ON THE FERMI LARGE AREA TELESCOPE SURPLUS OF DIFFUSE GALACTIC GAMMA-RAY EMISSION

Diffuse Galactic γ-ray emission (DGE) from the full sky has recently been analyzed and compared with observations by the Fermi Large Area Telescope (Fermi-LAT) for high energies (HE; 200 MeV ⩽ _γ ⩽ 100 GeV; Ackermann et al. 2012). The DGE was previously modeled using the GALPROP code (e.g., Moskalenko & Strong 1998; Strong et al. 2000; Porter et al. 2008); for a review, see Strong et al. (2007). These phenomenological models were constrained to reproduce directly measured cosmic ray (CR) data and were then used iteratively to calculate the DGE (e.g., Strong et al. 2004). To construct a model for the expected total γ-ray emission, the γ-ray emission from the resolved point sources together with the residual instrumental γ-ray background and the extragalactic diffuse γ-ray background—both assumed to be isotropic (Abdo et al. 2009b)—were added to the DGE model. In the inner Galaxy, the emission of the resolved sources apparently reaches a fraction of ∼50% of the expected overall spectral energy flux density at _γ ≈ 100 GeV (Ackermann et al. 2012).

These overall emission models describe the Fermi-LAT data well at high and intermediate latitudes and thereby show that the so-called EGRET GeV excess (e.g., Hunter et al. 1997) does not exist in the form previously inferred (Stecker et al. 2008; Abdo et al. 2009a; Ackermann et al. 2012). However, in the Galactic plane, these models systematically underpredict the data above a few GeV, and they do so increasingly above about 10 GeV until 100 GeV (see Figure 15 of Ackermann et al. 2012). In this paper, this difference between the data and model is called the "Fermi-LAT Galactic Plane Surplus" (FL-GPS) and it is most pronounced in the inner Galaxy. According to Ackermann et al. (2012), it can, however, also be seen in the outer Galaxy, with even a small excess at intermediate latitudes.³

The GALPROP code is constrained by the charged energetic particles directly measured in the neighborhood of the solar system, which are, by assumption, truly diffuse CRs. Therefore, the above discrepancy is not too surprising, because, in this comparison, the γ-ray emission from particles within the CR sources is only taken into account for those γ-ray sources that are resolved by the instrument.

As far as the Galaxy is concerned, the majority of the γ-ray sources resolved by the Fermi-LAT, with 1451 items listed in the Fermi-LAT 1FGL catalog and taken into account in the Ackermann et al. (2012) analysis, are pulsars. Except for the Crab Nebula and Vela X, the HE γ-ray emission from pulsar wind nebulae (PWNs) may actually be pulsar radiation, even though three additional PWNs have recently been identified with Fermi-LAT (Acero et al. 2013). For the purposes of their γ-ray emission, in this paper, we assume that these objects are sources of energetic electrons and positrons, but not sources of nuclear CRs. Of the latter, presumably only a handful have been resolved and are thus included in the overall Fermi-LAT emission model (Ackermann et al. 2012). In all probability, the majority of nuclear CR sources remains unresolved and is therefore excluded from that model. As a consequence, the FL-GPS can be expected to be a physical, not an instrumental, effect.

Independent of whether or not they are resolved, the nuclear CR sources are presumably concentrated in the Galactic disk if they are the consequence of star formation processes. In the present paper, they are assumed to be shell-type supernova remnants (SNRs), regardless of whether they are isolated or embedded in stellar associations, e.g., in superbubbles. The fact that the FL-GPS is concentrated in the inner Galaxy is then the result of the well-known concentration of supernova (SN) explosions in the inner Galaxy (e.g., Case & Bhattacharya 1998) and in the inner parts of other galaxies (Dragicevich et al. 1999). This concentration is also confirmed by the Galactic distribution of pulsars as compact remnants of core-collapse SN explosions (Lorimer 2004). The total γ-ray emission does not have such a strong radial gradient in the Galactic plane, as observed at comparatively low energies where the purely diffuse emission should dominate, by, e.g., the COS-B satellite for _γ > 150 MeV (Strong et al. 1988) and the EGRET instrument on the Compton Gamma Ray Observatory satellite for 100 MeV < _γ < 10⁴ MeV (Strong & Mattox 1996; Digel et al. 1996). This difference has also been discussed by Paul (2001).

A weak gradient of the diffuse emission has been theoretically interpreted as a consequence of preferential (faster) convective CR removal from the disk into the halo in the inner Galaxy, where the higher CR source density and the decrease of the Galactic escape velocity with increasing Galactic radius drive a faster Galactic wind (Breitschwerdt et al. 2002). This is a nonlinear propagation effect. Therefore, the concentration of the FL-GPS in the inner Galaxy is largely the result of the radial gradient in the CR source density because the diffuse CR density is largely independent of radius.⁴

The dependence of the FL-GPS on γ-ray energy is another aspect which is suggested to be due to the difference between the diffuse particle spectra and the particle source spectra. In a self-consistent model for energetic particle propagation in such a Galactic wind (Ptuskin et al. 1997), where nonlinear damping of the scattering magnetic irregularities balances their growth due to the outward CR streaming, this spectral difference is naturally explained.

The theoretical interpretation of the location of the FL-GPS in the Galaxy and of its energy dependence, presented here, is therefore entirely based on the propagation characteristics of the diffuse CR population in the Galaxy, both its dependence on the radial distance from the axis of rotation as well as its variation with particle energy.

From a purely phenomenological point of view, it is only necessary to assume that the DGE is essentially independent of radius in the Galactic plane and that the diffuse energy distribution of the charged energetic particles is significantly softer than the energy distribution in their sources. Indeed, starting with the work of Juliusson et al. (1972) on the decrease of the flux ratio of secondary to primary nuclear particles with increasing rigidity, as observed in the neighborhood of the solar system, the particle energy spectra inside these sources have been inferred to be substantially harder than the energy spectrum of the average CRs in the interstellar medium (ISM; see, e.g., Ptuskin 2012 for a recent review). Even though the volume fraction of the active sources is presumably quite small, the energy density of the energetic particles there is very high, coexisting together with a thermal gas density comparable to that in the average ISM. In this context, an active source is meant to be one that still basically contains the particles it has accelerated. These particles will be called source cosmic rays (SCRs). As demonstrated for the first time in Berezhko & Völk (2000, hereafter BV00), the contribution of SCRs to the DGE from accelerated CRs—still confined in their parent SNRs—progressively increases with energy due to their hard spectra. Therefore, this contribution becomes unavoidably significant at very high energies (VHEs). In a subsequent paper, the SCR contribution was studied in greater detail (Berezhko & Völk 2004, hereafter BV04). A later study of the contribution of unresolved source populations was done by Strong (2007).

Shell-type SNRs have been observed, especially at VHE (_γ ⩾ 100 GeV), to be important sources of γ-rays (e.g., Hinton & Hofmann 2009). It is less clear in which phase of their lifetimes they lose most of their accelerated particles. This is determined by a complex interplay of the dissipation of magnetic field fluctuations inside (e.g., Ptuskin & Zirakashvili 2003, 2005) and the hydrodynamic break up of the overall expanding SNR shell.

The shell-type SNRs are not the only VHE γ-ray sources in the Galaxy. In fact, the most abundant resolved Galactic γ-ray sources at VHE are PWNs (e.g., de Oña-Wilhelmi et al. 2013). In all likelihood these result from the winds of young, energetic pulsars with characteristic ages ≲ 10⁵ yr, and of the accelerated population of electrons in them which, in particular, emit inverse Compton (IC) γ-rays in an almost calorimetric fashion (de Oña-Wilhelmi et al. 2013). The particle acceleration and resulting emission is quite complex and poorly understood. PWN observations, including the radio and X-ray range, are often fitted by a phenomenological model going back to Aharonian & Atoyan (1999) which assumes an energy distribution of the radiating electrons in the form of a power law with a cutoff, in a uniform magnetic field and in a radiation field approximately equal to the cosmic microwave background (e.g., Abramowski et al. 2012). The γ-ray spectral energy densities of PWNs are therefore assumed to typically peak with a cutoff at a few TeV. The lifetime τ_PWN of TeV-emitting leptons in typical PWNs has been estimated to be ∼40 kyr (de Jager & Djannati-Atai 2009). The total number N_PWN of PWNs in the Galaxy is roughly N_PWN ∼ 800(τ_PWN/40 kyr)(ν_PWN/2), where ν_PWN denotes the Galactic VHE-emitting PWN rate in units of core-collapse events per century (de Oña-Wilhelmi et al. 2013). This is at least as large as the number of active shell-type SNRs (see below). As a result, the population of unresolved PWNs may contribute to the FL-GPS, and substantially to the total VHE emission, even though this is difficult to estimate for the reasons given above.

The relative size of the contributions of the unresolved parts of the various γ-ray source populations is an important question. Presumably, at energies beyond about 10 TeV, the PWN fraction is small due to the cutoff in the electron spectra as a result of radiative losses. The spectra of nuclear particles in shell-type SNRs can, on the other hand, extend to energies of many hundreds of TeV with differential energy spectra close to a power law ^−γ, where 2.0 ≲ γ ≲ 2.2. Such an extent in energy is particularly expected in young objects, such as the Type Ia objects SN1006 (Berezhko et al. 2012) or Tycho's SNR (Berezhko et al. 2013), where the escape of the highest-energy particles is arguably still weak. In very energetic young core-collapse SNe, γ-ray energies well in excess of 100 TeV might be reached (Bell et al. 2013). However, the corresponding SN rate is expected to be so low (Ptuskin et al. 2013, less than about 1% of the total SN rate) that their contribution is disregarded here.

In this paper, the contribution of unresolved shell-type SNRs to the Galactic γ-ray emission will be discussed in detail, following earlier work by BV00 and BV04. As will be shown, the radiation from these SCRs which extends with a hard power-law-type energy spectrum to multi-TeV γ-ray energies can readily explain the above-mentioned FL-GPS and predicts its monotonic increase from _γ ∼ 100 GeV into the multi-TeV region. The evaluation of the contribution of the unresolved PWN population is left for a future study.

The overall Fermi-LAT emission model corresponds to a spectral energy distribution (SED) I_tot = I_DGE + I_RS + I_IB, where I_DGE denotes the sum of the truly diffuse hadronic plus IC plus bremsstrahlung emission spectra, I_RS is the contribution from the resolved sources, and I_IB corresponds to that of the isotropic backgrounds. The resolved sources—which include a few shell-type SNRs, many pulsars, and, at energies _γ ≳ 100 GeV, the resolved PWNs—are expected to have a harder spectrum than the DGE. Therefore, I_tot corresponds to a harder spectrum than the approximate $\epsilon _{\gamma }^{-2.75}$-dependence of the modeled DGE in the inner Galaxy at energies _γ ≳ 10 GeV below any cutoff. To a large degree, this is due to the contribution of the IC component and especially that of the resolved sources.

Given the representation by the Fermi-LAT collaboration, the most reasonable initial extension beyond 100 GeV of I_tot appears to be an extrapolation by a power-law distribution $I_\mathrm{tot}\propto \epsilon _{\gamma }^{-\gamma }$ of the same spectral index γ which it has between _γ ∼ 10 GeV and 100 GeV. However, the γ-ray spectral energy densities of PWNs are assumed to typically peak with a cutoff at several TeV. The truly diffuse leptonic emission is expected to cut off at similar energies. Therefore, the component $I_\mathrm{tot}-I^{\pi }_\mathrm{GCR}$ is multiplied by an exponential term exp (− _γ/1TeV). I_tot extrapolated in this form is shown by the dashed curve in Figure 1.⁵ From its construction, I_tot is an approximate lower limit to the expected SED without unresolved sources.

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The aim of the present paper is then to calculate the contribution of the unresolved shell-type SNRs in order to predict a lower limit to the total emission in the inner Galaxy in the GeV range and above. As will be shown, this lower limit implies an increasing discrepancy between the measured and expected "diffuse Galactic emission" in the region of few GeV <_γ < 100 GeV, and thus can explain the FL-GPS. The discrepancy is predicted to increase up to energies 10 TeV in the VHE range. Beyond that, γ-ray energy the contribution of the unresolved shell-type SNRs decreases due to the energy-dependent escape of CRs from shell-type SNRs that makes for a shorter confinement time of CRs with higher energy.

Figure 1 presents a calculated γ-ray spectrum I_γ = I_tot + I_SCR of the low latitude inner Galaxy (−80° < l < 80°, |b| ⩽ 8°). Besides the overall emission model I_tot, discussed in the previous section, it includes the contribution I_SCR of SCRs confined within unresolved SNRs. The SCR contribution is calculated based on the approach developed in BV04:

$$\begin{equation*} I_\mathrm{SCR}=I_\mathrm{GCR}^{\pi }R(1+R_\mathrm{ep}), \end{equation*}$$

where R depends on the Galactocentric distance as a result of the non-uniform distribution of the Galactic SNRs (see the Introduction). I_SCR comprises the π⁰-decay component due to hadronic SCRs $I_\mathrm{SCR}^{\pi }=I_\mathrm{GCR}^{\pi }R$ and the IC component due to electron SCRs $I_\mathrm{SCR}^\mathrm{IC}=I_\mathrm{GCR}^{\pi }RR_\mathrm{ep}$. Here $I_\mathrm{GCR}^{\pi }$ is chosen as the π⁰-decay component due to hadronic Galactic cosmic rays (GCRs). At γ-ray energy _γ < 100 GeV, the quantity $I_\mathrm{GCR}^{\pi }(\epsilon _{\gamma })$, determined by Ackermann et al. (2012), and its power-law extrapolation, $I_\mathrm{GCR}^{\pi }\propto \epsilon _{\gamma }^{-2.75}$, to higher energies, _γ > 100 GeV, is used.

The dimensionless ratio $R=I_\mathrm{SCR}^{\pi }/I_\mathrm{GCR}^{\pi }$ according to BV04 has the form

$$\begin{equation} R(\epsilon _{\gamma })=0.07 \frac{N_\mathrm{g}^\mathrm{SCR}}{N_\mathrm{g}^\mathrm{GCR}} \left(\frac{T_\mathrm{p}}{10^5 \,{\rm yr}}\right) \left(\frac{\epsilon _{\gamma }}{1 \,{\rm GeV}}\right)^{0.6}, \end{equation} \tag{ 1 }$$

where $N_\mathrm{g}^\mathrm{GCR}$ and $N_\mathrm{g}^\mathrm{SCR}$ are the gas number density in the Galactic disk and inside SNRs, respectively. The first factor in this expression, $N_\mathrm{g}^\mathrm{SCR}/N_\mathrm{g}^\mathrm{GCR}$, represents the ratio of target nuclei for the two populations of CRs. The second factor, T_p, represents the fact that the number of SNRs, which confine SCRs with energy , is proportional to T_p(). The third factor, $\epsilon _{\gamma }^{0.6}$, is due to the harder SCR spectrum compared with the GCR spectrum.

Contrary to the earlier considerations of BV00 and BV04, where the interior SNR magnetic field was assumed to be time-independent, the proton confinement time T_p( ≈ 10_γ) inside the expanding SNR is determined, taking into account magnetic field amplification and its time dependence $B\propto (\rho V_\mathrm{S}^2)^{1/2}\propto t^{-3/5}$. Magnetic field amplification in strong shocks is a general theoretical concept arising from the expected nonlinear growth of unstable magnetic irregularities produced by the accelerating CRs themselves (Bell 2004). Field amplification is, however, also clearly observed in a number of SNR sources (e.g., Völk et al. 2005). Here it is assumed to be a universal effect in shell-type SNRs (e.g., Bell et al. 2013). In this case, the maximum energy of SCR protons, accelerated at a given SNR evolutionary epoch, decreases with time according to the relation ∝BR_SV_S∝t^−4/5. This leads to the following expression for the proton confinement time

$$\begin{equation} T_\mathrm{p}={\rm min}\lbrace T_\mathrm{SN},t_0(\epsilon /\epsilon _\mathrm{max})^{-5/4}\, {\rm yr}\rbrace , \end{equation} \tag{ 2 }$$

where T_SN is the time until which SNRs can confine the main fraction of accelerated particles and _max is the maximal energy of protons that can be accelerated in SNRs, achieved at the beginning of the Sedov phase t = t₀. The decrease of the proton confinement time T_p with energy is due to the diminishing ability of the SNR shock to produce HE CRs during the Sedov phase, which implies the escape of the highest-energy CRs from the SNR (Berezhko & Krymsky 1988; Berezhko 1996; Berezhko et al. 1996). Therefore, the description in terms of the escape time T_p is equivalent to an initial production of the SCR spectrum N_SCR∝^−γ for mc² < < _max in the evolving SNR, following the SN explosion proper. Subsequently, this spectrum is confined over the timescale T_p(). This weighs the fraction of the highest energies _max with the Sedov time t₀, lower-energy particles with a larger time that increases with decreasing , and "low-energy" particles with T_SN.

Note that, due to the considerably harder SCR energy spectrum compared with the GCR spectrum, the factor $R\propto \epsilon _{\gamma }^{0.6}$ grows with energy up to a maximal energy $\epsilon _{\gamma }^*\approx 0.1\epsilon _\mathrm{max}(t_0/T_\mathrm{SN})^{4/5}$ of γ-rays produced by SCRs confined in SNRs for the whole active evolutionary time T_p = T_SN. The spectrum I_SCR(_γ) at $\epsilon _{\gamma }<\epsilon _{\gamma }^*$ is predominantly produced by SNRs of ages t ≈ T_SN. Therefore, it is only weakly sensitive to the details of SNR evolution at the epochs t < T_SN. At higher energies $\epsilon _{\gamma }>\epsilon _{\gamma }^*$, the ratio $R\propto \epsilon _{\gamma }^{0.6-5/4}$ decreases with energy and the SCR contribution decreases due to the decrease of the confinement time T_p.

The ratio $R_\mathrm{ep}=I_\mathrm{SCR}^\mathrm{IC}/I_\mathrm{SCR}^{\pi }$ is calculated as described in BV04 with a time-independent interior magnetic field value B = 30 μG.

Since the dominant part of SNRs in our Galaxy is due to Type II SNe with a relatively low progenitor star mass, which do not form extended wind bubbles during their evolution, only the contribution of SNRs expanding into a uniform ISM will be discussed here.

The calculations presented in Figure 1 have been performed with the following values of the relevant physical parameters: sweep up time t₀ = 10³ yr, confinement time of SCRs inside SNRs T_SN = 3 × 10⁴ yr, electron to proton ratio K_ep = 10⁻², effective SCR power-law index γ = 2.15, and 10% efficiency of SCR production in SNRs.⁶ Since the SCRs are confined inside the SNRs within a thin shell behind the shock, which is also where the shock-compressed ISM gas is situated, the interior SNR gas number density $N_\mathrm{g}^\mathrm{SCR}=4\,N_\mathrm{g}^\mathrm{GCR}$ is used, where $N_\mathrm{g}^\mathrm{GCR}$ is the ISM gas number density. Since convincing observational evidence of considerable SNR magnetic field amplification (Völk et al. 2005) leading to the production of CRs with energy up to _max ≈ 2 × 10⁶ GeV (Berezhko & Völk 2007) has been found in the last 10 years, the value _max = 2 × 10⁶ GeV is used here. It is considerably larger compared with BV00 and BV04.

Due to the cutoff introduced for the component $I_\mathrm{tot}-I_\mathrm{GCR}^{\pi }$, the calculated SED $\epsilon _{\gamma }^2I_{\gamma }$ represents a lower limit for the expected γ-ray background at energies _γ > 1 TeV. It should nevertheless be emphasized here that $I_\mathrm{SCR}/I_\mathrm{GCR}^{\pi }$ is given in terms of the ratios $I_\mathrm{SCR}^{\pi }/I_\mathrm{GCR}^{\pi }$ and $I_\mathrm{SCR}^\mathrm{IC}/I_\mathrm{GCR}^{\pi }$, and that $I_\mathrm{GCR}^{\pi }$ is determined by the diffuse nuclear CR distribution which is rather well-known at least until the "knee" of the all-particle GCR spectrum. Therefore, I_SCR does not depend on the form of the extrapolation of I_tot beyond 100 GeV.

As demonstrated in BV04, there are two physical parameters that significantly influence the expected γ-ray emission from the ensemble of shell-type SNRs: the CR confinement time T_SN and the mean magnetic field strength B inside the SNRs. For a conventional value B = 10 μG, the expected γ-ray flux from SNRs exceeds the HEGRA upper limit considerably if the SCR confinement time is as large as 10⁵ yr. The contradiction can be resolved either if one suggests an appreciably higher postshock magnetic field B ≳ 30 μG or if the SCR confinement time is as small as T_SN ∼ 10⁴ yr. These possibilities can be attributed to field amplification by SCRs. In fact, nonlinear field amplification may also lead to a substantial decrease of the SCR confinement time: according to Ptuskin & Zirakashvili (2003), maximal turbulent Alfvén wave damping with its corresponding increase of CR mobility could make the SCR confinement time as small as T_SN ∼ 10⁴ yr.

The only physical parameter that strongly influences the IC γ-ray production rate for given synchrotron emission rate is the SNR magnetic field B: $I_\mathrm{SCR}^\mathrm{IC}\propto B^{-2}$. Due to the high magnetic field B = 30 μG, the overall contribution of the hadronic SCRs to the γ-ray flux dominates over that of the electron SCRs: at TeV-energies $I_\mathrm{SCR}^\mathrm{IC}\approx 0.3I_\mathrm{SCR}^{\pi }$.

As shown in Figure 1, the discrepancy between the observed "diffuse" intensity and standard model predictions at energies above a few GeV (Ackermann et al. 2012) can be attributed to the SCR contribution alone up to ∼100 GeV.

Unfortunately, existing VHE measurements do not coincide well in their longitude (l) and latitude (b) coverage with the corresponding ranges for Fermi-LAT that correspond to −80° < l < +80° and −8° < b < +8°. Therefore, the following comparisons must be taken with reservations, and for this reason the observational results from these measurements are indicated in gray in Figure 1: as one can see, the expected γ-ray flux at _γ = 1 TeV is consistent with the HEGRA upper limit, the Whipple upper limits (Reynolds et al. 1993; LeBohec et al. 2000), the Tibet-AS upper limits (Amenomori et al. 2005), and with the fluxes measured by the Milagro detector (Abdo et al. 2008) at _γ = 15 TeV, as well as with those obtained with the ARGO-YBJ detector (Ma et al. 2011) in the TeV range, taken at face value.⁷

These considerations demonstrate that the SCRs inevitably make a strong contribution to the "diffuse" γ-ray flux from the Galactic disk at all energies above a few GeV, if the population of shell-type SNRs is the main source of the GCRs. This explains the FL-GPS. The quantitative estimates show in addition that the SCR contribution dominates over the extrapolated Fermi-LAT model for the total diffuse emission—which includes detected sources and isotropic backgrounds—at energies greater than 100 GeV due to its substantially harder spectrum. The diffuse emission measured by Milagro at 15 TeV and by ARGO-YBJ at TeV energies provide limited evidence for that. It will be interesting to see if and to what extent TeV observations will show an even higher total emission than the one calculated here. If this was the case, it is suggested that such a difference should be attributed to the emission from PWNs, or even to other particle sources, some of which were mentioned above. A full consideration of the potential particle sources is beyond the scope of this paper.

The Galactocentric variation of any surplus over the purely diffuse emission above a few GeV should correspond to the Galactocentric variation of the ratio R between the π⁰-decay components of SCRs and truly diffuse GCRs as a result of the observed SNR distribution. This implies a concentration in the inner Galaxy.

The authors thank the referee, Olaf Reimer, for a number of critical and insightful comments that helped to improve the manuscript. This work has been supported in part by the Department of Federal Target Programs and Projects (grant 8404), the Russian Foundation for Basic Research (grants 13-02-00943 and 13-02-12036), and the Council of the President of the Russian Federation for Support of Young Scientists and Leading Scientific Schools (project NSh-1741.2012.2).