StellarICS: Inverse Compton Emission from the Quiet Sun and Stars from keV to TeV

The code package includes an example ($solar\_ic\_driver.cc$) containing the parameters editable by the users depending on their requirements. The parameters to be defined are: radius, temperature, distance of the Sun (star), energy range and grid factor for both CR electrons and positrons, gamma rays, grid spacing (linear or logarithmic) of the angle from the Sun, integration range and steps, all-electron spectra models including cases with free parameterizations.

In the public version many models for the electron and positron spectra are implemented to the $LeptonSpectrum$ class, including those based on existing publications [45, 46, 42, 2]. The model parameters are freely definable.

Additional examples are reported in Section 3. Moreover, additional electron spectra and positron spectra can be defined by the user by adding named models to the same class. The CR modulation due to the solar wind is described with the force-field approximation, but this can be freely extended to more realistic models that we plan to include in the future.

The black-body formulation for the solar (and stellar) radiation field as a function of the distance from the Sun (and star) is calculated in the class called $StarPhotonField$. The formulation is characterized by the effective temperature of the photosphere following the Stephan-Boltzmann equation. Following our previous approach [46], the distribution of photon density close to the Sun (star), as extended source where the simple inverse-square law is inappropriate, is given by integrating over the solid angle of the solar (stellar) surface (see Fig. 2 of [46] for details), resulting in:

with $R_{\odot}$ radius of the Sun (star), $E_{ph}$ solar (stellar) photon energy, and $n_{BB}$ black-body photon density. For large distances from the Sun, eq. 2.4 reduces to the usual inverse-square law.

Cross-sections and IC emissivity are calculated in class $InverseCompton$.
The IC emissivity for a given all-electron spectrum and radiation field is computed following the formulation in [45]:

where $N(E_{e},r)$ is the electron density, with $E_{e}$ kinetic energy of the particle as defined above, $n_{ph}(E_{ph},r)$ is the solar (or stellar) photon density as function of the distance from the Sun (or star) and of the solar (stellar) photon field $E_{ph}$ as in eq. 2.4, $\sigma_{KN}$ is the Klein-Nishina cross-section, $\gamma=E_{e}/m_{e}$, and $E_{\gamma}$ is the energy of the gamma-ray IC photons. The CR all-electrons are assumed to have an isotropic angular distribution everywhere in the heliosphere, as is the case for CRs in the Galaxy.

Both isotropic and anisotropic Klein-Nishina cross-sections are included, as described in [57] and in [46]. In particular for the anisotropic cross section we have:

with $E_{ph}$ target photon energy in lab; $E_{\gamma}$ gamma-ray energy in lab, $E_{ph}^{\prime}$ target photon energy in electron system with $E_{ph}^{\prime}=\gamma E_{ph}(1-\cos\eta)$, $E_{\gamma}^{\prime}$ gamma-ray energy in electron system with $E_{\gamma}=\gamma E_{\gamma}^{\prime}(1-\cos\eta^{\prime})$, $E_{e}$ electron energy, $\eta$ scatter angle of photon in lab, and $\eta^{\prime}$ scatter angle of photon in electron system.

The Thompson cross-section formulation is also included for a cross-check.
For comparison with other formulations of the IC, the reader is referred to the documentation in the StellarICS package (IC$\_$documentation.pdf). In particular, in that documentation, the form in terms of the energy transfer $v=E_{\gamma}/E_{e}$, as used in the code, is compared to other forms (e.g. as used in GALPROP [41, 56, 48, 24]).

The calculation of the IC emission integrated along the line-of-sight is performed in the class $SolarIC$. This class uses the input parameters from $solar\_ic\_driver$, the electron and positron spectrum parametrization from $LeptonSpectrum$, the photon field of the Sun (or star) from $StarPhotonField$ as a black-body source, and the IC cross-sections based on the Klein-Nishina formula from $InverseCompton$ which can be isotropic or anisotropic.

The IC intensity spectrum along the line-of-sight $s$ for a given all-electron spectrum and radiation field follows the formulation in [45, 46]:

where $ds$ is the infinitesimal increment along the line-of-sight and $\theta$ is the angular distance from the Sun (star); $s$ goes from 0 at the Earth, to the user’s defined maximum distance along the line-of-sight. In the direction to the Sun (star) surface $s_{max}=d-R$, with $d$ distance to the Sun (star) and $R$ solar (stellar) radius. The available option to use a logarithmic angular grid, instead of the default linear angular grid, is useful to resolve the rapid variation near the solar disk while still covering the full angular range required far from the Sun.

StellarICS also accounts for the following effect. The solar (or stellar) radiation field is anisotropic since the photons are traveling away from the surface, and the radiation is radial for distances $r~{}\gg~{}R$. The IC reaction rate depends on the relative velocity of electron and photon, $2c~{}(1-\cos\eta)$ where $\eta$ is the angle between electron and photon momenta. This causes larger variation with angle from the Sun (star) than would be the case for isotropic radiation. See also the discussion of the dip in the direction of the disk in Section 3.1.

The output is a FITS file with multiple extensions for a convenient use. Extensions names and contents of the output FITS file and respectively units are listed in Table LABEL:Table1.

In addition, the code outputs an $idl$ program which generates spectra and profiles directly.

Pamela [5] and AMS-02 [6] all-electron measurements show noticeably differences with respect to the measurements used in our previous works [45, 46, 2].

Here we present some examples of IC models that incorporate the local interstellar all-electron spectra that fit Pamela and AMS-02 measurements at 1 AU. In more detail, we define two baseline all-electron models: the first one mainly based on the Pamela all-electron spectrum for the period of 2008 [5], which we consider the representative model for periods of solar minimum; the second one mainly based on the AMS-02 all-electron spectrum for the period of 2013 [6], which we consider the representative model for periods of solar maximum. Both modeled spectra are fitted to AMS-02 data above 40 GeV, where AMS-02 data are more precise than Pamela and the solar modulation is not important. As shown in [46], all-electrons from  1 GeV to  1 TeV are responsible for generating the solar IC emission in the $\sim$30 MeV - 200 GeV Fermi-LAT energy range. Figure 1 shows the all-electron spectra used to generate the two baseline IC models that can be used for the Fermi-LAT studies. The blue solid line and the green dashed line represent the models that fit the all-electron spectra at 1 AU respectively for AMS-02 and Pamela data. At higher energies the models can reproduce also recent HESS⁴⁴4https://www.mpi-hd.mpg.de/hfm/HESS/pages/home/som/2017/09/ preliminary data [27]. DAMPE data [18] are also plotted for comparison.

As an example, the plot on the left of Figure 2 shows the IC spectral flux integrated in 5^∘ (solid lines), 10^∘ (dashed lines), and 20^∘ (dotted lines) around the Sun based on the two all-electron spectra of different solar conditions (green and blue spectra in Figure 1 respectively for solar minimum and maximum conditions). Spectral fluxes correspond to extension 8 in the output FITS file (see Table LABEL:Table1). Here and throughout the paper we assume the temperature of the Sun to be 5778 K, the solar radius to be 6.955 $\times$10¹⁰ cm, and the Sun’s distance to be 1 AU. The right plot in the same figure shows the spectral intensity (per steradian) for the same two all-electron spectra in Figure 1 for various angular distances from the Sun: 0.3^∘ (solid lines), 1^∘ (dashed lines), and 5^∘ (dotted lines). Intensities correspond to those in extension table 6 in the output FITS file (see Table LABEL:Table1). For illustration these two models in Figure 2 assume a constant modulation potential in the inner heliosphere, with the parameterization reported in eq. 2.3 for the outer heliosphere ($r$ > 1 AU); thus they do not assume any additional modulation in the inner heliosphere, i.e. $\Phi(r)=\Phi_{0}$ for $r$ < 1 AU in eq.2.3, with $\Phi_{0}=600~{}MV$ for solar maximum and $\Phi_{0}=350~{}MV$ for solar minimum. For comparison the isotropic diffuse gamma-ray background (IGRB) has spectral intensity of (9.5 $\pm$ 0.8) $\times$10^-4 MeV cm^-2s^-1sr^-1 [4] at 100 MeV.

Alternatively, we can extend eq.2.3 also to the inner heliosphere, when $\Phi(r)$ is given by eq.2.3 for any $r~{}\leq r_{b}$ for the same all-electron AMS02 and Pamela spectra in Figure 1. The angular profiles of the two IC models (extension 2 in the output FITS file as in Table LABEL:Table1) for these two cases are reported in Figure 3. These profiles are integrated in energies above 100 MeV and above 1 GeV for illustration. Note that the effect of solar modulation is smaller for $>$ 1 GeV than for $>$ 100 MeV, as expected. Emission from the disk extension between the Earth and the Sun, which is not occulted by the disk itself, is expected. However, the emission within the disk extension is substantially reduced. This is due to the Klein-Nishina formulation that decreases in the disk, going to 0 at $\theta=0$.

It is instructive to compare the IC emission with well-known reference levels: for example the IGRB integrated above 100 MeV has a total intensity of (7.2 $\pm$ 0.6) $\times$10^-6 cm^-2s^-1sr^-1 [4], which can be below the solar IC even up to 25^∘ – 40^∘ angular distance from the Sun, depending on the models.

We expect the Sun to emit IC gamma rays also at keV and MeV energies. KeV and MeV energies are of interest for various proposed missions, such as AMEGO, GECCO, and an e-ASTROGAM-like instrument. AMEGO is designed to detect photons from 200 keV to 10 GeV, while GECCO from 50 KeV to 10 MeV, and e-ASTROGAM from 300 keV to 3 GeV. Moreover, this could be of interest of current and future X-ray telescopes looking at the Sun or even detecting the diffuse emission at larger solar distances (e.g. eRosita). Therefore, this section presents our IC calculations down to keV energies.

Figure 4 shows the extended all-electron spectra below 1 GeV, which would be responsible for the IC emission at keV and MeV energies. Also in this case we show the two baseline IC models: the first one mainly based on the Pamela all-electron spectrum for the period of 2008, which we consider the representative model for periods of solar minimum (green line); the second one mainly based on the AMS-02 all-electron spectrum for the period of 2013, which we consider the representative model for periods of solar maximum (blue line). Both modeled spectra are fitted to available low-energy Balloon data from 1968 [14] and 1974 [25]. These are interplanetary all-electrons. The same figure also shows Voyager I all-electron measurements [17] in the interstellar space and a model from the local interstellar all-electron spectrum [50] that fits also Voyager I data.
The plot on the left of Figure 5 shows calculations for the IC spectral flux integrated in 5^∘, 10^∘, and 20^∘ around the Sun based on the two all-electron spectra of different solar conditions: solar minimum (green lines) and solar maximum (blue lines) that correspond to the all-electron spectra shown in Figure 4 with the same color coding. Spectral fluxes correspond to extension 8 in the output FITS file (see Table LABEL:Table1). AMEGO sensitivity (grey solid line, preliminary point source sensitivity for 5-year mission, private communication) and e-ASTROGAM-like instrument point source sensitivity [19] (grey dashed line, for 1-year effective exposure) are also shown. Spectral sensitivities are for a point source, which would be different from the sensitivity for an extended source. If we want to directly compare models of extended emissions with the instrumental sensitivity, this should be given for extended emission where the angular resolution of the instrument would be taken into account and instrumental simulations would be performed, which is beyond the present effort and our knowledges of the specific instruments. For a more informative comparison, AMEGO spectral sensitivity is 4$\times$10^-6 MeV cm^-2s^-1 at 1 MeV, 4.8$\times$10^-6 MeV cm^-2s^-1 at 10 MeV, and 1$\times$10^-6 MeV cm^-2s^-1 at 100 MeV, with an angular resolution of 3^∘ at 1 MeV and 10^∘ at 10 MeV ⁵⁵5https://asd.gsfc.nasa.gov/amego/technical.html. Below 10 MeV it is expected that GECCO has even better sensitivity being a Compton telescope down to 100 keV, even though it may not be suitable to directly point to the Sun. As reported in [19] an e-ASTROGAM-like instrument would have a sensitivity better than the following fluxes: 2$\times$10^-5 MeV cm^-2s^-1, 5$\times$10^-5 MeV cm^-2s^-1, and 3$\times$10^-6 MeV cm^-2s^-1, at 1 MeV, 10 MeV, and 500 MeV respectively for a 1-year effective exposure of a high Galactic latitude source.

The right plot in the same figure shows calculations for the spectral intensity (per steradians) based on the same two all-electron spectra in Figure 4 for various angular distances from the Sun: 0.3^∘ (solid lines), 1^∘ (dashed lines), 5^∘ (dashed-dotted lines), and 20^∘ (dotted lines). Intensities correspond to extension 6 in the output FITS file (see Table LABEL:Table1). For comparison the expected interstellar emission at intermediate latitudes (i.e. 10^∘$<|b|<$20^∘) from [50] is also plotted, together with the data of the total emission detected by Fermi-LAT at intermediate latitudes by [1], which includes the interstellar emission as the main component, and the extragalactic background and sources as minor components. Even though the expected solar IC emission above 100 MeV is comparable with the interstellar emission detected by Fermi LAT that acts as a confusing background, the solar IC component is definitively detected by Fermi LAT, and it has been currently studied in detail. As a consequence, because also below 100 MeV the expected solar IC emission is comparable with the expected interstellar emission, we expect the IC emission to be detectable by any instrument sensitive enough to diffuse emissions in this energy range.
This makes the solar IC emission very important for such proposed MeV missions as it is for Fermi-LAT. On the contrary, the extragalactic X-ray background is $\sim$ 10^-2 MeV cm^-2s^-1sr^-1 at 1 keV and $\sim$ 3$\times$ 10^-2 MeV cm^-2s^-1sr^-1 at 10 keV making the solar large-scale emission hardly detectable below 100 keV (and hence for eRosita).
For illustration, the models in Figure 5 assume no additional modulation from Earth to the Sun, which gives an upper limit to the IC emission produced by Galactic and interplanetary CR all-electrons. However, various modulation models can be assumed to test gamma-ray data at MeV when they will be collected by AMEGO, an e-ASTROGAM-like instrument, or GECCO. Hence, such data will enable us to study the low-energy part of the all-electron spectrum in close proximity of the Sun. As for the Fermi-LAT, given the broad distribution of the IC emission on the sky, the solar contribution to the diffuse MeV background cannot be neglected.

Investigating possible TeV emission from the quiet Sun has recently sparked interest, especially because of the HAWC, ARGO, LHAASO experiments and the forthcoming next generation SWGO [7]. In fact, upper limits to the emission from the solar disk have been recently obtained with HAWC [8] and ARGO-YBJ [12]. Hence, in this section we extend our calculations of the IC emission to the TeV energy band.

At TeV energy ranges the Klien-Nishina formulation would suppress the IC emission making the IC emission hardly observable by present and future telescopes. Hence in this example we calculate an upper limit to the expected IC emission for the case of the most extreme all-electron spectrum expected. In more details, Figure 6 shows an extreme all-electron spectrum that extends to 100 TeV (blue line). For energies below 1 TeV, the all-electron spectrum is constrained by all-electron CR measurements, whose details are provided in Figure 4. Recent HESS data [27] provide additional constraints to all-electrons up to 20 TeV. Because above 20 TeV there are no CR all-electron measurements, but there are instead CR proton measurements, we account for an extreme all-electron flux following the work in [8, 60]. In this extreme all-electron spectrum the upper limit to the flux is given by the CR proton flux and by gamma-ray observations of nearby sources. The consequence is that at $\sim$20 TeV this extreme model provides a spike, whose upper limit is given by the CR proton flux by assuming that CR all-electrons flux cannot be larger than the CR proton flux (usual all-electron to proton ratio is 1%). At even higher energies the upper limit to the CR all-electron flux is instead given by ground-based arrays observations of nearby sources (for example as reported in [32, 33]). As in [8, 60] this represents an extreme case and it would give an upper limit to the all-electron spectrum, and hence to the calculated IC emission at TeV energies. We are not investigating here the possible origin of such an extreme all-electron flux at these energies, if given by sources or dark matter (e.g. [53, 52, 28, 16]). Above 100 TeV, we assume a sharp cut-off with no all-electrons.

The resulting calculations of the IC emission at TeV energies for this all-electron spectrum are presented in Figure 7. The plot on the left of the figure shows calculations for the IC spectral flux integrated in 0.3^∘ (solid line), 1^∘ (dashed line) and 5^∘ (dotted line) around the Sun. Spectral fluxes correspond to extension 8 in the output FITS file (see Table LABEL:Table1). The same figure reports also recent upper limits from observations of the Sun with HAWC [8]. HAWC 3-year sensitivity (dashed grey line, [8]), LHAASO sensitivity (solid grey line, [10]), and SWGO sensitivity (solid black line, [7]) for point sources are also shown. Also in this case, there may not be direct comparison of point-source sensitivities with calculations of the extended emission, because spectral sensitivities are for a point source, which may differ from the sensitivity for an extended source. For a more informative comparison by accounting for the angular resolution of the instruments HAWC spectral sensitivity for a point source for 5-year observations is about 4 $\times$ 10^-6 MeV cm^-2s^-1 at 1 TeV, about 4 $\times$ 10^-7 MeV cm^-2s^-1 at 10 TeV, and about 2 $\times$ 10^-7 MeV cm^-2s^-1 at 30 TeV, with an angular resolution of 0.5^∘ at 1 TeV improving at higher energies⁶⁶6https://www.hawc-observatory.org/details/xtreme.php. LHAASO sensitivity for one year is 9 $\times$ 10^-7 MeV cm^-2s^-1 at 1 TeV, about 3 $\times$ 10^-7 MeV cm^-2s^-1 at 10 TeV, and about 10^-7 MeV cm^-2s^-1 at 30 TeV [10]. The 5-year spectral sensitivity of SWGO is 4 $\times$ 10^-7 MeV cm^-2s^-1 at 1 TeV, 5 $\times$ 10^-8 MeV cm^-2s^-1 at 10 TeV, and about 3 $\times$ 10^-8 MeV cm^-2s^-1 at 30 TeV with an angular resolution of 0.25^∘ at 1 TeV strongly improving up to 0.1^∘ at higher energies⁷⁷7https://www.swgo.org.
The consequence is that the solar IC emission will be barely observable even by most sensitive planned telescopes at TeV and for an extreme all-electron spectrum.
The right plot in the same figure shows calculations for the spectral intensity (per steradian) for 0.3^∘ (solid line), 1^∘ (dashed line) and 5^∘ (dotted line) angular distances from the Sun. This corresponds to extension table 6 in the output FITS file (see Table LABEL:Table1).