On the Jet Properties of γ-Ray-loud Active Galactic Nuclei

In the case of Thomson scattering, the SSC peak frequency follows

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{\mathrm{ssc}}^{{\prime} }}{{\nu }_{\mathrm{sy}}^{{\prime} }}=\displaystyle \frac{4}{3}{\gamma }_{0}^{2}.\end{eqnarray} \tag{ 8 }$$

Integrating the monochromatic luminosity (see Equations (4) and (5)), one can get the total synchrotron luminosity,

$$\begin{eqnarray}&&{L}_{\mathrm{sy}}^{{\prime} }={\nu }_{\mathrm{sy}}^{{\prime} p}{L}_{\mathrm{sy}}^{{\prime} }({\nu }_{\mathrm{sy}}^{{\prime} p})\sqrt{\displaystyle \frac{\pi \mathrm{ln}10}{b/4}}.\end{eqnarray} \tag{ 9 }$$

Within the one-zone model, the average synchrotron energy density is (see Chen 2017)

$$\begin{eqnarray}&&{U}_{\mathrm{sy}-{Th}}=\displaystyle \frac{9}{4}\displaystyle \frac{{L}_{\mathrm{sy}}^{{\prime} }}{4\pi {R}^{2}c}.\end{eqnarray} \tag{ 10 }$$

Therefore, the SSC peak luminosity follows

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{\mathrm{ssc}}^{{\prime} p}L^{\prime} ({\nu }_{\mathrm{ssc}}^{{\prime} p})}{{\nu }_{\mathrm{sy}}^{{\prime} p}L^{\prime} ({\nu }_{\mathrm{sy}}^{{\prime} p})}=\displaystyle \frac{{U}_{\mathrm{sy}-{Th}}}{{U}_{B}}=\displaystyle \frac{{\sigma }_{T}}{2}\sqrt{\displaystyle \frac{\pi \mathrm{ln}10}{b/4}}{{RN}}_{0}{\gamma }_{0}^{3}.\end{eqnarray} \tag{ 11 }$$

Combining Equations (6), (9), (8), (11), and

$$\begin{eqnarray}&&\left\{\begin{array}{l}\nu =\delta \nu ^{\prime} \\ \nu L(\nu )={\delta }^{4}\nu ^{\prime} L^{\prime} (\nu ^{\prime} )\\ R=c{\rm{\Delta }}t\delta /(1+z)\end{array}\right.,\end{eqnarray} \tag{ 12 }$$

we have

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\gamma }_{0}=8660.3{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}}{{10}^{23}\mathrm{Hz}}\tfrac{{10}^{15}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{1/2}\\ \delta =9.0365{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{45}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/4}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{15}\mathrm{Hz}}\right)}^{-1/2}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{-1/4}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}}{{10}^{23}\mathrm{Hz}}\tfrac{{10}^{15}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{1/2}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{-1/2}{\left(\tfrac{b}{2}\right)}^{-1/8}\\ B=0.39533{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{45}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-1/4}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{15}\mathrm{Hz}}\right)}^{3/2}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{1/4}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}}{{10}^{23}\mathrm{Hz}}\tfrac{{10}^{15}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{-3/2}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{1/2}{\left(\tfrac{b}{2}\right)}^{1/8}\\ R=2.3406\times {10}^{16}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{45}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/4}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{15}\mathrm{Hz}}\right)}^{-1/2}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{-1/4}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}}{{10}^{23}\mathrm{Hz}}\tfrac{{10}^{15}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{1/2}{\left(\tfrac{{\rm{\Delta }}t}{1\mathrm{day}{(1+z)}^{3}}\right)}^{1/2}{\left(\tfrac{b}{2}\right)}^{-1/8}\\ {N}_{0}=5.1991\times {10}^{-5}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{45}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-1/4}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{15}\mathrm{Hz}}\right)}^{1/2}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{5/4}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}}{{10}^{23}\mathrm{Hz}}\tfrac{{10}^{15}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{-2}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{-1/2}{\left(\tfrac{b}{2}\right)}^{5/8}\end{array}\right..\end{eqnarray} \tag{ 13 }$$

In the case of KN scattering (${\gamma }_{0}h{\nu }_{\mathrm{sy}}^{{\prime} p}\gtrsim {m}_{e}{c}^{2}$), the SSC peak frequency follows

$$\begin{eqnarray}&&{\nu }_{\mathrm{ssc}}^{{\prime} p}\approx \displaystyle \frac{2}{\sqrt{3}}\displaystyle \frac{{\gamma }_{0}{m}_{e}{c}^{2}}{h}.\end{eqnarray} \tag{ 14 }$$

The effective SSC seed photon energy density can be easily derived through integrating from the minimum frequency to the critical frequency ${\nu }_{\mathrm{sy}-c}^{{\prime} }=(\sqrt{3}/2)({m}_{e}{c}^{2}/h{\gamma }_{0})$,

$$\begin{eqnarray}&&{U}_{\mathrm{sy}-\mathrm{KN}}={U}_{\mathrm{sy}-{Th}}\displaystyle \frac{1}{\sqrt{\pi }}{\int }_{{t}_{c}}^{+\infty }{e}^{-{t}^{2}}{dt},\end{eqnarray} \tag{ 15 }$$

where

$$\begin{eqnarray}&&{t}_{c}=\sqrt{\displaystyle \frac{b}{4\mathrm{ln}10}}\mathrm{ln}\left(\displaystyle \frac{h{\nu }_{\mathrm{sy}}^{{\prime} p}}{{m}_{e}{c}^{2}}\displaystyle \frac{h{\nu }_{\mathrm{ssc}}^{{\prime} p}}{{m}_{e}{c}^{2}}\right).\end{eqnarray} \tag{ 16 }$$

In the case of the KN regime, the SSC peak luminosity follows

$$\begin{eqnarray}\displaystyle \frac{{\nu }_{\mathrm{ssc}}^{{\prime} p}L^{\prime} ({\nu }_{\mathrm{ssc}}^{{\prime} p})}{{\nu }_{\mathrm{sy}}^{{\prime} p}L^{\prime} ({\nu }_{\mathrm{sy}}^{{\prime} p})} & = & \displaystyle \frac{{U}_{\mathrm{sy}-\mathrm{KN}}}{{U}_{B}}\\ & = & \displaystyle \frac{{\sigma }_{T}}{2}\sqrt{\displaystyle \frac{\pi \mathrm{ln}10}{b/4}}{{RN}}_{0}{\gamma }_{0}^{3}\displaystyle \frac{1}{\sqrt{\pi }}{\displaystyle \int }_{{t}_{c}}^{+\infty }{e}^{-{t}^{2}}{dt}.\end{eqnarray} \tag{ 17 }$$

Because Equation (17) is actually an integral equation (t_c is a function of δ through ${\nu }_{\mathrm{sy},\mathrm{ssc}}^{{\prime} p}={\nu }_{\mathrm{sy},\mathrm{ssc}}^{p}/\delta$), the final estimated jet parameters cannot be analytically expressed as in Equation (13). The parameters (γ₀, N₀, B, δ, and R) can be derived through numerically solving the set of Equations (6), (9), (12), (14), and (17).

The FSRQs usually have very strong emission from the BLR and dust torus, and these photons can be IC scattered by relativistic electrons in the jet. In the frame of the jet, the external photon energy density will be enhanced and the frequency will be amplified: ${U}_{\mathrm{ext} \mbox{-} \mathrm{Th}}^{{\prime} }\approx (17/12){{\rm{\Gamma }}}^{2}{U}_{\mathrm{ext}}$ and ${\nu }_{\mathrm{ext}}^{{\prime} }\approx {\rm{\Gamma }}{\nu }_{\mathrm{ext}}$. In this case, the EC peak luminosity follows

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{{\rm{e}}{\rm{c}}}^{{\rm{{\prime} }}p}{L}^{{\rm{{\prime} }}}({\nu }_{{\rm{e}}{\rm{c}}}^{{\rm{{\prime} }}p})}{{\nu }_{{\rm{s}}{\rm{y}}}^{{\rm{{\prime} }}p}{L}^{{\rm{{\prime} }}}({\nu }_{{\rm{s}}{\rm{y}}}^{{\rm{{\prime} }}p})}=\displaystyle \frac{{U}_{{\rm{e}}{\rm{x}}{\rm{t}}{\textstyle \text{-}}{\rm{T}}{\rm{h}}}^{{\rm{{\prime} }}}}{{U}_{B}}\approx \displaystyle \frac{{U}_{{\rm{e}}{\rm{x}}{\rm{t}}}}{{U}_{B}}\displaystyle \frac{17}{12}{{\rm{\Gamma }}}^{2},\end{eqnarray} \tag{ 18 }$$

and the EC peak frequency,

$$\begin{eqnarray}&&{\nu }_{\mathrm{ec}}^{{\prime} p}\approx \displaystyle \frac{4}{3}{\gamma }_{0}^{2}{\rm{\Gamma }}{\nu }_{\mathrm{ext}}.\end{eqnarray} \tag{ 19 }$$

Combining Equations (6), (12), (18), and (19), we have (see also Chen & Bai 2011)

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{\mathrm{ec}}^{p}L({\nu }_{\mathrm{ec}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}=\displaystyle \frac{17{e}^{2}}{6\pi {m}_{e}^{2}{c}^{2}}\displaystyle \frac{{U}_{\mathrm{ext}}}{{\nu }_{\mathrm{ext}}^{2}}{\left(\displaystyle \frac{{\nu }_{\mathrm{ec}}^{p}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{2}.\end{eqnarray} \tag{ 20 }$$

This means that the external photon properties (i.e., ${U}_{\mathrm{ext}}/{\nu }_{\mathrm{ext}}^{2}$) will be determined when given the synchrotron/EC peak frequency and luminosity. For the BLR, the emission is mainly contributed by Lyα lines, ν_ext-BLR ≈ 2 × 10¹⁵ Hz (Ghisellini & Tavecchio 2009). The dust torus reprocesses the disk emission into the IR band. As indicated by Spitzer observations, the typical peak frequency of IR dust torus emission is around ν_ext-IR ≈ 3 × 10¹³ (Cleary et al. 2007; Ghisellini & Tavecchio 2009; Gu 2013). Following Ghisellini & Tavecchio (2009), the radiation from the reprocessed dust torus (IR) or BLR is described as a blackbody spectrum. The issue of whether the radiation regions are inside or outside the BLRs is highly debated. Considering the γ-ray photons can be absorbed via photon–photon pair production above ≳10 GeV of BLR emission (e.g., Liu & Bai 2006; Bai et al. 2009), some authors suggested that the emission region should be outside the BLR (e.g., Sikora et al. 2009; Tavecchio & Mazin 2009; Tavecchio et al. 2013). The broadband SED modeling of a large sample also suggests that the emission region may be outside the BLR for most sources (e.g., Kang et al. 2014). In this case, the IC/dust process will dominate the γ-ray emission (Ghisellini & Tavecchio 2009). In this paper, the IC/dust torus models are considered in our calculation.

We assume a Doppler factor of $\delta =1/{\rm{\Gamma }}(1-\beta \cos \theta )\approx {\rm{\Gamma }}$ for the relativistic jet close to the line of sight in blazars with a viewing angle θ ≲ 1/Γ (Jorstad et al. 2005; Pushkarev et al. 2009). Combining Equations (6), (9), (11), (12), (18) (or (20)), and (19), the jet parameters can be derived (the SSC emission in this case is within the Thomson regime),

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\gamma }_{0}=1641.5{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{14}\mathrm{Hz}}\right)}^{1/2}{\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{22}\mathrm{Hz}}\tfrac{{10}^{14}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{1/4}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{46}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-1/8}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{1/8}{\left(\tfrac{{\nu }_{\mathrm{ext}}}{3\times {10}^{13}\mathrm{Hz}}\right)}^{-1/4}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{1/4}{\left(\tfrac{b}{2}\right)}^{1/16}\\ \delta =9.6321{\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{22}\mathrm{Hz}}\tfrac{{10}^{14}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{1/4}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{46}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/8}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{-1/8}{\left(\tfrac{{\nu }_{\mathrm{ext}}}{3\times {10}^{13}\mathrm{Hz}}\right)}^{-1/4}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{-1/4}{\left(\tfrac{b}{2}\right)}^{-1/16}\\ B=1.0323{\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{22}\mathrm{Hz}}\tfrac{{10}^{14}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{-3/4}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{46}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/8}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{-1/8}{\left(\tfrac{{\nu }_{\mathrm{ext}}}{3\times {10}^{13}\mathrm{Hz}}\right)}^{3/4}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{-1/4}{\left(\tfrac{b}{2}\right)}^{-1/16}\\ R=2.4949\times {10}^{16}{\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{22}\mathrm{Hz}}\tfrac{{10}^{14}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{1/4}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{46}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/8}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{-1/8}{\left(\tfrac{{\nu }_{\mathrm{ext}}}{3\times {10}^{13}\mathrm{Hz}}\right)}^{-1/4}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{3/4}{\left(\tfrac{b}{2}\right)}^{-1/16}\\ {N}_{0}=7.1622\times {10}^{-3}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{14}\mathrm{Hz}}\right)}^{-3/2}{\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{22}\mathrm{Hz}}\tfrac{{10}^{14}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{-1}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{46}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/4}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{3/4}\left(\tfrac{{\nu }_{\mathrm{ext}}}{3\times {10}^{13}\mathrm{Hz}}\right){\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{-3/2}{\left(\tfrac{b}{2}\right)}^{3/8}\\ {U}_{\mathrm{ext}}=3.2260\times {10}^{-4}\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}L({\nu }_{\mathrm{ec}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right){\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{22}\mathrm{Hz}}\tfrac{{10}^{14}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{-2}{\left(\tfrac{{\nu }_{\mathrm{ext}}}{3\times {10}^{13}\mathrm{Hz}}\right)}^{2}\end{array}.\right.\end{eqnarray} \tag{ 21 }$$

In the case of the KN regime (γ₀Γhν_ext ≳ m_ec²), the EC peak frequency is

$$\begin{eqnarray}&&{\nu }_{\mathrm{ec}}^{{\prime} p}\approx \displaystyle \frac{2}{\sqrt{3}}\displaystyle \frac{{\gamma }_{0}{m}_{e}{c}^{2}}{h}.\end{eqnarray} \tag{ 22 }$$

The effective EC seed photon energy density can be easily calculated through integrating from the minimum frequency to the critical frequency ${\nu }_{\mathrm{ext} \mbox{-} {\rm{c}}}^{{\prime} }=(\sqrt{3}/2)({m}_{e}{c}^{2}/h{\gamma }_{0})$,

$$\begin{eqnarray}&&{U}_{\mathrm{ext} \mbox{-} \mathrm{KN}}^{{\prime} }={U}_{\mathrm{ext} \mbox{-} \mathrm{Th}}^{{\prime} }\displaystyle \frac{15}{{\pi }^{4}}{\int }_{0}^{{x}_{c}}\displaystyle \frac{{x}^{3}}{{e}^{x}-1}{dx},\end{eqnarray} \tag{ 23 }$$

where⁷

$$\begin{eqnarray}&&{x}_{c}=2.82\displaystyle \frac{{m}_{e}^{2}{c}^{2}}{h{\nu }_{\mathrm{ext}}}\displaystyle \frac{{m}_{e}^{2}{c}^{2}}{h{\nu }_{\mathrm{ec}}}.\end{eqnarray} \tag{ 24 }$$

In the case of the KN regime, the EC peak luminosity follows

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{\mathrm{ec}}^{{\prime} p}L^{\prime} ({\nu }_{\mathrm{ec}}^{{\prime} p})}{{\nu }_{\mathrm{sy}}^{{\prime} p}L^{\prime} ({\nu }_{\mathrm{sy}}^{{\prime} p})}=\displaystyle \frac{{U}_{\mathrm{ext} \mbox{-} \mathrm{KN}}^{{\prime} }}{{U}_{B}}=\displaystyle \frac{{U}_{\mathrm{ext}}}{{U}_{B}}\displaystyle \frac{17}{12}{{\rm{\Gamma }}}^{2}{\int }_{0}^{{x}_{c}}\displaystyle \frac{{x}^{3}}{{e}^{x}-1}{dx}.\end{eqnarray} \tag{ 25 }$$

Combining Equations (6), (9), (11), (12), and (22), we have (the SSC in this case is within the Thomson regime)

$$\begin{eqnarray}&&\left\{\begin{array}{l}{\gamma }_{0}=61460{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{46}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-1/8}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{1/8}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{17}\mathrm{Hz}}\right)}^{1/4}{\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{26}\mathrm{Hz}}\tfrac{{10}^{17}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{1/2}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{1/4}{\left(\tfrac{b}{2}\right)}^{1/16}\\ \delta =11.404{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{46}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/8}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{-1/8}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{17}\mathrm{Hz}}\right)}^{-1/4}{\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{26}\mathrm{Hz}}\tfrac{{10}^{17}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{1/2}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{-1/4}{\left(\tfrac{b}{2}\right)}^{-1/16}\\ B=0.62198{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{46}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/8}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{-1/8}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{17}\mathrm{Hz}}\right)}^{3/4}{\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{26}\mathrm{Hz}}\tfrac{{10}^{17}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{-3/2}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{-1/4}{\left(\tfrac{b}{2}\right)}^{-1/16}\\ R=2.9539\times {10}^{16}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{46}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/8}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{-1/8}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{17}\mathrm{Hz}}\right)}^{-1/4}{\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{26}\mathrm{Hz}}\tfrac{{10}^{17}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{1/2}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{3/4}{\left(\tfrac{b}{2}\right)}^{-1/16}\\ {N}_{0}=1.1526\times {10}^{-7}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}{{10}^{46}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/4}{\left(\tfrac{{\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})}{{\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})}\right)}^{3/4}{\left(\tfrac{{\nu }_{\mathrm{sy}}^{p}}{{10}^{17}\mathrm{Hz}}\right)}^{-1/2}{\left(\tfrac{{\nu }_{\mathrm{ec}}^{p}}{{10}^{26}\mathrm{Hz}}\tfrac{{10}^{17}\mathrm{Hz}}{{\nu }_{\mathrm{sy}}^{p}}\right)}^{-2}{\left(\tfrac{{\rm{\Delta }}t(1+z)}{1\mathrm{day}}\right)}^{-3/2}{\left(\tfrac{b}{2}\right)}^{3/8}\end{array}\right..\end{eqnarray} \tag{ 26 }$$

After getting these parameters, the external energy density, U_ext, can be easily derived through combining Equations (12) and (25).

As discussed above, we use the SSC model for all BL Lacs and the EC model for all FSRQs. The input parameters include ${\nu }_{\mathrm{sy}}^{{\rm{p}}}$, ${\nu }_{\mathrm{ssc}/\mathrm{ec}}^{{\rm{p}}}$, ${\nu }_{\mathrm{sy}}^{p}L({\nu }_{\mathrm{sy}}^{p})$, ${\nu }_{\mathrm{ssc}}^{p}L({\nu }_{\mathrm{ssc}}^{p})$, and P₁. The criterion from the Thomson to KN regimes in the SSC model is ${\gamma }_{0}h{\nu }_{\mathrm{sy}}^{{\prime} p}\gtrsim {m}_{e}{c}^{2}$ (see Section 3.2), and that in the EC model is ${\gamma }_{0}{\rm{\Gamma }}h{\nu }_{\mathrm{ext}}\gtrsim {m}_{e}{c}^{2}$ (see Section 3.4).

The most promising scenario for launching astrophysical relativistic jets involves large-scale magnetic fields anchored in rapidly rotating compact objects. The idea of driving outflows by rotating magnetic fields, originally invented by Weber & Davis (1967) to explain the spindown of young stars, was successfully applied to pulsar winds (Michel 1969) and became a dominant mechanism in theories of relativistic jets in AGNs (Lovelace et al. 1987; Li et al. 1992; Vlahakis & Königl 2004) and γ-ray bursts (e.g., Spruit et al. 2001; Vlahakis & Königl 2001). Powerful jets in AGNs can be powered by the innermost portions of accretion disks and/or by rapidly rotating BHs (Blandford & Znajek 1977; Blandford & Payne 1982; Spruit 2010). The correlation between the jet and the accretion disk/BLR has been presented in many studies (e.g., Cao & Jiang 1999; Ho & Peng 2001; Wang et al. 2003; Ghisellini et al. 2014; Liu et al. 2014; Du et al. 2016; Sbarrato et al. 2016). Because the disk continuum and line emission are usually missing in BL Lacs, we study the jet–disk connection only for FSRQs through comparing the accretion (e.g., accretion disk/BLR/BH mass) with the jet properties.

The disk bolometric luminosity L_bol can be derived from a continuum luminosity based on the empirical relation (Shen et al. 2011). In order to avoid contamination by the nonthermal continuum, the accretion disk luminosity can be derived from the total luminosity of broad emission lines as a proxy of L_disk ≈ 10L_BLR (Calderone et al. 2013), and the total luminosity of broad emission lines can be reconstructed through visible broad lines (see, e.g., Francis et al. 1991; Vanden Berk et al. 2001). Hβ, Mg ii, and C iv (Shaw et al. 2012) are widely used to scale to the quasar template spectrum of Francis et al. (1991) to calculate the total BLR luminosity L_BLR. In Francis et al. (1991), Lyα is used as a reference of 100, and the total relative BLR flux is 555.77, of which Hβ is 22, Mg ii is 34, and CIV is 63 (see also, e.g., Vanden Berk et al. 2001; Ghisellini et al. 2014). The BH mass and accretion rate are fundamental parameters of AGNs. The virial method is now the most widely used to estimate the BH mass (e.g., Gu et al. 2001; Wu et al. 2004; Greene & Ho 2005). The uncertainty of a BH mass derived in this way is a factor of ∼3–4 (Vestergaard & Peterson 2006). Other methods include using the relation between BH mass and stellar velocity dispersion (e.g., Wu et al. 2002) and between BH mass and host-galaxy bulge luminosity (e.g., Laor 2001). Shaw et al. (2012) reported the optical spectroscopy of a large sample of γ-ray-detected blazars and estimated their BH mass through the virial method. We collect their available data (BH mass and emission lines) for our sample and finally get 144 FSRQs. As in the method discussed above, we use their optical emission lines to scale to the quasar template spectrum to calculate the total BLR luminosity. Because the BLR luminosities derived from various emission lines are consistent with each other (e.g., Francis et al. 1991; Shen et al. 2011; Shaw et al. 2012), we use the average value of the BLR luminosity if more than one emission line is available. The disk luminosity is then derived by L_disk ≈ 10L_BLR. Shen et al. (2011) studied the properties of quasars in the Sloan Digital Sky Survey Data Release 7 catalog. They calculated the virial BH mass and bolometric luminosity L_bol through the correlation between L_bol and continuum luminosity as presented in Richards et al. (2006). The disk luminosity can be calculated L_disk = L_bol/2 (see Calderone et al. 2013). We further compile available data from Shen et al. (2011) and get another 46 FSRQs with disk luminosity and 44 FSRQs with BH mass. We note that some famous FSRQs are not included in these samples, e.g., 3C 273 and 3C 454.3. We then search the literature for FSRQs with well-measured disk luminosity and BH mass (Xiong & Zhang 2014). Finally, we have 232 FSRQs with measured disk luminosity and 229 with measured BH mass; 222 FSRQs have both values. The three FSRQs with Doppler factors δ < 1 are not included in these sources.

Integrating the observed broadband SED, including both synchrotron and IC components, we can calculate the jet bolometric luminosity L_obs. Figure 6 shows L_obs as a function of L_disk for the 232 FSRQs with well-measured disk luminosity. These show a robust correlation⁹ $\mathrm{log}{L}_{\mathrm{disk},46}=(0.788\,\pm 0.076)\mathrm{log}{L}_{\mathrm{obs},46}-(1.44\pm 0.14)$ (solid blue line) with a chance probability p = 4.76 × 10⁻¹⁵ (Pearson test). The black dotted line represents L_obs = L_disk and the black dashed line L_obs = 100L_disk, which indicates that the jet bolometric luminosity is of the order of ∼100 times the disk luminosity L_obs ≈ 100L_disk. The entire radiation power of the jet can be estimated through P_r = 2L_obsΓ²/δ⁴ ≈ 2L_obs/δ², which is the power that the jet expends in producing the nonthermal radiation. Based on VLBI-detected superluminal motion, or the broadband SED model, the jet Doppler factor is found to be an order of δ ≳ 10 (e.g., Lister et al. 2013; Ghisellini & Tavecchio 2015). Our estimation gives a median value of δ = 10.7 for FSRQs (see the discussion above and/or the upper left panel of Figure 4). In this case, the entire jet radiation power will be on the order of the disk luminosity P_r ≈ L_disk. The jet radiative efficiency is believed to be on the order of P_r/P_jet ∼ 10%, which holds for AGNs, γ-ray bursts, and even BH X-ray binaries (Nemmen et al. 2012; Zhang et al. 2013; Ma et al. 2014), which gives an inevitable consequence that the jet power, P_jet ≈ 10P_r, is larger than the disk luminosity, P_jet ≈ 10L_disk. This suggests that the jet-launching processes and the way of transporting energy from the vicinity of the BH to infinity must be very efficient.

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The relativistic jet may be driven from the rapidly rotating BH. Increasing the spin of the BH shrinks the innermost stable orbit while increasing the accretion disk radiative efficiency ${\eta }_{\mathrm{acc}}={L}_{\mathrm{disk}}/\dot{M}{c}^{2}$ to a maximum value η_acc ≈ 0.3 (Thorne 1974), where $\dot{M}$ is the mass accretion rate. Assuming the accretion disk radiative efficiency η_acc = 0.3, we calculate the accretion power ${L}_{\mathrm{acc}}=\dot{M}{c}^{2}={L}_{\mathrm{disk}}/{\eta }_{\mathrm{acc}}$ (same as that in, e.g., Ghisellini et al. 2014).

With the jet physical parameters, we calculate the jet power through¹⁰ ${P}_{\mathrm{jet}}=2\pi {R}^{2}c{{\rm{\Gamma }}}^{2}({U}_{{\rm{B}}}+{U}_{{\rm{e}}}+{U}_{{\rm{p}}})$, where U_e and U_p are the electron and proton energy density in the jet comoving frame (see, e.g., Celotti & Ghisellini 2008; Ghisellini et al. 2014). We adopt the conventional assumption that jet power is carried by electrons and protons (with one cold proton per emitting electron). In this case, U_e and U_p can be easily derived for the log-parabolic distribution of electrons (see Equation (5)): ${U}_{{\rm{e}}}=\int N(\gamma )\gamma {m}_{{\rm{e}}}{c}^{2}d\gamma ={N}_{0}{\gamma }_{0}^{2}{m}_{{\rm{e}}}{c}^{2}\sqrt{\pi \mathrm{log}10/b}{10}^{1/4b}$ and ${U}_{{\rm{p}}}\,={m}_{{\rm{p}}}{c}^{2}\int N(\gamma )d\gamma ={N}_{0}{\gamma }_{0}{m}_{{\rm{p}}}{c}^{2}\sqrt{\pi \mathrm{log}10/b}{10}^{1/b}$. The existence of electron–positron pairs would reduce the jet power. Note that the median value of γ₀ = 1167.8 is near the mass ratio of proton to electron, m_p/m_e, implying that the jet power will not reduce significantly when considering electron–positron pairs. In addition, pairs cannot largely outnumber protons, because otherwise the Compton rocket effect would stop the jet (see, e.g., Ghisellini & Tavecchio 2010).

In order to study the relations among the jet and accretion disk, we plot the accretion power as a function of the jet power in Figure 7 (upper panel). The solid blue line shows the best-fit relation¹¹ : $\mathrm{log}{L}_{\mathrm{acc},46}$ $=\,(2.25\pm 0.27)\mathrm{log}{P}_{\mathrm{jet},46}-(0.34\pm 0.12)$ (the Pearson test shows a chance probability p = 2.85 × 10⁻¹¹). The dashed black line is the equality line. This result shows that a significant number of FSRQs have jet power greater than the accretion power, supported by the distribution of the ratio of jet power to accretion power (P_jet/L_acc) as shown in the left panel of Figure 8, implying that accretion power is not sufficient to launch the jets. The production of such a large-power jet cannot be treated as a marginal by-product of the accretion disk flow and most likely is governed by the BZ mechanism (Blandford & Znajek 1977). The gravitational energy released from accretion cannot only be transformed into heat and radiation but also powers jets through the BP process (Blandford & Payne 1982). In addition, the gravitational energy can also amplify the magnetic field, allowing the field to access the rotational energy of the BH and transform part of it into powerful jets through the BZ process (Blandford & Znajek 1977). As revealed by recently general relativistic magnetohydrodynamical (MHD) numerical simulations (e.g., Tchekhovskoy et al. 2011; McKinney et al. 2012; Li 2014), the average outflowing jet/wind power can even exceed the total accretion power for the case of spin value a = 0.99 when the BZ process dominates. The magnetic flux required to explain the production of the most powerful jets has been found to agree with the maximum magnetic flux that can be confined on BHs by the ram pressure of "magnetically arrested disks" (MAD; Narayan et al. 2003). In recent years, the MAD scenario has been thoroughly investigated and is now considered to be the likely remedy for the production of very powerful jets (Tchekhovskoy et al. 2011; McKinney et al. 2012; Sikora & Begelman 2013). However, as numerical simulations suggest, the powers of the jets launched in the MAD scenario depend not only on the spin and magnetic flux but also on the disk's geometrical thickness (Avara et al. 2016), with the jet power scaling approximately quadratically with all of these quantities (Sikora 2016). This large power of the relativistic jet in some FSRQs suggests that the BZ process will dominate the jet launching, which confirms some previous studies (see, e.g., Ghisellini et al. 2014; Zamaninasab et al. 2014).

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The middle panel of Figure 7 indicates jet power as a function of Eddington luminosity, L_Edd = 1.26 × 10³⁸(M/M_⊙) erg s⁻¹, for 229 sources with measured BH mass, which presents a significant correlation with the best linear fitting¹² $\mathrm{log}{L}_{\mathrm{Edd},46}\,=(1.38\pm 0.14)\mathrm{log}{P}_{\mathrm{jet},46}+(0.220\pm 0.064)$ (solid blue line) and a chance probability p = 4.0 × 10⁻¹⁴ (the Pearson test). The black dashed line is the equality line, which shows that the jet power in units of Eddington luminosity is less than 1 for most sources. This is supported by the distribution of the ratio of jet power to Eddington luminosity, as shown in the right panel of Figure 8, with a median value of 0.382.

The bottom panel of Figure 7 shows a relation between the Eddington ratio (defined as λ = L_disk/L_Edd) and jet power, suggesting almost no correlation between these two parameters (the Pearson test gives r = 0.144 and p = 0.0325; see, e.g., Zhang et al. 2015 for similar results of FSRQs). This suggests that the accretion rate may not be very important for driving the jet. As discussed above, this is consistent with the idea that these jets may not be launched through the BP process, related to the accretion process, but rather through the BZ process extracting the rotational energy of the BH. Actually, BL Lacs are believed to have very weak disk continuum emission and low accretion rates compared with FSRQs (Cao 2002; Ghisellini et al. 2009; Xu et al. 2009; Falomo et al. 2014). BL Lacs have comparable (very slightly lower) jet powers with FSRQs (the median values P_jet = 20.0, 6.30, and 12.0 × 10⁴⁵ erg s⁻¹ for FSRQs, BL Lacs, and total blazars, respectively; see discussion next subsection, Table 2, and/or the bottom right panel of Figure 4). Therefore, one may expect that the noncorrelation between Eddington ratio and jet power will extend down to the low Eddington ratio tail when considering BL Lacs.

The distribution of the Eddington ratio (L_disk/L_Edd) is presented in the left panel of Figure 8 with a median value of 0.148, consistent with previous results (e.g., Ghisellini et al. 2014). This implies that FSRQs have the standard geometrically thin, optically thick accretion disk (Shakura & Sunyaev 1973), which can launch powerful jets, contrary to some expectations (e.g., Livio et al. 1999; Meier 2002).

The most diverse opinions about the nature of AGN relativistic jets concern their magnetic field and magnetization σ = P_B/P_kin, where P_B and P_kin are the magnetic and kinetic power, respectively. As discussed above, the current picture describing the powerful jet launching is that the jet can be produced from the central accretion system through the BZ mechanism by extracting BH rotation energy. This process is attributed to a key role of the dynamically important magnetic field, by means of which the BH spin energy is extracted and channeled into a Poynting/magnetic flux (Blandford & Znajek 1977; Tchekhovskoy et al. 2009, 2011). This magnetic flux cannot be developed by dynamo mechanisms in standard radiation-dominated accretion disks (Ghosh & Abramowicz 1997). However, such a flux is expected to be accumulated in the inner regions of the accretion flow and/or on the BH by the advection of magnetic fields from external regions (see Cao & Spruit 2013; Sikora & Begelman 2013 and references therein). In fact, the magnetic flux close to the BH horizon is so large that accretion likely occurs through an MAD flow, as discussed above (Narayan et al. 2003; Tchekhovskoy et al. 2011; McKinney et al. 2012). Starting as Poynting flux-dominated outflows, the jets are smoothly accelerated as magnetic power is being progressively converted to kinetic power and their magnetization drops, until a substantial equipartition between the magnetic and kinetic power is established (σ ≈ 1; see, e.g., Komissarov et al. 2007; Tchekhovskoy et al. 2009; Vlahakis 2015 and references therein). Therefore, at the end of this acceleration phase, the jet should still carry a substantial fraction (≈half) of its power in the form of a Poynting flux. The jet acceleration becomes inefficient when σ ≲ 1, from which point on, in the ideal MHD picture, σ will decrease logarithmically (Lyubarsky 2010).

However, even though blazars can have jets originating from MADs and be powered by the BZ mechanism, at the same time, they can have σ ≪ 1 at the radio core and/or the blazar zone (i.e., the region where most of the radiation is produced, as indicated by blazar models with jet opening angles 1/Γ; Nalewajko et al. 2014b; Janiak et al. 2015; Zdziarski et al. 2015). Zdziarski et al. (2015) suggested that the magnetic-to-kinetic energy flux conversion is assumed to result from the differential collimation of poloidal magnetic surfaces, which is the only currently known conversion mechanism in steady-state, axisymmetric, and nondissipative jets¹³ (the conversion process can initially proceed quite efficiently; see also Tchekhovskoy et al. 2009; Lyubarsky 2010). Hence, it is likely that other mechanisms are involved in the conversion process working at σ ≲ 1, such as MHD instabilities (see Komissarov 2011 and references therein), randomization of magnetic fields (Heinz & Begelman 2000), reconnection of magnetic fields (Drenkhahn & Spruit 2002; Lyubarsky 2010), and/or impulsive modulation of jet production (Lyutikov & Lister 2010; Granot et al. 2011). Little is known about the feasibility and efficiency of these processes in the context of AGNs. The MHD instabilities can develop when σ drops to unity or even earlier if stimulated by high-amplitude fluctuations of the jet power and direction, which are predicted in the MAD model (e.g., McKinney et al. 2012). In this case, the magnetization can reach σ ≪ 1 even prior to the radio core and/or the blazar zone.

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 and its efficiency. Dissipation of part of the kinetic (through shocks) and/or magnetic (through reconnection) power leads to the acceleration of particles up to ultrarelativistic energies, producing the nonthermal emission we observe from blazars. Both versions (the first/second order) of Fermi acceleration were, and still are, commonly invoked as responsible for the ultrarelativistic electrons in AGN jets (see, e.g., Hoshino et al. 1992; Sironi & Spitkovsky 2011; Marscher 2014). Based on so-called particle-in-cell simulations, it is shown that electrons can be efficiently heated in very low magnetized jets (σ ≪ 1; Madejski & Sikora 2016). The reconnection of magnetic energy is also a promising mechanism for particle acceleration in AGN jets (e.g., Giannios et al. 2009; Uzdensky 2011; Cerutti et al. 2012). A prediction of this scenario, resulting from detailed particle-in-cell simulations, is the substantial equipartition between the magnetic field and the accelerated electrons downstream of the reconnection site, where particles cool and emit the radiation we observe. For example, relativistic reconnection 2D models predict a lower limit of the order of σ ∼ 0.3 in the dissipation region (Sironi et al. 2015). Needless to say, the magnetic reconnection scenario requires that jets carry a sizable fraction of their power in the magnetic form up to the emission regions.

Based on the jet parameters estimated in this paper, the magnetization properties can be studied. Here we define a magnetization parameter as the ratio of energy densities between the magnetic field and relativistic electrons¹⁴ σ =U_B/U_e (also called the equipartition parameter). The distribution of σ is shown in Figure 9. It can be seen that FSRQs have a narrower distribution than BL Lacs, with median values σ = 6.42 and 0.0285 (and 0.640 for total blazars), respectively (see also Table 2). The Kolmogorov–Smirnov (KS) test yields the significance level probability for the null hypothesis that FSRQs and BL Lacs are drawn from the same distribution p = 4.67 × 10⁻⁷⁶ and the statistic D_KS = 0.503 (the maximum separation of the two cumulative fractions). Similar results have been presented on single sources (Abdo et al. 2011b; Acciari et al. 2011; Aleksić et al. 2015) and through modeling the SEDs of the large sample blazars (Zhang et al. 2013; Tavecchio & Ghisellini 2016), which demonstrate that the magnetization parameter of the large majority of BL Lacs is commonly at ∼10⁻². Costamante et al. (2017) studied six hard-TeV BL Lacs and found that the equipartition parameters range from 10⁻³ to 10⁻⁵. Kharb et al. (2012) presented deep Chandra/ACIS observations and Hubble Space Telescope (HST) Advanced Camera for Surveys observations of two FSRQs, PKS B0106+013 and 3C 345, and found the shocked jet regions upstream of the radio hot spots, the kiloparsec-scale jet wiggles, and a "nose cone"-like jet structure in PKS B0106+013, as well as the V-shaped radio structure in 3C 345, which suggest that the jet still has a large σ at these large scales. Modeling of FSRQ SEDs is also in agreement with this result (e.g., Ghisellini & Tavecchio 2015).

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The smaller σ in BL Lacs relative to FSRQs implies that BL Lacs jets would suffer deceleration more easily than those of FSRQs (Piner et al. 2008; Homan et al. 2015), although both present similar violent γ-ray emission from the central region. This is consistent with the idea that some BL Lacs show almost no superluminal motion in the Very Long Baseline Array (VLBA) scale (e.g., Mrk 501 and Mrk 421; Giroletti et al. 2004, 2006), while the multiwavelength SED and variability imply a highly Doppler-booting emission, which indicates that the jet has already been decelerated at that scale (e.g., Giroletti et al. 2004; Ghisellini et al. 2005). For some FSRQs (e.g., 3C 279 and 3C 345; Lobanov & Zensus 1999; Piner et al. 2003), the Monitoring of Jets in AGN with VLBA Experiments (MOJAVE) project shows that jets will still be accelerated at the VLBA scale (see, Homan et al. 2015). The low magnetization in BL Lacs is essential in order to avoid efficient cooling of relativistic electrons, which would lead to much larger values of γ₀ of BL Lacs relative to FSRQs (see Table 2 and/or Figure 4). This also seems to imply that the magnetic reconnection cannot be strong in these emission regions. The particle acceleration in situ may originate from the energy dissipated in shocks and/or in boundary shear layers. The spine/layer jet structure has been revealed through many observational techniques and can contribute significant X-ray emission due to the enhancement of the seed photon energy density by relative opposite motions between the spine and layer (see Chen 2017 for a recent review).

So far, there is no observational evidence supporting the theoretical prediction that a high-magnetization jet will more easily transport energy to a large scale than a low-magnetization jet. As found above, FSRQs have relatively larger values of σ than BL Lacs, which simply predicts that the jet may transport energy more efficiently to a large scale in FSRQs than BL Lacs. Therefore, we may expect a larger extended jet kinetic power (relative to the core SED jet power) in FSRQs than BL Lacs. To answer this question, we should estimate the (extended) jet kinetic power of these sources.

The jet power is basically important for understanding jet formation, the jet–disk relation (Blandford & Znajek 1977; Blandford & Payne 1982; Gu et al. 2009; Spruit 2010; Li & Cao 2012; Narayan & McClintock 2012; Wu et al. 2013; Cao 2014), and the AGN feedback on structure formation (Best et al. 2007; Magliocchetti & Brüggen 2007). It is generally believed that the X-ray cavities are direct evidence of AGN feedback and provide a direct measurement of the mechanical energy released by the AGN jets (jet kinetic power) through the work done on the hot, gaseous halos surrounding them (e.g., Bîrzan et al. 2004; Allen et al. 2006; Cavagnolo et al. 2010). It is also found that the cavity kinetic power is correlated with the radio extended power (Rawlings & Saunders 1991; Bîrzan et al. 2004; Cavagnolo et al. 2010), and the scaling relationship is roughly consistent with the theoretical relation (e.g., Willott et al. 1999; Cavagnolo et al. 2010; Meyer et al. 2011). Therefore, in systems where the X-ray cavities are lacking or not observed, the radio extended luminosity is used to estimate the jet kinetic power. The relations presented in Willott et al. (1999) are commonly used to do this estimation (Cavagnolo et al. 2010). Based on this method, Meyer et al. (2011) presented the largest blazar sample with measured (extended) jet kinetic power, until now.

In this paper, we collect the (extended) jet kinetic power, P_jet,ext, from Meyer et al. (2011), resulting in a total of 159 blazars, including 76 BL Lacs and 83 FSRQs, whose data are presented in Table 1. To study the jet energy transportation, we define η as a ratio of the jet SED power (from the central nuclei, P_jet) to the (extended) jet kinetic power (P_jet,ext), η = P_jet/P_jet,ext. The distribution of η is shown in Figure 10 with median values η = 57.2, 230, and 73.9 for FSRQs, BL Lacs, and total blazars, respectively, which implies that the jet SED power is significantly larger than the (extended) jet kinetic power. Two reasons may account for this: (1) the (SED) jet power may represent jet power measured in AGN active state, while the (extended) jet kinetic power is the historically average power, and (2) most of the energy is dissipated at the central region before being transported to a large scale. We note that FSRQs have an average smaller η than BL Lacs (by about one order of magnitude). The KS test gives the significance level probability that FSRQs and BL Lacs are drawn from the same distribution P = 2.58 × 10⁻¹⁰ and the statistic D_KS = 0.524. This suggests that relatively more energy is transported to a large scale in FSRQs than in BL Lacs. As discussed above, the central jets of FSRQs also have relatively higher magnetization (σ) than those of BL Lacs, suggesting that a higher-magnetization jet can more easily transport energy to a large scale. As discussed above, some BL Lacs have extremely estimated Doppler factors. We instead use the median value of δ to determine the other jet parameters for these sources. This uncertainty may affect our results. Therefore, we exclude all BL Lacs with δ > 100 or δ < 1 and find that the remaining 56 BL Lacs have median values of η = 230 and σ = 0.00389, which is also consistent with the above result that the higher-magnetization jet can more easily transport energy to a large scale. For further research, we plot η versus σ in Figure 11 for sources with 1 < δ < 100, which presents a significant anticorrelation. The solid black line shows the best linear fit, $\mathrm{log}\sigma =(8.54\,\pm 0.69)-(4.44\pm 0.33)\mathrm{log}\eta$, with a chance probability p = 1.23 × 10⁻²⁶ (Pearson test). This result confirms the above result, suggesting that a higher-magnetization jet can more easily transport energy to a larger scale, which, for the first time, offers supporting evidence for the jet energy transportation theory.

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Our sample include 393 unknown-redshift sources (157 BL Lacs and 236 BCUs), which is about 28% of the sample. We use the median values of known redshifts of blazars for these sources. In order to explore the possible effect of this replacement on our results, we recalculate the median values of all jet parameters of another two cases: case 1 uses known-redshift sources including BCUs, and case 2 uses known-redshift sources excluding BCUs. All of these values are presented in Table 2, where the label (z) is for case 1 and the label (cz) is for case 2. It can be seen that both cases are consistent with our above results.