DIFFERENCES OF LEPTONIC AND HADRONIC RADIATION PRODUCTION IN FLARING BLAZARS

The differential number density of relativistic electrons n_e(r, γ, t) and protons n_p(r, γ, t), respectively, obey the transport equation of ESR12 and takes spatial diffusion, continuous energy losses, and a separable injection term q₁(γ, t)q₂(r) into account:

$$\begin{equation} {\partial n_{e,p}(r,\gamma,t)\over \partial t}={1\over r^2}{\partial \over \partial r} \left(r^2D_1(\gamma){\partial n_{e,p}(r,\gamma,t)\over \partial r}\right)+ {\partial \over \partial \gamma }[|\dot{\gamma }|n_{e,p}(r,\gamma,t)]+q_1(\gamma,t)q_2(r). \end{equation} \tag{ 1 }$$

Here we idealized that the spatial diffusion is independent of its solid angle, as the anisotropy becomes negligible with increasing time and decreasing Lorentz factor, respectively. Furthermore, the captured relativistic electrons and protons may increase their energy in the emission knot by distributed stochastic acceleration (Schlickeiser & Dermer 2000) and diffusive shock wave acceleration processes (Ball & Kirk 1992; Kirk et al. 1998). However, we do not consider these processes here: (1) stochastic gyroresonant acceleration from dust induced turbulence (Schlickeiser & Dermer 2000) depends sensitively on the details of providing low-frequency MHD turbulence with multiple phase speeds, (2) a relativistic shock (driven by the emission knot) may form upstream of the emission knot by escaping charged particles from the knot (Gerbig & Schlickeiser 2011), but its collision-free properties will be different from pure hydrodynamical shocks.

Our energization of relativistic particles in the emission knot only relies on the relativistic pick-up process (Pohl & Schlickeiser 2000) with subsequent plateauing (Pohl et al. 2002).

According to ESR12 the initial spatial distribution of the injected particles has no significant influence on the temporal synchrotron flare behavior, so that we initially assume them to be homogeneously distributed in the whole emission volume, i.e., q₂(r) = 1. The injected particle beam excites electrostatic turbulences, whereby the particles' energy is quickly changed until a plateau distribution in p_∥ is established (Pohl et al. 2002), so that the injected particles can be considered as plateau distributed and we obtain

$$\begin{equation} q_1(\gamma, t)=\frac{q_0}{\gamma _0-1}\,H[\gamma _0-\gamma ]\,\delta (t-t_0). \end{equation} \tag{ 2 }$$

Additionally, electromagnetic turbulences (i.e., Alfvén waves) are excited, which result in pitch angle scattering of the injected charged particles with an energy dependent diffusion coefficient D₁(γ). We assume an isospectral power-law turbulence with a spectral index 1 ⩽ q ⩽ 3, so that the spatial diffusion coefficient is determined by (Schlickeiser 2002)

$$\begin{equation} D_1(\gamma)=\frac{c\sqrt{1-\gamma ^{-2}}}{2\pi (q-1)}\left(\frac{B}{\partial B}\right)^2\,R_L(\gamma)^{2-q}\,k_{{\rm min}}^{1-q}\,F(Z,H_c, \sigma _{f,b}), \end{equation} \tag{ 3 }$$

with the magnetic turbulence ∂B, the minimum wavenumber k_min, and the Larmor radius $R_L(\gamma)=mc^2\sqrt{\gamma ^{2}-1}/(ZeB)$ of the relativistic particle. The function F is only determined by the charge Z of the particle, as well as the spectral index q and the helicity parameters which are all independent of the energetic properties of the particle. In the same way, the minimum wavenumber k_min only refers to the maximum Alfvén wavelength λ_max = 2πk⁻¹_min, which has to be smaller than the size R of the emission knot. In general, the diffusion coefficient is also a function of r and t, as the magnetic field adjusts itself to the actual energy density of the radiating particles (Schlickeiser & Lerche 2008). But we ignore this nonlinear behavior in order to solve the differential equation (1). However, in the special case of q = 2 Equation (3) becomes independent of the Larmor radius and the diffusion coefficient is a constant, as we expect a similar spatial and temporal dependency of the magnetic field B and its perturbation δB. Since most of the microphysical details of the turbulence are unknown, we summarize some of the parameters of Equation (3) by the energy independent mean free path l_{e, p} of the electron and proton, respectively. Thus, in the full diffusion limit l_{e, p}γ^β₀ ≪ R the spatial diffusion coefficient at relativistic particle energies (γ ≫ 1) yields

$$\begin{equation} D_1(\gamma)={c\,l_{e,p}\over 3} \gamma ^{\beta }, \end{equation} \tag{ 4 }$$

where β = 2 − q.

Furthermore, the relativistic particles cool by different continuous energy loss effects. In case of the relativistic electrons ESR12 have shown that the energy loss is caused by synchrotron and EC effects at high energies and Coulomb interactions with the background plasma at small energies, so that

$$\begin{equation} |\dot{\gamma }|_e=D_{ex}+D_s\gamma ^2,\,\,\,\,D_s=1.3\times 10^{-9}\left(b^2+l_{{\rm EC}}\right)\,{\rm s}^{-1}\;\;{\rm and }\;\;D_{ex}=0.008\,N_{10}\,{\rm s}^{-1}. \end{equation} \tag{ 5 }$$

Here we assume a magnetic field of constant strength B = 1 b G, which is randomly distributed on scales larger than the Larmor radii of the injected electrons, as well as a constant density N_b = 10¹⁰N₁₀ cm⁻³ of the fully ionized background plasma. Furthermore, l_EC = 0.093 L₄₆ δ²₁ τ₋₂ R⁻²_pc refers to the EC losses due to an isotropic external photon field (Dermer & Schlickeiser 1994), which depends on the luminosity L_ad = 10⁴⁶ L₄₆ erg s⁻¹ of the accretion disk, the scattering optical depth τ_sc = 10⁻² τ₋₂ and the extension R_sc = 1 R_pc pc of the scattering gas, as well as the Doppler factor δ = 10 δ₁ of the emission volume.

In the case of relativistic protons the cross section of electromagnetic interactions is much smaller than that of electrons with the same Lorentz factor, as the Thomson cross section for a proton σ_{T, p} = 3 × 10⁻⁷ σ_{T, e}, where σ_{T, e} is the Thomson cross section for an electron. Thus, the relevant interaction processes are on the one hand hadron–photon interactions with the accretion disk photons with a thermal energy θ = 10θ₁ eV and on the other hand hadron–hadron interactions with the thermal background plasma. However, at a particle density N₁₀ > 2.7 × 10⁻⁸ δ²₁ L₄₆τ₋₂ R⁻²_pc θ⁻¹₁ and a Lorentz factor γ₆ < 7.3 K θ⁻¹₁, the dominant proton energy loss rate

$$\begin{equation} |\dot{\gamma }|_p\simeq D_c\,H[\gamma -1.30]\,\gamma \,,\quad D_c=7\times 10^{-6}\,N_{10}\,{\rm s}^{-1}\, \end{equation} \tag{ 6 }$$

results from inelastic interactions with the protons of the background plasma (see Appendix A.1).

In spite of the energy dependent diffusion coefficient D₁(γ), the radial diffusion operator is still a Sturm–Liouville type and thus we solve the kinetic transport Equation (1) according to ESR12 in the eigenspace of the operator by using the spatial boundary conditions:

• 1.  
  a finite density at the center of the knot; and
• 2.  
  an exclusively outward directed particle flux at the boundary surface of the knot:

  $$\begin{equation} \left[\frac{\partial n_{e,p}}{\partial r}+\frac{3}{2\,l_{e,p}\,\gamma ^{\beta }}n_{e,p}\right]_{r=R}=0. \end{equation} \tag{ 7 }$$

Hence, the particle density yields

$$\begin{equation} n_{e,p}(r, \gamma,t)=\sum _{k=1}^\infty n^{e,p}_k(\gamma,t){\sin (\lambda _kr)\over r}, \end{equation} \tag{ 8 }$$

where in the full diffusion limit l_{e, p} γ^β ≪ R of many relativistic particle scattering in the knot the eigenvalues are

$$\begin{equation} \lambda _k\simeq {k\pi \over R},\quad k=1,2,3,...,k_{{\rm max}}, \end{equation} \tag{ 9 }$$

with k_max ≪ R/(l_{e, p} γ^β). Using the ansatz

$$\begin{equation} n^{e,p}_k(\gamma,t)=r^{e,p}_k(\gamma,t)\,\exp \left({\lambda _k^2\,c\,l_{e,p}\over 3}\int d\gamma \,{\gamma ^\beta \over |\dot{\gamma }|_{e,p}}\right), \end{equation} \tag{ 10 }$$

the particle transport Equation (1) becomes

$$\begin{equation} \frac{\partial r^{e,p}_k(\gamma,t)}{\partial t}-\frac{\partial }{\partial \gamma }\left[ |\dot{\gamma }|_{e,p}\,r^{e,p}_k(\gamma,t)\right]=q_1(\gamma,t)\,\exp \left(-{\lambda _k^2\,c\,l_{e,p}\over 3}\int d\gamma \,{\gamma ^\beta \over |\dot{\gamma }|_{e,p}}\right). \end{equation} \tag{ 11 }$$

We solve the differential Equation (11) using the Green's function of ESR12, so that the expansion coefficients are

$$\begin{equation} n^{e,p}_k(\gamma,t)=b_k\,\int ^\infty _{-\infty }dt^{\prime }\int ^\infty _1 d\gamma ^{\prime }\,q_1(\gamma ^{\prime },t^{\prime })\,H[t-t^{\prime }]\,\delta (t^{\prime }-t+x_{e,p}(\gamma)-x_{e,p}(\gamma ^{\prime }))\,{E^{e,p}_k(\gamma,\gamma ^{\prime })\over |\dot{\gamma }|_{e,p}}, \end{equation} \tag{ 12 }$$

where

$$\begin{equation} b_k\simeq {2\over R} \int _0^Rdr\, q_2(r)r\sin (\lambda _kr), \end{equation} \tag{ 13 }$$

and the general functions

$$\begin{equation} x_{e,p}(\gamma)=-\int {d\gamma \over |\dot{\gamma }|_{e,p}}, \end{equation} \tag{ 14 }$$

as well as

$$\begin{equation} E^{e,p}_k(\gamma,\gamma ^{\prime })=\exp \left({\lambda _k^2\,c\,l_{e,p}\over 3}\left[\int d\gamma \,{\gamma ^\beta \over |\dot{\gamma }|_{e,p}}-\int d\gamma ^{\prime }\,{\gamma ^{\prime \beta }\over |\dot{\gamma ^{\prime }}|_{e,p}}\right]\right), \end{equation} \tag{ 15 }$$

specify the temporal development of the relativistic particle density depending on the particle species. In the case of relativistic electrons the cooling is determined by Equation (5) and thus we obtain

$$\begin{equation} x_e(\gamma)=-{1\over D_s\,\gamma _q}\,\arctan \left({\gamma \over \gamma _q}\right) \end{equation} \tag{ 16 }$$

with $\gamma _q=\sqrt{D_{ex}/D_s}$. By inserting Equation (12), the relativistic electron density (Equation (8)) yields

$$\begin{eqnarray} n_e(r,\gamma,t)=&{q_0\over \gamma _0-1}\,H[t-t_0]\,H[\gamma _t(\gamma,t)-1]\,H[\gamma _0-\gamma _t(\gamma,t)]\,{1+(\gamma _t(\gamma,t)/\gamma _q)^2\over 1+(\gamma /\gamma _q)^2} \nonumber \\ &\times \sum _{k=1}^\infty \,b_k\,{\sin (\lambda _kr)\over r}\,E_k^e(\gamma,\gamma _t(\gamma,t)), \end{eqnarray} \tag{ 17 }$$

in which

$$\begin{equation} \gamma _t(\gamma,t)={\gamma _q\gamma +\gamma _q^2\tan (D_s\gamma _q(t-t_0))\over \gamma _q-\gamma \tan (D_s\gamma _q(t-t_0))} \end{equation} \tag{ 18 }$$

and

$$\begin{eqnarray} E^e_k(\gamma,\gamma _t(t))&=\exp \Biggl ({\lambda _k^2\,c\,l_{e}\over 3D_{ex}(\beta +1)}\Biggl [\gamma ^{\beta +1} {\,}_{2}\!F_1\left(1,{\beta +1\over 2};{\beta +3\over 2};{\gamma ^2\over \gamma _q^2}\right) -\gamma _t(\gamma,t)^{\beta +1} {\,}_{2}\!F_1\left(1,{\beta +1\over 2};{\beta +3\over 2};{\gamma _t(\gamma,t)^{\,2}\over \gamma _q^2}\right)\Biggr ]\Biggr)\nonumber\\ \ms &=\left\lbrace \begin{array}{@{}l@{\quad }l@{}}\left({(\gamma /\gamma _q)^2\,+\,1\over (\gamma _t(\gamma,t)/\gamma _q)^2\,+\,1}\right)^{{\lambda _k^2\,c\,l_e\over 6\,D_s}},&{\rm for }\;\beta =1\\ \exp \left({\lambda _k^2\,c\,l_e\over 3\,D_s\,\gamma _q}\left[\arctan \left({\gamma \over \gamma _q}\right)-\arctan \left({\gamma _t(\gamma,t)\over \gamma _q}\right)\right]\right),&{\rm for }\;\beta =0\\ \left({(\gamma _q/\gamma _t(\gamma,t))^2\,+\,1\over (\gamma _q/\gamma)^2\,+\,1}\right)^{{\lambda _k^2\,c\,l_e\over 6\,D_{ex}}},&{\rm for }\;\beta =-1.\end{array}\right. \end{eqnarray} \tag{ 19 }$$

Equation (6) defines the cooling in the case of the relativistic protons and hence with

$$\begin{equation} x_p(\gamma)=-{\ln \gamma \over D_c} \end{equation} \tag{ 20 }$$

the relativistic proton density is calculated by

$$\begin{equation} n_p(r,\gamma,t)={q_0\over \gamma _0-1}\,H[t-t_0]\,H[\gamma _0-\gamma _r(\gamma,t)]\,{\gamma _r(\gamma,t)\over \gamma }\,\sum _{k=1}^\infty \,b_k\,{\sin (\lambda _kr)\over r}\,E_k^p(\gamma,\gamma _r(\gamma,t)), \end{equation} \tag{ 21 }$$

where

$$\begin{equation} \gamma _r(\gamma,t)=\gamma \,\exp (D_c\,(t-t_0)) \end{equation} \tag{ 22 }$$

and

$$\begin{equation} E^p_k(\gamma,\gamma _r(\gamma,t))=\left\lbrace \begin{array}{@{}l@{\quad }l@{}}\exp \left({\lambda _k^2\,c\,l_p\over 3\,D_c\,\beta }\,(\gamma ^\beta -\gamma _r(\gamma,t)^\beta)\right)\,, &{\rm for }\; \beta \ne 0\\ (\gamma /\gamma _r(\gamma,t))^{{\lambda _k^2\,c\,l_p\over 3\,D_c}}\,, &{\rm for }\; \beta =0.\end{array}\right.\, \end{equation} \tag{ 23 }$$

Figure 1 shows that the continuous energy losses of the relativistic particles result in an increase of the number density (at a Lorentz factor γ < γ₀) with increasing time. The electrons cool much faster than the protons by inelastic p–p interactions. In the case of the full diffusion limit (R ≫ γ^β₀l_{e, p}), the different spatial diffusion assumptions (4) have no influence on the particle density, so that the emergent intensity of the generated photons and neutrinos, respectively, is also independent of the spatial diffusion behavior of the relativistic particles. Only the cooling mechanisms determine the temporal development. Thus, the relativistic particles cool down to γ = 1 and due to the third Heaviside function in Equation (17) we obtain for γ₀ > γ_q the maximal electron-generated flare duration

$$\begin{equation} t^e_{{\rm ul}}\simeq {\pi \over 2\,D_s\,\gamma _q}+t_0={4.9\times 10^5\,{\rm s}\over (b^2+l_{{\rm EC}})^{1/2}\,N_{10}^{1/2}}+t_0. \end{equation} \tag{ 24 }$$

A similar result was already given by ESR12 and hence the energetic distribution of injected particles, as well as the spatial diffusion, has no influence on the upper limit (Equation (24)). In the case of the relativistic protons, the second Heaviside function in Equation (21) yields the maximal proton-generated flare duration

$$\begin{equation} t^p_{{\rm ul}}={\ln \gamma _0\over D_c}+t_0=2.0\times 10^{6}\,N_{10}^{-1}\,(1+\ln (\gamma _6)/\ln (10^6))\,{\rm s}+t_0. \end{equation} \tag{ 25 }$$

Consequently, an AGN flare generated by relativistic protons can last significantly longer than a flare that is produced by relativistic electrons when the background particle density N₁₀ < 16.7 (b² + l_EC) (1 + ln (γ₆)/ln (10⁶))². However, we have to consider the different particle production rates of each emission process in order to obtain a detailed description of the temporal flare behavior of photons and neutrinos.

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First, we calculate the synchrotron photon emission generated by the relativistic electron density distribution (Equation (17)). According to ESR12 the spontaneous synchrotron emission coefficient j_s(r, ν, t) that results from an isotropic relativistic electron distribution n_e(r, γ, t) yields

$$\begin{eqnarray} j_s(r,\nu,t)&=\frac{P_0}{4\pi }\left(\frac{\nu }{\nu _s}\right)^{\frac{1}{3}}\int ^{\infty }_{1}d\gamma \:\gamma ^{-\frac{2}{3}}\,\exp\left(-{\nu \over \nu _s\gamma ^2}\right)\,n_e(r,\gamma,t), \end{eqnarray} \tag{ 26 }$$

with P₀ = 2.647 × 10⁻¹⁰ eV s⁻¹ Hz⁻¹ and the characteristic frequency ν_s γ² = 3eBγ²/(4πm_ec). For an optically thin source, the synchrotron intensity on the surface (r = R) at a time t and an energy = hν is calculated by (Gould 1979)

$$\begin{equation} I_{{\rm sy}}(R,\epsilon,t)=\frac{1}{2\,R}\int ^R_0 dr^{\prime }\:r^{\prime } \, \int ^{R+r^{\prime }}_{R-r^{\prime }}ds\: \frac{\rho _{{\rm sy}}\left(r^{\prime },\epsilon,t-\frac{s}{c}\right)}{s}, \end{equation} \tag{ 27 }$$

with the omnidirectional photon production rate

$$\begin{equation} \rho _{{\rm sy}}\left(r^{\prime },\epsilon,t-\frac{s}{c}\right)=4\pi \exp \left(-\frac{g(x)\,s}{R}\right)\, j_s\left(r^{\prime },\epsilon,t-\frac{s}{c}\right), \end{equation} \tag{ 28 }$$

where the exponential term expresses the survival probability of traveling the distance $s=\sqrt{R^2+r^{\prime 2}-2Rr^{\prime }\mu ^{\prime }}$ from the initial spherical coordinates r', μ' = cos ϕ' to the boundary surface of the knot (Lightman & Zdziarski 1987). Here x = hν/(m_ec²) is the dimensionless photon energy and

$$\begin{eqnarray} g(x)=\left\lbrace \begin{array}{@{}ll} {[}1+\tau _{KN}]^{-1} & {\rm for}\quad x\le 0.1 \\ {[}1+\tau _{KN}\,(1-x)/0.9]^{-1} & {\rm for} \quad 0.1<x<1 \\ 1 & {\rm for}\quad x\ge 1\end{array}\right., \end{eqnarray} \tag{ 29 }$$

with τ_KN(x) = R N_b σ_KN(x)/3, in which σ_KN(x) denotes the total Klein–Nishina cross section. We use Equation (17) and account for the Heaviside functions in the first integrand, so that the synchrotron intensity (27) yields

$$\begin{eqnarray} I_{{\rm sy}}(R,\epsilon,t)=&{P_0\,q_0\over 2Rh(\gamma _0-1)}\left({\epsilon \over h\nu _s}\right)^{{1\over 3}}\,\sum ^\infty _{k=1}b_k\int ^R_0dr^{\prime }\,\sin (\lambda _kr^{\prime })\nonumber\\ &\times \int ^{R+r^{\prime }}_{R-r^{\prime }}{ds\over s}\,H[c(t-t_0)-s]\,H\left[s-c\left(t-t_0-{1\over D_s\gamma _q}\arctan \left({\gamma _0\gamma _q-\gamma _q\over \gamma _q^2+\gamma _0}\right)\right)\right]\nonumber\\ &\times \,e^{-\frac{g(x)\,s}{R}}\int _1^{\gamma _u(t-s/c)}d\gamma \,{1+(\gamma _t(\gamma,t)/\gamma _q)^2\over \gamma ^{{2\over 3}}+(\gamma ^{{4\over 3}}/\gamma _q)^2}\,\exp \left(-{\epsilon \over h\nu _s\gamma ^2}\right)\,E^e_k(\gamma,\gamma _t(t-s/c)), \end{eqnarray} \tag{ 30 }$$

where

$$\begin{eqnarray} &&\gamma _u(t-s/c)={\gamma _q\gamma _0-\gamma _q^2\tan (D_s\gamma _q(t-s/c-t_0))\over \gamma _q+\gamma _0\tan (D_s\gamma _q(t-s/c-t_0))}. \end{eqnarray} \tag{ 31 }$$

According to Appendix A.2 the Heaviside functions split the integral expression in several integrands, which are finally solved numerically by a 40-point Gaussian quadrature rule.

Due to the synchrotron cooling of the relativistic electrons the amplitude of the emergent intensity, as well as the total flare duration and the time of maximal synchrotron flare emission, increase with decreasing photon energy (see Figure 2).

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Second, we consider the inverse Compton scattering of low-energy photons by the relativistic electron density distribution (Equation (17)). Since we assume a low number density of injected particles (q₀ < 10⁹ R⁻¹₁₅ γ⁻²₀) to keep the SSC losses negligible, the seed photons for the inverse Compton scattering result from an external photon field. Here, we consider the accretion disk with a temperature θ = 10 θ₁ eV and a Planckian distributed spectral luminosity

$$\begin{equation} L(\epsilon)={15\,L_{{\rm ad}}\,m_ec^2\over \pi ^4\,\theta ^4}\,{\epsilon ^3 \over \exp \left({\epsilon \over \theta }\right)-1}, \end{equation} \tag{ 32 }$$

where the total luminosity L_ad = 10⁴⁶ L₄₆ erg s⁻¹. Assuming that the intergalactic gas scatters the thermal photons to an isotropic distribution, the external photon field in the frame of the emission knot yields (Dermer & Schlickeiser 1993)

$$\begin{equation} n_{{\rm ph}}(\epsilon)=n_0\,{\epsilon ^2\over \exp \left({\epsilon \over \theta }\right)-1}, \end{equation} \tag{ 33 }$$

with

$$\begin{equation} n_0=1.4\times 10^{10}\,{\tau _{-2}\,L_{46}\,\delta _1^2\over \theta _1^4\, R_{{\rm pc}}^2}\,{\rm eV}^{-2}\;{\rm cm}^{-3}\, \end{equation} \tag{ 34 }$$

and has a spectral maximum at

$$\begin{equation} \epsilon _{{\rm max}}\simeq 1.59\,\theta. \end{equation} \tag{ 35 }$$

Using the convenient δ-function approximation for inverse Compton scattering (Dermer & Schlickeiser 1993) in the limit γ ≫ 1 ≫ /(m_ec²) the spontaneous EC emission coefficient results in

$$\begin{equation} j_c(r,E_\gamma,t)={E_\gamma c \sigma _T\over 12\pi ^2\,m_ec^2}\int ^{\infty }_0 d\epsilon \,n_{{\rm ph}}(\epsilon)\int ^{m_ec^2/\epsilon }_0 d\gamma \,n_e(r,\gamma,t)\delta (E_\gamma -\gamma ^2\epsilon). \end{equation} \tag{ 36 }$$

Analogous to the previous section the emergent intensity of EC photons at a time t and an energy E_γ is calculated by

$$\begin{equation} I_{{\rm EC}}(R,E_\gamma,t)=\frac{1}{2\,R}\int ^R_0 dr^{\prime }\:r^{\prime } \, \int ^{R+r^{\prime }}_{R-r^{\prime }}ds\: \frac{\rho _{{\rm EC}}\left(r^{\prime },E_\gamma,t-\frac{s}{c}\right)}{s}, \end{equation} \tag{ 37 }$$

with the omnidirectional photon production rate

$$\begin{equation} \rho _{{\rm EC}}(r^{\prime },E_\gamma,t-\frac{s}{c})=4\pi \exp \left(-{g(x)\,s\over R}\right)\, j_c\left(r^{\prime },E_\gamma,t-\frac{s}{c}\right). \end{equation} \tag{ 38 }$$

Inserting the relativistic electron density (Equation (17)) and including the Heaviside functions in the innermost integral, the emergent EC intensity results in

$$\begin{eqnarray} I_{{\rm EC}}(R,E_\gamma,&t)={\sqrt{E_\gamma }\,c\,\sigma _T\,q_0\,n_0\over 12 \pi \,R\,m_ec^2(\gamma _0-1)}\sum ^\infty _{k=1}b_k\int ^R_0dr^{\prime }\,\sin (\lambda _kr^{\prime }) \int ^{R+r^{\prime }}_{R-r^{\prime }}{ds\over s}\,H[c(t-t_0)-s]\nonumber\\ &\times \,H\left[s-c\left(t-t_0-{1\over D_s\gamma _q}\arctan \left({\gamma _0-{E_\gamma \over m_ec^2}\over {E_\gamma \over m_ec^2}{\gamma _0\over \gamma _q}+\gamma _q}\right)\right)\right]\exp \left(-\frac{g(x)\,s}{R}\right)\nonumber\\ &\times \int _{\epsilon _1(E_\gamma,t-s/c)}^{\epsilon _2(E_\gamma)}d\epsilon \,\,\epsilon ^{{3\over 2}}\,{E^e_k(\sqrt{E_\gamma /\epsilon },\gamma _t(\sqrt{E_\gamma /\epsilon },t-s/c))\over \exp (0.1\epsilon /\theta _1)-1}\,{\gamma _q^2+\gamma _t^2(\sqrt{E_\gamma /\epsilon },t-s/c)\over \gamma _q^2+E_\gamma / \epsilon }, \end{eqnarray} \tag{ 39 }$$

when E_γ > m_e c². Here the Heaviside functions yield the integration limits

$$\begin{equation} \epsilon _1(E_\gamma,t-s/c)=E_\gamma \left({\gamma _0\gamma _q-\gamma _q^2\tan (D_s\gamma _q(t-s/c-t_0))\over \gamma _q+\gamma _0\tan (D_s\gamma _q(t-s/c-t_0))}\right)^2, \end{equation} \tag{ 40 }$$

as well as

$$\begin{equation} \epsilon _2(E_\gamma)=m_{e}^{2}c^{4}/E_\gamma . \end{equation} \tag{ 41 }$$

Using Appendix A.2 and a 40-point Gaussian quadrature rule, Equation (39) is finally solved analogous to the previous section. Figure 3 shows that the temporal development of the emergent EC intensity is similar to the case of synchrotron radiation and the time of maximal flare emission, and the flare duration increases with decreasing photon energy E_γ. Furthermore, with increasing disk temperature θ the amplitude of the light curves decreases and the time of maximal flare emission shifts to later times, and the flare duration enlarges (especially at low gamma energies E_γ) due to the spectral dependence (Equation (35)) of the maximal external photon density and the δ - approximation in Equation (36).

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The relativistic protons with an energy E_p = γ m_pc² collide inelastically with the non-relativistic protons of the background plasma of constant density N_b and generate, among other things, neutral and charged ultrarelativistic π-mesons that decay quasi-instantaneously into gamma-rays (l = γ) and neutrinos (l = {ν_e, ν_μ}), respectively, with an energy E_l. The energy losses of the intermediate particles are also negligible, according to the high mass (in reference to the electron mass) and the short mean lifetime of the pions and muons. In the case of the longer-living muons the mean lifetime in the rest frame of the plasmoid yields τ_μ ≃ 2.2γ_μ μs and the energy losses at high Lorentz factors are dominated by synchrotron cooling with the rate $|\dot{\gamma }|_\mu =1.3\times 10^{-9}\,b^2(m_e/m_\mu)^3\,\gamma _\mu ^2\,{\rm s}^{-1}$. Thus, the mean total muon energy loss Γ_μ can be approximated in the first order by

$$\begin{equation} \Gamma _\mu \simeq |\dot{\gamma }|_\mu \,\tau _\mu =3.3\times 10^{-22}\,b^2\,\gamma _\mu ^3\, \end{equation} \tag{ 42 }$$

so that the approach Γ_μ/γ_μ ≃ 0 is even accurate at the highest considered Lorentz factors when b ≪ 100.

Using the energy spectra F(E_l/E_p, E_p) of secondary particles in the energy range E_l ⩾ 100 GeV (Kelner et al. 2006), as well as the cross section of hadronic pion production

$$\begin{equation} \sigma ^{pp}_\pi (E_p)=(34.3+1.88\ln (E_p/ 1{\rm TeV})+0.25\ln ^2(E_p/ 1{\rm TeV}))\;{\rm mb}, \end{equation} \tag{ 43 }$$

the omnidirectional production rate of the particle species l is calculated by

$$\begin{equation} \rho ^{pp}_{l}(r,E_l,t)=\exp (-s/R)\,c\,N_b\int ^{\infty }_{E_l}{dE_p \over E_p}\,\sigma ^{pp}_{\pi }(E_p)\,n(r,E_p,t)\,F(E_l/E_p,E_p), \end{equation} \tag{ 44 }$$

where high-energy photons (x ⩾ 1) and neutrinos have the same escape probability exp (− s/R).

Consequently, the emergent intensity of gamma-rays and neutrinos by inelastic p–p collisions yields

$$\begin{eqnarray} I^{pp}_{l}(R,E_l,t)=&\,\frac{1}{2\,R}\int ^R_0 dr^{\prime }\:r^{\prime } \, \int ^{R+r^{\prime }}_{R-r^{\prime }}ds\: \frac{\rho ^{pp}_{l}\left(r^{\prime },E_l,t-\frac{s}{c}\right)}{s}\nonumber\\ \ms =&\,{c\,N_b\,q_0\over 2R(\gamma _0-1)}\sum _{k=1}^\infty b_k\,\int ^R_0dr^{\prime }\,\sin (\lambda _kr^{\prime })\int ^{R+r^{\prime }}_{R-r^{\prime }}{ds\over s}\,H[c(t-t_0)-s]\nonumber\\ \ms &\times H\left[s-c\left(t-t_0-{\ln (\gamma _0 m_pc^2/E_l)\over D_c}\right)\right]\,\exp (D_c(t-s/c-t_0)-s/R)\nonumber\\ \ms &\times \int _{E_l}^{E_u(t-s/c)}{dE_p\over E_p}\,\sigma ^{pp}_{\pi }(E_p)\,F(E_l/E_p,E_p)\,E^p_k\left({E_p\over m_pc^2},\gamma _r\left({E_p\over m_pc^2},t-{s\over c}\right)\right), \end{eqnarray} \tag{ 45 }$$

with the upper limit of integration

$$\begin{equation} E_u(t-s/c)=\gamma _0\,m_pc^2\,\exp (-D_c(t-s/c-t_0)). \end{equation} \tag{ 46 }$$

According to Appendix A.2 we finally compute the emergent intensity (Equation (45)) like in the previous sections.

Due to the slower cooling mechanism (Equation (6)) of the relativistic protons, the flare duration of gammas and neutrinos considerably increases with a decreasing number density N_b of background protons. Furthermore, the flare duration slightly increase with decreasing E_l, which refers to the energy spectra of the secondary particles, whose maximum is around E_l/E_p ≃ 0.05–0.1, so that the cooling time of relativistic protons increases with decreasing secondary particle energy in order to obtain this quotient for most of the primary particles. However, the time of maximal emission hardly increases with decreasing E_l (see Figure 4). Except for the amplitudes of the emergent intensities there is only a marginal difference in the emission of gammas and neutrinos.

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In order to observe the calculated neutrino flare with the half-completed IceCube 40 detector the muon neutrino flux has to be above $F_{\nu _\mu }=10^{-12}\,\phi ^{90}_{\nu _\mu }\,{\rm TeV}^{-1}\,{\rm cm}^{-2}\,{\rm s}^{-1}$, where the factor $\phi ^{90}_{\nu _\mu }$ has a 90% confidence level and depends on the declination of the neutrino source (Abbasi et al. 2011). Assuming a Doppler factor of δ = 10 δ₁, as well as the distance source-observer of D = 1 D_Mpc Mpc, the intensity of neutrinos with an energy E*_ν = δ E_ν in the observers frame yields I*_ν(E*_ν) = 1.1 × 10⁻¹⁶ δ³₁ R²₁₅ D⁻²_Mpc I_ν(E_ν = E*_ν/δ) at the detector. A crude comparison with the detection limit demonstrates that the maximal muon neutrino intensity $I_{\nu _\mu,{\rm max}}$ has to satisfy the following relation:

$$\begin{equation} I_{\nu _\mu,{\rm max}}\big(E_{\nu _\mu }=E_{\nu _\mu }^*/\delta\big)\ge 9.1\times 10^3\phi ^{90}_{\nu _\mu }\big(E^{*}_{\nu _\mu }/{\rm TeV}\big)\,\delta _1^{-3}\,D_{{\rm Mpc}}^2\,R_{15}^{-2}\,{\rm cm}^{-2}\,{\rm s}^{-1}. \end{equation} \tag{ 47 }$$

Using Figure 4 we reason that N₁₀ = 1 is the minimal background proton density in order to obtain a detectable neutrino flare with the energy $E^*_{\nu _\mu }=1\,{\rm TeV}$ of a source at 1 Mpc distance and optimal declination condition ($\phi ^{90}_{\nu _\mu }\simeq 1$). Furthermore, the emergent neutrino intensity also increases by increasing the injected proton density q₀, according to Equation (45) and the upper flux limit of the recently completed IceCube 86 detector is expected to decrease slightly (Argüelles et al. 2010).

Here we examine the different continuous energy losses of protons with thermal photon and proton fields. Due to the thermal radiation field of the accretion disk with a photon energy θ = 10θ₁ eV, the production of K pions by photohadronic interactions occur only for ultrarelativistic protons with a minimum Lorentz factor of (Mannheim & Schlickeiser 1994)

$$\begin{equation} \gamma ^{\pi }_{{\rm min}}\simeq {K\,m_\pi \,c^2\over 2\,\theta }\left(1+{K\,m_\pi \over 2m_p}\right)\simeq 7.3\times 10^6\,K\,\theta _1^{-1}. \end{equation} \tag{ A1 }$$

Hence, the photohadronic pion production is negligible for protons with a Lorentz factor γ₆ < 7.3 K θ⁻¹₁. The pair production in the field of the relativistic proton has a smaller minimal Lorentz factor of $\gamma ^{e^\pm }_{{\rm min}}\simeq 5.1\times 10^4\,\theta _1^{-1}$ and with the thermal isotropized radiation density of the accretion disk the pair production energy loss rate for protons at threshold yields

$$\begin{equation} |\dot{\gamma }|^{p\gamma }_{e^\pm }\simeq 9.5\times 10^{-9}\,{\delta _1^2\,L_{46}\tau _{-2}\over R_{\rm pc}^2\,\theta _1^2}\,{\rm s}^{-1}. \end{equation} \tag{ A2 }$$

The cross section of inelastic proton–proton interactions (Kelner et al. 2006)

$$\begin{equation} \sigma ^{pp}_\pi (\gamma)=34.3+1.88\ln (\gamma \,(m_pc^2/ 1\;{\rm TeV}))+0.25\ln ^2(\gamma \,(m_pc^2/ 1\;{\rm TeV}))\;{\rm mb} \end{equation} \tag{ A3 }$$

can be approximated by the first summand at a Lorentz factor of γ ≪ 10¹¹ and thus we obtain the energy loss rate

$$\begin{equation} |\dot{\gamma }|^{pp}_\pi =4.9\,c\frac{m_\pi }{m_p}n_b\,H[\gamma -1.30]\,\gamma \,\sigma ^{pp}_\pi (\gamma)\simeq D_c\,H[\gamma -1.30]\,\gamma \,,\quad D_c=7\times 10^{-6}\,N_{10}\,{\rm s}^{-1}\, \end{equation} \tag{ A4 }$$

of proton–proton pion production. We examine only the case of relativistic protons, so that the Heaviside function equals one. Comparing the rate (Equation (A2)) with (Equation (A4)) yields a dominant energy loss by the hadronic pion production at a density N₁₀ > 2.7 × 10⁻⁸ δ²₁ L₄₆τ₋₂ R⁻²_pc θ⁻¹₁. Furthermore, the Coulomb interactions with the fully ionized background plasma are negligible, as its loss rate

$$\begin{equation} |\dot{\gamma }|^{pp}_C\simeq 3.3\times 10^{-6}\,N_{10}\,{\rm s}^{-1}\, \end{equation} \tag{ A5 }$$

is independent of the parameters much smaller than $|\dot{\gamma }|^{pp}_\pi$.

In order to calculate the two-dimensional spatial integral expressions of the emergent intensity (30), (39), and (45), respectively, we have to take the limitation of the integrals by the Heaviside functions into account. Hence, we consider the general Heaviside functions H[A − s] and H[s − B], so that after multiple but simple calculations we obtain

$$\begin{eqnarray} \int ^R_0 dr^{\prime }\,\int ^{R+r^{\prime }}_{R-r^{\prime }}ds\,H[A-s]\,H[s-B] &=& H[A]\biggl (H[A-2R]\,H[2R-B]\biggl [H[R-B]\bigg (H[-B]\int ^R_0dr^{\prime }\int ^{R+r^{\prime }}_{R-r^{\prime }}ds\nonumber\\ &&+H[B]\biggl (\int ^{R-B}_0dr^{\prime }\int ^{R+r^{\prime }}_{R-r^{\prime }}ds+\int ^{R}_{R-B}dr^{\prime }\int ^{R+r^{\prime }}_{B}ds\biggr)\biggr)\nonumber\\ &&+H[B-R]\,H[B]\int ^{R}_{B-R}dr^{\prime }\int ^{R+r^{\prime }}_{B}ds\biggr ]\nonumber\\ &&+H[A-R]\,H[2R-A]\,H[A-B]\biggl [H[R-B]\biggl (H[2R-A-B]\int ^{A-R}_{0}dr^{\prime }\int ^{R+r^{\prime }}_{R-r^{\prime }}ds\nonumber\\ &&+H[A+B-2R]\biggl (\int ^{R-B}_{0}dr^{\prime }\int ^{R+r^{\prime }}_{R-r^{\prime }}ds+\int ^{A-R}_{R-B}dr^{\prime }\int ^{R+r^{\prime }}_{B}ds\biggr)\biggr)\nonumber\\ &&+H[B-R]\,H[A+B-2R]\int ^{A-R}_{B-R}dr^{\prime }\int ^{R+r^{\prime }}_{B}ds\nonumber\\ &&+H[2R-A-B]\biggl (H[-B]\int ^{R}_{A-R}dr^{\prime }\int ^{A}_{R-r^{\prime }}ds\nonumber\\ &&+H[B]\biggl (\int ^{R-B}_{A-R}dr^{\prime }\int ^{A}_{R-r^{\prime }}ds+\int ^{R}_{R-B}dr^{\prime }\int ^{A}_{B}ds\biggr)\biggr)\nonumber\\ &&+H[A+B-2R]\,H[B]\int ^{R}_{A-R}dr^{\prime }\int ^{A}_{B}ds \biggr ]\nonumber\\ &&+H[R-A]H[A-B]\biggl [H[-B]\int ^{R}_{R-A}dr^{\prime }\int ^{A}_{R-r^{\prime }}ds\nonumber\\ &&+H[B]\biggl (\int ^{R-B}_{R-A}dr^{\prime }\int ^{A}_{R-r^{\prime }}ds+\int ^{R}_{R-B}dr^{\prime }\int ^{A}_{B}ds\biggr)\biggr ]\biggr). \end{eqnarray} \tag{ A6 }$$

After specifying the parameters A and B the emergent intensities can be calculated by a numerical algorithm of integration.

There are several different matter and photon fields of the inter- or extragalactic medium, which can be discussed in reference to the attenuation of the emitted gamma-rays; however, only a temporal correlated and varying particle field can change the general temporal development (flare duration, time of maximal flare emission) of the high-energy photon intensity (39) or (45). Consequently, the number of relevant interaction particles reduces to those who are related to the initial pick-up of the relativistic electrons and protons, respectively. In the following, we examine the photon–photon pair attenuation with the low-energy synchrotron photons of Section 3.1. From Dermer & Schlickeiser 1994 the absorption coefficient for an isotropic radiation field n_ph(, r, t) can be approximated by

$$\begin{equation} \alpha _{\gamma \gamma }(E_\gamma,r,t)\simeq {2\over 3}{\,m_ec^2\over E_\gamma }\,\sigma _T\,n_{{\rm ph}}\left(\epsilon =2m_e^2c^4/ E_\gamma, r, t\right), \end{equation} \tag{ A7 }$$

and the optical depth is determined by

$$\begin{equation} \tau _{\gamma \gamma }(E_\gamma,r,t)=\int ^r_R dr^{\prime }\,\alpha _{\gamma \gamma }(E_\gamma,r^{\prime },t-(r^{\prime }-r)/c). \end{equation} \tag{ A8 }$$

According to ESR12, the emission knot is optically thin for gamma-rays independent of the temporal and spatial development of the synchrotron photon density. Here we confirm this result, when we neglect the spatial, temporal and energetic dependence of the synchrotron photon density I_sy(R, = 2m²_ec⁴/E_γ, t)/c and take its maximum I^max_sy/c, so that the maximal optical depth yields

$$\begin{equation} \tau ^{{\rm max}}_{\gamma \gamma }\simeq {2\,\sigma _T\,R\,m_ec^2\over 3\,E_\gamma \,c}\,I^{{\rm max}}_{{\rm sy}}. \end{equation} \tag{ A9 }$$

Based on observations of flaring blazars, we consider a maximal luminosity of L* ≃ 10⁴⁶ erg s⁻¹ at the low synchrotron energies and with a Doppler factor δ = 10 δ₁ the emergent synchrotron intensity in the frame of the emission knot becomes I^max_sy(R, = 2m_e²c⁴/E_γ, t) ≃ 9 × 10²³ (E_γ/1 TeV) R⁻²₁₅ δ⁻³₁ s⁻¹ cm⁻². Hence the optical depth yields

$$\begin{equation} \tau ^{{\rm max}}_{\gamma \gamma }\simeq 0.007\,R_{15}^{-1}\,\delta _1^{-3} \end{equation} \tag{ A10 }$$

and the emission knot is optically thin for gamma-rays independent of its energy. Consequently, there is no relevant systematic influence on the temporal behavior of the emergent gamma-ray intensity.

In the following we account for the non-vanishing neutrino mass, which retards the propagation speed and results in a delay between the photon and the neutrino observation. Due to the large distance between the emission knot and the observer, we have to consider the structure of the four-dimensional space-time. In the case of an homogenous and isotropic expansion of the universe, Einstein's field equations yield the Robertson–Walker metric, so that a particle k with the rest mass m_k and the energy E_k has at a distance z the delay (Visser 2004)

$$\begin{equation} \Delta t\simeq {15.42\,{\rm s}\over H_0/{{\rm km}\over {\rm sMpc}}}\left({m_k c^2\over {\rm eV}}\right)^2\left({{\rm GeV}\over E_k}\right)^2z\left[1-{3+q_0\over 2}z+{12-j_0+8q_0+3q_0^2\over 6}\,z^2+\mathcal {O}(z^3)\right], \end{equation} \tag{ A11 }$$

to a simultaneous produced massless particle. According to the analysis of 307 supernovae type Ia in the redshift range 0.015 ⩽ z ⩽ 1.62 by Guimarães et al. 2009, we assume the Hubble constant H₀ = 72 km s⁻¹ Mpc⁻¹, the deceleration parameter q₀ = −0.57 and the jerk parameter j₀ = −1.

The pion decay generates different neutrino flavors with the mixing ratio ν_e: ν_μ: ν_τ = 1: 2: 0; however, the ratio changes as a result of neutrino oscillation and we obtain ν_e: ν_μ: ν_τ = 1: 1: 1 at a distance much larger than the oscillation length L_osc = 4πℏcE Δm⁻²_ij. Consequently, we have to consider the mean neutrino mass $\bar{m}_\nu =\ssty\sqrt{{(1/3)}m_{\nu _e}^2\,+\,{(1/3)}m_{\nu _\mu }^2\,+\,{(1/3)}m_{\nu _\tau }^2}$. To date there are only upper limits of the three neutrino masses, so that we can just calculate an upper limit of the delay (Equation (A11)). The experimental observed upper limits (Altarelli & Winter 2003) yield the mean neutrino mass $\bar{m}_\nu ^{{\rm exp}}=10.5\,{\rm MeV}/c^2$, whereas theoretical considerations claim a maximal neutrino mass of 2.5 eV/c² independent of the flavor (Fukugita & Yanagida 2003). Figure 6 shows that in the case of the theoretical upper limit the delay is negligible even at large redshifts and small neutrino energies, but in the case of the experimental upper limit the delay (Equation (A11)) can increase to several months or years. Thus, the mean neutrino mass is the crucial parameter and only an improved accuracy of the muon and tau neutrino mass can resolve whether the delay (Equation (A11)) is significant or not. Vice versa, the observation of a delay between a correlated gamma and neutrino flare can be used to determine the mean neutrino mass.

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