Proton-Proton collisions

In collisions between high energy protons and nuclei, pions of all charges, $\pi^+$, $\pi^-$ and $\pi ^0$ are produced as secondaries. $\pi^+$ and $\pi^-$ generate electrons and positrons, while the neutral $\pi$ decades in a pair of gamma photons. The energy of the emitted gamma is, in the reference frame of the $\pi ^0$ , half of the $\pi ^0$ rest mass, and therefore the observed energy of the gamma ray depends on the momentum of $\pi ^0$ respect to the observer.If $f_p$ is the spectra of protons

$$f_p = \frac{dN_p}{dA \; dt \; d \Omega \; dE_p} \;\; \left[ \frac{proton}{cm^2 \; sec \; sr \; MeV} \right]$$ (1.8)

the $\pi$ emissivity per hydrogen atom is :

$$q_{\pi}(E_{\pi}) = 4 \pi \sigma_{pi} (E_{p}) f_p(E_{p}) \frac{1}{k_{\pi}} \;\; \left[ \frac{\pi}{sec \; MeV \; H} \right]$$ (1.9)

where $E_{\pi}$ is the energy of the emitted pion, $E_p$ is the energy of the proton, $\sigma_{\pi}$ is the cross section for $\pi$ production through proton-proton collision (see par. 1.5.1). $k_{\pi}$ is the mean fraction of proton kinetic energy transferred to the neutral pion, and is about 0.17 [Mori, 1997]. $E_{\pi}$ and $E_p$ then are related by:

$$E_{\pi}=\frac{E_p - m_p c^2}{K_{\pi}}$$ (1.10)

$m_p$ is the proton rest mass. The gamma-ray emissivity results to be:

$$q_{\gamma}(E_{\gamma}) = 2 \int^{\inf}_{E_{min}} \frac{q_{\pi}(E_{\pi})}{p_{\pi}c} dE_{\pi} \;\; \left[ \frac{ph}{sec \; MeV \; H} \right]$$ (1.11)

where $E_{\gamma}$ is the photon observed energy, $p_{\pi}$ is the pion momentum and $E_{min}$ is the minimal energy necessary for a pion to create a photon of energy $E_{\gamma}$ (in the observer reference system):

$$E_{min}=E_{\gamma} + \frac{m_{\pi}^2 c^4}{4 E_{\gamma}}$$ (1.12)

where $m_{\pi}$ is the neutral pion rest mass. The emissivity per unit volume of interstellar matter is then:

$$g_{pp}=q_{\gamma}n_{H} \;\; \left[ \frac{ph}{cm^3 \; sec \; MeV} \right]$$ (1.13)

where $n_{H}$ is the density of H atom (both atomic and molecular).