© ESO, 2011

1. Introduction

The non-thermal high-energy emission from active galactic nuclei (AGNs) has been widely studied in recent years at different wavelength ranges through both satellite-borne and ground-based detectors (e.g. Urry & Padovani 1995). Several theoretical models have been proposed to explain the electromagnetic emission of these objects. It is commonly accepted that the high-energy radiation is emitted by particles accelerated in the relativistic jets launched from the inner parts of an accretion disk that surrounds the central black hole. In general, the high-energy spectral energy distribution (SED) of AGNs presents two characteristic bumps. The lower energy bump, located at optical to X-ray energies, is usually explained as synchrotron emission of electrons while the origin of the high-energy peak in the SED is still under debate (see e.g. Böttcher 2007, for a review). Leptonic models attribute this component to inverse-Compton up-scattering off synchrotron or external photons from the disk and/or radiation reprocessed in nearby clouds (see e.g. Katarzyński et al. 2001; Lenain et al. 2008; Ghisellini et al. 2005). In hadronic models, interactions of highly relativistic protons in the jet with ambient matter and photon fields, proton-induced cascades, or synchrotron radiation of protons, are responsible for the high-energy photons (see e.g. Mücke et al. 2000; Mücke & Protheroe 2001; Reimer et al. 2004; Costamante et al. 2008; Hardcastle et al. 2009; Romero et al. 1996; Orellana & Romero 2009). There also exist models which are not based on the emission of accelerated particles in the relativistic jet and assume the production of TeV γ-rays in a pulsar-like cascade mechanism in the magnetosphere of the black hole (e.g. Neronov & Aharonian 2007; Rieger & Aharonian 2008b).

The recently reported detection by HESS of the nearby radiogalaxy Cen A (Aharonian et al. 2009) is of great relevance since it establishes radiogalaxies as VHE γ-ray emitters. Cen A is the second non-blazar AGN discovered at VHE, after the HEGRA detection of γ-rays from M 87 (Aharonian et al. 2003) and the later confirmation by HESS (Aharonian et al. 2006). A great variety of leptonic and hadronic models has been already applied to this kind of sources and a full review is beyond the scope of this work. During the first year of operation, Fermi/LAT has detected HE emission from Centaurus A (Abdo et al. 2009a) and M 87 (Abdo et al. 2009c), providing new constraints to the models.

In this work we present a lepto-hadronic model for the emission from FR I radiogalaxies. Section 2 contains a brief description of observational facts on this type of sources. In Sect. 3 we present the outline of our scenario, describing its most relevant characteristics. Section 4 is devoted to the description of the model. In Sect. 5 we present the application to Centaurus A, whereas in Sect. 6 the results for M 87 are given. Finally, in Sect. 7 we discuss the model implications and perspectives.

2. FR I radiogalaxies

According to the unification model of AGNs (e.g. Urry & Padovani 1995) FR I radiogalaxies, with their jet axis at a large angle with the line-of-sight, are the parent population of BL Lac objects whose jets are closely aligned to the line of sight. We concentrate here on the only two of them observed until now in the VHE range.

2.1. Cen A

Cen A is the closest FR I radiogalaxy (<4 Mpc, Harris et al. 1984; Hui et al. 1993) and its proximity makes it uniquely observable among such objects, eventhough its bolometric luminosity is not high as compared to other AGNs. It is very active at radio wavelengths presenting a rich jet structure. We can distinguish in its structure two components: inner jets at a kpc scale and giants lobes covering 10° in the sky. A detailed description of the radio morphology can be found in Meier et al. (1989). The inner kpc jet has also been detected in X-rays (Kraft et al. 2002) with an structure of knots and diffuse emission. Recently Croston et al. (2009) reported the detection of non-thermal X-ray emission from the shock of the southwest inner radio lobe from deep Chandra observations.

The supermassive black hole at the center of the active galaxy has an estimated mass of about 10⁷   M_⊙ (Neumayer et al. 2007; Cappellari et al. 2009; Israel 1998) to 10⁸   M_⊙ (Silge et al. 2005; Marconi et al. 2001). The black hole host galaxy is an elliptical one (NGC 5128) with a twisted disk which obscures the central engine at optical wavelengths.

Cen A was observed by the Compton Gamma Ray Observatory (CGRO) with all its instruments from MeV to GeV energies (Gehrels & Cheung 1992; Kinzer et al. 1995; Paciesas et al. 1993; Steinle et al. 1998; Thompson et al. 1995). In this period, this source exhibited X-ray variability (Bond et al. 1996) and also some soft γ-ray variability (Bond et al. 1996; Kinzer et al. 1995; Steinle et al. 1998). However, Sreekumar et al. (1999) found that the EGRET flux was stable during the whole period of CGRO observations.

In 1999 the new Chandra X-ray Observatory took images of Cen A with an unprecedented resolution. More than 200 X-ray point sources were identified in those images (Kraft et al. 2001).

Cen A as a possible source of UHE cosmic rays was early proposed by (Romero et al. 1996). Recently, the Pierre Auger Collaboration reported the existence of anisotropy on the arrival directions of UHE cosmic rays (Abraham et al. 2007), remarking that at least 2 of this events can be correlated with the Cen A position (3° circle). Further works have claimed that there are several events that can be associated with Cen A and its big radio lobes (Gorbunov et al. 2008; Fargion 2008; Wibig & Wolfendale 2007) but this correlation is still statistically weak.

Finally, Fermi/LAT has detected Cen A in the first three months of survey with a significance above 10σ (Abdo et al. 2009b).

2.2. M 87

The giant radiogalaxy M 87 is located at 16.7 Mpc within the Virgo cluster (Macri et al. 1999). It presents a one-sided jet which is inclined with respect to the line of sight an angle between 20° − 40° (Biretta et al. 1995, 1999). In addition to its bright and well resolved jet, M 87 harbors a very massive black hole (6.0  ±  0.5)  ×  10⁹   M_⊙ (Gebhardt & Thomas 2009) which is thought to power the relativistic outflow. Given its proximity, the substructures inside the jet could be resolved in the X-ray, optical, and radio wavebands (Wilson & Yang 2002). High frequency VLBI observations have resolved the inner jet up to about 70 Schwarzschild radii (Junor et al. 1999). Along the jet, nearly stationary components (Marscher et al. 2008) and features moving at superluminal speeds (Ly et al. 2007; Kovalev et al. 2007) were observed (100 pc-scale).

M 87 is also a well-known VHE γ-rays emitter (Aharonian et al. 2003, 2006; Acciari et al. 2008; Albert et al. 2008) showing a γ-ray flux variability on short time scales with flaring phenomena in VHE, radio, and X-ray wavebands simultaneously (Acciari et al. 2009). Recently, it was detected by Fermi/LAT with a significance greater than 10σ in 10 months of observations (Abdo et al. 2009c).

Rapid variability constrains the emission region extent to less than  ≈ 5δR_S, where δ is the relativistic Doppler factor. Some suggested explanations for the VHE γ-ray emission were ruled out (e.g. dark matter annihilation, Baltz et al. 2000). At the same time, various VHE γ-ray jet emission models were proposed: leptonic (Georganopoulos et al. 2005; Lenain et al. 2008) and hadronic (Reimer et al. 2004) ones. However, the location of the emission region is still unknown. The nucleus (Neronov & Aharonian 2007; Rieger & Aharonian 2008a), the inner jet (Tavecchio & Ghisellini 2008), or larger structures in the jet such as the knot HST-1, have been discussed as possible sites of particle acceleration (Cheung et al. 2007).

3. Basic scenario

We assume that a population of relativistic particles can be accelerated to very high energies close to the base of the AGN jet. These primary electrons and protons carry a fraction of the total kinetic power of the jet ${\mathit{L}}_{\mathrm{j}}^{\mathrm{\left(}\mathrm{kin}\mathrm{\right)}}$, and as they are dragged along with the jet, they cool giving rise to electromagnetic emission and neutrinos.

Assuming that a fraction q_j of the Eddington luminosity is carried by the jet and a counter jet, the jet kinetic power is $\begin{array}{ccc}{\mathit{L}}_{\mathrm{j}}^{\mathrm{\left(}\mathrm{kin}\mathrm{\right)}}\mathrm{=}\frac{{\mathit{q}}_{\mathrm{j}}}{\mathrm{2}}\frac{\mathrm{4}\mathit{\pi G}{\mathit{M}}_{\mathrm{bh}}{\mathit{m}}_{\mathrm{p}}\mathit{c}}{{\mathit{\sigma }}_{\mathrm{T}}}\mathrm{·}& & \end{array}$(1)This power can be very high if the jet is launched by a dissipationless accretion disk (Bogovalov & Kelner 2010).

Most of the jet content is in the form of a thermal plasma with a constant bulk Lorentz factor Γ_b. This plasma is initially in equipartition with a tangled magnetic field at the Alfv$\begin{array}{}{\mathrm{´}}_{}\\ \mathrm{e}\end{array}$n surface (z₀ = 50   R_g from the central black hole) (e.g. McKinney 2006). The highly disorganized magnetic field has a root mean square value B(z) at a distance z from the black hole in the observer frame, such that ${\mathit{B}}^{\mathrm{2}}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathrm{:}\mathrm{=}\mathrm{⟨}{\mathit{B}}_{\mathit{x}}^{\mathrm{2}}\mathrm{⟩}\mathrm{+}\mathrm{⟨}{\mathit{B}}_{\mathit{y}}^{\mathrm{2}}\mathrm{⟩}\mathrm{+}\mathrm{⟨}{\mathit{B}}_{\mathit{z}}^{\mathrm{2}}\mathrm{⟩}$. The magnetic energy density for z = z₀ is then ${\mathit{\rho }}_{\mathrm{m}}\mathrm{\left(}{\mathit{z}}_{\mathrm{0}}\mathrm{\right)}\mathrm{=}{\mathit{B}}_{\mathrm{0}}^{\mathrm{2}}\mathit{/}\mathrm{\left(}\mathrm{8}\mathit{\pi }\mathrm{\right)}$ with B₀ = B(z₀). Equating the magnetic to the kinetic energy density, yields: $\begin{array}{ccc}{\mathit{B}}_{\mathrm{0}}\mathrm{=}\sqrt{\frac{\mathrm{8}{\mathit{L}}_{\mathrm{j}}^{\mathrm{\left(}\mathrm{kin}\mathrm{\right)}}}{{\mathrm{[}{\mathit{r}}_{\mathrm{j}}\mathrm{(}{\mathit{z}}_{\mathrm{0}}{\mathrm{)}}^{\mathrm{]}}}^{\mathrm{2}}{\mathit{v}}_{\mathrm{b}}}}\mathit{,}& & \end{array}$(2)where z₀ is the distance to the black hole, v_b is the jet velocity, and r_j(z₀) = z₀tanξ_j is the radius of the jet assuming that it has a conical shape with half-opening angle ξ_j. A widely accepted view is that jets are accelerated through the conversion of magnetic energy into kinetic energy (e.g. Komissarov et al. 2007). We adopt a phenomenological dependence on the distance to the black hole for the magnetic field (e.g. Krolik 1999), $\begin{array}{ccc}\mathit{B}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathrm{=}{\mathit{B}}_{\mathrm{0}}{\left(\frac{{\mathit{z}}_{\mathrm{0}}}{\mathit{z}}\right)}^{\mathit{m}}\mathrm{·}& & \end{array}$(3)Since the density of cold material within the jet decays as z^-2, using an exponent m ∈ (1,2) in the above expression implies that, as z increases, the magnetic energy decreases more rapidly than the kinetic one. The corresponding increase in the bulk Lorentz factor is taken into account as described in Appendix A. In the following, we will write simply Γ_b, but it actually depends on z.

The particle acceleration takes place in a compact but inhomogeneous region of size Δz < z_acc near the base of the jet, at a distance z_acc away from the black hole. The value of z_acc is fixed by requiring the magnetic energy density to be in sub-partition with the jet kinetic energy density. This condition enables strong shocks to develop (Gaisser 1990). Assuming that at z_acc the magnetic energy is a fraction q_m of its value at z₀, it follows that $\begin{array}{ccc}{\mathit{z}}_{\mathrm{acc}}\mathrm{=}{\mathit{z}}_{\mathrm{0}}{\mathit{q}}_{\mathrm{m}}^{\frac{\mathrm{1}}{\mathrm{2}\mathrm{-}\mathrm{2}\mathit{m}}}\mathit{.}& & \end{array}$(4)For example, if M_bh = 10⁸   M_⊙, m = 1.5, and z₀ = 50   R_g, using q_m = 0.38 yields a distance z_acc = 132   R_g.

The power injected in the form of relativistic particles (L_rel) is considered to be a small fraction q_rel of the total jet kinetic power and the relation between proton and electron powers is given by the parameter a such that L_p = a  L_e (see e.g. Romero & Vila 2008; Vila & Romero 2010).

These parameters are constrained by the available observational data on each source and reasonable theoretical considerations. In the following section, we describe the procedure used to obtain the particle distributions along the jet and the radiative output that then arises.

The existence of heavy jets magnetically driven in AGN and microquasars has been supported by several scientific works in the last few years. For example, Sa¸dowski & Sikora (2010) conclude that mildly relativistic proton-electron jets might be formed by magnetocentrifugal launching by inner portion of magnetized disks around rotating black holes. This leads to a triple-component jet structure: proton-electron component sourrounded by the relativistic pair-dominated sheat. The final speed of this centrifugal outflow depends strongly on the disk vertical structure (see e.g. Wardle & Koenigl 1993) which is certainly unknown. However, for barionic outflows to reach mildly relativistic speeds, some inital boost would be neccesary, which would be produced by heating or mechanically, by flaring activity and/or by radiation pressure. Previous work by Heinz (2008) has proved the existence of a relativistic and baryon-loaded jet in Cygnus X-1 and that the bulk of the kinetic energy is carried by cold protons, as in the case of SS 433. For further information about magnetically launched jets see (Spruit 2010), where the production, acceleration, collimation, and composition of jets are well explained.

4. Description of the model

The model developed for this work is based on the energy distribution of the different particle populations along the jet. These are obtained as solutions of a 1-dimensional steady-state transport equation that includes the relevant cooling terms and a convective one. The radiative output is obtained in the jet reference frame, where the particle distributions are isotropic, and the result is transformed back to the observer frame.

The procedure begins with the calculation of the distribution of primary electrons along the jet taking into account synchrotron and adiabatic cooling. After that, the synchrotron radiation emitted by the primary electrons can be calculated. To check that the electron distribution is consistent with the energy loss mechanisms operating, the synchrotron cooling rate must be much greater than the inverse Compton (IC) one due to electrons interacting with the synchrotron photons (SSC). If this is the case, it means that the main cooling is due to synchrotron radiation, and neglecting the IC energy loss is a valid approximation to obtain the electron distribution. If the SSC cooling can not be neglected, then the transport equation becomes more complicated and a different approach is needed (e.g. Schlickeiser 2009).

Having obtained the electron distribution, the next step is to calculate the distribution of primary protons taking into account the cooling due to synchrotron emission, adiabatic expansion, pp and pγ interactions. The latter two types of interactions yield the production of secondary pions, muons, and electron-positron pairs. These three populations of particles are also described with the transport equation, and the radiative output that they produce is also considered. According to, e.g. Khangulyan et al. (2008) and Pellizza et al. (2010), it can be seen that in the present scenario, IC cascades are suppressed by the synchrotron cooling of secondary e ^± , since the magnetic field is greater than 10 G in the regions of the jet where emission takes place. Therefore, we neglect the effect of IC cascading and calculate the synchrotron emission of the secondary electrons and positrons.

In this section we present all the relevant expressions used in this model. We discuss on the injection of primary particles, the relevant cooling rates, the transport equation used, the injection of secondary particles, and the emission of photons and neutrinos.

4.1. Injection of primary particles

At a distance z to the black hole taken in the reference frame of the observer, we adopt an injection function in terms of the particle energy E′ in the co-moving reference frame of the jet: $\begin{array}{ccc}{\mathit{Q}}_{\mathit{i}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{:}\mathrm{=}& \\ & \mathrm{=}& \end{array}$Here, represents the total number of relativistic electrons (i = e) or protons (i = p), and the cut-off energy is obtained from the balance of particle gains and losses. These processes are described below.

The injection function can be transformed to the observer frame by taking into account that $\begin{array}{ccc}\frac{{\mathit{E}}^{\mathrm{\prime }}}{{{\mathit{p}}^{\mathrm{\prime }}}^{\mathrm{2}}}\frac{\mathrm{d}{\mathrm{𝒩}}_{\mathit{i}}}{\mathrm{d}{\mathit{V}}^{\mathrm{\prime }}\mathrm{d}{\mathit{p}}^{\mathrm{\prime }}\mathrm{d}{\mathrm{\Omega }}^{\mathrm{\prime }}\mathrm{d}{\mathit{t}}^{\mathrm{\prime }}}& & \end{array}$(7)is a Lorentz invariant (e.g. Dermer & Schlickeiser 2002; Torres & Reimer 2011). From this, it follows that the particle injection in the observer frame is given by $\begin{array}{ccc}{\mathit{Q}}_{\mathit{i}}\mathrm{\left(}\mathit{E,z}\mathrm{\right)}\mathrm{=}{\left[\frac{{\mathit{E}}^{\mathrm{2}}\mathrm{-}{\mathit{m}}_{\mathit{i}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}}{{{\mathit{E}}^{\mathrm{\prime }}}^{\mathrm{2}}\mathrm{-}{\mathit{m}}_{\mathit{i}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}}\right]}^{\mathrm{1}\mathit{/}\mathrm{2}}{\mathit{Q}}_{\mathit{i}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,}{\mathit{z}}^{\mathrm{\prime }}\mathrm{\right)}\mathit{,}& & \end{array}$(8)where the energy in the jet frame is given in terms of the energy in the observer frame E, and the angle θ between the particle momentum p and the bulk velocity of the jet v_b = cβ_bẑ: $\begin{array}{ccc}{\mathit{E}}^{\mathrm{\prime }}\mathrm{=}{\mathrm{\Gamma }}_{\mathrm{b}}\left(\mathit{E}\mathrm{-}{\mathit{\beta }}_{\mathrm{b}}\cos \mathit{\theta }\sqrt{{\mathit{E}}^{\mathrm{2}}\mathrm{-}{\mathit{m}}_{\mathit{i}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}}\right)\mathit{.}& & \end{array}$(9)The normalization constant K_i is found in each case using the power in relativistic species L _{ e,p } , integrating in the energy, solid angle, and volume of the acceleration zone (z ∈  [z_acc,z_acc + Δz] ): $\begin{array}{ccc}{\mathit{L}}_{\mathit{i}}\mathrm{=}{\mathrm{\int }}_{\mathrm{\Delta }\mathit{E}}\mathrm{d}\mathit{E}{\mathrm{\int }}_{\mathrm{4}\mathit{\pi }}\mathrm{d\Omega }{\mathrm{\int }}_{\mathrm{\Delta }\mathit{V}}\mathrm{d}\mathit{VE}{\mathit{Q}}_{\mathit{i}}\mathrm{\left(}\mathit{E,z}\mathrm{\right)}\mathit{.}& & \end{array}$(10)

4.2. Accelerating and cooling processes

To determine the maximum energies, it is necessary to account for the particles acceleration and cooling rates t′^-1 = E′^-1|dE′/dt′|.

We assume that in the acceleration zone, particles are accelerated by a diffusive shock-modulated mechanism with a rate (e.g. Begelman et al. 1990): $\begin{array}{ccc}{{\mathit{t}}^{\mathrm{\prime }}}_{\mathrm{acc}}^{-1}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{=}\mathit{\eta }\frac{\mathit{ce}{\mathit{B}}^{\mathrm{\prime }}\mathrm{\left(}\mathit{z}\mathrm{\right)}}{{\mathit{E}}^{\mathrm{\prime }}}\mathit{,}& & \end{array}$(11)where 0 < η ≪ 1 is the efficiency of the mechanism. The magnetic field in the jet frame is assumed to be random and with no preferred direction. B′ represents the root mean square field in the jet frame: ${{\mathit{B}}^{\mathrm{\prime }}}^{\mathrm{2}}\mathrm{=}\mathrm{⟨}{{\mathit{B}}_{{\mathit{x}}^{\mathrm{\prime }}}^{\mathrm{\prime }}}^{\mathrm{2}}\mathrm{⟩}\mathrm{+}\mathrm{⟨}{{\mathit{B}}_{{\mathit{y}}^{\mathrm{\prime }}}^{\mathrm{\prime }}}^{\mathrm{2}}\mathrm{⟩}\mathrm{+}\mathrm{⟨}{{\mathit{B}}_{{\mathit{z}}^{\mathrm{\prime }}}^{\mathrm{\prime }}}^{\mathrm{2}}\mathrm{⟩}$. As considered also in Heinz & Begelman (2000), the average of each component of the magnetic field is supposed to vanish in the jet frame: $\mathrm{⟨}{\mathit{B}}_{\mathit{i}}^{\mathrm{\prime }}\mathrm{⟩}\mathrm{=}\mathrm{0}$. Assuming that there is no electric field in the jet frame, the components of the magnetic field in the observer frame are related to the ones in the jet frame as ${\mathit{B}}_{\mathit{x}}\mathrm{=}{\mathrm{\Gamma }}_{\mathrm{b}}{\mathit{B}}_{{\mathit{x}}^{\mathrm{\prime }}}^{\mathrm{\prime }}$, ${\mathit{B}}_{\mathit{y}}\mathrm{=}{\mathrm{\Gamma }}_{\mathrm{b}}{\mathit{B}}_{{\mathit{y}}^{\mathrm{\prime }}}^{\mathrm{\prime }}$, and ${\mathit{B}}_{\mathit{z}}\mathrm{=}{\mathit{B}}_{{\mathit{z}}^{\mathrm{\prime }}}^{\mathrm{\prime }}$, which yields¹$\begin{array}{ccc}{\mathit{B}}^{\mathrm{\prime }}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathrm{=}\sqrt{\frac{\mathrm{3}}{\mathrm{2}{\mathrm{\Gamma }}_{\mathrm{b}}^{\mathrm{2}}\mathrm{+}\mathrm{1}}}\mathit{B}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathit{.}& & \end{array}$(12)where B(z) is given by Eq. (3).

As for the energy loss processes, particles emit synchrotron radiation at a rate $\begin{array}{ccc}{{\mathit{t}}^{\mathrm{\prime }}}_{\mathrm{sync}}^{-1}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{=}\frac{\mathrm{4}}{\mathrm{3}}{\left(\frac{{\mathit{m}}_{\mathrm{e}}}{\mathit{m}}\right)}^{\mathrm{3}}\frac{{\mathit{\sigma }}_{\mathrm{T}}{{\mathit{B}}^{\mathrm{\prime }}}^{\mathrm{2}}}{{\mathit{m}}_{\mathrm{e}}\mathit{c}\mathrm{8}\mathit{\pi }}\frac{{\mathit{E}}^{\mathrm{\prime }}}{\mathit{m}{\mathit{c}}^{\mathrm{2}}}\mathrm{·}& & \end{array}$(13)The lateral expansion of the jet implies an adiabatic cooling rate (Bosch-Ramon et al. 2006) in the observer frame given by $\begin{array}{ccc}{\mathit{t}}_{\mathrm{ad}}^{-1}\mathrm{\left(}\mathit{E,z}\mathrm{\right)}\mathrm{=}\frac{\mathrm{2}}{\mathrm{3}}\frac{{\mathit{v}}_{\mathrm{b}}}{\mathit{z}}\mathrm{·}& & \end{array}$(14)The density of cold matter at a distance z from the black hole is, in observer frame, $\begin{array}{ccc}{\mathit{n}}_{\mathrm{c}}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathrm{=}\frac{\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{q}}_{\mathrm{rel}}\mathrm{)}\mathit{ṁ}\mathrm{j}}{{\mathit{m}}_{\mathrm{p}}\mathit{\pi }{\mathit{z}}^{\mathrm{2}}{\tan }^{\mathrm{2}}{\mathit{\xi }}_{\mathrm{j}}{\mathit{v}}_{\mathrm{b}}}\mathit{,}& & \end{array}$(15)where the mass loss rate in the jet is $\begin{array}{ccc}\mathit{ṁ}\mathrm{j}\mathrm{=}{\frac{{\mathit{L}}_{\mathrm{j}}^{\mathrm{\left(}\mathrm{kin}\mathrm{\right)}}\mathrm{(}{\mathrm{\Gamma }}_{\mathrm{b}}\mathrm{-}\mathrm{1}\mathrm{)}}{{\mathit{c}}^{\mathrm{2}}}|}_{{\mathit{z}}_{\mathrm{0}}}\mathrm{·}& & \end{array}$In the jet frame, the cold matter density is $\begin{array}{ccc}{\mathit{n}}_{\mathrm{c}}^{\mathrm{\prime }}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathrm{=}\frac{{\mathit{n}}_{\mathrm{c}}\mathrm{\left(}\mathit{z}\mathrm{\right)}}{{\mathrm{\Gamma }}_{\mathrm{b}}}\mathrm{·}& & \end{array}$Relativistic protons in the jet undergo pp collisions with these cold protons at a rate $\begin{array}{ccc}{{\mathit{t}}^{\mathrm{\prime }}}_{\mathit{pp}}^{-1}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{=}{\mathit{n}}_{\mathrm{c}}^{\mathrm{\prime }}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathit{c}{\mathit{\sigma }}_{\mathit{pp}}^{\mathrm{\left(}\mathrm{inel}\mathrm{\right)}}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{K}}_{\mathit{pp}}\mathit{.}& & \end{array}$(16)Here the inelasticity coefficient is K_pp ≈ 1/2, the corresponding cross section for inelastic pp interactions can be approximated by (Kelner et al. 2006, 2009) $\begin{array}{ccc}{\mathit{\sigma }}_{\mathit{pp}}^{\mathrm{\left(}\mathrm{inel}\mathrm{\right)}}\mathrm{\left(}{\mathit{E}}_{\mathrm{p}}\mathrm{\right)}& \mathrm{=}& \mathrm{(}\mathrm{34.3}\mathrm{+}\mathrm{1.88}\mathit{L}\mathrm{+}\mathrm{0.25}{\mathit{L}}^{\mathrm{2}}\mathrm{)}\\ & & \mathrm{\times }\hspace{0.17em}{\left[\mathrm{1}\mathrm{-}{\left(\frac{{\mathit{E}}_{\mathrm{th}}}{{\mathit{E}}_{\mathrm{p}}}\right)}^{\mathrm{4}}\right]}^{\mathrm{2}}\mathrm{\times }{\mathrm{10}}^{-27} {\mathrm{cm}}^{\mathrm{2}}\mathit{,}\end{array}$(17)where L = ln(E_p/1000   GeV) and E_th = 1.2   GeV.

4.2.1. Synchrotron radiation and Inverse Compton interactions

To consider IC interactions of primary electrons with synchrotron photons in the jet (SSC), it is necessary to know the particle distribution of the synchrotron emitting particles, ${\mathit{N}}_{\mathrm{e}\mathit{,}\mathrm{p}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}$, which is obtained by solving the transport equation described below. Actually, the synchrotron emission of electrons dominates the low energy photon background which is a good target for IC and pγ interactions. Since in the cases considered here, IC cooling is much less efficient than synchrotron cooling, the electron distribution can be obtained to a good approximation without considering IC cooling. In the jet frame, the background radiation density, in units [GeV^-1   cm^-3], has been approximated locally as $\begin{array}{ccc}{\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{\approx }\frac{{\mathit{\epsilon }}_{\mathrm{syn}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}}\frac{{\mathit{r}}_{\mathrm{j}}\mathrm{\left(}\mathit{z}\mathrm{\right)}}{\mathit{c}}\mathit{,}& & \end{array}$(18)where ${\mathit{\epsilon }}_{\mathrm{syn}}^{\mathrm{\prime }}$ is the power per unit energy per unit volume of the synchrotron photons, $\begin{array}{ccc}{\mathit{\epsilon }}_{\mathrm{syn}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{=}\left(\frac{\mathrm{1}\mathrm{-}{\mathit{e}}^{\mathrm{-}{\mathit{\tau }}_{\mathrm{SSA}}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}}{{\mathit{\tau }}_{\mathrm{SSA}}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}\right){\mathrm{\int }}_{{\mathit{m}}_{\mathrm{e}}{\mathit{c}}^{\mathrm{2}}}^{\mathrm{\infty }}\mathrm{d}{\mathit{E}}^{\mathrm{\prime }}\mathrm{4}\mathit{\pi }{\mathit{P}}_{\mathrm{syn}}{\mathit{N}}_{\mathrm{e}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathit{.}& & \end{array}$(19)Here, the synchrotron power per unit energy emitted by the electrons is given by (Blumenthal & Gould 1970): $\begin{array}{ccc}{\mathit{P}}_{\mathrm{syn}}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{=}\frac{\sqrt{\mathrm{2}}{\mathit{e}}^{\mathrm{3}}{\mathit{B}}^{\mathrm{\prime }}\mathrm{\left(}\mathit{z}\mathrm{\right)}}{{\mathit{m}}_{\mathrm{e}}{\mathit{c}}^{\mathrm{2}}\mathit{h}}\frac{{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}}{{\mathit{E}}_{\mathrm{cr}}}{\mathrm{\int }}_{{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{/}{\mathit{E}}_{\mathrm{cr}}}^{\mathrm{\infty }}\mathrm{d}\mathit{\zeta }{\mathit{K}}_{\mathrm{5}\mathit{/}\mathrm{3}}\mathrm{\left(}\mathit{\zeta }\mathrm{\right)}\mathit{,}& & \end{array}$(20)where K_5/3(ζ) is the modified Bessel function of order 5/3 and $\begin{array}{ccc}{\mathit{E}}_{\mathrm{cr}}\mathrm{=}\frac{\sqrt{\mathrm{6}}\mathit{he}{\mathit{B}}^{\mathrm{\prime }}\mathrm{\left(}\mathit{z}\mathrm{\right)}}{\mathrm{4}\mathit{\pi }{\mathit{m}}_{\mathrm{e}}\mathit{c}}{\left(\frac{{\mathit{E}}^{\mathrm{\prime }}}{{\mathit{m}}_{\mathrm{e}}{\mathit{c}}^{\mathrm{2}}}\right)}^{\mathrm{2}}\mathrm{·}& & \end{array}$The factor in parenthesis in Eq. (19) accounts for the effect of synchrotron self-absorption (SSA) within the jet, with an optical depth $\begin{array}{ccc}{\mathit{\tau }}_{\mathrm{SSA}}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{=}{\mathrm{\int }}_{\mathrm{0}}^{\frac{{\mathit{r}}_{\mathrm{j}}\mathrm{\left(}\mathit{z}\mathrm{\right)}}{\sin {\mathit{\theta }}^{\mathrm{\prime }}}}\mathrm{d}{\mathit{l}}^{\mathrm{\prime }}{\mathit{\alpha }}_{\mathrm{SSA}}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,}{\mathit{z}}^{\mathrm{\prime }}\mathrm{\right(}{\mathit{l}}^{\mathrm{\prime }}\mathrm{\left)}\mathrm{\right)}\mathit{,}& & \end{array}$where θ′ is such that tanθ′ = Γ_btanθ, and the SSA coefficient is given by (Rybicki & Lightman 1979): $\begin{array}{ccc}{\mathit{\alpha }}_{\mathrm{SSA}}& \mathrm{=}& \mathrm{-}\frac{{\mathit{h}}^{\mathrm{3}}{\mathit{c}}^{\mathrm{2}}}{\mathrm{8}\mathit{\pi }{{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}}^{\mathrm{2}}}{\mathrm{\int }}_{{\mathit{m}}_{\mathrm{e}}{\mathit{c}}^{\mathrm{2}}}^{\mathrm{\infty }}\mathrm{d}{\mathit{E}}^{\mathrm{\prime }}{{\mathit{E}}^{\mathrm{\prime }}}^{\mathrm{2}}{\mathit{P}}_{\mathrm{syn}}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,}{\mathit{E}}^{\mathrm{\prime }}\mathrm{\right)}\\ & & \mathrm{\times }\hspace{0.17em}\frac{\mathit{\partial }}{\mathit{\partial }{\mathit{E}}^{\mathrm{\prime }}}\left[\frac{{\mathit{N}}_{\mathrm{e}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,}{\mathit{z}}^{\mathrm{\prime }}\mathrm{\right(}{\mathit{l}}^{\mathrm{\prime }}\mathrm{\left)}\mathrm{\right)}}{{{\mathit{E}}^{\mathrm{\prime }}}^{\mathrm{2}}}\right]\mathrm{·}\end{array}$(21)After all these considerations, the cooling rate due to IC scattering for electrons of energy E′ in the jet frame can be obtained by integrating in the target photon energy ${\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}$ and in energy of the scattered photon ${\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}$ (Blumenthal & Gould 1970): $\begin{array}{ccc}{{\mathit{t}}^{\mathrm{\prime }}}_{\mathrm{IC}}^{-1}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \frac{\mathrm{3}{\mathit{m}}_{\mathrm{e}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}{\mathit{\sigma }}_{\mathrm{T}}}{\mathrm{4}{{\mathit{E}}^{\mathrm{\prime }}}^{\mathrm{3}}}{\mathrm{\int }}_{{{\mathit{E}}^{\mathrm{\prime }}}_{\mathrm{ph}}^{\mathrm{\left(}\min \mathrm{\right)}}}^{{\mathit{E}}^{\mathrm{\prime }}}\mathrm{d}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\frac{{\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{{{\mathit{E}}^{\mathrm{\prime }}}_{\mathrm{ph}}}\\ & & \end{array}$(22)where ${{\mathit{E}}^{\mathrm{\prime }}}_{\mathrm{ph}}^{\mathrm{\left(}\min \mathrm{\right)}}$ is the lowest energy of the available background of synchrotron photons, and $\begin{array}{ccc}\mathit{F}\mathrm{\left(}\mathit{q}\mathrm{\right)}\mathrm{=}\mathrm{2}\mathit{q}\ln \mathit{q}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{+}\mathrm{2}\mathit{q}\mathrm{)}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{q}\mathrm{)}\mathrm{+}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{q}\mathrm{)}\frac{\mathrm{(}\mathit{q}{\mathrm{\Gamma }}_{\mathrm{e}}^{\mathrm{\prime }}{\mathrm{)}}^{\mathrm{2}}}{\mathrm{1}\mathrm{+}{\mathrm{\Gamma }}_{\mathrm{e}}^{\mathrm{\prime }}}\mathit{,}& & \end{array}$(23)with ${\mathrm{\Gamma }}_{\mathrm{e}}^{\mathrm{\prime }}\mathrm{=}\mathrm{4}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}{\mathit{E}}^{\mathrm{\prime }}\mathit{/}\mathrm{\left(}{\mathit{m}}_{\mathrm{e}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}\mathrm{\right)}$ and $\begin{array}{ccc}\mathit{q}\mathrm{=}\frac{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}{{\mathrm{\Gamma }}_{\mathrm{e}}^{\mathrm{\prime }}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{/}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{)}}\mathrm{·}& & \end{array}$

4.2.2. Proton-photon interactions

Proton interactions with the photon background can be an important cooling process, with a rate appoximated as $\begin{array}{ccc}{{\mathit{t}}^{\mathrm{\prime }}}_{\mathrm{p}\mathit{\gamma }}^{-1}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \frac{\mathit{c}}{\mathrm{2}{\mathit{\gamma }}_{\mathrm{p}}^{\mathrm{2}}}{\mathrm{\int }}_{\frac{{\mathit{ϵ}}_{\mathrm{th}}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}}{\mathrm{2}{\mathit{\gamma }}_{\mathrm{p}}}}^{\mathrm{\infty }}\mathrm{d}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\frac{{\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{{{\mathit{E}}^{\mathrm{\prime }}}_{\mathrm{ph}}^{\mathrm{2}}}\\ & & \end{array}$(24)Here, ${\mathit{ϵ}}_{\mathrm{th}}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}\mathrm{=}\mathrm{150}$ MeV and we use the expressions for the cross section ${\mathit{\sigma }}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}$ and the inelasticity ${\mathit{K}}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}$ given in Begelman et al. (1990).

4.3. Particle distributions in the jet

The energy distribution for each particle population is obtained as a solution to the following 1D stationary transport equation (e.g. Ginzburg & Syrovatskii 1964; Khangulyan et al. 2008): $\begin{array}{ccc}{\mathit{v}}_{\mathrm{b}}\frac{\mathit{\partial }{\mathit{N}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{\mathit{\partial z}}\mathrm{+}\frac{\mathit{\partial }\left[\mathit{b}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}{\mathit{N}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\right]}{\mathit{\partial }{\mathit{E}}^{\mathrm{\prime }}}\mathrm{+}\frac{{\mathit{N}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{{\mathit{T}}_{\mathrm{d}}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathrm{\right)}}\mathrm{=}{\mathit{Q}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathit{.}& & \end{array}$(25)This equation includes the effects of convection with a speed v_b ~ c, and particle cooling with an energy loss b(E′,z) = dE′/dt. Particle decay with a timescale T_d(E′) is also considered in the cases of secondary pions and muons. We note that the energy E′ in the above equation corresponds to the co-moving frame, where the particle distributions are isotropic, while the spatial coordinate z corresponds to the observer frame (e.g. Jokipii & Parker 1970; Kirk et al. 1988). The source term is given by a function Q′(E′,z), which in the case of primary particles is given by Eq. (6), while for secondaries it is obtained using the parent particle distribution (N′(E′,z)) along with the secondary particle production rates (see Sect. 4.3.1).

We can solve the transport equation using the method of characteristics, i.e., writing $\begin{array}{ccc}\frac{\mathrm{d}\mathit{z}}{{\mathit{v}}_{\mathrm{b}}}\mathrm{=}\frac{\mathrm{d}{\mathit{E}}^{\mathrm{\prime }}}{\mathit{b}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}\mathrm{=}\frac{\mathrm{d}{\mathit{N}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{{\mathit{Q}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{-}\mathrm{[}\frac{\mathrm{1}}{{\mathit{T}}_{\mathrm{d}}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathrm{\right)}}\mathrm{+}{\frac{\mathit{\partial b}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{\mathit{\partial E}}}^{\mathrm{]}}\mathit{N}\mathrm{\left(}\mathit{E,z}\mathrm{\right)}}\mathit{,}& & \end{array}$(26)where the first two terms allow us to find a characteristic curve z_c(E_c) for each pair (E′,z) of interest. Equating the second and third members it follows that $\begin{array}{ccc}{\mathit{N}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \int \begin{array}{}\mathrm{\infty }\\ \mathit{E}\end{array}\mathrm{d}{\mathit{E}}_{\mathrm{c}}\frac{{\mathit{Q}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{c}}\mathit{,}{\mathit{z}}_{\mathrm{c}}\mathrm{\right)}}{\mathrm{\left|}\mathit{b}\mathrm{\right(}{\mathit{E}}_{\mathrm{c}}\mathit{,}{\mathit{z}}_{\mathrm{c}}\mathrm{\left)}\mathrm{\right|}}\\ & & \mathrm{\times }\hspace{0.17em}\exp {\mathrm{\int }}_{{\mathit{E}}^{\mathrm{\prime }}}^{{\mathit{E}}_{\mathrm{c}}}\mathrm{d}{\mathit{E}}^{\mathrm{\prime \prime }}\left[\frac{{\mathit{T}}_{\mathrm{d}}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime \prime }}\mathrm{\right)}\mathrm{\left|}\frac{\mathrm{d}\mathit{b}}{\mathrm{d}\mathit{E}}\mathrm{\right(}{\mathit{E}}^{\mathrm{\prime \prime }}\mathit{,}{\mathit{z}}^{\mathrm{\prime \prime }}\mathrm{\left)}\mathrm{\right|}\mathrm{-}\mathrm{1}}{{\mathit{T}}_{\mathrm{d}}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime \prime }}\mathrm{\right)}\mathrm{\left|}\mathit{b}\mathrm{\right(}{\mathit{E}}^{\mathrm{\prime \prime }}\mathit{,}{\mathit{z}}^{\mathrm{\prime \prime }}\mathrm{\left)}\mathrm{\right|}}\right]\mathrm{·}\end{array}$(27)

4.3.1. Pion and muon production

As high energy protons interact with background matter and radiation, they produce pions.

The pion injection due to pp interactions in the jet frame is calculated as $\begin{array}{ccc}{\mathit{Q}}_{\mathit{\pi ,pp}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& {\mathit{n}}_{\mathrm{c}}^{\mathrm{\prime }}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathit{c}{\mathrm{\int }}_{\mathrm{0}}^{\mathrm{1}}\frac{\mathrm{d}\mathit{x}}{\mathit{x}}{\mathit{N}}_{\mathrm{p}}^{\mathrm{\prime }}\left(\frac{{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}}{\mathit{x}}\mathit{,z}\right)\\ & & \end{array}$(28)where $\begin{array}{ccc}{\mathit{F}}_{\mathit{\pi }}\left(\mathit{x,}\frac{{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}}{\mathit{x}}\right)& \mathrm{=}& \mathrm{4}\mathit{\alpha }{\mathit{B}}_{\mathit{\pi }}{\mathit{x}}^{\mathit{\alpha }\mathrm{-}\mathrm{1}}{\left(\frac{\mathrm{1}\mathrm{-}{\mathit{x}}^{\mathit{\alpha }}}{\mathrm{1}\mathrm{+}{\mathit{r}}^{\mathrm{\prime }}{\mathit{x}}^{\mathit{\alpha }}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{x}}^{\mathit{\alpha }}\mathrm{)}}\right)}^{\mathrm{4}}\\ & & \mathrm{\times }\hspace{0.17em}\left(\frac{\mathrm{1}}{\mathrm{1}\mathrm{-}{\mathit{x}}^{\mathit{\alpha }}}\mathrm{+}\frac{{\mathit{r}}^{\mathrm{\prime }}\mathrm{(}\mathrm{1}\mathrm{-}\mathrm{2}{\mathit{x}}^{\mathit{\alpha }}\mathrm{)}}{\mathrm{1}\mathrm{+}{\mathit{r}}^{\mathrm{\prime }}{\mathit{x}}^{\mathit{\alpha }}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{x}}^{\mathit{\alpha }}\mathrm{)}}\right){\left(\mathrm{1}\mathrm{-}\frac{{\mathit{m}}_{\mathit{\pi }}{\mathit{c}}^{\mathrm{2}}}{{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}}\right)}^{\mathrm{1}\mathit{/}\mathrm{2}}\end{array}$(29)is the distribution of pions produced per pp collision, with $\mathit{x}\mathrm{=}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathit{/}{\mathit{E}}^{\mathrm{\prime }}$, B_π = a′ + 0.25, a′ = 3.67 + 0.83L + 0.075L², ${\mathit{r}}^{\mathrm{\prime }}\mathrm{=}\mathrm{2.6}\mathit{/}\sqrt{{\mathit{a}}^{\mathrm{\prime }}}$, and $\mathit{\alpha }\mathrm{=}\mathrm{0.98}\mathit{/}\sqrt{{\mathit{a}}^{\mathrm{\prime }}}$ (see Kelner et al. 2006, 2009).

In the same way, the source function for charged pions produced by pγ interactions is $\begin{array}{ccc}{\mathit{Q}}_{\mathit{\pi ,p\gamma }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \int {\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathrm{d}{\mathit{E}}^{\mathrm{\prime }}{\mathit{N}}_{\mathrm{p}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}{{\mathit{\omega }}^{\mathrm{\prime }}}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\\ & & \mathrm{\times }\hspace{0.17em}{\mathrm{𝒩}}_{\mathit{\pi }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathrm{\right)}\mathit{\delta }\mathrm{(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathrm{-}\mathrm{0.2}{\mathit{E}}^{\mathrm{\prime }}\mathrm{)}\\ & \mathrm{=}& \end{array}$(30)Here ${{\mathit{\omega }}^{\mathrm{\prime }}}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}$ is the pγ collision frequency defined as (Atoyan & Dermer 2003): $\begin{array}{ccc}{{\mathit{\omega }}^{\mathrm{\prime }}}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \frac{\mathit{c}}{\mathrm{2}{\mathit{\gamma }}_{\mathrm{p}}^{\mathrm{2}}}{\mathrm{\int }}_{\frac{{\mathit{ϵ}}_{\mathrm{th}}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}}{\mathrm{2}{\mathit{\gamma }}_{\mathrm{p}}}}^{\mathrm{\infty }}\mathrm{d}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\frac{{\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{{{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}}^{\mathrm{2}}}\\ & & \mathrm{\times }\hspace{0.17em}{\mathrm{\int }}_{{\mathit{ϵ}}_{\mathrm{th}}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}}^{\mathrm{2}\mathit{ϵ}{\mathit{\gamma }}_{\mathrm{p}}}\mathrm{d}{\mathit{ϵ}}^{\mathrm{\prime }}{\mathit{\sigma }}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}\mathrm{\left(}{\mathit{ϵ}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{ϵ}}^{\mathrm{\prime }}\mathit{,}\end{array}$(31)and the mean number of positive or negative pions is $\begin{array}{ccc}{\mathrm{𝒩}}_{\mathit{\pi }}\mathrm{\approx }\frac{{\mathit{p}}_{\mathrm{1}}}{\mathrm{2}}\mathrm{+}\mathrm{2}{\mathit{p}}_{\mathrm{2}}\mathit{.}& & \end{array}$(32)This number depends on the probabilities of single pion and multi-pion production p₁ and p₂ = 1 − p₂. Using the mean inelasticity function $\mathit{K̅}{}_{\mathrm{p}\mathit{\gamma }}\mathrm{=}{\mathit{t}}_{\mathrm{p}\mathit{\gamma }}^{-1}\mathit{/}{\mathit{\omega }}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{\pi }\mathrm{\right)}}$, the probability p₁ is $\begin{array}{ccc}{\mathit{p}}_{\mathrm{1}}\mathrm{=}\frac{{\mathit{K}}_{\mathrm{2}}\mathrm{-}\mathit{K̅}\mathrm{p}\mathit{\gamma }}{{\mathit{K}}_{\mathrm{2}}\mathrm{-}{\mathit{K}}_{\mathrm{1}}}\mathit{,}& & \end{array}$(33)where K₁ = 0.2 and K₂ = 0.6.

The injection functions of charged pions given by Eqs. (28), (30) are used to work out the distribution ${\mathit{N}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}$ by solving the transport Eq. (25).

For muon injection, we proceed as (Lipari et al. 2007) and we consider the production of left handed and right handed muons separately, which have different decay spectra: $\begin{array}{ccc}& & \frac{\mathrm{d}{\mathit{n}}_{{\mathit{\pi }}^{\mathrm{-}}\mathrm{\to }{\mathit{\mu }}_{\mathit{L}}^{\mathrm{-}}}}{\mathrm{d}{\mathit{E}}_{\mathit{\mu }}}\mathrm{\left(}{\mathit{E}}_{\mathit{\mu }}\mathrm{;}{\mathit{E}}_{\mathit{\pi }}\mathrm{\right)}\mathrm{=}\frac{{\mathit{r}}_{\mathit{\pi }}\mathrm{(}\mathrm{1}\mathrm{-}\mathit{x}\mathrm{)}}{{\mathit{E}}_{\mathit{\pi }}\mathit{x}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{r}}_{\mathit{\pi }}{\mathrm{)}}^{\mathrm{2}}}\mathrm{\Theta }\mathrm{(}\mathit{x}\mathrm{-}{\mathit{r}}_{\mathit{\pi }}\mathrm{)}\\ & & \frac{\mathrm{d}{\mathit{n}}_{{\mathit{\pi }}^{\mathrm{-}}\mathrm{\to }{\mathit{\mu }}_{\mathit{R}}^{\mathrm{-}}}}{\mathrm{d}{\mathit{E}}_{\mathit{\mu }}}\mathrm{\left(}{\mathit{E}}_{\mathit{\mu }}\mathrm{;}{\mathit{E}}_{\mathit{\pi }}\mathrm{\right)}\mathrm{=}\frac{\mathrm{(}\mathit{x}\mathrm{-}{\mathit{r}}_{\mathit{\pi }}\mathrm{)}}{{\mathit{E}}_{\mathit{\pi }}\mathit{x}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{r}}_{\mathit{\pi }}{\mathrm{)}}^{\mathrm{2}}}\mathrm{\Theta }\mathrm{(}\mathit{x}\mathrm{-}{\mathit{r}}_{\mathit{\pi }}\mathrm{)}\mathit{,}\end{array}$\arraycolsep1.75ptwith x = E_μ/E_π and r_π = (m_μ/m_π)².

The injection function of negative left handed and positive right handed muons is $\begin{array}{ccc}{\mathit{Q}}_{{\mathit{\mu }}_{\mathit{L}}^{\mathrm{-}}\mathit{,}{\mathit{\mu }}_{\mathit{R}}^{\mathrm{+}}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \int \begin{array}{}\mathrm{\infty }\\ {\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}\end{array}\mathrm{d}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}{\mathit{T}}_{\mathit{\pi ,}\mathrm{d}}^{-1}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathrm{\right)}\\ & & \mathrm{\times }\hspace{0.17em}({\mathit{N}}_{{\mathit{\pi }}^{\mathrm{-}}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\frac{\mathrm{d}{\mathit{n}}_{{\mathit{\pi }}^{\mathrm{-}}\mathrm{\to }{\mathit{\mu }}_{\mathit{L}}^{\mathrm{-}}}}{\mathrm{d}{\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}}\mathrm{\left(}{\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}\mathrm{;}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathrm{\right)}\\ & & \mathrm{+}\hspace{0.17em}{\mathit{N}}_{{\mathit{\pi }}^{\mathrm{+}}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\frac{\mathrm{d}{\mathit{n}}_{{\mathit{\pi }}^{\mathrm{+}}\mathrm{\to }{\mathit{\mu }}_{\mathit{R}}^{\mathrm{-}}}}{\mathrm{d}{\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}}\mathrm{\left(}{\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}\mathrm{;}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathrm{\right)})\mathit{.}\end{array}$(36)Given that CP invariance implies that $\mathrm{d}{\mathit{n}}_{{\mathit{\pi }}^{\mathrm{-}}\mathrm{\to }{\mathit{\mu }}_{\mathit{L}}^{\mathrm{-}}}\mathit{/}\mathrm{d}{\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}\mathrm{=}\mathrm{d}{\mathit{n}}_{{\mathit{\pi }}^{\mathrm{+}}\mathrm{\to }{\mathit{\mu }}_{\mathit{R}}^{\mathrm{+}}}\mathit{/}\mathrm{d}{\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}$, and since the total distribution obtained for all charged pions is ${\mathit{N}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{=}{\mathit{N}}_{{\mathit{\pi }}^{\mathrm{+}}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{+}{\mathit{N}}_{{\mathit{\pi }}^{\mathrm{-}}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}$, it follows that the injection for left handed muons is $\begin{array}{ccc}{\mathit{Q}}_{{\mathit{\mu }}_{\mathit{L}}^{\mathrm{-}}\mathit{,}{\mathit{\mu }}_{\mathit{R}}^{\mathrm{+}}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \int \begin{array}{}\mathrm{\infty }\\ {\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}\end{array}\mathrm{d}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}{\mathit{T}}_{\mathit{\pi ,}\mathrm{d}}^{-1}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathrm{\right)}\\ & & \end{array}$(37)And in a similar way, the right handed muons injection is $\begin{array}{ccc}{\mathit{Q}}_{{\mathit{\mu }}_{\mathit{R}}^{\mathrm{-}}\mathit{,}{\mathit{\mu }}_{\mathit{L}}^{\mathrm{+}}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \int \begin{array}{}\mathrm{\infty }\\ {\mathit{E}}_{\mathit{\mu }}^{\mathrm{\prime }}\end{array}\mathrm{d}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}{\mathit{T}}_{\mathit{\pi ,}\mathrm{d}}^{-1}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathrm{\right)}\\ & & \end{array}$(38)Again, the injection functions of muons given by Eqs. (37), (38) are used to work out the distribution ${\mathit{N}}_{\mathit{\mu }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}$ by solving the transport Eq. (25).

4.4. Electromagnetic radiation

The main radiative processes considered in this work are: synchrotron radiation, IC emission, pp and pγ interactions. For each emission process we can calculate the injection of photons or radiation emissivity, which is given by the general expression of Eq. (5) and represents the number of photons produced per time unit, per volume unit, per photon energy, and per solid angle. In the observer frame, the corresponding emissivity is given by $\begin{array}{ccc}{\mathit{Q}}_{\mathit{\gamma }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}\mathit{,z}\mathrm{\right)}\mathrm{=}\mathit{D}\hspace{0.17em}{\mathit{Q}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathit{,}& & \end{array}$(39)where the Doppler factor is $\mathit{D}\mathrm{=}{\mathrm{\Gamma }}_{\mathrm{b}}^{-1}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{\beta }}_{\mathrm{b}}\cos \mathit{\theta }{\mathrm{)}}^{-1}$ and E_γ  =  $\mathit{D}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}$.

In the case of synchrotron radiation, the emissivity in the jet frame is $\begin{array}{ccc}{\mathit{Q}}_{\mathit{\gamma ,}\mathrm{syn}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{=}\frac{{\mathit{ϵ}}_{\mathrm{syn}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{\mathrm{4}\mathit{\pi }{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}\mathrm{·}& & \end{array}$(40)and the IC emissivity is $\begin{array}{ccc}{\mathit{Q}}_{\mathit{\gamma ,}\mathrm{IC}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \frac{{\mathit{r}}_{\mathrm{e}}^{\mathrm{2}}\mathit{c}}{\mathrm{2}}{\mathrm{\int }}_{{{\mathit{E}}^{\mathrm{\prime }}}_{\mathrm{ph}}^{\mathrm{\left(}\min \mathrm{\right)}}}^{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}\mathrm{d}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\frac{{\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{{\mathit{E}}_{\mathrm{ph}}}\\ & & \mathrm{\times }\hspace{0.17em}{\mathrm{\int }}_{{\mathit{E}}_{\min }^{\mathrm{\prime }}}^{{\mathit{E}}_{\max }^{\mathrm{\prime }}}\mathrm{d}{\mathit{E}}^{\mathrm{\prime }}\frac{{\mathit{N}}_{\mathrm{e}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{{\mathit{\gamma }}_{\mathrm{e}}^{\mathrm{2}}}\mathit{F}\mathrm{\left(}\mathit{q}\mathrm{\right)}\mathit{,}\end{array}$(41)where r_e is the classical electron radius, and we integrate in the target photon energy ${\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}$ and in the electron energies E′ between $\begin{array}{ccc}{\mathit{E}}_{\min }^{\mathrm{\prime }}\mathrm{=}\frac{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}{\mathrm{2}}\mathrm{+}\frac{{\mathit{m}}_{\mathrm{e}}{\mathit{c}}^{\mathrm{2}}}{\mathrm{2}}\sqrt{\frac{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}{\mathrm{2}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}}\mathrm{+}\frac{{{\mathit{E}}^{\mathrm{\prime }}}_{\mathit{\gamma }}^{\mathrm{2}}}{\mathrm{2}{\mathit{m}}_{\mathrm{e}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}}}& & \end{array}$(42)and $\begin{array}{ccc}{\mathit{E}}_{\max }^{\mathrm{\prime }}\mathrm{=}\frac{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}{\mathrm{1}\mathrm{-}\frac{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}{{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}}}\mathrm{·}& & \end{array}$(43)With respect to the hadronic contribution to the total emission, the photon emissivity due to pp interactions is obtained as $\begin{array}{ccc}{\mathit{Q}}_{\mathit{\gamma ,pp}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& {\mathit{n}}_{\mathrm{c}}^{\mathrm{\prime }}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathit{c}{\mathrm{\int }}_{\mathrm{0}}^{\mathrm{1}}\frac{\mathrm{d}\mathit{x}}{\mathit{x}}{\mathit{N}}_{\mathrm{p}}^{\mathrm{\prime }}\left(\frac{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}{\mathit{x}}\mathit{,z}\right)\\ & & \mathrm{\times }\hspace{0.17em}{\mathit{F}}_{\mathit{\gamma }}\left(\mathit{x,}\frac{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}{\mathit{x}}\right){\mathit{\sigma }}_{\mathit{pp}}^{\mathrm{\left(}\mathrm{inel}\mathrm{\right)}}\left(\frac{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}{\mathit{x}}\right)\mathit{,}\end{array}$(44)where the function F_γ(x,E′) is the same as defined by Kelner et al. (2006), for a proton energy ${\mathit{E}}^{\mathrm{\prime }}\mathrm{=}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{/}\mathit{x}$.

For the γ-ray emissivity due to pγ interactions we use the expression $\begin{array}{ccc}{\mathit{Q}}_{\mathit{\gamma ,p\gamma }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \frac{\mathrm{1}}{\mathrm{4}\mathit{\pi }}{\mathrm{\int }}_{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}\frac{\mathrm{d}{\mathit{E}}^{\mathrm{\prime }}}{{\mathit{E}}^{\mathrm{\prime }}}{\mathit{N}}_{\mathrm{p}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\\ & & \mathrm{\times }\hspace{0.17em}{\mathrm{\int }}_{{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\left(}\min \mathrm{\right)}}}^{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}\mathrm{d}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}{\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{\Phi }\mathrm{(}{\mathit{\eta }}_{\mathrm{p}\mathit{\gamma }}\mathit{,}{\mathit{x}}^{\mathrm{)}}\mathit{,}\end{array}$(45)where ${\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathrm{=}\mathit{x}{\mathit{E}}^{\mathrm{\prime }}$, ${\mathit{\eta }}_{\mathrm{p}\mathit{\gamma }}\mathrm{=}\frac{\mathrm{4}{\mathit{E}}^{\mathrm{\prime }}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}}{{\mathit{m}}_{\mathrm{p}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}\mathit{x}}$, $\begin{array}{ccc}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\left(}\min \mathrm{\right)}}\mathrm{=}\frac{{\mathit{m}}_{\mathrm{p}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}\mathit{x}}{\mathrm{4}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}}\left[\mathrm{2}\left(\frac{{\mathit{m}}_{{\mathit{\pi }}_{\mathrm{0}}}}{{\mathit{m}}_{\mathrm{p}}}\right)\mathrm{+}{\left(\frac{{\mathit{m}}_{{\mathit{\pi }}_{\mathrm{0}}}}{{\mathit{m}}_{\mathrm{p}}}\right)}^{\mathrm{2}}\right]\mathit{,}& & \end{array}$(46)and with the function $\mathrm{\Phi }{}^{\mathrm{(}}\mathit{\eta }_{\mathrm{p}\mathit{\gamma }}\mathit{,}{\mathit{x}}^{\mathrm{)}}$ as tabulated in Kelner & Aharonian (2008).

4.5. Internal and external photon absorption

We consider the absorption of the radiation due to photoionization processes at eV energies, and also due to e ⁻ e ⁺  creation by γγ and γN interactions at high energies in the source.

For γN interactions, we take as target the material along the line of sight corresponding to each particular object, which is measured through the column density of neutral hydrogen N_H. The absorption cross section σ_γN in this case is taken as in Ryter (1996) for E_γ < 1 keV, assuming that the medium is composed by atomic hydrogen and dust within galactic abundances (see Fig. 1). This cross section includes the effects of photoionizaton for E_γ > 13.6 eV and scattering with dust below this energy. In the present work we do not consider recombination, the inverse process of photoionization (see Appendix B). The resulting optical depth for γN interactions is approximated as $\begin{array}{ccc}{\mathit{\tau }}_{\mathit{\gamma }\mathrm{H}}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}\mathrm{\right)}\mathrm{=}{\mathit{N}}_{\mathrm{H}}{\mathit{\sigma }}_{\mathit{\gamma N}}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}\mathrm{\right)}\mathit{.}& & \end{array}$(47)It can be seen from Fig. 1 that besides the large absorption edge corresponding to the ionization energy of hydrogen, three aditional absorption edges are included in the cross section. They correspond to different ionization energies of helium, K-shell electrons of oxygen, and iron. In the present context, as can be seen below, we find no significant features associated with these aditional absorption edges (e.g. Cruddace et al. 1974; Ghisellini et al. 1999). The possible re-emission of lines is beyond the scope of this work, and could be studied, e.g., as in Ghisellini et al. (1999). For energies above 1 keV we take the γN cross section from Amsler et al. (2008), to account for Compton scattering and e ^±  production. In Fig. 1 we show the cross section used for γ   N interactions.

Fig. 1 Cross section for γN interactions. The absorption edges corresponding to the different elements are indicated.

In the case of γγ interactions, the main radiation target is considered to be the synchrotron photons inside the jet, which are characterized by a radiation density ${\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}$. If a dissipationless accretion disk is present, no other significant photon field is relevant for gamma-ray absorption. The cross section is given by $\begin{array}{ccc}{\mathit{\sigma }}_{\mathit{\gamma \gamma }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\right)}& \mathrm{=}& \frac{\mathrm{3}{\mathit{\sigma }}_{\mathrm{T}}}{\mathrm{16}}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{\beta }}_{\mathrm{e}}^{\mathrm{2}}\mathrm{)}\\ & & \mathrm{\times }\hspace{0.17em}\left[\mathrm{2}{\mathit{\beta }}_{\mathrm{e}}\mathrm{(}{\mathit{\beta }}_{\mathrm{e}}^{\mathrm{2}}\mathrm{-}\mathrm{2}\mathrm{)}\mathrm{+}\mathrm{(}\mathrm{3}\mathrm{-}{\mathit{\beta }}_{\mathrm{e}}^{\mathrm{4}}\mathrm{)}\ln \left(\frac{\mathrm{1}\mathrm{+}{\mathit{\beta }}_{\mathrm{e}}}{\mathrm{1}\mathrm{-}{\mathit{\beta }}_{\mathrm{e}}}\right)\right]\mathit{,}\end{array}$with $\begin{array}{ccc}{\mathit{\beta }}_{\mathrm{e}}\mathrm{=}{\left[\mathrm{1}\mathrm{-}\frac{\mathrm{2}{\mathit{m}}_{\mathrm{e}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}}{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{(}\mathrm{1}\mathrm{-}\cos {\mathit{\theta }}_{\mathit{\gamma \gamma }}\mathrm{)}}\right]}^{\mathrm{1}\mathit{/}\mathrm{2}}\mathit{,}& & \end{array}$(48)where θ_γγ is the angle of interaction between the incident photons. For γ-rays produced at a position z in the jet, we assume that they undergo collisions over a length r_j(z)/sinθ′ inside the jet with a target radiation field that is isotropic in the jet frame. The corresponding optical depth is calculated using the general expression of Gould & Schréder (1967) to obtain $\begin{array}{ccc}{\mathit{\tau }}_{\mathit{\gamma \gamma }}^{\mathrm{\left(}\mathrm{int}\mathrm{\right)}}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \mathrm{2}\mathit{\pi }{\mathrm{\int }}_{\mathrm{0}}^{\frac{{\mathit{r}}_{\mathrm{j}}\mathrm{\left(}\mathit{z}\mathrm{\right)}}{\sin {\mathit{\theta }}^{\mathrm{\prime }}}}\mathrm{d}{\mathit{l}}^{\mathrm{\prime }}{\mathrm{\int }}_{-1}^{\mathrm{1}}\mathrm{d}{\mathit{x}}^{\mathrm{\prime }}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{x}}^{\mathrm{\prime }}\mathrm{)}\\ & & \mathrm{\times }\hspace{0.17em}{\mathrm{\int }}_{\frac{\mathrm{2}{\mathit{m}}_{\mathrm{e}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}}{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{x}}^{\mathrm{\prime }}\mathrm{)}}}^{\mathrm{\infty }}\mathrm{d}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}{\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,}{\mathit{z}}^{\mathrm{\prime }}\mathrm{\right(}{\mathit{l}}^{\mathrm{\prime }}\mathrm{\left)}\mathrm{\right)}{\mathit{\sigma }}_{\mathit{\gamma \gamma }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\right)}\mathit{.}\end{array}$(49)In Fig. 2 we show the optical depths obtained for the case of Cen A and M 87, τ_γN and τ_γγ, the latter evaluated within the injection zone and also outside it. The absorbing photons outside the injection zone are those of synchrotron emission of protons and secondary e ^± , which in that case are also included in ${\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}$. It can be seen from this plot that very important absorption occurs at low energies by photoionization, and also at VHE through internal γγ interactions. The latter effect takes place mainly at the injection zone, since for z > z_acc + Δz, the optical depth is much lower.

Fig. 2 Optical depth for Cen A and M 87 at different distances form the black hole.

Taking into account the total optical depth τ_γ(E_γ,z) = τ_γγ(E_γ,z) + τ_γN(E_γ), the differential photon flux at Earth, i.e., the number of photons with energy E_γ, per unit energy, per unit area, per unit time, can be calculated as $\begin{array}{ccc}\frac{\mathrm{d}{\mathrm{\Phi }}_{\mathit{\gamma }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}\mathrm{\right)}}{\mathrm{d}{\mathit{E}}_{\mathit{\gamma }}}\mathrm{=}\frac{\mathrm{1}}{{\mathit{d}}^{\mathrm{2}}}\mathrm{\int }\mathrm{d}\mathit{V}{\mathit{Q}}_{\mathit{\gamma }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}\mathit{,z}\mathrm{\right)}\exp \mathrm{[}\mathrm{-}{\mathit{\tau }}_{\mathit{\gamma }}\mathrm{(}{\mathit{E}}_{\mathit{\gamma }}\mathit{,z}{\mathrm{)}}^{\mathrm{]}}\mathit{,}& & \end{array}$(50)where d is the distance from the source to Earth.

4.6. Neutrino emission

Neutrinos arise from direct pion decays plus muon decays with a total emissivity $\begin{array}{ccc}{\mathit{Q}}_{{\mathit{\nu }}_{\mathit{\mu }}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{=}{\mathit{Q}}_{\mathit{\pi }\mathrm{\to }{\mathit{\nu }}_{\mathit{\mu }}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{+}{\mathit{Q}}_{\mathit{\mu }\mathrm{\to }{\mathit{\nu }}_{\mathit{\mu }}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathit{,}& & \end{array}$which correspond to the observer frame and transforms according to Eq. (39). The contribution from pion decays (π ⁺  → μ ⁺ ν_μ, ) is represented in the co-moving frame by $\begin{array}{ccc}{\mathit{Q}}_{\mathit{\pi }\mathrm{\to }{\mathit{\nu }}_{\mathit{\mu }}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \int \begin{array}{}\mathrm{\infty }\\ {\mathit{E}}^{\mathrm{\prime }}\end{array}\mathrm{d}{\mathit{E}}_{\mathit{\pi }}{\mathit{T}}_{\mathit{\pi ,}\mathrm{d}}^{-1}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}\mathrm{\right)}{\mathit{N}}_{\mathit{\pi }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\pi }}\mathit{,z}\mathrm{\right)}\\ & & \mathrm{\times }\hspace{0.17em}\frac{\mathrm{\Theta }\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{r}}_{\mathit{\pi }}\mathrm{-}\mathit{x}\mathrm{)}}{{\mathit{E}}_{\mathit{\pi }}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{r}}_{\mathit{\pi }}\mathrm{)}}\mathit{,}\end{array}$(51)with x = E′/E_π and T_π,d = γ_π   2.6  ×  10^-8   s.

The neutrino emissivity for muon decays (${\mathit{\mu }}^{\mathrm{-}}\mathrm{\to }{\mathit{e}}^{\mathrm{-}}\mathit{\nu ̅}{}_{\mathrm{e}}\mathit{\nu }_{\mathit{\mu }}$, ) can be calculated as $\begin{array}{ccc}{\mathit{Q}}_{\mathit{\mu }\mathrm{\to }{\mathit{\nu }}_{\mathit{\mu }}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \sum _{\mathit{i}\mathrm{=}\mathrm{1}}^{\mathrm{4}}{\mathrm{\int }}_{{\mathit{E}}^{\mathrm{\prime }}}^{\mathrm{\infty }}\frac{\mathrm{d}{\mathit{E}}_{\mathit{\mu }}}{{\mathit{E}}_{\mathit{\mu }}}{\mathit{T}}_{\mathit{\mu ,}\mathrm{d}}^{-1}\mathrm{\left(}{\mathit{E}}_{\mathit{\mu }}\mathrm{\right)}{\mathit{N}}_{{\mathit{\mu }}_{\mathit{i}}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\mu }}\mathit{,z}\mathrm{\right)}\\ & & \mathrm{\times }\hspace{0.17em}\left[\frac{\mathrm{5}}{\mathrm{3}}\mathrm{-}\mathrm{3}{\mathit{x}}^{\mathrm{2}}\mathrm{+}\frac{\mathrm{4}}{\mathrm{3}}{\mathit{x}}^{\mathrm{3}}\right]\mathit{.}\end{array}$(52)In this expression, x = E′/E_μ, ${\mathit{\mu }}_{\mathrm{1}\mathit{,}\mathrm{2}}\mathrm{=}{\mathit{\mu }}_{\mathit{L}}^{\mathrm{-}\mathit{,}\mathrm{+}}$, T_μ,d = γ_μ   2.2  ×  10^-6   s, and ${\mathit{\mu }}_{\mathrm{3}\mathit{,}\mathrm{4}}\mathrm{=}{\mathit{\mu }}_{\mathit{R}}^{\mathrm{-}\mathit{,}\mathrm{+}}$ (see Lipari et al. 2007). The differential neutrino flux can be computed using a expression analogous to Eq. (50).

The synchrotron cooling of μs and πs, which is taken into account in our model, will be traduced on a lower neutrino emissivity as it was previously shown in Reynoso & Romero (2009).

4.7. Secondary e ⁺ e ⁻  production

We consider the production of secondary electrons and positrons in the jet through pγ, pp, and γγ interactions. The inelastic pγ collisions with can produce e ⁻ e ⁺  pairs apart from pions as discussed above. We consider this direct production of pairs through the injection function in the jet frame $\begin{array}{ccc}{\mathit{Q}}_{{\mathrm{e}}^{\mathrm{\pm }}\mathit{,}\mathrm{p}\mathit{\gamma }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \mathrm{2}{\mathrm{\int }}_{{\mathit{m}}_{\mathrm{p}}{\mathit{c}}^{\mathrm{2}}}^{\mathrm{\infty }}\mathrm{d}{\mathit{E}}_{\mathrm{p}}^{\mathrm{\prime }}{\mathit{N}}_{\mathrm{p}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{p}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\\ & & \mathrm{\times }\hspace{0.17em}{{\mathit{\omega }}^{\mathrm{\prime }}}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{e}\mathrm{\right)}}\mathrm{\left(}{\mathit{E}}_{\mathrm{p}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\hspace{0.17em}\mathit{\delta }\left({\mathit{E}}^{\mathrm{\prime }}\mathrm{-}\frac{{\mathit{m}}_{\mathrm{e}}}{{\mathit{m}}_{\mathrm{p}}}{\mathit{E}}_{\mathrm{p}}^{\mathrm{\prime }}\right)\mathit{,}\end{array}$(53)where the ${\mathit{\omega }}_{\mathrm{p}\mathit{\gamma ,}{\mathrm{e}}^{\mathrm{\pm }}}\mathrm{\left(}{\mathit{E}}_{\mathrm{p}}^{\mathrm{\prime }}\mathrm{\right)}$ is the collision frequency given by $\begin{array}{ccc}{{\mathit{\omega }}^{\mathrm{\prime }}}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{e}\mathrm{\right)}}\mathrm{\left(}{\mathit{E}}_{\mathrm{p}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \frac{\mathit{c}}{\mathrm{2}{\mathit{\gamma }}_{\mathrm{p}}^{\mathrm{2}}}{\mathrm{\int }}_{\frac{{\mathit{ϵ}}_{\mathrm{th}}^{\mathrm{\left(}\mathit{e}\mathrm{\right)}}}{\mathrm{2}{\mathit{\gamma }}_{\mathrm{p}}}}^{\mathrm{\infty }}\mathrm{d}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\frac{{\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{{{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}}^{\mathrm{2}}}\\ & & \mathrm{\times }\hspace{0.17em}{\mathrm{\int }}_{{\mathit{ϵ}}_{\mathrm{th}}^{\mathrm{\left(}\mathit{e}\mathrm{\right)}}}^{\mathrm{2}\mathit{ϵ}{\mathit{\gamma }}_{\mathrm{p}}}\mathrm{d}{\mathit{ϵ}}^{\mathrm{\prime }}{\mathit{\sigma }}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{e}\mathrm{\right)}}\mathrm{\left(}{\mathit{ϵ}}^{\mathrm{\prime }}\mathrm{\right)}{\mathit{ϵ}}^{\mathrm{\prime }}\mathit{.}\end{array}$(54)Here, the photopair cross section ${\mathit{\sigma }}_{\mathrm{p}\mathit{\gamma }}^{\mathrm{\left(}\mathit{e}\mathrm{\right)}}$ is considered as in (Begelman et al. 1990).

As discussed above, the inelastic pp and pγ collisions produce pions that decay giving muons. The decay products of muons include electrons and positrons. The decay spectrum of the electrons and positrons are taken to be equal to that of muon neutrinos and anti-neutrinos (see e.g. Gaisser 1990). Thus, the injection of electrons and positrons from muon decay after pp and pγ interactions is considered following Schlickeiser (2002).

As for the e ⁺ e ⁻  pairs produced in γγ interactions, we consider the injection according to Aharonian et al. (1983), $\begin{array}{ccc}{\mathit{Q}}_{{\mathrm{e}}^{\mathrm{\pm }}\mathit{,\gamma \gamma }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \frac{\mathrm{3}}{\mathrm{32}}\frac{{\mathit{\sigma }}_{\mathrm{T}}\mathit{c}}{{\mathit{m}}_{\mathrm{e}}{\mathit{c}}^{\mathrm{2}}}{\mathrm{\int }}_{{\mathit{\gamma }}_{\mathrm{e}}}^{\mathrm{\infty }}\mathrm{d}{\mathit{ϵ}}_{\mathit{\gamma }}{\mathrm{\int }}_{\frac{{\mathit{ϵ}}_{\mathit{\gamma }}}{\mathrm{4}{\mathit{\gamma }}_{\mathrm{e}}\mathrm{(}{\mathit{ϵ}}_{\mathit{\gamma }}\mathrm{-}{\mathit{\gamma }}_{\mathrm{e}}\mathrm{)}}}^{\mathrm{\infty }}\mathrm{d}\mathit{\omega }\frac{{\mathit{n}}_{\mathit{\gamma }}\mathrm{\left(}{\mathit{ϵ}}_{\mathit{\gamma }}\mathit{,z}\mathrm{\right)}}{{\mathit{ϵ}}^{\mathrm{3}}}\\ & & \mathrm{\times }\frac{{\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}\mathit{\omega }{\mathit{m}}_{\mathrm{e}}{\mathit{c}}^{\mathrm{2}}\mathit{,z}\mathrm{\right)}}{{\mathit{\omega }}^{\mathrm{2}}}\{\frac{\mathrm{4}{\mathit{ϵ}}_{\mathit{\gamma }}^{\mathrm{2}}}{{\mathit{\gamma }}_{\mathrm{e}}\mathrm{(}{\mathit{ϵ}}_{\mathit{\gamma }}\mathrm{-}{\mathit{\gamma }}_{\mathrm{e}}\mathrm{)}}\ln \left[\frac{\mathrm{4}{\mathit{\gamma }}_{\mathrm{e}}\mathit{\omega }\mathrm{(}{\mathit{ϵ}}_{\mathit{\gamma }}\mathrm{-}{\mathit{\gamma }}_{\mathrm{e}}\mathrm{)}}{{\mathit{ϵ}}_{\mathit{\gamma }}}\right]\\ & & \mathrm{-}\hspace{0.17em}\mathrm{8}{\mathit{ϵ}}_{\mathit{\gamma }}\mathit{\omega }\mathrm{+}\frac{\mathrm{4}{\mathit{ϵ}}_{\mathit{\gamma }}^{\mathrm{3}}\mathit{\omega }\mathrm{-}\mathrm{2}{\mathit{ϵ}}_{\mathit{\gamma }}^{\mathrm{2}}}{{\mathit{\gamma }}_{\mathrm{e}}\mathrm{(}{\mathit{ϵ}}_{\mathit{\gamma }}\mathrm{-}{\mathit{\gamma }}_{\mathrm{e}}\mathrm{)}}\mathrm{-}\left(\mathrm{1}\mathrm{-}\frac{\mathrm{1}}{{\mathit{ϵ}}_{\mathit{\gamma }}\mathit{\omega }}\right)\frac{{\mathit{ϵ}}_{\mathit{\gamma }}^{\mathrm{4}}}{{\mathit{\gamma }}_{\mathrm{e}}^{\mathrm{2}}\mathrm{(}{\mathit{ϵ}}_{\mathit{\gamma }}\mathrm{-}{\mathit{\gamma }}_{\mathrm{e}}{\mathrm{)}}^{\mathrm{2}}}\begin{array}{}\mathrm{⎫}\\ \mathrm{⎪}\\ \mathrm{⎪}\\ \mathrm{⎬}\\ \mathrm{⎪}\\ \mathrm{⎪}\\ \mathrm{⎭}\end{array}\mathit{,}\end{array}$(55)where γ_e = E′/(m_ec²), $\mathit{\omega }\mathrm{=}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{/}\mathrm{\left(}{\mathit{m}}_{\mathrm{e}}{\mathit{c}}^{\mathrm{2}}\mathrm{\right)}$, and $\mathit{ϵ}\mathrm{=}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{/}\mathrm{\left(}{\mathit{m}}_{\mathrm{e}}{\mathit{c}}^{\mathrm{2}}\mathrm{\right)}$. The soft photon density ${\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}$ refers to the synchrotron radiation of electrons, while the density of higher energy photons from IC, pp, and pγ interactions, is given by $\begin{array}{ccc}{\mathit{n}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{\approx }\mathrm{4}\mathit{\pi }\frac{{\mathit{Q}}_{\mathit{\gamma ,}\mathrm{IC}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{+}{\mathit{Q}}_{\mathit{\gamma ,pp}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}\mathrm{+}{\mathit{Q}}_{\mathit{\gamma ,p\gamma }}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}{\mathit{c}\mathit{/}\mathrm{\left(}\mathit{z}\tan {\mathit{\xi }}_{\mathrm{j}}\mathrm{\right)}\mathrm{+}{\mathit{t}}_{\mathit{\gamma \gamma }}^{-1}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}}\mathrm{·}& & \end{array}$Here, the annihilation rate of gamma-rays due to pair production is given by $\begin{array}{ccc}{\mathit{t}}_{\mathit{\gamma \gamma }}^{-1}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}& \mathrm{=}& \mathrm{2}\mathit{\pi }\hspace{0.17em}\mathit{c}{\mathrm{\int }}_{-1}^{\mathrm{1}}\mathrm{d}{\mathit{x}}^{\mathrm{\prime }}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{x}}^{\mathrm{\prime }}\mathrm{)}\\ & & \mathrm{\times }\hspace{0.17em}{\mathrm{\int }}_{\frac{\mathrm{2}{\mathit{m}}_{\mathrm{e}}^{\mathrm{2}}{\mathit{c}}^{\mathrm{4}}}{{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{x}}^{\mathrm{\prime }}\mathrm{)}}}^{\mathrm{\infty }}\mathrm{d}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}{\mathit{n}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\left(}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathit{,z}\mathrm{\right)}{\mathit{\sigma }}_{\mathit{\gamma \gamma }}\mathrm{\left(}{\mathit{E}}_{\mathit{\gamma }}^{\mathrm{\prime }}\mathit{,}{\mathit{E}}_{\mathrm{ph}}^{\mathrm{\prime }}\mathrm{\right)}\mathit{.}\end{array}$(56)The distribution of secondary electrons and positrons in the jet is found solving the transport equation. This allows us to calculate the synchrotron and IC emission produced by these particles.