Implications of loop-top origin for microwave, hard X-ray, and low-energy gamma-ray emissions from behind the limb flares

The Fermi gamma-ray Space Telescope (Fermi) has detected hard X-ray (HXR) and gamma-ray photons from three flares, which according to \stereo occurred in active regions behind the limb of the Sun as delineated by near Earth instruments. For two of these flares \r has provided HXR images with sources located just above the limb, presumably from the loop top (LT) region of a relatively large loop. Fermi-Gamma-ray Burst Monitor has detected HXRs and gamma-rays, and RSTN has detected microwaves emissions with similar light curves. This paper presents a quantitative analysis of these multi-wavelength observations assuming that HXRs and microwaves are produced by electrons accelerated at the LT source, with emphasize on the importance of the proper treatment of escape of the particles from the acceleration-source region and the trans-relativistic nature of the analysis. The observed spectra are used to determine the magnetic field and relativistic electron spectra. It is found that a simple power-law in momentum (with cut off above a few 100 MeV) agrees with all observations, but in energy space a broken power law spectrum (steepening at rest mass energy) may be required. It is also shown that the production of the $>100$ MeV photons detected by The Fermi-Large Area Telescope at the LT source would require more energy compared to photospheric emission. These energies are smaller than that required for electrons, so that the possibility that all the emissions originate in the LT cannot be ruled out on energetic grounds. However, the differences in the light curves and emission centroids of HXRs and $>100$ MeV gamma-rays favour a different source for the latter.


INTRODUCTION
Fermi Gamma Ray Observatory (Fermi; Atwood et al. 2009) observes the Sun once every other orbit. During the past solar active phase its Large Area Telescope (LAT) has detected > 100 MeV photons from more than 40 solar flares. Few of these are detected only during the impulsive phase coincident with hard X-rays (HXRs) produced as nonthermal electron bremsstrahlung (NTB) and nuclear gamma-ray lines excited by high energy ions (mostly protons) (Ackermann et al. 2012). There is considerable evidence that the electrons are accelerated in a reconnection region near the looptop (LT) of the flaring loops (Masuda et al. 1994;Petrosian et al. 2002;Nitta et al. 2010;Krucker et al. 2010;Liu et al. 2013), and it is generally assumed that this is the site of acceleration of the impulsive phase protons (and ions) as well. But a majority of the LAT detected flares show only long duration emission (extending up to 10's of hours) usually rising after the impulsive phase . Some stronger flares show both impulsive and gradual emission (Ackermann et al. 2014). Almost all LAT flares are associated with relatively fast (> 1000 km/s) Coronal Mass Ejections (CMEs) and are often accompanied with gradual Solar Energetic Particle (SEP) events. This may indicate that the high energy particles responsible for the LAT gamma-rays are accelerated in the CME shock environment where the SEPs are produced. However, while SEPs are particles escaping the upstream region of the CME shock the gamma-ray producing particles, if originating at the CME, most likely come from the downstream region of the shock, with magnetic connection to the higher density solar atmosphere, which is the only place such high energy radiation can be pro-duced. This scenario has received further support from Fermi -LAT detection of three flares which, as observed by STEREO, originate from active regions (ARs) behind the limb (BTL) of the Sun as delineated by near-Earth instruments. The analysis and some preliminary interpretation of the data from Fermi and other instruments on the BTL flares are presented in Pesce-Rollins et al.(2015) and Ackermann et al. (2017;Ack17).
Our aim here is a more detailed modeling of the BTL flares with the particular focus on the determination of electron spectra and energy contents required to produce the multiwavelength radiations seen in two of these flares. It should be noted that flares, such as these with occulted foot points, provide a clearer view of the coronal LT source, which may be the site of particle acceleration. Thus, the analysis presented below provides the most direct information on the acceleration process. There are several reports of observations of partially occulted flares in HXRs (see, e.g. Frost & Dennis 1971;) and in gamma-ray emissions (Vestrand & Forrest 1993;Barat et al. 1994;Vilmer et al. 1999). More recently Effenberger et al. (2017) have provided a complete list of RHESSI observed partially occulted flares combining those from cycle 24 with the earlier list by Krucker & Lin (2008) from cycle 23. Analysis similar to that presented here can be carried out for any of these flare with contemporaneous microwave coverage.
The next section presents a summary of the relevant observational characteristics of these flares (all taken from Ack17). §3, provides a description of the main focus of this paper, which is to describe the emission processes and to determine the characteristics of the nonthermal electrons required for their production. §4 contains a brief discussion of the possibility of LT origin of > 100 MeV gamma-rays detected by the LAT. A summary and conclusions are presented in §5.

REVIEW OF RELEVANT OBSERVATIONS
Multiwavelength observations of the BTL flares and their analysis were presented in Ack17, the main source of the data used here. In Table 1 we reproduce some of these, and few new result from further analysis of the radio observations, relevant for our modeling, in particular for the determination of the broadband spectra and numbers (or energy contents) of the accelerated particles. Only two of the three BTL flares, namely SOL2013-10-11 and SOL2014-09-0, had complete sets of HXR, radio, and gamma-ray data. (For the sake of brevity, hereafter in the text we will refer to these as Oct13 and Sep14 flares, respectively). For each flare we give spectral parameters averaged over the duration ∆T of the flare (25 and 18 min, respectively) covering most of the impulsive phase. For HXRs we give the νf (ν) energy flux. 1 in units of erg cm −2 s −1 at 30 keV (above which the emission is dominated by NTB), the photon number spectral index, γ X , and a high energy exponential cutoff energy, ǫ c,X . Most of these are obtained from Fermi -GBM data, which agree with RHESSI and Konus-WIND data. The same parameters are also given for the LAT > 100 MeV gamma-rays. These are fits to the photon counts and can be used for modeling these observations by either a relativistic NTB or by a pion decay model. In Ack17 the photon counts were fitted directly to the thick-target pion decay model giving the time averaged simple power law (accelerated) proton indexes of 4.4 and 4.6 for these two flares, respectively.
The radio spectral parameters were obtained using the radio spectra shown in Figure 12 of Ack17 (also shown below; Fig. 4). These spectra appear to peak at a frequency ν p falling as a power law, f (ν) ∝ ν −γr , above the peak and decrease relatively steeply below it. These are clearly portions of optically thin and thick gyrosynchrotron emission with optical depth τ νp ∼ 1 at the peak. 2 In Table 1 we give our best estimates for the peak frequency ν p , νf (ν) flux at ν p and at optically thin part ν = 10 GHz, and the spectral index γ r . There are many causes of absorption of microwave radiation in solar flares (see Ramaty & Petrosian 1972) but the most common cause is synchrotron self absorption that gives a spectrum f (ν) ∝ ν 5/2 for ν ≪ ν p . The extant data is not accurate enough to distinguish among the various possibilities. In what follows, we will consider free-free and self absorptions. We note that the microwave spectra used for these estimation are for the one minute interval around the peak of the light curve where the particle and photon spectra are generally harder. This should be kept in mind when comparing the radio with the HXR and γ-ray spectra that are integrated over longer times used for HXRs.
In addition to the spectral observations given in Table 1, we need a few other properties of the emission site for detailed modeling of these flares. Table 2 gives some of these properties. For each flare, we give the angular size in sr (based on RHESSI images), height above the photosphere (based on the position of the AR BTL as determined by STEREO), the distance between the centroids of RHESSI and LAT sources (in arc seconds), emission measure EM (usually obtained from fits to the lower energy HXR thermal component), density as n = EM/V ( with V the volume of the source; see also footnote 5), the above mentioned durations and the magnetic field estimates based on the spectral fits to the optically thick radio spectra as described in the next section. For the Oct13 flare the EM value obtained from RHESSI thermal component by Fatima Da Costa Rubio (private communication) and determine the volume, V , from the source area times an assumed depth comparable to the width of the source. For Sep14 flare we do not have access to the thermal component so we assume an upper limit for the EM which gives a lower density, appropriate for its height above the photosphere.

MODELING OF THE LOOPTOP SOURCE
We assume that particles of energy E (in units m e c 2 ) are either accelerated outside the looptop (LT) source and injected into it at a rate ofQ(E, t) or they are accelerated in this source region with a spectrum N (E, t). As shown below, in either case, because the particle energy loss time τ L ≫ T esc (E, t), the time spend traversing the LT source, they lose a small fraction of their energy and produce thin-target radiation. In the first case, the spectrum of particles integrated over the source region would be N (E, t) =Q(E, t)T esc (E, t). If the acceleration and emission sites are the same thenQ(E, t) = N (E, t)/T esc (E, t) will represent the flux of the escaping particles. As evident the difference between these two scenarios is a matter of semantics, so in what follows we will use the first scenario which gives the number (and energy) flux of particles that escape the LT region (essentially at the injection rate) to the footpoints (FPs) of the AR located BTL, where they lose all their energy and produce the usual thick target FP radiations. These emissions, the usual focus for disk flares, are obscured by the optically thick solar gas from near Earth instruments for a BTL flare. STEREO , the only satellite with a direct view of the AR detected EUV radiation from these flares.
Our goal is to use the observations to obtain the spectrum of the injected flux,Q(E), and accelerated particle number, N (E). Over the small range of energies commonly provided by observations, one can uses a simple power law to describe this spectrum. However, for modeling the combined HXR and microwave data (and the gamma-rays in case of Sep14), we must consider electron spectra spanning a wide range of energies; from nonrelativistic (for production of HXRs) to extreme relativistic (for production of radio and gamma-rays). In this case, a broken power law, or a power law with an exponential cut off, could provide a better fit. If accelerated protons are responsible for LAT gamma-rays we need their spectra from 300 MeV to tens of GeV, again straddling the trans-relativistic range. This raises two important issues.
1. When dealing with trans-relativistic spectra one must distinguish between spectra in the energy and mo-  1 X-and gamma-ray Flux's refer to ǫ 2 J(ǫ ) in erg cm −2 s −1 , with J(ǫ ) as the number flux averaged over the durations given in Table 2. Radio fluxes F (ν) are in erg cm −2 s −1 Hz −1 averaged over over one minute around the peak. 2 X-ray and gamma-ray indexes refer to photon numbers J(ǫ ) ∝ ǫ −γ ; radio index is photon energy index F (ν) ∝ ν −γr . 3 High energy cutoffs in MeV. 4 Radio peak flux frequency in GHz. mentum spaces. A simple power law in energy space [N (E) ∝ E −δ ] will turn into a broken power law in the momentum space [N (p) ∝ p (1−2δ) at nonrelativistic and N (p) ∝ p −δ at extreme relativistic momenta], and vice versa, with a break at p ∼ mc or E ∼ mc 2 . This should be distinguished from the actual breaks determined by the interplay between the parameters of the acceleration and energy transport and loss mechanisms. In what follows, we will consider spectra in both momentum and energy.
2. The energy dependence of T esc (E), or the time the particles spend in the LT source. The usual assumption of the thin target model is that particles cross the length L of the source freely with T esc (E, t) ∼ τ cross ∼ L/v. However, this is the shortest possible escape time. In general, T esc (E, t) > τ cross because the source region is highly magnetized and may contain turbulence, in which case magnetic mirroring or scattering by turbulence become important. 3 . As a result the energy dependence of T esc (E) is more complex, and here also, it could change across the trans-relativistic energy. (Note that even though T esc (E, t) > τ cross the thin-target assumption is still valid because as shown below T esc (E, t) < τ L .) In the strong diffusion case, i.e., when scattering time τ sc ≪ τ cross the accelerated particles random walk across the source so that T esc ∼ τ 2 cross /τ sc and the magnetic field variations on the scale L have a small effect. On the other hand, in the weak diffusion limit with τ sc ≫ τ cross , and for a converging field geometry, the escape time is 3 Scattering by Coulomb collisions cannot be the agent here because then energy loss time, which is comparable to scattering time, will also be shorter than the crossing time and we will no longer be in the thin target regime (Petrosian & Donaghy, 1999). This may be the case at electron energies < 25 keV (see, Fig. 1  below) determined by how fast particles are scatted into the loss cone, in which case (for injected particle pitch angle distribution not highly beamed along the field lines) T esc ∝ τ sc , with the proportionality constant increasing with increasing field convergence. In summary we have if τ sc ≫ τ cross , Converging field τ cross /τ sc if τ sc ≪ τ cross , Strong diffusion.
Combining these three cases we obtain (see, Malyshkin & Kulsrud, 2001;Fig. 2 in Petrosian, 2016;[P16]) where η is a measure of the convergence rate of the field lines (e.g. the inverse of the ratio of magnetic field at the top of the loop to where they exit the LT source. In what follows we will use η = 3 (see, e.g. McTiernan & Petrosian 1991). In either case, the scattering by turbulence plays a crucial role. Relativistic particles scatter primarily by large scale fast mode or Alfvén waves, with τ sc ∝ E αer=(2−q) , where q is the spectral index of the turbulence; for Kolmogorov spectrum α er = 1/3. For semi-relativistic and nonrelativistic particles this relation is more complicated and does not fit a simple power law (see Pryadko &Petrosian, 1997 andLiu, 2004). CP13), applying the inversion method proposed by Piana et al. (2003) to two flares, find energy dependences for the escape, energy loss and acceleration times, empirically and directly from RHESSI data, in the nonrelativistic regime. These results indicate that we are dealing with the middle case in the above equation with α nr ∼ 0.8 and 0.2. This is in good agreement with the distribution of α nr determined (also empirically) based on comparison of SEP and HXR producing electron spectral indexes (see Fig. 4 in P16). In what follows we will treat α nr as a free parameter and use α er = 1/3 (or q = 5/3).
These two energy dependencies mold the thin target spectra. In Figure 1 we show the energy dependences of τ cross , τ sc , T esc (in cyan), where we have used a normalization (i.e. τ sc,0 ) that gives them the same relative value with respect to crossing time and Coulomb energy losstime determined in CP13. Here we also give energy loss times, defined as τ L = E/Ė L , for the energy loss ratesĖ L ) due to Coulomb, bremsstrahlung, synchrotron and inverse Compton (IC), 4 As evident for electron energies of > 10 keV of interest here the total energy loss time (solid black) is longer than the escape time justifying the thin-target assumption. To have a thick-target LT source we need T esc ≤ τ L . For Oct13 flare this would require an escape time that is 10 or 100 times longer, for HXR and microwave ranges, respectively. This means a 10-100 times shorter τ sc or a 10-100 times higher field convergence parameter η, for strong and weak diffusion cases, respectively. The requirement is more severe for Sep14 flare where the emission extends to relativistic regime so we need T esc ∼ 10 4 s (i.e. τ sc ∼ 10 −5 s or η > 10 4 , for strong and weak diffusion cases, respectively.) As also evident from these figures for most of the relevant energies we are in the weak diffusion limit. Thus, in order to simplify the analysis, in what follows, we will ignore the transition from weak to strong diffusion case and set T esc (E) ∝ τ sc (E).
In what follows we will deal mainly with νf (ν) spectra integrated over the LT source region and the specified duration ∆T around the peak of the impulsive phase emission so that Q(E) = ∆TQ (E, t)dt (or N (E) = Q/T esc (E)) is the total number of injected (or accelerated) particles, and T esc (E) will be the time averaged escape time.

Electron Bremsstrahlung and HXRs
The NTB νf (ν) spectrum of photons with energy ǫ (in units of m e c 2 ) produced by nonthermal electrons (interacting with background ions at nonrelativistic energies but with both electrons and ions in the relativistic regime) is obtained using differential cross section (integrated over angles) dσ/dǫ [see Eq. (3BN) of Koch & Motz (1959;KM59)] as: -Energy loss times for Coulomb (dashed-black), Bremstrahlung (dashed-green), synchrotron (solid-blue), IC scattering by optical photons (dotted-blue), total radiative loss (red) and total loss (solid-black). We use magnetic field values appropriate for these flares based on the analysis of the radio data in §3.2. In cyan we show the crossing (dotted), scattering (dashed) and escape (solid) times based on Eqs. (2) and (1) (for η = 3 and for two sets of parameters: αnr = 1.0, τsc 0 = 3.0 and αnr = 0.0, τsc 0 = 0.3) showing transition from strong to weak diffusion at τsc = τcross . Note that, for energies of interest here (30 keV to 100 MeV), energy losses can be neglected and we have a thin target situation.
Thus, given the T esc (E) as described above we can use Eq. (3) to obtain the total flux (in and out of the source), Q(E), of the accelerated electrons during the impulsive phase. In general, for most solar flares the HXR spectra J(ǫ ) decrease rapidly with energy (powerlaw index γ X > 3), with most of the emission in the nonrelativistic regime, so that the lowest energy value of ǫ 2 0 J(ǫ 0 ) provides a good estimate of the total photon energy integrated over the duration ∆T of the flare: where C = 4πd 2 ∆T and 4πd 2 = 2.8 × 10 27 cm 2 for distance d of 1 AU. This is the case for Oct13 flare with γ X = 3.2 but not Sep14 where the HXR νf (ν) spectrum is flat (i.e. γ X ∼ 2 or ǫ 2 J(ǫ ) ∼ const.) over several decades in energy, ∆ ln ǫ ∼ 7 giving the total In what follows we relate the photon energies to the total flux and energy of electrons SOL2013-10-11: As mentioned above the Oct13 flare has a well defined nonthermal spectrum, a simple power law with γ X = 3.2, between 30 and 100 keV (based on RHESSI and Fermi-GBM data). Thus, we can use the nonrelativistic approximations (β 2 ∼ 2E, T esc = T esc,0 E αnr , f nr ) and for a power law electron spectrum Q(E) = Q 0 E −δ obtain (see, e.g. Lin & Hudson 1971;Brown 1972;Petrosian 1973) and γ X = δ + 0.5 − α nr , where where Γ's stand for the gamma function. From these and 5 For fully ionized plasma the effective density n eff,nr = Σ i (Z 2 i n i ) at nonrelativistic energies and n eff,rel = n eff,nr + ne, where ne = (1 + X)/(2X)np and n i = X i /(XA i )np. Here Z i , A i and X i are the charge, atomic number and fractional mass of ions, and np is the proton density with X 1 = X. The background densities are usually obtained from the emission measure of the thermal HXR component, EM = V Σ i Z 2 i n i ne = V n eff,nr ne. For solar abundances (X 1 = X ∼ 0.74, X 2 = Y ∼ 0.25 and Z = Σ i>2 X i ∼ 0.01) it is easy to show that n eff,nr = 2 EM/V /(1 + X) So that for an average Z 2 i /A i ∼ 4 we get n eff,rel = 2.1 EM/V . In what follows we will use n eff = (1 and 2) EM/V for nonrelativistic and relativistic regimes, respectively. the observed photon flux J 0 and γ X = 3.2, we can obtain injected electron flux at E = mc 2 where we have defined η O ≡ 10 10 cm −3 n eff 5s Tesc,0 , or the average accelerated electron spectrum Note that the energy dependence of the escape time does not enter in the determination of the spectrum of the accelerated electrons, δ N ≡ −d ln N/d ln E = γ X − 0.5 = 2.7, while the index of the total flux Q is δ = 2.7 + α nr so for the range 0 < α nr < 1.0 we have 2.7 < δ < 3.7. Fig. 2 shows the calculated photon spectra for simple power law electron spectra (in both momentum and energy) for the flux Q (with very high energy exponential cutoff not relevant here) and for escape time index α nr as a free parameter. (We use the exact bremsstrahlung cross section; formula 3BN, KM59). As expected, in the nonrelativistic range the two spectra agree with each other (but, of course, with different indexes) and with the observations based on Fermi-GBM data (which agrees with RHESSI data for this flare). However, the two model spectra begin to diverge in the relativistic regime (with the power law in energy predicting higher emission). These deviations are beyond the observed HXR range, where we have only upper limits (open circles; (except two possible detection with large error bars). As shown below radio observations shed light on the spectra at these energies. Note also that the values of index obtained from numerical fits, δ e = α nr +3.0±0.05 (or δ p = 2α nr + 5.0 ± 0.05) are slightly different than the above values (δ e = 2.7 + α nr ) which assume the nonrelativistic approximations (e.g. δ ln β/d ln E = 1/[(E + 1)(E + 2)] ∼ 0.5) Using these fit parameters, the observed ǫ 2 0 J(ǫ 0 ) = 9 × 10 −8 erg cm −2 s −1 at ǫ 0 = 0.06, we can obtain the total number, and energy of the injected (or escaping) electrons for the ∆T = 25 min duration of the flare as follows. Using the normalization value (for energy fit) of 0.25 shown in Fig. 2 Given Q 0 we then obtain the total electron number and energy (above E 0 = 30 keV) as: and 6 If we use the approximate nonrelativistic relations in Eqs. (6) and (8) and the fact that where we have used the fitted index δ e = 3 − α nr , with α nr = 0.0. For α nr = 0.5 the number and energy values will be larger by factors of 5.5 and 8.2, respectively.  (from Ack17). The calculated spectra are for power law electron flux in momentum (blue) and energy (red) with a high energy cut off, and for the shown escape time parameters. Note that δe = (δp + 1)/2 as expected for nonrelativistic regime. The calculated spectra are based on the numerical integration of Eq. (3) with the exact function f (ǫ , E) from KM59. ǫ 2 0 J(ǫ 0 ) = 9 × 10 −8 erg cm −2 s −1 for ǫ 0 = 0.059, or 30 keV, as observed (see Table 1). Note that 0 < αnr < 1 is a free parameter and the calculated fluxes in the observed range are not affected by the relativistic index αer and the cutoff energy (or momentum).
SOL2014-09-01: We can carry out a similar analysis for Sep14 flare as well. However, because here we have a nearly flat νf (ν) flux extending over three decades in energy from nonrelativistic to extreme relativistic regime (30 keV to 30 MeV) we need to rely on numerical solutions. In fact as shown in appendix A it is difficult to obtain such a spectrum via bremsstrahlung emission because of the changes in the energy-momentumvelocity relation and the bremsstrahlung cross section across the trans-relativistic region. For a simple power law spectrum of the accelerated particles, N (E) = Q(E)T esc (E) ∝ E −δNe , and using the nonrelativistic and extreme relativistic forms of the function f (ǫ , E) in Eq. (3), it is easy to show that one obtains, respectively, photon spectra J nr (ǫ ) ∝ ǫ −(δNe+0.5) and J er ∝ ǫ −δNe (ln ǫ +c 1 ) (with c 1 a constant of order unity), which indicates spectral hardening of √ ǫ ln ǫ or photon index change of γ nr X − γ er X = 0.5 + 1/(ln ǫ + c 1 ). Thus, to get a power law photon spectrum we need a BPL spectrum of accelerated electrons, N (E), that steepens for E > 1. However, this spectral hardening can be compensated by a break in T esc (E), which, as can be seen from Eq. (2), is the case for α nr > α er = 1/3, so that a simple power law of injected electrons Q(E) can reproduce the observations. As shown in the top panel of Fig. 3, this is the case for α nr = 1.0, α er = 1/3 and δ e = 2.5 ± 0.1.
Similarly, for a simple power law in momentum, N (p) ∝ p −δNp , we have J nr (ǫ ) ∝ ǫ −(1+δNp/2) and J er ∝ ǫ −δNp (ln ǫ + c 1 ); again with spectral index γ X changing from 1 + δ p /2 to δ p − 1/(ln ǫ + c 1 ). In this case we have a spectral softening (or steepening) for δ p > 3 (which is usually the case). Thus, in momentum space we need an electron spectrum that gets harder (flattens) in the relativistic range. However, as shown in Appendix A, for 2.3 < δ p < 2.5 the logarithmic part can compensate for this steepening and give a nearly flat ǫ 2 J(ǫ ) spectrum across the trans-relativistic range. Again, for the injected (or escaping) spectrum, Q(p), we need to include the energy dependence of T esc . The above discussion implies that we need a weaker (or no) energy dependence for T esc . As shown in the bottom panel of Fig. 3, we obtain acceptable fits for α nr = 0.3 and δ p = 2.8 ± 0.1.
In summary, power-law injected spectra (with exponential cut off at above few 100 MeV) both in momentum and energy space can explain the observations with different values of index α nr but well within the range obtained empirically by CP13 and P16. Note however that, if we include the transition from weak to strong diffusion the photon spectra will be steeper than shown in the above figures at (low) energies below the observed range.
Following the same procedure as above, we can also derive the total number and energy flux of the electrons. We will use the fit parameters in the energy space which is simpler. The fitted index δ e = 2.5 and normalization Q 0 T esc,0 /τ brem = 2.8 used in obtaining the fit (top panel Fig. 3) implies where we set ǫ 2 0 J(ǫ 0 ) = 4 × 10 −7 erg cm −2 s −1 ), C = 4πd 2 ∆T = 3.0 × 10 30 cm 2 s (for duration ∆T = 18 min), and we have defined η S ≡ 10 9 cm −3 n eff 10s Tesc,0 . From this we can get the total number and energy flux of injected electrons above energy E 0 = 0.059 (30 keV) as Q(> E 0 ) = 2.1×10 37 η S and E e (> E 0 ) = 3.0×10 30 η S erg.
(14) Here we have ignored the exponential cut off which will reduce these numbers by a factor < 1 − E c /E 0 ∼ 0.99. Fig. 4 shows the observed microwave spectra (points) of the two flares. Synchrotron emission by relativistic electrons is the most likely mechanism of these emissions. The high frequency optically thin portion is observed over only one decade (∼ 1 < ν <∼ 10 GHz, with the spectral index γ r ) so that only a fit to a simple power law electron density spectrum (n(E) = n 0 E −δ ) is possible. In addition, because of the unusually large height (above the photosphere) of these sources, we most likely are dealing with lower than usual magnetic fields, lower gyro-frequencies, ν B = 2.8 × 10 6 (B/G), high harmonic ν/ν B > (300G)/B and Lorentz factor (γ ∼ 14 G/B). Thus, we are most likely in the relativistic regime, with no difference between the spectra in energy and momentum spaces, and we can use the usual relativistic formulation of the synchrotron emission and absorption coefficients J(ν) and κ(ν) (see, e.g. Rybicki & Lightman 1979):

General Synchrotron Spectra
and where a(δ) and b(δ) are slowly varying functions of order unity (see Appendix B). From these we get the source term and the spatially integrated radio flux F (ν) = S(ν)Ωf (τ ν ), where Ω is the angular size (in sr) and is the optical depth (integrated over the source depth along the line of sight, L = V /A). The function f (τ ) depends on the source shape and size, and magnetic field geometry. However, as shown in Appendix B, the spatially integrated results depend weakly on the exact form of this function. In what follows we will use primarily the plane parallel radiative transfer relation f (τ ) = 1 − e −τ . In general however, in the optically thin (τ ν ≪ 1) regime f (τ ) = τ and and in the optically thick (τ ν ≫ 1) regime f (τ ) = 1 and we get with a peak flux F p = S(ν p )Ωf (τ p ∼ 1), at frequency ν p . Fig. 4.-Observed (points; from Ack17) and self-absorbed fitted (curves) spectra for Oct13 (red) and Sep14 (blue) BTL flares. The dashed green curve includes free-free absorption, which provides a better fit for Oct13 flare (see Appendix B).

Electron Characteristics
From the observed spectral index γ r of microwave flux in the optically thin regime, we determine the electron index δ = 2γ r + 1 (and hence a(δ), b(δ)). 7 As is well known flux measurements in this regime is not sufficient to determine the number (or energy) of the electrons because of the degeneracy between n 0 and magnetic field B (or ν B ). Observations in the optically thick regime [Eq. (20)] provide the second datum which allows us to break this degeneracy and determine both n 0 and B. Using the expression for the source in Eq. (17), it is easy to show that we can write (see Appendix B for more details) which then can be used in Eq. (19) along with flux measurements in the optically thin regime to determine n 0 , or the spatially and temporally integrated number N 0 = dt n 0 ( r, t)dV as where we have used V /(L∆Ω) = d 2 andF is the average flux for the duration of the microwave flare. The spectra shown in Fig. 4 are for about one minute duration around the peak of the radio light curve. For the purpose of comparing with electron numbers and spectra obtained from the analysis of the NTB emission, we need the value of flux averaged over the same durations used above (∆T = 25 and 18 min for Oct13 and Sep14 flares, respectively). Since the radio light curve are almost triangular (see Figs. 2 and 5 in Ack17), we estimate average fluxes of 1/2 and 3/4 of the peak-time fluxes shown in Fig. 4 and given in Table 1, for Oct13 and Sep14, respectively.
In Fig. 4 we show self-absorbed spectra based on the above equations superimposed on the RSTN observations of the two flares from which we can determine ν p and F p . These are not very accurate fits, especially for Oct13 flare, but allow us to obtain a rough estimates of the required quantities. In particular, the value of B thus obtain is very uncertain for several reasons. One, as evident from Eq. 21, ν B is very sensitive to the measured parameters; it depends on the fifth power of ν p and square of F p . Two, inhomogeneities in the source can bias the result. Three, there may be other absorption processes, in particular as shown by Ramaty & Petrosian (1972) freefree absorption may be important in a high elevation, low magnetic field situation. In fact, the spectrum of the Oct13 flare in Fig. 4 shows some flattening around 5 GHz, perhaps due to free-free absorption, with possible emergence of self-absorption around 1 GHz. As described in Appendix B, and shown by the dashed green curve, inclusion of free-free absorption improves the fit consid-7 This and all of the above relativistic relations are valid for low magnetic fields (ν B ≪ ν) (and hence high Lorentz factors γ ∼ ν/ν B ). As shown in Petrosian (1981) (see also Petrosian & McTiernan, 1983), in the semi-relativistic regime these relations are more complicated. In general, for a power law electron index the synchrotron spectra steepen at lower frequencies (see, e.g. Ramaty, 1969), so that the relation between δ and γr varies slowly with frequency (see, Ramaty & Petrosian, 1972). Using numerical results, Dulk (1985) gives the semi-relativistic relation δ ∼ 1.11γr + 1.36, which is an approximate average value.
erably. As also indicated in Appendix B, this model also implies presence of optically thin free-free emission from 5 GHz to soft X-rays of < 10 keV well below the observed microwave fluxes and in rough agreement with the thermal bremsstrahlung flux observed below 10 keV (see, Pesce-Rollins et al. 2015).
In Appendix B using a self-absorbed model for the Sep14 flare we obtain magnetic field values ranging from 2 − 20 G. The fact that for this flare with a height of 10 10 cm we get magnetic fields lower than the usual B ∼ 100 G associated with low lying (∼ 10 9 cm) LT sources is encouraging. A self-absorbed fit to Oct13 flare gives B values in the range 300 to 3000 G. This is most likely not correct because of the poor fit. Using the fit parameters including free-free absorption yields a more reasonable value of ∼ 200 G. We use these values of B (or ν B ) and the fluxes at ν = 10 GHz (in the optically thin range) in Eq. (22) to calculate the number of electrons required for the production of the microwaves. For the Oct13 flare with fit parameters δ = 5.2, a(δ) = 2.6 and B = 200 G (obtained from the fit including free-free absorption) and the observed fluxF (ν = 10 GHz)=10 SFU we obtain N 0 or Q 0 = N 0 /T esc,0 to be This should be compared with 2.5 × 10 34 obtained in Eq. (10). There are however two uncertain parameters; n eff and B. For example the two estimates would agree for n eff = 5 × 10 10 cm −3 and B = 70 G. Note that for this magnetic field ν B = 0.2 GHz, and the lowest reliable observed microwave point of ν = 0.6 GHz is produced roughly by electrons with Lorentz factor γ ∼ (ν/ν B ) 1/2 = 1.7 so that relativistic expressions used here begin to break down and one should use the semirelativistic expressions. However, at such low frequencies we are in the optically thick regime, while the values of B and n 0 are determined by higher frequency data points. For the Sep14 flare using self-absorbed fit parameters δ = 2.7, B = 2.5 G, a(δ) = 0.1 andF (ν = 10 GHz)=10 SFU we obtain which is somewhat fortuitously exactly what was obtained from X-and gamma-ray observations given in Eq. (13).

Combined Electron Spectra
We now combine the results obtained for the electron characteristics from HXR and microwave data. In Table  3 we summarize our results on electron spectral indexes assuming Kolmogorov turbulence with α er = 1/3. For the Oct13 flare the index of 5.1 − 5.5 obtained from the microwave data agrees with momentum index based on HXRs with α nr = 0.05 − 0.25, but agreement with the energy index requires either an unusually large α nr > 2 or a spectral steepening (by 1 to 2 units) above E ∼ mc 2 . For the Sep14 flare the radio index of 2.8 for N (E) and ∼ 3.1 ± 0.1 for Q(E) is closer to the HXR momentum index of 2.8 ± 0.1 than the energy index of 2.5 ± 0.1. For this flare the values of Q 0 (or number of electrons at E = mc 2 ) obtained from radio and x-gamma-ray data are in excellent agreement. But as mentioned above some adjustments of uncertain parameter values (such as n eff , T esc , etc.) is needed for an acceptable agreement for the Oct13 flare. Figure 5 summarizes these findings. In addition to the preliminary analysis in Ack17 mentioned at the outset, there have been similar determination of electron spectra based on HXRs Plotnikov et al. 2017) assuming both thin and thick-target, based only on electron energy spectra, and without consideration of the energy dependence of the escape time. As expected the electron indexes derived in these papers are different than those presented here, which not only are for a thin target model but also include the energy dependence of the escape time. And in the case of Sep14 flare the analysis here includes the exact relativistic bremsstrahlung cross section. These factors can account for such differences. (red) assuming B = 70 G and Sep14 (black) assuming B = 2.5 G, using spectral parameters obtained from fits to HXR data (solid) and radio data (dashed). As evident there is excellent agreement between radio and X-ray based spectra. Blue curves use parameters based on fits to HXRs and power law in energy space, which shows deviation from spectra based on radio data. Dotted sections are extrapolations.

Emissions by Escaping Electrons
Some of the particles escape along closed field lines to the FPs to the AR located BTL and visible onle to STEREO . They lose all their energy at the FPs and produce thick target HXRs and microwaves. Some escape out of the corona along open field lines and eventually reach the Earth and are detected as SEPs by near-Earth instruments. As shown in P16, the escape times up, T u esc (E), and down, T d esc (E), will most likely have different values and energy dependences, so that the flux of SEPs will be different than those traveling to the FPs and produce HXRs. As shown in , observations indicate that most of the particles are directed downward and produce thick-target HXR and microwave emission more efficiently than in the LT region. For example the NTB spectrum would be which is similar to the thin target expression given in Eq.
(3) but with two differences. The first is that, instead of Q(E) we now have the effective electron spectrum given by the integral in the parenthesis, which for a power-law injected spectrum is equal to Q(E)/(δ − 1). The second is that, instead of escape time, the integrand contains the energy loss time τ L (E) = E/Ė L shown by the solid black lines in Fig. 1. In the nonrelativistic limit (e.g. for Oct13 flare) with T esc (E) ∝ E αnr and τ L ∝ E 1.5 this will lead to a FP photon spectrum with index γ FP X = δ − 1 instead of γ LT X = δ + 1/2 − α nr , implying that γ FP X = γ LT X − 1.5 + α nr = 1.7 + α nr . As shown in Fig. 1 for the energy range of 10 to few 100 keV T esc ∼constant (α nr = 0) so that the FP HXR emission will be much harder. Also, since loss time is about 10 times larger than the escape time in this energy range, the FP flux will be correspondingly larger (modulo the factor δ − 1 ∼ 2). These relations are more complicated for Sep14 flare with HXRs extending into relativistic range, but in general we would expect even a harder and higher flux of FT emission. The same is true for synchrotron emission by relativistic electrons where one must also consider the synchrotron emission, absorption and loss process in higher magnetic fields at the FPs, which affect both the emission and energy loss rates. This implies that 10 to 100 times higher fluxes of HXRs and microwaves are emitted from the FPs (in the AR BTL) than those emitted from the LT.
The above equation is also applicable if the LT source was a thick rather than a thin-target source. As stated in §3 this will require an unusually short scattering mean free path (i.e. short τ sc ) or highly converging magnetic field structure. but if these were the case it would require a steeper accelerated electron spectra. For example, in the nonrelativistic HXR emission case, instead of δ thin = γ X + α nr − 0.5 [see discussion related to Eqs. (6) and (7)] one needs δ thick = γ X + 1 which is steeper by (index higher by 1.5, for α nr ∼ 0). Similarly the required energy fluxes of electrons will be lower by a factor equal to the average value of T esc (E)/τ L (E) in the relevant energy range. Thus, all the curves in Fig. 5 would be lower and steeper and the transitions from nonrelativistic to relativistic range would be somewhat different.

LAT GAMMA-RAYS AND ACCELERATED PROTONS
The Fermi-LAT emission of > 100 MeV photons is different from the impulsive emissions considered in this paper in two important ways. The first difference is that centroids of the LAT sources are located ∼ 65 ′′ and 275 ′′ away from the centroids of the RHESSI LT sources for the Oct13 and Sep14 flares, respectively. The second is that, like most flares detected by the Fermi-LAT, the LAT light curves of the flares under consideration are very different than the light curves of impulsive emissions. They rise somewhat later and decay much more slowly with a duration more similar to gradual SEPs that are believed to be accelerated in the CME environment. Since Fermi-LAT flares are almost always associated with fast CMEs, the possibility that the LAT emission is produced by particles accelerated in the CME-shocks and escape from the shock downstream toward the Sun has gained some momentum. For the BTL flares under consideration here this scenario will require a magnetic connection between the downstream region and areas in the photosphere in the visible disk far away from the AR where these flares originated. Recent simulation (Jin et al. 2018;Plotnikov et al. 2017) indicate that this is a likely scenario.
These two differences point to a different origin for the LAT observations than the LT source considered above. However, based on the localization data alone, the possibility that the LAT gamma-rays may also be a thintarget emission coming from the LT RHESSI location cannot be ruled out with high confidence. So it is important to explore this possibility as well. Just as in the case of HXRs described above, a thin-target LT emission would require higher energy contents for the accelerated protons by a factor equal to τ L /T esc at the LT. The Coulomb loss time for 500 MeV protons is τ L ∼ 10 5 s (for n = 10 10 cm −3 ), but, unlike for electrons, we have no empirically based information on the escape time. Assuming the same (theory based) relativistic approximation used for electrons, we estimate escape times of 10 to 100 s. This means that the production of the LAT gamma-rays at the LT would require 10 4 to 10 5 times more energy for protons than that required for the thicktarget photospheric emission. Ack17, assuming thick target photospheric emission estimate proton energies of E p (E > 500 MeV) ∼ 1 and 7 × 10 25 erg, for Oct13 and Sep14 flares, respectively. This means that the LT thintarget model would require proton energies in the range of 10 28−29 erg. These, though larger are still about 10 times smaller than the energies of the electrons shown in Fig. 5. The proton energies would become comparable and could exceed the electron energies if their spectra are extrapolated to 10s of MeV. However, absence of a strong signature of nuclear de-excitation lines rules out this possibility. We therefore conclude that the possibility of a thin target LT source for gamma-rays cannot be ruled out with high confidence on energetic grounds alone. However, the differences in the light curves and centroids of HXR and > 100 MeV emissions favors a dif-ferent acceleration site and mechanisms for protons than HXR-microwave producing electrons.
Finally, we consider the possibility of the LAT emission being due to electron bremsstrahlung from a second relativistic electron component with 0.1 < E < 5 GeV. This component cannot be due to electrons accelerated at (and emitting from) the LT source because they will produce microwaves of ν > 10 GHz, and with a flux much higher than that observed. On the other hand, if the emission comes from the photosphere (produced, for example, by electrons that are accelerated at the CME and find their way to the photosphere) then such energetic electrons penetrate to very high densities just below the photosphere and lose almost all their energies via bremsstrahlung emission. In that case the required energy of electrons would be slightly larger than the observed energies of γ-rays of ∼ 1.4 × 10 24 and ∼ 1.2 × 10 25 ergs, for Oct13 and Sep14 flares, respectively) which are about 5 times lower than the energy of proton given in Ack17. However, acceleration of electrons to tens of GeV and their transport over large distances requires a very high acceleration or a very low energy loss rate. This fact also favors pion decay production of > 100 MeV photons.

SUMMARY AND CONCLUSIONS
In this paper we present a detailed analysis of HXR and microwave spectra of two solar flares (Oct13 and Sep14) which, based on STEREO observations, originated 10 and 40 degrees BTL of the Sun, but were detected by Fermi, RHESSI, SDO Konus-WIND and ground based radio telescopes. The relevant observed characteristics are summarized in Tables 1 and 2. The 20-30 min HXR light curves observed by RHESSI, Fermi-GBM and Konus-WIND are almost identical and similar to the radio light curves for both flares. The Fermi-LAT light curves are somewhat delayed and last longer. RHESSI images (up to 25-50 keV for Oct13 and 6-12 keV for Sep14) show sources (of size ∼ 50 ′′ ) at the limb presumably the top of a relatively large flare loop peeking over the limb. The LAT localizations puts the centroid of gamma-ray emission 65 ′′ and 275 ′′ away from the RHESSI source for Oct13 and Sep14 flares, respectively.
Based on the similarity of light curves we assume a co-spatial emission of HXR and microwave emissions and determine accelerated electron characteristics over a broad range of energies from sub-relativistic regime (based on bremsstrahlung emission of low HXRs) to extreme relativistic regime (based on bremsstrahlung emission of gamma-rays and synchrotron emission of microwaves). In case of Sep14 flare the measured electron bremsstrahlung emission extends from 30 keV to ∼ 100 MeV. This requires careful consideration of two important aspects. The first is the question of the time accelerated particles spend in the source region, which we call the escape time, and the second is that, because the observations span the trans-relativistic region, we should distinguish between spectra in momentum and energy space. Using empirically determined values and energy dependence of the escape time in 10-100 keV range by CP13 and P16, and their extension to relativistic energies based on theoretical considerations, we show that we are dealing with a thin target processes which then allows us to get the electron characteristics. Our results can be summarized as follows.
1. From modeling of the NTB emission of Oct13 flare we find that simple power law electron spectra in both momentum and energy space can reproduce the observed HXRs. For Sep14 flare a simple power law in momentum can describe the broad range of the observed HXRs more readily and with more reasonable values for the escape time index than simple power law in energy. From these fits we determine the spectral index, numbers and energy content of accelerated electrons.
2. The radio spectra for both flares show a distinct optically thin emissions that peak around 1 GHz and a well defined turnover at lower frequencies indicating emergence of a optically thick spectrum. Selfabsorbed synchrotron spectrum provides an adequate fit for the Sep14 flare, but for the Oct13 flare a self-absorbed synchrotron spectrum does not fit the observations in the range 0.5 < ν < 5 GHz.
We show that a model whereby free-free absorption starts at about 7 GHz with self-absorption becoming dominant below 2 GHz provides an acceptable fit. These modelings allow us to determine both the spectrum and numbers of relativistic electrons and the magnetic field (that turn out to be lower than usual appropriate for the large height of the source).
3. We then compare the two electron spectra obtained by these two methods. We show that for both flares extrapolation of spectra based on HXRs to the relativistic regime agree with those based radio data assuming a simple power law (with exponential cut off at several 100 MeV) in momentum but not in energy space. The latter require a broken power law with a break at E < mc 2 . The numbers and energy content of these flares are in the right ball park and allow us to predict the FP emissions from AR located BTL.
4. We also consider the possibility of thin target LT emission of the LAT gamma-rays and find that this requires 100 to 1000 time more energy of accelerated protons compared to thick target photospheric emission. However, even these energies are less than those of the electrons so that this scenario of high energy gamma-ray LT emission cannot be ruled out on energetic grounds. This is also true for production of these higher energy gamma-rays by GeV electrons at the photosphere. Nevertheless, because of the difficulty of acceleration of electrons to several GeV, pion decay scenario is favored, and the differences in the light curves and centroids of HXR and > 100 MeV emissions indicates a different acceleration site and mechanisms for (pion producing) protons than (HXR-microwave producing) electrons.
5. The radiative signatures of occulted flares, such as those considered here, provide the most direct information on spectra and energy content of accelerated particles, and hence on the acceleration mechanism, uncontaminated by the stronger FP emission. For example, the differences between the required spectra in energy and momentum spaces can shed light on the details of the acceleration process. This important aspect of the problem will be dealt with in subsequent papers.

Acknowledgements:
This work is supported by NASA LWS grant NNX13AF79G, H-SR grant NNX14AG03G and Fermi-GI grant NNX12AO78G. I would like to thank the Fermi colaboration, in particular the corresponding authors of Ack17 (A. Allafort, M. Pesce-Rollins, N. Omodei, F. Rubio and W. Liu) for help in preparation of this paper. I would also like to thank anonymous referees for many helpful comments.
6. APPENDIX A: SOME ASPECTS OF BREMSSTRAHLUNG EMISSION 1. Approximate Cross Section: The nonrelativistic and extreme relativistic approximations given after Eq. (3) (same as expressions 3BNa and 3BNb of KM59, respectively) can be combined as (26) Fig. 6 compares this cross section (dashed-green) with the exact (3BN) cross section of KM59 (solid-black). As evident the above simpler expression agrees with the exact values very well with largest deviation of less than few % around energies ǫ = mc 2 and E = mc 2 . This expression can be used for analytic derivation of photon spectra. To include the contribution of relativistic electronelectron bremsstrahlung one should change E → 2E in the numerator. 2. Flat νf (ν) Bremsstrahlung Spectra: Fig. 7 shows ǫ 2 J(ǫ ) ∝ ǫ ∞ ǫ 2 (dσ/dǫ )β(E)Q(E)dE NTB photon spectra obtained for a power law (with exponential cutoff) electron spectra in energy and momentum space. As evident flat photon spectra extending over several decades in photon energy is not possible for such electron spectra in the energy space but can be achieved for a power-law in momentum space for δ ∼ 2.3. Fig. 7.-Bremsstrahlung energy spectra for power laws in energy (top) and momentum (bottom) with exponential cut offs. For power laws in E flat spectra can be obtained for δe ∼ 1.6 only in the relativistic regime and for δe ∼ 2.1 only in the nonrelativistic regime. This due to logarithmic dependence of J(ǫ ) ∝ (ln ǫ + a) in the relativistic regime. However, for power-law spectra in momentum this term is compensated by the steeping of the spectra in the relativistic regime and fairly flat spectra can be obtained in both regimes for δp ∼ 2.3. 7. APPENDIX B: SOME DETAILS OF SYNCHROTRON EMISSION 1. Numerical coefficients: In §3.2 we introduced two coefficients which depend only on the spectral index of the electrons. In the relativistic regime they are (see, e.g. Rybicki & Lightman 1979)  (28) where Γ stands for the Gamma function and θ is the angle between the line of sight and the B field. For the LT sources the magnetic field may be radial or horizontal with respect to the limb so that we have θ = π/2 and the angular terms are equal to one. In the opposite case of chaotic field lines the last terms in the above equations are equal to ( √ π/2)Γ[(δ + 5)/4))]/Γ[(δ + 7)/4] and ( √ π/2)Γ[(δ + 6)/4))/Γ[(δ + 8)/4], respectively. An accurate determination of these coefficients is important because the magnetic field estimates are sensitive to their values. Table 3 gives the values of these and other parameters for the range 3 < δ < 5 of interest here.

Optical Depths and Magnetic Fields:
We are interested in the spatially integrated flux where S(ν) is the average source term and f (τ ) depends on the shape and geometry of the source. For example, for the plane-parallel approximation f (τ ) = 1 − e −τ and for a spherically symmetric source f (τ ) = 1 − 2/τ + 2(1 − e −τ )/τ 2 . Setting the derivative of the flux to zero we get d ln f (τ )/d ln τ = 5/(δ + 4), and peak optical depths τ p , f (τ p ) and c(δ)f (τ p ) shown in Table 3 for plane parallel and spherical (in parenthesis) geometries. Inserting these values in Eqs. (29) and using Eq. (17) we calculate gyrofrequency as Inserting the observed values shown in Tables 1 and 2 and the coefficients in Table 3 we find gyro-frequencies and magnetic field values of 1.0(0.6) GHz and 360(220) G for Oct13, and 10(4) MHz and 3.6(1.5) G for Sep14 flares (spherical geometry in parenthesis). (Note that c(δ) = a(δ)/b(δ), and hence the B field, varies more slowly with the spectral index δ [than a(δ) and b(δ)] and the angle θ (it would change by 10% going from random field to ordered field with θ = π/2).
We can obtain the B field with an alternative method which is independent of F (ν p ), the most uncertain observationally determined parameter. In this method we first eliminate one of the unknowns, namely n 0 using Eqs. (18) and (19) to obtain ν B . From the first equation evaluated at ν p and the second equation at any frequency in the optically thin regime we get which then gives ν B = ν p [c(δ)τ p Ωmν 2 /F (ν)] 2 (ν p /ν) 3+δ .
3. Free-free absorption: As mentioned above, free-free absorption with the absorption coefficient (see, e.g. Benz 1993) κ f f = 0.2ν −2 T −1.5 (EM/V ) 1 + 0.05 ln T /10 7 K ν/GHz can be important for high densities and low magnetic fields. For the Oct13 flare with large emission measure EM ∼ 1.3 × 10 48 cm −3 at T = 0.63 × 10 7 K (obtained from RHESSI data; Fatima Rubio 2017, private communication) we get an optical depth of τ f f = 50(GHz/ν) 2 , where we have used an area A = V /L ∼ 10 18 cm 2 , so that free free absorption can be important below ∼ 7 GHz. The dashed green curve in Fig. 4 shows a model spectrum that includes both free-free and synchrotron self-absorption with a total optical depth τ (ν) = (ν/ν f f ) −2 + (ν/ν sy ) −(2+δ)/2 , so that τ f f = 1 at ν f f = 7.5 GHz and self absorption, with optical depth of unity at ν sy = 2.7 GHz, becomes dominant for ν < 1.2 GHz for electron index δ = 5.2. This will require EM/(T 1.5 A) ∼ 3 × 10 20 which is within a factor of 3 of values quoted above. This will change the required magnetic field to a lower value. Following the steps of the second (alternative) method used above, we can again eliminate n 0 and obtain ν B and B. After some algebra we get or B = 210 G, similar to the lower values obtained above.
Since the inclusion of free-free absorptions improves the fit to the data we will use this value of the magnetic field.
In summary, averaging the above result we get magnetic field values of 2 to 10 G (Sep14) and 200-500 G for Oct13 flares which we have entered in the last column of Table 2.
It is interesting to note that this emission, extrapolated to few keV range should also agree with the thermal HXR flux. This involves extrapolation over a large frequency range (from ∼ 10 10 to 10 18 Hz) and differences between the Gaunt factors at microwaves of ∼ 15 and HXRs of unity. Nevertheless, dividing the above flux by this factor and the Boltzmann factor e ǫ /kT ∼ 10 4.5 (for ǫ = 10 keV and T = 10 7 K), we get 10 keV thermal bremsstrahlung flux of F (ǫ = 10keV ) ∼ ×10 −23 erg cm −2 s −1 , Hz −1 or νf (ν) flux of 10 −3 vs the observed value of ∼ 10 −4 erg cm − 2 s −1 (see, Fig. 3 in Pesce-Rollins 2015). Considering the scale of the extrapolation this is a satisfactory agreement.