The AGILE diffuse gamma-ray emission model

The AGILE diffuse gamma-ray emission model is a 3D grid with bins of $0.25^{\circ}$ in galactic longitude and latitude and 0.2 kpc in distance along the line of sight. The gamma-ray emissivity is supposed to be produced by the interaction of cosmic rays with the interstellar medium through these physical processes: (1) protons interacting with the interstellar-medium nuclei (pp scattering) and producing $\pi ^0$ which in turn decay in a gamma-ray pair. (2) Electrons interacting with the diffuse matter by Bremsstrahlung and with the interstellar radiation field by inverse Compton scattering.
Therefore, to build a gamma-ray emissivity map of the Galaxy it is necessary to know the 3D distribution both of cosmic rays and of cosmic-ray targets, namely the interstellar matter, given mainly by atomic hydrogen and molecular clouds, and the interstellar radiation field. The distributions of the interstellar diffuse matter determine the spatial features of the gamma-ray emissivity model at small angular scale. The spectrum of the gamma emissivity is instead mainly determined by the cosmic rays model assumed (even if the ratio between the diffuse matter density and the interstellar radiation field can influence the spectrum at high energy). The spatial variation at large scale, instead, depend both on the cosmic ray model and on the cosmic rays target distribution.
In order to model the Galaxy matter distribution we use the HI and CO radio surveys recently terminated (described in section 2.8) which are more accurate than those used in previous works (as for the EGRET model, or [Pohl & Esposito, 1998], or [Strong et al., 2000]). These new radio data allow to derive a matter distribution with a finer angular resolution; moreover, the good sensitivity of the new surveys allows mapping the Galaxy up to large distances.
For the cosmic-ray model we have used the results of the GALPROP code (with the hard electron spectrum described in [Strong et al., 2000]), which is able to reproduce the GeV-excess.
It is convenient to express the interstellar-medium gamma-ray emissivity $g(l,b,r,E)$ by separating the terms depending on the cosmic rays from those related to their target density.

$$g(l,b,r,E) = [ q_{pp}(l,b,r,E) + q_{br}(l,b,r,E) ] [ n_{H_2}(l,b,r) + n_{HI}(l,b,r) ] +$$ (2.1)

$$+ q_{iC}(l,b,r,E) n_{ph}(l,b,r)$$

where $q_{pp}$ and $q_{br}$ are the gamma-ray emissivity per hydrogen atom due to either pp scattering or Bremsstrahlung, while $q_{iC}$ is the gamma emissivity per target photon due to inverse Compton. $n_{HI}$, $n_{H_2}$ and $n_{ph}$ are, respectively, the density of molecular hydrogen, the density of atomic hydrogen and the density of ISRF photons.