Magnetization of jets in luminous blazars

Abstract

The luminosities of many powerful blazars are strongly dominated by γ-rays, which most likely result from Comptonization of radiation produced outside a jet. This observation sets certain constraints on the composition and energetics of the jet, as well as the surrounding quasar environment. We study the dependence of Compton dominance on jet magnetization (the magnetic-to-matter energy flux) and on the location of the ‘blazar zone’. Calculations are performed for two geometries of broad emission-line and hot dust regions: spherical and planar. The jet magnetization corresponding to the large observed Compton dominance is found to be ∼0.1(θ_jΓ)² for spherical geometries and ∼0.01(θ_jΓ)² for planar geometries, where θ_j is the jet half-opening angle and Γ is the jet Lorentz factor. This implies that jets in luminous blazars are matter-dominated and that this domination is particularly strong for the flattened geometry of external radiation sources.

acceleration of particles, radiation mechanisms: non-thermal, quasars: general

1 INTRODUCTION

As indicated by the Compton Gamma-Ray Observatory (CGRO) Energetic Gamma Ray Experiment Telescope (EGRET: von Montigny et al. 1995) and confirmed by the Fermi Large Area Telescope (LAT: Abdo et al. 2010; Ackermann et al. 2011), the apparent luminosities of blazars associated with flat-spectrum radio quasars (FSRQ) are often dominated by γ-rays. For most of them, the ratio of the γ-ray luminosity to the synchrotron luminosity is larger than 4 and in many cases it exceeds 10 (see fig. 22 of Giommi et al. 2012). A dense radiative environment in quasar nuclei strongly favours the external radiation Compton (ERC) mechanism of γ-ray production (see Sikora et al. 2009, and references therein). In such a case, with the assumption of the ‘one-zone’ model, the γ-to-synchrotron luminosity ratio can be approximated by the ratio of the external radiation energy density, |$u_{\rm ext}^{\prime }$|⁠, to the internal magnetic energy density, |$u_{\rm {B}}^{\prime }$|⁠, both as measured in the jet comoving frame. Hence, by combining knowledge about external radiation fields, the kinematics of the jet and the observationally determined Compton dominance, one may estimate the intensity of the magnetic field in the blazar zone and then the flux of magnetic energy L_B. Comparison of L_B with the total jet energy flux, L_j ∼ L_γ/(η_radη_eη_dissΓ²), can then be used to determine the sigma parameter, σ, defined to be the ratio of the magnetic energy flux to the matter energy flux, i.e. σ ≡ L_B/L_kin = (L_B/L_j)/[1 − (L_B/L_j)], where η_diss is the fraction of L_j dissipated in the blazar zone, η_e is the fraction of dissipated energy channelled to accelerate electrons and η_rad is the average radiative efficiency of relativistic electrons (Sikora et al. 2013).

Studies of the σ parameter are important, not only for better understanding of the dynamical structure and evolution of relativistic jets but also because its value determines the dominant particle acceleration mechanism (shock versus reconnection) and its efficiency; therefore, such studies should be performed in a more systematic and complete manner than so far. In particular, one should take into account uncertainties such as the location of the blazar zone and the geometry of the external radiation fields. In both cases, uncertainties are still very large. The blazar zone deduced by some models is located at ∼ 300R_g (see Stern & Poutanen 2011), while in others it can be up to thousands of times farther (see Marscher & Jorstad 2010). Also, the geometry of the broad-line region (BLR) and hot-dust region (HDR) is often considered to be spherical and not stratified, while in reality it may be very flat and significantly radially extended (for the BLR, see Vestergaard, Wilkes & Barthel 2000; Decarli et al. 2008; Decarli, Dotti & Treves 2011; for the HDR, see Wilkes et al. 2013; Roseboom et al. 2013). In this article, we present the results of such studies by mapping theoretical blazar spectral features as a function of distance, geometry of external photon sources and jet magnetization and comparing them with observations.

Our theoretical models of broad-band spectra are constructed using knowledge of the typical parameters of radio-loud quasars: black hole (BH) masses, Eddington ratios and jet powers. We assume strong coupling between protons and electrons, as indicated by particle-in-cell (PIC) simulations (see Sironi & Spitkovsky 2011). Detailed model assumptions are specified and discussed in Section 2. Results of our modelling of blazar spectra and their features (bolometric apparent luminosities, locations of the spectral peaks, Compton dominance, external photon source energy densities), including dependence on distance from the BH for different σ values and different geometries of external radiation fields, are presented in Section 3. These results are discussed and summarized in Section 4.

2 MODEL ASSUMPTIONS

2.1 Dissipation region

The jet is assumed to propagate with a constant bulk Lorentz factor Γ and to diverge conically with a half-opening angle θ_jet = 1/Γ. Energy dissipation and particle acceleration are assumed to take place within a distance range r₁ − r₀ = r₀ and to proceed in the steady-state manner (Sikora et al. 2013). Radiation production is followed up to distance r₂ = 10r₁. The emitting jet volume is divided into s cells, each with the same radial size δr = (r₂ − r₀)/s. The number of cells used in calculations is not a physical model property, but just a parameter responsible for numerical precision. We assumed s = 100, which is a fair compromise between accuracy and calculation time. Assuming uniformity of matter, magnetic fields and external photon fields across the jet and within the cell thickness δr, the emitting jet volume is approximated as a sequence of ‘point sources’ (the real cell emission volume is used only to compute the density of the synchrotron radiation needed to calculate the synchrotron self-Compton (SSC) luminosity). Jet radiation spectra are computed for different r₀ and presented as a function of the parameter r = 1.5r₀.

Fig. 1 presents a sketch of model geometry.

Figure 1.

A sketch of the blazar model geometry used in the calculations.

2.2 Electron acceleration and cooling

We follow the evolution of the electron energy distribution in the region of interest by solving the kinetic equation for relativistic electrons (Moderski, Sikora & Błażejowski 2003), which can be presented in the form

\begin{equation} \frac{{\mathrm{\partial} }N_{\gamma ,i} (r)}{{\mathrm{\partial} }r} = - \frac{{\mathrm{\partial} }}{{\mathrm{\partial} }\gamma } \left( N_{\gamma ,i}(r) \frac{{\rm d} \gamma }{{\rm d} r} \right) + \frac{ Q_{\gamma ,i}(r) }{ c \beta \Gamma } \,, \end{equation}

(1)

where N_{γ, i} is the number of electrons per energy and cell volume, |$\beta = \sqrt{\Gamma ^2-1}/\Gamma$|⁠, dγ/dr = (dγ/dt^′)/(βcΓ) and dγ/dt^′ are the electron energy-loss rates as measured in the jet comoving frame and Q_{γ, i}(r) is the electron injection function, assumed to take the form

\begin{equation} Q_{\gamma ,i}(r) = K_i f(\gamma ) \,, \end{equation}

(2)

where

\begin{equation} f(\gamma ) = \left\lbrace \begin{array}{ll}\gamma _{\rm b}^{p_{1}-p_{2}} \gamma ^{-p_{1}}\qquad & \mbox{for } \gamma _{\rm min} \le \gamma \le \gamma _{\rm b},\\ \gamma ^{-p_{2}}\qquad & \mbox{for } \gamma _{\rm b} \le \gamma \le \gamma _{\rm max}. \end{array} \right. \end{equation}

(3)

Assuming that each electron is involved in the acceleration process, one can relate the normalization of the injection function, K_i, to the jet power using the following formula:

\begin{equation} K_i = \left( \frac{\delta r}{r_{1}-r_{0}} \right) \left( \frac{\eta _{\rm e} \eta _{\rm diss} L_{{\rm j},0}}{(\bar{\gamma }_{\rm inj}-1) m_{\rm e} c^2 \int _{\gamma _{\rm min}}^{\gamma _{\rm max}} f(\gamma ) \, {\rm d} \gamma } \right) \,, \end{equation}

(4)

where

\begin{equation} \bar{\gamma }_{\rm inj} \equiv \int _{\gamma _{\rm min}}^{\gamma _{\rm max}} \gamma f(\gamma ) \, {\rm d} \gamma /\int _{\gamma _{\rm min}}^{\gamma _{\rm max}} f(\gamma )\, {\rm d} \gamma \end{equation}

(5)

is the average electron energy Lorentz factor and L_{j, 0} is the total jet power prior to the dissipation region. Equating this to the expression for an average injected electron energy (Sikora et al. 2013),

\begin{equation} \bar{\gamma }_{\rm inj} = 1+\frac{m_{\rm p}/m_{\rm e}}{n_{\rm e}/n_{\rm p}} \frac{\eta _{\rm e} \eta _{\rm diss}}{(1-\eta _{\rm diss})} (1-1/\Gamma )(1+\sigma ) , \end{equation}

(6)

gives the value of the break energy in the electron spectrum γ_b, where m_e and m_p are the electron and proton masses, respectively, and n_e/n_p is the pair content. For p₁ < 1 (as expected in the case of strong coupling between electrons with protons heated in the dissipation zone), p₂ > 2 (as indicated by Fermi/LAT observations) and γ_min ≪ γ_b ≪ γ_max, γ_b can be of the order of |$\bar{\gamma }_{\rm inj}$|⁠.

Synchrotron, SSC and adiabatic electron energy-loss rates are calculated using the procedure presented by Moderski et al. (2003), but with the energy density of synchrotron radiation now given by the formula

\begin{equation} u_{{\rm syn},i}^{\prime } = \frac{L_{{\rm syn},i}^{\prime }}{2 \pi R \delta r c \Gamma } \,, \end{equation}

(7)

where R = rθ_j. The ERC electron losses are computed using the approximate formula (Moderski et al. 2005)

\begin{equation} \left|\frac{{\rm d}\gamma }{{\rm d}t^{\prime }} \right|_{\rm ERC} = {4 \sigma _{\rm T} \over 3 m_{\rm e} c^2} \frac{ (\beta _{\rm el} \gamma )^2 u_{\rm ext}^{\prime }}{(1+b)^{3/2}} \,, \end{equation}

(8)

where |$b=4 \gamma {\rm h}\nu _{\rm ext}^{\prime }/{\rm m_{\rm e} c^2}$|⁠. For spherical geometry of external sources |$\nu _{\rm ext}^{\prime } = \Gamma \nu _{\rm ext}$|⁠, while for planar geometry |$\nu _{\rm ext}^{\prime } = \Gamma (1-\beta \cos \theta _{\rm m}) \nu _{\rm ext}$|⁠, where ν_ext and |$\nu _{\rm ext}^{\prime }$| are the characteristic external photon frequencies in the external and jet comoving frames, respectively, and θ_m is the angle from which the contribution to the energy density in the jet comoving frame is maximal. Radiation energy densities, |$u_{\rm ext}^{\prime }$|⁠, of external spherical sources are calculated using the approximate formulae given in Appendix A; those of planar sources are calculated using the equations presented by Sikora et al. (2013) in their appendix A.1.

2.3 Radiation

In the jet comoving frame, radiation produced within the ith cell is

\begin{equation} \left( \frac{{\mathrm{\partial} } L_{\nu ^{\prime }}^{\prime }}{{\mathrm{\partial} } \Omega ^{\prime }} \right)_i = \int {\left( \frac{{\mathrm{\partial} } P_{\nu ^{\prime }}^{\prime }(\gamma )}{{\mathrm{\partial} } \Omega ^{\prime }} \right)_i} N_{\gamma ,i} \, {\rm d}\gamma \,, \end{equation}

(9)

where the single electron radiative power, |${\mathrm{\partial} } P_{\nu ^{\prime }}^{\prime }(\gamma )/{\mathrm{\partial} } \Omega ^{\prime }$|⁠, is superposed from synchrotron, SSC and ERC components, the latter consisting of radiation from the accretion disc, broad emission region and hot dust. We assume that a conical jet is transversly uniform, so that evolution of the electron energy distribution N_{γ, i} is the same along all equally normalized θ angles within the jet. Therefore, radiation produced at a given distance from the central black hole can be approximated as being independent of θ.¹ The total apparent luminosity is then a sum of radiation from s cells:

\begin{equation} \nu L_{{\rm app},\nu } = 4 \pi \left(\nu \frac{{\mathrm{\partial} } L_{\nu }}{{\mathrm{\partial} } \Omega } \right) = 4 \pi \frac{{\cal D}^3}{\Gamma } \left( \nu ^{\prime } \frac{{\mathrm{\partial} } L_{\nu ^{\prime }}^{\prime }}{{\mathrm{\partial} } \Omega ^{\prime }} \right) \,, \end{equation}

(10)

where

\begin{equation} \frac{{\mathrm{\partial} } L_{\nu ^{\prime }}^{\prime }}{{\mathrm{\partial} } \Omega ^{\prime }} = \sum _{i=1}^s{ \left( \frac{{\mathrm{\partial} } L_{\nu ^{\prime }}^{\prime }}{{\mathrm{\partial} } \Omega ^{\prime }} \right)_i } \end{equation}

(11)

and |${\cal D} = [\Gamma (1-\beta \cos {\theta _{\rm obs}})]^{-1}$|⁠.

The synchrotron and SSC luminosities are calculated using the procedure presented in Moderski et al. (2003). The ERC luminosities are computed using the equations presented in Appendix B.

3 MODEL PARAMETERS

We compute theoretical blazar spectra as a function of the distance from the central black hole, r, for three different jet magnetization values σ = 1.0, 0.1 and 0.01 and for two geometries of external photon sources. We cover four distance decades, from 10¹⁶–10²⁰ cm. We assume a central black hole mass M_BH = 10⁹ M_⊙, accretion rate |$\dot{M} = 3 L_{\rm Edd}/c^2$| and accretion disc radiative efficiency |$\eta _{\rm d} \equiv L_{\rm d}/\dot{M} c^2 = 0.1$|⁠, which gives a total accretion disc luminosity L_d ≈ 4 × 10⁴⁶ erg s⁻¹. With such a value, the sublimation radius r_sub ≈ 1  pc (see equation A3). Following the results of numerical simulations of magnetically arrested discs (McKinney, Tchekhovskoy & Blandford 2012) and noticing the observed energetics of jets in radio-loud quasars (see Sikora & Begelman 2013, and references therein), we set the jet production efficiency |$\eta _{\rm j} \equiv L_{\rm j}/\dot{M} c^2 = 1$|⁠, which leads to L_{j, 0} ≈ 2 × 10⁴⁷ erg s⁻¹.

The total efficiency of energy dissipation η_diss must be high, because of the high observed γ-ray luminosities in FSRQs, but should not exceed ∼ 0.5, as a substantial part of the jet energy needs to be transported to the radio lobes of Fanaroff–Riley class II (FRII) radio sources associated with radio-loud quasars. We set η_diss = 0.3. We use Γ = 15 (Hovatta et al. 2009) and the jet opening angle θ_j = 1/Γ. Noting the strong coupling between electrons and protons (Sironi & Spitkovsky 2011) indicated by particle-in cell (PIC) simulations of shocks, we assume that the dissipated energy is equally distributed between these particles by setting η_e = 0.5.

The injected electron energy spectrum is assumed to be a broken power law, with spectral indices p₁ = −1 for γ ≤ γ_b and p₂ = 2.5 for γ > γ_b. The choice of a very hard, low-energy injection function is dictated by the aforementioned energetic coupling between electrons and protons, while a much steeper high-energy portion of the electron injection function is required to reproduce the typical slopes of γ-ray spectra observed by Fermi/LAT (Ackermann et al. 2011). The break energy γ_b is calculated using equations (3), (5) and (6).

The detailed parameters used in our modelling are summarized in Table 1.

Table 1.

Parameters used in numerical simulations.

Parameter	Value
Black hole mass MBH	109 M⊙
Accretion rate |$\dot{M}$|	3LEdd/c2
Accretion disc radiative efficiency ηd	0.1
Jet production efficiency ηj	1.0
Energy dissipation efficiency ηdiss	0.3
Jet Lorentz factor Γ	15
Fraction of energy transferred to electrons ηe	0.5
Jet magnetization σ	0.01, 0.1, 1.0
Pair content ne/np	1.0
Electron injection function indices p1, p2	−1.0, 2.5
Min. and max. injection energies γmin, γmax	1, 4 × 104
Jet opening angle θj	1/Γ
Observing angle θobs	1/Γ
BLR photon energy	10 eV
HDR photon energy	0.06–0.6 eV
BLR radius	0.1 rsub
HDR radius	1.0 rsub
BLR covering factor ξBLR	0.1
HDR covering factor ξHDR	0.3

It should be emphasized here that, in contrast to most studies of blazar spectra, we use the magnetization parameter, σ, as an input parameter, instead of magnetic field intensity B^′ or its energy density |$u_{\rm B}^{\prime } = {B^{\prime }}^2/(8\pi )$|⁠. With our input parameters, the value of |$u_{\rm B}^{\prime }$| is determined by the following relation:

\begin{equation} u_{\rm B}^{\prime }= \frac{L_{\rm B}}{\kappa \pi r^2 (\theta _{\rm j} \Gamma )^2 c} = \frac{\sigma }{1+\sigma } \, \frac{(1-\eta _{\rm diss}) L_{{\rm j},0}}{\kappa \pi r^2 (\theta _{\rm j}\Gamma )^2 c}, \end{equation}

(12)

where |$L_{{\rm j},0} = 0.5 \eta _{\rm j} \dot{m} L_{\rm Edd}$| (the value of κ depends on the ratio of the chaotic to the toroidal magnetic component intensity and is enclosed between 4/3 and 2).

4 RESULTS

4.1 Energy densities of external radiation fields

If spherically isotropized at a distance r, the entire disc radiation would have (in a jet comoving frame) an energy density ≃ L_dΓ²/(4πr²c). Hence, this value sets an upper limit for any contribution to |$u_{\rm ext}^{\prime }$| from disc radiation and its fractions ξ_ext reprocessed in the BLR and HDR. We visualize these contributions in Fig. 2 as a function of distance, using the parameter

\begin{equation} \zeta = \frac{4\pi r^2 {\rm c} u_{\rm ext}^{\prime }}{L_{\rm d} \Gamma ^2} \,. \end{equation}

(13)

For BLR and HDR, ζ = g_uξ_ext, where g_u accounts for the geometry of a source and was introduced and preliminarily studied by Sikora et al. (2013) and Nalewajko, Sikora & Begelman (2014b). Fig. 2 presents the ζ parameter for spherical and planar geometries of the BLR and HDR and a geometrically thin accretion disc.

Figure 2.

Values of ζ for spherical and flat geometries of the BLR and HDR and a flat accretion disc. The black bold solid line is the total ζ for the spherical case and the grey bold solid line presents the total ζ for planar geometry.

Close to the black hole, at r < 0.01 pc, the contribution to ζ is dominated by the accretion disc and was considered by Dermer & Schlickeiser (1993) to be the dominant source of seed photons for the ERC process. However, the absence of a bulk Compton feature in the X-ray band indicates that jets at such distances are not yet accelerated enough and therefore radiation produced at such distances is not sufficiently Doppler-boosted to explain very large luminosities of FSRQs (Sikora et al. 2005; Celotti, Ghisellini & Fabian 2007). Furthermore, γ-rays produced too close to the BH would be absorbed by X-rays from the accretion disc corona (see Ghisellini 2012, and references therein).

At larger distances, the accretion disc contribution to |$u_{\rm ext}^{\prime }$| drops quickly and the BLR starts to dominate. For the spherical model of the BLR, its contribution is already comparable to that of the accretion disc at 0.01 pc, while for the planar model that distance is about three times larger. The maximal BLR contribution to ζ in both cases is at r_BLR ∼ 0.1 pc, which corresponds to the distance of maximal bolometric BLR luminosity in the calculated spectra. At r > 0.3 pc, the contribution to ζ starts to be dominated by the HDR and reaches a maximum around 1 pc, just beyond r_sub. Then the value of ζ drops steeply, for the spherical model due to the assumed gradient of the dust radiation luminosity and for the planar model due to the assumed presence of the outer edge of the HDR (see Appendix A).

It is interesting to note that, due to stratification of the BLR and HDR, the value of ζ_BLR + ζ_HDR within a distance range 0.03–3 pc does not vary strongly with distance, oscillating around 0.1 for spherical geometry and around 0.01 for planar geometry.

4.2 Bolometric apparent luminosities

We calculated apparent bolometric luminosities L_bol = ∫L_ν dν for each radiation process separately and their sum, i.e. the total radiative power. Fig. 3 presents their dependence on distance from the central black hole r for three different magnetization parameters and two extreme geometries. Studying bolometric luminosities is a useful tool to investigate radiative efficiency and the dominant radiative mechanism where dissipated energy is deposited. For our models, the total value of dissipated energy channelled to relativistic electrons is L_el ≈ 6 × 10⁴⁸  erg s⁻¹. If the total cooling rate is high enough to cool electrons effectively down to energies γ < γ_b, then the total radiative efficiency is very high, of the order of 70–90 per cent. This means that the total radiation power is of the order of 10⁴⁸ erg s⁻¹, as typically observed in FSRQs with M_BH = 10⁹ M_⊙. Such a high radiative efficiency is achievable up to distances ∼3 pc for spherical models and ∼1 pc for planar models. At larger distances, γ_cool > γ_b (where γ_cool is the electron energy at which the time-scale of radiative energy losses equals the time-scale of adiabatic energy losses) and radiation production efficiency drops quickly. Noting that at such distances ζ_HDR decreases faster than ζ_syn, the dominance of HDR luminosities drops even faster.

Figure 3.

Apparent bolometric luminosities calculated for spherical (left column) and flat (right column) geometry of the BLR and HDR and a flat accretion disc for σ =0.01 (1st row), 0.1 (2nd row) and 1.0 (3rd row). ‘BLR’ and ‘HDR’ stand for ‘ERC(BLR)’ and ‘ERC(HDR)’, respectively. The black solid line corresponds to the total radiative power and the horizontal black dotted line shows the total dissipated power transferred to electrons.

4.3 Spectral peaks

The location of the luminosity peaks of synchrotron and ERC spectral components in the νL_ν representation is presented in Fig. 4. The synchrotron peak is located at |$\nu ^{\rm peak}_{\rm syn} = 3.7 \times 10^{6} \hat{\gamma }^2 B^{\prime } \Gamma$|⁠, where |$B^{\prime }= \sqrt{8\pi u_{\rm B}^{\prime }}$| and |$u_{\rm B}^{\prime }$| is given by equation (12); the ERC peak is at |$\nu ^{\rm peak}_{\rm ERC} = (16/9) \hat{\gamma }^2 \Gamma ^2 \nu _{\rm ext}$|⁠, where

\begin{equation} \hat{\gamma } = \left\lbrace \begin{array}{llll}\gamma _{\rm b} & \textrm {for} & \gamma _{\rm cool} < \gamma _{\rm b} & \textrm {({i.e.} {the fast-cooling regime}),}\\ \gamma _{\rm cool} & \textrm {for} & \gamma _{\rm cool}>\gamma _{\rm b} & \textrm {({i.e.} {the slow-cooling regime}),}\\ \end{array} \right. \end{equation}

(14)

where γ_b is given by equations (3), (5) and (6). As long as electrons with energies γ_b cool efficiently, the synchrotron peak location drops with distance like magnetic field strength, i.e. r⁻¹, from 10¹⁴–10¹⁵ Hz at 10¹⁶ cm to 10¹²–10¹³ Hz at ∼1 pc. At larger distances, the synchrotron peaks are produced by electrons with energies γ_cool and, therefore, their location shifts very fast to higher frequencies, until γ_cool = γ_max at r ∼ 10 pc.

Figure 4.

Location of spectral peaks calculated for synchrotron (left panel) and ERC components (right panel) for different values of σ and for both planar and spherical geometries of BLR and HDR and a flat accretion disc.

The location of ERC luminosity peaks depends on the dominant source of seed photons at a particular distance r and their characteristic energy, hν_ext. The largest values are observed at distances of ERC(BLR) domination and γ-ray peaks result from Compton upscattering of photons with energies ∼10 eV of most luminous spectral lines. At r < 0.03 pc, for planar BLR geometry, we can see the effect of ERC(disc) domination. A much lower – but increasing with decreasing distance – ERC peak location is determined by the temperature of those portions of the accretion disc that contribute most to |$u_{\rm ext}^{\prime }$| at a given distance along the jet. At distances between 0.3 and 3 pc, the γ-ray luminosity peak is set by ERC(HDR), with energies of the seed photons hν_ext ∼ 0.3 eV. At r > 3 pc, we see the same effect as in the case of the synchrotron peak, i.e. the luminosity peak is produced by electrons with energies γ_cool and therefore its location increases with distance very rapidly up to the distance where γ_cool = γ_max and then drops sharply. For all models taken into consideration, |$10^{20}\,{\rm Hz} < \nu ^{\rm peak}_{\rm ERC} < 10^{23}\,$|Hz.

4.4 Compton dominance

The Compton dominance parameter q is defined as the ratio of the ERC to synchrotron peak luminosities, |$(\nu L_{\nu })_{\rm ERC}^{\rm peak}/ (\nu L_{\nu })_{\rm syn}^{\rm peak}$|⁠. Its dependence on distance from the BH for different parameters σ and different external radiation source geometries is shown in Fig. 5. One can easily note that the shape of q(r) is very similar to the shape of ζ(r) (see Fig. 2). This can be explained by using approximate scalings of peak luminosities |$(\nu L_{\nu })_{\rm ERC}^{\rm peak} \propto u_{\rm ext}^{\prime }$| and |$(\nu L_{\nu })_{\rm syn}^{\rm peak} \propto u_{\rm B}^{\prime }$|⁠. Using equations (12) and (13), one can approximate q by the formula

\begin{equation} q \sim \frac{u^{\prime }_{\rm ext}}{u^{\prime }_{\rm B}} = \frac{(1+\sigma )\Gamma ^2 (\theta _{\rm j} \Gamma )^2 \zeta \eta _{\rm d}}{4\sigma \eta _{\rm j}} \,. \end{equation}

(15)

Hence, for fixed values of Γ, θ_j, η_j and η_d, the approximate dependence of q on distance is the same as that of ζ, while its normalization depends on σ. We can see in Fig. 5 that, for planar geometries, values of q > 4, typical for FSRQs, are achievable only for σ < 0.03. In the case of spherical models and r > 0.01 pc, large values of Compton dominance require σ < 0.3. All the spectral features discussed for the whole range of distances and parameters studied are presented in Fig. 6, where examples of computed broad-band spectra are presented.

Figure 5.

Compton dominance parameter q for different external source geometries and values of σ.

Figure 6.

Spectral energy distributions calculated for spherical (left column) and flat (right column) geometry of the BLR and HDR and a flat accretion disc for σ =0.01 (1st row), 0.1 (2nd row) and 1.0 (3rd row). The solid lines correspond to the sum of all radiation components and the grey dash–dotted line shows the radio-loud quasar radiation template (Elvis et al. 1994).

5 DISCUSSION

The most diverse opinions about the nature of active galactic nucleus (AGN) relativistic jets concern their magnetization, σ. As they are initially dominated by the Poynting flux, relativistic jets are thought to be converted at some distance to matter-dominated flows (see Sikora et al. 2005, and references therein). The conversion process can initially proceed quite efficiently, even if the jet is stable and in a steady state (Komissarov et al. 2009; Tchekhovskoy, McKinney & Narayan 2009; Lyubarsky 2010a). However, after σ drops to unity, the conversion process becomes inefficient and magnetization decreases much more slowly with distance, unless supported by processes such as magnetohydrodynamic instabilities, randomization of magnetic fields (Heinz & Begelman 2000), reconnection of magnetic fields (Drenkhahn & Spruit 2002; Lyubarsky 2010b) and/or impulsive modulation of jet production (Lyutikov & Lister 2010; Granot, Komissarov & Spitkovsky 2011). Little is known about the feasibility and efficiency of these processes in the context of AGNs and the only chance to verify the evolution of magnetization in relativistic jets is by investigating their observational properties over different spatial scales. Close to the jet base, such studies can only be performed by analysing the broad-band spectra in blazars.

Results obtained using the ERC models for γ-ray production in FSRQs indicate that jets in the blazar zones of these objects are dominated by an energy flux of cold protons (see Ghisellini et al. 2010, 2014, and references therein). These results have been obtained by recovering the jet parameters and the location of the blazar zone from fits of the observed spectra. However, noticing the very poor coverage of blazar spectra in the far-IR and 10 keV–100 MeV band, where the synchrotron and ERC luminosity peaks in FSRQs are usually located, the quality of such fits is very limited.

In this article, we present the results of modelling blazar spectra with parameters typical for radio-loud quasars (see Table 1) and assuming (suggested by PIC simulations) strong coupling between the electrons and protons heated in shocks (Sironi & Spitkovsky 2011). The blazar spectra were computed as a function of distance for three different values of σ: 1, 0.1 and 0.01. Since the geometry of the BLR and HDR is not well determined, we took into consideration both flat and spherical geometry with radial stratification. Any kind of ‘real’ geometry of the BLR and HDR is thought to be in between those two extreme cases. As a source of seed photons for the ERC process, we also included the accretion disc.

We would like to stress that the model and calculations presented do not use any ‘best-fitting’ values. All parameters used are either observationally determined or predicted by other models and we do not use any model fine-tuning. In fact, equation (15) shows that the only free parameters on which magnetization is dependent are ζ (calculated numerically) and θ_jΓ. Since causality conditions imply θ_jΓ ≤ 1 (Clausen-Brown et al. 2013), while constraints imposed by the synchrotron-self-Compton process (Nalewajko, Begelman & Sikora 2014a) and observations of radio cores (Pushkarev et al. 2009) indicate that θ_jΓ ≥ 0.1, in the calculations presented we adopted a value of θ_jΓ = 1; in the discussion below we show the approximate dependence of magnetization constraints on θ_jΓ using equation (15). Other parameters mentioned in Table 1 have no significant impact on these constraints.

The computed models allowed us to calculate constraints on the jet magnetization as imposed by the spectral features produced at distances within the range 10¹⁶–10²⁰ cm. As typical for FSRQs, the values of Compton dominance, 4 < q < 30, are found to be reproducible for σ ∼ 0.1(θ_jΓ)² in the case of spherical geometries of the BLR and HDR and for σ ∼ 0.01(θ_jΓ)² in case of planar geometries (see Fig. 5 and equation 15). Noting that the real geometries of the BLR and HDR are intermediate between spherical and planar, one may conclude that typical values of Compton dominance imply σ ∼ 0.03(θ_jΓ)². This result indicates that conversion of the initially Poynting-flux-dominated jet to a matter-dominated jet takes place in a region located closer to the BH than the blazar zone.

Our studies also provide interesting constraints on other model parameters, particularly concerning the maximal and minimal distances of the blazar zone location and pair contents. Due to the fast drop of radiation efficiency at large distances, the blazar zone should not be located farther than ∼3 pc (see Section 4.2 and Figs 3 and 6); due to an increase of synchrotron peak frequency with decreasing distance, the typical observed value ν_{syn, peak} ∼ 10¹³ Hz (see Ackermann et al. 2011, fig. 6) is achievable at r > 10¹⁷ cm, unless n_e/n_p ≫ 1 (see the left panel in Fig. 4 and equation 6). In contrast, in the ERC(HDR) dominance region the pair content is required to be negligible in order to keep the location of the ERC spectral peaks within the observationally acceptable range (see the right panel in Fig. 4 and Giommi et al. 2012, fig. 18).

6 CONCLUSIONS

Our main results can be summarized as follows.

• The large |$\nu _{\rm syn}^{\rm peak}$| at distances r < 0.03  pc (Section 3.3 and Fig. 4) and low radiative efficiencies at r > 3  pc (Section 3.2 and Fig. 3) seem to favour the location of the blazar zone in FSRQs being within a distance range 0.03–3  pc; this range is similar to the one obtained by Nalewajko et al. (2014a) using variability and compactness constraints.

• Typical values of the FSRQ Compton dominance parameter, q ∼ 10, can be recovered within a distance range 0.03–3  pc for σ ∼ 0.1(θ_jΓ)² in the case of spherical BLR and HDR and σ ∼ 0.01(θ_jΓ)² for their planar geometries (see Fig. 5 and equation 15).

• Due to the value of γ_b being fixed by the fixed dissipation efficiency η_diss (see Section 2.2) and ν_BLR/ν_HDR ∼ 30, the spectral peak of ERC(HDR) is located at ∼30 times lower energy than the spectral peak of ERC(BLR) (see Section 4.3 and Fig. 4); noting that |$\nu _{\rm HDR}^{\rm peak} \sim (n_{\rm p}/n_{\rm e})^2\,$|MeV, ERC(HDR) models with significant pair content are excluded.

• Noting that observations suggest the real geometries of the BLR and HDR to be significantly flattened (see Section 2), one may conclude that typical values of Compton dominance imply σ ≪ 1 and, therefore, that the conversion of an initially Poynting-flux-dominated jet to a matter-dominated jet takes place in a region located closer to the BH than the blazar zone.

The constraints obtained on jet magnetization can be relaxed, at least quantitatively, if one notes the possibility that jets in blazars are magnetically very inhomogeneous and that most blazar emission takes place in more weakly magnetized sites associated with reconnection layers and/or the jet spine region. Such a possibility was investigated by Nalewajko et al. (2014b), following claims that the value of σ in radio cores is of the order of unity (Zamaninasab et al. 2014); however, this value of σ was found by Zdziarski et al. (2014) to be overestimated.

We thank the referee for critical comments, which helped to improve the article. We acknowledge financial support by the Polish NCN grants DEC-2011/01/B/ST9/04845, DEC-2012/07/N/ST9/04242 and DEC-2013/08/A/ST9/00795, the NSF grant AST-0907872 and the NASA ATP grant NNX09AG02G.

1

A non-zero jet opening angle θ_j implies a spread of Doppler factors for radiation produced by jet elements moving at different angles to the observer's line of sight. It has significant effect on the time profiles of the flares produced (Sikora et al. 2001), yet does not affect the time-averaged spectra much.

APPENDIX A: RADIATION ENERGY DENSITIES OF EXTERNAL SOURCES

A1 Spherical geometry of external sources

In the case of spherical geometry, we assume approximate stratification of BLR emission by two functions, ∂L_BLR/∂lnr∝r for r < r_BLR and ∂L_BLR/∂lnr∝1/r for r > r_BLR, where r_BLR is the distance at which the maximum BLR luminosity is produced. For such a stratification, the energy density of the BLR radiation field on the jet axis in the jet comoving frame is

\begin{equation} u_{\rm BLR}^{\prime } = \frac{\xi _{\rm {BLR}} L_{{\rm d}}}{6 \pi {\rm c} r_{\rm BLR}^2} \times \left\lbrace \begin{array}{lcl}\left( \frac{r}{r_{\rm {BLR}}} \right)^{-1} & \rm {for} & r \le r_{\rm {BLR}},\\ \left( \frac{r}{r_{\rm {BLR}}} \right)^{-3} & \rm {for} & r > r_{\rm {BLR}}, \end{array} \right. \end{equation}

(A1)

where ξ_BLR ≡ L_BLR/L_d.

For the HDR, we assume that ∂L_HDR/∂lnr = 0 for r < r_sub and ∂L_HDR/∂lnr∝1/r for r > r_sub. The jet comoving energy density of such a radiation field is

\begin{equation} u_{\rm HDR}^{\prime } = \frac{\xi _{\rm {HDR}} L_{{\rm d}}}{3 \pi {\rm c} r_{\rm sub}^2} \times \left\lbrace \begin{array}{lcl}1 & \rm {for} & r \le r_{\rm {sub}} ,\\ \left( \frac{r}{r_{\rm {sub}}} \right)^{-3} & \rm {for} & r > r_{\rm {sub}}, \end{array} \right. \end{equation}

(A2)

where ξ_HDR ≡ L_HDR/L_d and

\begin{equation} r_{\rm sub} = 1.6 \times 10^{-5} L_{\rm d}^{1/2} \end{equation}

(A3)

is the graphite sublimation radius (Sikora et al. 2013).

A2 Planar geometry of external sources

The energy density of the radiation field on the jet axis in the jet comoving frame is

\begin{eqnarray} u_{\rm ext}^{\prime } &=& \frac{1}{c} \int {I_{\rm ext}^{\prime } \,{\rm d}\Omega _{\rm ext}^{\prime }} = \frac{1}{c} \int {\frac{I_{\rm ext}}{{\cal D}_{\rm ext}^2} \,{\rm d}\Omega _{\rm ext}} \nonumber\\ &&= \frac{\Gamma ^2}{4\pi c} \int _{R_1}^{R_2} { \frac{(1-\beta \cos \theta _{\rm ext})^2 f_{\rm d}(\theta _{\rm ext})}{r^2 + R^2} \, \frac{{\mathrm{\partial} } L_{\rm ext}}{{\mathrm{\partial} } R}\, {\rm d} R} \,, \end{eqnarray}

(A4)

where

\begin{equation} {\cal D}_{\rm ext} \equiv \nu _{\rm ext}/\nu _{\rm ext}^{\prime } = [\Gamma (1-\beta \cos \theta _{\rm ext})]^{-1} \,, \end{equation}

(A5)

\begin{equation} {\rm d} \theta _{\rm ext} = \frac{\cos \theta _{\rm ext} \,{\rm d} R}{\sqrt{r^2+R^2}} = \frac{r \,{\rm d} R}{r^2 + R^2} \,, \end{equation}

(A6)

\begin{equation} {\rm d} \Omega _{\rm ext} = \sin \theta _{\rm ext} \, {\rm d}\theta _{\rm ext}\, {\rm d}\phi _{\rm ext} = \frac{r R \, {\rm d}R \, {\rm d}\phi _{\rm ext}}{(r^2 + R^2)^{3/2}} \,. \end{equation}

(A7)

For optically thick sources (the case of the accretion disc), f_d(θ_ext) = 2cos θ_ext, while for optically thin sources (assumed to be the case of the BLR and HDR) f_d = 1. The accretion disc is assumed to extend from R₁ = R_g (where R_g is the gravitational radius) to R₂ = r_sub, the BLR from R₁ = 0.1r_sub to R₂ = r_sub and the HDR from R₁ = r_sub to R₂ = 10r_sub (Sikora et al. 2013).

APPENDIX B: ERC LUMINOSITIES

B1 Spherical geometry of external sources

The ERC luminosity for a spherical external radiation field is (see equation 15 in Moderski et al. 2005)

\begin{equation} \nu ^{\prime } \frac{{\mathrm{\partial} } L_{\nu ^{\prime }}^{\prime }}{{\mathrm{\partial} } \Omega ^{\prime }} = {3 {\rm c} \sigma _T \over 16\pi } u_{\rm {\rm ext}}^{\prime } \left( \nu ^{\prime } \over \nu _{\rm ext} \right)^2 \int {\frac{N_{\gamma }}{\gamma ^2} \, f_{\rm sc} \, {\rm d} \gamma } \,, \end{equation}

(B1)

where |$u_{\rm ext}^{\prime }$| for the BLR and HDR are given in our Appendix A, while function f_sc is specified by equation (A3) in Moderski et al. (2005).

B2 Planar geometry of external sources

We adopt the formalism presented in Dermer & Schlickeiser (2002) and Dermer et al. (2009) to calculate the ERC radiation for seed photons coming from a flat accretion disc. This work extends it to include seed photons from planar BLR and HDR.

The ERC luminosity produced at a distance r by Comptonization of radiation reaching a jet from different directions is

\begin{eqnarray} \nu ^{\prime } \frac{{\mathrm{\partial} } L_{\nu ^{\prime }}^{\prime }}{{\mathrm{\partial} } \Omega ^{\prime }} ={3\sigma _T \over 16\pi } {\nu ^{\prime }}^2 \int _{4\pi } \int \frac{I_{\nu _{\rm ext}}}{{\nu _{\rm ext}}^2} \, \left[ \int \frac{N_{\gamma }}{\gamma ^2} \,f_{\rm sc} \,{\rm d}\gamma \right]{\rm d}\nu _{\rm ext} \,{\rm d}\Omega _{\rm ext} \,,\nonumber\\ \end{eqnarray}

(B2)

where dΩ_ext = dcos θ_ext dϕ_ext and θ_ext and ϕ_ext are the polar and azimuthal angles of the rays approaching a jet at a given distance r determined in the external (BH) frame. This equation comes from the generalization of equation (15) in Moderski et al. (2005) to include external radiation entering a jet comoving frame from all directions and using the Lorentz invariant |$I_{\nu _{\rm ext}} \,{\rm d}\nu _{\rm ext} \,{\rm d}\Omega _{\rm ext} /\nu _{\rm ext}^2$|⁠. For planar external sources with dR rings approximated to radiate mono-energetically but with ν_ext being, in general, dependent on R, the above expression for ERC luminosities takes the form

\begin{eqnarray} \nu ^{\prime } \frac{{\mathrm{\partial} } L_{\nu ^{\prime }}^{\prime }}{{\mathrm{\partial} } \Omega ^{\prime }} &=& {3\sigma _T \over 8} {\nu ^{\prime }}^2 \int _{R_1}^{R_2} \frac{I_{\rm ext}}{(\nu _{\rm ext})^2} \, \frac{rR}{(r^2+R^2)^{3/2}} \nonumber\\ &&\times \left[\int _{\gamma _{\rm m}}^{\gamma _{\rm max}} \frac{N_{\gamma }}{\gamma ^2}\,f_{\rm sc} \,{\rm d}\gamma \right] \, {\rm d}R \,, \end{eqnarray}

(B3)

where

\begin{equation} I_{\rm ext} = \frac{f_{\rm d}(\theta _{\rm ext})}{8\pi ^2 R \cos \theta _{\rm ext}} \frac{{\mathrm{\partial} } L_{\rm ext}}{{\mathrm{\partial} } R} \, \end{equation}

(B4)

and luminosities ∂L_ext/∂R are specified by Sikora et al. (2013) in their Appendix A.2.