1
$\begingroup$

How does one calculate the the energy spectrum of gamma-rays produced by a proton-proton collision? I'm at a complete loss.

$\endgroup$
3
$\begingroup$

Most gamma rays from $pp$ collisions come from neutral pions ($p+p\to p+p+\pi^0$), you'd first have to do some relativistic momentum & energy conservation to determine the energy of the neutral pion. It's easiest if you consider the two subsequent reactions: $$ p+p\to p+\Delta^+ \\ \Delta^+\to p+\pi^{0} $$ (it's up to you to figure out the kinematics). The $\pi^0$ quickly decays (with $\langle\tau\rangle\sim10^{-16}\,{\rm s}$) into two photons: $\pi^0\to2\gamma$.

Fortunately, the source function of pions is symmetric about $\frac12m_{\pi^0}\simeq70\,{\rm MeV}$ ($c=1$ units) so that the pion source function for a power-law distribution, $n(E)\sim\mathbf n_pE_i^{s-1}E^{-s}$, is given by Edmon et al (2011) (cf. Section 3.3.3), \begin{align} q_{\pi^0}&\approx\frac{\sigma_{pp}c}{m_pc^2}2^{2-s}\frac{4}{3s}\left(\frac{m_\pi}{m_p}\right)^{-s}\\ &\qquad\times\left[\left(\frac{2E_\gamma}{m_\pi c^2}\right)^\delta+\left(\frac{2E_\gamma}{m_\pi c^2}\right)^{-\delta}\right]^{-s/\delta}n\left(s-1\right)E_i^{s-1}\mathbf n_p \end{align} where $\sigma_{pp}\sim3\times(0.96+\exp\left[4.4-2.4s\right])\times10^{-26}$ cm$^2$ is the cross-section, $E_i$ the cosmic ray proton energy, $\delta=0.14s^{-1.6}+0.44$, and $n$ is the number density of thermal protons. This has asymptotic limits of

$$ q_{\pi^0}\left(E_\gamma,\mathbf{r}\right)\propto K_{\pi^0}\left(\mathbf{r}\right)\begin{cases}\left(E_\gamma/m_{\pi^0}\right)^{s} & \quad E_\gamma\ll m_{\pi^0} \\ \left(m_{\pi^0}/E_\gamma\right)^{s} & \quad E_\gamma\gg m_{\pi^0} \end{cases} $$ if you are interested in only those limits.

Once you have this source function, you need to integrate it over all-space to get the differential flux of $\gamma$-rays (the energy spectrum most people see) is found via: $$ \frac{dN}{dt\,dE_\gamma\,d\Omega}=\frac{1}{4\pi}\int dr\,q_{\pi^0}\left(E_\gamma,\mathbf{r}\right) $$

For more in-depth covering of $\gamma$-ray spectrum from $pp$ reactions, see pretty much anything by Charles Dermer (especially his manuscript, High Energy Radiation from Black Holes: Gamma Rays, Cosmic Rays, and Neutrinos) or Reinhard Schlickeiser (especially his book Cosmic Ray Astrophysics). The aforementioned Edmon et al paper covers all $\gamma$-ray emissions from cosmic rays (inverse-Compton & bremsstrahlung). Kamae et al (2006) discusses the $\gamma$-ray and neutrino signals from the p-p collisions.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.