Fermi-LAT Observations of High-energy Behind-the-limb Solar Flares

SOL2013-10-11: On 2013 October 11 (Oct13), between 07:05:51 and 07:10:51 UT, STEREO-B detected a solar flare with an AR located at N21E103. GOES detected an M1.5 class flare starting at around the same time as STEREO-B. However, based on the STEREO-B 195 Å emission and the method described in Nitta et al. (2013), we estimate that the GOES class of this flare had the AR not been occulted would have been M4.9 with an uncertainty within a factor of three. This method utilizes the pre-flare background-subtracted, full-disk integrated EUV intensity, as shown in Figure 1(b). A fast CME was observed by LASCO with a reported first appearance by the white light coronagraph C2 (imaging from 2–6 solar radii) at 07:24:10 UT and a linear speed of 1200 km s⁻¹. Fermi coverage started at 07:08:00 UT and continued for more than 30 minutes. Fermi-LAT detected >100 MeV emission for ∼30 minute with the maximum of the flux occurring between 07:20:00–07:25:00 UT.⁵³ The Fermi-GBM detection of HXRs began a few minutes before Fermi-LAT and peaked earlier (∼07:10 above 50 keV). RHESSI coverage was from 07:08:00–07:16:40 UT, overlapping with Fermi for 9 minutes. Konus-Wind, located at Lagrangian point L1, detected emission in the 20–78 keV and 78–310 keV energy bands simultaneously with RHESSI and Fermi-GBM. This flare was also observed in radio by the RSTN and the Nobeyama radio polarimeter (Nakajima et al. 1994) at frequencies up to 9 GHz (see Section 3.4).

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The LCs for the GOES, STEREO-B, RHESSI, Fermi-GBM, Konus-Wind, and Fermi-LAT are shown in Figure 1. The microwave (MW) emission is compared with the HXR LC from Konus-Wind in Figure 2.

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SOL2014-01-06: On 2014 January 6 (Jan14), between 07:35:46 and 07:45:46 UT, a solar flare erupted from an AR located at S8W110. Both STEREO spacecraft had a full view of the AR and detected a large filament eruption from the AR at approximately 07:50:00 UT. The tip of this filament was seen from the visible solar disk by SDO/AIA. There is a hint of detection by GOES of a smaller-than-C-class flare. However, the peak rate of 2.5 × 10⁵ photons s⁻¹ detected by STEREO-B in its 195 Å channel indicates that the flare would have been classified as GOES X3.5 had it not been occulted (Nitta et al. 2013). LASCO detected a halo CME with a first C2 appearance at 08:00:05 UT with a linear speed of 1400 km s⁻¹.

Upon exiting the South Atlantic Anomaly (SAA)⁵⁴ at 07:55:00 UT, both instruments on board Fermi detected emission associated with this flare. Fermi-LAT detected >100 MeV emission for approximately 20 minutes (with no evidence of temporally extended emission after 08:15:00 UT), and Fermi-GBM detected emission in the tens of keV range. RHESSI detected emission starting at ∼08:18:00 UT (upon exiting spacecraft night) also in the tens of keV energy range for over 40 minutes. Konus-Wind had a full view of the flare and detected emission only in the softest energy band, 20–78 keV, starting at 07:43:00 UT. Radio data above 1.3 GHz from RSTN for this flare did not indicate a detection. The GOES, RHESSI, Konus-Wind, Fermi-GBM, and Fermi-LAT LCs for this flare are shown in Figure 3.

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SOL2014-09-01: On 2014 September 1 (Sep14), between 10:55:56 and 11:00:56 UT, a bright solar flare occurred in an AR located at N14E126. There was no GOES signal but STEREO-B had an unblocked view of the entire flare and detected a maximum rate of 1.7 × 10⁷ photons s⁻¹ in its 195 Å channel, indicating an unocculted GOES X2.4 class flare (Nitta et al. 2013). LASCO detected a halo CME with first C2 appearance at 11:12:05 UT with a linear speed of 1900 km s⁻¹. A type II radio burst with an estimated velocity of 2079 km s⁻¹ was reported by NOAA Space Weather Alerts in association with this flare. SDO/AIA reported a coronal wave from this AR starting during 10:45:35–12:21:35 UT. This wave was seen to propagate along the limb and over onto the visible disk.⁵⁵

Fermi-LAT detected emission from this flare for ∼2 hr, peaking between 11:10:00–11:15:00 UT.⁵⁶ The GBM detected emission up to a few MeV in temporal coincidence with the Fermi-LAT emission in both the BGO and NaI detectors. RHESSI was in the SAA from 10:55:00 to 11:11:00; upon exiting the SAA it detected emission up to 30 keV. Konus-Wind detected emission in all three energy bands, 20–78 keV, 78–310 keV, and 310–1180 keV, in temporal coincidence with Fermi-GBM. A significant radio flux at frequencies up to 16 GHz was detected by the San Vito station of the RSTN simultaneously with the HXR emission peak detected by both Fermi-GBM and Konus-Wind. Figure 4 shows the LCs from GOES, STEREO, RHESSI, Fermi-GBM, Konus-Wind, and Fermi-LAT. The LCs of the microwave emission (MW) are compared with the HXR LC from Konus-Wind in Figure 5. The MW intensity distribution with frequency has a maximum at about 1 GHz (see Section 3.4). Thus, we show the time profile of only this frequency for this flare. As evident from Figure 5, there is good agreement between these LCs.

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We present composite images of the Sun as seen by STEREO and SDO and, whenever available, the position of HXR emission based on RHESSI and the >100 MeV gamma-ray emission centroid based on Fermi-LAT data. We used the FITS World Coordinate System software package (Thompson & Wei 2010) to co-register the locations of the flares between STEREO and SDO images. We applied the CLEAN imaging algorithm (Hurford et al. 2002) to RHESSI data using detectors 3–9 to reconstruct the X-ray images. The centroid for the >100 MeV gamma-ray emission is determined using the gtfindsrc tool, which performs a likelihood analysis of the average position in the time-integrated data set.

Figure 6 displays the composite image for the Oct13 flare showing the STEREO-B 195 Å image of the AR located about 10° behind the limb, the SDO/AIA 193 Å emission peaking above the limb, a contour of the RHESSI image, and results from reanalysis of the flare with Pass 8 data. The new emission centroid is located in heliocentric coordinates [−880'', 290''], about 200'' closer to the RHESSI centroid, and with a 68% error radius of 190'', which is ∼20% smaller than the value we reported in Pesce-Rollins et al. (2015) using Pass7_REP data. In addition, in the Pass 8 data set, the total number of >1 GeV events measured from this flare increased from four to seven. The highest-energy photon detected from this flare was 3.4 GeV and the arrival was 07:19:00 UT.

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For the Jan14 flare, the Fermi-LAT photon statistics were not sufficient to provide an emission localization error circle smaller than 05. However, we can still conclude that the emission detected by Fermi-LAT was consistent with the position of the Sun. In Figure 7 we show the STEREO-A and SDO images of this event at two different times. The top panels of Figure 7 present SDO 171 Å (left) and STEREO-A 195 Å (right) image at 07:55:46 UT and show the filament eruption, while the bottom panels show the SDO 193 Å (left) and STEREO-A 195 Å (right) images at 08:25:46 UT with the RHESSI 6–12 and 25–50 keV contours of this flare.

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The Fermi-LAT >100 MeV emission centroid of Sep14 is located at heliocentric coordinates [−720'', 610''] with a 68% error radius of 100''. RHESSI imaging shows a 6–12 keV source located above the visible limb slightly offset from the Fermi-LAT centroid, both shown in Figure 8. If the RHESSI source is the loop top of the BTL flare, then the minimum height needed for this source to be visible from ∼40° BTL would be ∼10¹⁰ cm. HXR loop-top emission from a flare located ∼40° BTL has been detected before by RHESSI (Krucker et al. 2007). Fermi-LAT measured 17 photons with energies >1 GeV; 15 of these (including a 3.5 GeV photon with an arrival time of 11:16:01 UT) arrived during the first 20 minutes of Fermi-LAT detection.

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We performed an unbinned likelihood analysis of the Fermi-LAT data with the gtlike program distributed with the Fermi ScienceTools.⁵⁷ We selected Pass 8 Source class events from a 10° circular region centered on the Sun and within 100° from the local zenith (to reduce contamination from the Earth limb).

We fit three models to the Fermi-LAT gamma-ray spectral data. The first two, a pure power law (PL) and a power law with an exponential cutoff (PLEXP), are phenomenological functions that may describe bremsstrahlung emission from relativistic electrons. The third model uses templates based on a detailed study of the gamma-rays produced from the decay of pions originating from accelerated protons with an isotropic pitch angle distribution in a thick-target model (updated from Murphy et al. 1987).

We rely on the likelihood ratio test and the associated test statistic TS (Mattox et al. 1996) to estimate the significance of the detection. The TS is defined as twice the increment of the logarithm of the likelihood obtained by fitting the data with the source and background model components simultaneously. Because the null hypothesis (i.e., the model without an additional source) is the same for the PL and PLEXP models, the increment of the TS (ΔTS = TS_PLEXP– TS_PL) is equivalent to the corresponding difference of the maximum likelihoods computed between the two models. Note that the significance in σ can be roughly approximated as $\sqrt{\mathrm{TS}}$ for two degrees of freedom.

In Table 1 we list the TS_PL, ΔTS, photon index Γ for the best-fit model (PL when ΔTS < 25 or PLEXP when ΔTS ≥ 25), and PLEXP cutoff energy. For several intervals, ΔTS > 25, indicating that PLEXP provides a significantly better fit than PL. For these intervals we fit the pion-decay models to the data to determine the best proton spectral index following the same procedure described in Ajello et al. (2014). In particular, we performed a series of fits with the pion-decay template models calculated for a range of proton spectral indices. We then fit the resulting profile of the log-likelihood with a parabolic function of the proton index. The minimum gives the most likely index for the pion-decay model. Note that the TS values for the PLEXP and pion-decay fits cannot be directly compared (Wilks 1938) because they are not nested models. However, PLEXP approximates the shape of the pion-decay spectrum; thus, we expect the pion-decay models to provide a similarly acceptable fit.

Table 1.  Fermi-LAT Spectral Analysis of the Solar Flares Considered in this Work

Time Interval	TSPL	ΔTSa	Photon Indexb	Cutoff Energyc	Fluxd	Proton Index
(UT)				(MeV)	(×10−5 ph cm−2 s−1)
SOL2013-10-11
07:10:00–07:15:00	20	3	−2.0 ± 0.4	⋯	0.9 ± 0.1	⋯
07:15:00–07:18:00	487	43	−0.4 ± 0.4	130 ± 30	17 ± 1	4.5 ± 0.4
07:18:00–07:20:00	846	77	−0.1 ± 0.4	112 ± 21	49 ± 2	4.3 ± 0.2
07:20:00–07:22:00	776	76	−0.3 ± 0.3	128 ± 23	46 ± 2	4.4 ± 0.3
07:22:00–07:24:00	677	58	−0.2 ± 0.4	153 ± 31	29 ± 2	3.7 ± 0.2
07:24:00–07:26:00	345	25	−0.9 ± 0.4	208 ± 61	21 ± 2	3.9 ± 0.3
07:26:00–07:28:00	219	34	0.6 ± 0.7	86 ± 22	16 ± 1	4.5 ± 0.4
07:28:00–07:40:00	283	29	0.1 ± 0.7	86 ± 26	7 ± 1	5.3 ± 0.4
SOL2014-01-06
07:55–08:15	67	20	−2.4 ± 0.2	⋯	0.6 ± 0.1	⋯
SOL2014-09-01
11:02:00–11:04:00	42	9	−2.5 ± 0.3	⋯	11 ± 3	⋯
11:04:00–11:06:00	321	41	−0.2 ± 0.5	117 ± 27	117 ± 10	4.7 ± 0.4
11:06:00–11:08:00	1070	120	−0.5 ± 0.3	116 ± 15	360 ± 17	5.2 ± 0.2
11:08:00–11:10:00	1549	126	−0.9 ± 0.2	158 ± 20	477 ± 19	4.9 ± 0.2
11:10:00–11:12:00	2549	115	−1.2 ± 0.2	194 ± 28	565 ± 21	4.8 ± 0.2
11:12:00–11:14:00	6394	157	−0.8 ± 0.1	167 ± 10	522 ± 20	4.4 ± 0.1
11:14:00–11:16:00	2430	159	−0.6 ± 0.2	148 ± 17	540 ± 22	4.5 ± 0.2
11:16:00–11:18:00	1069	32	−1.9 ± 0.2	488 ± 90	465 ± 23	4.2 ± 0.2
11:18:00–11:20:00	1047	49	−1.0 ± 0.3	182 ± 37	396 ± 27	4.5 ± 0.2
12:29:00–12:34:00	56	12	−2.2 ± 0.2	⋯	3 ± 1	⋯
12:34:00–12:39:00	133	3	−2.3 ± 0.2	⋯	3 ± 1	⋯
12:39:00–12:44:00	130	5	−2.2 ± 0.2	⋯	2 ± 1	⋯
12:44:00–12:49:00	97	11	−2.3 ± 0.2	⋯	2 ± 1	⋯
12:49:00–12:54:00	84	3	−2.4 ± 0.2	⋯	2 ± 1	⋯

Notes.

^aΔTS = TS_PLEXP– TS_PL. ^bPhoton index from the best-fit model. The PL is defined as $\tfrac{{dN}(E)}{{dE}}={N}_{0}{E}^{{\rm{\Gamma }}}$ and the PLEXP as $\tfrac{{dN}(E)}{{dE}}={N}_{0}{E}^{{\rm{\Gamma }}}\exp (-\tfrac{E}{{E}_{c}})$, where E_c is the cutoff energy. ^cFrom the fit with the PLEXP model. ^dIntegrated flux between 100 MeV and 10 GeV calculated for the best-fit model.

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The main contributions to the systematic uncertainties are uncertainties in the effective area; in the energy range 100 MeV to ∼100 GeV, these are of the order of ±5%. This uncertainty applies directly to the flux values and from our previous studies on LAT-detected solar flares (Ajello et al. 2014) we find that the systematic uncertainties on the cutoff energy and photon index are also of the order of ±5%.⁵⁸

3.1.1. Comparison of BTL and Disk Flare Characteristics

We compare the characteristics of the >100 MeV emission associated with the three BTL flares with three disk flares with similar GOES classifications and temporally extended emissions and are described in Ackermann et al. (2014) and Ajello et al. (2014). In addition to spectral parameters, we also compare the total >100 MeV emitted energies, and the total energy released by protons with energy >500 MeV needed to produce the detected gamma-ray emission, based on the templates of (updated from) Murphy et al. (1987). We present these numbers along with the date, estimated GOES class, CME speed, AR position, and Fermi-LAT detection duration in Table 2. The proton indexes are very similar whereas the on-disk flares appear to have more energy; this is most likely because we observed the on-disk flares over longer timescales. Peak fluxes and the total energy released by protons with E > 500 MeV for Sep14 and the impulsive phase of SOL2012-03-07 are comparable.

Table 2.  Comparison Between Behind-the-limb and On-disk Flare Quantities

Date	GOESa	CME Speedb	AR	Duration	Peak Fluxc	Eγ > 100 MeVd	Protone	Ep > 500 MeVf
(UTC)	Class	(km s−1)	Position	(minutes)	(10−5 ph cm−2 s−1)	(erg)	Index	(erg)
2013-10-11	M4.9	1200	N21E103	30	49 ± 2	1.5 × 1023	4.3 ± 0.1	9.8 × 1024
2014-01-06	X3.5	1400	S8W110	20	0.8 ± 0.1	4.2 × 1021	5.3 ± 0.4g	3.5 × 1023
2014-09-01	X2.4	2000	N14E126	113	565 ± 14	1.4 × 1024	4.7 ± 0.1	7.0 × 1025
On-disk flares
2011-03-07	M3.7	2125	N30W48	798	3 ± 1	5.1 × 1023	4.7 ± 0.2	3.6 × 1025
2011-06-07	M2.5	1255	S22W53	38	3 ± 1	3.2 × 1022	5.0 ± 0.3	2.5 × 1024
2012-03-07Ih	X5.4	2684	N16E30	45	417 ± 13	3.9 × 1024	3.90 ± 0.02	2.1 × 1026
2012-03-07Ei	X5.4	2684	N16E30	1068	97 ± 2	1.4 × 1025	4.3 ± 0.1	9.0 × 1026

Notes.

^aGOES class for BTL flares is estimated based on the STEREO 195 Å flux. ^bSpeed is the linear speed reported by the LASCO CME catalog. ^cFor photon energies >100 MeV. ^dTotal energy released in >100 MeV gamma-rays integrated over the time interval when ΔTS > 25. ^eProton index in the same time interval as (d). ^fTotal energy released by protons with E > 500 MeV estimated over the same interval as in (d). Values may be underestimated for flares with centroids at heliocentric angles >75°. ^gThe ΔTS value for this flare is 20, so the improvement over a power law is marginal. ^hImpulsive phase of the flare, Fermi-LAT detection from 00:38:52-01:23:52 UT. Note that the peak of the GOES X-ray flare occurred ∼6 minutes prior to Fermi orbital sunrise. ⁱExtended phase of the flare, starting from 02:27 UT.

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Figure 9 shows the proton index as a function of time for the Oct13 and Sep14 BTL flares. We fit a constant (red dashed line) and a first-degree polynomial (blue dashed line) to the data. In the case of the constant fit, we list the best-fit value in the upper left corner, whereas for the straight-line fit, we list the value of the slope. The temporal variation over tens of minutes is not sufficient to conclude whether a softening or hardening is present for these BTL flares. In Ajello et al. (2014) we found that for the on-disk flare SOL2012-03-07 the spectrum softened with a timescale of a few hours.

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As described above we have HXR data from three instruments: RHESSI, Fermi-GBM, and Konus-Wind. Konus-Wind was in waiting mode during the Jan14 and Sep14 flares; therefore, these events were detected in only three broad energy channels, namely ∼20–78, 78–310, and 310–1180 keV, with 2.944 s time resolution. RHESSI provides only a limited coverage of these flares. For the Oct13 flare there is some overlap with Fermi-GBM during the rise of the impulsive phase. As shown in Pesce-Rollins et al. (2015) the HXR spectra of RHESSI and Fermi-GBM agree well. Fermi-GBM was in the SAA and RHESSI was in spacecraft night during the impulsive phase of the Jan14 flare and detected the flare during the decay phase in only two low-energy channels. Thus, we do not have sufficient data for a spectral fit. For the Sep14 flare, RHESSI gives information only on the decay phase of the flares. In the following we show results from the analysis of the Fermi-GBM data for the Oct13 and Sep14 flares.

For both Oct13 and Sep14, we performed a combined Fermi-GBM/LAT fit using the XSPEC package (Arnaud 1996). The spectral fits were done by minimizing PGSTAT, a profile likelihood statistic that takes into account Poisson error on the total count spectrum and Gaussian error on the background. In order to obtain the background-subtracted spectra of the Fermi-GBM data for Sep14, we used both Fermi-GBM-NaI and Fermi-GBM-BGO spectra accumulated before the flare (from 10:54 UT to 10:57 UT) and after the flare (from 11:42-11:50 UT). For Oct13, because there was a minor on-disk flare whose onset time was 7:01 UT, we used the procedure described in Pesce-Rollins et al. (2015), which consists of using the background estimation tool developed in Fitzpatrick et al. (2012) with an additional 5% systematic error. For the LAT, we follow the procedure used in Ajello et al. (2014) and Pesce-Rollins et al. (2015) that consists of deriving the background spectrum directly from the model of the background used in the standard LAT likelihood analysis (first using gtlike and then gtbkg), and obtaining the response of the LAT using gtrspgen (all of these tools are available in the Fermi ScienceTools). The combined spectral energy distribution (SED) derived from the Fermi-GBM and Fermi-LAT data extending from 30 keV to 10 GeV for these flares is shown in Figures 10 and 11, respectively, and the parameters for the spectral fit for both solar flares are listed in Table 3.

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The Oct13 flare had a very weak signal in the BGO, and the best-fit model (M₀) consisted of a single PL and the pion-decay templates to describe the bremsstrahlung and >60 MeV emission detected by  Fermi-GBM and Fermi-LAT, respectively. For the Sep14 flare, the best-fit model (M₀) consisted of a single PL with an exponential cutoff at high energies to describe the bremsstrahlung emission, and the pion-decay templates to describe the >60 MeV emission detected by Fermi-LAT, similar to what was done for the 2010 June 12 Fermi flare (Ackermann et al. 2012). We also tested the statistical significance of an extra PL with a high-energy cutoff (model M₁). To this end, we performed Monte Carlo simulations (using XSPEC fakeit) by generating a spectrum described by the best-fit model (M₀). We then re-optimized the parameters of M₀ by fitting it to the simulated data. We finally compared the improvement of PGSTAT when fitting the simulated data with the model M₁. The improvement of PGSTAT for the Oct13 and Sep14 flares is Δ_PGSTAT ≈ 408 and ≈109, respectively. Despite these high values of Δ_PGSTAT, dedicated Monte Carlo simulations indicate that they correspond to significances of 2.5σ and 2.0σ, respectively.

We then checked whether the addition of the 2.223 MeV line (on top of the model M₀) significantly improved the fit. For the Oct13 flare, the improvement was negligible (Δ_PGSTAT ≈ 0), while for the Sep14 the improvement was Δ_PGSTAT ≈ 25, corresponding to a significance of 2.0σ, estimated from Monte Carlo simulations. We tested whether adding nuclear de-excitation narrow lines and continua provided an improvement to the fit for both flares and found that it was not significant. The nuclear line and continua templates used here are based on a detailed study of the nuclear gamma-ray production from accelerated ion interactions with elements found in the solar atmosphere (Murphy et al. 2009).

The radio spectra obtained at the peaks of the Oct13 and Sep14 events are shown in Figure 12. The spectrum was obtained by using the RSTN one-second resolution fluxes. The spectra were background subtracted and the fluxes were integrated within the selected time intervals for each frequency. For each spectrum we detected the frequency where the maximum flux was observed. The spectra of the Oct13 and Sep14 flares are similar to those expected from the gyro-synchrotron (GS) mechanism, with peaks that separate the optically thin and thick parts (for GS spectrum properties see, for example, Dulk 1985). The values for the microwave photon spectral index, α, for both flares are shown in Figure 12. Unfortunately, the RSTN data for the time period of the Jan14 flare are not sufficient to obtain radio spectra.

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Since the spectral information available for the Jan14 flare is limited, we cannot constrain the acceleration, transport, or radiation processes. However, this flare is remarkable because the AR, being located beyond the western limb, provided good magnetic connection with the Earth and as a result it was associated with a very strong SEP event and increased the neutron flux in the South Pole neutron monitor. As presented in Figure 13, the acceleration time at the Sun for the SEP protons is 07:55 ± 0:05 UT, which is in agreement with the SPR time of 07:47 ± 0:08 UT reported by Thakur et al. (2014). Unfortunately, Fermi-LAT was in the SAA at the SPR time and exited at 07:55 UT. Therefore, we can only provide an upper limit on the time of acceleration of gamma-ray-producing particles (presumably protons) at the Sun. However, Konus-Wind detected emission in the 20–78 keV band starting at 07:43 UT, implying that the electrons were accelerated at the Sun at 07:35 UT, roughly 12 minutes prior to the SPR.

Due to the location of the AR, the other two flares were not magnetically well connected to the Earth. For the Sep14 flare, the GOES proton fluxes begin to increase roughly 9 hr after the flare start time. STEREO-B observed the flare in its entirety but provides proton fluxes only in three energy bands, which is not sufficient to estimate the acceleration time at the Sun as we did for Jan14. Thus, we do not have reliable SEP data to test whether the onset times of the SEP and gamma-rays coincide; however, we can compare the height of the CME at the time of the gamma-ray onset with the CME height at SPR found for Jan14. Based on the second-order fit of height as a function of time (from the LASCO CME online catalog), we estimated that the CME was at ∼2.5 R_⊙ at the onset time of the Fermi-LAT emission. We found a similar value for Oct13. These heights are significantly smaller than the predicted values reported in Thakur et al. (2014) for the CME heights at SPR as a function of source longitudes from Solar Cycle 23. To test whether these CME heights are compatible with the gamma-ray emission detected from the BTL flares, detailed simulations on particle transport in the solar magnetic field are required. This study is beyond the scope of this paper and will be addressed in a future work.

The RHESSI images show that the 20–50 keV HXRs, due to electron bremsstrahlung, are emitted from ≈30'' (L ∼ 2 × 10⁹ cm) and ≈50'' (L ∼ 3.5 × 10⁹ cm) sources from the tops of loops extending to heights of ∼4 × 10⁹ and ∼2 × 10¹⁰ cm above the photosphere for the Oct13 and Sep14 flares, respectively. The column depth required for a particle of mass m and kinetic energy E (in units of mc²), or Lorentz factor $\gamma =E+1={(1-{\beta }^{2})}^{-1/2}$, to lose all its energy via Coulomb interactions is

$$\begin{eqnarray}{N}_{\mathrm{Coul}} & = & {(4\pi {r}_{0}^{2}\mathrm{ln}{\rm{\Lambda }})}^{-1}(m/{m}_{e}){\displaystyle \int }_{0}^{E}{\beta }^{2}{dE}\\ & = & 5\times {10}^{22}(m/{m}_{e}){E}^{2}/\gamma \,{\mathrm{cm}}^{-2},\end{eqnarray} \tag{ 2 }$$

where r₀ is the classical electron radius and $\mathrm{ln}{\rm{\Lambda }}\sim 20$ is the Coulomb logarithm. From Equation (2) we find that the column depth required to stop the >50 keV electrons that produce the HXRs detected by RHESSI is ∼5 × 10²⁰ cm⁻². Assuming a density of n < 10¹⁰ cm⁻³, the sizes of the loop-top sources imply a column depth of N_LT < 2 and 3.5 × 10¹⁹ cm⁻² for the two HXR sources, respectively. This means that the energy-loss time ${\tau }_{\mathrm{Coul}}\sim 2{N}_{\mathrm{Coul}}/({vn})$ is more than 10 times longer than the crossing time ${\tau }_{\mathrm{cross}}\equiv L/v={N}_{\mathrm{LT}}/({vn})$. These values, together with the results reported in Chen & Petrosian (2013), suggest that the HXR emission from these two BTL flares are thin-target sources.

If these loop-top sources were to be produced in a thick target, which can come about in two ways, it would be under extreme conditions not likely to be the case here. The first is in the strong diffusion limit, i.e., if the electrons are trapped because the scattering time, τ_sc, is smaller than the crossing time, τ_cross. Note that because ${\tau }_{\mathrm{Coul}}\gt {\tau }_{\mathrm{cross}}$, the scattering cannot be due to Coulomb interactions but it could be due to turbulence. The second possibility is in the weak diffusion limit ${\tau }_{\mathrm{sc}}\gt {\tau }_{\mathrm{cross}}$, where trapping can occur if the field lines converge strongly. In this case we need ${\tau }_{\mathrm{sc}}\gt {\tau }_{\mathrm{Coul}}/\eta$ ⁵⁹ (with proportionality constant η increasing with increasing field convergence (Petrosian 2016)). These are somewhat extreme conditions requiring either a very high density (therefore increasing the scattering, presumably turbulence) or a somewhat strong field convergence and a "clean" loop with low density and low level of turbulence.

Thus, the HXR emission is most likely due to electron bremsstrahlung from a thin-target loop-top source.

Based on the positions of the Fermi-LAT >100 MeV emission centroids alone, we cannot exclude that the gamma-rays come from the loop-top source. In order to investigate this possibility, we rely on Equation (2) to find the column depth required to stop >350 MeV protons (which produce >100 MeV photons) and find ${N}_{\mathrm{Coul}}(350\,\mathrm{MeV})\sim {10}^{25}$ cm⁻². This is much larger than the loop-top column depth N_LT ∼ 2–3.5 × 10¹⁹ cm⁻² so that, in the absence of trapping, we are again dealing with a thin-target loop-top source and a small energy loss compared to what would be the case for emission from a thick-target photospheric source. Consequently, we would need much larger energies of accelerated protons than those given in Table 2 assuming a thick-target scenario. The condition for proton trapping in the loop top (or any coronal trap region, for that matter) is also more extreme than that described above for electrons, and we would need escape times 10⁵ times longer than the crossing time. Thus, the emission detected by the Fermi-LAT is most likely due to decay of pions produced by energetic protons interacting in a thick-target photospheric source.

As shown in Lin & Hudson (1971) and McTiernan & Petrosian (1990), in a thin-target scenario the relation between the HXR spectral index, γ, and the number index of the energy spectrum of the emitting electrons⁶⁰ through bremsstrahlung processes, δ, is

$$\begin{eqnarray}\delta =\left\{\begin{array}{ll}\gamma -0.5 & \text{non-relativistic case},\\ \gamma & \mathrm{relativistic}\ \mathrm{case}.\end{array}\right.\end{eqnarray} \tag{ 3 }$$

For optically thin synchrotron radiation, the relation between δ and the microwave photon spectral index, α, can be described by the following relations for the relativistic (e.g., Rybicki & Lightman 1979) and semirelativistic (e.g., Dulk 1985) cases

$$\begin{eqnarray}\delta =\left\{\begin{array}{ll}1.1\alpha +1.36 & \text{semi-relativistic case},\\ 2.0\alpha +1 & \mathrm{relativistic}\ \mathrm{case}.\end{array}\right.\end{eqnarray} \tag{ 4 }$$

We will use these relations in the following for the interpretation of the detected emission from the two brightest BTL flares reported in this work. The spectra of the two flares in the 30 keV–10 MeV energy range are somewhat different so we discuss them separately.

Oct13: As already shown in Pesce-Rollins et al. (2015), both RHESSI and Fermi-GBM detect HXR spectra for the Oct13 flare that are relatively steep (the best-fit value of the power-law photon index γ = 3.2 ± 0.1). However, as shown in Figure 10, there is a tentative indication that the spectrum may harden into a nearly flat part in the 1–10 MeV range, requiring either a similar hardening of the spectrum of the accelerated electrons, as, e.g., described in Park et al. (1997), or a contribution from de-excitation nuclear lines. We found that the inclusion of a 2.223 MeV broad neutron capture line and the nuclear de-excitation lines did not improve the fit significantly and that the best-fit model for the emission observed by Fermi-GBM is a single PL.

As described in Section 5.2, the thin target is most likely, in which case following Equation (3) for the nonrelativistic case (<100 keV) the index of the electron number density δ ∼ 2.7. The radio spectrum observed by RSTN, shown in Figure 12, indicates microwave flux of 60 solar flux units (6 × 10⁵ Jy), which peaks at frequency ν ∼ 5 GHz and declines as ${\nu }^{-\alpha }$ with the index α ∼ 2 above this frequency. Based on the relations described in Equation (4), we find that the required power-law index for the number density of electrons to be δ ∼ 5 or δ ∼ 3.5 in the relativistic or semirelativistic regime, respectively.

These values are steeper than the index δ ∼ 2.7 required in the nonrelativistic HXR range, indicating a steepening of the electron spectrum above 1 MeV. We therefore conclude that the GBM flux in the energy range 300 keV–10 MeV is probably not due to the accelerated electrons and no extra component is expected.

Sep14: The best-fit model for the combined HXR Fermi-GBM gamma-ray emission from the Sep14 flare is a single PL with an exponential decay with photon index 2.06 ± 0.01 and a cutoff energy of 90 ± 7 MeV. This is an extremely hard spectrum for a solar flare and if it is produced by a thin-target electron bremsstrahlung model as described above, it would require an electron number spectrum with index δ ∼ 1.5 in the nonrelativistic range steepening to an index δ ∼ 2 in the relativistic range. However, as noted above, inclusion of the 2.223 MeV line improves the fit slightly, indicating a possible contribution from nuclear de-excitation lines so the steepening of the electron spectrum could be larger.

The radio spectrum of this flare, shown in Figure 12, is also harder. The emission detected by the San Vito station has an index α ∼ 1 above 1 GHz. Again assuming optically thin synchrotron emission would require semirelativistic electrons with electron power-law index δ ∼ 2.5, indicating a slightly larger steepening at high energies. If the magnetic field is lower, which could be the case because of the larger loop, the steepening would be even larger (δ ∼ 3). Thus, for the Sep14 flare a power-law spectrum, steepening at higher energies, is required to explain the HXR and microwave observations from the loop-top region. This, plus a pion-decay component, provides an acceptable fit to the Fermi data from 30 keV to several GeV.

The above interpretation of the HXR and MWs indicates that the electrons responsible for these emissions are most likely accelerated in the reconnection region above the loop-top source either by turbulence (Petrosian 2012) or in merging islands manifested in the particle-in-cell simulations (Drake & Swisdak 2012). The electrons are trapped in the loop-top region long enough (longer than the crossing time of a fraction of second) to produce detectable bremsstrahlung and synchrotron radiation, but most of their energy is lost in thick-target footpoints hidden from near-Earth instruments because of high optical depths (Pesce-Rollins et al. 2015). This scenario can also explain the HXR emission from a BTL flare reported by Krucker et al. (2007). The steepening of the electron spectrum in the semirelativistic regime is due to a decrease in the acceleration rate and not to transport or energy-loss effects.

If the Fermi-LAT emission is from a thick-target photospheric site, then this emission site is located on the visible part of the disk. It cannot come from the occulted AR. The fact that accelerated protons reach the on-disk emission site provides strong evidence that the acceleration site is the CME environment, as suggested by Cliver et al. (1993) and Pesce-Rollins et al. (2015). They would diffuse across a wide range of magnetic fields, some connected to the AR behind the limb and some to the visible side of the Sun. These ions will most likely come from the downstream region of the CME shock while SEPs escape the upstream region. This can account for the differences in their spectral and temporal characteristics. It is then possible that the discrepancy between the spectral characteristics in 1–10 MeV for the GBM and radio observations can be explained by lower-energy 1–100 MeV protons or electrons precipitating to the visible disk side from the downstream region and producing the 1–10 MeV nuclear de-excitation line or bremsstrahlung emission below the chromosphere. These and other possibilities, for example, the production of Fermi-LAT emission by relativistic electrons either in the CME shock, in reconnection regions on current sheets behind the CME, or trapped in a large loop with strong convergence, will be discussed in future papers.