Australia Telescope National Facility

On photohadronic processes in astrophysical environments
A. Mücke, J.P. Rachen, Ralph Engel, R.J. Protheroe, Todor Stanev, PASA, 16 (2), in press.

Next Section: Astrophysical applications
Title/Abstract Page: On photohadronic processes in
Previous Section: Introduction

Contents Page: Volume 16, Number 2

Cross section and kinematics

Consider a relativistic proton with energy

$E_p=\gamma m_p$ and rest mass m_p, which interacts with a photon of energy $\epsilon$ at an angle $\theta$ . The square of the total CM frame energy of the interaction is given by

$\begin{displaymath} s = m_p^2 + 2E_p\epsilon(1-\beta\cos{\theta}) = m_p^2 + 2m_p\epsilon^\prime, \end{displaymath}$

(1)

where

$\beta=\sqrt{1-\gamma^{-2}}$ is the velocity of the proton in terms of the velocity of light and

$\epsilon^{\prime}$ the photon energy in the nucleon rest frame (NRF). In SOPHIA, proton and neutron induced interactions are distinguished - for simplicity we quote only proton interactions in the following.

The partial cross sections for resonance excitation, direct (non-resonant) single-pion production, and diffractive scattering are determined by fits to exclusive data (Fig. 1). We consider the 9 most important resonances (

$\Delta^+(1232)$ , N⁺(1440), N⁺(1520), N⁺(1535), N⁺(1650), N⁺(1680),

$\Delta^+(1700)$ ,

$\Delta^+(1905)$ and

$\Delta^+(1950)$ )¹, and use the Breit-Wigner formula together with their known properties of mass, width and decay branching ratios to determine their contribution to the individual interaction channels. The direct channel is defined as the residual, non-resonant contribution to the channels

$p\gamma \to n\pi^+$ ,

$p\gamma \to \Delta^{++}\pi^-$ , and

$p\gamma \to\Delta^{0}\pi^+$ . The method is described in detail in Rachen (1996). Note that, although the $\Delta(1232)$ -resonance has the highest cross section at low interaction energies, the direct channel dominates near threshold. The direct channel can be important for proton interactions in soft photon spectra since it produces exclusively charged pions. Diffractive scattering is due to the coupling of the photon to the vector mesons $\rho^0$ and $\omega$ , which are produced at very high interaction energies with the ratio 9:1 and a cross section proportional to the total cross section. Finally, the residuals to the total cross section are fitted by a simple model and treated as statistical multipion production which is simulated by a QCD string fragmentation model (Andersson et al. 1983). After the decay of all intermediate states, considering basic kinematical relations and accelerator data on rapidity distributions, the resulting distributions of protons, neutrons and pions in the source determine the measurable astrophysical quantities, for example, the cosmic ray, neutrino and $\gamma$ -ray emission. Details of the simulation techniques are described in Mücke et al. (1999a).

**Figure 1:** The total $p\gamma$ cross section with the contributions of the baryon resonances considered in this work, the direct single-pion production, diffractive scattering, and the multipion production as a function of the photon's NRF energy (1 $\mu$ barn = 10^-34 m²). Data are from Baldini et al. (1988).
$\begin{figure} \centerline{\psfig{figure=fig1.ps,height=9cm}}\end{figure}$

In the $\Delta$ -approximation (Stecker 1973, Gaisser et al. 1995) the total cross section is given as

$\sigma_{\Delta} = 500 ~\mu\rm {barn} \, \Theta(\sqrt{s} - m_\Delta+\Gamma_\Delta/2) \cdot \Theta(m_\Delta + \Gamma_\Delta/2-\sqrt{s})$ , where $m_\Delta$ = 1.232 GeV is the mass and

$\Gamma_\Delta = 0.115$ GeV is the width of the $\Delta(1232)$ -resonance, and $\Theta$ is the Heaviside step function. The $\Delta$ -approximation uses the branching ratios of the

$\Delta^+(1232)$ -resonance to determine the number ratio $\pi^0$ to $\pi^+$ of 2:1. Photohadronic $\nu$ -production is the result of the decay of charged secondary pions (

$\pi^+\rightarrow e^+\nu_{\mu}\bar\nu_{\mu}\nu_e$ ,

$\pi^-\rightarrow e^-\nu_{\mu}\bar\nu_{\mu}\bar\nu_e$ ). Gamma rays are produced via neutral pion decay (

$\pi^0\rightarrow\gamma\gamma$ ) and synchrotron/Compton emission from the resulting relativistic leptons. In the $\Delta$ -approximation this leads to a ratio of the energy content in $\gamma$ -rays to neutrinos

$\sum E_\gamma:\sum E_\nu \equiv {\cal E_\gamma}:{\cal E_\nu} = 3:1$ . The decay kinematics of this resonance decay predicts a nucleon inelasticity

$K_p = \Delta E_p/E_p\approx 0.2$ .

Next Section: Astrophysical applications
Title/Abstract Page: On photohadronic processes in
Previous Section: Introduction

Contents Page: Volume 16, Number 2

ASKAP

Public