1 Introduction

A number of investigations have identified a close connection between coronal mass ejections (CMEs) and the associated flares: i) the CME acceleration is synchronized with the rise time of the associated flare (Zhang et al. 2001; Zhang and Dere 2006; Gopalswamy et al. 2012), ii) the CME kinetic energy and soft X-ray peak flux are correlated (Gopalswamy 2009), iii) the CME width is determined by the flare magnetic field (Moore, Sterling, and Suess 2007), iv) flare reconnection (RC) and flux rope formation are related (Leamon et al. 2004; Longcope and Beveridge 2007; Qiu et al. 2007; Hu et al. 2014), v) the CME nose is directly above the flare location (Yashiro et al. 2008), and vi) the high charge state of minor ions in interplanetary coronal mass ejections (ICMEs) is a consequence of the heated flare plasma entering the CME flux rope during the eruption (Lepri et al. 2001; Reinard 2008; Gopalswamy et al. 2013a). One of the key aspects of CMEs is their flux rope nature, which has been considered extensively from theory and observations (see e.g., Mouschovias and Poland 1978; Burlaga et al. 1981; Marubashi 1997; Gibson et al. 2006; Linton and Moldwin 2009). The flux rope nature of CMEs provides an important eruption scenario that can be tested using remote-sensing and in-situ observations.

A number of investigations have also shown that the flare RC process results in the simultaneous formation of a post-eruption arcade (PEA) and a flux rope during solar eruptive events (Leamon et al. 2004; Longcope and Beveridge 2007; Qiu et al. 2007; Hu et al. 2014). At present, it is not fully understood whether CME flux ropes exist before eruption or if they are formed during eruption. The reality may be something in between: flux may be added to a pre-existing flux rope via flare RC (see e.g., Lin, Raymond, and van Ballegooijen 2004). If the flux rope is formed by reconnection, the flux rope is twisted by one turn for each flare loop that forms, as explained in Longcope et al. (2007); Longcope and Beveridge (2007). The flare loops are rooted in the flare ribbons on either side of the polarity inversion line. Thus flare reconnection flux \(\Phi_{\mathrm{r}}\) and the poloidal flux \(\Phi_{\mathrm{p}}\) of the associated flux rope at 1 AU are expected to be equal. On the other hand, \(\Phi_{\mathrm{r}} < \Phi_{\mathrm{p}}\) might indicate that flux is added to preexisting flux of the flux rope. For a set of nine eruptions, Qiu et al. (2007) computed \(\Phi_{\mathrm{r}}\) from the Transition Region and Coronal Explorer (TRACE: Strong et al. 1994) 1600 Å flare-ribbon areas and photospheric/chromospheric magnetic field strength. They compared \(\Phi_{\mathrm{r}}\) with \(\Phi_{\mathrm{p}}\) obtained by fitting a flux rope to the in-situ observations of the associated CMEs and found an approximate equality between \(\Phi_{\mathrm{r}}\) and \(\Phi_{\mathrm{p}}\). Qiu et al. (2007) analysis resulted in the relation

$$ \Phi_{\mathrm{p}} = 1.12(\Phi_{\mathrm{r}})^{0.82}, $$
(1)

suggesting that \(\Phi_{\mathrm{r}}\) is approximately equal to \(\Phi_{\mathrm{p}}\). One of the reasons for the small number of events is that complete observations of flare ribbons are rare. The purpose of this article is to report on a new technique to compute \(\Phi _{\mathrm{r}}\) from PEAs and also to test the \(\Phi_{\mathrm{r}}\mbox{--}\Phi_{\mathrm{p}}\) relationship with a larger data set.

2 Description of the New Technique and Data

The post-eruption arcade (PEA) technique to determine \(\Phi_{\mathrm{r}}\) makes use of the close relationship between flare ribbons and PEAs because the ribbons essentially mark the footprints of PEAs. Instead of counting the ribbon pixels in a series of images, we demarcate the area under PEAs, \(A_{\mathrm{P}}\), using a polygon, overlplot the polygon on a magnetogram obtained around the time of the PEA image, and sum the absolute value of the photospheric magnetic flux in all the pixels within the polygon. The resulting RC flux is then half of the total flux through the polygon. The justification for this technique is that the ribbon separation ends after the flare maximum and the accumulated pixel area should roughly correspond to the area below the PEA on one side of the polarity inversion line.

2.1 Illustrative Examples

As an example we consider the 12 May 1997 eruption, which resulted in a magnetic cloud (MC) detected by the Wind and Advance Composition Explorer (ACE) spacecraft (Gopalswamy et al. 2010). The eruption occurred in NOAA Active Region (AR) 8038 located at N21W08. Various aspects of this event have been studied extensively using remote observations (Thompson et al. 1998; Webb et al. 2000), in-situ observations (Baker et al. 1998), and modeling (Titov et al. 2008). Figure 1 shows the evolution of the eruption in a series of \(\mathrm{H}\upalpha\) pictures from the Domeless Solar Tower Telescope of the Kyoto University’s Hida Observatory in Japan (Nakai and Hattori 1985) revealing the evolution of the flare ribbons. Initially, the ribbons are thin and close to each other. As time progresses, the ribbons separate and their areas increase. The RC flux in the ribbons is the magnetic flux computed over the cumulative area of the ribbons on one side of the polarity inversion line (PIL) in the flaring region. Ideally, the ribbons are well defined over the whole event and either side of the PIL can be used. In practice, the ribbons can have different morphology on either side of the PIL. If the measurements are from both sides of the ribbon, we divided the flux by two. Sometimes, we were able to measure the area only from one side of the PIL. In computing the cumulative area, we eliminated the overlapping area in successive frames to avoid double counting. In the case of the 12 May 1997 event, both ribbons are well defined, so that we computed the total cumulative area below both ribbons. The outline of the cumulative ribbon areas are superposed on a magnetogram obtained by the Michelson Doppler Imager (MDI: Scherrer et al. 1995) onboard the Solar and Heliospheric Observatory (SOHO: Domingo, Fleck, and Poland 1995), as shown in Figure 1. The cumulative ribbon area was \(1.7\times10^{19}~\mbox{cm}^{2}\) (both ribbons) and the average field strength was 171.5 G. The RC flux corresponds to the flux in one ribbon, which means that we obtain an average \(\Phi_{\mathrm{rR}} = 1.46\times10^{21}~\mbox{Mx}\). We denote the RC flux from the ribbon method by \(\Phi_{\mathrm{rR}}\) to distinguish it from the flux obtained using the arcade method (\(\Phi_{\mathrm{rA}}\)).

Figure 1
figure 1

A series of \(\mathrm{H}\upalpha\) images (a – e) from the Hida Observatory of Kyoto University in Japan showing flare ribbons for the 12 May 1997 (event 2) eruption from 04:46 to 05:31 UT. The cumulative ribbon areas denoted by the green contours in panel (f) are superposed on an MDI magnetogram obtained at 04:52 UT (white is the positive and black is the negative magnetic field in the magnetogram). The total ribbon area is \(1.7\times10^{19}~\mbox{cm}^{2}\). With the average field strength of 171.5 G within the ribbon area, the RC flux becomes \(1.46\times10^{21}~\mbox{Mx}\) (average below one ribbon).

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In order to compute \(\Phi_{\mathrm{rA}}\), we consider a 195 Å EUV image at 16:50 UT obtained by the Extreme-ultraviolet Imaging Telescope (EIT: Delaboudiniére et al. 1995) onboard SOHO. By visual examination, we marked the edges of the PEA as a polygon, which we superposed on an MDI magnetogram obtained at 16:04 UT (see Figure 2). The flux in each MDI pixel within the polygon is summed to obtain the total flux in the arcade area as \(4.72\times10^{21}~\mbox{Mx}\). Half of this quantity is the flare RC flux from the arcade method, \(\Phi_{\mathrm{rA}} = 2.36\times10^{21}~\mbox{Mx}\). We see that \(\Phi_{\mathrm{rA}}\) is larger than \(\Phi_{\mathrm{rR}}\) by 38%. Given the uncertainties in identifying the edges of the ribbons, the agreement is reasonable.

Figure 2
figure 2

(a) The post-eruption arcade imaged by SOHO/EIT at 195 Å at 16:50 UT for the 12 May 1997 (event 2) eruption. The solid lines are drawn near the feet of the arcade, while the dashed lines complete the polygon. The EUV images have a pixel size of \({\sim}\,2.6~\mbox{arcsec}\). (b) The polygon from panel (a) is superposed on an MDI magnetogram at 16:04 UT. Summing the flux from each pixel within the area occupied by the arcade (corrected for projection effects) and dividing by two, we obtain the total flux as \(2.36 \times10^{21}~\mbox{Mx}\) (the arcade area is \(4.25\times10^{19}~\mbox{cm}^{2}\)).

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Since there were not many events with good \(\mathrm{H}\upalpha\) flare observations, we also considered flare ribbons observed with TRACE at 1600 Å. Figure 3 shows an event with PEAs from SOHO/EIT 195 Å and TRACE 171 Å, while the ribbons are from TRACE 1600 Å. The eruption is the famous Bastille Day (14 July 2000) event, which had severe space weather impact because of a large solar energetic particle event that included ground-level enhancement (Gopalswamy et al. 2004; Gopalswamy et al. 2012; Mewaldt et al. 2012) and a super-intense geomagnetic storm (Zhang et al. 2007). Various aspects of this eruption have been extensively studied (Reiner et al. 2001; Aschwanden and Alexander 2001; Yan and Huang 2003 and references therein). As in Figure 1, we mark the PEA area by a polygon drawn on the SOHO/EIT 195 Å image. The TRACE image has a spatial resolution of \({\sim}\,1~\mbox{arcsec}\) compared to \({\sim}\,5~\mbox{arcsec}\) in the EIT image, which is evident in the detailed structure of the arcade in the TRACE image. We also see some difference at the edges of the arcade, where TRACE observes additional faint structures. Despite this difference, we use the EIT images for identifying the arcade area because they are full-disk images and hence most of the eruptions are observed. Furthermore, not all arcades were observed by TRACE because of its limited FOV, and it was generally pointed at intense flaring regions. The RC flux estimated from the EIT area of the arcade and the MDI pixels within the area gives a flux of \(13.1\times10^{21}~\mbox{Mx}\). The additional structures to the right edge of the polygon contribute only \({\sim}\,5.2\times10^{19}~\mbox{Mx}\), which is only about 0.4% of the arcade flux and hence negligible. Figure 3 also shows that the ribbon is very extensive and well defined for measurements of the RC flux from flare ribbons. We used 71 of the 381 TRACE 1600 Å images available between 10:00 UT and 14:59 UT to obtain the ribbon area as \(5.64\times10^{19}~\mbox{cm}^{2}\). The average magnetic field strength below the ribbon area was 446.9 G. The average RC flux from the ribbons on one side of the neutral line was obtained as \(12.61\times10^{21}~\mbox{Mx}\). This value is also close to the RC flux from the arcade method, differing only by \({\sim}\,3.7\%\). We note that the RC flux in the 14 July 2000 event is much higher than the flux in the 12 May 1997 event because of the high magnetic field strength. Only one \(\mathrm{H}\upalpha\) image at 10:03 UT was available for this event. The ribbon area was \(1.69\times10^{19}~\mbox{cm}^{2}\) and the average field strength was 557.5 G, giving a partial RC flux of \(9.44\times10^{21}~\mbox{Mx}\), which is an underestimate, but consistent with the RC flux from the arcade method.

Figure 3
figure 3

PEA of the 14 July 2000 (event 19) eruption as observed by (a) SOHO/EIT at 195 Å and (b) TRACE at 171 Å. The TRACE image in panel (c) taken at 10:39 UT shows the ribbons at 1600 Å. The green contour in (c) corresponds to the cumulative ribbon area (TRACE 1600 Å) between 10:00 and 14:59 UT.

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2.2 The Data Set

We are interested in solar eruptions that result in an MC at Earth so that we can make quantitative comparison between the flare RC flux and the poloidal flux of the associated MCs. For this purpose, we started with the list of 54 eruptions in Solar Cycle 23 considered for the Flux Rope CDAW workshops (Gopalswamy et al. 2013b). The eruptions occurred from within \({\pm}\,15^{\circ}\) in longitude from the disk center. The longitudinal criterion was imposed to ensure that the associated CMEs observed by the Large Angle and Spectrometric Coronagraph (LASCO: Brueckner et al. 1995) arrived at Earth as interplanetary CMEs (ICMEs), as detected in situ by spacecraft located at Sun–Earth L1. The PEAs in the eruptions were observed in EUV or X-rays, and their measurable properties have already been reported (e.g., Yashiro et al. 2013). Yashiro and colleagues compared PEAs associated with CMEs that ended up at 1 AU as MCs and non-cloud ICMEs. We are interested in MCs because their flux rope structure allows us to determine the poloidal flux of the MCs at 1 AU (e.g., Lepping, Burlaga, and Jones 1990) to compare them with the RC flux. The fitted parameters of the MCs are available online ( http://wind.nasa.gov/mfi/mag_cloud_S1.html ).

Of the 54 ICMEs, only 23 were MCs. For one of the MCs, there was no EIT observations. For another event, the arcade at the Sun was not well defined. Excluding these two events, we have listed the remaining 21 events in Table 1. The event number (column 1) is taken from the CDAW list (Gopalswamy et al. 2013a). We have retained the original event identifiers to facilitate comparison. The date and time of the MC and the associated CME at the Sun are listed in columns 2 and 3, respectively. The start time of the associated soft X-ray flare, the flare class, and the flare location in heliographic coordinates are listed in columns 4, 5, and 6, respectively. The area below the EUV arcade (\(A_{\mathrm{a}}\) in units of \(10^{19}~\mbox{cm}^{2}\), column 7), the average field strength (\(B_{\mathrm{a}}\) in Gauss, in column 8) and half of the total unsigned magnetic flux below the arcade (\(\Phi_{\mathrm{rA}}\) in units of \(10^{21}~\mbox{Mx}\), column 9) are derived with the PEA method described in Figures 1 and 2. \(\mathrm{H}\upalpha\) observations were generally not uniform and were also not available for many events. In some cases, we were able to measure the ribbon area only on one side of the neutral line. In some cases, only a single frame of the \(\mathrm{H}\upalpha\) image was available; the ribbon area in such cases represents a lower limit to the area. In all, there was at least one \(\mathrm{H}\upalpha\) picture for 12 events. Of these, only eight had more than one \(\mathrm{H}\upalpha\) frames; the number of frames were sufficient to obtain the RC flux only in six cases. For the remaining events, the ribbon areas and hence the RC flux were underestimated. The ribbon area (\(A_{\mathrm{R}}\), in units of \(10^{19}~\mbox{cm}^{2}\)), the average magnetic field strength (\(B_{\mathrm{R}}\) in units of Gauss), and the RC flux from the ribbon method are listed in columns 10, 11, and 12, respectively. When we searched for ribbon observations in TRACE 1600 Å data, we found eight events (events 19, 21, 32, 43, 45, 46, 49, and 53 noted with a superscript “T” in column 10). For event 43 there was only one frame in the rise phase, therefore the ribbon area is an underestimate. The remaining seven events had usable TRACE data. In column 10, the source of the ribbon data (H – \(\mathrm{H}\upalpha\); T – TRACE) is noted. The poloidal flux \(\Phi_{\mathrm{p}}\) of MCs computed from the Lundquist solution (Lepping, Burlaga, and Jones 1990) is listed in column 13. Of the many output parameters obtained by the flux rope fitting, we use the axial field strength (\(B_{0}\)) and the flux rope radius (\(R_{0}\)) at 1 AU to derive the poloidal flux of the MC,

$$ \Phi_{\mathrm{p}} = L (B_{0} R_{0} )/x_{01}, $$
(2)

assuming that the flux rope extends up to the radius where the axial field component vanishes. Here \(x_{01}\) is the first zero (2.4048) of the Bessel function \(J_{0}\) and \(L\) is the total length of the flux rope, taken as 2 AU following Nindos, Zhang, and Zhang (2003).

Table 1 List of solar eruptions that were associated with magnetic clouds at Earth.
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3 Analysis and Results

3.1 Comparison Between RC Flux from the Arcade and Ribbon Methods

We extend the case studies presented in Section 2.1 to all Cycle 23 eruptions that originated within \({\pm}\,15^{\circ}\) from the disk center and resulted in MCs at 1 AU (see Table 1). First we check whether the arcade and ribbon methods yield consistent results. To do this, we extracted the 15 events with ribbon information (H\(\alpha\) or TRACE) and listed them in Table 2. For six events, there were no usable ribbon data from either \(\mathrm{H}\upalpha\) or from TRACE 1600 Å (events 23, 24, 27, 39, 44, and 54 in Table 1). We separated \(\mathrm{H}\upalpha\) and TRACE data on ribbons along with the number of frames available for each event. We also list the number of frames available for each event in \(\mathrm{H}\upalpha\) and TRACE. In the last column, we list the name of the observatory that provided the \(\mathrm{H}\upalpha\) images. When the number of frames was insufficient to make complete ribbon measurements in an event, we considered the computed RC flux be an underestimate. Only three events (events 32, 49 and 53) had data both in \(\mathrm{H}\upalpha\) and TRACE. Table 2 shows that there were only a total of ten events with RC flux computed from flare ribbons. In the remaining five events, ribbons were observed, but not in a sufficiently large number of frames to compute the cumulative ribbon areas. However, we used these events to show that they provide lower limits to the RC flux and check whether they have the correct trend.

Table 2 List of events with RC flux computed from \(\mathrm{H}\upalpha\) or TRACE 1600 Å ribbons.
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The RC fluxes from the arcade and ribbon methods are shown in Figure 4a as a scatter plot, indicating a high correlation (\(r=0.94\)). A least-squares fit gives the regression equation,

$$ \Phi_{\mathrm{rA}} = 1.24(\Phi_{\mathrm{rR}})^{0.99}. $$
(3)
Figure 4
figure 4

(a) Scatter plot between the RC fluxes obtained from the arcade method and the ribbon method for ten events identified by the event numbers. The RC flux from \(\mathrm{H}\upalpha\) ribbons (red symbols) and TRACE 1600 Å ribbons (blue symbols) are distinguished. Data points connected by a horizontal line represent ribbon measurements from \(\mathrm{H}\upalpha\) and TRACE. In these cases, the TRACE and \(\mathrm{H}\upalpha\) data were combined to obtain the cumulative area, and the resulting flux is indicated by a black square (events 32, 49, and 53). The dotted line corresponds to equal arcade and ribbon fluxes. The solid line is the least-squares fit to the data points (\(\mathrm{H}\alpha + \mbox{TRACE}~1600~{{\mathring{\mathrm{A}}}}\)). The correlation coefficient (0.94) is higher than the critical value (0.549) of the Pearson correlation coefficient for ten data points at 95% confidence level. In events with both \(\mathrm{H}\upalpha\) and TRACE measurements, the combined flux (black squares) is used in the correlation. (b) The same scatter plot, but this time, it includes events with incomplete ribbon data (open circles and diamonds). The plotted fluxes are lower limits for these events. Event 33 discussed in the text is pointed at by an arrow.

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This result is significant because it shows that the simpler arcade method works well. The regression line deviates only slightly from the equal-flux line (\(\Phi_{\mathrm{rA}} = \Phi_{\mathrm{rR}}\)). The largest deviation is for event 37, in which \(\Phi_{\mathrm{rA}}\) is greater than \(\Phi_{\mathrm{rR}}\) by a factor of \({\sim}\,2.4\). In Figure 4b we include events that did not have complete ribbon information (open symbols). The RC fluxes from the ribbon method are underestimated in these events because not enough frames are available. However, the open symbols are consistent with the trend that the RC flux from the arcade method correlates with that from the ribbon method. If these events had complete observations, the open symbols would move closer to the equal-flux line. We also note that five of the ten data points in Figure 4a are on the line of equal fluxes. The fact that the equal-flux line needs to be multiplied by 1.24 to obtain the regression line indicates that the RC flux is overestimated by the arcade method and underestimated by the ribbon method, or both. Possibilities for the overestimate of the RC flux from the arcade method include i) the arcade is observed in the corona, resulting in a slight overestimate of the arcade area compared to the actual area at the photospheric level and hence a higher RC flux, and ii) the arcade flux might include contributions from very close to the neutral line, while the ribbons may start from a finite distance from the neutral line. These fluxes that do not participate in the flare reconnection can also contribute to an overestimate of the arcade flux.

Event 37 had 11 \(\mathrm{H}\upalpha\) frames covering most of the flare duration. There was no data coverage for the last 2 hours in the decay phase of the flare. Most likely, the RC flux from the ribbon method was underestimated. Except for this event, we see from Table 2 that the ratio \(\Phi_{\mathrm{rA}} /\Phi_{\mathrm{rR}}\) is in the range 0.89 to 1.6 for all events with well-defined \(\Phi_{\mathrm{rR}}\). One of the events with the highest ratio (\({\sim}\,1.6\)) is the 29 October 2003 eruption (event 46). This well-known eruption is from AR 10486 that resulted in a magnetic cloud (Gopalswamy et al. 2005). Figure 5 shows an overview of the event with the EUV arcade and the PIL involved. The ribbon on the positive side was fragmented because of a narrow lane of negative field region at the northern end of the arcade. We used the ribbon on the negative side of the PIL, which was not fragmented. The times of the available \(\mathrm{H}\upalpha\) and TRACE 1600 Å frames are marked in the GOES soft X-ray plot in Figure 5; these frames were used to compute the RC flux using the ribbon method. The early part of the flare ribbons was observed with high cadence, while the late-decay phase was observed with lower cadence by TRACE. In \(\mathrm{H}\upalpha\), there were only three frames in the decay phase. Using the TRACE observations alone, we obtained an RC flux of \({\sim}\,13.82\times10^{21}~\mbox{Mx}\). The \(\mathrm{H} \alpha\) observation alone gave a lower value (\(5.99\times10^{21}~\mbox{Mx}\)) because of the incomplete observations. On the other hand, the \(\Phi_{\mathrm{rA}}\) of this event was \(21.6\times10^{21}~\mbox{Mx}\). It seems unlikely that \(\Phi_{\mathrm{rA}}\) in this event is overestimated. An underestimate of the RC flux using the ribbon method is possible because of the uneven coverage. For example, there was a gap of \({\sim}\,30~\mbox{min}\) between the last TRACE observation and the first \(\mathrm{H} \alpha\) observation. Figure 5 also shows that the arcade extends farther to the south than the cumulative ribbon area, suggesting a possible underestimate of the RC flux. This shows that using the ribbon method can be subjective because it depends on the threshold used in defining the ribbons. The Halloween event on 28 October 2003 (event 45) was also an event in the list of Qiu et al. (2007), who estimated the RC flux from TRACE ribbons as \(18.8\times10^{21}~\mbox{Mx}\), which is similar to our value (\(20.88\times 10^{21}~\mbox{Mx}\)). Both values are in good agreement with the RC flux from the arcade method.

Figure 5
figure 5

The 29 October 2003 eruption (event 46) that showed a large deviation between RC fluxes from the arcade and ribbon methods for the group of ten cases in which \(\Phi_{\mathrm{rR}}\) could be estimated. (a) PEA at 00:49 UT on 30 October 2003 with the edges of the arcade marked by the red lines. The white box placed at S15W08 marks the area where another flare occurred, but it did not affect the flux computations. (b) The arcade area superposed on an MDI magnetogram taken at 20:47 UT. (c) The combined ribbon area superposed on the MDI magnetogram. (d) The GOES soft X-ray light curve (black) with the times of TRACE (green) and \(\mathrm{H}\upalpha\) (red) images used in computing the ribbon areas. The second flare can be seen as a small increase in the GOES intensity around 21:30 UT. The main part of the flare had high-cadence TRACE observations, while in the late-decay phase the cadence dropped to below 10 min. Only three \(\mathrm{H}\upalpha\) images were available for this event.

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The event on 26 April 2001 (event 33) was a clear case of an overestimated arcade flux; it is marked in Figure 4b. A large bipolar active region was located near the PIL of the arcade, but did not participate in the flare reconnection and hence contributed to overestimating \(\Phi_{\mathrm{rA}}\). We estimate the unsigned flux that is due to the bipole as \({\sim}\,15.2\times10^{21}~\mbox{Mx}\) from the MDI magnetogram. Since the bipole did not participate in the eruption process, it needs to be subtracted from the total arcade flux (\(39.2 \times10^{21}~\mbox{Mx}\)). The corrected arcade flux is \(24.0\times10^{21}~\mbox{Mx}\), and the RC flux accordingly is \(12.0\times10^{21}~\mbox{Mx}\). The corrected value is now closer to the equal-flux line. Unfortunately, there were \(\mathrm{H}\upalpha\) frames only in the decay phase of the flare, so the RC flux is underestimated. Using the nine available frames, the RC flux from the ribbon method was estimated as \(6.45\times10^{21}~\mbox{Mx}\). The true value is expected to be higher, therefore the data point will move to the right, closer to the equal fluxes line. This event demonstrates that large deviations can be understood by examining the magnetograms and the PEAs.

Qiu et al. (2007) may have misidentified the solar eruption in this event as described in Figure 6, which shows the PEA and the source region of the eruption from the Yohkoh/Soft X-ray Telescope, SOHO/EIT, MDI magnetogram, and the GOES light curve. This event has large uncertainties both at the Sun and at 1 AU. The CME of interest was a halo (likely to reach Earth) and first appeared in the LASCO FOV at 12:30 UT on 26 April 2001. The CME was associated with a gradual flare with a soft X-ray flare class of M1.5 that resulted in a large PEA. The centroid of the arcade was at N20W05. Roughly after the peak of the M1.5 gradual flare, an impulsive flare (M8.9) started at \({\sim}\,13{:}04\) UT and peaked at 13:13 UT. The M8.9 flare originated about \(26^{\circ}\) to the west of the centroid of the arcade. The M8.9 flare was not associated with the halo CME. It was associated with a narrow CME that first appeared in the LASCO FOV at 13:31 UT and was heading to the west. Qiu et al. (2007) used this flare to derive the flare RC flux, but it is unlikely that the 13:31 UT CME would have arrived at Earth. Both the M1.5 and M8.9 eruptions were from the same magnetic complex, which consisted of AR 9433 surrounded by weaker magnetic flux regions. Figure 6 shows that AR 9433 was located between the arcade centroid and the impulsive flare source. AR 9433 was located below the PEA, but close to its western part. The RC flux from the impulsive source was reported by Qiu et al. (2007) as \(0.7\times10^{21}~\mbox{Mx}\), more than an order of magnitude smaller than that of the M1.5 gradual flare (\(12.0\times10^{21}~\mbox{Mx}\) after eliminating the flux due to AR 9433).

Figure 6
figure 6

(a) A Yohkoh/Soft X-ray Telescope (SXT) image showing the PEA associated with the 26 April 2001 (event 33) eruption. The arcade was associated with the M1.5 gradual flare from N20W05. There was another impulsive flare starting around 13:04 UT, but to the west at N17W31 (inside the box shown). The full-disk SXT images did not capture the impulsive event, but the SOHO/EIT image shows the compact event at 13:13 UT enclosed by the box in (b). (c) MDI magnetogram of the extended magnetic region that hosted both the gradual and the impulsive eruptions. NOAA AR 9433 (pointed at by arrow) was located between the two eruptions, but did not participate in the eruption. The feet of the arcade (red lines) and the box within which the impulsive flare occurred are superposed on the magnetogram. (d) The GOES soft X-ray light curve showing the gradual M1.5 flare and the impulsive M8.9 flare.

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A similar case of multiple events was observed on 11 April 2004 (event 49). At the solar source (AR 10696), three M-class flares occurred in quick succession (see http://www.lmsal.com/solarsoft/last_events_20041106_1019/index.html for details). The first was an M9.3 impulsive flare (00:11 to 00:42 UT) from N08E05 with no obvious CME association. The second was an M5.9 flare (00:44 to 01:10 UT) that occurred almost in the same location (N10E06) as the first, but was associated with a halo CME (Gopalswamy, Yashiro, and Akiyama 2006, their Figure 4) with a speed of \({\sim}\,818~\mbox{km}\,\mbox{s}^{-1}\) appearing at 01:31 UT in the LASCO FOV. The third flare was of X-ray class M3.6 (01:40 to 02:08 UT) and occurred to the west and south of the previous flares, at N07E00. This flare was associated with a faster (\(1111~\mbox{km}\,\mbox{s}^{-1}\)) partial-halo CME, which appeared at 02:06 UT in the LASCO FOV. It quickly caught up with the previous CME. The 1 AU MC has been determined to be due to the second CME (Gopalswamy et al. 2010). Although the arcade appears to be a single one, it is possible to separate it into individual sections by looking at the time evolution. The arcade corresponding to the second CME had an RC flux of \(6.70\times10^{21}~\mbox{Mx}\), while the combined arcade yielded a flux of \(14.85\times10^{21}~\mbox{Mx}\). Both \(\mathrm{H}\upalpha\) and TRACE observations were available for this event, which yielded reasonable values of the RC flux, especially when the two sets of observations were combined (see Figure 4a).

3.2 Comparison Between the RC Flux and the MC Poloidal Flux

We now consider how the RC flux from the arcade method relates to the poloidal flux of the associated MCs listed in Table 1. Figure 7a shows a scatter plot between \(\Phi_{\mathrm{r}}\) and \(\Phi_{\mathrm{P}}\) (we drop the subscript “A” in \(\Phi_{\mathrm{rA}}\) for simplicity). We see a modest correlation, with an equal number of data points on either side of the equal-flux (\(\Phi_{\mathrm{r}} = \Phi_{\mathrm{P}}\)) line. A linear least-squares fit to \(\Phi_{\mathrm{r}}\mbox{--}\Phi_{\mathrm{P}}\) pairs (on a logarithmic scale) gives the regression equation,

$$ \Phi_{\mathrm{P}} = 1.20(\Phi_{\mathrm{r}})^{0.85}. $$
(4)
Figure 7
figure 7

(a) Scatter plot between the RC flux from the arcade method (\(\Phi_{\mathrm{r}}\)) and the 1 AU poloidal flux \((\Phi_{\mathrm{P}})\) in the associated magnetic cloud. The dotted line denotes equal fluxes (\(\Phi_{\mathrm{P}} = \Phi_{\mathrm{r}}\)). The correlation coefficient \(r = 0.43\) is higher than the critical value of the Pearson correlation coefficient (0.378 at 95% confidence level). (b) Same as (a) with the data points from Qiu et al. (2007) superposed (black squares), showing that the two sets of data yield a similar relation between \(\Phi_{\mathrm{r}}\) and \(\Phi_{\mathrm{P}}\). The vertical black lines indicate the range of values over which the average flux was computed in Qiu et al. (2007). We have shown event 33 with the uncorrected (open circle) and corrected (filled circle) fluxes. Although this event follows the trend of the scatter plot, we did not include it for the reasons given in the text.

Full size image

The correlation coefficient (\(r = 0.43\)) is higher than the critical value for the Pearson correlation (0.378 at 95% confidence level). Thus the \(\Phi_{\mathrm{P}}\mbox{--}\Phi_{\mathrm{r}}\) relationship is very similar to Equation (1), obtained by Qiu et al. (2007) for nine events. The coefficient of \(\Phi_{\mathrm{r}}\) in Equation (4) is only 7.1% higher than that in Equation (1). Similarly, the exponent of \(\Phi_{\mathrm{r}}\) is only 3.6% larger than that in Qiu et al. (2007) result. More importantly, the RC flux used in Equation (1) was from the ribbon method, whereas it is from the arcade method in Equation (4). Thus we confirm the result presented in Figure 4, viz., the RC fluxes obtained from the arcade and ribbon methods lead to similar correlation with the MC poloidal flux. In addition, we see that the flare RC flux at the Sun is closely related to the poloidal flux of the associated MC, especially close to the \(\Phi_{\mathrm{r}} = \Phi_{\mathrm{P}}\) line. Webb et al. (2000) considered the axial flux of the MC and related it to the flux in the dimming region for event #2 and found good agreement. Axial field is related to, but smaller than, the poloidal flux in a force-free flux rope. These authors obtained a dimming flux of \(\sim 1.0\times 10^{21}~\mbox{Mx}\), which is smaller than the RC flux (see Table 1).

The close correspondence between the results of Qiu et al. (2007) and our result may be due to a combination of several circumstances. Since Qiu et al. used a flux rope length of 1 AU instead of the 2 AU used by us, we may be overestimating \(\Phi_{\mathrm{P}}\). If we also overestimated the RC flux in the arcade method, then these two might have balanced to yield the same relationship between \(\Phi_{\mathrm{P}}\) and \(\Phi_{\mathrm{r}}\). However, Qiu et al. (2007) used \(\Phi_{\mathrm{P}}\) values that are taken from three different sources. Sometimes the values differed by an order of magnitude for a given event. Therefore, the comparison between our \(\Phi_{\mathrm{P}}\) and that of Qiu et al. (2007) may not be appropriate. When we checked our \(\Phi_{\mathrm{P}}\) values with those in Qiu et al. (2007) for five events that are also in our list (events 21, 23, 32, 45, and 53), we found that our values are only slightly higher, by a factor 1.3. The reconnection flux from the ribbon method obtained by Qiu et al. (2007) in three of the five cases was lower than our reconnection flux from the ribbon method. Therefore it is possible that this underestimate and their use of 1 AU for the flux rope length balanced out. We also found that one of the data points (10 April 2001) differs between their Table 4 and their Figure 8c. When we used their table values, we obtained a new line with the same exponent, but with a slightly higher coefficient (1.24 vs. 1.12). Because of the small sample available for comparison, it is difficult to derive strong conclusions on the similarity between Equations (1) and (4).

It must be noted that sometimes the association between the solar and interplanetary events may be incorrect. For instance, in the case of the 10 April 2001 event (event 33) discussed in Section 3.1, the low value of the RC flux obtained by Qiu et al. (2007) matched the MC poloidal flux at 1 AU (\(1.25\times10^{21}~\mbox{Mx}\)). This is fortuitous because the CME–MC association was not correct. Moreover, the MC was quite complex, with different authors giving different durations ranging from 11 to 60 hours (see Table 1 of Qiu et al. 2007). In the CDAW list, the MC was reported to have a duration of \({\sim}\,11~\mbox{hours}\). However, there was ICME material with a low proton temperature before and after the MC interval. Our poloidal flux value was \(2.5\times10^{21}~\mbox{Mx}\) (since we use a flux rope length of 2 AU), which is lower by a factor of 4.8 than the RC flux from the arcade. Because of the uncertainties surrounding the 26 April 2001 event, we did not include it in the correlation, although the data points shown in Figure 7 are consistent with the rest of the events.

4 Discussion

We presented a new method to estimate the flare RC flux based on post-eruption arcades and the photospheric magnetic field underlying the arcades. We have shown that i) half the magnetic flux below post-eruption arcades is a reasonable representation of the flare RC flux (Figure 4), and ii) the RC flux obtained from the arcade method and the poloidal flux of the associated magnetic cloud at 1 AU are correlated significantly (Figure 7). The relationship is very similar to the one obtained by Qiu et al. (2007).

One of the major advantages of the new method is that it requires only one image that shows the “mature” arcade in EUV, X-rays, or even microwaves (see e.g., Hanaoka et al. 1994; Gopalswamy et al. 2013a). Typically, this image corresponds to the decay phase of flares. Observations of post-eruption arcades are generally better available than that of flare ribbons. We see that only ten of the 23 eruptions from Cycle 23 had usable ribbon observations (from \(\mathrm{H}\upalpha\) and TRACE 1600 Å). On the other hand, 21 of the 23 eruptions had usable arcade observations. Thus \({\sim}\,3\) times more events are usable for the arcade method than for the ribbon method in Cycle 23. The reliability of the arcade method is important for space weather applications.

Hu et al. (2014) extended the study of Qiu et al. (2007) to include ten more events from Cycle 24, thus doubling their sample. They did find the continued correlation between the flare RC flux and MC poloidal flux, but with a pronounced deviation from the equal-flux line. Their sample size (19) is now similar to ours (21), but their solar source locations do not meet our longitude criterion (within \({\pm}\,15^{\circ}\)) for all but two events. One of the two events did not have a computed RC flux. The only event matching our criterion (23 May 2010 eruption from N19W12) was reported to have \(\Phi_{\mathrm {r}} = 0.3 \times10^{21}~\mbox{Mx}\) and \(\Phi_{\mathrm{P}} = 0.83\times10^{21}~\mbox{Mx}\) (Hu et al. 2014). Substituting this \(\Phi_{\mathrm{r}}\) into Equation (4), we obtain \(\Phi_{\mathrm{P}} = 0.43\times10^{21}~\mbox{Mx}\), which is within a factor of two from Hu et al.’s (2014) \(\Phi_{\mathrm{P}}\) value. We note that Hu et al. (2014) considered only the Grad–Shafranov (GS) method of flux rope fitting rather than the force-free (FF) fitting we used. In Table 4 of Qiu et al. (2007), all the poloidal flux values from the FF method were higher than those from the GS method – by a factor of \({\sim}\,1.6\) on average. In our case, we were able to estimate the poloidal flux using the GS method for 13 of the 21 events (Möstl 2014, private communication). In all 13 cases, the FF method gave a poloidal flux higher by a factor of five on average. Similar differences were also found in the five events in Qiu et al. (2007) that overlapped with ours: the GS fluxes were lower by factors of 5.8 than the Qiu et al. FF values and by a factor of 7.6 than our values. Since we have good information on magnetic clouds of Cycle 24 (Gopalswamy et al. 2015), we will add more data points to the current 21 for better statistics and report the results elsewhere.

We considered only eruptions from the disk center that were observed as MCs at 1 AU. However, there are many disk-center eruptions that are not observed as MCs. Propagation effects such as deflection in the corona (Xie, Gopalswamy, and St. Cyr 2013; Mäkelä et al. 2013) seem to be responsible for the non-cloud appearance of these events at 1 AU, even though there is no difference in the source properties of cloud and non-cloud ICMEs (Gopalswamy et al. 2013a; Yashiro et al. 2013). Therefore, it is possible to estimate the expected poloidal flux of the non-cloud ICMEs at 1 AU from the RC flux. Marubashi et al. (2015) were able to fit a flux rope to all but three of the non-cloud ICMEs in the CDAW list. A comparison between the RC flux and the poloidal flux from Marubashi et al. (2015) may provide a clue to understand why the Lepping, Burlaga, and Jones (1990) force-free fitting did not recognize flux rope signatures.

5 Summary and Conclusions

We investigated a set of 21 solar eruptions originating from within \({\pm}\,15^{\circ}\) of the disk center and the associated magnetic clouds from Solar Cycle 23 to demonstrate a new method of measuring the flare RC flux. We measured the RC flux by combining observations of post-eruption arcades in EUV and line-of-sight photospheric magnetic fields. We also measured the RC flux from \(\mathrm{H}\upalpha\) and TRACE 1600 Å observations using the flare ribbon method. We found that the RC flux obtained from the two methods agreed quite closely. The RC flux from the arcade method is slightly higher than the flux from the ribbon method. This is mostly due to insufficient flare ribbon data. Occasionally, there may be high flux close to the polarity inversion line that does not participate in the reconnection and hence can cause an overestimate of the arcade RC flux. We also computed the poloidal flux of the associated magnetic clouds at 1 AU and found it to be approximately equal to the RC flux. This result is consistent with the idea that the flux ropes are formed during eruptions and any pre-existing flux rope needs to be rather small in the set of events we considered.