TeV solar gamma rays from cosmic-ray interactions

TeV solar gamma rays from cosmic-ray interactions

Abstract

The Sun is a bright source of GeV gamma rays, due to cosmic rays interacting with solar matter and photons. Key aspects of the underlying processes remain mysterious. The emission in the TeV range, for which there are neither observational nor theoretical studies, could provide crucial clues. The new experiments HAWC (running) and LHAASO (planned) can look at the Sun with unprecedented sensitivity. In this paper, we predict the very high-energy (up to 1000 TeV) gamma-ray flux from the solar disk and halo, due to cosmic-ray hadrons and electrons (), respectively. We neglect solar magnetic effects, which is valid at TeV energies; at lower energies, this gives a theoretical lower bound on the disk flux and a theoretical upper bound on the halo flux. We show that the solar-halo gamma-ray flux allows the first test of the –70 TeV cosmic-ray electron spectrum. Further, we show that HAWC can immediately make an even stronger test with nondirectional observations of cosmic-ray electrons. Together, these gamma-ray and electron studies will provide new insights about the local density of cosmic rays and their interactions with the Sun and its magnetic environment. These studies will also be an important input to tests of new physics, including dark matter.

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I Introduction

The Sun is a passive detector for cosmic rays in the inner Solar System, where direct measurements are limited. It shines in gamma rays from its disk and from a diffuse halo Seckel et al. (1991); Moskalenko et al. (2006); Orlando and Strong (2007, 2008); Abdo et al. (2011); Ng et al. (2016). Disk emission is expected due to cosmic-ray hadrons interacting with solar matter, which produces pions and other secondaries of which the decays and interactions lead to gamma rays. Halo emission is expected due to cosmic-ray electrons () interacting with solar photons via inverse-Compton scattering. There are no other important astrophysical mechanisms for steady solar gamma-ray production; solar-flare gamma rays are episodic, and are observed up to only a few GeV Schneid et al. (1996); Ajello et al. (2014); Ackermann et al. (2014a); Pesce-Rollins et al. (2015).

Gamma-ray observations thus open the possibility of detailed cosmic-ray measurements near the Sun. The hadronic and leptonic components can be distinguished because the disk and halo emission can be separated by direction. Further, the energy spectra of the cosmic rays can be inferred from the gamma-ray spectra, which can be measured over a wide energy range. This would give a significant advance compared to typical satellite detectors in the inner Solar System, which only measure the energy-integrated all-particle flux (e.g., Refs Rodgers et al. (2015); Lawrence et al. (2016)), and are thus dominated by low-energy particles. Further, gamma-ray data can trace the full solar cycle, testing how solar modulation of cosmic rays depends on energy and position Jokipii (1971); Potgieter (2013).

Figure 1 shows that the prospects for measuring TeV solar gamma rays are promising. The solar-disk fluxes measured in the GeV range with Fermi data Abdo et al. (2011); Ng et al. (2016) are high,

Figure 1: Prospects for TeV solar gamma-ray observations, illustrated with the disk emission (details in Fig. 5). Points: observations with Fermi Abdo et al. (2011); Ng et al. (2016), where the flux difference is due to time variation. Green band: the only theoretical prediction that includes magnetic effects Seckel et al. (1991). Dashed lines: the estimated differential point-source sensitivity of HAWC Abeysekara et al. (2017) (scaled to one year) and LHAASO Cao (2014); He (2016).

and naive extrapolation suggests that HAWC and LHAASO may detect gamma rays in the TeV range. Further, the GeV observations are significantly higher than the theoretical prediction of Seckel et al. Seckel et al. (1991), who proposed a compelling mechanism by which the solar-disk gamma-ray flux could be enhanced by magnetic effects. Evidently, even this expected enhancement is not enough, which increases the need for new observations to reveal the underlying physical processes. Even if HAWC and LHAASO only set limits on the TeV gamma-ray flux, that would be important.

Our goal here is to provide a theoretical foundation to quantitatively assess the TeV detection prospects. At low energies, cosmic rays are affected by magnetic modulation in the inner Solar System, as well as by magnetic fields in the solar atmosphere, all of which are complicated Seckel et al. (1991). At high energies, where magnetic effects can be neglected, the calculations are relatively straightforward but have not been done before this paper. The energy separating the two regimes is not known. We estimate that neglecting magnetic effects is appropriate for TeV–PeV gamma rays and show that it leads to useful benchmarks for GeV–TeV gamma rays. In future work, we will treat magnetic effects in detail. For a broader context on our program of work on understanding the gamma-ray emission of the Sun — aimed toward eventual new measurements of cosmic rays, among other goals — see Ref. Ng et al. (2016).

We now provide more information about gamma-ray observations and prospects. Over the 0.1 GeV–TeV range, the Sun has been well observed. Following the upper limits given by EGRET Thompson et al. (1997) and the first detection using EGRET archival data Orlando and Strong (2008), more detailed measurements were reported in Ref. Abdo et al. (2011) by the Fermi Collaboration, based on 1.5 years of data. Over the range 0.1–10 GeV, they separately measured the disk and halo fluxes, finding spectra , plus a hint of time variation in the disk flux. In Ref. Ng et al. (2016), where we used six years of Fermi data and a newer version of the data processing (Pass 7 vs Pass 6), we detected the disk flux up to 100 GeV, finding that its spectrum falls more steeply than . We also made the first robust detection of time variation, showing that the disk flux decreased by a factor of 2.5 from solar minimum to maximum. While the solar-halo gamma-ray flux is reasonably well understood, our results deepen the mysteries of the solar-disk gamma-ray flux. New observations are needed, especially at higher energies, which will critically test emissions models. However, this is difficult with Fermi due to the low gamma-ray flux.

In the TeV–PeV range, the only ground-based gamma-ray experiments that can observe the Sun are those that directly detect shower particles. (For air-Cherenkov detectors, based on detecting optical photons, the Sun is too bright.) The HAWC experiment began full operations in 2015, and is now reporting first results. The LHAASO experiment, under construction, is expected to begin operations in 2020. These experiments will greatly improve upon the energy range and flux sensitivity of their predecessors, e.g., Milagro Atkins et al. (2000), ARGO-YBJ  Aielli et al. (2006), and Tibet AS-gamma Hibino et al. (1989). Those and other experiments have observed the “Sun shadow,” a deficit of shower particles caused by the solar disk blocking cosmic-ray hadrons Amenomori et al. (2013); Enriquez-Rivera and Lara (2016), but none have detected an excess gamma-ray emission from the Sun. The shadow is displaced by by from the Sun’s position due to magnetic deflections of cosmic rays en route to Earth, but the gamma-ray excess will be centered on the Sun. HAWC and LHAASO observations in the TeV range, combined with Fermi observations in the GeV range, will provide a long lever arm to test models of solar gamma-ray emission.

This paper makes steps toward a comprehensive understanding of solar gamma rays. In Sec. II, we discuss the effects of magnetic fields and justify why we can neglect them here. The next three sections are ordered by the directionality of the signals. In Sec. III, we detail our calculation of the hadronic gamma-ray emission from the limb of the solar disk. This calculation has not been done before. We also estimate the flux of other secondary products (electrons, positrons, and neutrons), discussing if they are significant background for the gamma rays. In Sec. IV, we detail our calculation of the leptonic gamma-ray emission from the solar halo. We extend earlier calculations to higher energies and are the first to include allowed new contributions to the electron spectrum. In Sec. V, we discuss the all-sky signal of directly detected cosmic-ray electrons. Our points about these prospects are new and exciting. In Sec. VI, we present our conclusions and the outlook for further work.

Ii Interplanetary and Solar Magnetic Fields

The flux of cosmic rays near the Sun is altered by magnetic effects. Throughout the Solar System, there are magnetic disturbances sourced by the Sun and carried by the solar wind Solanki et al. (2006); Priest (2014). These form an interplanetary magnetic field (IMF) that repels galactic cosmic rays (“solar modulation”) Jokipii (1971); Potgieter (2013); the effects and their uncertainties increase at low energies and at small distances from the Sun. In addition, near the Sun, within approximately  AU, there are solar magnetic fields (SMF) that are quite strong, especially in the photosphere and corona Wiegelmann et al. (2014). Because the SMF are complex and not completely measured, their effects may be varied and are quite uncertain.

In this paper, we focus on gamma-ray signals in the energy regime where magnetic effects can be neglected. When this is appropriate for the solar-disk signal, it will be even more so for the solar-halo signal, for which cosmic rays interact farther from the Sun. We begin by discussing SMF effects on the solar-disk signal, as these turn out to be dominant over IMF effects.

SMF effects enhance gamma-ray production from the solar disk. A likely physical mechanism was proposed in Ref. Seckel et al. (1991), although the authors’ predictions still fall far below observations Abdo et al. (2011); Ng et al. (2016). The enhancement is due to the mirror effect of solar magnetic flux tubes on charged hadronic cosmic rays, which can reverse the directions of cosmic rays before they interact, thus producing outgoing gamma rays that are not absorbed by the Sun. At high enough energies, this mirroring becomes ineffective, and the enhancement ends. To estimate the critical energy for this transition, where magnetic-field effects on cosmic rays can be neglected, we compute the Larmor radius, , using the typical SMF strength near the Sun, G, and the solar radius, cm Solanki et al. (2006); Priest (2014), finding

(1)

A similar value is obtained for a single flux tube, for which the magnetic field strength can be times larger and the distance scale times smaller Solanki et al. (2006); Priest (2014). (Ref. Seckel et al. (1991) estimated to be between  GeV and  GeV, so our choice is conservative.) Because for typical hadronic interactions, SMF effects should therefore be negligible for gamma-ray energies above about 1 TeV. However, SMF models are uncertain, and it is important to test them with new data.

IMF effects reduce gamma-ray fluxes. Near Earth, IMF effects on the cosmic-ray spectrum are well described by the widely used force-field approximation Gleeson and Axford (1968); Abdo et al. (2011); Cholis et al. (2016) and detailed simulations Bobik et al. (2012); Miyake and Yanagita (2006), which are informed by extensive measurements. For cosmic rays in the inner Solar System, both modeling and data are sparse. A key clue is that the MESSENGER probe to Mercury found only modulation of the cosmic-ray spectrum above 0.125 GeV near solar distances around 0.4 AU Rodgers et al. (2015); Lawrence et al. (2016). Using a force-field model with appropriate parameters to be consistent with these data (potentials ), we find that IMF effects can be neglected for cosmic rays with energies above 100 GeV (and thus gamma rays above 10 GeV), even near the solar surface. However, IMF models are also uncertain, heightening the need for new data.

At energies where magnetic effects can be neglected, the solar-disk signal should thus be wholly due to the limb contribution. This emission is caused by cosmic rays that graze the Sun, encountering a column density that is large enough for them to interact but small enough for their gamma rays to escape. Because this signal can be calculated with minimal uncertainty, a gamma-ray measurement consistent with its flux prediction would confirm that magnetic effects are negligible. In principle, this could also be checked by the angular distribution of the signal, where the Sun would appear as a bright ring with a dark center, although planned TeV–PeV experiments may not have adequate angular resolution Abeysekara et al. (2013, 2017); Cao (2014); He (2016). Finally, tests could also be made by the time variation, as there should be none.

At lower energies, where magnetic effects are important, several distinctive signatures of the solar-disk signal should emerge. The flux should be larger, as SMF effects that enhance the gamma-ray flux dominate over IMF effects that decrease it Seckel et al. (1991). That is, our solar-disk prediction neglecting magnetic effects provides a theoretical lower bound on the disk flux, which is especially interesting at GeV–TeV energies. The angular variation of the signal should tend toward illumination of the full disk. And there should be time variations that reveal the nature of the dominant magnetic effects. IMF effects decrease gamma-ray production near solar maximum, due to cosmic-ray modulation Abdo et al. (2011); Ng et al. (2016). Perhaps surprisingly, SMF effects must act in the same sense, as the IMF effects are too small to explain the observed time variation Ng et al. (2016).

For the solar-halo signal, IMF effects dominate over SMF effects Abdo et al. (2011), except perhaps very near the Sun. The comparison of disk and halo signals will thus help disentangle IMF and SMF effects. It also means that neglecting magnetic effects provides a theoretical upper bound on the halo flux.

Iii Hadronic Gamma Rays

iii.1 Calculational framework

In the direction of the solar disk, the dominant source of gamma rays is the interactions of hadronic cosmic rays with matter in the solar atmosphere Seckel et al. (1991); Abdo et al. (2011); Ng et al. (2016). Of these interactions, the most important are inelastic proton-proton collisions that produce neutral pions, which promptly decay to gamma rays. (In Sec. IV, we calculate gamma-ray production by leptonic cosmic rays, including near the direction of the solar disk, although the interactions occur well away from the solar surface.)

Here we calculate the gamma-ray emission from the solar limb — the small fraction of the Sun encountered by cosmic rays that just graze its surface on trajectories toward Earth. We use the straight-line approximation, where gamma rays maintain the direction of their parent hadrons, appropriate because the particle energies are so high. We ignore emission from the disk because we neglect magnetic effects that can reverse the directions of cosmic rays before they interact Seckel et al. (1991) and because the contributions of back-scattered pions are tiny Ambrosio et al. (1998). As the ingredients of the calculation are reasonably well known, the predicted limb emission is robust and, as noted, sets a theoretical lower bound on the solar-disk flux.

We calculate the total flux from the limb, integrating over its solid angle. Here we assume that it cannot be resolved, as single-shower angular resolution of HAWC and LHAASO near 1 TeV is comparable to the -diameter of the Sun Abeysekara et al. (2013, 2017); Cao (2014); He (2016). The solid angle of the limb is tiny, of that of the solar disk. If the limb could be resolved, it would appear as a thin, bright ring, with the intensity (flux per solid angle) enhanced by over the intensity averaged over the solar disk. The angular resolutions of HAWC and LHAASO improve at higher energies, which may allow partial resolution of the limb, especially with stricter cuts to select events with the best angular resolution. In the long term, hardware upgrades to improve this should be considered.

We begin in Sec. III.2 by discussing gamma-ray production in a simplified case — proton-proton production of neutral pions in the thin-target limit — which can be handled semianalytically, following Ref. Kelner et al. (2006). Then, in Sec. III.3, we include the effects of multiple scattering and absorption, cascade processes, and nuclear composition through a simulation using GEANT4 Agostinelli et al. (2003); Gea (). In ths simplified case, the flux is

(2)

where is the number density of target protons at the line-of-sight coordinate , is the cosmic-ray proton intensity, is the inelastic proton-proton scattering cross section, and is the spectrum of gamma rays per interaction. The length of the chord through the solar atmosphere is  km, where the 8 comes from geometry,  km is the radius of the Sun, and km is the scale height of the solar matter density in the photosphere. In the realistic case, the most important interactions occur at proton optical depths , so this simplified case is not adequate for our full results, although it does introduce the framework well.

iii.2 Calculation for the simplified case

Figure 2 shows the solar mass density from Refs. Baker and Temesváry (1966); Vernazza et al. (1973). Above the photosphere, the density declines exponentially, following the Boltzmann distribution of gravitational potential energy in the nearly isothermal atmosphere. Figure 2 also shows the proton optical depth as a function of height above the photosphere. The cross section for inelastic proton-proton collisions changes only modestly with energy and is  mb for proton energies GeV Olive et al. (2014). In the optically thin limit, gamma-ray production is dominated by the decay of neutral pions, which, at the low densities considered here, always decay in flight before interacting. Kelner, Aharonian, and Bugayov Kelner et al. (2006) have extensively studied the yields of secondaries in proton-proton collisions, in which their results are based on a fit to data and to particle-interaction simulations. The yield of gamma rays has a broad energy spectrum, but the most important gamma ray typically has . The shape of closely follows that of , due to the exponential dependence, with the conversion factor .

The cosmic-ray flux can be taken to be that at Earth, as we neglect magnetic effects. (Technically, the flux at Earth includes some modulation effects, but these are negligible at such high energies.) Up to 1 TeV, we use the precisely measured proton spectrum from the Alpha Magnetic Spectrometer (AMS-02) Aguilar et al. (2015a). At higher energies, it is sufficient to extrapolate this using cm s sr GeV Olive et al. (2014).

Figure 3 shows the resulting gamma-ray spectrum for the case where we integrated over , up to roughly the largest value for which an optically thin calculation is appropriate (the probability for a proton to interact twice is then , so the gamma-ray spectrum scales linearly with ). We checked the results of our semianalytic calculation by a Monte Carlo simulation with the particle-interaction code GEANT4 Agostinelli et al. (2003); Gea (), for which the results matched to within . This shows that effects beyond those in Ref. Kelner et al. (2006), e.g., particle cascades in the medium, are unimportant.

Lastly, compared to the Sun, the gamma-ray flux from the limb of the Earth’s atmosphere has been measured by Fermi up to nearly 1 TeV and compared to simulations, finding good agreement with predictions, which demonstrates the robustness of theoretical calculations Abdo et al. (2009); Ackermann et al. (2014b). In principle, in the thin-target limit, the limb flux from the Sun could simply be expressed in terms of the limb flux from Earth, nullifying several potential uncertainties, such as the energy spectrum, composition, and cross section.

Figure 2: Solar mass density as function of height above the photosphere (left axis, blue dashed), as well as the same for optical depth for inelastic proton-proton collisions (right axis, red solid).

iii.3 Calculation for the realistic case

To include proton-proton interactions in the optically thick case, we use GEANT4 Agostinelli et al. (2003); Gea (). This allows protons to interact several times, and takes into account their particle and energy losses from all processes. It also includes gamma-ray production by cascade processes, such as bremsstrahlung by electrons. The number density of target photons is times smaller than that of solar matter so energy losses and gamma-ray production by inverse-Compton processes can be neglected Rott et al. (2013). The density is low enough that charged pions below 1 PeV will typically decay in flight before interacting. Neutrons and muons may escape, and the neutrons may survive to Earth without decay.

Figure 4 shows the range of values that contribute most to gamma-ray production, based on our GEANT4 simulation. The y axis is weighted to properly compare different logarithmic ranges of . The peak is near , where of cosmic-ray protons will interact at least once. To the left of the peak, the linear decline is due to reduced optical depth. To the right, the exponential decline is due to proton cooling and especially gamma-ray absorption, which happen to have similar interaction lengths (for pion production and electron-positron pair production, respectively). In combination, about 90% of the total flux arises in the range .

Figure 3: Solar-limb gamma-ray spectrum produced by hadronic cosmic rays. Red dotted line: semianalytic result for proton-proton interactions with . Green dash-dotted line: GEANT4 results for the full range of ; the gradual cutoff is because it cannot simulate proton interactions above 100 TeV. Blue dashed line: our empirical fit to the GEANT4 results, extrapolated to higher energies. Black solid line: our full prediction, including a correction factor for nuclei. The light grey shading approximately indicates the energies at which magnetic effects, neglected here, should be included.

Using this range of values, we use Fig. 2 to determine the corresponding range of heights above the photosphere and corresponding mass densities, finding  km and  g cm. This leads to important insights about the physical conditions in which interactions occur. In this range, the solar properties are reasonably well known and are stable in time. The conditions for the production of solar atmospheric gamma rays are quite different from those for Earth atmospheric neutrinos Gaisser and Honda (2002); for the latter, g cm at an altitude of 10 km, and the distance scales are short but the proton optical depth is high (). Lastly, this information will be useful for assessing interactions in the presence of magnetic effects, which we will consider in future work.

With GEANT4, we can simulate proton interactions only up to a laboratory energy of 100 TeV, which leads to a gradual cutoff of the gamma-ray spectrum near 10 TeV. To extend our results to higher energies, we develop an empirical fit to the GEANT4 results at lower energies. We modify our semianalytic approach, Eq. (2), by including a correction factor, , that only becomes important in the optically thick regime. For the free parameter , we find that 0.65 gives a good match to the GEANT4 results. This is shown in Fig. 3.

Figure 4: Normalized relative contributions of different values to the predicted gamma-ray flux, based on our GEANT4 simulation. We show the example of  TeV; other energies give similar results.

Finally, we consider the effect of nuclei in the cosmic rays and in the solar atmosphere. Besides protons, the only important constituent is helium, which has a relative number abundance in both the beam and target Baker and Temesváry (1966); Vernazza et al. (1973); Olive et al. (2014). We use the cosmic-ray helium data from Ref. Aguilar et al. (2015b) up to 1 TeV. Above that, we use a power law and extrapolate up to PeV with spectral index 2.7, which roughly describes the data compilation in Ref. Olive et al. (2014). Following Ref. Kachelriess et al. (2014), we calculate the gamma-ray flux enhancement factor due to cosmic-ray helium. We find that the gamma-ray flux is increased by an overall factor , with a small energy dependence due to a slightly different spectral shape between the proton and helium. We also consider the case in which the helium spectrum may be harder than 2.7 at high energies Ahn et al. (2010); Yoon et al. (2011); Atkin et al. (2017). If we use a spectral index of 2.58 Yoon et al. (2011) for extrapolation, our result changes by less than 20% near 10 TeV. Thus, we can safely ignore the spectral hardening.

Figure 3 shows our full prediction for the gamma-ray spectrum from the solar limb. The gamma-ray spectrum closely follows the cosmic-ray proton spectrum Kelner et al. (2006). This is because the pions and gamma rays typically carry fixed fractions of the parent proton energy, the cross sections and multiplicities for (high-energy) pion production and gamma-ray absorption have only mild energy dependence, and the pions decay before interacting. (For the same reasons, Earth atmospheric neutrinos at sub-TeV energies also follow the cosmic-ray spectrum Gaisser and Honda (2002).) In Fig. 3, the gamma-ray flux has normalization sr times the proton intensity (flux per solid angle). This factor can be roughly reproduced using , where sr is the relevant solid angle of the limb, is a typical value, and the last factor comes from assuming that each proton produces gamma-ray at .

The hadronic-interaction processes discussed here also produce neutrinos, electrons (including positrons), and neutrons Seckel et al. (1991). The neutrino flux Moskalenko et al. (1991); Moskalenko and Karakula (1993); Ingelman and Thunman (1996); Fogli et al. (2006) is an important background for dark matter searches with neutrino telescopes Tanaka et al. (2011); Aartsen et al. (2013); Rott et al. (2011), and constitutes a sensitivity floor Argüelles et al. (2017); Ng et al. (2017); Edsjo et al. (2017). The other species could be useful messengers to study cosmic-ray interactions with the Sun, using detectors such as Fermi Ackermann et al. (2012), AMS-02 Aguilar et al. (2013), CALET Torii (2011), and DAMPE Chang (2014). A dedicated study of their detectability, is beyond the scope of this paper, and will be considered elsewhere. Here we briefly comment on their relevance to gamma-ray observations.

Electrons can be effectively separated from gamma rays in space-borne detectors. However, this separation is difficult for ground-based experiments, as both particles induce electromagnetic showers in the atmosphere. In principle, the inclusion of electrons enhances the detectability of the Sun for ground-based experiments. The flux of the electrons can be estimated similarly to that of gamma rays, described above, also by first ignoring magnetic-field effects. The electron flux is found to times lower than that of the gamma rays, due to receiving a smaller fraction of the pion energy. Further, the detection of these secondary electrons with ground-based experiments is more complicated than gamma rays, as the effects of solar, interplanetary, and Earth magnetic fields all need to be taken into account, demonstrated by cosmic-ray shadow studies Amenomori et al. (2013); Enriquez-Rivera and Lara (2016). The deflections and diffusion they cause will reduce the electron flux per solid angle. Therefore, for the current study, we neglect the addition of the electron flux to the total electromagnetic signal observable by ground-based experiments.

Neutrons, the most important secondary hadrons, travel without being affected by the magnetic fields. The Sun is therefore a point source of neutrons, and could in principle be detectable by ground-based experiments. Compared to gamma-ray production in pionic processes, secondary neutrons carry a smaller fraction of the primary energy. However, spallation of helium is efficient at producing secondary neutrons. Combining these two factors, the limb neutron flux is comparable to that of the gamma rays (also the disk flux Seckel et al. (1991)). In practice, it is difficult for these neutrons to be confused with gamma rays by ground-based experiments, due to the excellent hadron rejection factor, . The detection in the hadron channel is also likely to be difficult due to the much higher background, compared to that of gamma rays and electrons. A more careful treatment of hadrons, in particular at lower energies, is the subject of a separate paper (Zhou et al., in prep.).

Iv Leptonic Gamma Rays

In directions away from the solar disk, there is a solar halo of gamma-ray emission, of which the dominant source is the interactions of cosmic-ray electrons () with solar photons Moskalenko et al. (2006); Orlando and Strong (2007, 2008); Abdo et al. (2011). Of these interactions, the most important is inverse-Compton scattering. There is also a contribution in the direction of the solar disk. We estimate that other interactions with solar photons are irrelevant; these include Bethe-Heitler Heitler (1954); Berezinsky et al. (2006) and photo-pion interactions of protons Andersen and Klein (2011) and deexcitation interactions of nuclei following photodisintegration Karakula et al. (1994); Anchordoqui et al. (2007a, b); Murase and Beacom (2010).

Here we calculate this leptonic gamma-ray emission, mostly following prior work Moskalenko and Strong (2000); Moskalenko et al. (2006); Orlando and Strong (2007, 2008). For the first time, we calculate results up to 1 PeV and show that uncertainties in the electron spectrum at very high energies allow larger signals than in the nominal case (a broken power-law spectrum for cosmic-ray electrons). As above, we neglect magnetic effects and assume straight-line propagation for the parent-daughter kinematics. Although the solar halo flux is present in all directions, its intensity (flux per solid angle) is greatest near the Sun, falling approximately as  Moskalenko et al. (2006); Orlando and Strong (2007, 2008) , where is the angle away from the center of the Sun. The flux within a given angle thus grows as , but the backgrounds — especially significant for ground-based detectors — grow as . Therefore, the solar-halo signal is most interesting at relatively small angles. We calculate the leptonic signal within degrees of the solar center; this value matches what we used for our Fermi analysis Ng et al. (2016) and will allow HAWC and LHAASO to treat it as a near-point source.

In the optically thin regime, the gamma-ray flux from the inverse-Compton interactions of cosmic-ray electrons is

(3)

where is the number-density spectrum of target photons at the line-of-sight coordinate , is the cosmic-ray intensity, and is the electron-photon differential cross section including Klein-Nishina effects.

Figure 5: Gamma-ray spectrum of the Sun. Points: disk observations with Fermi Abdo et al. (2011); Ng et al. (2016), where the flux difference is due to time variation. Green band: the predicted disk flux Seckel et al. (1991). Dotted lines: the estimated differential point-source sensitivity of HAWC Abeysekara et al. (2017) (scaled to one year) and LHAASO Cao (2014); He (2016). Our new prediction of the solar-disk signal due to cosmic-ray hadrons (from the limb) is shown by the green solid line. Our new prediction of the solar-halo signal due to inverse-Compton scattering of cosmic-ray electrons is shown by the black solid line for the nominal case and by the dashed lines for enhanced cases from Fig. 6.

The column density of the solar photon field is for small angles  Orlando and Strong (2008), where is the number density of photons at the solar surface and AU. For electron energies below about 0.25 TeV, the inverse-Compton cross section is in the Thompson regime, where the total cross section is constant with energy. At higher energies, it is in the Klein-Nishina regime, where the total cross section falls with increasing energy. An electron passing close to the Sun has an optical depth of (in the Thompson regime; less at higher energies), so the optically thin assumption of Eq. (3) is appropriate. To calculate the gamma-ray spectrum, we use the StellarICs code Orlando and Strong (2013a, b), slightly modified to include a parametrization of the electron spectrum at the highest energies. The solar photons are taken to have a blackbody spectrum with temperature 5780 K and corresponding typical energy of  eV. The photon density falls as distance squared far from the Sun but less quickly near its surface, where it varies as with radial distance as  Moskalenko et al. (2006); Orlando and Strong (2008). The cosmic-ray electron flux has been precisely measured by AMS-02 up to almost 1 TeV Aguilar et al. (2014), and measured moderately well by H.E.S.S. Aharonian et al. (2008, 2009) and VERITAS Staszak (2016) up to 5 TeV. We use a broken power-law fit to these data. As discussed in detail in Sec. V, the electron spectrum at very high energies might be much larger than expected from this nominal case, in which the flux above 5 TeV is assumed to fall off quickly. Our calculation is the first to show how allowed contributions to the electron spectrum above 5 TeV would enhance the solar-halo gamma-ray signal.

V Cosmic-Ray Electrons

Figure 6: Diffuse flux (weighted with ) of cosmic-ray electrons. Below about 5 TeV, there are measurements (points, as labeled Aguilar et al. (2014); Aharonian et al. (2009, 2008); Staszak (2016)). Above about 70 TeV, there are limits (gray region, which combines many experiments Kistler and Yuksel (2009); Ahlers and Murase (2014)). In between, the spectrum could be as large as the blue solid line, allowing enhanced contributions (pulsar or dark matter; details are in the text). HAWC should be able to immediately improve sensitivity down to (hadronic rejection) of the proton spectrum (red dashed line).

Figure 5 shows our results for the leptonic gamma-ray emission in the nominal case plus some enhanced cases. (Below 10 GeV, where there are measurements from Fermi Abdo et al. (2011), not shown here, our prediction is consistent.) In the Thomson regime, the gamma-ray spectrum is less steep than the electron spectrum due to the nature of the differential cross section. In the Klein-Nishina regime, the gamma-ray spectrum steepens sharply due to the suppression of the total cross section (in addition to the steepening electron spectrum). The nominal predictions are not detectable with HAWC and LHAASO. In fact, only the most extreme enhanced scenarios — with the cosmic-ray electron flux as large as the proton flux — are (lines labeled “Max” in Figs. 5 and 6). If no solar-halo signals are detected, as is likely, that will make it easier to isolate hadronic gamma-ray flux in the direction of the solar disk. Section V introduces a better way to probe cosmic-ray electrons.

Figure 5 also recaps our result for the hadronic gamma-ray emission from the solar limb. This is well below the leptonic gamma-ray emission from the solar halo near the disk (below about 1 TeV), as well as the sensitivity of HAWC and LHAASO. However, this prediction leads to several important points. The gamma rays observed from the solar disk must be hadronic, with their flux enhanced by magnetic effects, and the ratio of the data to our limb prediction provides a first direct measure of the strength of that enhancement. The hadronic gamma-ray spectrum must eventually bend toward and join with our limb prediction. Until the energy at which that occurs, there is positive evidence for interesting processes (magnetic effects) beyond the limb emission. It may be that the leptonic gamma-ray emission is never dominant in the data, despite its apparent dominance in Fig. 5.

Here we show that HAWC and LHAASO can directly measure the cosmic-ray electron () spectrum, which is of great interest Chang et al. (2008); Aharonian et al. (2008); Adriani et al. (2009); Profumo (2011); Aharonian et al. (2009); Kistler and Yuksel (2009); Hinton et al. (2011); Ackermann et al. (2012); Torii (2011); Adriani et al. (2011); Aguilar et al. (2013); Ahlers and Murase (2014); Chang (2014); Staszak (2016). Compared to the method of Sec. IV, this is simpler and more powerful. The flux is expected to be isotropic. If a nearby pulsar or dark matter halo contributes significantly, the resulting anisotropy would enhance the detection prospects, but we neglect this possibility. Because cosmic-ray electrons lose energy quickly, by synchrotron and inverse-Compton processes, the highest-energy electrons must come from quite nearby, e.g., a few hundred pc at 10 TeV.

Figure 6 summarizes present knowledge of the cosmic-ray electron spectrum. Below 5 TeV, there are measurements from various detectors, including AMS-02 Aguilar et al. (2014), H.E.S.S. Aharonian et al. (2008, 2009), and VERITAS Staszak (2016). Above 70 TeV, there are strong limits from ground-based arrays (summarized in Refs. Kistler and Yuksel (2009); Ahlers and Murase (2014)). Importantly, at 5–70 TeV, there have been no experimental probes, as emphasized in Ref. Kistler and Yuksel (2009). At those energies, the only limit, which is quite weak, comes from requiring that the electron flux not exceed the all-particle flux. New sensitivity is needed to probe the electron spectrum in this energy range, where new components could appear. Intriguingly, there are hints of a new component starting to emerge at 5 TeV, seen by both the southern-sky H.E.S.S. Aharonian et al. (2008) and the northern-sky VERITAS Staszak (2016).

HAWC and LHAASO detect electrons and gamma rays with comparable efficiency BenZvi et al. (2016). However, the flux sensitivity for electrons is worse because, like the background protons, they are isotropic. The sensitivity depends on just the hadronic rejection factor. (Gamma rays are not a background, except in the direction of point sources; the diffuse flux of TeV electrons, even in the nominal case, exceeds that of gamma rays, even in the direction of the Milky Way plane Prodanovic et al. (2007).) We assume a hadronic rejection factor of , which should be reachable (Segev BenZvi, private communication). Performance close to this has been demonstrated by some analyses with a partially complete HAWC detector Ayala Solares et al. (2016); Pretz (2016a). More importantly, HAWC has already shown preliminary limits that approach our estimated sensitivity Pretz (2016b).

Figure 6 shows the estimated HAWC sensitivity to the electron flux (LHAASO’s will likely be similar), along with possible enhancements to the 5–70 TeV electron spectrum. HAWC and LHAASO can reach higher energies than air-Cherenkov detectors because of their huge advantages in field of view and uptime.

Probing the 5–70 TeV cosmic-ray electron spectrum for the first time will allow interesting tests of pulsars, dark matter, and possible surprises. For pulsars, we use predictions from Refs. Profumo (2011); Hinton et al. (2011), which may explain the positron excess Adriani et al. (2009); Aguilar et al. (2013). (Even larger fluxes can be found in Ref. Fang et al. (2016).) For dark matter, we use the PPPC4DMID code Cirelli et al. (2011); Buch et al. (2015) to calculate the electron spectra from dark matter decay, in this case with a mass of 100 TeV and a lifetime of  s, which is comparable to current constraints Murase and Beacom (2012); Esmaili and Serpico (2013); Feldstein et al. (2013).

While simple, our results are important. Although the gap in coverage of the cosmic-ray electron spectrum was known Kistler and Yuksel (2009), as was the possibility of using HAWC to detect electrons BenZvi et al. (2016), this paper is the first to combine those points and quantify the prospects. In the near future, there will be good sensitivity to high-energy cosmic-ray electrons from the CALET Torii (2011), DAMPE Chang (2014) and CTA Actis et al. (2011) experiments. Even so, they may only reach  TeV. With more than a year of data already collected, HAWC has a unique opportunity now, and we encourage swift action to complete an analysis.

Vi Conclusions and Outlook

The Sun’s high-energy gamma-ray emission — seemingly due to irradiation by cosmic rays — is not well understood. Above 10 GeV, the Sun is one of the brightest sources detected by Fermi, and its disk emission is nearly an order of magnitude brighter Abdo et al. (2011); Ng et al. (2016) than predicted Seckel et al. (1991). In the TeV range, there have been no theoretical or observational studies.

Now there is a convergence of two opportunities: the recognition that the high-energy Sun can reveal important physics and the unprecedented sensitivity of the already running HAWC experiment. These opportunities will be enhanced by ongoing theoretical work and the sensitivity gain due to the coming LHAASO experiment.

This paper has three main results.

The first calculation of the gamma-ray emission due to hadronic cosmic rays interacting with the solar limb. At high enough energies ( TeV), magnetic effects can be neglected, and the complete emission from the solar disk should be from only the thin ring of the limb. This flux can be robustly calculated. Further, it serves as an important theoretical lower bound on the solar-disk emission at all energies. The enhancement of the disk flux by magnetic fields can be deduced by the ratio of the observed flux to this prediction. In the GeV range, this is a factor . As illustrated in Fig. 1, HAWC and LHAASO will provide new sensitivity to solar gamma rays in the TeV range, and can test if this enhancement continues, plus if there are new contributions, e.g., due to dark matter. (Limits from ARGO-YBJ Aielli et al. (2006) are already in preparation zhe (), and is about one order of magnitude weaker than HAWC sensitivity at TeV energies.) Finally, the limb flux would be significantly more detectable if the solar disk could be resolved, due to lower backgrounds per solid angle. Although we have conservatively neglected this possibility, it seems attainable.

New results on the gamma-ray emission due to cosmic-ray electrons interacting with solar photons. This emission forms a gamma-ray halo around the Sun, and the intensity peaks near the disk. For the first time, we calculate the TeV–PeV gamma-ray flux, including the possibility of new components in the 5–70 TeV electron spectrum. HAWC and LHAASO can at least set constraints at these energies, where there are no measurements.

A new perspective on allowed enhancements to the cosmic-ray electron spectrum and direct tests of such. Lastly, we show that direct observations of electromagnetic showers by HAWC and LHAASO can provide unprecedented sensitivity to the 5–70 TeV cosmic-ray electron spectrum. This search, based on nondirectional signals, will be a powerful probe of the high-energy electron spectrum, testing some realistic models.

This paper is part of a larger program of work to develop the Sun as a new high-energy laboratory (see Ref. Ng et al. (2016) for further discussion). With a good theoretical understanding of magnetic effects, the Sun could be used as a passive detector for cosmic rays in the inner Solar System, allowing measurements that are differential in particle type and energy, a capability unmatched by any existing or planned detector. Currently, the major roadblock to this goal is taking into account the complicated magnetic field effects, but this problem is tractable in principle, and progress is being made (Zhou et al., in preparation). The Sun is already a calibration source for direction, and could become one for flux. Interestingly, unlike any other astrophysical source, the Sun’s hadronic and leptonic emission can be clearly separated using angular information alone. Finally, a thorough understanding of cosmic-ray interactions with the Sun is crucial for testing dark matter and neutrino physics Leane et al. (2017); Arina et al. (2017).

Acknowledgements.
We thank Andrea Albert, Mauricio Bustamante, Rebecca Leane, Shirley Li, Shoko Miyake, Carsten Rott, Qingwen Tang, and especially Segev BenZvi, Igor Moskalenko, Elena Orlando, and Andrew Strong for helpful discussions. BZ was supported by Ohio State University’s Fowler and University Fellowships. KCYN was supported by NASA Grant No. NNX13AP49G, Ohio State’s Presidential Fellowship, and NSF Grant No. PHY-1404311. JFB was supported by NSF Grant No. PHY-1404311. AHGP was supported by NASA Grant No. NNX13AP49G.

Footnotes

  1. thanks: http://orcid.org/0000-0003-1600-8835
  2. thanks: http://orcid.org/0000-0001-8016-2170
  3. thanks: http://orcid.org/0000-0002-0005-2631
  4. thanks: http://orcid.org/0000-0002-8040-6785

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