On the Factors Determining the Eruptive Character of Solar Flares

Solar flares often occur in combination with coronal mass ejections (CMEs). It is believed, that both phenomena are closely related and different manifestations of the same underlying physical processes (Priest & Forbes 2002; Shibata & Magara 2011). Flares associated with a CME are usually referred to as eruptive events, while flares that lack associated ejections of coronal plasma are called confined or "CME-less" events (Svestka & Neidig 1986). However, we lack a deeper understanding of the physical mechanisms behind flares and CMEs, which knowingly condition our Space Weather on Earth (e.g., Gosling et al. 1991) To date, we suffer from moderate abilities to predict (i) when a solar flare will happen and (ii) whether or not it will develop an associated CME.

Wang & Zhang (2007) analyzed eight large flares (≥GOES class X1.0) and found that confined/eruptive flares tend to occur closer to/farther from the flux-weighted center of their host AR. They also analyzed the horizontal magnetic flux that penetrates a vertical plane, aligned with the flare-relevant photospheric polarity inversion line (PIL). They defined a lower (1.0–1.1 R_⊙) and an upper (1.1–1.5 R_⊙) height regime within this plane, and the corresponding fluxes as F_low and F_high. They evaluated the "flux ratio" as F_low/F_high, to quantify the relative strength of the horizontal magnetic field strength as a function of height in the solar corona. They found lower/higher flux ratios for confined/eruptive flares, indicating that a less strongly decaying magnetic field with height may have hindered the ejection of an associated CME.

Alternative measures to quantify the decay of the horizontal field as a function of coronal height have been used, e.g., in the form of the decay index, $n=-d\,\mathrm{ln}\,{B}_{\mathrm{hor}}/d\,\mathrm{ln}\,h$, with B_hor denoting the constraining (external) magnetic field and h the height in the corona. It is based on the torus instability model by Kliem & Török (2006), which predicts that a flux rope becomes unstable if the central axis reaches a height (h_crit) at which n reaches a critical value (n_crit). In other words, a system becomes unstable when the overlying horizontal field strength decayes sufficiently fast. Based on magnetohydrodynamic simulations involving different geometrical assumptions on the current channel mimicking the flux rope (e.g., Fan & Gibson 2007; Török & Kliem 2007; Démoulin & Aulanier 2010; Zuccarello et al. 2015) and prominence observations (e.g., Zuccarello et al. 2016), n_crit is found in the range ≃1.0–2.0. Despite this extended range of critical values, n_crit = 1.5 is often used in observation-based studies that aim at assessing the likelihood of a CME to occur. As a common practice, the constraining magnetic field above the potentially unstable flux rope is approximated by the horizontal component of an associated potential (current-free) magnetic field model.

Recently, Wang et al. (2017) presented a statistical analysis of 60 flares. They found that in ARs that host eruptive flares, h_crit is reached at significantly lower coronal heights. They also found that n increases monotonously with height in 86% (84%) of the eruptive/confined events studied, whereas the rest of the analyzed events exhibited a saddle-like profile (see also, e.g., Cheng et al. 2011). They found the saddle-like profiles to be characteristic of an underlying "multipolar" photospheric magnetic flux concentration, i.e., when the line connecting the opposite-polarity centroids of the flaring AR did not pass through the flare-relevant PIL or was oriented almost parallel to it. Careful examination of the saddle-like n(h) profiles also revealed significantly lower values of n at the saddle bottoms in confined events.

In this paper, we also investigate the large-scale magnetic field structure in flaring ARs, in context with the occurrence of eruptive or confined events. We study 44 large (≥M5.0) flare events that occurred during the Solar Dynamics Observatory (SDO) era, between 2011 January and 2015 December. In contrast to earlier works, we seek to pin down the magnetic field related parameter that discriminates best between confined and eruptive events. Therefore, we assess the performance of different known measures that characterize the decay of the confining field above flaring ARs (decay index and flux ratio).

Going beyond, we also analyze the change of orientation of the flare-related magnetic field as a function of height in the corona. Ideally, if the photospheric magnetic field of an AR is approximated by an east–west aligned symmetric potential (current-free) dipole field, the corresponding PIL would run in the north–south direction, both at low as well as large heights. In reality, however, the orientation of PILs at low heights are often found to deviate strongly from such an idealized north–southward direction, but to transit toward the north–south direction with increasing coronal height (see, e.g., Figure 2 of Jing et al. 2015). This motivates us to present here a first systematic study of the orientation of the flare-relevant PIL as a function of height in the corona, for a large number of flares.

In Section 2, we describe our event sample as well as the data and methods used. The details of the analysis are described in Section 3, and the obtained results are presented in Section 4 and discussed in Section 5.

2.1. Event Sample

2.2. Data and Methods

The physical and geometrical parameters described in the following are illustrated using the confined X1.0 flare that occurred on 2014 October 25 and are visualized in Figure 1. The analyzed parameters for all events are listed accordingly in Table 1.

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3.1. Photospheric Magnetic Field Structure and Flare Location

3.2. Orientation of the Flare-relevant PIL as a Function of Height

3.3. Decay Index and Flux Ratio

In Figure 2(a), the flare distance (d_FC) is shown with respect to the size of the AR (d_PC). Notably, eruptive flares appear to occur in ARs of all sizes, from compact (d_PC ≲ 60 Mm) to extended ARs (d_PC up to ≳100 Mm). Confined flares, on the other hand, are found to predominantly originate from extended underlying bipolar field configurations (d_PC ≳ 60 Mm).

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Eruptive flares are found at a large range of distances (5 ≲ d_FC ≲ 80 Mm). In contrast, confined flares originate mainly from close to the flux-weighted magnetic center of the host AR (d_FC ≲ 25 Mm). This is not the case in only two confined events: SOL20110730T02:09M9.3 (event no. 21) and SOL20120510T04:18M5.7 (event no. 29), which can be explained based on the fact that the flares originated from a "spot-satellite" region (see Figure 6 of Toriumi et al. 2017). That means that they originated from a minor ("satellite") bipolar magnetic flux concentration located in the periphery of an AR. This naturally results in a larger value of d_FC, if measured with respect to the flux-weighted center of the main AR.

For the eruptive events, we find a clear dependency between d_PC and d_FC, with a correlation of cc ≃ 0.6 ± 0.1, independent of the magnitude of the flares (compare size of plot symbols, where smaller/larger symbols denote M-/X-flares). The trend of the flare distance to increase with the size of the host AR might not be a real trend; however, since a larger d_FC can be expected for a more extended host AR (with a larger d_PC).

We fitted a linear regression to the values found for eruptive events in the form y = a + b · x, where we proceeded in the following way. We randomized the x  and y  values around the respective values of d_PC and d_FC within their uncertainty ranges and fitted the result using the IDL function LADFIT. Based on n = 100.000 realizations of this scheme, we determined the mean offset $a=\langle [{a}_{1},\,\ldots ,\,{a}_{n}]\rangle$ and mean slope $b=\langle [{b}_{1},\,\ldots ,\,{b}_{n}]\rangle$, together with their corresponding uncertainties. For the eruptive events, we find a = 4 ± 6 and b = 0.5 ± 0.1. The same approach as outlined here was used to describe the dependencies of the analyzed parameters throughout this section.

In order to account for the extent of the underlying magnetic dipole field, we show the normalized flare distance d_FC/d_PC in Figure 2(b), which also allows us to compare the flare location within an AR for different events. A value d_FC/d_PC ≤ 0.5 then refers to a location underneath the magnetic field connecting the opposite magnetic polarity centers of the AR. In case of magnetically compact ARs, this may imply a strong influence of the overlying dipole field. This complies with the fact that we find all but one confined flare at normalized flare distances d_FC/d_PC ≤ 0.5 in Figure 2(b).

In contrast, a value d_FC/d_PC > 0.5 refers to a location in the periphery of the AR, where the overlying magnetic field should be weaker. And, indeed, a large number of eruptive flares are found at such normalized flare distances (red crosses above horizontal dashed line in Figure 2(b)). Note also that a fraction of the eruptive events originates from close to the magnetic center of their host AR (red crosses below horizontal dashed line in Figure 2(b)). This is mostly the case for compact ARs (d_PC ≤ 60 Mm), however. This is in contrast to confined flares, which tend to predominantly occur close to the flux-weighted magnetic AR center (d_FC/d_PC ≤ 0.5) of extended ARs (d_PC ≳ 60 Mm).

Only the confined events no. 29 and 44 do not follow this trend (d_PC ≳ 60 Mm but d_FC/d_PC ≳ 0.5), as well as event no. 21 (d_PC ≲ 60 Mm and d_FC/d_PC < 0.5). As explained above, the spacial organization of the host AR in the form of a spot-satellite region explains the deviation from the general tendency.

Figure 3 shows the orientation of the flare-relevant PIL as a function of the normalized height (h/d_PC) in our magnetic field models, i.e., with respect to the vertical height of the underlying dipole field. We find that in confined flares, the flare-relevant PIL at photospheric levels is inclined by $| \varphi | \gtrsim 50^\circ$ (Figure 3(b)). In contrast, the flare-relevant PIL in AR's hosting eruptive flares, appear at a large range of possible inclinations (Figure 3(a)). Notably, for confined events, we find the PIL to align with the solar north–south direction (indicated by the gray-shaded area in Figures 3(a) and (b)) rather quickly with height, mostly within a height of h/d_PC ≃ 0.5, which corresponds to the apex of the surrounding magnetic dipole field. This is not the case for ARs hosting eruptive flares, where the flare-relevant PIL may still span a large angle with respect to the north–south direction (φ ≳ 10° at h/d_PC = 0.5 in Figure 3(a)). In other words, the PIL orientation changes much faster with height above ARs that host confined flares than above those that host eruptive flares.

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In Figures 3(c) and (d), we plot the change of the orientation of the flare-relevant PIL with respect to the photospheric level, ${\rm{\Delta }}\varphi =| \varphi (h)-\varphi (0)|$, i.e., with respect to the orientation of the underlying dipole field. For confined events, we find the most prominent changes (Δφ ≃ 40°–60°) of the PIL orientation to occur over a height characteristic of the underlying dipole field (h/d_PC ≲ 0.5; Figure 3(d)). This is not the case for eruptive flares, where the changes range from a few degrees up to about 60° (Figure 3(c)). Moreover, the orientation of the flare-relevant PIL may still undergo strong changes above the apex of the underlying dipole field (above h/d_PC = 0.5). In Figure 4(c), we show the distribution of Δφ, separately for confined and eruptive flares (blue and red histograms, respectively). We find the distributions of the two groups of events as distinct, with the confined events joining the distribution of eruptive events at large values (Δφ ≳ 40). In particular, we find Δφ ≲ 40° for most of the ARs that host eruptive events and Δφ ≳ 40° for those that host confined events (compare also Figures 3(c) and (d)). We find mean values of $\langle {\rm{\Delta }}\varphi \rangle =26\buildrel{\circ}\over{.} 1\pm 16\buildrel{\circ}\over{.} 4$ and $\langle {\rm{\Delta }}\varphi \rangle =53\buildrel{\circ}\over{.} 1\pm 8\buildrel{\circ}\over{.} 2$, respectively.

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Note that five confined flares in our event sample (events no. 13–16 and 37) originated from a single AR (NOAA 12192) over four days, and thus exhibit very similar values of the deduced parameters. In order to compensate for this over-representation in our confined events statistics, we computed the mean values of Δφ from the individual values of the five events and show the corresponding occurrences as filled histograms in Figure 4(c) (and in a similar way for h_crit and the flux ratio in Figures 4(a) and (b), respectively). When considering this "reduced" set of values, we obtain $\langle {\rm{\Delta }}\varphi \rangle =55.5\pm 5\buildrel{\circ}\over{.} 5$.

In Figure 4(a), we show the distribution for h_crit which well separates between AR's that host confined (h_crit ≳ 40 Mm) and eruptive (h_crit ≲ 40 Mm) events, with only a small overlap of the populations. We find mean values of $\langle {h}_{\mathrm{crit}}\rangle =52.7\pm 16.7$ Mm and $\langle {h}_{\mathrm{crit}}\rangle =20.9\pm 9.5$ Mm for ARs that hosted confined and eruptive events, respectively. Compensation for the over-represented NOAA 12192 gives $\langle {h}_{\mathrm{crit}}\rangle =47.4\pm 16.0$ Mm. The corresponding histogram of the flux ratio (Figure 4(b)) shows confined and eruptive events as two distinct but overlapping populations, with the confined events residing at the lower range of values (flux ratio ≲25). Only a few confined events, no. 21 (flux ratio ≃ 58), 29 (flux ratio ≃ 49), and 44 (flux ratio ≃ 67) show larger values. Again, these are the confined flare events that originated from spot-satellite ARs.

In order to test whether or not the distributions of h_crit, flux ratio, and Δφ for confined and eruptive events can be modeled to originate from a single, normally distributed population of events, we performed an Anderson–Darling two-sample test (using the package "kSamples" in R). The p-values obtained for the three considered quantities are found not to exceed 0.01. In other words, we may reject the corresponding null-hypothesis with 95% confidence and consider the results obtained for confined and eruptive events as distinctly different.

Last, we check for possible relationships between the different deduced quantities. We find a clear correspondence between h_crit and the flux ratio (cc ≃ −0.7 ± 0.1; see Figure 5(a)). Since the data points align well with the linear fit (black line), we may safely assume that both parameters are robust indicators of the decay of the horizontal magnetic field overlying a flaring AR. However, h_crit appears to be a much better discriminator between ARs that host confined or eruptive flares. Also h_crit and Δφ are correlated (cc ≃ 0.6 ± 0.1; see Figure 5(b)). In that case, however, the correlation appears to result mainly from the apparent clustering of the parameters deduced for confined and eruptive flares in the distinct parts of the scatter plot. While confined events are concentrated in the upper right quadrant (h_crit ≳ 40 Mm, Δφ ≳ 40°), the parameters for the majority of eruptive cases is found in the lower left quadrant.

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After compensation for NOAA 12192 is over-represented, we find cc(h_crit, flux ratio) ≃ −0.4 ± 0.1 and cc(h_crit, Δφ) ≃ 0.2 ± 0.1.

We aimed for a better understanding of the magnetic field structure in and above flaring ARs in terms of the type of flare activity—confined or eruptive. We analyzed 44 large flares (GOES class M5.0 and larger) that occurred between 2011 January and 2015 December, which were observed close to the solar disk center, and for which the flare type could be unambiguously determined.

Our goal was to deepen our insights regarding the discriminating factors in the flare-associated magnetic field, both at photospheric and coronal levels. For this purpose, we constructed a 3D potential field magnetic field model for each flare AR, based on a pre-flare photospheric SDO/HMI vector magnetic field map. Importantly, we put all considerations in context with the characteristic length scale of the bipolar magnetic field configuration of the host AR, which we approximated by a symmetric dipole with a typical length scale equal to d_PC/2, both in the horizontal and the vertical direction (height). Here, d_PC denotes the distance between the flux-weighted centers of opposite magnetic polarities within the AR.

At photospheric levels, we investigated the distance of the flare site to the flux-weighted AR polarity center (d_FC), as well as the size of the host AR (given by d_PC). At coronal levels, we analyzed quantities characterizing the relative strength of the confining magnetic field (decay index n and flux ratio) and the change of orientation of the flare-relevant PIL as a function of height (Δφ). To our knowledge, Δφ, has not been investigated so far systematically for a large number of flare events.

Perhaps our most important finding is a distinctly different spatial organization of the magnetic field above ARs that host confined or eruptive flares. In ARs that host confined flares, the orientation of the flare-relevant PIL changes much faster with height toward a direction representative of the underlying dipole magnetic field. The orientation of the PIL changes quickly from an initial high inclination at photospheric levels to a direction representing the underlying dipole field, within a typical length scale of the underlying bipolar magnetic field (Δφ ≳ 40° until a height h ≃ d_PC/2; Figure 3(d)). This is not the case for ARs that host eruptive flares. There, the transition toward the characteristic orientation of the underlying dipole field involves much larger heights (Figure 3(c)). Δφ discriminates to a certain degree between ARs that host confined and eruptive flares, showing two populations in the corresponding histogram that overlap at large values (40° ≲ Δφ; see Figure 4(c)). This indicates that the relative orientation of the coronal field at different heights above a flare site may play a role in determining whether a confined or eruptive flare will occur.

Equally important is the finding that the flare type can be easily understood in terms of the flare site with respect to the size of the dipole magnetic field of the host AR. We found that confined flares mainly originate from extended ARs (d_PC ≳ 60 Mm), while eruptive flares originate from ARs of very different sizes (40 ≲ d_PC ≲ 140 Mm; see Figure 2). Also, confined flares primarily originate from locations close to the flux-weighted AR center (d_FC ≲ 25 Mm), while eruptive flares may originate from basically everywhere within an AR (5 ≲ d_FC ≲ 80 Mm). In order to address the importance of the location of the flare with respect to the underlying bipolar magnetic field, we considered normalized flare distances (d_FC/d_PC). This allowed us, first, to place the flare site with respect to the possibly weak or strong confinement of the surrounding magnetic field and, second, to make the measured distances and extensions of the considered ARs comparable to each other.

The use of normalized distances places the confined events clearly inside of the underlying magnetic dipole field (d_FC/d_PC < 0.5; Figure 2(b)). Since the overlying magnetic field in extended ARs (d_PC ≳ 60 Mm) decays slowly with height, a correspondingly strong confinement exists that likely prohibits the escape of a possibly associated CME. In contrast, a large number of eruptive flares originates from the periphery of ARs (d_FC/d_PC > 0.5) where the overlying field is weaker. Notably, eruptive flares that originate from within the strong confinement of the underlying AR dipole field (d_FC/d_PC < 0.5) mostly do so in compact ARs (d_PC ≲ 60 Mm). Here, we may argue that in compact ARs, the confining magnetic field decays rapidly with height, i.e., the critical height for torus instability h_crit, resides at lower coronal heights, allowing for flare-associated CMEs.

Correspondingly, we find clearly lower values of h_crit in ARs that host eruptive flares ($\langle {h}_{\mathrm{crit}}\rangle \simeq 21\pm 10$ Mm) than for those that host confined flares ($\langle {h}_{\mathrm{crit}}\rangle \simeq 53\pm 17$ Mm), in accordance with earlier studies (e.g., Cheng et al. 2011; Wang et al. 2017). Wang et al. (2017) suggested an empirical relation of the form h_crit ≃ 0.5 · d_PC (see their Figure 2 and see the gray dotted line in Figure 6). Based on our event sample and analysis, we find h_crit ≃ 0.4 ± 0.1 · d_PC (black solid line in Figure 6), which agrees with the trend found in Wang et al. (2017) within our uncertainty range. As a reason for the slightly different result, we suspect the different flare sizes covered in our study. Wang et al. (2017) analyzed 60 flares down to GOES class M1.0, where the majority of events (75%) were classified as <M5.0. In our study, we considered exclusively large events, i.e., flares of GOES class M5.0 and larger.

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Another important finding in our study is that h_crit discriminates better between ARs that host confined and eruptive flares than the flux ratio, an alternative quantity to measure the decay of the horizontal field (Figure 5(a)), which has been introduced by Wang & Zhang (2007). Importantly, h_crit shows two distinct populations, and clearly separates between confined (h_crit ≳ 40 Mm) and eruptive flares (h_crit ≲ 40 Mm; Figure 4(a)). The flux ratio, on the other hand, shows a less clear distinction (compare Figure 4(b)). We suspect the reason for this is the higher sensitivity of h_crit toward the nonlinear decay of the horizontal field with height in the corona.

In order to substantiate whether or not the studied parameters deduced from ARs hosting confined and eruptive events can be used as discriminating factors at all (i.e., whether they stem from a single normally or non-normally distributed population of events), we performed an Anderson–Darling two-sample test for h_crit, flux ratio, and Δφ. This allows us to state with 95% confidence that the studied parameters do not fit a normal distribution, i.e., can be considered to be distinctly different for confined and eruptive events.

Wang & Zhang (2007), furthermore, reported a clear segregation of confined and eruptive events in a flux ratio versus d_FC diagram (see their Figure 5). More precisely, they found the confined/eruptive events to reside at lower/higher values of both, d_FC and flux ratio. We do recover this trend only to a certain degree, and with a large overlap of the two populations (see Figure 7). A reason for the differing results may again be the particular event selection, on which the different studies are based. While Wang & Zhang (2007) analyzed only eight flares, we considered a much larger number of events (44 in total). Also, while Wang & Zhang (2007) studied only major (X-class) flares, we included large flares (M5.0-class and larger). Importantly, the overlap in our diagram is not solely caused by the eruptive M-flares included in our study, i.e., we also find eruptive X-flares at low values of both, d_PC and flux ratio. Consequently, the different finding cannot be simply attributed to the larger range of flare sizes included in our study.

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We analyzed 44 flares (GOES class ≥M5.0) that occurred between 2011 January and 2015 December. Based on SDO/HMI vector magnetic field maps, we modeled the 3D potential magnetic field above the flare ARs. We studied the vertical decay and orientation of the flare-relevant 3D magnetic field, in relation to the associated flare type, confined or eruptive. Our most important findings include:

• 1.  
  The flare type may be well understood in terms of the flare site within an AR, measured by the flare distance from the flux-weighted AR center, d_FC. This is especially true if d_FC is interpreted with respect to the characteristic length scale of the underlying photospheric magnetic dipole (approximated by the half distance between the centers of opposite magnetic polarity, d_PC/2). If located underneath the confining field spanned by the magnetic dipole (i.e., at a distance d_FC/d_PC ≤ 0.5), flares tend to be eruptive when hosted by a compact (d_PC ≲ 60 Mm) AR and confined when launched from an extended AR (d_PC ≳ 60 Mm). Flares that originate from the periphery of the confining bipolar field (d_FC/d_PC > 0.5) tend to be eruptive, no matter whether the AR is compact or not.
• 2.  
  The critical height for torus instability, h_crit = h(n_crit = 1.5), is the most robust measure to discriminate between ARs that host confined and eruptive flares. ARs exhibiting values of h_crit ≲ 40 Mm above the flare-relevant PIL are likely to produce an eruptive flare. It segregates confined from eruptive flares better than alternative measures for the vertical decay of the magnetic field (the flux ratio), most likely due to its higher sensitivity to the nonlinear decay of the confining magnetic field above ARs.
• 3.  
  The orientation of the magnetic field is organized distinctly differently in the corona above ARs that host confined or eruptive flares. In ARs that host confined flares, the orientation of the flare-relevant PIL, Δφ, quickly adjusts to the underlying photospheric AR dipole field with height. The adjustment (Δφ ≳ 40°) is completed already at the apex of the confining dipole field, i.e., at a height h ≃ d_PC/2. This is different for ARs that host eruptive flares, where the flare-relevant PIL might still be considerably inclined near the apex of the underlying magnetic dipole field.
• 4.  
  Though there exists a moderate degree of correlation between Δφ and h_crit, it appears that the critical height for torus instability is a stronger measure for the type of the upcoming flaring. This implies that the vertical decay of the confining field plays a stronger role than the particular orientation of the field at different heights.
• 5.  
  The parameters deduced for ARs that host confined and eruptive events (h_crit, flux ratio, Δφ) cannot be modeled to originate from a single normally distributed population of events, i.e., they can be considered to be distinctly related to the particular flare type.

The authors acknowledge support from the Austrian Science Fund (FWF): P27292-N20. We are thankful for helpful discussions with F. P. Zuccarello. We acknowledge constructive comments from an anonymous referee and helpful suggestions from a statistics review. SDO data are courtesy of the NASA/SDO AIA and HMI science teams. SDO is a mission for NASA's Living With a Star (LWS) Program. GOES is a joint effort of NASA and the National Oceanic and Atmospheric Administration (NOAA).