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

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Title/Abstract Page: On photohadronic processes in
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Contents Page: Volume 16, Number 2

Astrophysical applications

In this section we illustrate the importance of using the complete photohadronic cross section and final state composition and kinematics for a number of astrophysical applications. In particular, we compare the average proton inelasticity in a photopion production K_p and the ratio

${\cal E}_\gamma/{\cal E}_{\nu}$ from simulations with SOPHIA to the $\Delta$ -approximation. We assume that due to radiative processes electrons convert all their kinetic energy into photons, and so $e^{\pm}$ are counted in

${\cal E_\gamma}$ . The total neutrino energy

${\cal E}_{\nu}$ is the sum of the energies of $\nu_e$ , $\nu_\mu$ and their antiparticles.

Power-law seed photon spectra

In order to illustrate the contribution of the different interaction processes in various astrophysical environments we convolve the cross section with proton and seed photon spectra. Proton spectra as well as non-thermal seed photon spectra can be well approximated by power laws. Figure 2 shows the distribution of the number of interactions with CMF energy squared, s, for power-law seed photon spectra

$n_\epsilon \sim \epsilon^{-\alpha_\gamma}$ and a proton spectrum

$n_E \sim E_p^{-\alpha_p}$ with index $\alpha_p = 2$ . Spectra with $\alpha_p = 2$ are typical for shock accelerated particles. Shock acceleration is believed to be the dominant process for acceleration of galactic cosmic rays (Jones & Ellison 1991). It may also be present in extragalactic particle accelerator (Biermann & Strittmatter 1987), in particular AGN (e.g. Stecker et al. 1991, Mannheim 1993, Protheroe 1997) and in GRBs (e.g. Waxman 1995, Vietri 1998a,b).

**Figure 2:** Interaction probability distribution as function of the CMF energy with photon index
$\alpha_{\gamma}$ and proton spectrum

$n_E \sim E^{-2}$ .
$\begin{figure} \centerline{\psfig{figure=fig2.ps,height=8cm,width=12cm}}\end{figure}$

It has been proposed that the high energy gamma ray emission from blazar jets as observed in the GeV to TeV energy range results from photopion production in a proton-electron plasma where the relativistic electrons provide a synchrotron seed photon field for the high energy protons (``Proton Blazar'' model, Mannheim 1993). The synchrotron photon spectrum is described by a power law which is often observed in jets of AGN. Photopion production is followed by electromagnetic cascades reprocessing the injected power of the pion decay products (Mannheim et al. 1991). Therefore, Fig. 2 may approximate the situation in AGN jets. While for steep photon spectra the prominent resonance region near threshold is the dominating contributor to the interaction rate, the high energy region of the cross section gains importance for flat seed photon spectra. Radio spectra with

$\alpha_\gamma\approx1$ (

$F_\nu \propto {\rm const.}$ ) up to a break frequency $\nu_b$ are, for example, observed in flat spectrum radio quasars (FSRQs) and BL Lac objects. Such spectra are believed to originate from a superposition of several self-absorbed synchrotron components (Cotton et al. 1980, Shaffer & Marscher 1979). Above $\nu_b$ the spectrum is loss dominated (i.e., the cooling time scale is shorter than the dynamical time scale) and steepens to

$\alpha_{\gamma} \approx 2$ . Recent observations of blazars have revealed synchrotron peaks ranging from the optical/IR-band up to UV/soft X-rays, especially in the flaring state (see, e.g., Fossati et al. 1998). Thus flat power law photon spectra exist in AGN jets and would shift the average CMF energy of photopion production towards high values. For protons with energies below 10¹⁶eV only the infrared and higher frequency photons are relevant for photopion production due to the threshold condition. For blazars with a synchrotron peak at rather low frequencies the target photon spectrum for photomeson production may be rather steep, and photohadronic interactions would mainly occur near the threshold.

In both, flat and steep photon spectra, the simulations with SOPHIA show that the $\gamma$ -ray-to- $\nu$ -energy ratio is

${\cal E}_{\gamma}/{\cal E}_{\nu} \approx 1$ . The difference with the $\Delta$ -approximation (

${\cal E}_{\gamma}/{\cal E}_{\nu} = 3$ ) is due to the contribution of the second resonance and the multipion production regions in the case of flat photon spectra. In steep spectra, the difference is explained by the influence of the immediate threshold region, where the direct channel dominates over the Delta resonance (Mücke et al. 1999b). Therefore, the $\gamma$ -ray and neutrino outputs are approximately equal. This may have significant impact on estimates of the predicted neutrino fluxes from AGN jets which are usually normalized to the observed $\gamma$ -ray luminosity.

The absolute $\gamma$ -ray and neutrino energy outputs depend on the total proton energy loss K_p E_p. For photon power law indices

$\alpha_\gamma \geq 2$ we find

$K_p \approx 0.2$ , as in the $\Delta$ -approximation. The inelasticity increases rapidly for flatter photon spectra to reach

$K_p \approx 0.6$ for

$\alpha_\gamma = 1$ . This shows that the $\Delta$ -approximation underestimates significantly the fractional energy loss of the incident nucleon in environments with flat seed photon spectra, where protons cool much faster.

Proton energy losses, however, limit the maximum possible proton energy which can be achieved in AGN jets (Biermann & Strittmatter 1987). Setting the time scales for proton acceleration and energy loss equal, Mannheim (1993) obtained cutoff energies of $\sim 10^{16}$ eV. This estimate is based on a mean cross section for photopion production

$\langle K_p \sigma_{p\gamma}\rangle=50\mu\rm {barn}$ and a mean inelasticity

$\langle K_p\rangle=0.25$ . Noting that for flat seed photon spectra SOPHIA simulations give proton inelasticities about a factor of 2 larger than Mannheim (1993), we expect slightly lower maximum proton energies. Similar corrections could apply to any proton acceleration model where the maximum proton energy is limited by photohadronic interactions. In contrast, models in which Larmor radius constraints and finite acceleration time scales limit the proton energies, would not be affected (e.g. proton acceleration in hot spots of radio galaxies, Rachen & Biermann 1993).

**Figure 3:** Interaction probability distribution as function of the CMF energy for a broken power-law seed photon spectra as observed in GRBs (
$n_{\epsilon} \sim \epsilon^{-2/3}$ for

$10^{-8}eV\leq\epsilon\leq 1$ keV and

$n_{\epsilon} \sim \epsilon^{-2}$ for 1 keV

$\leq\epsilon\leq$ 100 keV and proton spectrum

$n_E \sim E^{-2}$ with

$E_{\rm{min}}\leq E \leq 10^{21}$ eV and different minimum energies

$E_{\rm{min}}=10^{12}$ eV (solid line), 10¹⁴eV (dashed line), 10¹⁶eV (dashed-dotted line) and 10¹⁸eV (dashed-dashed line). All quantaties are in the fluid frame comoving with a small element of the blast wave region in GRB fireballs.
$\begin{figure} \centerline{\psfig{figure=fig3.ps,height=9cm,width=12cm}}\end{figure}$

An ideal environment for proton acceleration may be found in gamma-ray bursts which have a broken power-law photon spectrum steepening from

$\alpha_\gamma\approx1$ to

$\alpha_{\gamma} \approx 2$ at keV energies (in the comoving frame). Some GRB models involve the acceleration of ultrahigh energy cosmic rays. Cascades initiated by photopion production follow (Waxman & Bahcall 1997, Vietri 1998a,b, Böttcher & Dermer 1998, Rachen & Meszaros 1998). Figure 3 shows the probability of photoproduction interaction at squared CMF energy s for a E^-2-proton spectrum with a photon spectrum with

$\alpha_\gamma = {2/3}$ below 1 keV and

$\alpha_\gamma =2$ above that energy. The curves are calculated for different parts of the proton spectrum, extending from minimum energies of

10¹²-10¹⁸eV to a high energy cutoff of 10²¹eV. While for a proton spectrum with

$E_{\rm{min}}=10^{12}$ eV most interactions occur on the $\epsilon^{-2}$ -part of the photon field, for higher $E_{\rm {min}}$ proton-photon interactions involve the flat part of the seed photon spectrum leading to higher average s values. Consequently, a $\gamma$ -ray-to neutrino energy ratio of 1 and inelasticities of the order of 0.5-0.7 are expected for the production of the highest neutrino and secondary photon energies (Mücke et al., 1999b).

Recently, Vietri (1998a,b) has claimed that in situ photoproduced neutrinos in the blast-waves associated with GRBs can reach energies up to

${\approx}10^{19}$ eV. In order to calculate the maximum neutrino energy, he assumed that individual neutrinos typically carry 5% of the incident proton energy which is derived from the $\Delta$ -approximation. However, SOPHIA simulations show that this value is rather about 1% due to the increasing neutrino multiplicity, constraining the maximum neutrino energies to lower values (Mücke et al. 1999b).

Since the average interaction energies are well beyond the resonance region in GRB we expect that detailed calculations with SOPHIA will change significantly the predictions of hadronic GRB models.

Thermal seed photon spectra

By contrast to power-law photon spectra, thermal (black body) photon fields are strongly concentrated about a characteristic energy k T. This concentration of the seed photons leads to an emphasis of one specific photoproduction energy range. We consider the interactions of protons with different power law spectra with photons from T = 2.73 K and k T = 10eV blackbody spectra. The former case is of particular astrophysical interest for cosmic ray (CR) propagation through the cosmic microwave background. The latter case may approximately describe the situation in AGN jets with the accretion disk photons being the target for proton photon interactions. We show the interaction probability as a function of s in Fig. 4. In the case of CR transport through the CMBR the $\Delta(1232)$ -resonance region plays the dominant role. SOPHIA simulations give a photon-to-neutrino energy ratio

${\cal E}_{\gamma}/{\cal E}_{\nu} \approx 1.7$ and inelasticities $\approx 0.2$ . We note that the neutrino component dominates directly at the threshold with

${\cal E}_{\gamma}/{\cal E}_{\nu} \approx 1/2$ due to the dominance of the direct channel.

Recent AGASA measurements seem to indicate an extension of the CR spectrum beyond 10²⁰eV (Takeda et al. 1998). It is generally assumed that acceleration scenarios are limited to maximum energies of 10²¹eV. If the CR proton spectrum extends above 10²¹eV, one may have to consider so called top-down (TD) models where the highest energy cosmic rays are decay products of GUT scale ultraheavy (10²⁵eV) particles (e.g., Hill 1983; Sigl et al. 1994; Protheroe & Stanev 1996). While for TD models the photon-to-neutrino energy ratio would reach unity and proton inelasticities can rise up to 0.6, in case of acceleration scenarios the $\Delta$ -approximation describes sufficiently well the proton propagation through the CMBR.

In previous Monte Carlo calculations of cosmic ray pion photoproduction interactions (e.g. Protheroe & Johnson 1996), an earlier photoproduction event generator based on the observed inclusive cross sections for

$p\gamma \to \pi^0p$ , and for

$p\gamma \to \pi^+n$ , together with a semi-empirical model for multi-pion production (with equal numbers of $\pi^+$ , $\pi^-$ and $\pi^0$ produced in the central region, and charge ratios in the fragmentation regions based on naive quark model arguments; Szabo & Protheroe 1994) was used. The results from this previous event generator are not too different, and give

${\cal E}_{\gamma}/{\cal E}_{\nu} \approx 1.6$ and inelasticity $\approx 0.2$ at the peak of the resonance, and

${\cal E}_{\gamma}/{\cal E}_{\nu} \approx 1.0$ and inelasticity ${\approx}0.55$ in the extreme multipion region.

For photopion production in an isotropic UV radiation field, the high energy and the resonance regions of the cross section are of equal importance (see Fig. 4). Thermal radiation peaking in the UV-range of the electromagnetic spectrum is observed from some radio-loud AGN (for example 3C273). Therefore a hadronic blazar model has been proposed in which protons accelerated in some regions of the jet interact with accretion disk photons (``Accretion Disk Proton Blazar Model'', Protheroe 1997). In this model the protons, being charged, are isotropised in the jet frame and readily interact with the almost black-body photons coming from the inner part of the accretion disk near the base of the jet, provided the disk is sufficiently luminous or the emission region is not too far away from the disk. The model is tuned such that the resulting $\gamma$ -rays, which are relativistically beamed along the jet direction, undergo a pair-synchrotron cascade in the accretion disk radiation and jet magnetic field such that the emerging radiation can reproduce features of the observed $\gamma$ -ray emission of blazars. The physics implemented in SOPHIA allows for a detailed investigation of the parameter space of this model.

**Figure 4:** Interaction probability distribution as a function of the CMF energy, for black body seed photon spectra and proton input spectra
$n_E \sim E_p^{-\alpha_p}$ in the energy range

$E_p = 10^{6\ldots12}$ GeV.
$\begin{figure} \centerline{\psfig{figure=fig4.ps,height=8cm,width=12cm}}\end{figure}$

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Title/Abstract Page: On photohadronic processes in
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Contents Page: Volume 16, Number 2

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Astrophysical applications

Power-law seed photon spectra

Thermal seed photon spectra