DIFFUSE EMISSION MEASUREMENT WITH THE SPECTROMETER ON INTEGRAL AS AN INDIRECT PROBE OF COSMIC-RAY ELECTRONS AND POSITRONS

We exclude data taken during viewing periods (exposures) that have high background contamination or are dominated by short transient sources which are not useful for our study of the diffuse emission. For example, viewing periods containing solar flares and periods when the spacecraft enters the radiation belts are excluded from the analysis in order to remove periods when the data were dominated by high backgrounds. For energies around 1 MeV, high-energy particles saturate the electronics and can generate false triggers. However, it is possible to use these data thanks to other electronics (via pulse shape discriminators or PSDs) not affected by the saturation problem. These electronics operate in parallel to the fast pre-amplifier output, generating an independent trigger for photon energies between 650 keV and 2.2 MeV. The trigger signal issued by the PSD electronics is used to confirm and select events between 650 keV and 2.2 MeV. The procedure is explained in more detail in Jourdain & Roques (2009).

A further selection based on the χ² between the sky model and the data was made. For a few viewing periods, the sky modeling does not correspond very well with the data. This is due to the point-source treatment: some very short transient sources that are difficult to identify are missed in our sky model and/or there is inaccurate modeling of source variability (see Section 3.2.1). The affected viewing periods, which have relatively high χ² values, contribute only to the systematics for the sky model and hence do not bring useful information on the diffuse emission. These viewing periods account for only ∼2% of the whole data set. Therefore, we do not use those exposures in the present analysis. After their removal, the data set contains 38,699 exposures for the analysis below 650 keV, which corresponds to a livetime of ∼1.1 × 10⁸ s. Figure 1 shows the resulting 25–600 keV exposure map. For analysis above 650 keV, the PSD electronics did not operate during some of the viewing periods. Therefore, we also excluded these viewing periods, with the total number in our data set for this energy range then further reduced to 36,486. The global χ², and the maximum χ² per pointing, are given in Table 1 for the best model (see Section 4.2).

The data contained in the 196–200 keV and 1336–1342 keV band correspond to strong instrumental background lines and are not used in the analysis. The first band contains the 198 keV line due to the de-excitation of an isomeric state of ⁷¹Ge. The second band contains the 1337 keV line related to ⁶⁰Ge K-shell electron capture, which may blend with the 1333 keV ⁶⁰Fe line. We bin the data between 20 keV and 2.4 MeV into 24 energy bands for the spectral analysis, and into 7 large energy bands for imaging or diffuse continuum morphology analysis. For the latter analysis, the data contained in the 1170–1176, 1330–1336, and 1806–1809 keV bands are removed because they contain counts from the ⁶⁰Fe and ²⁶Al radioactive lines (see Section 6).

The modeling of the instrumental background is an important issue and challenge for the data analysis. The distribution of the instrumental background in the detector plane changes significantly during the period spanned by the observations because two detectors failed early in the mission (detector 2 on 2003 December 7 and detector 17 on 2004 July 17). The "uniformity" map or background pattern (Equation (2)) should ideally be derived from "empty field" observations. However, the dedicated SPI "empty fields" are too rare to derive suitable and precise "uniformity" maps for the large data set used in the present analysis. Another way to determine the background pattern is to solve Equation (1) for the intensities of both sources, background, and detector pattern simultaneously.

The resulting patterns are therefore model dependent because of the unavoidable cross-talk between the extended source emission and the background pattern (and sources). These effects have been intensively studied above and around 500 keV, where many high latitude exposures can be considered as "empty field" exposures (see Bouchet et al. 2010) because they contained no detectable sources. For the present study, this has a minor impact because the sky model is sufficiently well approximated and hence the derived intensities are not significantly affected.

We treat the background intensity as varying with time and test several timescales. Timescales above 6 hr give poor χ². In general, the F-test shows that using a background intensity, A_p (Equation (2)), which varies on an exposure timescale (∼2800 s), 2 or 4 hr does not improve the fit compared to using a background timescale of ∼6 hr. The ∼6 hr timescale analysis also produces slightly reduced error bars because there are less free parameters for background and source components. However, the intensities are perfectly compatible with those obtained with a more variable background (timescale <6 hr). We therefore use a background timescale of ∼6 hr throughout the following analysis.

Finally, the standard configuration used for the analysis consists of computing the background pattern per period between two detector annealings (performed every ∼6 months to restore the detector energy resolution) with an intensity normalization that varies with a 6 hr timescale (leading to ∼6000 intensities to be fitted for the 38,699 exposures).

The data contain contributions from point sources, positron annihilation radiation (511 keV line plus positronium continuum) and from the process of interest in this study, the "diffuse" (non-annihilation) continuum emission, in addition to backgrounds. To extract the diffuse flux with SPI, information for the source positions along with their variability timescales is required. We therefore need to build a sky model to derive the corresponding fluxes.

3.2.1. Point-source Emission

3.2.2. Annihilation Radiation Spectrum

3.2.3. "Diffuse" Emission

The tools used for background modeling, imaging, and model fitting were specifically developed for the analysis of SPI data and described in Bouchet et al. (2008). To determine the sky model parameters, we adjust the data through a multi-component fitting algorithm, based on the likelihood test statistic. The core algorithm to handle such a large, but sparse, system is based on MUMPS ⁶ (Amestoy et al. 2006) together with an error computation method dedicated and optimized for the SPI response matrix structure (Rouet 2009; Tzvetomila 2009).

To determine the (model independent) spatial distribution of the Galactic ridge emission, the entire sky is divided into pixels. The sky exposure (Figure 1) is extremely non-uniform. To compensate, the pixels are chosen to have different sizes in latitude and longitude to extract fluxes with a comparable signal-to-noise ratio in each pixel over the whole sky. Because the point-source intensities are derived simultaneously, the pixel sizes also have to be chosen large enough to avoid "cross-talk" with point sources. The region |l| < 100°, |b| < 30° is divided into cells of size δl × δb = 15° × 26 (40° × 55 for the 600–1800 keV band). Outside of this region, the pixels have larger sizes depending on their locations. The pixel sizes are chosen a posteriori to optimize the signal-to-noise ratio per cell while being sufficiently small to follow the observed diffuse spatial variations. The number of unknowns (the diffuse part of the sky to be "imaged" contains 201 pixels for E < 600 keV and 121 pixels above) is high but reasonable compared to the available data, and the problem is easily tractable using a simple likelihood maximization to determine all the corresponding intensities and error estimates. For this particular analysis, the fitted intensities are constrained to be positive. This constraint stabilizes the "imaging" system of equations to be solved. Figure 3 shows the all-sky intensity images of the diffuse emission in several energy bands. These images and profiles give a qualitative view of the diffuse emission, but very localized structures outside the region delimited by |l| > 100° and |b| > 30° may be missed, due to the sizes of the pixel in this region δl ⩾ 15° × δb ⩾ 52.

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To quantify more easily the behavior of the diffuse emission, we present our results in terms of longitude and latitude profiles in Figures 4 and 5. These figures are obtained by integrating the flux measured for |b| ⩽ 65 in longitude or |l| ⩽ 23° in latitude, and |b| ⩽ 82 and |l| ⩽ 60° for the 600–1800 keV band.

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Because the images and profiles have a low signal-to-noise ratio, in order to have more quantitative measurements a complementary analysis was performed in which the diffuse continuum spatial morphology is modeled using several template maps (Table 3), separately or in combination.

Several DIRBE emission maps have been tested as tracers of the diffuse emission: the NIR/DIRBE map (1–10 μm) related to the stellar emission, the MIR/DIRBE map (10–30 μm) related to dust nanograins and PAHs heated to high temperatures, and the FIR/DIRBE map (30–240 μm) related to ∼ micron-sized dust grains emitting in thermal equilibrium with the heating Galactic ISRF. We have also tested 21 cm H i (Dickey & Lockman 1990) and ¹²CO (Dame et al. 2001) maps. The CO templates trace the molecular gas of the Galaxy (which can be related to the distribution of the young stellar populations), and are used to represent the spatial distribution for the nucleosynthesis lines.

Our analysis employs extinction-corrected IR maps, where the method is described in Krivonos et al. (2007), which produces corrected maps to an accuracy of ∼10%. We denote these maps as A_λ where λ is the wavelength. For example, the A_4.9 μm map is motivated by the probable stellar origin for the Galactic ridge emission below 100 keV (Krivonos et al. 2007).

The IC maps are based on a physical model for the IC emission generated with the GALPROP code (Section 5.1). Note that the IC map spatial morphology is energy dependent, whereas the spatial morphology for the other templates is independent of energy band.

All template sky maps are convolved with the instrument response and compared to the data. The resulting fit parameters are given in Table 3.

If a single template is fitted (with intensity as free parameter), the IC templates (models 54z04LMS or 54z04LMS-efactor, described in Section 5.3) are the best tracers for the spatial distribution of the diffuse continuum. They provide a better fit to the whole-sky distribution because they account for the emission at high latitudes, but are not always the best fit in the ridge region, where the NIR/DIRBE maps fit the emission better. The best non-IC map is the A_4.9 μm, especially for the low energy range (E < 200 keV), consistent with the results of Krivonos et al. (2007) who found that this map traces the diffuse stellar emission distribution at low energies. The IC templates constitute the best model of the whole-sky "diffuse" emission, but our derived map contains some unresolved source component in the ridge region. Therefore, an additional second map is needed to model more accurately the excess in the region |b| < 10°.

Above 600 keV, the IC map fits the data very well, although a marginally better fit can be obtained with the 1.25 μm map. However, the improvement over the IC map is small and is restricted to this energy range. A combination of IC and 1.25 μm maps is also marginally a better fit to the data in the 600–1800 keV band, but, as similarly for the other possible combinations of maps, does not really improve the fit, considering the number of extra degrees of freedom introduced. Furthermore, when the maps are combined the contribution of each separate map is difficult to measure, due to both the statistics and/or the unavoidable "cross-talk." So, above 600 keV, qualitatively, the best diffuse spatial emission model is the IC map.

At energies below 600 keV, the situation is more complex. In addition to the diffuse continuum, the annihilation radiation component is modeled as described in Section 3.2.2. A single IC map is not sufficient to fit the rest of the emission distribution because it does not fit the peak along the Galactic plane well. An additional map such as DIRBE, CO, or H i is needed to reproduce this structure. Table 3 indicates that below 600 keV, better fits to the spatial morphology emission can be obtained by combining two maps. For the 25–600 keV band, a combination of IC and the 4.9 μm, 3.5 μm, or A_4.9 μm maps give the best fits. For the 25–50 keV band, an overall best fit can be obtained with a combination of the IC and the 4.9 μm maps. This is consistent with the stellar origin for the Galactic ridge emission in the X-ray domain proposed by Krivonos et al. (2007).

Based on these results, and to simplify the statistical analysis, the emission profiles are finally fitted with a combination of the A_4.9 μm plus IC and bulge maps below 600 keV, and with a pure IC map above 600 keV; see Figures 4 and 5, and Table 3. With this method the signal-to-noise ratio of the derived diffuse total emission is optimized. Using the absolute best-fit model for each narrow energy band (bold italic in Table 3) instead of the above adopted model (bold in Table 3) does not change the results significantly.

In the previous section, we considered the spatial morphology and choice of skymap templates. We now turn to the spectral analysis. To extract spectra, we fix the sky model so that the diffuse spectrum morphology below 732 keV (the relevant upper spectral channel above 600 keV), is modeled with a linear combination of A_4.9 μm and an IC map computed using GALPROP, plus two Gaussians representing the bulge annihilation radiation component. For energies above 732 keV, the morphology is modeled by the IC map plus three spectral lines (Section 6). The intensities of the spatial components related to the diffuse emission together with sources and background intensities are adjusted to the data as described in Section 3.3.

Details on the individual component separation procedure are given in Appendix A. The bulge morphology differs significantly from the A_4.9 μm and IC map components and there is very little "cross-talk," hence this component is easily extracted separately (Table 4). On the other hand, IC and A_4.9 μm components are more difficult to disentangle due to their rather similar morphologies. Nevertheless, it is possible to separate the contribution of each map (Table 4). To take into account the "cross-talk" between the IC and A_4.9 μm components, we add, in addition to statistical errors on fluxes, artificial systematic errors.⁷ When deriving these components, the data above ∼1 MeV are not used because the lower statistics do not allow for the distinction. The IC and A_4.9 μm spatially derived maps are then fitted as a superposition of three expected physical spectral components:

• 1.  
  emission by "unresolved sources," dominating below ∼50 keV, modeled by an exponential cutoff power-law spectrum;
• 2.  
  annihilation radiation spectrum modeled using a Gaussian centered at 511 keV with 2.5 keV FWHM plus positronium continuum (Ore & Powell 1949); and
• 3.  
  diffuse continuum mainly attributed to interstellar emission, modeled by a power law.

The diffuse spectral fitted parameters following the above procedure for the central radian defined by |l| < 30° and |b| < 15° are given in Table 4.

The IC extracted component is found to have a power-law index ∼1.8 with a 100 keV flux 9 × 10⁻⁵ photons cm⁻² s⁻¹ keV⁻¹. The A_4.9 μm tracer component is found to have a much harder power-law index of ∼1 with an intensity significantly lower than the IC component. In other words, the IC component dominates the flux over the sky. For the full sky, the ratio of IC to A_4.9 μm components is 14, 8, 1.8, and 0.95 at 50, 100, 500, and 1 MeV, respectively. When considering only the central radian, the IC component contains 15% at 20 keV rising to 25% at 1 MeV of the total IC sky flux. The 20–100 keV flux ratio of IC to A_4.9 μm component is ∼9 for the whole sky and 2.6 for the central radian.

The final fit is made for the total spectrum (sum of all components) by combining the spectral information obtained from each spatial morphology component fit. For this step, the A_4.9 μm and IC power-law indices and intensities are fixed while fitting the other components. We also introduce a cutoff in the A_4.9 μm component to steepen its contribution at higher energies (above 1 MeV) where it would be inconsistent with the data from the higher energy instruments otherwise. We model the component related to unresolved sources with a single exponential cutoff spectrum.

If the summed spectral components for IC, A_4.9 μm, and annihilation radiation are directly decomposed into the three spectral components described above, then the resulting spectral shape is similar to that described in the previous paragraph. The interstellar emission component is then best fit by a power law of index ∼1.4–1.5. It results in a diffuse continuum spectrum very similar to that of the summed power laws of the IC and A_4.9 μm components. The total diffuse spectrum for the central radian region is presented in Figure 6.

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These results are consistent with our previous work (Bouchet et al. 2008), except that the diffuse continuum power law obtained with the IC maps has a higher intensity (20%–30% at 50 keV). This is due to the large latitude extent of this model, which contributes significant integrated flux very far from the Galactic ridge, and is not included in Bouchet et al. (2008).

Above 1 MeV, the decomposition of the emission into two spatial components gives too much uncertainty for each component. We therefore show our determination of the diffuse emission based on estimating a minimum and maximum extracted intensity for each component. To do this, we fitted the spatial morphology with the A_4.9 μm tracer, which gives nearly the minimum extracted flux, while the IC map gives the maximum extracted flux. A similar spectral analysis as that described above was then performed. The range of uncertainty is shown as a shaded area in Figure 6.

The conventional coded-mask system provides, by construction, flux free from extragalactic background (EGB) contamination, but for SPI we need a background model (see Section 3.1). Due to the background modeling, there is uncertainty associated with the level of diffuse EGB flux in our diffuse emission determination. The instrument background is modeled with a background pattern whose intensity varies on a ∼6 hr timescale (Section 3.1). Alternatively, it can also be modeled as above but with an additional isotropic term, which can be either stable in time (to represent the EGB) or variable in time (related to long term variations of the background component). Nevertheless, it is difficult to distinguish between the two, and the first one is preferred for its simplicity. In our analysis, for the region |l| ⩽ 30° and |b| ⩽ 15° we find that EGB contamination is negligible (<3%) below 600 keV. For energies above 600 keV, because of the lower statistics, if we assume that the measured diffuse emission is EGB-dominated, then the diffuse component is reduced by at most ∼28% in the region |l| < 30° and |b| < 15°. This is within the error bars and does not affect our conclusions associated with the modeling (see below).

The GALPROP code (Strong & Moskalenko 1998; Strong et al. 2000, 2004, 2007) including a new model for the Galactic ISRF is the basis for predicting Galactic diffuse emission in the energy range from keV to TeV energies, thus covering more than 10 orders of magnitude in energy (Porter et al. 2008). The goal of the GALPROP project is to combine CR and broadband diffuse emission data from radio to γ-rays into a single interpretative framework. Therefore, while the modeling and interpretation has most recently focused on γ-ray data from the Fermi mission, it is also applicable to other experiments like the Wilkinson Microwave Anisotropy Probe (WMAP), Planck, INTEGRAL, and Milagro.

In Porter et al. (2008), we used the so-called EGRET-excess-based model (Strong et al. 2004), which invoked CR proton and electron spectra different from those measured directly in the solar system, in order to account for the γ-ray spectrum measured by EGRET. Now that the "GeV excess" in this spectrum has been shown to be absent in Fermi Large Area Telescope (LAT) data (Abdo et al. 2009), being presumably an EGRET instrumental effect, we use in the present work the "conventional model," which requires consistency of the modeled CR intensities and spectra with those directly measured. We use the model (GALPROP ID 54_z04LMS) described in Strong et al. (2010), which reproduces the electron (plus positron) spectrum measured by Fermi-LAT (Abdo et al. 2010), but is not fitted to Fermi-LAT γ-ray data. It has a halo height of 4 kpc and includes CR reacceleration; for further details see Strong et al. (2010).

The calculations presented in Strong et al. (2004), Porter et al. (2008), and Strong et al. (2010) show the importance of secondary leptons in CRs for the proper calculation of the diffuse emission. Secondary CR positrons and electrons produced via interactions of energetic nucleons with interstellar gas are usually considered as a minor CR component. However, the secondary positron and electron flux is comparable to the primary electron flux around ∼1 GeV in the interstellar medium (ISM), providing diffuse emission in addition to that from primary CR electrons. The enhancement is ∼1.2–1.4 times higher in the IC γ-rays up to MeV energies relative to that from pure primary electrons. This leads to a considerable contribution of secondary positrons and electrons to the diffuse γ-ray flux via IC and bremsstrahlung and to a significant increase of the Galactic diffuse flux below 100 MeV. For a detailed breakdown of the primary and secondary leptonic components as a function of energy, see Porter et al. (2008) and Strong et al. (2010). Secondary positrons and electrons are, therefore, indirectly traced by hard X-rays and γ-rays. The spectrum of secondary positrons and electrons depends only on the ambient spectrum of nucleons, the interstellar gas, and the adopted propagation model.⁸ Figure 7 shows the components of the ISRF that contribute to the IC emission in different energy ranges. The scattering of optical photons provides the majority contribution for ≳ 10 MeV, while the far-IR dominates in the ∼0.1–10 MeV range, and the cosmic microwave background (CMB) is dominant below ∼0.1 MeV.

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Figure 7 compares our baseline GALPROP model with the spectrum measured by SPI. The agreement with the spectral shape is reasonable but the overall intensity is slightly lower than the data. Better agreement with the SPI data is obtained by considering a model with a higher normalization for the primary electron spectrum, which is illustrated in Figure 8 where the total electrons are increased by a factor of two over the baseline model. An interpretation for this increase can be that the locally measured spectrum is not typical of the global average at the solar position, or that the CR source distribution is more peaked toward the inner Galaxy than that used in the baseline model. Alternatively, Figure 9 shows other possibilities to increase the IC component, either by increasing the Galactic CR halo height from 4 kpc to 10 kpc, or by increasing the input luminosity of the bulge component for the ISRF by a factor of 10. The uncertainty associated with the bulge input luminosity, metallicity gradient, and other factors all contribute so that the ISRF in the inner Galaxy is not as constrained as observed locally. Currently, the radial distribution of CR sources is not well known or constrained, and a factor two increase of the electrons in the inner few kpc is possible. The same effect could be obtained with secondary CR positrons and electrons if their CR nuclei progenitors were increased by a similar factor, but then this could be inconsistent with the diffuse γ-ray emission measured by the Fermi-LAT, which is mainly produced by the same CR nuclei. A larger halo is suggested by analysis of CR data (Trotta et al. 2010) and γ-rays (Strong 2010). Unfortunately, the morphology of these components is not distinguishable by SPI over the spatial region currently covered. Future observations at higher latitudes would allow discrimination between the different halo/enhanced ISRF combinations, but this will be the subject of future work.

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As a further check, we compare power-law approximations of the GALPROP models with those derived from template fitting. The best power-law fit to the GALPROP model spectrum shown in Figure 7 (ID 54z04LMS) is N(E) = 0.16 × E^−1.76, where E is the energy in keV. The power-law fit to the SPI-extracted IC template is N(E) = 0.34 × E^−1.79, which is two times higher than the GALPROP model intensity. For the model shown in Figure 8 (model 54_z04LMS_efactorS, which has the primary electrons increased by a factor of two), the best power-law fit to the model in the range 20 keV–5 MeV is N(E) = 0.29 × E^−1.76. Meanwhile, the power-law fit to the SPI-extracted IC template is now N(E) = 0.30 × E^−1.76, which is completely consistent with the model spectrum. Similar results can be obtained for the other models involving modifications to the CR confinement volume, or the intensity of the ISRF in the inner Galaxy. Thus, it seems that at least some enhancement in the inner Galaxy is required for the diffuse emission, but it can be due to a combination of effects. These we will explore in a subsequent work.

The photon energy range from 50 keV to 2 MeV is sensitive to IC from electrons below about 5 GeV. To illustrate this, Figure 10 shows GALPROP model calculations as in Figure 7, but for primary electron source spectra cutoff below 1 GeV and 5 GeV, respectively. The 5 GeV cutoff removes most of the emission in the SPI range, which comes mainly from CMB and far-IR component of the ISRF. With the 1 GeV cutoff much of the emission is restored, showing that most of the IC in the energy range considered in this paper comes from electrons between 1 and 5 GeV. For these energies, the locally measured CR electron spectrum is strongly affected by heliospheric modulation and the SPI measurements can allow indirect probing of the interstellar spectrum of these CRs. In turn, understanding the heliospheric transport of CRs could be improved because uncertainties on the true interstellar CR spectra directly affect the heliospheric model predictions.

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Using Fermi-LAT data, a claim has been made for two large γ-ray "bubbles," extending 50° above and below the Galactic center, with a width of about 40° in longitude (Su et al. 2010; Dobler et al. 2010). The γ-ray emission associated with these bubbles appears to have a harder spectrum (dN/dE∝E⁻²) than the γ-rays produced by π⁰-decay by CR nuclei interacting with the ISM, or IC emission from CR electrons and positrons modeled using GALPROP. It has also been suggested that the "bubbles" are spatially partly correlated with the hard-spectrum microwave excess known as the WMAP haze. If the features are real, and are associated, the IC γ-ray emission could extend down into the SPI energy range.

We modeled the "bubbles" using circular regions centered at b = 305 and b = −305 with a radius of 22° having a uniform emissivity, and tested various combinations of the maps (see Table 3) along with the "bubble" templates. For the 25–50 keV range, it is not possible to find a unique combination of maps including the "bubble" template that do not result in it being assigned a negative coefficient in the fit. Imposing the positivity constraint for the fluxes, we find no detection in this energy range, as indicated in the table. Similarly, above 50 keV there is no emission found corresponding to the "bubbles." The 2σ upper limits in several energy bands above 50 keV, assuming an emission with a power-law index −2, are given in Table 5.

The increased electron flux in the inner Galaxy could also be related to past activity in the Galactic center, as has been proposed for example for X-ray fluorescence (Terrier et al. 2010; Capelli et al. 2011). However, this would be much more localized near the Galactic center, so that a direct connection seems unlikely. Other more exotic possibilities for increased electron fluxes, like hypernovae (e.g., related to γ-ray bursts) cannot be ruled out with the present data.

To reduce the dimension of the problem to be solved, a flux extraction in counts space is first done and the resulting source counts are then converted into an incident photon spectrum. In this case only, the photopeak response part is used.

$$\begin{equation} C(E,d)= C^{(1)}(E,\theta,d) \times (1+\frac{C^{(2)}(E,\theta,d)+C^{(3)}(E,\theta,d)}{C^{(1)}(E,\theta,d)}. \end{equation} \tag{ A7 }$$

The term IRF⁽¹⁾ is photopeak efficiency (and omitting the energy resolution term for simplicity)

$$\begin{eqnarray} {\rm RMF}^{(1)}(E,E_{{\rm ph}}) &=& \delta (E,E_{{\rm ph}}) \end{eqnarray} \tag{ A8 }$$

$$\begin{eqnarray} C^{(1)}(E,\theta,d) &=& {\rm IRF}^{(1)}(E,d) \times S(E,\theta). \end{eqnarray} \tag{ A9 }$$

Then

$$\begin{equation} C(E,d)={\rm IRF}^{(1)}(E,d) \times S(E,\theta) \times \beta (E,\theta,d). \end{equation} \tag{ A10 }$$

For a fixed energy E, making the approximation that whatever the event types, the spatial distribution of counts over the detector plane (in function of the detector number d) is similar and hence does not depend on the detector number d (β(E, θ, d) ≃ β(E, θ)), then

$$\begin{eqnarray} C(E,d)&=&{\rm IRF}^{(1)}(E,\theta,d) \times S(E,\theta) \times \beta (E,\theta)\nonumber\\ &=& {\rm IRF}^{(1)}(E,\theta,d) \times S^{{\rm counts}}(E,\theta). \end{eqnarray} \tag{ A11 }$$

Here S^counts is called the flux in counts space.

It is possible to extract the source photon spectrum directly from the data. For that the emission spectrum's spectral shape is assumed to be known and to be continuous in energy. This shape can, for example, be extracted after a first flux extraction in counts space and conversion into incident photon spectrum. S^Fitted being the fitted photon spectrum, then

$${\fontsize{7.3}{8.3}\selectfont \begin{equation} C(E,d)= \frac{\sum _{i=1}^{3} \int {\rm IRF}^{(i)}(E_{{\rm ph}},\theta,d) \times {\rm RMF}^{(i)}(E,E_{{\rm ph}}) S^{{\rm Fitted}}(E_{{\rm ph}},\theta) {\it dE}_{{\rm ph}} }{S^{{\rm Fitted}}(E,\theta)} \times S(E,\theta). \end{equation}} \tag{ A12 }$$

If S^Fitted(E, θ) is sufficiently close to the incident spectrum S(E, θ) then the above formula predicts counts in data space. It can be rewritten as

$$\begin{equation} C(E,d)=R^{{\rm Pseudo}}(E,\theta,d) \times S(E,\theta). \end{equation} \tag{ A13 }$$

R^Pseudo is the response of the instrument assuming a continuous photon emission spectrum of known spectral shape (for one incident photon in each energy band).

Finally, for N_s sources located in the FoV, the data D(d, p, E) obtained during an exposure (pointing p) in the detector d for given energy E is

$$\begin{equation} D_{dp}=\sum _{j=1}^{N_s} R_{dp,j} S_{p,j} + B_{dp}. \end{equation} \tag{ A14 }$$

B_dp is the background obtained during an exposure (pointing p) in detector d for given energy E. R can be either IRF⁽¹⁾orR^Pseudo. The equation will be solved in this latter form (Equation (1)).

If the source emission spectrum follows a known spectral shape, then the counts predicted by the pseudo-efficiency method is more realistic. Anyway, this method can be more sensitive to errors in the simulated response matrix and storage simplification made, and requires the knowledge of the incident photon spectrum which might be continuous in energy. On the other hand, the flux extraction in counts space may predict inaccurate source counts as the photopeak-like assumed response is not the true response.