Comparing Long-duration Gamma-Ray Flares and High-energy Solar Energetic Particles

Among the more unusual solar phenomena are the long-duration gamma-ray flares (LDGRFs). The prime feature of these events is delayed and prolonged γ-ray (>100 MeV) emission after the impulsive phase (Ryan 2000). As we discuss below, with the exception of the 2.223 MeV line and other associated emission, there is no clear signal at other wavelengths while this high-energy emission persists. The LDGRF emission is believed to originate from the decay of pions produced by ≳300 MeV protons and ≳200 MeV α particles. Typically, LDGRFs are associated with fast coronal mass ejections (CMEs) and large solar energetic particle (SEP) events,  often exceeding  the energies of ground-level enhancements (GLEs; ≳500 MeV). However, any direct connection between energetic GLE-type particles observed in space and the accelerated ion population producing the high-energy γ-ray emission is unclear (Cliver 1989).

During the Solar Maximum Mission and the Compton Gamma Ray Observatory (CGRO) mission, only 12 LDGRFs were observed (Forrest et al. 1985; Ryan 2000; Chupp & Ryan 2009). The LDGRFs of 1991 June observed with CGRO and GAMMA-1 exhibited nuclear emissions for many hours after the impulsive phase (Akimov et al. 1991; Kanbach et al. 1993; Rank et al. 2001). Notably, the 1991 June 11 flare exhibited emission for nearly 11 hr after the impulsive phase with >50 MeV emission detected with the Energetic Gamma Ray Experiment Telescope (EGRET; Schneid et al. 1994). Mandzhavidze & Ramaty (1992) analyzed the EGRET data for the extended phase of the flare. They found a best fit to the emission spectrum with a combination of pion decay radiation and primary electron bremsstrahlung. However, this was later contraindicated by the Imaging Compton Telescope (COMPTEL) data (Rank et al. 2001) showing the 2.223 MeV neutron-capture line synchronized with the >100 MeV emission, consistent with a 100% nuclear origin.

More recently, the Fermi Large Area Telescope (LAT) has observed dozens of LDGRFs (Ackermann et al. 2012, 2014, 2017; Share et al. 2018). These peculiar events share some common characteristics: (1) the extended γ-ray emission is often delayed by several minutes from the impulsive hard X-ray emission and, in some cases, lasts for as long as tens of hr; (2) it appears to lack temporal structure on scales much less than the overall decay time, suggesting that the acceleration takes place over large volumes (≳10⁵ km), smoothing over the details of the dynamics; and (3) the emission is primarily from high-energy ion interactions, consistent with an origin from π⁰ decay (Ackermann et al. 2017). The most intense and longest-duration example of an LDGRF is the 2012 March 7 event, for which >100 MeV emission was observed for nearly 20 hr (Ajello et al. 2014). The long duration of the emission cannot be explained by passive trapping without continued production (such as occurs in the Earth's radiation belts), because even the minutest pitch angle scattering would cause particles to escape into the loss cone on much shorter timescales. To produce a 10 hr particle lifetime in a radiation belt–like loop configuration implies scattering mean free paths of tens of au (Ryan 2000) that are not even observed in the most anisotropic GLE events and would require a magnetic fluctuation level δB/B ∼ 0.05, far quieter than the typical solar wind turbulence.

Several scenarios have been posited to explain the many hours' duration of high-energy emission. These include (1) particle trapping with and without continuous acceleration within large coronal loops, characterized by delayed onset due to the need for ion energies to exceed the pion production threshold (Ryan & Lee 1991; Mandzhavidze & Ramaty 1992; Chupp & Ryan 2009); (2) backward precipitation of particles accelerated at a CME-driven shock (Cliver et al. 1993; Kocharov et al. 2015); and (3) particle acceleration by large-scale electric fields (Akimov et al. 1991, 1996; Litvinenko 2006). Early models considered the injection of particles into large loops with a single phase of acceleration followed by precipitation resulting in the observed γ-ray emission. Murphy et al. (1987) modeled the emission with two particle populations, while Mandzhavidze & Ramaty (1992) included the production of γ-rays from charged pions, the effects of passive particle trapping, and pitch angle scattering into an infinitesimal loss cone. Post-flare loops may provide the magnetic structures necessary for continuous acceleration, in which pitch angle scattering from magnetic turbulence within the loops serves to further accelerate the ion population (Ryan & Lee 1991). Loops of the necessary length appear during the gradual phase of two-ribbon flares and CME liftoff as field line reconnection gives rise to hot flare loops that cool within a few hours. The Hα loops may be as large as 5 × 10⁵ km, creating a system of arches that can last for several hours. Such structures may not be visible in soft X-rays, however. Recent microwave observations from the Expanded Owens Valley Solar Array (EOVSA) suggest the presence of an extended, static loop, with a circular length of  1.4 R_s (Gary et al. 2018), for the 2017 September 10 solar flare that produced significant >100 MeV γ-ray emission.

On the other hand, CME shock–accelerated protons could make their way back from the shock front to the photosphere, radiating γ-rays, as first proposed by Cliver et al. (1993). The CME-driven shocks are widely accepted to accelerate the majority of large gradual SEP events (Reames 1999; Cliver 2016) and, in so doing, transfer typically ≤10% of the CME kinetic energy to energetic particles (Mewaldt et al. 2008; Aschwanden et al. 2017). The highest-energy events, reaching GeV energies, originate within a few solar radii, where efficient shock acceleration can take place (Zank et al. 2000; Berezhko & Taneev 2003; Lee 2005; Afanasiev et al. 2015). Ng & Reames (2008) showed that GeV energies can be reached within ∼10 minutes for a shock speed of 2500 km s⁻¹ with typical coronal conditions. Large-scale quiet loops that could efficiently transport particles may connect the shock front (for extended periods) and lead particles back to the photosphere. The idea that the same CME-driven shock is responsible for accelerating SEPs and protons producing LDGRFs is supported by recent studies investigating statistical correlations (Gopalswamy et al. 2018; Winter et al. 2018). However, particle diffusion through the turbulent sheath downstream of the shock is expected to be inefficient; furthermore, the entire particle population is being rapidly convected outward. Kocharov et al. (2015) examined the feasibility of back precipitation and could achieve only up to 1% of GLE particles precipitating back to the Sun. Notably, Fermi/LAT observations revealed three behind-the-limb LDGRFs (Ackermann et al. 2017), suggesting a much larger longitudinal extent to the solar source. Plotnikov et al. (2017) investigated the 3D reconstruction of coronal shocks from these backside events and concluded that LDGRF emission begins when the magnetic connectivity to the shock reaches the solar surface facing the Earth. Indeed, the highest fluxes in terms of both LDGRF emission and in situ SEPs were observed for the backside event on 2014 September 1, with the fastest-moving shock. Jin et al. (2018) simulated this event using a global magnetohydrodynamic (MHD) model and also found that particles could escape downstream of the shock along magnetic field lines that connect to the solar surface facing Earth.

A CME-driven shock origin is not without challenges. Hudson (2017) pointed out the importance of magnetic mirroring as particles propagate sunward in preventing back precipitation to the photosphere, consistent with the conclusions of Kocharov et al. (2015). He proposed two alternative scenarios, including the "Lasso" scenario, where particles are trapped in a magnetic structure that subsequently retracts back to the solar surface. The second alternative is a coronal "thick target" scenario, where protons are trapped in a static volume for several hours. Depending on the level of turbulence in this volume, these same particles may be accelerated concomitant with the trapping, extending the potential duration for trapping (Ryan et al. 2018a, 2018b). Grechnev et al. (2018) found the detectable emission from the backside event of 2014 September 1 to be consistent with flare-accelerated particles trapped in static coronal loops and possibly reaccelerated in these loops by a shock wave excited by the initial eruption.

High-energy charged-particle events detected by in situ spacecraft at large distances from the Sun and LDGRFs share similar energy ranges for the protons/ions responsible for them. High-energy SEPs and LDGRFs are due to ions rather than electrons, and both are delayed by several minutes from the associated X-ray event (Kahler et al. 1984; Ryan 2000), suggesting a linkage. A potential association between LDGRFs and GLE-type particles has been investigated, but the results are inconclusive (Ramaty & Murphy 1987; Ryan 2000; Chupp & Ryan 2009; Ackermann et al. 2017). Absent accompanying signals at other wavelengths, understanding the LDGRF emission is difficult. Only a few options seem to survive scrutiny: (1) LDGRFs contribute directly to SEPs, (2) SEPs produce LDGRFs through back precipitation of particles accelerated at the CME-driven shock, and (3) LDGRFs and high-energy SEP events are correlated but not causally (e.g., the two phenomena may be linked to M- and X-class flares but are otherwise separate processes with no exchange of particles).

One way to constrain the possible scenarios is to compare the number of protons interacting at the Sun above the pion production threshold (∼300 MeV) inferred from the extended γ-ray emission with the number of SEPs in space above the same energy. This is now possible for the first time with the Payload for Matter-Antimatter Exploration and Light Nuclei Astrophysics (PAMELA) and the accompanying γ-ray observations with Fermi/LAT. In particular, we calculate the total number of >500 MeV protons at 1 au, N_SEP, taking advantage of the PAMELA and Solar Terrestrial Relations Observatory (STEREO) data with the aid of transport simulations, and compare it with the number of high-energy protons at the Sun, N_LDGRF, as deduced from Fermi/LAT data. The ions producing the emission detected with Fermi are in the same energy range as those observed with PAMELA (see next section), presenting an opportunity for a proper comparison of the two particle populations, which is key to determining the mechanism responsible for the extended duration of the γ-ray emission.

The paper is structured as follows. In Section 2 we present the SEP observations from PAMELA and LDGRF data from Fermi/LAT for the events used in our analysis. Section 3 describes how we derived information on the spatial extent of SEP events from combined spacecraft data and how simulations of SEP propagation were used to estimate the number of times particles cross 1 au. Our main results on comparing N_LDGRF and N_SEP are presented in Section 4 and discussed in Section 5. Finally, Section 6 reports our summary and conclusions.

PAMELA is a space experiment designed to measure the charged cosmic radiation (protons, electrons, their antiparticles, and light nuclei) in the energy range from several tens of MeV up to several hundred GeV. The instrument consists of a magnetic spectrometer equipped with a silicon tracking system, a time-of-flight system shielded by an anticoincidence system, an electromagnetic calorimeter, and a neutron detector. The Resurs-DK1 satellite carrying the apparatus was launched into a semipolar 70° inclination and elliptical (350–610 km) orbit on 2006 June 15. PAMELA provided comprehensive observations of the galactic, solar, and magnetospheric radiation in the near-Earth environment (Adriani et al. 2011, 2014; Bruno et al. 2016a, 2016b; Adriani et al. 2017). The mission lifetime was extended beyond 2015, in part due to the promise of new PAMELA SEP science, such as that presented in this paper. On 2016 January 24, the spacecraft lost contact with ground stations. The PAMELA team has recently published the detailed spectra of 26 high-energy SEPs between 2006 December and 2014 September (Bruno et al. 2018). These observations span a broad range in energy from ∼80 MeV to 1–2 GeV, encompassing both the low-energy measurements of in situ spacecraft and the ground-based observations of the neutron monitor network. The reported spectra are consistent with diffusive shock acceleration with clear exponential rollovers attributed to particle escape from within the shock region during acceleration. The absence of qualitative differences between the spectra of GLE and non-GLE events suggests that GLEs are not a separate class of SEP events but rather are the extreme end of a continuous spectral distribution. The PAMELA observations have been used by Bruno (2017) to calibrate the >80 MeV proton channels of the Energetic Proton, Electron, and Alpha Detectors (EPEADs) and the High Energy Proton and Alpha Detectors (HEPADs) on board GOES-13 and -15, bringing the detected spectral intensities in line with those registered by PAMELA and thus enabling a more reliable spectroscopic measurement up to ∼1 GeV for SEP events occurring during periods when PAMELA was not acquiring data or after the mission termination (Bruno et al. 2019).

Fermi/LAT is a pair-conversion telescope with sensitivity to γ-rays between ∼20 MeV and 300 GeV (Atwood et al. 2009) and a duty cycle for solar events of only ∼15%–20% due to frequent occultation of the Sun by the Earth and passages through the South Atlantic Anomaly (SAA). However, the Fermi satellite is able to perform pointed ("target-of-opportunity") observations increasing the exposure to a particular part of the sky, including the Sun. Dozens of LDGRFs have been reported since the launch of the spacecraft on 2008 June 11 (Ackermann et al. 2012, 2014, 2017; Share et al. 2018). To derive events from the data set, a point source is placed at the location of the Sun and a power law with an exponential cutoff is assumed as the best-fit spectral model. All events with photon energies above 100 MeV and directions within 12° of the Sun are included in the analysis. The γ-ray background from the Earth's atmosphere is reduced by restricting the allowable events to zenith angles <100°. A solar flux is obtained using a "maximum likelihood" analysis (https://hesperia.gsfc.nasa.gov/fermi/lat/qlook/max_likelihood/) that compares the likelihood obtained by fitting the data with the solar source included with the likelihood of the null hypothesis (no solar source). Details of the analysis of LAT solar flares were published by Ackermann et al. (2013). The number of protons inferred from the γ-ray emission used in this study relies on the observations of Share et al. (2018) based on a "light-bucket" approach (https://hesperia.gsfc.nasa.gov/fermi/lat/qlook/light_bucket/), implementing a less accurate but faster algorithm to identify intervals of transient excess high-energy solar emission. It should be noted that, with respect to the maximum likelihood method, the background is not fitted, and the exposure is calculated with an assumed spectral model.

Table 1 reports the SEP events with an associated LDGRF detected by Fermi/LAT above 100 MeV between 2008 August and 2019 April. The LDGRF list is based on Share et al. (2018), except for events 15, 16, 19, and 20, identified with the maximum likelihood analysis by Winter et al. (2018), and the two events occurring in 2017 (22 and 23), derived from the light-bucket list of LAT observations (https://hesperia.gsfc.nasa.gov/fermi/lat/qlook/lat_events.txt). The first five columns provide the event number, the SEP event onset times (UT), and the event-integrated intensities of protons above 80, 300, and 500 MeV, based on the extrapolation of fits of the PAMELA spectra (Bruno et al. 2018); data from GOES-13/15 are used below 80 MeV to constrain the fits at low energies, based on the mean energies provided by Sandberg et al. (2014). For events not registered by PAMELA (the two events on 2011 August, the 2012 March 7 event, and those occurring after 2014), the fits are based on the EPEAD and HEPAD spectral points as described in Bruno et al. (2019), using the mean energies derived by Bruno (2017). The uncertainties on the event-integrated intensities are computed from the covariance matrix of the fits. For the four eruptions occurring in 2012 March (no. 8), a single SEP intensity value is provided. Columns 6–8 list the parent flare onset time (UT), location (degrees), and soft X-ray class from the GOES X-ray archive (ftp://ftp.ngdc.noaa.gov/STP/space-weather/solar-data/solar-features/solar-flares/x-rays/goes/). For the 2014 January 6 and September 1 events, originating on the far side of the Sun, the flare size was estimated by Ackermann et al. (2017) from observations made by the Extreme Ultraviolet Imager (EUVI) on board the STEREO spacecraft based on the method by Nitta et al. (2013). Columns 9–11 provide the associated CME first appearance time (UT) and space (3D) velocity (km s⁻¹) according to the Coordinated Data Analysis Workshops (CDAW; https://cdaw.gsfc.nasa.gov/CME_list/halo/) catalog of the Large Angle and Spectrometric Coronagraph (LASCO) on board the Solar and Heliospheric Observatory and the CME direction from the Database Of Notifications, Knowledge, Information (DONKI; https://kauai.ccmc.gsfc.nasa.gov/DONKI/); for the two 2017 September CMEs, the space speeds are not available, so the corresponding projected (plane-of-the-sky) velocities are reported. Both flare locations and CME directions are expressed in terms in the heliocentric Earth equatorial (HEEQ) coordinate system.

All of the events are associated with ≥M-class flares with hard X-ray emission extending above 100 keV (Share et al. 2018) and full halo CMEs in the CDAW catalog. In addition, they were linked to long-duration type II and III radio bursts, indicating the presence of a shock and open field lines, respectively. In particular, the measured type II emission ranges from metric to decameter-hectometric wavelengths for most events (Miteva et al. 2017), suggesting that the shocks accelerating particles formed close to the Sun (Gopalswamy et al. 2017). The heliographic locations of the flares associated with the SEP events listed in Table 1 are shown in Figure 1 for both front- and backside events. The color code indicates the SEP event-integrated intensity above 500 MeV. As expected based on magnetic connectivity considerations, most eruptions populate the western hemisphere; however, eruptions in the eastern hemisphere also contribute to the SEP flux detected near the Earth; in particular, two long-duration high-energy events from near the east limb were observed on 2014 February 25 and September 1.

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Figure 2 displays the heliographic distribution of solar events accompanied by LDGRFs registered by Fermi/LAT. In particular, the red circles show the locations of eruptions associated with SEP events measured by near-Earth spacecraft (see Table 1), while the blue squares indicate the events for which no significant SEP signal was detected. The latter set, listed in Table 2, essentially corresponds to poorly connected events concentrated in the eastern hemisphere. In particular, columns 9–11 indicate the energy range of SEP observations made by STEREO-A/B and near-Earth spacecraft. The high background from a previous event (referred as "bkgr?") may have obscured any SEP signal in the cases of the eruptions on 2011 September 7 and 24, 2012 March 5, and 2013 May 14. In addition, for five events—2011 June 2, 2012 June 3, 2013 May 13, 2013 October 11, and 2013 October 25—STEREO-B reported some moderate or large SEP enhancements. However, for the two remaining events on 2012 October 23 and November 27, no SEP signal was observed by STEREO and near-Earth spacecraft; above all, these events are peculiar because they were not linked to CMEs. Share et al. (2018) reported an association with magnetic eruptions that may indicate failed CMEs. Nevertheless, such events suggest that the association with fast CMEs is not a necessary requirement for LDGRFs.

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For comparison, the green triangles in Figure 2 denote three eruptions without Fermi/LAT detections linked to SEP events with a statistically significant proton signal with energies in excess of 500 MeV: the 2012 March 13, 2012 July 8, and 2015 October 29 events. The first two originated close to the western limb and were associated with an M7.9 flare and a full halo CME with a 1884 km s⁻¹ space speed and an M6.9 flare and a partial halo CME with a 1497 km s⁻¹ projected speed, respectively (Bruno et al. 2018). However, any firm conclusion about attendant >100 MeV γ-ray emission is precluded by the limited exposure of the LAT instrument, which can monitor the Sun for 20–40 contiguous minutes every 1–2 hr depending on the precession of the orbit and the variation of the Sun's latitude (https://hesperia.gsfc.nasa.gov/fermi/lat/lat_solar_exposure_times.txt). To illustrate this point, the time profiles of soft X-ray emission associated with the 2012 March 13 and 2012 July 8 flares are shown in Figure 3. The vertical dashed lines refer to the related CME first appearance times; the horizontal blue lines indicate the average effective area of Fermi/LAT, while the hatched regions show the time intervals in which the Sun was occulted in the instrument field of view. Consequently, it can be speculated that Fermi/LAT may have missed the detection of relatively short-duration (≲1 hr) γ-ray flares. The third eruption producing a >500 MeV SEP event measured by near-Earth spacecraft without Fermi/LAT detection occurred on 2015 October 29; however, in this case, it was located well behind the western limb (S20W150). Finally, no LDGRF was detected during the PAMELA SEP events with proton energies in excess of 300 MeV but without a statistically appreciable signal above 500 MeV (not shown in Figure 2), such as the events on 2012 July 19, 2013 May 22, and 2014 April 18; these were linked to moderately bright X-ray flares (M7.7-, M5.0-, and M7.3-class, respectively) and, interestingly, fast (1631, 1491, and 1359 km s⁻¹ space speed, respectively) halo CMEs (Bruno et al. 2018).

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3.1. Evaluation of the SEP Spatial Distribution

Estimating the total proton number in space from near-Earth measurements of SEP fluxes requires knowledge of the SEP spatial distribution at 1 au and the mean free path during transport. The longitudinal spread of SEPs has generally been attributed to the large spatial extent of the associated CME-driven shock (Mason et al. 1984; Cane et al. 1988) with the earliest estimates inferred from single spacecraft measurements and comparisons with the associated active region (Van Hollebeke et al. 1975; Cane et al. 1986; Reames 1999). However, examples exist of events whose extent is wider than that suggested by the CME shock (e.g., Cliver et al. 1995, 2005), as well as examples of highly prompt intensity increases at widely separated spacecraft (e.g., the 2011 November 3 event; Mewaldt et al. 2013; Richardson et al. 2014). Propagation across the nominal Parker spiral magnetic field is supposed to be enhanced by transport effects in interplanetary space, including cross-field diffusion and early-time nondiffusive propagation in turbulent fields along meandering field lines (see Desai & Giacalone 2016; Laitinen et al. 2018 and references therein). However, the mechanisms that lead to rapid and efficient transport of SEPs within the heliosphere are still not well understood.

Recently, several authors have investigated the SEP spatial distributions by taking advantage of multipoint observations (Lario et al. 2006, 2013; Richardson et al. 2014, 2017; Cohen et al. 2017). These studies show that the SEP peak or event-integrated intensity decreases with increasing longitudinal separation between the solar source and the footpoints of the interplanetary magnetic field (IMF) line connected to the observing spacecraft. With the limited number of observation points available, the longitudinal distribution cannot be determined directly but is usually assumed to be Gaussian. Richardson et al. (2014) obtained an average standard deviation σ = 43° for protons between 14 and 25 MeV, consistent with a separate analysis by Lario et al. (2013) for the peak intensity of protons between 25 and 53 MeV. Lario et al. (2006) compared the longitudinal spread of peak and event-integrated intensities and found that their Gaussian standard deviations were not appreciably different. Cohen et al. (2017) performed a fit of event-integrated intensities to periodic Gaussian distributions at two- and three-spacecraft locations and determined an average standard deviation of 43° ± 1° that decreases with increasing energy. Richardson et al. (2017) bracketed the most intense of the nearly 1000 SEP events in their study with a Gaussian standard deviation of σ = 43° and suggested that this width is a good indicator of the upper limit of the intensity of 25 MeV proton events as a function of longitude of the solar event with respect to the observer.

In this study, the SEP longitudinal distribution is inferred by combining the event-integrated intensities measured by near-Earth spacecraft (PAMELA and GOES-13/15) and STEREO-A/B. In the case of STEREO, the SEP spectra are evaluated by using the data from the Solar Electron and Proton Telescope (SEPT; 0.084–6.5 MeV, 10 minutes resolution), the Low Energy Telescope (LET; 4–12 MeV, 10 minutes resolution), and the High Energy Telescope (HET; 13.6–100 MeV, 15 minutes resolution), according to the procedure described in Bruno et al. (2019); upper limits on the event-integrated intensities are computed by extrapolating the spectral fits to higher energies. Since STEREO measurements are limited to 100 MeV, using the intensities extrapolated to above 500 MeV would result in large uncertainties, producing a significant overestimate of the SEP longitudinal spread. Consequently, we evaluate upper limits for the >500 MeV event-integrated intensity distributions by examining the values obtained for energies higher than 80 MeV (PAMELA threshold).

Specifically, we use a periodic Gaussian function with the form

$$\begin{eqnarray}&&G(\delta )=\displaystyle \frac{1}{3}\exp \left(-\displaystyle \frac{{\delta }^{2}}{2{\sigma }^{2}}\right)+\displaystyle \frac{1}{3}\exp \left(-\displaystyle \frac{{\delta }_{+}^{2}}{2{\sigma }^{2}}\right)+\displaystyle \frac{1}{3}\exp \left(-\displaystyle \frac{{\delta }_{-}^{2}}{2{\sigma }^{2}}\right),\end{eqnarray} \tag{ 1 }$$

with δ given by the great-circle or orthodromic distance from the peak of the SEP spatial distribution (β_sep, α_sep),

$$\begin{eqnarray}\begin{array}{rcl}\delta & = & \arccos \left[\sin (\alpha )\sin ({\alpha }_{\mathrm{sep}})\right.\\ & & +\,\left.\cos (\alpha )\cos ({\alpha }_{\mathrm{sep}})\cos (\beta -{\beta }_{\mathrm{sep}})\right],\end{array}\end{eqnarray} \tag{ 2 }$$

where β and α are the HEEQ longitude and latitude, σ is the distribution standard deviation, and the terms associated with

$$\begin{eqnarray}\begin{array}{rcl}{\delta }_{\pm } & = & \arccos \left[\sin (\alpha )\sin ({\alpha }_{\mathrm{sep}})\right.\\ & & +\left.\,\cos (\alpha )\cos ({\alpha }_{\mathrm{sep}})\cos (\beta -{\beta }_{\mathrm{sep}}\pm 2\pi )\right]\end{array}\end{eqnarray} \tag{ 3 }$$

account for the possible contribution from particles propagating at angles >180° from the center of the distribution, ensuring that G(δ) = G(δ ± 2π) (Cohen et al. 2017). The use of a periodic function is important, especially for SEP events characterized by a broad longitudinal distribution. In previous studies, the latitudinal magnetic connectivity to the SEP sources (Dalla & Agueda 2010; Gopalswamy & Mäkelä 2014) was typically neglected compared to longitudinal variations, and the inclination of the propagation axis was ignored when fitting spatial distributions. However, we note that, unless α_sep is null, the function describing the distribution projection on the X–Y plane is not Gaussian but includes a factor depending on α_sep,

$$\begin{eqnarray}&&G(\beta )=\displaystyle \frac{1}{3}\exp \left\{-\displaystyle \frac{{\arccos }^{2}\left[\cos ({\alpha }_{\mathrm{sep}})\cos (\beta -{\beta }_{\mathrm{sep}})\right]}{2{\sigma }^{2}}\right\}+...,\end{eqnarray} \tag{ 4 }$$

where the two terms centered at β_sep ± 2π are omitted for brevity. Consequently, fitting the particle intensities measured by spacecraft at 1 au with a Gaussian function will result in a systematically larger standard deviation for higher-latitude events; in particular, for 2012 January 23 and 27, we estimate an ∼3° (∼8%) difference. To a first approximation, α_sep is assumed to coincide with the latitudinal angle α_flare describing the parent flare location (see column 7 in Table 1).

Particle transport through the IMF is accounted for by computing the footpoints of the Parker spiral field lines crossed by the spacecraft, which are mapped ballistically back to 30 R_s and down to the photosphere based on measured plasma speed data by using the Predictive Science web tools (http://www.predsci.com/stereo/spacecraft_mapping.php and http://www.predsci.com/hmi/spacecraft_mapping.php); as results are available with a 1 hr resolution, the calculation is performed for the hour before the flare onset. However, we do not use magnetic footpoints at the photosphere for the estimate of N_SEP because the results would be model-dependent (see, e.g., Lario et al. 2017). Consistent with Cohen et al. (2017), the differences between footpoints obtained at 30 R_s and at the photosphere are of the order of ∼10°, which can be assumed as an indication of the associated uncertainty. On the other hand, significantly larger deviations are found for the 2014 February 25 event, for which the derived STEREO-A/B footpoints at the photosphere are very close (see Figure 18 in Cohen et al. 2017), in contrast to the large discrepancy in terms of detected particle intensities. Because the footpoints of the field lines connecting the spacecraft generally do not lie in the X–Y plane, and in order to apply Equation (4) to the measured intensities, we apply a correction factor to account for the spacecraft footpoint latitude α_sc,

$$\begin{eqnarray}&&{K}_{\alpha }=\exp \left(\displaystyle \frac{{\delta }_{\mathrm{sc}}^{2}-{\delta }_{\mathrm{sc},0}^{2}}{2{\sigma }^{2}}\right),\end{eqnarray} \tag{ 5 }$$

where δ_sc and ${\delta }_{\mathrm{sc},0}$ are the great-circle distances from the peak of the SEP spatial distribution evaluated with respect to the spacecraft footpoints and their projection on the X–Y plane (α_sc = 0), respectively. Since σ is unknown a priori, we perform an iterative procedure until the fit results become stable. We note that, even though α_sc is typically a few degrees, the differences in terms of great-circle distances are much larger, and thus the associated correction can be significant. Finally, the small discrepancies (<5%) regarding radial distances are neglected, and all of the spacecraft are assumed to be located exactly at 1 au.

3.2. Estimate of Transport Effects in Interplanetary Space

The transport of SEPs in the interplanetary medium is governed not only by the large-scale magnetic field geometry but also by small-scale scattering from turbulence. One consequence is that particles may pass the distance of the observing spacecraft several times (Zank et al. 2006), requiring a correction of the flux and, in turn, the local number density. Zank et al. (2006) modeled the number of crossings at 1 au, N_cross(E), for SEPs accelerated by both weak and strong shocks and found an ${\overline{N}}_{\mathrm{cross}}\sim 2\mbox{--}4$ mean value above 100 MeV, but, on occasion, high-energy particles experience multiple crossings as high as 15. Chollet et al. (2010) used numerical simulations of particle transport to determine ${\overline{N}}_{\mathrm{cross}}\sim 6\mbox{--}7$ for 100 MeV.

Since SEP intensity and anisotropy distributions depend on the scattering mean free path λ parallel to the magnetic field (Palmer 1982), we estimate ${\overline{N}}_{\mathrm{cross}}$ by means of simulations of relativistic proton propagation under a variety of scattering conditions. We consider two separate test particle simulation codes, the first (simulation code 1 (SC1)) presented by Chollet et al. (2010) and the second (simulation code 2 (SC2)) by Battarbee et al. (2018). For both codes, we assumed an impulsive injection of monoenergetic isotropic particles at 0.1 au and followed them for 10 days, including magnetic focusing and scattering off an unspecified plasma turbulence field. Particle crossings over the entire 1 au sphere are added together and averaged over the monoenergetic population considered.

Using SC1, we let the turbulence take one of two forms: either uniform from the launch radius to 1 au (turbulence model 1) or varying in proportion to the gyrocyclotron radius r_g (turbulence model 2). The resulting transport calculations predict the time-dependent development and decay of the intensity at 1 au. The mean numbers of crossings as a function of energy are reported in Table 3 for both turbulence models. A flat heliospheric current sheet (HCS) is assumed, and the results are valid for a positive solar magnetic field polarity in the northern hemisphere. It should be noted that ${\overline{N}}_{\mathrm{cross}}$ distributions are very broad, as the associated rms values (not shown in the table) are of the same magnitude as the mean values.

Table 3.  Average Number of 1 au Crossings at Sample Proton Energies as Predicted by SC1 (Chollet et al. 2010) for Different Turbulence Models

Energy	Turbulence Model 1		Turbulence Model 2
(MeV)	λ = const., λ0 = 0.1 au	λ = const., λ0 = 0.5 au	$\lambda \propto {r}_{g}$, λ0 = 0.1 au	$\lambda \propto {r}_{g}$, λ0 = 0.5 au
500	18.4	7.0	14.5	4.7
1000	16.5	5.9	11.8	4.1
2000	17.2	5.0	10.6	3.6

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The average number of 1 au crossings was also calculated by means of SC2 (Battarbee et al. 2018; Marsh et al. 2013), a code that can include the effects of a wavy HCS. We use λ = const. (turbulence model 1) for these simulations. Table 4 shows how the number of crossings varies for different configurations of the HCS, including no, flat, or wavy HCS. Using the standard galactic cosmic-ray definition, A⁺ refers to a situation in which the polarity of the IMF is positive (outward) in the northern hemisphere and negative (inward) in the southern hemisphere, with the opposite for A⁻. The number of crossings for E = 1000 MeV, λ₀ = 0.1 au, flat HCS, and A⁺ compares well with that of SC1, as shown in Table 3 (corresponding to a run with flat HCS and A⁺), showing that the two codes are in good agreement. Table 4 shows that at particle energies of 1 GeV, there is a large difference in the number of crossings for the A⁺ and A⁻ cases. The difference is similar for the cases of both flat and wavy HCS. The cause of this difference is the particle drift along the HCS, which for the A⁺ case helps to move protons outward from the inner heliosphere faster. A full simulation of the 2012 May 17 event, with the initial proton distribution given by a power law, shows good agreement between the SC2 and PAMELA intensity time profiles when λ₀ = 0.3 au (S. Dalla et al. 2019, in preparation).

Table 4.  Average Number of 1 au Crossings at Sample Proton Energies as Predicted by SC2 (Battarbee et al. 2018) for Different Types of HCS Configurations, Mean Free Path Values, and Solar Magnetic Polarities

Energy	λ0	HCS	Magnetic	${\overline{N}}_{\mathrm{cross}}$
(MeV)	(au)	Configuration	Polarity
1000	0.1	No HCS	+ both poles	31
1000	0.1	No HCS	− both poles	28
500	0.1	Flat HCS	A+	19
500	0.1	Flat HCS	A−	29
500	0.5	Flat HCS	A+	8
500	0.5	Flat HCS	A−	11
1000	0.1	Flat HCS	A+	15
1000	0.1	Flat HCS	A−	28
500	0.1	Wavy HCS	A+	21
500	0.1	Wavy HCS	A−	30
500	0.5	Wavy HCS	A+	8
500	0.5	Wavy HCS	A−	11
1000	0.1	Wavy HCS	A+	17
1000	0.1	Wavy HCS	A−	29
1000	0.5	Wavy HCS	A+	7
1000	0.5	Wavy HCS	A−	11

Note. Here A⁺ corresponds to a positive polarity north pole and negative polarity south pole, while the opposite holds for A⁻; "+/− both poles" denote a positive/negative polarity for both poles.

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The solar polarity reversal during cycle 24 was unusually complex. In fact, based on the measurements of the Wilcox Solar Observatory, the northern polar field changed polarity in 2012 June, while the southern polar field reversed in 2013 July; however, additional analyses of solar data suggest that the field reversal was completed in 2014–2015 (see, e.g., Janardhan et al. 2018 and references therein).

Figure 4 shows decay times deduced from the time-intensity profiles for the SEPs observed by PAMELA. With the exception of the two eastern events on 2014 February 25 and 2014 September 1, for which we expect significant cross-field diffusion, the decay times all show similar energy dependence. Table 5 gives the values of the decay times from SC1 for the two turbulence models considered. As expected from diffusive transport theory, a shorter mean free path results in a longer decay time. The SC1 values that best match the PAMELA observations are those for turbulence model 1 (λ = constant), with a mean free path at 1 au of λ₀ = 0.1 au. The simulation results are in better agreement with the predictions of model 1 for λ = constant. In particular, the analysis of the 2012 May 17 event suggests a value of ∼0.3 au for the mean free path. For the purpose of this work, because we compute upper limits on the number of SEPs at 1 au (see below), we conservatively assume λ₀ = 0.5 au, and we use the number of crossings estimated at 500 MeV with the wavy HCS configuration and A⁻/A⁺ magnetic polarity for SEP events occurring before/after 2012 June, respectively (see Table 4). Finally, we note that model calculations do not include possible effects related to local solar wind structures, including magnetic mirroring from nearby reflecting boundaries (see, e.g., Tan et al. 2009).

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Table 5.  Decay Times for 1 au Intensity Time Profiles at Sample Proton Energies Predicted by SC1 (Chollet et al. 2010) for Different Turbulence Models

	Turbulence Model 1		Turbulence Model 2
Energy	λ = const., λ0 = 0.1 au	$\lambda =\mathrm{const}$., ${\lambda }_{0}=0.5$ au	$\lambda \propto {r}_{g}$, ${\lambda }_{0}=0.1$ au	$\lambda \propto {r}_{g}$, λ0 = 0.5 au
(MeV)	τ (hr)	τ (hr)	τ (hr)	τ (hr)
100	>15.5	2.8	10.6	1.5
200	13.1	2.1	8.7	1.3
500	6.5	1.5	5.7	1.1
1000	4.8	1.4	3.7	1.0
2000	2.6	1.1	2.6	1.0

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3.3. Assessment of the Number of SEP Protons at 1 au

After estimating the SEP spatial distribution at 1 au and the mean free path during transport, we can derive the total number of solar protons above 500 MeV using the following relation:

$$\begin{eqnarray}&&{N}_{\mathrm{SEP}}={\overline{N}}_{\mathrm{cross}}^{-1}{\int }_{4\pi }d{\rm{\Omega }}{\int }_{S}{dS}\left({\boldsymbol{J\cdot n}}\right)\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&=\,{\overline{N}}_{\mathrm{cross}}^{-1}{\int }_{4\pi }d{\rm{\Omega }}{\int }_{S}{dS}\cos (\theta )\,J({\rm{\Omega }},S),\end{eqnarray} \tag{ 7 }$$

where S is the heliocentric spherical surface with radius R_o = 1 au, dΩ = dϕ dθ sin(θ) is the solid-angle element with polar angles ϕ and θ defining the particle velocity direction in the reference frame centered at a point on the sphere, and J = J(Ω, S) is the event-integrated intensity for energies >500 MeV; the dot product accounts for the fact that the flux at an angle θ with respect to the local normal to the sphere surface (n is the unit vector) is proportional to $\cos (\theta )$. Consequently, for an isotropic flux, $J({\rm{\Omega }},S)$ = J(S) (independent on Ω), the integration over the 4π detection solid angle subtended at a point on the sphere gives

$$\begin{eqnarray}&&\begin{array}{l}{\displaystyle \int }_{4\pi }d{\rm{\Omega }}\,\cos (\theta )\,J({\rm{\Omega }},S)\\ \quad =\,{\displaystyle \int }_{0}^{2\pi }d\phi {\displaystyle \int }_{0}^{\pi }d\theta \,\sin (\theta )\cos (\theta )\,J(\phi ,\theta ,S)=2\pi J(S).\end{array}\end{eqnarray} \tag{ 8 }$$

Integrating Equation (8) over the whole spherical surface and assuming a Gaussian spatial distribution J(S) = ${J}_{\max }G(\delta )$, where the amplitude J_max is the maximum particle intensity at 1 au (corresponding to the SEP propagation axis) and G(δ) is given by Equation (1), Equation (6) reduces to

$$\begin{eqnarray}&&{N}_{\mathrm{SEP}}=2\pi \,{\overline{N}}_{\mathrm{cross}}^{-1}\,{J}_{\max }\,{\int }_{S}{dS}\,G(\delta )\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&=\,2\pi \,{\overline{N}}_{\mathrm{cross}}^{-1}\,{J}_{E}\,{S}_{J}\,{C}_{\mathrm{spa}},\end{eqnarray} \tag{ 10 }$$

where J_E is the >500 MeV event-integrated intensity measured by PAMELA and S_J is the spherical area weighted by the particle spatial distribution,

$$\begin{eqnarray}&&{S}_{J}=2\pi {R}_{o}^{2}{\int }_{0}^{\pi }d\delta \,\sin \delta \,G(\delta ),\end{eqnarray} \tag{ 11 }$$

where δ is the great-circle distance with respect to the peak of the SEP spatial distribution (Equation (2)). The Gaussian standard deviation is derived by fitting the longitudinal distribution given by >80 MeV particle intensities measured by STEREO and PAMELA by means of the function given by Equation (4). The integration of Equation (11) is then performed by means of numerical techniques. Finally, C_spa accounts for the fact that, in general, PAMELA's observations are not made on IMF lines that connect with the peak of the particle distribution,

$$\begin{eqnarray}&&{C}_{\mathrm{spa}}=\exp \left(\displaystyle \frac{{\delta }_{{pam}}^{2}}{2{\sigma }^{2}}\right),\end{eqnarray} \tag{ 12 }$$

where δ_pam is the great-circle distance between PAMELA's magnetic footpoints and the peak of the SEP spatial distribution, so that

$$\begin{eqnarray}&&{J}_{\max }={J}_{E}\,{C}_{\mathrm{spa}}.\end{eqnarray} \tag{ 13 }$$

As a final remark, we note that the mathematical formulation described in this section provides a significantly more robust basis to our calculation, representing a major improvement with respect to the simpler approaches adopted in previous studies.

4.1. SEP Spatial Distributions

The numbers of protons N_SEP and N_LDGRF were derived for the SEP events in common with the "light-bucket" analysis by Share et al. (2018; see Table 1), with the exception of the 2013 May 13 event, for which no appreciable >500 MeV SEP signal was measured by near-Earth spacecraft; the 2013 May 15 eruption, due to the high background from a previous event that may have obscured any clear SEP enhancement in the measurements of STEREO-B, the best-connected spacecraft; and the 2015 June 21 event, occurring after the loss of communications with STEREO-B in 2014 October. The selected sample consists of 14 events mostly located in the visible western hemisphere, but it also includes poorly connected events such as the 2014 February 25 eastern limb and 2014 September 1 backside events. The PAMELA and STEREO-A/B event-integrated intensities are displayed in Figure 5 as a function of the longitudinal difference ${\rm{\Delta }}\beta \,={\beta }_{\mathrm{sc}}$ − β_sep (connection angle) between the spacecraft magnetic footpoints at 30 R_s and the location of the parent flare; thus, positive (negative) angles correspond to footpoints west (east) of the flare.

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The SEP distributions are influenced by several factors. For the 2011 June 7 event, the intensities measured by STEREO-A/B are dominated by the background from a previous event. Here we assume a standard deviation of 40.4°, which, as discussed below, represents the mean σ that characterizes the connection-angle dependence of the event-integrated >80 MeV intensity spectra measured by PAMELA. The same standard deviation is assumed for the 2012 July 7 event, for which no significant SEP signal was measured by STEREO-B; the integration interval used for the intensities measured by PAMELA and STEREO-A is limited by the onset of a following SEP event on July 8, resulting in intensities that are underestimated. Similarly, the time integration over the 2012 January 23 event is limited by the onset of the January 27 event, and the intensities measured during the latter include a component from the previous event. In the case of the 2014 September 1 event, large gaps present in the STEREO-A data preclude constructing event-integrated intensities; based on the comparison of the respective time profiles, the SEP flux is assumed to be equal to the one measured by STEREO-B. The time-integrated intensities measured during the 2011 March 7 event include particles injected at three different eruptions. In particular, those registered by STEREO-B comprise a significant component from the previous well-connected eruption, originating a few hours before. This translates to a longitudinal distribution that is narrower with a peak closer to the Earth's magnetic footpoints. Similarly, SEP intensities measured by STEREO-B and, to a lesser extent, also by STEREO-A during the 2013 October 28 event include a contribution from another eruption occurring 3 days before, resulting in an overestimate of the total number of protons N_SEP at 1 au.

Some studies reported that the centers of the derived SEP distributions tend to lie west of the flare location (Lario et al. 2006, 2013; Cohen et al. 2017), although this trend was not confirmed by other analyses (e.g., Richardson et al. 2014). Lario et al. (2014) suggested that this displacement would be likely due to the fact that the maximum peak intensity is observed at some helioradial distance for which the eruption occurs eastward with respect to the footpoints of the nominal field line connecting the spacecraft with the Sun. The effect is expected to be larger at lower energies as particles escape from the acceleration region at higher heights (Cohen et al. 2017). The SEP event sample analyzed in this work is characterized by significant variability, with eight (six) event distributions centered at positive (negative) connection angles Δβ, resulting in an ∼0.7° mean angle between the flare and the peak of the SEP distribution and an ∼22° rms value. In addition, effects related to the multiple injections discussed above are also likely to contribute. A majority of the SEP events with intensities reaching 1 GeV are confined to a narrow swath of longitudes, suggesting that the longitudinal spread at higher energies is even narrower than at 80 MeV, consistent with the energy dependence reported by Cohen et al. (2017) below 10 MeV. Thus, the longitudinal distribution inferred from >80 MeV protons is an upper bound to the longitudinal extent at higher energies. This upper limit is also justified by the fact that we assume the same angular distribution for the latitudinal extent as the longitudinal dependence (i.e., σ = σ_α = σ_β). The normalized SEP spatial distributions G(δ) = G(β, α) (Equation (1)) are shown in Figure 6 as a function of HEEQ coordinates. The stars show the parent flare locations, the circles indicate PAMELA's magnetic footpoints, and the upward- and downward-pointing triangles show those of STEREO-A and -B, respectively. Only SEP events with three spacecraft measurements are displayed. The angles defining the direction of the peak of the SEP spatial distribution are reported at the top of the panels.

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The results of the calculation of the SEP parameters at 1 au are shown in Table 6. The first two columns give the SEP event number and date. Columns 3–13 list the longitudinal and latitudinal connection angles between the flare location and the spacecraft magnetic footpoints at 30 R_s and the event-integrated intensities measured by the three spacecraft above 80 MeV. For STEREO-A/B, upper limits are provided, and the spacecraft location is also given; "bkgr" indicates that a high background from a previous event may have obscured any SEP signal. Columns 14–15 display the Gaussian fit parameters (peak longitude and standard deviation), while the values of the weighted spherical surface S_J and the spatial correction factor C_spa (see Equations (11) and (12)) are reported in columns 16 and 17. Column 18 gives the number of crossings used for the calculation. Column 19 lists the numbers of SEP protons above 500 MeV at 1 au. For comparison, the numbers of protons interacting at the Sun inferred from Fermi/LAT observations by Share et al. (2018) are shown in column 20; they include the correction factor accounting for a downward isotropic angular distribution (see Table 3 in Share et al. 2018).

Table 6.  SEP Longitudinal Extent at 1 au

1		2				3			4				5		6	7	8	9	10
SEP Event		STEREO-B				PAMELA			STEREO-A				Fit Param.		SJ	Cspa	${\overline{N}}_{\mathrm{cross}}$	NSEP	NLDGRF
No.	Date	Loc.	Δβ	Δα	Intensity	Δβ	Δα	Intensity	Loc.	Δβ	Δα	Intensity	βsep	σ
1	2011 Mar 7	−93.9	−91.0	−23.8	1.6 × 103a	5.6	−33.0	1.4 × 103b	88.7	97.3	−36.1	3.5 × 101	2.0	44.5	6.97 × 1026	2.42	11	1.18 × 1026	(1.51 ± 0.44) × 1029
2	2011 Jun 7	−91.9	−108.2	13.8	bkgr	0.5	21.0	1.5 × 105	96.0	67.4	28.2	bkgr	54.5	41.0	6.08 × 1026	1.14	11	1.93 × 1029	(2.20 ± 0.80) × 1028
3	2011 Aug 4	−91.9	−88.8	−23.5	4.4 × 103	22.9	−11.9	4.8 × 104	101.3	115.7	−6.8	9.6 × 102	40.4	39.5	5.72 × 1026	1.21	11	5.04 × 1027	(2.52 ± 0.63) × 1028
4	2011 Aug 9	−92.0	−105.4	−26.2	8.1 × 10−1	−33.5	−15.8	2.7 × 104	101.8	68.7	−10.1	2.6 × 102	65.9	21.0	1.80 × 1026	3.69	11	1.41 × 1028	(5.60 ± 1.40) × 1027
5	2011 Sep 6	−94.0	−69.7	−19.7	1.0 × 104	35.2	−11.1	1.8 × 104a	103.9	155.5	−8.3	4.4 × 102a	17.2	54.2	9.42 × 1026	1.27	11	7.09 × 1028	(5.06 ± 0.92) × 1028
6	2012 Jan 23	−113.0	−69.3	−17.7	1.1 × 104c	24.7	−29.6	1.5 × 105c	108.8	144.0	−36.1	6.7 × 103c	53.4	42.0	6.35 × 1026	1.30	11	2.89 × 1027	(6.60 ± 1.32) × 1028
7	2012 Jan 27	−113.6	−134.5	−17.9	3.4 × 103a	−32.5	−28.2	5.6 × 105a	108.9	86.0	−34.7	4.2 × 104a	64.8	36.9	5.09 × 1026	1.70	11	9.74 × 1029	(1.87 ± 1.10) × 1028
8	2012 Mar 7	−116.7	−28.9	−9.4	1.3 × 107	81.1	−20.0	5.3 × 106	110.6	−162.7	−21.6	1.8 × 105	−24.1	52.6	9.03 × 1026	3.14	11	1.75 × 1031	(3.81 ± 0.68) × 1030
9	2012 May 17	−116.8	−141.6	−14.7	2.7 × 10−1	−17.6	−12.2	3.0 × 105	116.0	72.5	−5.8	2.4 × 100	46.0	20.8	1.78 × 1026	1.41	11	7.56 × 1029	(2.10 ± 1.40) × 1027
10	2012 Jul 7	−114.5	−132.5	4.7	bkgr	−12.3	14.0	1.5 × 104c	120.7	110.2	20.0	2.8 × 102c	51.8	41.0	6.08 × 1026	1.07	8	7.10 × 1027	(1.89 ± 0.63) × 1028
11	2013 Apr 11	−140.7	−65.4	−5.4	3.4 × 104	65.8	−11.5	1.1 × 105	134.8	−175.7	−7.7	2.1 × 103	13.6	54.1	9.41 × 1026	1.35	8	8.47 × 1028	(1.70 ± 0.69) × 1028
12	2013 Oct 28	−141.2	−50.8	4.5	9.3 × 104	103.3	9.6	1.8 × 104	149.5	−133.6	3.7	3.4 × 103a	−26.9	48.1	7.87 × 1026	9.73	8	4.41 × 1029	(8.80 ± 4.40) × 1026
13	2014 Feb 25	−159.3	−38.0	15.7	1.1 × 106	130.9	9.5	1.3 × 105	153.7	−67.9	11.8	5.9 × 104	−47.1	29.7	3.45 × 1026	174.24	8	3.77 × 1031	(1.14 ± 0.52) × 1030
14	2014 Sep 1	−159.8	20.5	−19.2	4.9 × 107	173.4	−11.7	1.8 × 105	167.8	−16.2	−15.1	4.7 × 107	−122.9	48.1	7.89 × 1026	261.00	8	2.35 × 1032	(1.99 ± 0.90) × 1030

Notes. The first two columns report the SEP event number and date. Columns 3–13 give the HEEQ longitude (degrees) of the STEREO-A/B location ("Loc."), the longitudinal (Δβ, degrees) and latitudinal (${\rm{\Delta }}\alpha$, degrees) deviation between the footpoints of PAMELA or STEREO-A/B at 30 R_s, and the parent flare location, along with the event-integrated intensities (sr⁻¹ cm⁻²) above 80 MeV measured by the three spacecraft. Upper limits are provided for the STEREOs; "bkgr" indicates that a high background from a previous event may have obscured any SEP signal. Columns 14–17 indicate the peak longitude β_sep (degrees) and standard deviation σ (degrees) of the SEP spatial distribution derived from the fit of measured intensities (Equation (4)), the weighted surface (cm²), and the spatial correction factor defined by Equations (11) and (12). Column 18 gives the number of crossings used for the calculation. Finally, columns 19–20 report the total number of SEP protons above 500 MeV at 1 au and the number of protons interacting at the Sun inferred from Fermi/LAT observations by Share et al. (2018).

^aComponent from a previous SEP event. ^bUpper limit. ^cIntegration interval limited by the onset of a successive SEP event.

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The average σ value estimated for the analyzed SEP event sample is ∼41°, characterized by a large rms value (∼11°). For two SEP events, the one on 2011 August 9 and the GLE event on 2012 May 17, the standard deviation is small (∼21°), resulting in a narrow distribution that peaks close to the Earth's magnetic footpoints. On the other hand, the three events on 2011 September 6, 2013 April 11, and 2014 September 1 are very broad, with a σ value larger than 50°; however, as aforementioned, the integrated intensities measured during the 2011 September 6 event by PAMELA and STEREO-A include a component from the previous SEP event, probably resulting in an overestimate of the spatial extent. For the other two events, it can be speculated that the broad distributions are due to significant transport effects, such as cross-field diffusion and IMF corotation in combination with the extended SEP source provided by the CME-driven shock. The rest of the SEP sample is characterized by a standard deviation between 30° and 50°.

Three of the considered events, the 2012 January 23 and 27 events and the 2014 February 25 event, were also investigated by Cohen et al. (2017) at much lower energies (≤10 MeV). In general, the standard deviation of the spatial distribution is expected to decrease with increasing energy as a consequence of several acceleration and/or transport related phenomena. This can reasonably explain the relatively small differences regarding the 2014 February event, for which the derived standard deviations are consistent (33° at 10 MeV versus 30° above 80 MeV). On the other hand, they found significantly larger σ values for the 2012 January 23 and 27 events (respectively, 45° and 49° at 10 MeV versus 42° and 37° above 80 MeV). As discussed in Section 3, projection effects have a significant impact for high-latitude events such as those in 2012 January, and the use of a Gaussian function to fit the particle intensities neglecting the latitude of the SEP propagation axis results in some overestimate of the distribution standard deviation. The correction given by Equation (4) likely constitutes a major source of discrepancy between the two calculations.

4.2. Comparison between N_LDGRF and N_SEP

The comparison between proton numbers derived from the >100 MeV γ-ray emission (N_LDGRF) and SEPs at 1 au (N_SEP) is displayed in Figure 7. The solid and dashed lines mark the 1-to-1 and 1-to-100 correspondences, respectively. The Fermi/LAT proton numbers are from Share et al. (2018) and corrected for anisotropic effects characterized by the downward proton distribution (Mandzhavidze & Ramaty 1992). The vertical error bars include the uncertainties on LDGRF proton numbers from Share et al. (2018). For the quantity N_SEP, upper limits are provided accounting for the assumptions made in the spatial distribution estimate, as discussed earlier. A large scatter among the points can be observed, with no significant correlation, as confirmed by the low values of the Kendall's τ and Spearman rank correlation coefficients, reported on the plot along with the corresponding p-values (significance level). The ${N}_{\mathrm{SEP}}/{N}_{\mathrm{LDGRF}}$ ratio spans more than 5 orders of magnitude, ranging from ∼7.8 × 10⁻⁴ to ∼5.0 × 10², with a mean value of ∼78.

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The lowest ratio values are obtained for the 2011 March 7 and 2012 January 23 SEP events, which were characterized by very soft energy spectra in both PAMELA and STEREO observations. Gopalswamy et al. (2018) suggested that the relatively low SEP intensities measured during the 2011 March 7 event are probably due to the poor latitudinal magnetic connectivity. As our calculation accounts for these effects by means of the spatial factor given by Equation (12), we conclude that the very low ${N}_{\mathrm{SEP}}/{N}_{\mathrm{LDGRF}}$ ratio cannot be explained by connectivity arguments. Indeed, the ∼33° latitudinal deviation between the PAMELA magnetic footpoints and the flare location for this event is, for instance, comparable to the corresponding angle for the 2012 January 27 event (∼28°; see Table 6), which, however, extended above 500 MeV. An alternative explanation for the soft spectra is provided by the values of the shock formation height, a key factor in determining the maximum energy of SEPs, as the acceleration efficiency strongly depends on the coronal magnetic field strength, which decreases with increasing heliocentric distance (Zank et al. 2000). Most events shown in Figure 7 were accompanied by type II radio bursts with high (>100 MHz) starting frequencies (see, e.g., Miteva et al. 2017), suggesting high local plasma densities and thus small (<1.6 R_s) shock formation heights based on the empirical formula by Gopalswamy et al. (2013). In contrast, using STEREO coronagraphic and EUV observations close to the solar surface, Gopalswamy et al. (2013) estimated a 1.93 R_s height for the 2011 March 7 event, whose type II emission was limited to relatively lower frequencies (f_max ∼ 50 MHz). A similar height (2 R_s) was computed by Joshi et al. (2013) for the 2012 January 23 event; they also proposed that the unusually high f_max value (∼200 MHz) associated with the eruption was probably due to the fact that the shock formed in the body of a previous CME.

The highest ${N}_{\mathrm{SEP}}/{N}_{\mathrm{LDGRF}}$ ratio values correspond to the 2012 May 17 GLE event and the 2013 October 28 event, for which the detected >100 MeV γ-ray emission was rather low. The N_SEP is somewhat overestimated for the latter event due to the presence of a previous event, but this alone is insufficient to account for the high N_SEP/N_LDGRF ratio. Similar circumstances in other events also do not produce the large scatter in the values of this ratio.

Another way of comparing the proton numbers is to calculate the precipitation fraction, reported in Figure 8 by event number. This is the fraction ${N}_{\mathrm{LDGRF}}/({N}_{\mathrm{LDGRF}}+{N}_{\mathrm{SEP}})$ of the total number of protons accelerated (those that escape as SEPs plus those that produce the LDGRFs) that would have to precipitate to account for the LDGRFs. For five events, only a relatively small fraction of particles (≲3%) are required to precipitate back to the Sun, while for the rest of the sample, the fraction is larger than 10%, with three events that require ≳83% to precipitate back to the Sun. We conclude that the large fraction of precipitating protons required to explain the LDGRF emission in several of the events surveyed and the lack of any pattern in the ratio of N_LDGRF over the sum of N_LDGRF and N_SEP pose significant challenges to the CME-driven shock scenario as the source of LDGRF emission.

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4.3. Uncertainties on the SEP Propagation Axis

4.4. Comparison with Previous Works

One of the prominent explanations for the extended duration of LDGRFs is continuous back precipitation of energetic protons accelerated at a CME-driven shock as it propagates outward from the Sun. In this model, both the LDGRF and 1 au SEP signatures are due to protons drawn from the same population. However, because of the strong mirroring effect impeding the transport of protons back to the Sun (Hudson 2017), this scenario would require N_SEP to be significantly larger than N_LDGRF. The picture that emerges from our analysis is not consistent with a source rooted in back precipitation from CMEs. For example, even with large systematic uncertainties, the 2012 May 17 event misses the equality line in Figure 7 by a factor of ∼10³ in γ-ray-producing protons, and it differs from the 2012 March 7 event by a factor of >10³ for a similar integrated particle number at 1 au. This could be explained by the likely sporadic and unpredictable magnetic connection between the shock front and the Sun. Nevertheless, it would follow that the intensity time profile in any given LDGRF would be similarly wildly varying, atypical of most well-measured LDGRFs observed by Fermi/LAT.

The events to the upper left of the equality are doubly problematic because they represent events where the particle number at the flare exceeds, sometimes considerably, the particle number in space. The implications of this pose a serious challenge to theory. If the particles above 500 MeV in space were from the same population as those responsible for the γ-ray emission at the Sun, it would imply that in some cases, more than ∼80% of this population must be extracted from the acceleration process to produce the γ radiation. In particular, although the estimate of N_SEP is subject to large uncertainties related to multiple injections, the number of protons inferred from the Fermi/LAT detection during the 2011 March 7 event, associated with the second-longest duration γ-ray emission, is more than 4 orders of magnitude larger. Losses of this magnitude from the shock would certainly quench the particle acceleration process long before particles of sufficient energy and number could be produced by the shock. Furthermore, because the shock formation height estimated for the 2011 March 7 and 2012 January 23 events is relatively large (∼2 R_s; Gopalswamy et al. 2013; Joshi et al. 2013), the challenge to get particles back from a weakening shock to the Sun is even greater.

The problem is exacerbated for events such as those of 2012 March 7 and 2014 February 25 because the SEP spectra fall off exponentially above a few hundred MeV (Bruno et al. 2018), while there are abundant γ-rays exceeding 1 GeV from the flare (Share et al. 2018). Consequently, a particle number plot similar to that for Figure 7, but for higher energies, would push the points much farther upward and to the left of an equality line, aggravating the problem. Although CME-driven shock protons exhibit a prolonged, delayed, and high-energy behavior, we find it difficult to reconcile the back-precipitation scenario with the production of the high-energy γ radiation at the Sun because it seems to badly fail the total particle number test, implying a separate process associated with the CME only in a general order-of-magnitude sense, consistent with the trend in the data.

Complications arise in other ways, in particular from two Fermi/LAT detections on 2012 October 23 and 2012 November 27 that were not linked to CME eruptions, suggesting that a fast CME is not a necessary requirement of LDGRFs. In addition, while it has been shown that many bright X-ray flares are not associated with LDGRFs (Winter et al. 2018), there are also counterexamples of very fast front-side CMEs that were not accompanied by >100 MeV γ-ray emission, such as the two full halo CMEs observed at 10:48 UT on 2011 September 22 (1905 km s⁻¹) and 12:48 UT on 2011 September 24 (2018 km s⁻¹) and the partial halo CME registered at 03:12 UT on 2013 June 21 (1900 km s⁻¹). Similarly, while all reported LDGRFs are associated with impulsive-flare hard X-rays with energies above 100 keV, the latter is not a sufficient condition for the production of LDGRFs (Share et al. 2018). Finally, no long-duration γ-ray observation was reported by Fermi/LAT during the two front-side eruptions producing >500 MeV SEPs measured by PAMELA on 2012 March 13 and 2012 July 8. However, firm conclusions about the absence of >100 MeV γ-ray emission in such events are complicated by the limited duty cycle of the LAT when viewing the Sun.

An alternative to back precipitation is the scenario where particles are accelerated via second-order Fermi mechanism and trapped locally within extended coronal loops, after which (or concurrently) they diffuse to the denser photosphere to radiate (Ryan & Lee 1991). The favorable conditions can be provided by the magnetic structures appearing during the gradual phase of two-ribbon flares and CME liftoff, as field line reconnection gives rise to hot flare loops whose size can exceed 1 R_s, creating a system of arches that can persist for several hours. Based on observations of gyrosynchrotron (GS) emission from the Nancay Radioheliograph (NRH), Grechnev et al. (2018) provided evidence that the behind-the-limb flare of 2014 September 1 involved two distinct quasi-static loops, one that is associated with an initial hard X-ray flare and another, larger one associated with a second hard X-ray flare. Comparing these observations with simulations, they noted that the time profiles of the GS, hard X-ray, and, to some extent, >100 MeV γ-ray emission from Fermi/LAT of the latter microwave source are consistent with prolonged confinement (and perhaps reacceleration) of particles injected within a magnetic trap.

A compelling association of a large, extended loop with >100 MeV γ-ray emission is also provided by recent microwave observations from EOVSA (Gary et al. 2018). During the 2017 September 10 solar eruption that produced a GLE and an LDGRF, EOVSA microwave observations identified the footpoints of a large coronal loop with a circular length of ∼1.4 R_s. The microwave emission persisted well into the extended phase of the >100 MeV γ-ray emission. The robust and smooth exponential decay of the latter argues for the coronal trap scenario, with spatial and momentum diffusion governing the precipitation of high-energy particles (Ryan et al. 2018b), as seen in every other well-measured LDGRF (Share et al. 2018). This approach conveniently decouples the acceleration of the γ-producing particles from the acceleration and transport of the SEPs, which, in turn, allows for different spectral shapes, as well as different energetic particle productivities, as seen in this analysis. The intrinsic prolonged durations of the γ-ray signatures from both processes speak to the large requisite spatial scales for both processes and the resulting high energies. Because the loop scenario is local and diffusive in nature, it would naturally produce smooth exponential decays, since no sequence of magnetic connects or disconnects would occur, as would be expected for a propagating large-scale feature like a CME.

A problem encountered in more widespread acceptance of this scenario is that although large loop structures (≳1 R_s) are common, they are often difficult to visualize, not filled with hot plasma or enough 100 keV electrons to be visible in soft X-ray or radio emission. For the purpose of accelerating protons, only magnetic turbulence or Alfvén waves are necessary with δB/B ∼ 10% (Ryan et al. 2018a). New radio observations can help to place constraints on loop size and the ambient conditions within the loop that will improve future modeling of LDRGFs within the context of the continuous acceleration and trapping scenario.

Taking advantage of the unique high-energy observations from PAMELA, we conducted the first direct comparison of the number of interacting protons at the Sun and the number of SEP protons in the energy range above the ∼300 MeV pion production threshold. Several key factors contribute to the analysis, including the ability to obtain SEP energy spectra up to a few GeV, firmly establishing spectral rollovers for all high-energy SEP events observed (Bruno et al. 2018). In particular, we calculate the total number of >500 MeV protons at 1 au by combining PAMELA and STEREO data with the aid of transport simulations and compare it with the number of high-energy protons at the Sun, as deduced from Fermi/LAT data (Share et al. 2018). The results of our analysis show that the two proton numbers are uncorrelated such that their ratio spans more than 5 orders of magnitude. The lack of correlation, and in particular, several extreme cases where the number of protons required to account for LDGRFs far exceeds the number of SEP protons, suggests that the LDGRF emission is probably not due to the back precipitation of particles accelerated at CME-driven shocks. Moreover, as demonstrated by the two LDGRF events occurring on 2012 October 23 and November 27, the association with fast CMEs does not appear to be a necessary requirement for high-energy γ-ray emission. In fact, while bright flares and impulsive >100 keV hard X-rays are not sufficient conditions for LDGRFs, there also several counterexamples of fast halo CMEs that were not accompanied by >100 MeV γ-ray emission, though the limited exposure of the LAT instrument complicates the interpretation. An alternative explanation for LDGRFs based on continuous particle acceleration and trapping within large coronal structures that are not causally connected to the CME shock is discussed, and new remote observations of these loops, such as those provided by EOVSA, may help to constrain the role of such acceleration in producing LDGRF emission.

The authors thank the ACE, GOES, and STEREO teams for making their data publicly available. G.A. de N. acknowledges support from Fermi/GI grant NNH10ZDA001N-FERMI, NASA/HSR grant NNH13ZDA001N-HSR, and NASA/ISFM grant HISFM18. A.B. acknowledges support by an appointment to the NASA postdoctoral program at the NASA Goddard Space Flight Center, administered by Universities Space Research Association under contract with NASA. S.D. acknowledges support from the UK Science and Technology Facilities Council (STFC; grant ST/R000425/1). J.G. acknowledges support from NASA grant NNX15AJ 71G and NSF grant 1735422. I.G.R. acknowledges support from NASA Living With a Star grant NNG06EO90A.