# Magnetic Flux and Magnetic Non-potentiality of Active Regions in Eruptive
and Confined Solar Flares

Ting Li11affiliation: CAS Key Laboratory of Solar Activity, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101,
China<EMAIL_ADDRESS>22affiliation: School of Astronomy and Space Science,
University of Chinese Academy of Sciences, Beijing 100049, China , Anqin
Chen33affiliation: Key Laboratory of Space Weather, National Center for Space
Weather, China Meteorological Administration, Beijing 100081, China , Yijun
Hou11affiliation: CAS Key Laboratory of Solar Activity, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China;
<EMAIL_ADDRESS>22affiliation: School of Astronomy and Space Science,
University of Chinese Academy of Sciences, Beijing 100049, China , Astrid M.
Veronig44affiliation: Institute of Physics & Kanzelhöhe Observatory for Solar
and Environmental Research, University of Graz, A-8010 Graz, Austria , Shuhong
Yang11affiliation: CAS Key Laboratory of Solar Activity, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China;
<EMAIL_ADDRESS>22affiliation: School of Astronomy and Space Science,
University of Chinese Academy of Sciences, Beijing 100049, China & Jun
Zhang55affiliation: School of Physics and Materials Science, Anhui University,
Hefei 230601, China

###### Abstract

With the aim of understanding how the magnetic properties of active regions
(ARs) control the eruptive character of solar flares, we analyze 719 flares of
Geostationary Operational Environmental Satellite (GOES) class $\geq$C5.0
during 2010$-$2019\. We carry out the first statistical study that
investigates the flare-coronal mass ejections (CMEs) association rate as
function of the flare intensity and the AR characteristics that produces the
flare, in terms of its total unsigned magnetic flux ($\Phi$AR). Our results
show that the slope of the flare-CME association rate with flare intensity
reveals a steep monotonic decrease with $\Phi$AR. This means that flares of
the same GOES class but originating from an AR of larger $\Phi$AR, are much
more likely confined. Based on an AR flux as high as 1.0$\times$$10^{24}$ Mx
for solar-type stars, we estimate that the CME association rate in X100-class
“superflares” is no more than 50%. For a sample of 132 flares $\geq$M2.0
class, we measure three non-potential parameters including the length of steep
gradient polarity inversion line (LSGPIL), the total photospheric free
magnetic energy (Efree) and the area with large shear angle (AΨ). We find that
confined flares tend to have larger values of LSGPIL, Efree and AΨ compared to
eruptive flares. Each non-potential parameter shows a moderate positive
correlation with $\Phi$AR. Our results imply that $\Phi$AR is a decisive
quantity describing the eruptive character of a flare, as it provides a global
parameter relating to the strength of the background field confinement.

Sun: activity—Sun: coronal mass ejections (CMEs)—Sun: flares

##  1 Introduction

Solar flares and coronal mass ejections (CMEs) are the most catastrophic
phenomena in the present solar system, driven by a sudden release of magnetic
energy stored in the solar corona. Large solar flares are often, but not
always, associated with CMEs. We dub flares with a CME “eruptive” and flares
without a CME “confined” (Moore et al. 2001). There are two main factors that
are considered to determine whether a flare event is CME-eruptive or not. One
factor is the constraining effect of the background magnetic field overlying
the flaring region, i.e., the strength of magnetic field or its decay with
height (Wang & Zhang 2007; Yang et al. 2014; Thalmann et al. 2015; Baumgartner
et al. 2018). Li et al. (2020) analyzed 322 large flares and found that the
flare-CME association rate decreases with the increasing magnetic flux of the
active region (AR) that produces the flare, implying that large magnetic flux
generally tends to confine eruptions.

Another factor determining the eruptive character of solar flares is thought
to be related to magnetic complexity and non-potentiality of ARs (Sun et al.
2015; Liu et al. 2016; Jing et al. 2018; Duan et al. 2019), such as free
magnetic energy, relative helicity, magnetic twists, etc. Numerous statistical
studies have revealed that strong flares mostly occur in ARs with a complex
configuration and high non-potentiality of magnetic fields (Mayfield &
Lawrence 1985; Falconer et al. 2002; Chen & Wang 2012, 2020; Su et al. 2014).
However, there are only a few statistical studies focusing on magnetic non-
potential measures in confined versus eruptive flares (Nindos & Andrews 2004;
Cui et al. 2018; Vasantharaju et al. 2018). Nindos & Andrews (2004) and Gupta
et al. (2021) found that in a statistical sense the pre-flare coronal magnetic
helicity of ARs producing confined large flares is smaller than that of ARs
producing eruptive large flares. Bobra & Ilonidis (2016) used machine-learning
algorithms to predict CME productivity and found that the “intensive”
parameters (those not scaling with the AR size) distinguish between eruptive
and confined flares. Until now, the key non-potential parameters of ARs
governing the eruptive character of solar flares are still unknown based on
statistical results. Moreover, the access to open flux (open to the
interplanetary space) is also thought to influence whether an X-class flare is
likely to be eruptive (DeRosa & Barnes 2018).

In this Letter, we carry out the first statistical study that investigates the
flare-CME association rate R as a function of the AR characteristics that
produces the flare, in terms of its total magnetic flux ($\Phi$AR). Our
findings reveal clear differences of R, with the slope of R as function of
flare intensity being a monotonically decreasing function of $\Phi$AR. This
result has important implications for the prediction of CMEs occurring in
association with large flares as well as for the solar-stellar connection,
where the solar flare-CME association rates are used to estimate stellar CME
occurrence frequencies. Moreover, we also find the distinct differences of
non-potential parameters characterizing ARs in eruptive and confined large
flares.

##  2 Observational Data and Analysis

###  2.1 Data and Event Selection

We check for the soft X-ray (SXR) flare catalog recorded by the Geostationary
Operational Environmental Satellite (GOES) system and select flare events
$\geq$C5.0 occurring within $45^{\circ}$ from the central meridian, from June
2010 to June 2019. A total of 719 events are selected, including 322 M-class
(Li et al. 2020) and 397 C-class flares. To determine whether a flare is
associated with a CME, we use the CME
catalog111https://cdaw.gsfc.nasa.gov/CME_list/ of the Solar and Heliospheric
Observatory (SOHO)/Large Angle and Spectrometric Coronagraph (LASCO; Brueckner
et al. 1995). The observations from the Atmospheric Imaging Assembly (AIA;
Lemen et al. 2012) on board the _Solar Dynamics Observatory_ (_SDO_ ; Pesnell
et al. 2012) and the twin _Solar Terrestrial Relations Observatory_ (_STEREO_
; Kaiser et al. 2008; Howard et al. 2008) are also used to help determining
the CME association (see detailed description in Li et al. 2020). Out of these
719 flares, 251 events are eruptive and 468 are confined (see Table 1 and the
database FlareC5.0222http://dx.doi.org/doi:10.12149/101067). For each event,
we calculated $\Phi$AR before the flare onset (within 30 min) by using the
available vector magnetograms from Space-Weather Helioseismic and Magnetic
Imager (HMI; Scherrer et al. 2012) AR Patches (SHARP; Bobra et al. 2014). Only
pixels that host a radial component of the magnetic field $|$$B_{r}$$|$$>$100
G are considered (Kazachenko et al. 2017).

###  2.2 Calculation of Magnetic Non-potential Parameters of ARs

For a subset of 132 flare events $\geq$M2.0 (86 eruptive and 46 confined), we
calculated three non-potential parameters before the flare onset (within 30
min) including the length of polarity-inversion lines (PILs) with steep
horizontal magnetic gradient (LSGPIL), the total photospheric free magnetic
energy (Efree) and the area with strong magnetic shear (AΨ). Magnetic PILs
mark the separation between positive and negative magnetic flux in the
photosphere of ARs. Properties of PILs in ARs have been found to be strongly
correlated to solar flare and CME occurrences (Falconer et al. 2002;
Vasantharaju et al. 2018; Kontogiannis et al. 2019; Wang et al. 2020). High-
gradient, strong-field PILs are proxies of (near-)photospheric compact
electrical currents and the occurrence of major flares was often associated
with the emergence of flux with high-gradient, strong-field PILs (Schrijver
2007; Toriumi & Wang 2019). According to the method of Chen & Wang (2012), we
measured the length of the PILs with a steep horizontal magnetic gradient
($\geq$ 300 G Mm-1) for each flare event based on the SHARP vector
magnetograms.

The energy that is released during a flare is generally believed to originate
from the free magnetic energy stored primarily in ARs, which is the amount of
magnetic energy in excess of the minimum energy attributed to the potential
field (Molodensky 1974). It was found that the higher the free magnetic energy
stored in an AR, the larger the size (magnitude) of upcoming flares (Jing et
al. 2010; Su et al. 2014). We use a proxy for the total photospheric free
magnetic energy (Wang et al. 1996; Chen & Wang 2012), which can be calculated
as

$E_{free}=\Sigma\rho_{free}dA,$ (1)

where $\rho_{free}$ is a proxy for the density of the free magnetic energy in
the photosphere, defined as

$\rho_{free}=\frac{|\textbf{B}_{o}-\textbf{B}_{p}|^{2}}{8\pi},$ (2)

where Bo and Bp are the observed and the potential magnetic fields,
respectively. Bp was derived from the observed Br component using the Fourier
transform method. $\rho_{free}$ and Efree are in units of [erg cm-3] and [erg
cm-1], respectively. We measured Efree by only considering the pixels with
$\rho_{free}$ $\geq$4.0$\times$104 erg cm-3.

Magnetic shear, defined as the angular difference between the measured field
and the calculated potential field, is another commonly used parameter in
describing the magnetic complexity and non-potentiality (Wang et al. 1994;
Zhang et al. 2007). We measured the area with shear angle $\geq$80∘, AΨ (Chen
& Wang 2012). The shear angle is given by

$\Psi=\arccos\frac{\textbf{B}_{o}\cdot\textbf{B}_{p}}{|B_{o}B_{p}|}.$ (3)

##  3 Statistical Results

###  3.1 Relations of Flare-CME Association Rate with Flare Intensity and
Magnetic Flux of ARs

We investigate the flare-CME association rate R as function of both the flare
class and the total flux of the source AR for 719 flares ($\geq$C5.0 class).
Figure 1(a) shows the scatter plot of $\Phi$AR versus flare peak SXR flux
(FSXR). Blue (red) circles are the eruptive (confined) flares. Obviously, when
$\Phi$AR is large enough ($>$1.0$\times$$10^{23}$ Mx; black dashed line), an
overwhelming majority (about 97%) of flares do not generate CMEs (57 of 59
flares are confined). Out of the flare events of
$\Phi$AR$>$1.0$\times$$10^{23}$ Mx (a total of 59 events), 32 flares occurred
in AR 12192, the huge AR known as flare-rich but CME-poor (Sun et al. 2015),
and the fraction is about 54%. If we remove the events in AR 12192, almost all
the events are confined (26 of 27 flares are confined).

The value of $\Phi$AR for the 719 flares ranges from 8.5$\times$$10^{21}$ Mx
to 2.3$\times$$10^{23}$ Mx, and we divide $\Phi$AR into five subintervals.
Figure 1(b) shows the relations of the association rate R with FSXR within the
five $\Phi$AR subintervals. For each subinterval, R clearly increases with
FSXR. Each straight line in Figure 1(b) shows the linear fit

$R=\alpha\log{F_{SXR}}+\beta,$ (4)

where R is in percentage and FSXR is in units of W m-2. For the smallest
$\Phi$AR subinterval ($\leq$ 2.0$\times$$10^{22}$ Mx), the slope $\alpha$ is
113.8$\pm$13.1 and R reaches 100% when the flare is $>$M1.3 class (red
straight line). For the subinterval of
2.0$<$$\Phi$AR$\leq$3.5$\times$$10^{22}$ Mx, the slope $\alpha$ decreases to
82.0$\pm$10.6 (green straight line). It can be seen that in ARs with a small
$\Phi$AR, about 50% C5.0-class flares have associated CMEs. For the moderate
$\Phi$AR subintervals (blue and orange lines), the slopes $\alpha$ are
48.2$\pm$5.4 and 38.4$\pm$6.3, respectively. The Spearman rank order
correlation coefficients rs are 0.96 and 0.94, respectively. In ARs with the
largest $\Phi$AR ($>$ 9.0$\times$$10^{22}$ Mx), R decreases significantly when
compared to subintervals characterized by smaller $\Phi$AR (black straight
line; also see Table 1). The relation between R and FSXR in the largest
$\Phi$AR subinterval is

$R=(22.9\pm 3.8)\log{F_{SXR}}+(125.7\pm 17.9).$ (5)

Based on Eq. 5, only 20% of all M-class flares originating from the largest
ARs have associated CMEs and the rate R reaches about 40% for flares $>$X2.
Almost all of C5.0-class flares are confined due to the strong constraining
fields in the largest ARs.

Figure 1(c) shows the relation of the slope $\alpha$ with $\Phi$AR. $\Phi$AR
is here defined as the mean of the individual log values in each $\Phi$AR
subinterval. The plot shows that the slope $\alpha$ decreases monotonically
with increasing $\Phi$AR. We assume ARs in solar-type stars of $\Phi$AR $\sim$
1.0$\times$$10^{24}$ Mx (Maehara et al. 2012; Shibata et al. 2013). For there
are only five known slope values, it is difficult to fit the plot and make an
extrapolation. We use the slope in the largest solar ARs, i.e., 22.9, minus
the average error estimate of five known slope values ($\sim$7.8,
corresponding to the average error of five diamonds), that is 22.9 minus 7.8
equals 15.1 as the slope $\alpha$ for stellar ARs of $\Phi$AR $\sim$
1.0$\times$$10^{24}$ Mx. We estimate that the slope $\alpha$ might be no more
than 15.1$\pm$7.8 (red circle). If C5.0-class flares are all confined on
solar-type stars (similar to the subinterval of $\Phi$AR $>$
9.0$\times$$10^{22}$ Mx), we can extrapolate the flare-CME association rate
for solar-type stars is given as

$R=(15.1\pm 7.8)\log{F_{SXR}}+80.0.$ (6)

Thus, for X100-class superflares in solar-type stars, the estimated
association rate R is no more than 50%.

###  3.2 Magnetic Non-potentiality of ARs in Eruptive and Confined Flares

We calculate three non-potential parameters for 132 flares $\geq$M2.0. Figure
2 shows an example of an eruptive X2.2-class flare occurring on 2011 February
15 in AR NOAA 11158. The SGPIL is located between two flare ribbons, and the
distributions of the photospheric free energy density $\rho_{free}$ and shear
angle $\Psi$ show similar patterns. In Figure 3, we display the scatter plots
and histograms of the three non-potential measures for the whole set of 132
flare events. It can be seen that the distributions of LSGPIL show evident
differences between confined and eruptive cases (Figures 3(a)-(b)). For
LSGPIL$<$22 Mm (black dash-dotted line in Figure 3(a)), an overwhelming
majority (about 90%) of flares are eruptive. The log-mean value of LSGPIL for
confined flares is 40.2 Mm (red dotted line in panel (b)), much larger than
that for eruptive events (20.8 Mm, blue dotted line in panel (b)). The
distributions of Efree are similar to those of LSGPIL. For Efree
$<$1.5$\times$$10^{23}$ erg cm-1 (black dash-dotted line in Figure 3(c)), 48
in 58 flares are eruptive. For AΨ$<$60 Mm2, about 79% (30 in 38) flares are
eruptive (Figure 3(e)). The log-mean values of Efree and AΨ for confined
flares are much larger than those for eruptive events (Figures 3(d) and (f)).

In Figure 4, we investigate the relationship between the three non-potential
parameters and $\Phi$AR in Figure 4. It can be seen that each two non-
potential parameters have strong correlations at rs $\sim$ 0.69$-$0.85
(Figures 4(a)-(c)). The high correlation coefficients between LSGPIL, Efree
and AΨ imply that ARs with long SGPIL tend to store more free magnetic energy
and have strong shearing. The scatter plot of LSGPIL with $\Phi$AR illustrates
that they have a moderate correlation at rs $\sim$ 0.4 (Figure 4(d)). About
89% of the confined flares occur in ARs with $\Phi$AR $>$ 3.5$\times$$10^{22}$
Mx and SGPIL longer than 22 Mm. Moderate correlations were also obtained for
Efree versus $\Phi$AR and AΨ versus $\Phi$AR (Figures 4(e)-4(f)). Their
moderate correlations with $\Phi$AR indicate that there is some trend that the
more magnetic flux of an AR, the higher non-potentiality of the AR.

##  4 Summary and Discussion

In this study, we have examined 719 flares $\geq$C5.0-class that were observed
on-disk from June 2010 to June 2019. We investigate for the first time the
flare-CME association rate R as function of both the flare class FSXR and the
total flux $\Phi$AR of the AR that produces the flare. We find that, for each
$\Phi$AR subinterval, R clearly increases with FSXR, i.e., larger flares are
more likely associated with a CME. This result is in agreement with previous
findings studying CME-flare association rates (Andrews 2003; Yashiro et al.
2006), who reported overall CME associations of about 60% for M-class flares
and 90% for X-class flares. However, what is new and particularly important in
our study is that we considered not only the relation to the intensity of the
flare but also to the characteristics of the AR in terms of its total magnetic
flux. Our results show that the slope of the flare-CME association rate
depends on the total flux of the AR that produces the flare, and reveals a
steep monotonic decrease with $\Phi$AR (Figure 1(c)). This means that flares
of the same GOES class but originating from an AR of larger $\Phi$AR, are much
more likely confined. Within the smallest $\Phi$AR subinterval ($\leq$
2.0$\times$$10^{22}$ Mx), all flares $>$M1.3 are all eruptive. On the other
end of the distribution for the largest $\Phi$AR subinterval ($>$
9.0$\times$$10^{22}$ Mx), only about 20% M-class flares have associated CMEs
and the association rate R reaches about 40% for those flares $>$X2.

Our results imply that $\Phi$AR is a key factor determining the eruptive
character of solar flares, consistent with our previous studies (Li et al.
2020). $\Phi$AR can be considered to be both a measure of the total flux that
is in principle available for flaring as well as being a measure of the
background field confinement overlying the flaring region. Our findings imply
that the latter is the more important factor here. Large $\Phi$AR means a
strong confinement and thus the flare-CME association rate is relatively low
compared to small $\Phi$AR. Moreover, based on solar observations, we can
speculate the associate rate R on solar-type stars by assuming $\Phi$AR of
1.0$\times$$10^{24}$ Mx (Maehara et al. 2012; Shibata et al. 2013). For
X100-class “superflares” on solar-type stars, no more than 50% flares can
generate stellar CMEs. This may provide an explanation why the detection of
stellar CMEs is rare (e.g. Leitzinger et al. 2014, Vida et al. 2019, Argiroffi
et al. 2019, Moschou et al. 2019, Veronig et al. 2021), while extrapolating
current flare-CME relations to solar-type stars leads to unphysically high CME
rates (Drake et al. 2013; Odert et al. 2017). Our findings provide an
important contribution to revise the flare-CME association rates for solar-
type stars, by including the distinct differences in these relations in
dependence of the AR magnetic flux.

Using HMI vector magnetograms, we also have studied the relation between the
degree of magnetic non-potentiality and the eruptive character of 132 flares
$\geq$M2.0, finding distinct differences between eruptive and confined flares
for all three non-potentiality parameters derived (Figures 3-4). LSGPIL, Efree
and AΨ all give smaller log-mean values for eruptive than non-eruptive flares.
Each non-potential parameter shows a moderate correlation with $\Phi$AR. Our
study shows that the three “extensive” parameters (those scaling with the AR
size) can only discriminate the flares with small non-potential parameters. As
seen in Figure 3, 42 out of 47 events with LSGPIL$<$22 Mm are eruptive.
However, for the remanent 85 flares with LSGPIL$>$22 Mm, it is difficult to
tell whether a flare is accompanied by a CME or not. The appearance of other
two parameters is similar to that of LSGPIL. Only a fraction of flare events
can be discriminated between confined and eruptive events. Thus we speculate
that if we use these “extensive” parameters to predict the CME productivity,
the True Skill Score (TSS) value is not high. This is consistent with the
study of Bobra & Ilonidis (2016), who shows that TSS for predicting the CME
productivity is low based on “extensive” parameters. They show that
“intensive” parameters can predict the CME productivity. In our study, we did
not calculate “intensive” parameters. In our future work, we will consider
“intensive” parameters in confined and eruptive large flares based on our
database.

In the statistical results of Cui et al. (2018), confined flares have larger
values of $\Phi$AR and the gradient-weighted area of the polarity inversion
region than eruptive flares, which are in agreement with our results. It was
found that large confined flares tend to occur in large ARs (Toriumi et al.
2017; Li et al. 2020), and thus LSGPIL, Efree and AΨ are larger for confined
than eruptive flares due to their positive correlations with $\Phi$AR. Our
results imply that $\Phi$AR is a key factor in determining whether a flare is
eruptive or confined, as it provides a global parameter relating to the
strength of the background field confinement.

This work is supported by the National Key R&D Program of China
(2019YFA0405000), the B-type Strategic Priority Program of the Chinese Academy
of Sciences (XDB41000000), the National Natural Science Foundations of China
(11773039, 11903050, 12073001, 11790304, 11873059 and 11790300), Key Programs
of the Chinese Academy of Sciences (QYZDJ-SSW-SLH050), the Youth Innovation
Promotion Association of CAS (2014043 and 2017078), Yunnan Academician
Workstation of Wang Jingxiu (No. 202005AF150025) and NAOC Nebula Talents
Program. A. Q. Chen is supported by the Strategic Priority Program on Space
Science, Chinese Academy of Sciences, Grant No. XDA15350203. Astrid M. Veronig
acknowledges the support by the Austrian Science Fund (FWF): P27292-N20. _SDO_
is a mission of NASA’s Living With a Star Program.

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Table 1: Number of Eruptive and Confined Flares in all ARs and ARs with largest $\Phi$AR Class | Eruptive11For all ARs | Confined11For all ARs | R22Flare-CME association rate | Eruptive33For ARs with largest $\Phi$AR $>$ 9.0$\times$$10^{22}$ Mx | Confined33For ARs with largest $\Phi$AR $>$ 9.0$\times$$10^{22}$ Mx | R22Flare-CME association rate
---|---|---|---|---|---|---
C | 82 | 315 | 21% | 1 | 40 | 2%
M | 154 | 147 | 51% | 6 | 30 | 17%
X | 15 | 6 | 71% | 1 | 4 | 20%
Total | 251 | 468 | 35% | 8 | 74 | 10%
Figure 1: Relations of flare-CME association rate (R) with flare peak soft
X-ray flux (FSXR) and total unsigned magnetic flux of ARs ($\Phi$AR). (a)
Scatter plots of $\Phi$AR vs. FSXR. Blue (red) circles are the eruptive
(confined) flares ($\geq$ C5.0-class). Black dashed line corresponds to
$\Phi$AR of 1.0$\times$$10^{23}$ Mx. (b) Association rate R as a function of
FSXR separately for five different subintervals of $\Phi$AR. The colored
straight lines show the results of linear fitting, and slopes $\alpha$ and
Spearman rank order correlation coefficients rs are shown at the bottom right.
(c) Plot of slope $\alpha$ vs. $\Phi$AR (plotted at the average of the log
values in each subinterval). Colored diamonds denote the slopes $\alpha$ in
five different $\Phi$AR subintervals. The red circle is the estimated value of
slope $\alpha$ (about 15.1) for solar-type stars by assuming $\Phi$AR of
1.0$\times$$10^{24}$ Mx (Maehara et al. 2012; Shibata et al. 2013).

Figure 2: Tracing the magnetic polarity inversion line with the steep
horizontal magnetic gradient (SGPIL), photospheric free magnetic energy
density $\rho_{free}$ and the magnetic shear angle $\Psi$ in AR 11158 on 2011
February 15. (a) SDO/AIA 1600 Å image showing the flare ribbon brightenings of
an eruptive X2.2-class flare. (b) SDO/HMI vector magnetogram with horizontal
magnetic field vectors (red and blue arrows) overplotted on the Br map. SGPIL
($\geq$ 300 G Mm-1) is shown as green lines. Yellow contours are the ribbon
brightenings. (c) $\rho_{free}$ distribution with contours of the Br component
and flare ribbon brightenings. The white (green) contours represent the
positive (negative) polarity. (d) $\Psi$ distribution with contours of the Br
component and ribbon brightenings.

Figure 3: Scatter plots and histograms of three different non-potentiality
measures for eruptive (blue) and confined (red) flares ($\geq$ M2.0-class).
Top: scatter plot of FSXR vs. SGPIL length (LSGPIL) and the histogram of
LSGPIL. Black dashed-dotted line in panel (a) refers to LSGPIL of 22 Mm.
Dotted vertical lines in panel (b) indicate the means of the log values.
Middle: scatter plot of FSXR vs. the total free magnetic energy Efree and the
histogram of Efree. Black dashed-dotted line in panel (c) refers to Efree of
1.5$\times$$10^{23}$ erg cm-1. Bottom: scatter plot of FSXR vs. the area with
strong magnetic shear AΨ and the histogram of AΨ. Black dashed-dotted line in
panel (e) refers to AΨ of 60 Mm2.

Figure 4:  Relations between LSGPIL, Efree, AΨ and $\Phi$AR for eruptive
(blue) and confined (red) flares. The black solid lines show the results of a
linear fitting, and slopes $\alpha$ and Spearman rank order correlation
coefficients rs are shown at the bottom right of each panel. Orange dash-
dotted lines in panels (d)-(f) correspond to $\Phi$AR of 3.5$\times$$10^{22}$
Mx. Orange dashed lines respectively denote LSGPIL of 22 Mm, Efree of
1.5$\times$$10^{23}$ erg cm-1 and AΨ of 60 Mm2.