT. Before the flare started, a filament was located along the PIL at the flaring region (left panel in Figure 6(a)). It did not show any rise process during the flare and was stably present after the flare (right panel in Figure 6(a)). A comparison of the 304 Å image with the HMI LOS magnetogram showed that the flare consisted of two positive-polarity ribbons PR1–PR2 and two negative-polarity ribbons NR1–NR2 (middle panel in Figure 6(a)). Ribbons PR1 and NR1 were located at two ends of the filament and PR2 and NR2 resided at both sides of the main body (axis) of the filament. High-temperature flare loops at 94 Å displayed notable dynamic evolution (Figure 6(b)). To display the fine structures of the EUV images, the 94 Å filter channel data have been processed using the multiscale Gaussian normalization (MGN) method (Morgan & Druckmüller 2014). At the start of the flare, two sets of loop bundles L1 (connecting PR2–NR1) and L2 (connecting PR1–NR2) overlying the stable filament became bright. Starting from about 05:10 UT, a group of brightened short loops L3 was formed connecting PR2 and NR2; meanwhile another longer loop bundle L4, linking PR1–NR1, can be discerned. During the flare, the north parts of loop bundles L2 and L3 exhibited apparent bidirectional slipping motions along ribbon NR2. Finally, strongly sheared post-flare loops (PFLs) appeared above the noneruptive filament. Based on the dynamic evolution of flare loops and their relations with flare ribbons, we suggest that slipping magnetic reconnection (Priest & Démoulin 1995; Aulanier et al. 2006) between loop bundles L1 and L2 occurred and led to the formation of L3 and L4. We estimated the inclination angles θ of PFLs with respect to the PIL, corresponding to the angle between the tangents of the PFL and PIL at their intersection (left panel in Figure 6(c)). The complementary angle of θ has been referred to as the shear angle in previous studies (Su et al. 2007; Aulanier et al. 2012). We derive small θ values, ranging from 10° to 30°, indicative of a high nonpotentiality in the form of a strong shear. Then more high-temperature PFLs gradually cooled down and formed PFLs overlying the stable filament at 171 Å (Figure 6(c)).

Using the photospheric vector magnetic field observed by SDO/HMI at 04:24 UT, we make a nonlinear force-free extrapolation by applying the optimization method (Wheatland et al. 2000; Wiegelmann 2004) and obtain the 3D coronal magnetic fields. There are two sets of sheared magnetic systems (MS1 and MS2 in Figure 7(a)) overlying a twisted flux rope (FR) prior to the flare onset. By comparing the AIA observations with the extrapolation results, we suggest that the two magnetic systems MS1 and MS2 approximately correspond to two sets of loop bundles L1 and L2 (Figure 6(b)) and the flux rope FR bears a good resemblance to the observed noneruptive filament (Figure 6(a)). Based on the extrapolated 3D coronal magnetic field, we calculated the squashing factor Q (Liu et al. 2016b), which defines the locations of the quasiseparatrix layers (QSLs; Démoulin et al. 1996; Titov et al. 2002). As seen from the distribution of Q (Figure 7(b)), the observed flare ribbons roughly match the locations of high Q values, implying that the magnetic reconnection involved in the flare probably occurs in regions of very strong magnetic connectivity gradients, i.e., QSLs.

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Figure 8 shows another confined M1.3-class flare on 19 December 2014 in AR 12242, which has a large ${{\rm{\Phi }}}_{\mathrm{AR}}$ of $\sim 1.11\times {10}^{23}$ Mx. The flare was initiated at 09:31 UT and peaked at 09:44 UT. It occurred in the northwest of AR 12242 and a filament was present along the PIL at the flaring region (Figure 8(a)). During the flare process, the main body of the filament did not show any rise phase except for the mild activation at its south part. After the flare, the filament remained stabilized, similar to the filament in the confined M1.7-class flare in AR 11520 (Figure 6(a)). Two quasi-parallel flare ribbons were distinguished from AIA 304 Å images, including ribbon PR in the leading positive-polarity sunspot and the negative-polarity ribbon NR. As shown from the high-temperature 131 Å observations, the flare loops were composed of two sets of magnetic systems S1 and S2 overlying the noneruptive filament, displaying a clear "X-shape" structure (Figure 8(b)). The south ends of systems S1 and S2 were anchored in ribbon PR and their north ends were in ribbon NR. During the flare evolution, S1 and S2 exhibited apparent slipping motions along ribbons PR and NR, and more flare loops successively appeared. In the gradual phase of the flare, low-temperature PFLs were formed, as best observed in the AIA 171 Å channel (Figure 8(c)). Similarly, the early formed PFLs also displayed an "X-shape" structure. We estimated the inclination angles θ of PFLs with respect to the PIL and found θ values in the range of 20°–28°. The small θ values imply that the PFLs are strongly sheared and have a higher nonpotentiality.

The apparent slipping motions of the fine structures within flare ribbons are further displayed in Figure 9. Ribbon PR was composed of numerous bright dot-like substructures, corresponding to the footpoints of high-temperature flare loops. These substructures exhibited apparent slipping motions in opposite directions (Figure 9(a)). We followed the trails of three different substructures within ribbon PR. From 09:33:10 UT, the bright knot "1" slipped toward the east with a displacement of about 3.8 Mm in 110 s (with a velocity of ∼30 km s⁻¹). Meanwhile, another bright knot "2" displayed a rapid slipping motion in the opposite direction at a velocity of ∼130 km s⁻¹. At 09:34:58 UT, the bright knot "3" at the middle part of PR underwent a fast slippage toward the northeast. In order to analyze the slipping motions of the substructures, we create a stack plot (Figure 9(c)) along slice "C–D" in the AIA 131 Å images (blue curve in Figure 9(a)). As seen from the stack plot, the slipping motions along ribbon PR were in both directions with speeds of 20–150 km s⁻¹. Figure 9(b) shows the stack plot of the other ribbon NR along slice "A–B" (green dashed–dotted curve in Figure 8(b)). Similarly, the slippage along ribbon NR was bidirectional, with apparent speeds of 20–30 km s⁻¹, smaller than those of ribbon PR.

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A large majority of flares (26 of 28 events) originating from ARs of ${{\rm{\Phi }}}_{\mathrm{AR}}\geqslant 1.0\times {10}^{23}$ Mx are confined; however, there are two special eruptive flares among the 28 events. One event is the M4.0 flare on 2014 October 24 and the other is the X1.8 event on 2014 December 20. The X1.8 flare caused a sympathetic eruption of a nearby quiescent filament and generated a CME. Figure 10 displays the appearance of the eruptive M4.0 flare on 2014 October 24. The flare was initiated at 07:37 UT and peaked at 07:48 UT. It was located in the southeast of AR 12192, far away from the main PIL of the AR. The flare was triggered by a blowout jet, as seen in AIA 304 and 131 Å images, and produced a CME at 08:12 UT observed by LASCO/C2. It was suggested that the eruptive flare on the southern border of the AR was close to neighboring open field regions (Thalmann et al. 2015) and thus the jet successfully escaped from the solar surface and formed a CME.

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About 92% of events occurring in ARs with ${{\rm{\Phi }}}_{\mathrm{AR}}$ smaller than $3.0\times {10}^{22}$ Mx are eruptive. Here, we present an eruptive M3.9 event on 2011 October 2 as an example (Figure 11). The flare originated from AR 11305 (N09°, W12°) with a smaller ${{\rm{\Phi }}}_{\mathrm{AR}}$ of ∼$1.67\times {10}^{22}$. It started at 00:37 UT and reached its peak at 00:50 UT. A high-temperature flux rope erupted toward the southwest in the 131 Å image. The angle of separation between SDO and STEREO B on 2011 October 2 was around 97°. An erupting CME bubble can be observed at the west limb in the STEREO B/EUVI 195 Å image. Starting from about 01:05 UT, an Earth-directed CME was observed by the COR1 coronagraph on board STEREO B.

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When ${{\rm{\Phi }}}_{\mathrm{AR}}$ is between $3.0\times {10}^{22}$ and $1.0\times {10}^{23}$ Mx, almost half of flares are confined. Here, we show two examples from the same AR 11429 on 2012 March 6. At 07:52 UT, an eruptive M1.0-class flare occurred in AR 11429 with ${{\rm{\Phi }}}_{\mathrm{AR}}$ of about $6.78\times {10}^{22}$ Mx measured before the flare onset (Figure 12). A reverse-S shaped filament was located along the PIL. During the flare, the middle part of the filament erupted and caused a CME. The AR was emerging persistently and the magnetic flux increased to ∼$7.96\times {10}^{22}$ Mx at 12:00 UT. Then another confined M2.1-class flare occurred at 12:23 UT and peaked at 12:41 UT (Figure 13). The filament in the AR did not erupt except for the activation at the north part. A flux rope was illuminated and started to rise at 12:26 UT as observed in 131 Å images. The rise lasted for about 10 minutes and ceased at 12:37 UT. Then the flux rope seemed to stay at a certain height and faded away gradually. The eruption of the flux rope failed and did not generate any CME.

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