SNRs are thought to be the dominant contributors to the Galactic CR flux up to the ‘knee’ at E = 1015 eV. The Galactic production rate of CRs can be explained if ηCR ∼ 10 per cent of the SN energy (ηCR is the cosmic ray acceleration efficiency and ESN ∼ 1051 erg) goes into accelerating CRs at SN blast waves (e.g. Aharonian 2004). Many middle-age SNRs (tSNR ∼ 105 yr) found in dense environments hosting GMCs have been observed as bright GeV and TeV sources. In fact, SNRs interacting with GMCs, as inferred from the detection of OH masers (e.g. Frail et al. 2013), constitute the dominant fraction of Galactic GeV SNRs (e.g. Thompson, Baldini & Uchiyama 2012). An estimate of the GeV/TeV flux from the hadronic component can be obtained with simple arguments. At high energies (Eγ > 1 GeV), the γ-ray spectrum is spectrally similar to the parent distribution of CR protons (e.g. Aharonian & Atoyan 1996). Then, for a power-law distribution of CR protons

(96)

\begin{equation}
n_{\rm p}(E_{\rm p}) = K_{\rm p}E_{\rm p}^{-s_{\rm p}}
\end{equation}

where the normalization Kp is obtained by assuming that the CR energy density $U_{\rm CR}\sim 3\eta _{\rm CR}E_{\rm SN}/4\pi R_{\rm SNR}^3$ (e.g. Drury, Aharonian & Völk 1994), the γ-ray photon emissivity [phs−1 cm−3 erg−1] is (e.g. Aharonian 2004)

(97)

\begin{equation}
J_{\gamma} (E_\gamma ) = \int _{E_{\rm p, min}}^\infty \frac{2cn_{\rm GMC}\sigma _{\rm pp}(E_{\rm p})\eta _An_{\rm p}(E_{\rm p})}{\sqrt{(E_{\rm p}-m_{\rm p}c^2)^2\kappa _{\pi} ^2-m_{\pi} ^2c^4}}{\rm d}E_{\rm p}
\end{equation}

where the threshold proton energy for π0 production is $E_{\rm p, min} = m_{\rm p}c^2+ \kappa _{\pi} ^{-1}(E_\gamma + m_{\pi} ^2c^4/4E_\gamma )$, mp and mπ are the proton and π0 masses, ηA ≃ 1.5 includes the contribution of nuclei other than protons towards the production of γ-rays (Dermer 1986), κπ = 0.17 is the mean fraction of proton's kinetic energy transferred to π0-meson per collision, and nGMC is the target proton number density (assumed uniform) in the GMC. The p–p inelastic collision cross-section is given by (e.g. Cheng & Romero 2004)

(98)

\begin{equation}
\sigma _{\rm pp}(E_{\rm p}) \approx 30 \left[0.95+0.06\ln \left(\frac{E_{\rm p}}{\rm GeV}\right)\right]\,{\rm mb}
\end{equation}

which is assumed to vanish below proton kinetic energy Ep − mpc2  1 GeV. Since σpp only has a weak logarithmic dependence on proton energy, the γ-ray spectrum is expected to reproduce the spectrum of the parent proton population. The integrated γ-ray photon flux in the GeV-region for sp = 2.15, E1 = 1 GeV and E2 = 100 GeV is

(99)

\begin{equation}
\Phi _\gamma = \int _{E_1}^{E_2}\frac{J_\gamma (E^{\prime }_\gamma )V_{\rm TeV}}{4\pi d^2} {\rm d}E_\gamma ^{\prime } \simeq 8.16\times 10^{-10} A\,{\rm cm}^{-2}\,{\rm s}^{-1}
\end{equation}

where $A = \eta _{\rm CR} E_{\rm SN,51}n_{\rm GMC}d_4^{-2}$. This agrees with the photon flux measured by Fermi (H.E.S.S. Collaboration 2015) in the (1–100) GeV region Φ ≈ 8.93 × 10−9 cm−2 s−1 for ηCR ∼ 0.1 and nGMC ∼ 109 cm−3. This is a reasonable estimate of the target proton number density in light of the fact that Tian et al. (2007) find a number density of ∼103 cm−3 for the GMC associated with W41 from 13CO observations.
Citation: Drury, Aharonian & Völk 1994
Citation function: Uses