DISCERNING THE GAMMA-RAY-EMITTING REGION IN THE FLAT SPECTRUM RADIO QUASARS

In the model, the relativistic plasma propagates with an associated bulk Lorentz factor Γ in a stationary funnel whose structure is constant with time in the lab frame, and the geometry is a truncated cone of length Γ L, where the length of the jet in the lab frame is related to that in the fluid frame by a simple Lorentz contraction. We define the dynamic variable x as the length along with the jet axis in the fluid frame, where x = 0 is the base of the jet and x = L is the end of the jet, so we can parameterize the geometry of the jet as follows:

$$\begin{eqnarray}&&R(x)={R}_{0}+x\tan {\theta }_{\mathrm{open}},\end{eqnarray} \tag{ 1 }$$

where R(x) is the radius of the jet at the length x, R₀ is the radius at the base of the jet, and ${\theta }_{\mathrm{open}}$ is a half-opening angle of the cone. We note that the jet opening angle ${\theta }_{\mathrm{open}}^{\prime }$ in the lab frame is related to the fluid-frame opening angle ${\theta }_{\mathrm{open}}$ via ${\rm{\Gamma }}\tan {\theta }_{\mathrm{open}}^{\prime }=\tan {\theta }_{\mathrm{open}}$. The stationary jet structure and magnetic energy conservation in each segment determined by Equation (1) could have the following relation:

$$\begin{eqnarray}&&B(x)={B}_{0}\displaystyle \frac{{R}_{0}}{R(x)},\end{eqnarray} \tag{ 2 }$$

where B(x) is the magnetic field of the jet at the length x, and B₀ is the magnetic field at the base of the jet. The relation describes a pure toroidal field distribution in which the magnetic flux is parallel to the jet axis that increases with x.

It is well known that both the particle energy and field energy dominate the total energy of the jet plasma (e.g., Celotti & Fabian 1993; Celotti et al. 2007; Finke et al. 2008). The energy equipartition could provide a minimum power solution for the emissions from the blazar jet. In the black hole–jet system, the relation between the particle energy and field energy depends on the uncertain jet formation, particle acceleration, and radiation mechanisms (e.g., Dermer et al. 2014). However, the baryon composition of the system is poorly known (e.g., Reynolds et al. 1996; Kataoka et al. 2008; Ghisellini 2012; Kino et al. 2012). In our model, we assume that a condition of the equipartition holds between the magnetic field energy density U_B and the non-thermal electron energy density U_e. In order to establish a parameter connection between the U_B and U_e, we assume a equipartition fraction ${A}_{\mathrm{equi}}$,

$$\begin{eqnarray}&&{A}_{\mathrm{equi}}=\displaystyle \frac{{U}_{B}}{{U}_{e}}.\end{eqnarray} \tag{ 3 }$$

From this equipartition fraction, a relation between B₀ and R₀ is found as (Potter & Cotter 2012)

$$\begin{eqnarray}&&{R}_{0}=\sqrt{\displaystyle \frac{2{E}_{j}{A}_{\mathrm{equi}}{\mu }_{0}}{{{\rm{\Gamma }}}^{2}(\pi {B}_{0}^{2})(1+{A}_{\mathrm{equi}})}},\end{eqnarray} \tag{ 4 }$$

where ${E}_{j}={P}_{j}/c$ is the energy contained in a section of plasma with the width of 1 m and P_j is the total jet power in the x-direction in the lab frame.

The present model is to evolve the electron population dynamically along with the jet by taking into account energy losses from synchrotron emission and ICS. We consider the electron population in a slab with the width of 1 m in the comoving frame and the jet structure is relative motion to the slab. Since a section of the jet with the width of dx at any length x containing n meter slabs travels with the velocity of light c toward the slabs, a slab takes n/c seconds to cross the section in the fluid frame. It is noted that the electrons in a slab lose an amount of the energy which is equal to the total power ${P}_{\mathrm{tot}}(x,{dx},{E}_{e})$ emitted by the section in the comoving frame divided by c (Potter & Cotter 2012). In this scenario, the evolution of the electron population along with the jet due to energy losses can be found by solving:

$$\begin{eqnarray}&&{N}_{e}({E}_{e},x+{dx})={N}_{e}({E}_{e},x)-\displaystyle \frac{{P}_{\mathrm{tot}}(x,{dx},{E}_{e})}{{{cE}}_{e}},\end{eqnarray} \tag{ 5 }$$

where E_e is the electron energy in comoving frame. Assuming that electrons emit all of their energy at a critical frequency ${\nu }_{\mathrm{cr}}(x)=3{{eE}}_{e}^{2}B(x)/4\pi {m}_{e}^{3}{c}^{5}$, we can obtain

$$\begin{eqnarray}&&{P}_{\mathrm{tot}}(x,{dx},{E}_{e})={P}_{\mathrm{syn}}(x,{dx},{E}_{e})+{P}_{\mathrm{ic}}(x,{dx},{E}_{e}),\end{eqnarray} \tag{ 6 }$$

where ${P}_{\mathrm{syn}}(x,{dx},{E}_{e})$ is the synchrotron emission power and ${P}_{\mathrm{ic}}(x,{dx},{E}_{e})$ is the ICS power corresponding to a critical energy ${E}_{\mathrm{cr}}(x)=h{\nu }_{{cr}}(x)$ at the jet length x. In order to solve Equation (5), a initial injected electron distribution with the cutoff energy ${E}_{{\rm{e}},\mathrm{cut}}$ has been assumed,

$$\begin{eqnarray}&&{N}_{e}({E}_{e},0)\simeq {A}_{0}{E}_{e}^{\alpha }{e}^{-{E}_{e}/{E}_{{\rm{e}},\mathrm{cut}}},\end{eqnarray} \tag{ 7 }$$

where, ${A}_{0}\simeq (2-\alpha ){E}_{j}/[{{\rm{\Gamma }}}^{2}(1+{A}_{\mathrm{equi}})({E}_{{\rm{e}},\mathrm{cut}}^{2-\alpha }-{E}_{e,\min }^{2-\alpha })]$ (Potter & Cotter 2012) with the initial injected electron minimum energy ${E}_{e,\min }$, and α is the electron spectral index.

In order to produce the synchrotron photons through the jet fluid, we must know the opacity of the plasma. Assuming that the photons are observed at an angle smaller than the opening angle of the jet in the lab frame, Potter & Cotter (2012) argued that the observer sees through to the layer within the opening angle where the total column optical depth is approximately 1, and beyond this there is an exponentially decreasing contribution from further layers. This suggests that the total absorption optical depth ${\tau }_{\mathrm{tot}}({E}_{s},x)$ is the summation of the optical depths from each segment. In this scenario, we could obtain the ${\tau }_{\mathrm{tot}}({E}_{s},x)$ in the comoving frame using the Lorentz transform,

$$\begin{eqnarray}&&{\tau }_{\mathrm{tot}}({E}_{s},x)={\int }_{x}^{L}k({E}_{s},x){{\rm{\Gamma }}}^{2}\left(\displaystyle \frac{1}{\cos {\theta }_{\mathrm{observe}}}-\beta \right){dx},\end{eqnarray} \tag{ 8 }$$

where E_s is the energy of the synchrotron photon, $k({E}_{s},x)$ is the synchrotron absorption coefficient at the jet length x, ${\theta }_{\mathrm{observe}}$ is the angle of the observer's line of sight to the jet axis in the lab frame, and β is the speed of jet material in units of c. Introducing a full integration with the modified Bessel functions of the order of 5/3 to substitute for an approximate single energy emitting in Equation (6), we write the synchrotron emission photon producing rate using an individual segment with the width dx in the comoving frame as

$$\begin{eqnarray}{w}_{\mathrm{syn}}(x,{dx},{E}_{s}) & = & \displaystyle \frac{\sqrt{3}{e}^{3}B(x)}{{{hm}}_{e}{c}^{2}}{\displaystyle \int }_{{E}_{e,\min }}^{{E}_{e,\mathrm{cut}}}{N}_{e}({E}_{e},x)\\ & & \times F\left[\displaystyle \frac{{E}_{s}}{{E}_{\mathrm{cr}}(x)}\right]{{dE}}_{e}.\end{eqnarray} \tag{ 9 }$$

We could approximate the energy density of the synchrotron photon field ${u}_{\mathrm{syn}}(x)$ in the comoving frame by setting

$$\begin{eqnarray}{u}_{\mathrm{syn}}(x) & = & \displaystyle \frac{1}{2\pi R(x)c}{\displaystyle \int }_{{E}_{s,\min }}^{{E}_{s,\max }}{w}_{\mathrm{syn}}(x,{dx},{E}_{s})\\ & & \times {e}^{-{\tau }_{\mathrm{tot}}({E}_{s},x)}{{dE}}_{s}.\end{eqnarray} \tag{ 10 }$$

We consider the contribution of the external ambient field that includes both BLR and MT. We assume that (i) the BLR is a shell located at a characteristic distance ${R}_{\mathrm{BLR}}\sim {10}^{17}{({L}_{d}/{10}^{45}\mathrm{erg}{{\rm{s}}}^{-1})}^{1/2}$ cm (e.g., Ghisellini & Tavecchio 2008), and its spectral shape observed in the comoving frame is a Maxwellian distribution peaking at photon energies of Lyα with ${E}_{\mathrm{BLR}}\sim 10\,\mathrm{eV}\times {\rm{\Gamma }}$; and (ii) there is a MT (e.g., Blazejowski et al. 2000; Sikora et al. 2002) at a characteristic distance ${R}_{\mathrm{MT}}\sim 2.5\times {10}^{18}{({L}_{d}/{10}^{45}\mathrm{erg}{{\rm{s}}}^{-1})}^{1/2}$ cm (e.g., Ghisellini & Tavecchio 2008), and its spectral shape observed in the comoving frame is also a Maxwellian distribution peaking at photon energies of dust with ${E}_{\mathrm{MT}}\sim 0.3\,\mathrm{eV}\times {\rm{\Gamma }}$. Where L_d is the luminosity of an accretion disk. We note that the defined initial location (x = 0) of the γ-ray production site in the model would not show the real distance between the SMBH and the base of the jet. In order to derive a correlation between the external ambient field and the energy dissipation region, we use the distance r between the position of the dissipation region and the SMBH (e.g., Dermer et al. 2014) to parameterize the energy density of BLR and MT in the jet comoving frame (e.g., Hayashida et al. 2012; Zheng & Yang 2016),

$$\begin{eqnarray}&&{u}_{\mathrm{BLR}}(r)=\displaystyle \frac{{\tau }_{\mathrm{BLR}}{{\rm{\Gamma }}}^{2}{L}_{d}}{3\pi {R}_{\mathrm{BLR}}^{2}c[1+{(r/{R}_{\mathrm{BLR}})}^{{\beta }_{\mathrm{BLR}}}]},\end{eqnarray} \tag{ 11 }$$

and

$$\begin{eqnarray}&&{u}_{\mathrm{MT}}(r)=\displaystyle \frac{{\tau }_{\mathrm{MT}}{{\rm{\Gamma }}}^{2}{L}_{d}}{3\pi {R}_{\mathrm{MT}}^{2}c[1+{(r/{R}_{\mathrm{MT}})}^{{\beta }_{\mathrm{MT}}}]},\end{eqnarray} \tag{ 12 }$$

where, $r={x}_{0}+x$ with the real distance x₀ between the SMBH and the base of the jet in the comoving frame, ${\tau }_{\mathrm{BLR}}$ and ${\tau }_{\mathrm{MT}}$ are the fractions of the disk luminosity L_d reprocessed into a broad-line region and hot dust radiation, respectively. In our calculation, we adopt the radiation density profiles ${\beta }_{\mathrm{BLR}}=3$ (e.g., Sikora et al. 2009) and ${\beta }_{\mathrm{MT}}=4$ (e.g., Hayashida et al. 2012), respectively. It can be seen that the BLR component is generated at distances ∼0.1 pc, much smaller than the MT that is produced at distances ∼1–10 pc. In both cases the photon densities decrease rather steeply with radius at distances beyond their characteristic radii, and achieve uniform densities at smaller distances.

We consider that target photons dominating ICS originate from both synchrotron photon and external ambient photon fields. In the model, we can write the total comoving target photon field ${u}_{\mathrm{tar}}(x)$ at the jet length x through the expression

$$\begin{eqnarray}{u}_{\mathrm{tar}}(x) & = & {u}_{\mathrm{syn}}(x)+{u}_{\mathrm{BLR}}(r={x}_{0}+x)\\ & & +{u}_{\mathrm{MT}}(r={x}_{0}+x).\end{eqnarray} \tag{ 13 }$$

The Klein–Nishina (KN) effect is properly considered in the ICS using a full KN cross section (e.g., Blumenthal & Gould 1979; Rybicki & Lightman 1979). We derived the total synchrotron emission and ICS contribution through integrating every segment from x = 0 to x = L. Then we could reproduce the observed spectra at the Earth.

Using the synchrotron radiation and ICS solution for the conical jet structure, we can reproduce the SED for each source in the sample. We note that the spectrum derived from the model is described by the geometry and physical parameters of the jet (P_j, L, B₀, x₀, ${A}_{\mathrm{equi}}$, ${\theta }_{\mathrm{open}}$, ${\theta }_{\mathrm{obs}}$ and Γ), the initial injected electron distribution (α, ${E}_{\min }$ and ${E}_{\mathrm{cut}}$), and the external ambient fields (L_d, ${\tau }_{\mathrm{BLR}}$ and ${\tau }_{\mathrm{MT}}$). It is well known that these parameters are not directly observable. On the other hand, they are coupled with each other in the framework of the length-dependent conical jet model. In these scenarios, if we allow all of the parameters to be free, it will take too long to reproduce the best SED. In order to determine the parameters' reliability, we list some constraints that characterize the parameters:

• (1)  
  Statistical results show that the typical jet powers of FSRQs are in the range of ${10}^{45}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\leqslant {P}_{{\rm{j}}}\leqslant {10}^{48}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (e.g., Ghisellini & Tavecchio 2008; Ghisellini 2010; Kang et al. 2014);
• (2)  
  We can provide a constraint on the length of the jet L and the magnetic field B₀ at the base of the jet from a radio spectrum (Zheng & Yang 2016);
• (3)  
  The angle of the observer's line of sight to the jet axis in the lab frame is ${\theta }_{\mathrm{obs}}\leqslant 5^\circ$ (Urry & Padovani 1995);
• (4)  
  The electron spectra index α is thought to be between 1 and 3 from the theory of the shock acceleration (Bell 1978; Bell et al. 2011; Summerlin & Baring 2012);
• (5)  
  We adopt a simplified Shakura–Sunyaev disk spectrum (Dermer et al. 2014) as follows:

  $$\begin{eqnarray}&&\epsilon {L}_{d}(\epsilon )=1.12{L}_{d}{\left(\displaystyle \frac{\epsilon }{{\epsilon }_{\max }}\right)}^{4/3}\exp \left(-\displaystyle \frac{\epsilon }{{\epsilon }_{\max }}\right),\end{eqnarray} \tag{ 14 }$$

where is the photon energy that is emitted by the accretion disk. While the value of ${\epsilon }_{\max }$ depends on the spin of the black hole and relative Eddington luminosity, we set a typical characteristic temperature of the UV bump in Seyfert galaxies with ${\epsilon }_{\max }\sim 10\,\mathrm{eV}$. The accretion disk produces a total luminosity ${L}_{d}=\eta \dot{M}{c}^{2}$, where $\dot{M}$ is the accretion rate and η is the accretion efficiency. It is known that the efficiency of conversion for a nuclear reaction is $\eta =0.007$ and that for pure accretion it is η = 0.1. If we set η = 0.08 (e.g., Ghisellini 2010), and we adopt an Eddington accretion rate $\dot{M}=1.39\times {10}^{18}(M/{M}_{\odot })\,{\rm{g}}\,{{\rm{s}}}^{-1}$ with $M\sim {10}^{9}\,{M}_{\odot }$, we could estimate an accretion disk luminosity ${L}_{d}\lesssim {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$.

Following the above constraints, we could search for the electron and photon spectra along the x-axis of the jet. Assuming a steady geometry of the jet structure for a source, and an immovable position of the observer during observed epochs, we calculate the electron and photon spectrum in every segment from x = 0 to x = L. We consider that the observed spectrum is a summation of each segment. Therefore, we can calculate the observed spectrum using the photon spectrum in every segment from x = 0 to x = L. We list these parameters in Table 1. The columns in this table are as follows:

• 1.  
  source name;
• 2.  
  z, redshift;
• 3.  
  P_j, total jet power, in units of $\mathrm{erg}\,{{\rm{s}}}^{-1};$
• 4.  
  L, the length of the jet, in units of pc;
• 5.  
  B₀, the magnetic field at the base of jet, in units of G;
• 6.  
  x₀, the distance between the SMBH and the base of the jet, in units of pc;
• 7.  
  ${E}_{{\rm{e}},\min }$, the minimum energy of the initial injected electron, in units of MeV;
• 8.  
  ${E}_{{\rm{e}},\mathrm{cut}}$, the cutoff energy of the initial injected electron, in units of of MeV;
• 9.  
  L_d, the luminosity of an accretion disk, in units of $\mathrm{erg}\,{{\rm{s}}}^{-1};$
• 10.  
  ${\tau }_{\mathrm{BLR}}$, the fraction of the disk luminosity reprocessed into the broad-line region;
• 11.  
  ${\tau }_{\mathrm{MT}}$, the fraction of the disk luminosity L_d reprocessed into hot dust radiation;
• 12.  
  α, the electron spectra index;
• 13.  
  ${\theta }_{\mathrm{open}}^{\prime }$, the half-opening angle of the cone in units of degrees;
• 14.  
  ${\theta }_{\mathrm{obs}}$, the angle of the observer's line of sight to the jet axis in units of degrees;
• 15.  
  Γ, the bulk Lorentz factor;
• 16.  
  N, the number of simultaneous observational data points;
• 17.  
  ${\chi }^{2}$, the ${\chi }^{2}=\tfrac{1}{N-\mathrm{dof}}{\sum }_{i=1}^{N}{\left(\tfrac{{\hat{y}}_{i}-{y}_{i}}{{\sigma }_{i}}\right)}^{2}$, where dof are the degrees of freedom, i.e., the number of free parameters used for the model. The ${\hat{y}}_{i}$ are the expected values from the model and the y_i are the observed data. ${\sigma }_{i}$ is the standard deviation for each data point. We take 1% of the observed radio and optical flux and take 2% of the observed UV and x-ray flux as the errors of the data points whose errors are available (e.g., Zhang et al. 2012; Aleksic et al. 2014).

In Figures 3–8, we show the multi-wavelength spectra of the sources in the sample, respectively. For comparison, the observed data of the sources are also shown. In these figures, the simultaneous data are shown in red; the quasi-simultaneous data including Fermi-LAT data over two months, Planck ERCSC and non-simultaneous ground-based observations are shown in green; the Fermi-LAT data integrated over 27 months are shown in blue; and the literature or archival data are shown in gray. The dashed line represents the synchrotron emission, the dotted line represents the ICS on the seed photons of the synchrotron, BLR and MT, and the thick solid line represents the total spectrum by summation of all the emission components, respectively. It can be seen that the observed data can be reproduced in the model.

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We emphasize that the characteristic distances ${R}_{\mathrm{BLR}}$ and ${R}_{\mathrm{MT}}$ could not depict the BLR and MT. In order to determine the boundaries of the BLR and MT, we take an interest in the energy density of BLR (Equation (11)) and MT (Equation (12)) in the jet comoving frame. It can be seen that the ${u}_{\mathrm{BLR}}(r)$ and ${u}_{\mathrm{MT}}(r)$ are decreased significantly when the distance r satisfies $r\gt {R}_{\mathrm{BLR}}$ and $r\gt {R}_{\mathrm{MT}}$, respectively. The numerical results indicate that the contribution of BLR and MT could be neglected when the ${u}_{\mathrm{BLR}}(r)$ decreases to 10% of ${u}_{\mathrm{BLR}}$ and the ${u}_{\mathrm{MT}}(r)$ decreases to 1% of ${u}_{\mathrm{MT}}$, where, the ${u}_{\mathrm{BLR}}$ is the BLR energy density at the characteristic distance R_BLR, and the ${u}_{\mathrm{MT}}$ is the MT energy density at the characteristic distance R_MT, respectively. In this scenario, we could set the outer boundary of the BLR and MT at the location where the ${u}_{\mathrm{BLR}}(r)$ decreases to 10% of ${u}_{\mathrm{BLR}}$ and the ${u}_{\mathrm{MT}}(r)$ decreases to 1% of ${u}_{\mathrm{MT}}$, respectively. These results give the energy density at the outer boundary ${R}_{\mathrm{BLR},\mathrm{out}}$ of the BLR ${u}_{\mathrm{BLR},{\rm{c}}}=0.1{u}_{\mathrm{BLR}}$ and the energy density at the outer boundary ${R}_{\mathrm{MT},\mathrm{out}}$ of MT ${u}_{\mathrm{MT},{\rm{c}}}=0.01{u}_{\mathrm{MT}}$. Replacing these results in Equations (11) and (12), we find

$$\begin{eqnarray}&&\displaystyle \frac{{u}_{\mathrm{BLR},{\rm{c}}}}{{u}_{\mathrm{BLR}}}=\displaystyle \frac{2}{1+{\left(\tfrac{{R}_{\mathrm{BLR},\mathrm{out}}}{{R}_{\mathrm{BLR}}}\right)}^{{\beta }_{\mathrm{BLR}}}},\end{eqnarray} \tag{ 15 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{u}_{\mathrm{MT},{\rm{c}}}}{{u}_{\mathrm{MT}}}=\displaystyle \frac{2}{1+{\left(\tfrac{{R}_{\mathrm{MT},\mathrm{out}}}{{R}_{\mathrm{MT}}}\right)}^{{\beta }_{\mathrm{MT}}}}.\end{eqnarray} \tag{ 16 }$$

Taking into account the radiation density profiles ${\beta }_{\mathrm{BLR}}=3$ and ${\beta }_{\mathrm{MT}}=4$ in the model, we could deduce the outer boundary radii of the BLR as ${R}_{\mathrm{BLR},\mathrm{out}}=2.7{R}_{\mathrm{BLR}}$ and the MT as ${R}_{\mathrm{MT},\mathrm{out}}=4.0{R}_{\mathrm{MT}}$. In our model, since we assume the external ambient fields ${u}_{\mathrm{ex}}(r)={u}_{\mathrm{BLR}}(r)+{u}_{\mathrm{MT}}(r)$, we argue that the choices of the inner boundary radii of the BLR and the MT have a negligible influence.

The numerical results show that the energy dissipation region could be determined by parameter $r={x}_{0}+x$ through reproducing the γ-ray spectra. It can be seen from Table 1 that modeling the SEDs of the sample give ${x}_{0}\gt 0.1$ pc. Since the numerical results show that the MeV–GeV γ-ray emission region depends on the parameters x₀ with a constraint of x ≲ x₀, we could use the parameter r ∼ x₀ to define the radius of the γ-ray emission region, that is, the parameter x₀ could trace the location of γ-ray emission site from SMBH. Due to the characteristic distance ${R}_{\mathrm{BLR}}$ and ${R}_{\mathrm{MT}}$ being assumed to relate to the luminosity of an accretion disk L_d, we show the location of the γ-ray emission site from SMBH (x₀) as a function of the luminosity of an accretion disk (L_d) for the sample sources in Figure 9, where the sample sources are exhibited as open circles, the red solid and blue dashed lines show the characteristic distances of BLR and MT, and the shaded regions show the extended ranges of BLR and MT, respectively. It can be seen that (1) the γ-ray-emitting region is located at the range from 0.1 to 10 pc; (2) the γ-ray-emitting region is located outside the BLR and within the MT; and (3) the γ-ray-emitting region is located closer to the MT range than the BLR.

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