A Database of Flare Ribbon Properties from the Solar Dynamics Observatory. I. Reconnection Flux

The key technical challenge in defining the set of pixels corresponding to the flare ribbon location, dS_ribbon, in the AIA image sequences is the correction of the saturated pixels caused by CCD saturation and pixel bleeding, and of the diffraction patterns from the EUV-telescope entrance filter. Unfortunately, existing software packages for automatic de-saturation of AIA images such as DESAT (Schwartz et al. 2015) are not applicable to the 1600 Å channel (G. Torre 2017, private communication). Here we present our own empirical approach to correct the intensities of "bloomed" pixels.

To describe the details of our saturation-correction approach, we use the SDO AIA observations of the well-known "Valentine's Day" flare as a representative example. This flare occurred in NOAA AR 11158 on 2011 February 15, 01:44 UT (Schrijver et al. 2011). The SDO AIA observations of this event were saturated during the impulsive phase, from 01:49 UT to 02:10 UT in the UV 1600 Å continuum as well as in other AIA bands. We re-examine this event in UV 1600 Å observations with 24 s cadence and 061 pixel resolution with the objective of removing the saturated pixels and reconstructing the evolution of the UV ribbons from the earlier and later (unsaturated) phases of the flare. We process the UV 1600 Å images in IDL using the aia_prep.pro SolarSoft package and co-align the AIA image sequence in time with the first frame.

Our saturation-correction approach includes the following steps. We first select the pixels above saturation level, ${I}_{\mathrm{sat}}=5000$ counts s⁻¹, and the pixels surrounding them within 2 and 10 pixels in the x- and y-directions. We then replace each saturated pixel intensity with the value linearly interpolated in time between the individual pixel's previous and subsequent unsaturated values that bracket the saturation duration. Figure 2, left column, shows a sequence of original AIA 1600 Å images on 2011 February 15: top panel, before the impulsive phase when AIA observations had no saturated pixels (01:47 UT); middle panel, at the peak of the impulsive phase with the largest number of saturated pixels (01:52 UT); and lower panel, during the gradual phase with no saturated pixels (02:11 UT). Figure 2, right column, shows the saturation-corrected images which differ from the original images only in the location of the saturation-corrected pixels. This empirical approach, while not suitable for photometric analysis of the corrected images, does allow one to identify flare ribbon locations (compare the original and corrected panels of the middle row). Thus, the saturation-corrected 1600 Å image sequence provides sufficient information to determine the reconnected flux using Equation (2).

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Figure 3 shows the area-integrated light curves of the AIA 1600 Å image sequence at each step of the saturation removal procedure. The dashed−triple-dotted curve, labeled "Original saturated," plots the total number of counts in the saturated AIA 1600 Å image sequence. The period from 01:49 to 02:10 UT, shown with vertical dotted lines, indicates the duration of the pixel saturation in the sequence. The dashed curve, labeled "Saturated pixels set to zero," plots the total number of counts after the saturated pixels above the threshold level of I_sat and the adjacent pixels have been removed. The dashed−dotted curve, labeled "Corrected image," plots the total counts after the saturated pixel values have been replaced by the interpolation in time between the unsaturated values of those pixels in the image sequence. Finally, the solid curve, labeled "Corrected image, ribbons," plots the light curve of the flare ribbons alone—the set of pixels that are identified by the ribbon search algorithm described in the next section. Note that the "Corrected image, Ribbons only" light curve is offset from the "Corrected image" curve by the total flux of the background that remains nearly constant during the flare. To summarize, Figure 3 quantitatively describes how much of the original image intensity is affected by the saturation and what fraction of this intensity is attributed to the flare ribbons using our saturation-correction approach.

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To find the reconnection flux as defined by Equation (2), we need to know the flare ribbon location and the normal component of the magnetic field.

To identify the flare ribbon locations, we use the AIA 1600 Å saturation-corrected image sequence from Section 2.1 and the methodology of Qiu et al. (2002, 2004, 2007). We define an instantaneous flare ribbon pixel mask ${N}^{({{\rm{I}}}_{c})}({x}_{i},{y}_{j},{t}_{k})$ in each pixel $({x}_{i},{y}_{j})$ at each time step t_k in the sequence with a value of 1 if the 1600 Å intensity is greater than an empirical ribbon-edge cutoff intensity, I_c, and with a value of zero if the intensity is below I_c. The cutoff intensity I_c for identifying the flare ribbon pixels ranges from the cutoff threshold c = 6 to c = 10 times the median image intensity at each time t_k. This range is consistent with the range previously used for TRACE 1600 Å UV data (Kazachenko et al. 2012). Since the cutoff threshold for the "steady-state" UV brightening associated with plage regions is typically $c\approx 3.5$ (see Figure 3 of Qiu et al. 2010), our empirical threshold range, $c\in [6,10]$, is significantly greater than typical non-flare-related UV emission and is appropriate to capture the flare ribbons.

To find the normal component of the magnetic field, B_n, we use the full-disk HMI vector magnetogram data series (hmi.B_720s) in the form of field strength, inclination, and azimuth in the plane-of-sky coordinate (Hoeksema et al. 2014). We perform a coordinate transformation and decompose the magnetic field vectors into three components in spherical coordinates⁴ (Sun 2013). The derived radial component is the normal component B_n that we need in Equation (2). To avoid noisy magnetic fields, we only use magnetic fields greater than the flux density threshold $| {B}_{n}| \gt 100\,{\rm{G}}$ (see Figure 2 in Kazachenko et al. 2015).

The unsigned reconnection flux or unsigned magnetic flux swept up by the flare ribbons up to time t_k is then calculated using the discrete observations as

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{ribbon}}^{({{\rm{I}}}_{c})}({t}_{k})=\mathop{\int }\limits_{{{\rm{I}}}_{c}}| {B}_{{\rm{n}}}({t}_{k})| \ {dS}({t}_{k})\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\approx \sum _{i,j}| {B}_{{\rm{n}}}({x}_{i},{y}_{j},{t}_{\mathrm{HMI}})| \ {M}^{({{\rm{I}}}_{c})}({x}_{i},{y}_{j},{t}_{k})\ {{ds}}_{{ij}}^{2},\end{eqnarray} \tag{ 4 }$$

where t_HMI corresponds to the time of the measurement of the normal component of the magnetic field B_n, and the ribbon area ${dS}({t}_{k})$ above the ribbon-edge cutoff intensity, I_c, is given by the discrete cumulative ribbon pixel mask ${M}^{({{\rm{I}}}_{c})}({x}_{i},{y}_{j},{t}_{k})$ multiplied by the pixel area ds_ij². We correct each individual pixel area for foreshortening: ${{ds}}_{{ij}}^{2}={{ds}}_{{ij},\mathrm{obs}}^{2}/\cos {\theta }_{j}$, where ${\theta }_{j}$ is the angular distance between the central meridian and the pixel $({x}_{i},{y}_{j})$. The cumulative ribbon pixel mask ${M}^{({{\rm{I}}}_{c})}({x}_{i},{y}_{j},{t}_{k})$ is related to the instantaneous ribbon pixel mask ${N}^{({{\rm{I}}}_{c})}({x}_{i},{y}_{j},{t}_{k})$ in the following way:

$$\begin{eqnarray*}&&{M}^{({{\rm{I}}}_{c})}({x}_{i},{y}_{j},{t}_{k})={M}^{({{\rm{I}}}_{c})}({x}_{i},{y}_{j},{t}_{k-1})\cup {N}^{({{\rm{I}}}_{c})}({x}_{i},{y}_{j},{t}_{k}).\end{eqnarray*}$$

Thus, the cumulative ribbon pixel mask represents the time integral (accumulation) of every flare ribbon pixel $({x}_{i},{y}_{j})$ that exceeds the 1600 Å intensity threshold at some instance from the first frame of the image sequence to t_k. Figure 4 shows the cumulative ribbon pixel mask ${M}^{({{\rm{I}}}_{8})}({x}_{i},{y}_{j},{t}_{k})$ consisting of pixels above c = 8 times the background median value from the sequence of corrected images at the times shown in Figure 2.

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In Figure 5, we summarize the temporal evolution of the flare ribbons associated with an X2.2 flare on 2011 February 15. The left panel shows the contours of the maximum flare ribbon area at the end of the AIA image sequence (${t}_{\mathrm{final}}$) superimposed on the co-aligned HMI magnetogram before the flare at ${t}_{\mathrm{hmi}}$. The middle panel shows the temporal evolution of the ribbons, with the blue and red colors corresponding to the early and late stages of the flare respectively. Note that the ${M}^{({{\rm{I}}}_{8})}({x}_{i},{y}_{j},{t}_{k})$ color-coded bitmaps are plotted in reverse temporal order so that every individual pixel in the cumulative ribbon pixel mask is colored according to the time of its initial brightening.

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The right panel of Figure 5 shows the evolution of the magnetic fluxes swept up by ribbons in positive and negative polarities, the signed reconnection fluxes, ${{\rm{\Phi }}}_{\mathrm{ribbon}}^{+}$ and ${{\rm{\Phi }}}_{\mathrm{ribbon}}^{-}$, respectively. To reflect the uncertainty in these estimates due to the ribbon area identification, we perform the entire pixel mask area calculation twice: once with the cutoff threshold c = 6, and again with c = 10. Then, the signed reconnection fluxes in each polarity at time t_k are

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{ribbon}}^{+}=\displaystyle \frac{{{\rm{\Phi }}}_{\mathrm{ribbon}}^{+({I}_{6})}+{{\rm{\Phi }}}_{\mathrm{ribbon}}^{+({I}_{10})}}{2},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{ribbon}}^{-}=\displaystyle \frac{{{\rm{\Phi }}}_{\mathrm{ribbon}}^{-({I}_{6})}+{{\rm{\Phi }}}_{\mathrm{ribbon}}^{-({I}_{10})}}{2},\end{eqnarray} \tag{ 6 }$$

where "+" and "−" refer to integration over the positive and negative polarities, respectively. In Figure 5, the reconnection fluxes in both polarities evolve nearly simultaneously. By t_final = 02:16 UT, the positive and negative reconnection fluxes are ${{\rm{\Phi }}}_{\mathrm{ribbon}}^{+}\ ({{\rm{\Phi }}}_{\mathrm{ribbon}}^{-})=5.67\ (-5.92)\times {10}^{21}\,\mathrm{Mx}$, and the total unsigned reconnection flux is ${{\rm{\Phi }}}_{\mathrm{ribbon}}=| {{\rm{\Phi }}}_{\mathrm{ribbon}}^{+}| +| {{\rm{\Phi }}}_{\mathrm{ribbon}}^{-}| \,=1.16\times {10}^{22}\,\mathrm{Mx}$. Theoretically, equal amounts of positive and negative flux should be reconnected. Hence, the balance between the two increases the credibility of the applied technique.

We estimate the errors in ${{\rm{\Phi }}}_{\mathrm{ribbon}}^{+}$ and ${{\rm{\Phi }}}_{\mathrm{ribbon}}^{-}$ at time t_k using the uncertainty in the ribbon area (see error bars in Figure 5):

$$\begin{eqnarray*}{\rm{\Delta }}{{\rm{\Phi }}}_{\mathrm{ribbon}}^{+} & = & \displaystyle \frac{{{\rm{\Phi }}}_{\mathrm{ribbon}}^{+({I}_{6})}-{{\rm{\Phi }}}_{\mathrm{ribbon}}^{+({I}_{10})}}{2},\\ {\rm{\Delta }}{{\rm{\Phi }}}_{\mathrm{ribbon}}^{-} & = & \displaystyle \frac{{{\rm{\Phi }}}_{\mathrm{ribbon}}^{-({I}_{6})}-{{\rm{\Phi }}}_{\mathrm{ribbon}}^{-({I}_{10})}}{2}.\end{eqnarray*}$$

Typically, these range within 10% to 20% of ${{\rm{\Phi }}}_{\mathrm{ribbon}}^{\pm }$. Further in the text we do not take into account the uncertainty associated with the physical height of ribbon formation. The 1600 Å UV emission corresponds to the upper chromosphere and transition region whereas the photospheric magnetic field measurements are ∼2 Mm below this. The differences in normal field strength at the photosphere and at the ribbon formation height typically lead to a maximum decrease in the total reconnection flux of 10%–20% (Qiu et al. 2007; Kazachenko et al. 2012).

The flare ribbon catalog RibbonDB contains properties of the ARs and flare ribbons for all well-observed flares of GOES class C1.0 and larger in the SDO era, from 2010 April until 2016 April. We used the existing Heliophysics Event Catalog (HEC) maintained by the INAF-Trieste Astronomical Observatory to select the events for our flare ribbon catalog. We chose flares within 45° from the central meridian to minimize projection effects. We also excluded events that were missing the AR number. We used get_nar.pro from SolarSoft to obtain the AR location coordinates for events with missing AR coordinates. These criteria resulted in 3137 flares for our RibbonDB catalog, including 17 X-class, 250 M-class, and 2870 C-class flares (see Table 1). Each entry contains the following information from the HEC: flare start time, flare peak time, flare end time, flare peak X-ray flux (flare class), flare heliographic longitude and latitude, and AR number. Figure 6 shows the monthly international sunspot number and the number of C-, M-, and X-class flares selected for the RibbonDB catalog as a function of time (upper panel) and also the time and location of ARs (lower panel). RibbonDB covers the first half of solar cycle 24, including its maximum around 2014 April. In the lower panel of Figure 6, the radius of each AR circle is proportional to the cumulative peak X-ray flux over the AR's lifetime. The ARs with the largest cumulative peak X-ray fluxes are NOAA AR 12192 (24 flares equivalent to 9 X1 flares in peak X-ray flux), NOAA AR 11429 (13 flares equivalent to 6 X1 flares), and NOAA AR 11515 (26 flares equivalent to 3 X1 flares).

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Figure 7 shows four representative events from our RibbonDB catalog, ranging from GOES C1.6 to X5.6 class, in the same format as Figure 5.

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For each event in the database, we use the AIA 1600 Å image sequence, of 24 s cadence and $0\buildrel{\prime\prime}\over{.} 61$ spatial resolution, and a pre-flare HMI vector field magnetogram, of 12 minute cadence and $0\buildrel{\prime\prime}\over{.} 5$ spatial resolution, to derive the flare-ribbon-pixel masks and the normal component of the magnetic field, B_n, respectively (see Section 2.2 for details).⁵ We use these observations to compute the following large-scale, event- and area-integrated quantities:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{AR}}=\mathop{\int }\limits_{\mathrm{AR}}| {B}_{n}| \ {dS},\quad {S}_{\mathrm{AR}}=\mathop{\int }\limits_{\mathrm{AR}}{dS},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{ribbon}}=\displaystyle \frac{\mathop{\int }\limits_{{{\rm{I}}}_{6}}| {B}_{n}| \ {dS}+\mathop{\int }\limits_{{{\rm{I}}}_{10}}| {B}_{n}| \ {dS}}{2}=\displaystyle \frac{{{\rm{\Phi }}}_{\mathrm{ribbon}}^{({I}_{6})}+{{\rm{\Phi }}}_{\mathrm{ribbon}}^{({I}_{10})}}{2},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{S}_{\mathrm{ribbon}}=\displaystyle \frac{\mathop{\int }\limits_{{{\rm{I}}}_{6}}\ {dS}+\mathop{\int }\limits_{{{\rm{I}}}_{10}}\ {dS}}{2}=\displaystyle \frac{{S}_{\mathrm{ribbon}}^{({I}_{6})}+{S}_{\mathrm{ribbon}}^{({I}_{10})}}{2},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\overline{B}}_{\mathrm{AR}}=\displaystyle \frac{{{\rm{\Phi }}}_{\mathrm{AR}}}{{S}_{\mathrm{AR}}},\quad {\overline{B}}_{\mathrm{ribbon}}=\displaystyle \frac{{{\rm{\Phi }}}_{\mathrm{ribbon}}}{{S}_{\mathrm{ribbon}}},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{R}_{{\rm{\Phi }}}=\displaystyle \frac{{{\rm{\Phi }}}_{\mathrm{ribbon}}}{{{\rm{\Phi }}}_{\mathrm{AR}}}\times 100 \% ,\quad {R}_{S}=\displaystyle \frac{{S}_{\mathrm{ribbon}}}{{S}_{\mathrm{AR}}}\times 100 \% ,\end{eqnarray} \tag{ 11 }$$

where ${{\rm{\Phi }}}_{{AR}}$ and ${S}_{\mathrm{AR}}$ are the unsigned AR magnetic flux and AR area, ${{\rm{\Phi }}}_{\mathrm{ribbon}}$ and ${S}_{\mathrm{ribbon}}$ are the unsigned flare ribbon reconnection flux and flare ribbon area, ${\overline{B}}_{\mathrm{AR}}$ and ${\overline{B}}_{\mathrm{ribbon}}$ are the mean AR and ribbon field strengths, and ${R}_{{\rm{\Phi }}}$ and R_S are the percentages of the ribbon-to-AR magnetic fluxes and ribbon-to-AR areas, respectively. We calculate all the ribbon-related quantities at the flare end time. The integration ${\int }_{\mathrm{AR}}{dS}$ means the summation over the AR region of interest, and ${\int }_{{I}_{6}}{dS}$ and ${\int }_{{I}_{10}}{dS}$ are the summations over the ribbon cumulative pixel mask at the c = 6 and c = 10 ribbon cutoff thresholds at ${t}_{\mathrm{final}}$. The AR region of interest is defined as an 800 × 800 pixel (400 × 400 arcsecond) rectangle centered on the AR. The coordinates of the AR center are derived from the HEC mentioned in Section 3.1.

As discussed in Section 2.2, the uncertainties in the ribbon reconnection flux and the ribbon area were obtained by varying the threshold of the minimum ribbon brightness c from 6 to 10 times the median background intensity. The errors in the unsigned reconnection flux and ribbon area are then

$$\begin{eqnarray}&&{\rm{\Delta }}{{\rm{\Phi }}}_{\mathrm{ribbon}}=\displaystyle \frac{{{\rm{\Phi }}}_{\mathrm{ribbon}}^{({I}_{6})}-{{\rm{\Phi }}}_{\mathrm{ribbon}}^{({I}_{10})}}{2},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{S}_{\mathrm{ribbon}}=\displaystyle \frac{{S}_{\mathrm{ribbon}}^{({I}_{6})}-{S}_{\mathrm{ribbon}}^{({I}_{10})}}{2}.\end{eqnarray} \tag{ 13 }$$

Table 2 summarizes the event information included in the RibbonDB catalog that is available online.

To quantitatively describe the relationship between the different properties of flares and ARs, e.g., ${\mathbb{X}}$ and ${\mathbb{Y}}$, we use the Spearman ranking correlation coefficient, ${r}_{s}({\mathbb{X}},{\mathbb{Y}})$. Unlike the Pearson correlation coefficient that measures the linear relationship between two variables—and therefore is not suitable for nonlinearly related variables—the Spearman rank correlation provides a measure of the monotonic relationship between two variables. To estimate the errors in the correlation coefficient due to sampling, we use a bootstrap method: for that we repeatedly calculate ${r}_{s}$ for N out of N randomly selected data pairs with replacement, and then estimate the mean and standard deviation as ${\bar{r}}_{s}\pm {\rm{\Delta }}{r}_{s}$ (Wall & Jenkins 2012, Chapter 6). From this point forward, we will refer to the mean Spearman correlation ${\bar{r}}_{s}$ as simply the correlation coefficient ${r}_{s}$ and its standard deviation ${\rm{\Delta }}{r}_{s}$ as the correlation's statistical uncertainty.

We describe the qualitative strength of the correlation using the following guide for the absolute value of r_s: ${r}_{s}\in [0.2,0.39]$—weak, ${r}_{s}\in [0.4,0.59]$—moderate, ${r}_{s}\in [0.6,0.79]$—strong, and ${r}_{s}\in [0.8,1.0]$—very strong. When the correlation coefficient is moderate or greater (${r}_{s}({\mathbb{X}},{\mathbb{Y}})\gt 0.4$), we fit the relationship between ${\mathbb{X}}$ and ${\mathbb{Y}}$ with a power-law function,

$$\begin{eqnarray}&&{\mathbb{Y}}=a{{\mathbb{X}}}^{b}.\end{eqnarray} \tag{ 14 }$$

We use the Levenberg–Marquardt nonlinear least-squares minimization method to find the scaling factor a and exponent b.

Table 3 summarizes the properties of the ARs and flare ribbons listed in Table 2, their range, and correlation coefficient with the GOES peak X-ray flux. The "Active Regions" column lists the AR unsigned magnetic flux, area, and the mean magnetic field: ${{\rm{\Phi }}}_{{AR}}$, S_AR, and ${\overline{B}}_{{AR}}$. The "Flare Ribbons" column lists the reconnection flux, flare ribbon area, and the mean magnetic field swept by the ribbons: ${{\rm{\Phi }}}_{\mathrm{ribbon}}$, ${S}_{\mathrm{ribbon}}$, and ${\overline{B}}_{\mathrm{ribbon}}$. The bottom row shows the fractions of magnetic flux and area of the whole AR involved in the flare reconnection, ${R}_{{\rm{\Phi }}}$ and R_S. We discuss each of these relationships further in the text and in Figures 8–11.

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Table 3.  Active Region and Flare Ribbon Properties, ${{\mathbb{X}}}_{\mathrm{AR}}$ and ${{\mathbb{X}}}_{\mathrm{ribbon}}$

	Active Regions		Flare Ribbons
Quantity	Typical Range	Correlation	Typical Range	Correlation	${I}_{{\rm{X}},\mathrm{peak}}$	Figure
${\mathbb{X}}$	${{\mathbb{X}}}_{\mathrm{AR}}[{P}_{20},{P}_{80}]$	${r}_{s}({{\mathbb{X}}}_{\mathrm{AR}},{I}_{{\rm{X}},\mathrm{peak}})$	${{\mathbb{X}}}_{\mathrm{ribbon}}[{P}_{20},{P}_{80}]$	${r}_{s}({{\mathbb{X}}}_{\mathrm{ribbon}},{I}_{{\rm{X}},\mathrm{peak}})$	∝
Φ	[28, 64] × 1021 Mx	0.22 ± 0.01	[5.4, 21] × 1021 Mx	0.66 ± 0.01	${{\rm{\Phi }}}_{\mathrm{ribbon}}^{1.53}$	Figure 8
S	[78, 183] × 1018 cm2	0.14 ± 0.02	[1.1, 3.7] × 1018 cm2	0.68 ± 0.01	${S}_{\mathrm{ribbon}}^{1.57}$	Figure 9
$\overline{B}$	[310, 442] G	0.21 ± 0.02	[408, 675] G	0.24 ± 0.02	...	Figure 10
RS	...	...	[0.9, 3.4]%	0.53 ± 0.01	${R}_{{\rm{\Phi }}}^{1.7}$	...
RΦ	...	...	[1.3, 5.1]%	0.54 ± 0.01	${R}_{{\rm{\Phi }}}^{1.9}$	Figure 11

Note.  Quantity X is either the active region or flare ribbon magnetic flux Φ, area S, mean magnetic Field B, ribbon-to-AR fractions of the area RS, or magnetic flux RF. The typical range of each quantity is described as the 20th to 80th percentile ${\mathbb{X}}[{P}_{20},{P}_{80}]$. The relationship between ${\mathbb{X}}$ and the peak X-ray flux is characterized by the correlation coefficient ${r}_{s}({\mathbb{X}},{I}_{{\rm{X}},\mathrm{peak}});$ for variables with ${r}_{s}\gt 0.4$ we find the coefficient b in the fit ${I}_{{\rm{X}},\mathrm{peak}}=a{{\mathbb{X}}}^{b}$. For more details, see the figures.

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Figure 8 shows the scatter plots of the flare peak X-ray flux versus the total AR unsigned magnetic flux and the flare ribbon reconnection flux at ${t}_{\mathrm{final}}$: ${I}_{{\rm{X}},\mathrm{peak}}$ versus ${{\rm{\Phi }}}_{\mathrm{AR}}$, left panel, and ${I}_{{\rm{X}},\mathrm{peak}}$ versus ${{\rm{\Phi }}}_{\mathrm{ribbon}}$, right panel. While the flare peak X-ray flux has very little correlation with the AR magnetic flux, ${r}_{s}({I}_{{\rm{X}},\mathrm{peak}},{{\rm{\Phi }}}_{\mathrm{AR}})=0.22\pm 0.01$, it is strongly correlated with the flare ribbon reconnection flux, ${r}_{s}({I}_{{\rm{X}},\mathrm{peak}},{{\rm{\Phi }}}_{\mathrm{ribbon}})=0.66\,\pm 0.01$. The correlation is strong: ${r}_{s}=0.66\pm 0.01$. The power-law fit to ${I}_{{\rm{X}},\mathrm{peak}}$ versus ${{\rm{\Phi }}}_{\mathrm{ribbon}}$ yields ${I}_{{\rm{X}},\mathrm{peak}}\propto {{\rm{\Phi }}}_{\mathrm{ribbon}}^{1.5}$.

If we restrict our analysis to stronger flares, M1 class and above, we find a weaker correlation coefficient with a larger standard deviation: ${r}_{s}({I}_{{\rm{X}},\mathrm{peak}},{{\rm{\Phi }}}_{\mathrm{ribbon}},\gt M1.0)=0.51\pm 0.05$. This weakening is a result of the well-known problem in the statistics of range restriction (Pearson 1903), rather than a consequence of different physical processes governing smaller and larger flares. This effect reduces the correlation coefficients for flares in a smaller range of flare classes, e.g., flares larger than M1 or flares smaller than C5. For the same reason, we would expect a larger correlation between the flare peak X-ray flux and the reconnection flux if we include flare classes beyond the RibbondDB range.

In addition, we investigated the difference between using the normal component of the magnetic B_n field derived from the line of sight (LOS) versus the full vector magnetograms (as we do here). The normal component is then derived as ${B}_{{\rm{n}}}={B}_{\mathrm{LOS}}/\cos {\theta }_{j}$, where ${\theta }_{j}$ is the angular distance between the central meridian and the pixel $({x}_{i},{y}_{j})$. We find that the relationships between ${I}_{{\rm{X}},\mathrm{peak}}$ and ${{\rm{\Phi }}}_{\mathrm{AR}}$ and ${{\rm{\Phi }}}_{\mathrm{ribbon}}$, their correlation coefficients, and the power-law exponents using B_LOS are within the uncertainties of the estimates using the vector magnetic fields shown above.

Figure 9 shows the scatter plots of the flare peak X-ray flux versus total AR area (${I}_{{\rm{X}},\mathrm{peak}}$ versus S_AR, left panel) and cumulative flare ribbon area at ${t}_{\mathrm{final}}$ (${I}_{{\rm{X}},\mathrm{peak}}$ versus S_ribbon, right panel). Here, we find an even weaker correlation between the peak X-ray flux and the AR area (${r}_{s}=0.14\pm 0.02$) and a slightly stronger correlation between the peak X-ray flux and the cumulative flare ribbon area (${r}_{s}=0.68\pm 0.01$) than the AR flux and the ribbon reconnection flux, respectively.

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We further examine whether the mean magnitude of the normal magnetic field swept by the flare ribbons, ${\overline{B}}_{\mathrm{ribbon}}$, is substantially different from the mean field of the whole AR, ${\overline{B}}_{\mathrm{AR}}$. Neither ${\overline{B}}_{\mathrm{AR}}$ nor ${\overline{B}}_{\mathrm{ribbon}}$ shows anything more than a weak correlation with the flare peak X-ray flux (0.21 ± 0.02 and 0.24 ± 0.02, respectively). This however does not contradict the known association between the strong gradients in the normal magnetic field across the AR polarity inversion line and the AR's flare and CME productivity (e.g., Welsch & Li 2008, and references therein). Figure 10 plots the distributions of ${\overline{B}}_{{AR}}$ and ${\overline{B}}_{\mathrm{ribbon}}$ for 3137 ribbon events. We find that the average of the ${\overline{B}}_{\mathrm{ribbon}}$ field strength distribution is 100 to 200 G higher than the average of the ${\overline{B}}_{{AR}}$ distribution. The range between the 20th–80th percentiles for ${\overline{B}}_{{AR}}$ is 310–442 G (light blue shaded region), whereas the corresponding percentile range for ${\overline{B}}_{\mathrm{ribbon}}$ is 408–675 G (light red shaded region). Figure 10 results confirm, in a statistical sense, that the magnetic fields that participate in the flare reconnection tend to be stronger fields than the AR as a whole.

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Lastly, in Figure 11, we examine the relationship between the fraction of the AR magnetic flux that participates in the flare reconnection and the flare peak X-ray flux: ${I}_{{\rm{X}},\mathrm{peak}}$ versus ${R}_{{\rm{\Phi }}}$ (see Equation (11)). The left panel of Figure 11 shows that the flare peak X-ray flux exhibits a moderate correlation with ${R}_{{\rm{\Phi }}}$ (${r}_{s}=0.54\pm 0.01$). The fit to the power-law relationship between the two quantities yields ${I}_{{\rm{X}},\mathrm{peak}}\propto {R}_{{\rm{\Phi }}}^{1.9}$. The right panels of Figure 11 show the histogram distributions of ${R}_{{\rm{\Phi }}}$ in four ranges of flare classes: C1–M1 (dark blue), M1–M2.5 (light blue), M2.5-X1 (green), and X1-X5 (red). The distribution mean (vertical dashed line) and standard deviation are listed in each panel with the total number of events in the class range. For C1–M1 flares, the mean and standard deviation of ${R}_{{\rm{\Phi }}}$ is 3% ± 2%; for M1–M2.5 flares, 8% ± 4%; for M2.5-X1 flares, 12% ± 6%; and for X-class flares, 21% ± 10%. To summarize, both the mean and the width of the distributions of the reconnected flux fraction increase with the flare class strength.

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To describe the statistical properties of solar and stellar flares, many authors have looked at the frequency distributions of various flare and CME parameters, such as flare duration, peak hard X-ray and soft X-ray fluxes, and total magnetic energy and its thermal, radiative, and the electron contributions (Drake et al. 2013; Maehara et al. 2015; Harra et al. 2016; Notsu et al. 2017). Since all of these quantities are products of the flare reconnection process (Forbes 2000), it is imperative to understand the quantitative distribution of flare reconnection fluxes and their associated magnetic energy release.

Following Aulanier et al. (2013), we estimate the energy released in the flare in terms of the properties of the reconnected magnetic field as

$$\begin{eqnarray}&&{E}_{\mathrm{flare}}\sim {{fE}}_{\mathrm{mag}}\sim f\displaystyle \frac{{\overline{B}}_{\mathrm{ribbon}}^{2}}{8\pi }V\sim f\displaystyle \frac{{\overline{B}}_{\mathrm{ribbon}}^{2}}{8\pi }{\displaystyle \frac{{S}_{\mathrm{ribbon}}}{2}}^{3/2},\end{eqnarray} \tag{ 15 }$$

where f is the fraction of magnetic energy released as flare energy and V is the volume of reconnecting magnetic fields expressed in terms of the flare ribbon area $V\sim {({S}_{\mathrm{ribbon}}/2)}^{3/2}$. We divide S_ribbon by a factor of 2 to derive the signed from unsigned quantities. We also assume that all non-potential magnetic energy is released by the flare, i.e., f = 1. We note that Equation (15) differs from the flare energy estimates of Shibata et al. (2013) and Maehara et al. (2015) for stellar flares. Here, instead of the properties of the whole AR, we only use the flaring portion of the AR, as defined by the flare ribbons.

Figure 12 shows the occurrence frequencies for the GOES peak X-ray flux, ribbon unsigned reconnection flux, and the flare magnetic energy proxy: ${I}_{{\rm{X}},\mathrm{peak}}$, ${{\rm{\Phi }}}_{\mathrm{ribbon}}$, and E_flare. The number of solar flares, proportional to their occurrence frequency, decreases dramatically as the flare energy increases. The fit to the power-law dependence of the peak X-ray flux distribution yields ${dN}/{{dI}}_{X,\mathrm{peak}}\propto {I}_{X,\mathrm{peak}}^{-1.8}$ for ${I}_{X,\mathrm{peak}}\,\in [{10}^{-6},5\,\times {10}^{-4}]\,{\rm{W}}\,{{\rm{m}}}^{-2}$. The fit to the power-law dependence of the reconnection flux distribution yields ${dN}/d{{\rm{\Phi }}}_{\mathrm{ribbon}}\propto {{\rm{\Phi }}}_{\mathrm{ribbon}}^{-1.7}$ for ${{\rm{\Phi }}}_{\mathrm{ribbon}}\in [8\times {10}^{20},{10}^{22}]\,\mathrm{Mx}$. The fit to the power-law dependence of the distribution of flare energy, E_flare, yields ${dN}/{{dE}}_{\mathrm{flare}}\propto {E}_{\mathrm{flare}}^{-1.6}$ for ${E}_{\mathrm{flare}}\in [{10}^{31},{10}^{33}]\,\mathrm{erg}$.

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We also compare the distribution of our flare energy proxy derived from the reconnection flux with previous results for both solar and stellar flares. For solar flares, Crosby et al. (1993), Shimizu (1995), Aschwanden et al. (2000), and Qiu et al. (2004) have examined the occurrence frequency as a function of flare energy over ${E}_{\mathrm{flare}}\sim {10}^{24\mbox{--}32}\,\mathrm{erg}$, which includes micro-flares and nano-flares. These distributions, obtained from EUV and soft and hard X-ray emission, are generally well-described by a simple power-law function, ${dN}/{{dE}}_{\mathrm{flare}}\propto {E}_{\mathrm{flare}}^{-\alpha }$, with an index, α, of 1.5–1.9. For much larger stellar flares, ${E}_{\mathrm{flare}}\sim {10}^{32\mbox{--}36}$ erg, Maehara et al. (2015) and Notsu et al. (2017) found a power-law dependence, $\alpha =1.5\pm 0.1$. In these estimates, the area of the starspot group was used as a proxy of stored magnetic energy, E_flare. Combining the solar and stellar flare results, Shibata et al. (2013) suggested that the frequency distribution should follow a universal power-law index, $\alpha =1.8$, for $E\sim {10}^{24\mbox{--}36}\,\mathrm{erg}$.

In this paper, we use high spatial resolution observations of flares on the Sun to estimate only the magnetic energy released during the flare, not the total AR magnetic energy. As shown in Figure 8, the total AR flux or area is only weakly correlated with the flare class. Hence, the total AR flux and area are, physically, not the best proxy for E_flare. The exponent for our distribution of flare energies is $\alpha =1.6$, well within the range of exponents previously found for both stellar and solar flares (1.5–1.9).