RECONCILING MODELS OF LUMINOUS BLAZARS WITH MAGNETIC FLUXES DETERMINED BY RADIO CORE-SHIFT MEASUREMENTS

Relativistic jets of active galactic nuclei (AGNs) are launched by spinning black holes (BHs) or accretion disks (Blandford & Znajek 1977; Blandford & Payne 1982). In radio galaxies and radio-loud quasars, they are relativistic and reach powers comparable to or sometimes even exceeding the accretion powers (Rawlings & Saunders 1991; Ghisellini et al. 2011; Punsly 2011). Launching such powerful jets requires magnetic fluxes that cannot be developed by dynamo mechanisms in standard, radiation-dominated accretion disks (Ghosh & Abramowicz 1997). However, such fluxes are expected in the magnetically arrested disk (MAD) scenario (Narayan et al. 2003; McKinney et al. 2012, and references therein). In this case, jets are most likely produced by the Blandford–Znajek mechanism, and the required strong net magnetic flux is expected to be accumulated onto the BH by the advection of magnetic fields from external regions (Sikora & Begelman 2013, and references therein). Starting as Poynting-flux-dominated outflows, these jets are smoothly accelerated as they convert energy from magnetic to kinetic, and their magnetization σ = B'²/(4πw), where w is the relativistic enthalpy density, drops with distance (Komissarov et al. 2007; Tchekhovskoy et al. 2009, and references therein). Jet acceleration becomes inefficient when σ ≲ 1, at which point, in the ideal magnetohydrodynamical picture, σ decreases logarithmically with distance (Lyubarsky 2010). This transition between the two acceleration regimes may already occur at a distance of 10³ R_g  ∼  0.05 pc (here R_g = GM_bh/c² is the gravitational radius of the BH of mass M_bh  ∼  10⁹ M_☉; Komissarov et al. 2007).

AGN jets produce large amounts of non-thermal radiation that is relativistically enhanced by two to three orders of magnitude in blazars where the jets are oriented close to the line of sight (Blandford & Königl 1979). This radiation is generally thought to be produced at a distance scale of 0.01 pc  <  r  <  10 pc, but in many cases it can be narrowed to 0.1 pc  <  r  <  1 pc (see Nalewajko et al. 2014 for a recent discussion). This main emission and dissipation region is referred to as the blazar zone. Modeling of the spectral energy distributions (SEDs) of blazars can be used to constrain the jet composition in the blazar zone (e.g., Sikora & Madejski 2000). The most luminous blazars, belonging to the class of flat-spectrum radio quasars (FSRQs), are strongly dominated by gamma-ray emission, which is thought to be produced by the external radiation-Comptonization (ERC) mechanism (Dermer et al. 1992; Sikora et al. 1994, 2009), the efficiency of which depends on the energy density of the external radiation fields, mainly the broad emission lines (BELs) and the thermal radiation of the dusty torus. The very high apparent gamma-ray luminosities of FSRQs, at times exceeding L_γ  ∼  10⁴⁸ erg s⁻¹ (Abdo et al. 2011), call for very high total jet powers L_j  ∼  10^46–47 erg s⁻¹ and high radiative efficiencies $\eta = L_\gamma /(\Gamma _{\rm j}^2L_{\rm j})\,{\sim}\,0.1$ (Nemmen et al. 2012), where Γ_j  ∼  20 is the jet Lorentz factor. On the other hand, most of the infrared (IR)/optical emission of blazars is due to the synchrotron mechanism, the efficiency of which depends on the local magnetic field strength. Observations of blazars where the ERC component is highly dominant over synchrotron emission place strong constraints on σ within the blazar zone, which we discuss in Section 2.

Recently, a magnetic field strength scaled to the distance of 1 pc was estimated for a large sample of blazars and radio galaxies by using the core-shift technique (Pushkarev et al. 2012). In this technique, the position of the radio core, assumed to be a photosphere due to the synchrotron self-absorption process (Blandford & Königl 1979), is measured in relation to sharp optically thin jet features as a function of observing frequency (Lobanov 1998). These magnetic field values were used to estimate the magnetic fluxes of jets Φ_j, which were compared to the theoretical magnetic fluxes threading the BHs Φ_bh as predicted by the MAD scenario (Zamaninasab et al. 2014). The close agreement between Φ_j and Φ_bh strongly supports the MAD scenario for the production of powerful AGN jets. In Section 3, we show that this is equivalent to the relation L_B  ∼  L_d between the magnetic jet power and accretion disk luminosity.

We identified a possible tension between the magnetic field strengths estimated from core-shift measurements and the magnetic field strengths estimated from modeling the emission of the most Compton-dominated FSRQs. The latter tend to be lower by a factor of ∼3; therefore, in Section 4, we consider dissipation sites that involve lower than average local magnetic field strengths: (1) magnetic reconnection layers and (2) weakly magnetized jet spines. We also emphasize the importance of the geometric distribution of external radiation sources, in particular, that a flat geometry of the broad-line region (BLR) and/or the dusty torus makes the problem much worse. Our main results are summarized in Section 5.

The SEDs of FSRQs are strongly dominated by the high-energy component peaking in the 10–100 MeV range, which is most naturally explained by the ERC model (Sikora et al. 2009). We define the Compton dominance parameter as q = L_ERC/L_syn, where L_ERC and L_syn are the apparent luminosities of the ERC and synchrotron components, respectively, at their spectral peaks. Numerous observations indicate that quite often q ≳ 10 for the brightest blazars (Abdo et al. 2010; Arshakian et al. 2012; Giommi et al. 2012).

On the other hand, if the ERC and synchrotron components are produced by the same population of electrons,⁷ then we can write $q \simeq u_{\rm ext}^{\prime } / u_{\rm B}^{\prime }$, where $u_{\rm ext}^{\prime }$ and $u_{\rm B}^{\prime }$ are the energy densities of the external radiation field and the magnetic field in the jet comoving frame, respectively. The external radiation density can be parameterized as $u_{\rm ext}^{\prime } = \zeta \Gamma ^2L_{\rm d}/(4\pi cr^2)$, where ζ is a dimensionless parameter representing the details of reprocessing and beaming of the external radiation (see below), Γ is the Lorentz factor of the emitting region, L_d is the accretion disk luminosity, and r is the distance of the emitting region from the supermassive BH. The magnetic energy density can be related to the jet magnetic power $L_{\rm B} = 2\pi R^2\Gamma ^2u_{\rm B}^{\prime }c$, where R = θ_jr is the jet radius and θ_j is the half-opening angle of the jet. Gathering these relations together, we obtain the following constraint:

$$\begin{equation} q = \left(\frac{\zeta }{0.005}\right)\left(\frac{\Gamma }{20}\right)^2(\Gamma \theta _{\rm j})^2\left(\frac{L_{\rm d}}{L_{\rm B}}\right)\,. \end{equation} \tag{ 1 }$$

Written in such a form, the above equation suggests the typical parameter values that we adopt as the starting point for further discussion.

The parameter ζ = ξg_u includes the traditional covering factor ξ and the geometric factor g_u (Sikora et al. 2013). The covering factor determines the total luminosity of the reprocessed accretion disk radiation, e.g., L_BLR = ξ_BLRL_d. Typically, it is assumed that ξ ≃ 0.1, although there are many indications that it can be as high as ξ  ∼  0.4 for both the BLR (Dunn et al. 2007; Gaskell 2009) and the dusty tori (Roseboom et al. 2013; Wilkes et al. 2013). The geometric factor depends on the geometric distribution of the reprocessing medium and on the radial stratification of the covering factor. As we demonstrate in the Appendix, for a spherical distribution g_u  <  0.7 and for flattened distributions g_u  <  0.1. Recently, there has been increasing interest in flattened distributions of the BLR (Tavecchio & Ghisellini 2012), motivated mainly by observations of rapidly variable very high energy emission from quasars (Aleksić et al. 2011) and supported by direct observations (Vestergaard et al. 2000; Decarli et al. 2011). The half-opening angle of the dusty tori is estimated at ∼30° (Wilkes et al. 2013). Assuming that g_u = ξ = 0.1, we expect that ζ can be as low as 0.01. However, in the case of a quasi-spherical reprocessing medium with high covering factor, we may expect ξ ≃ 0.4 and g_u ≃ 0.5, and hence ζ ≃ 0.2. High values of q may thus require the presence of a dense, quasi-spherical medium reprocessing the central AGN radiation.

The Lorentz factors Γ of blazar jets can be estimated from interferometric observations of apparent superluminal motions of radio features. Typical values for FSRQs are 10  <  Γ  <  40 (Hovatta et al. 2009). The jet collimation parameter Γθ_j should not exceed unity on both theoretical (Komissarov et al. 2009) and observational (Jorstad et al. 2005; Pushkarev et al. 2009) grounds. Therefore, it is very unlikely that we could obtain q > 10 by increasing either the Lorentz factor or the collimation parameter.

Finally, the parameter q can be increased by decreasing the magnetic jet power so that L_B  <  L_d. If the jets are significantly magnetized, with σ ≃ L_B/(L_j − L_B) > 1, then we would expect that L_B ≲ L_j, where L_j is the total jet power. Observational evidence suggests that for the most powerful jets $L_{\rm j}\,{\sim}\,\dot{M}c^2 > L_{\rm d}$ (see Section 1). This would also be consistent with the MAD scenario, in which it was demonstrated numerically that $L_{\rm j} \gtrsim \dot{M}c^2$ (Tchekhovskoy et al. 2011). As we show in the next section, the requirement that L_B  ∼  L_d is equivalent to the relation between the two magnetic fluxes Φ_j  ∼  Φ_bh (Zamaninasab et al. 2014); therefore, increasing q by decreasing L_B globally means a departure from the MAD scenario (in addition to departing from the core-shift measurements). However, one can still consider a local decrease in the magnetic field strength in order to obtain a high q (see Section 4).

In this section, we analyze the sample of blazars compiled by Zamaninasab et al. (2014), for which magnetic field estimates $B_{\rm 1pc}^{\prime }$ from core-shift measurements are available (Pushkarev et al. 2012), as well as accretion disk luminosities L_d and BH masses M_bh.

First, we estimate the magnetic jet power as $L_{\rm B} \simeq (c/4)(1\,{\rm pc})^2B_{\rm 1pc}^{\prime 2}(\Gamma \theta _{\rm j})^2$. In Figure 1, we show the distribution of L_B versus L_d for the case of Γθ_j = 1. We note a substantial scatter in the L_B values, most of them falling in the range 0.2  <  L_B/L_d  <  20. The sources with L_B  <  L_d may have q > 1, according to Equation (1). However, very few sources in this sample can have q > 10 solely due to the low value of L_B/L_d. Since the magnetic jet power is a steep function of the jet collimation parameter Γθ_j, allowing for Γθ_j  <  1 can substantially reduce L_B. However, since q∝(Γθ_j)²/L_B (Equation (1)), the Compton dominance would not be affected by adopting a different value of Γθ_j.

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The correlation between L_B and L_d is much worse that the correlation between the two magnetic fluxes Φ_j and Φ_bh identified by Zamaninasab et al. (2014). Those magnetic fluxes can be written as

$$\begin{eqnarray} \Phi _{\rm j} &\simeq & 8\pi (\Gamma \theta _{\rm j})f(a)R_{\rm g}B_{\rm 1pc}^{\prime }(1\,{\rm pc}) \propto (\Gamma \theta _{\rm j})f(a)L_{\rm B}^{1/2}M_{\rm bh}\,, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&\Phi _{\rm bh} \simeq 50R_{\rm g}\left(\frac{L_{\rm d}}{\eta c}\right)^{1/2} \propto L_{\rm d}^{1/2}M_{\rm bh}, \end{eqnarray} \tag{ 3 }$$

where η is the radiative efficiency of the accretion disk, f(a) = [1 + (1 − a²)^1/2]/a, and a is the dimensionless BH spin. The very good correlation between the magnetic fluxes for Γθ_j = 1, a = 1, and η = 0.4 can be partially explained by the fact that both fluxes are proportional to the BH mass M_bh. Because of the wide range of M_bh (about three orders of magnitude, 10⁷–10¹⁰ M_☉), the relatively poor correlation between L_B and L_d is efficiently stretched along the lines of constant L_B/L_d. Also, since Φ_j/Φ_bh ≃ (L_B/L_d)^1/2, the scatter between the Φ_j/Φ_bh values is smaller than the scatter between the L_B/L_d values.

In Figure 2, we show the relation between L_B/L_Edd and L_d/L_Edd, where L_Edd = 1.6 × 10³⁸(M_bh/M_☉) erg s⁻¹ is the Eddington luminosity. We note that the blazars in the sample compiled by Zamaninasab et al. (2014) occupy a narrow range of Eddington luminosity ratios, with 0.1 ≲ L_d/L_Edd ≲ 2. All sources in the sample must have prominent BELs in order to calculate both L_d and M_bh. Because of this selection effect, we effectively obtain L_d∝M_bh, L_B∝M_bh, and the magnetic fluxes scale as $\Phi _{\rm j} \simeq \Phi _{\rm bh} \propto M_{\rm bh}^{3/2}$.

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We consider two potential mechanisms for obtaining reduced local magnetic field strengths in jets with a typical magnetization of σ  ∼  1: one associated with reconnection layers and one related to the radial structure of magnetic fields across the jets. Magnetic reconnection events are likely to be triggered in mildly relativistic turbulent plasma which is expected to be driven by current-driven instabilities (Begelman 1998). Meanwhile, stratification of the toroidal magnetic component across the jet may result from balancing the magnetic stresses by the pressure of protons heated, e.g., by internal shocks.

4.1. Reconnection Layers

4.2. Central Core/Spine

Magnetic fluxes Φ_j derived by measurements of radio core shifts in blazars (Pushkarev et al. 2012) are consistent with the maximum magnetic fluxes Φ_bh predicted by the MAD model to thread the BH (Zamaninasab et al. 2014). As we have shown in Section 3, this is equivalent to the statement that magnetic jet power L_B is comparable to the accretion disk luminosity L_d, which for total jet power L_j  ∼  L_d implies typical jet magnetizations of σ  ∼  1. However, as we demonstrated in Section 2, even in the case of a geometrically thick distribution of external radiation sources, significantly lower magnetization values are required by radiation models of FSRQs in order to reproduce the high ratios q of gamma-ray to synchrotron luminosities.

This inconsistency can be resolved by noting that blazar jets need not be magnetically homogeneous and uniform across the jet as commonly assumed. As discussed in Section 4, in realistic jet models there may exist regions with lower magnetization. They can be generated by reconnection driven in mildly relativistic turbulence. They may also be associated with jet cores filled with hot protons heated by internal shocks. High values of q are achievable in both cases, and the energy of the gamma-ray luminosity peaks can be reproduced—in the shock scenario with a proton–electron plasma and in the reconnection scenario with significant pair content.

We also tentatively considered the possibility that the jet magnetic fields obtained from the radio core-shift measurements are overestimated. This could be the case if the radio cores are not photospheres due to the synchrotron self-absorption process, but rather they are due to a low-energy break in the electron distribution function. This idea will be developed in a future study.

We thank the reviewers and Andrzej Zdziarski for helpful comments on the manuscript. M.S. thanks the JILA Fellows for their hospitality during the early stages of this project. This project was partly supported by the NASA Fermi Guest Investigator program, NASA Astrophysics Theory Program grant NNX14AB375, and Polish NCN grant DEC-2011/01/B/ST9/04845. K.N. was supported by NASA through Einstein Postdoctoral Fellowship grant number PF3-140130 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060.

In Section 2, we parameterize the energy density of the external radiation fields using a geometrical correction factor g_u that was first introduced in Sikora et al. (2013). Given a particular geometrical model of the distribution of the medium producing the external radiation, we can integrate the external energy density $u_{\rm ext}^{\prime }(r)$ along the jet in the comoving frame, taking into account the exact distance and beaming factor for each volume element of the medium. Then, we calculate $g_u = 4\pi cr^2u_{\rm ext}^{\prime }/(\xi \Gamma ^2L_{\rm d})$.

Here, we adopt a specific geometry for the reprocessing medium (either BLR or the dusty torus), presented in Appendix A of Nalewajko et al. (2014). The optical depth gradient dτ/dr is assumed to scale as r⁻² for the BLR, and roughly as r⁻¹ for the dusty torus. We also adopt the covering factors ξ_BLR = ξ_IR = 0.1, and typical values for the inner radii r_BLR and r_IR of the BLR and the torus, respectively, from Sikora et al. (2009). The main variable is the half-opening angle α_max of the medium measured from the accretion disk plane.

In Figure 3, we show the functions g_u(r) for several values of α_max. We find that close to the characteristic radii g_u ≃ 0.04 for α_max = 10°, g_u ≃ 0.08 for α_max = 45°, and g_u ≃ 0.2 for α_max = 75°. This indicates that widely different geometries of the reprocessing medium may change g_u, and thus q, by a factor ∼5.

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