2.1 Derivation of the seed factor

In this work, we adopt the seed factor approach to distinguish the location of $\gamma$ -ray emission region. Following Georganopoulos et al. (Reference Georganopoulos, Meyer and Fossati2012), we have peak energies of synchrotron radiation and EC scattering in the observer’s frame,

(1) \begin{align} \epsilon^{\mathrm{obs}}_{\mathrm{syn}} = \frac{B}{B_{\mathrm{cr}}}\gamma^2_{\mathrm{b}}\delta/(1+z), \end{align}

(2) \begin{align} \epsilon^{\mathrm{obs}}_{\mathrm{EC}} = \frac{4}{3}\epsilon_{\mathrm{0,ext}}\gamma^2_{\mathrm{b}}\delta^2/(1+z), \end{align}

respectively (Coppi & Blandford Reference Coppi and Blandford1990; Tavecchio et al. Reference Tavecchio, Maraschi and Ghisellini1998; Ghisellini & Tavecchio Reference Ghisellini and Tavecchio2008a), where B is the magnetic field strength in units of Gauss; $\gamma_{\mathrm{b}}$ is the break Lorentz factor of relativistic electrons responsible for the SED peaks; $\epsilon_{\mathrm{0,ext}}$ is the dimensionless energy of ambient soft photons in the AGN frame; $B_{\mathrm{cr}} = m_e^2c^3/e\hbar$ is the critical magnetic field strength; $\delta = [\Gamma(1-\beta \cos \theta_{\mathrm{obs}})]^{-1}$ is the Doppler factor, where $\Gamma$ is the bulk Lorentz factor, $\beta c$ is the jet speed, and $\theta_{\mathrm{obs}}$ is the viewing angle. In this paper, by assuming $\theta_{\mathrm{obs}} \lesssim 1 / \Gamma$ , we have $\delta \approx \Gamma$ . It is worth noting that equation (2) is only applicable within the Thomson regime.

Dividing equation (1) by equation (2), we obtain

(3) \begin{align} \frac{B}{\delta} = \frac{4B_{\mathrm{cr}} \epsilon_{\mathrm{0,ext}} \epsilon^{\mathrm{obs}}_{\mathrm{syn}}}{3\epsilon^{\mathrm{obs}}_{\mathrm{EC}}}. \end{align}

And the peak luminosities of synchrotron radiation and EC scattering in the observer’s frame can be written as

(4) \begin{align} L^{\mathrm{obs}}_{\mathrm{syn,p}} = \frac{4}{3}\sigma_{\mathrm{T}}c\beta \gamma_{\mathrm{b}}^2n(\gamma_{\mathrm{b}})U_{\mathrm{B}}\delta^4, \end{align}

(5) \begin{align} L^{\mathrm{obs}}_{\mathrm{EC,p}} = \frac{4}{3}\sigma_{\mathrm{T}}c\beta \gamma_{\mathrm{b}}^2n(\gamma_{\mathrm{b}})U_{\mathrm{ext}}\delta^4, \end{align}

respectively (Blumenthal & Gould Reference Blumenthal and Gould1970; Rybicki, Lightman, & Tayler Reference Rybicki, Lightman and Tayler1981), where $\sigma_{\mathrm{T}}$ is the Thomson cross section; $n(\gamma_{\mathrm{b}})$ is the electron density distribution at $\gamma_{\mathrm{b}}$ ; $U_{\mathrm{B}} = B^2/8\pi$ is the energy density of the magnetic field. Here, the energy density of the ambient photon fields in the comoving frame can be calculated as

(6) \begin{align} U_{\mathrm{ext}} = \frac{17}{12}U_{\mathrm{0,ext}}\Gamma^2, \end{align}

where $U_{\mathrm{0,ext}}$ is the energy density in the AGN frame (Ghisellini & Tavecchio Reference Ghisellini and Tavecchio2008c; Ghisellini & Tavecchio Reference Ghisellini and Tavecchio2008b).

Take the ratio of equations (4) and (5), we then get

(7) \begin{align} \frac{B^2}{\delta^2} = \frac{34 \pi U_{\mathrm{0,ext}}}{3\textit{CD}}, \end{align}

where $\textit{CD} = \textit{L}^{\mathrm{obs}}_{\mathrm{EC,p}}/\textit{L}^{\mathrm{obs}}_{\mathrm{syn,p}}$ is the Compton dominance.

Combining equations (3) and (7), we ultimately derive the seed factor as

(8) \begin{align} {\textit{SF}} = \log(\frac{\sqrt{\textit{U}_{\mathrm{0,ext}}}}{\epsilon _{\mathrm{0,ext}}}) \approx \log(9\,863 \times \frac{\nu ^{\mathrm{obs}}_{\mathrm{syn,13}}}{\nu ^{\mathrm{obs}}_{\mathrm{EC,22}}}\sqrt{\textit{CD}}). \end{align}

Here, $\nu^{\mathrm{obs}}_{\mathrm{syn,13}}$ is the peak frequency of synchrotron radiation in units of 10¹³ Hz and $\nu^{\mathrm{obs}}_{\mathrm{EC,22}}$ is the peak frequency of EC scattering in units of 10²² Hz.

2.2 Characteristic values of the seed factor

As the ambient photon fields, the BLR and DT were discussed in the former seed factor approach (Georganopoulos et al. Reference Georganopoulos, Meyer and Fossati2012; Harvey et al. Reference Harvey, Georganopoulos and Meyer2020; Huang et al. Reference Huang, Li, Liao, Huang, Li, Qian, Pei and Fan2022). In addition, some studies revealed the significance of CMB and starlight (Böttcher et al. Reference Böttcher, Dermer and Finke2008; Potter & Cotter, Reference Potter and Cotter2013a,b,c). In this work, we comprehensively consider the seed factors of BLR, DT, CMB, and starlight. The accretion disc is out of consideration, because it is not suitable to this method.

To calculate the characteristic seed factor of BLR, the energy density $U_{\mathrm{0,ext}}$ and the dimensionless energy $\epsilon_{\mathrm{0,ext}}$ of the soft photons need to be determined. The typical size of BLR is $R_{\mathrm{BLR}} \approx 1 \times 10^{17} L_{\mathrm{d,45}}^{1/2} \ \mathrm{cm}$ , where $L_{\mathrm{d, 45}}$ is the luminosity of the accretion disc in units of $10^{45} \mathrm{erg \ s^{-1}}$ (Kaspi et al. Reference Kaspi, Brandt, Maoz, Netzer, Schneider and Shemmer2007; Bentz et al. Reference Bentz, Peterson, Netzer, Pogge and Vestergaard2009). The covering factor of BLR (the fractions of the disc luminosity $L_{\mathrm{d}}$ reprocessed into the BLR radiation) is $\xi_{\mathrm{BLR}} = 0.1$ (Ghisellini & Tavecchio Reference Ghisellini and Tavecchio2009). We then obtain the energy density $U_{\mathrm{0, BLR}} = \xi_{\mathrm{BLR}}L_{\mathrm{d}}/4\pi R_{\mathrm{BLR}}^2c = 2.65\times10^{-2}\mathrm{erg \ cm^{-3}}$ within the characteristic distance. The BLR can be regarded as a grey body with peak frequency of $1.5 {\nu}_{{\mathrm{Ly}}_{\alpha} }$ , then the dimensionless photon energy is $\epsilon_{\mathrm{0,BLR}} = 3\times10^{-5}$ (Tavecchio & Ghisellini Reference Tavecchio and Ghisellini2008). Finally, with a 5% uncertainty, we derive the characteristic seed factor of BLR as ${\textit{SF}}_{\mathrm{BLR}} = 3.74 \pm 0.19$ .

The typical size of DT is found to be $R_{\mathrm{DT}} \approx 2.5\times10^{18} L_{\mathrm{d,45}}^{1/2} \ \mathrm{cm}$ (Ghisellini & Tavecchio Reference Ghisellini and Tavecchio2009; Pei et al. Reference Pei, Fan, Yang, Huang and Li2022). In this work, we set the covering factor of DT as $\xi_{\mathrm{DT}} = 0.5$ (Ghisellini & Tavecchio Reference Ghisellini and Tavecchio2009). Then the energy density of DT within the characteristic distance is $U_{\mathrm{0, DT}} = \xi_{\mathrm{DT}} L_{\mathrm{d}}/4\pi R_{\mathrm{DT}}^2 c = 2.12 \times 10^{-4} \mathrm{erg\ cm^{-3}}$ . In the studies, the DT, which can also be approximated by a grey body, is endowed with three different temperatures, for example, 80K (Lopez-Rodriguez et al. Reference Lopez-Rodriguez2018), 370K (Ghisellini & Tavecchio Reference Ghisellini and Tavecchio2009), 1 500K (Almeyda et al. Reference Almeyda, Robinson, Richmond, Vazquez and Nikutta2017; Lyu & Rieke Reference Lyu and Rieke2018). Then we obtain three dimensionless photon energies for the DT, which are $\epsilon_{\mathrm{0,DT}}^{\mathrm{80\,K}} = 5.30 \times 10^{-8},\ \epsilon_{\mathrm{0,DT}}^{\mathrm{370K}} = 2.45\times 10^{-7}, \ \epsilon_{\mathrm{0,DT}}^{\mathrm{1\,500\,K}} = 9.94\times 10^{-7}$ . Considering the 5% uncertainty, three distinct characteristic seed factors of DT can be described as follows: ${\textit{SF}}_{\mathrm{DT}}^{\mathrm{80\,K}} = 5.44 \pm 0.27, \ {\textit{SF}}_{\mathrm{DT}}^{\mathrm{370\,K}} = 4.77\pm 0.24,\ {\textit{SF}}_{\mathrm{DT}}^{\mathrm{1\,500\,K}} = 4.17 \pm 0.21$ .

For CMB, the energy density is $U_{\mathrm{CMB}} = 4.02\times10^{-13}\mathrm{erg\ cm^{-3}}$ and the typical temperature is $T_{\mathrm{CMB}} = 2.72\,{\mathrm{K}}$ in the observer’s frame, respectively (Böttcher et al. Reference Böttcher, Dermer and Finke2008). In this case, the characteristic seed factor of CMB with 5% uncertainty is ${\textit{SF}}_{\mathrm{CMB}} = 2.55\pm0.13$ . The energy density for starlight is $U_{\mathrm{0,SL}} = 8.01 \times 10^{-13} \mathrm{erg\ cm^{-3}}$ and the typical temperature is $T_{\mathrm{SL}} = 30\,{\mathrm{K}}$ . Then we get the characteristic seed factor of starlight with 5% uncertainty ${\textit{SF}}_{\mathrm{SL}} = 1.65\pm0.08$ (H. E. S. S. Collaboration et al. 2017).

When applying the seed factor approach, there are several caveats that should be kept in mind. Firstly, the preceding derivation of the seed factor is within the Thomson regime. As a result of $\gamma \epsilon \lt 1/4$ (Moderski et al. Reference Moderski, Sikora, Coppi and Aharonian2005), the corresponding peak frequency of the EC radiation belonging to BLR, DT, CMB and starlight must be less than $1.03\times 10^{25}[\epsilon_0(1+z)/10^{-6}]^{-1}\,\mathrm{Hz}$ . Since high-energy component of LSP usually peaks around 1 GeV, the EC process associated to BLR occurs in the Klein-Nishina regime, while others are cooling in the Thomson regime. Then the available energy density of BLR could reduce and the actual characteristic seed factor of BLR would be smaller than the above derived one. Secondly, the energy density of the BLR and DT could be smaller at farther site as proposed by Hayashida et al. (Reference Hayashida2012), that is,

(9) \begin{align} U_{\mathrm{0,BLR}}(r) = \frac{\xi_{\mathrm{BLR}} L_{\mathrm{d}}}{4\pi R^2_{\mathrm{BLR}}c[1+(r/R_{\mathrm{BLR}})^3]}, \end{align}

and

(10) \begin{align} U_{\mathrm{0,DT}}(r) = \frac{\xi_{\mathrm{DT}} L_{\mathrm{d}}}{4\pi R^2_{\mathrm{DT}}c[1+(r/R_{\mathrm{DT}})^4]},\end{align}

where r is the distance between the dissipation region and the central black hole, both energy densities of the BLR and DT have been transformed into the AGN frame (see also Fig. 1). Then the actual characteristic seed factor could also be smaller if the emission region is beyond the typical distance. Since the above derived characteristic seed factor of DT is the largest one among these four photon fields, the actual seed factor of DT covers that of the others. For example, the actual seed factor of DT could decrease to about 3.5 and equal to the actual seed factor of BLR. Therefore, the above derived seed factor of DT but not of BLR, CMB, or starlight is effective. If the observed seed factor is approximated to the derived $\textit{SF}_{\mathrm{DT}}$ , it can be ascertained that the DT dominates the soft photon fields.

Figure 1. Energy density distribution of the broad line region (BLR) and the dusty torus (DT). $L_{\mathrm{d}} = 1\times 10^{45} {\mathrm{erg\ s^{-1}}}$ is adopted.

2.3 The fitting of spectral energy distribution

As shown in equation (8), the observed seed factor can be determined by extracting the peak frequencies and luminosities associated with the two humps. Then we fit each SED by the quadratic and cubic functions, respectively. Namely,

(11) \begin{align} \left\{ \begin{aligned} &\log(\nu F _{\nu}) = a_2(\log \nu)^2 + b_2\log \nu + c_2,\\ &\log(\nu F _{\nu}) = a_3(\log \nu)^3 + b_3(\log \nu)^2 + c_3\log \nu + d_3. \end{aligned}\right.\end{align}

We use two kinds of functions because some SEDs possess high symmetry but others do not, which causes difference on the parameters (Xue et al. Reference Xue2016). Markov-chain Monte Carlo (MCMC) analysis is employed since it returns robust uncertainties on the parameters (Speagle Reference Speagle2019). It works by randomly sampling from the posterior distribution, which are derived from the product of prior distribution and likelihood function. We use an uninformative uniform prior distribution because we have little knowledge about the parameters of quadratic and cubic functions in advance. Conservatively, it is expressed by

(12) \begin{align}p(m) = \left\{ \begin{aligned} &\frac{1}{1\,000}, &{\mathrm{if}}\ m_0-500 \lt m \lt m_0+500\\ &0, &\mathrm{otherwise} \end{aligned}\right.\end{align}

where m denotes the paremeters in both quadratic and cubic functions, such as $a_2, b_2 ...$ And $m_0$ is the preprocessed m derived by numpy.polyfit.Footnote ^a The likelihood function is written as (Yamada et al. Reference Yamada, Uemura, Itoh, Fukazawa, Ohno and Imazato2020)

(13) \begin{align} {\textit{L}} = \prod_{i=1}^n \frac{1}{\sqrt{2 \pi {\sigma_i}^2}} \exp{\left(-\frac{ ({\nu} F _{\nu,i} - \nu F _{\nu}(\nu_i))^2}{2\sigma^2_i} \right)}.\end{align}

Here, $\sigma_i$ is the Gaussian error of data point i and n is the number of data points in each energy hump. The emcee Python packageFootnote ^b (version 3.1.2, Foreman-Mackey et al. Reference Foreman-Mackey, Hogg, Lang and Goodman2013) is utilised to perform the MCMC algorithm. This package employs an affine invariant MCMC ensemble sampler with interdependent chains (Goodman & Weare Reference Goodman and Weare2010). While there is no fixed number of samples needed to make the convergence, we evaluate the convergence by inspecting the corner plot of parameters. The autocorrelation time, also the time that the chain ‘forgets’ where it started, range from 35 to 250 in our Python program. Conservatively, we adopt 32 walkers (chains) initialised by the above preprocessed values with a $10^{-10}$ Gaussian error, run 17 000 steps, burn 2 000 steps, and thin by 25. The results of posterior distribution are presented in Fig. 2.

While fitting the SEDs, we also compare the goodness of the two different function models with modified Akaike Information Criterion (AICc; Akaike Reference Akaike1974; Burnham & Anderson Reference Burnham and Anderson2002), written as

(14) \begin{align} {\mathrm{AICc}} = -2\ln(\hat{L}) + 2k + \frac{2k^2+2k} {n-k-1}.\end{align}

In this formula, $\hat{L}$ is the maximum likelihood, which corresponds to the maximum posterior since we used an uniform prior, and k is the number of free parameters. This criterion is chosen because the sample size of data points is small. AICc evaluates the loss of information during the fitting. The smaller AICc means the better model. Actually, the additional parameters in a model should improve the fitting to the data points, but we should also consider the increase in model complexity which causes overfitting. In this case, the AICc includes a penalty.

Figure 2. Posterior distribution from the Markov-chain Monte Carlo analysis of OD 166’s low-energy (synchrotron) hump with quadratic function. We present only one corner plot here, those of other SEDs are available in machine-readable form.

2.4 Parameters of the emission region

By fitting the SEDs of blazars, the observed peak frequencies and peak luminosities are determined. Then we select the blazars whose seed factors fall within the $\textit{SF}_{\mathrm{DT}}$ . With the gathered information on variability timescales and Doppler factors, we can deduce the other physical parameters of the $\gamma$ -ray emission region. For blazars, we have such a formula (Tavecchio et al. Reference Tavecchio, Maraschi and Ghisellini1998; Nalewajko, Begelman, & Sikora Reference Nalewajko, Begelman and Sikora2014):

(15) \begin{align} \frac{L^{\mathrm{obs}}_{\mathrm{syn}}}{U_{\mathrm{B}}} = \frac{L^{\mathrm{obs}}_{\mathrm{EC}}}{U_{\mathrm{ext}}} = \frac{L^{\mathrm{obs}}_{\mathrm{SSC}}}{U_{\mathrm{syn}}}.\end{align}

Here, $L^{\mathrm{obs}}_{\mathrm{syn}}$ , $L^{\mathrm{obs}}_{\mathrm{EC}}$ , $L^{\mathrm{obs}}_{\mathrm{SSC}}$ are the luminosities of the synchrotron, EC, and SSC radiations, respectively. $L^{\mathrm{obs}}_{\mathrm{syn}}$ and $L^{\mathrm{obs}}_{\mathrm{EC}}$ are determined by the integral of the low-energy and high-energy humps, respectively. For simplicity, we boldly assume that $L^{\mathrm{obs}}_{\mathrm{SSC}} = 10 \times L^{\mathrm{obs}}_{\mathrm{X}}$ , where $L^{\mathrm{obs}}_{\mathrm{X}}$ is the maximum luminosity in X-ray band. $U_{\mathrm{syn}} = L^{\mathrm{obs}}_{\mathrm{syn}}/( 4\pi R^2c\delta^4 )$ is the energy density of the synchrotron radiation. The radius of the $\gamma$ -ray emission region in the comoving frame can be estimated as

(16) \begin{align} R \approx ct_{\mathrm{var}}\frac{\delta}{1+z},\end{align}

where $t_{\mathrm{var}}$ is the variablility timescale.

If we have measured the variability timescale, we can derive the following parameters by combining the above formulas:

(17) \begin{align} \begin{cases} \delta = 6.39 \left( \frac{L^{\mathrm{obs}}_{\mathrm{syn}} L^{\mathrm{obs}}_{\mathrm{EC}}}{L^{\mathrm{obs}}_{\mathrm{SSC}} \cdot 10^{45} \mathrm{erg \ s^{-1}}} \frac{\mathrm{10^{-4}erg\ cm^{-3}}}{U_{\mathrm{0,ext}}}\right)^{1/8} \left(\frac{t_{\mathrm{var}}}{1\mathrm{d}(1+z)} \right)^{-1/4},\\[5pt] R = 1.66\times 10^{16} \left( \frac{L^{\mathrm{obs}}_{\mathrm{syn}} L^{\mathrm{obs}}_{\mathrm{EC}}}{L^{\mathrm{obs}}_{\mathrm{SSC}} \cdot 10^{45} \mathrm{erg \ s^{-1}} } \frac{\mathrm{10^{-4}erg\ cm^{-3}}}{U_{\mathrm{0,ext}}} \right)^{1/8} \left( \frac{t_{\mathrm{var}}}{1 \mathrm{d}(1+z)} \right)^{3/4},\\[5pt] B = 0.382 \left( \frac{L^{\mathrm{obs} \ 5}_{\mathrm{syn}}}{L^{\mathrm{obs} \ 3}_{\mathrm{EC}} L^{\mathrm{obs}}_{\mathrm{SSC}} \cdot 10^{45}\mathrm{erg\ s^{-1}}} \right)^{1/8} \left( \frac{U_{\mathrm{0,ext}}}{\mathrm{10^{-4}erg\ cm^{-3}}}\right) ^{3/8} \left( \frac{t_{\mathrm{var}}}{\mathrm{1d}(1+z)} \right)^{-1/4}.\end{cases}\end{align}

When the Doppler factor is measured, we obtain:

(18) \begin{align}\begin{cases} t_{\mathrm{var}} = 1.44\times10^{4} \left( \frac{L^{\mathrm{obs}}_{\mathrm{syn}} L^{\mathrm{obs}}_{\mathrm{EC}}}{L^{\mathrm{obs}}_{\mathrm{SSC}} \cdot 10^{45} \mathrm{erg\ s^{-s}}} \frac{\mathrm{10^{-4}erg\ cm^{-3}}}{U_{\mathrm{0,ext}}} \right)^{1/2} \left( \frac{\delta}{10} \right)^{-4} (1+z),\\[5pt] R = 4.33 \times 10^{15} \left( \frac{L^{\mathrm{obs}}_{\mathrm{syn}} L^{\mathrm{obs}}_{\mathrm{EC}}}{L^{\mathrm{obs}}_{\mathrm{SSC}} \cdot \mathrm{10^{45}erg\ s^{-1}}} \frac{\mathrm{10^{-4}erg\ cm^{-3}}}{U_{\mathrm{{0,ext}}}}\right) ^{1/2} \left( \frac{\delta}{10} \right) ^{-3},\\[5pt] B = 0.597 \left( \frac{L^{\mathrm{obs}}_{\mathrm{syn}}}{L^{\mathrm{obs}}_{\mathrm{EC}}} \frac{U_{\mathrm{0,ext}}}{\mathrm{10^{-4}erg\ cm^{-3}}}\right)^{1/2} \frac{\delta}{10}.\end{cases}\end{align}

Besides, $\gamma_{\mathrm{b}}$ can be calculated by Tavecchio et al. (Reference Tavecchio, Maraschi and Ghisellini1998)

(19) \begin{align}\gamma_{\mathrm{b}} = 5.2\times10^{-4} \left( \frac{\nu^{\mathrm{obs}}_{\mathrm{syn}}(1+z)}{B\delta} \right)^{1/2}.\end{align}

2.5 Constraint of the internal $\gamma \gamma$ -absorption

To make a comprison with the above derivation of physical parameters, we further make a constraint on $\delta$ through $\gamma\gamma$ -absorption (see also Dondi & Ghisellini Reference Dondi and Ghisellini1995). Due to the $\gamma\gamma$ annihilation, electron-positron pairs are generated. Applying Delta-approximation, the corresponding optical depth can be calculated as (Foffano et al. Reference Foffano, Vittorini, Tavani and Menegoni2022):

(20) \begin{align} \tau_{\gamma\gamma} = \sigma_{\gamma\gamma}n_{\mathrm{soft}}R,\end{align}

where $n_{\mathrm{soft}} = U_{\mathrm{soft}}/h\nu_{\mathrm{soft}}$ is the number density of the soft photons and $\sigma_{\gamma\gamma} = 1.68 \times 10^{-25} \mathrm{cm ^2}$ is the $\gamma \gamma$ -absorption cross section, which is assumed to be a constant when such a condition is fulfilled:

(21) \begin{align}\epsilon_{\mathrm{soft}}\epsilon_{\gamma} =2,\end{align}

where $\epsilon_{\mathrm{soft}}$ and $\epsilon_{\gamma}$ are the dimensionless energies of the soft photons and $\gamma$ -ray photons (comoving frame), respectively (Dermer & Menon Reference Dermer and Menon2009).

Since the $\gamma$ -ray is detected, optical depth must be less than 1. In order to solve the optical depth, energy density need to be determined. For internal soft photon fields such as synchrotron and IC radiation, we employ

(22) \begin{align} U_{\mathrm{soft}} = \frac{\nu L^{\mathrm{obs}}_{\nu, {\mathrm{soft}}}}{4\pi R^2c\delta^4}, \end{align}

where $\nu L^{\mathrm{obs}} _{\nu, {\mathrm{soft}}} = 4\pi d_{\mathrm{L}}^2 \nu F^{\mathrm{obs}}_{\nu,{\mathrm{soft}}}$ is the observed luminosity of the soft photons, $d_{\mathrm{L}}$ is the luminosity distance, and $\nu F^{\mathrm{obs}}_{\nu,{\mathrm{soft}}}$ is the observed flux. Then we derive

(23) \begin{align} \tau_{\gamma\gamma} = \frac{\sigma_{\gamma\gamma}d^2_{\mathrm{L}}\nu F^{\mathrm{obs}}_{\nu,{\mathrm{soft}}(1+z)}}{hc^2t_{\mathrm{var}}\delta^5\nu_{\mathrm{soft}}} \lt 1.\end{align}

Here, $\nu_{\mathrm{soft}}$ could be derived from equation (21) and expressed by

(24) \begin{align} \nu_{\mathrm{soft}} = \frac{2(m_{\mathrm{e}}c^2)^2}{h^2\nu_{\gamma}},\end{align}

where $\nu_{\gamma} = \nu_{\gamma}^{\mathrm{obs}}(1+z)/\delta$ is the frequency of $\gamma$ -ray in the comoving frame. Then we obtain the lower limit of $\delta$ :

(25) \begin{align} \delta \gt \left( \frac{h\sigma_{\gamma\gamma}d_{\mathrm{L}}^2\nu F_{\nu,{\mathrm{soft}}}^{\mathrm{obs}}\nu_{\gamma}^{\mathrm{obs}}(1+z)^2}{2m_{\mathrm{e}}^2c^6t_{\mathrm{var}}} \right)^{1/6}. \end{align}

Not only do internal photon fields constrain the physical parameter, but external photon fields also give an additional constraint on r. The absorption of DT could be omitted according to equation (21), since the corresponding $\gamma$ -ray ( $\nu_{\gamma}^{\mathrm{obs}} = 10^{27}(\nu_{\mathrm{DT}}/10^{13} \mathrm{Hz})^{-1}\mathrm{Hz}$ ) is beyond detection in our collected SEDs. However, the $\gamma$ -ray up to $10^{25}(\nu_{\mathrm{BLR}}/10^{15} \mathrm{Hz})^{-1}\mathrm{Hz}$ that is detectable could be absorbed by BLR. Therefore, we inspect the $\gamma\gamma$ -absorption of BLR by unfolding its frequency spectrum. Given that the BLR is a grey body, we have

(26) \begin{align} \frac{\mathrm{d}\textit{U}}{{\mathrm{d}} \nu} = \frac{8\pi h \nu^3}{c^3}(e^{h\nu / k_{\mathrm{B}}T}-1)^{-1}, \end{align}

where $T = h\nu_{\mathrm{BLR}}/3.93k_{\mathrm{B}}$ is the characteristic temperature of BLR and $k_{\mathrm{B}}$ is the Boltzmann constant (Ghisellini & Tavecchio Reference Ghisellini and Tavecchio2009). From the combination of equations (9) and (26), we derive the energy density of BLR as a function of both r and $\nu$ :

(27) \begin{align} U_{\mathrm{BLR}}(r,\nu) = \Gamma^2 U_{\mathrm{0,BLR}}(r)\nu \frac{\mathrm{d} \textit{U}}{{\mathrm{d}} \nu} / \int \frac{\mathrm{d} \textit{U}}{{\mathrm{d}} \nu}{\mathrm{d}}\nu.\end{align}

With the same condition about $\tau \lt 1$ and lower limit of $\delta$ derived from equation (25), we could get the lower limit of r:

(28) \begin{align} r \gt R_{\mathrm{BLR}} \left( \frac{\sigma_{\gamma\gamma}R}{h} \frac{\xi_{\mathrm{BLR}} \Gamma^2L_{\mathrm{d}}}{3\pi R^2_{\mathrm{BLR}}c} \frac{\mathrm{d} \textit{U}}{{\mathrm{d}} \nu} \bigg|_{\nu = \nu_{\mathrm{soft}}}/ \int\frac{\mathrm{d} \textit{U}}{{\mathrm{d}} \nu}\mathrm{d}\nu -1 \right)^{1/3}.\end{align}