# On a Possible Origin of the Gamma-ray Excess around the Galactic Center
## 2 Modeling the CR Interaction with Molecular Clouds
\(M\) is the total mass of of clouds emitting gamma rays, \(R_{GC}=8\) kpc is the distance to the galactic center, and \(m_{p}\) is the mass of the proton. The differential cross section of the photon production in proton-proton interactions is taken from [PERSON] et al. [20], and the bremsstrahlung cross section is from [PERSON] and [PERSON] [21]. Equation (9) assumes that the CR spectrum does not change significantly throughout the cloud. This assumption is based on the fact that clouds absorb only a small portion  
Figure 1: Gamma-ray emission from particles inside the clouds for different values of the cloud column density. (**a**) Emission from relativistic protons. (**b**) Bremsstrahlung emission from relativistic electrons. ICS emission due to scattering of optical photons is also displayed for comparison. The data points represent the GC excess spectrum according to [PERSON] et al. [15]. of the penetrating relativistic protons and electrons and that their density is therefore almost constant in the dense regions of the clouds. CR spectra experience significant spatial variations in the diffuse envelope, where the modulation process takes place. However, the mass of that envelope is negligible as compared to the total mass of the cloud, and gamma-ray emission from this envelope can be ignored. We did not implement any spatial modeling of the gamma rays in the current work. The main reason was that, to produce a correct spatial distribution of the gamma-ray emission, we need to know how clouds with different masses and different column densities are distributed in the galaxy. Indeed, for each cloud, there should be a specific value of \(E_{ex}\), and therefore, to get a correct spectral shape, we need to integrate over the distribution of all values of \(E_{ex}\). We intend to do this in the future; however, here we only want to show possible effects of the CR modulation on the gamma-ray spectrum. We assumed that the CR density in the GC is the same as local CR density [22] and took the analytical forms of CR spectra from [PERSON] et al. [23]. We considered four representative values of the cloud column density: 0 (marked in Figure 1 as ISM), \(10^{23}\), \(10^{24}\), and \(10^{25}\) cm\({}^{-2}\), which, according to [PERSON] et al. [18], correspond to the following values of \(E_{ex}\): 0, 3.4, 12, and 43 GeV. These column densities can be observed in the following clouds. \(N_{H_{2}}=10^{23}\) cm\({}^{-2}\) corresponds to the average column density of CMZ [24] or to the envelope of Sgr B2 [25]. \(N_{H_{2}}=10^{24}\) cm\({}^{-2}\) and \(N_{H_{2}}=10^{25}\) cm\({}^{-2}\) correspond to the moderate-density region and core of Sgr B2, accordingly [25]. One should notice, however, that Sgr B2 is one of the most massive molecular clouds in the galaxy; therefore, column density as high as \(10^{24}\)-\(10^{25}\) cm\({}^{-2}\) should be considered as a rare occurrence. The parameters of the ISM were taken as follows: magnetic field strength is \(B=10\)\(\mu\)G, and the diffuse gas density is \(n=10\) cm\({}^{-3}\)[24; 26]. We note that exact value of the magnetic field strength is uncertain, especially in the central regions of the galaxy; however, its value does not affect the value of \(E_{ex}\) significantly. Similarly to [PERSON] et al. [17], we assumed that the most abundant ion in the envelope is \(C^{+}\), with density of \(3\times 10^{-3}\) cm\({}^{-3}\). The corresponding value of the Alfven velocity in the envelope is \(v_{A}=10^{7}\) cm/s. To reproduce the curves shown in Figure 1 we need the following values of the total mass. For proton-proton collisions it is equal to \(M_{pp}=2\times 10^{8}M_{\odot}\) and for the electron bremsstrahlung it equals \(M_{br}=2\times 10^{9}M_{\odot}\). These values exceed the total mass of CMZ and even the mass of the galactic circumnuclear disk (CND) [24]. The reason for this discrepancy may be due to the influence of the local clouds along the line of sight.