Characterizing the emission region property of blazars

In the leptonic frame, the higher energy emission results from the inverse Compton scattering of internal or external soft photons, namely the synchrotron self-Compton (SSC) and external Compton (EC) process. The SSC process is believed to dominate the radiation in BL Lacs jets, while the EC process dominates over the SSC process for FSRQs jets. The synchrotron power is mainly produced by those electrons with Lorentz factor of $\gamma_{\rm p}$ and contribute most to the synchrotron peak, and the synchrotron peak frequency in the observer frame is given by

where $B$ in units of Gs (Tavecchio et al., 1998). Correspondingly, the synchrotron peak luminosity in the observer frame is expressed as (Chen, 2018)

where $\sigma_{\rm T}$ is the Thomson cross section, $U_{\rm B}$ is the magnetic field energy density ($U_{\rm B}=B^{2}/8\pi$), $N_{\rm 0}$ is the normalization parameter of the EED. Following Chen (2018), a three-parameter log-parabolic function is employed to describe the EED in this work

$b$ is the curvature and has a relation with the synchrotron bump, $b\simeq 5P_{\rm 1}$ (see, eg., Massaro et al., 2006; Chen, 2014). This EED is only phenomenologically assumed to follow the log-parabola shape of SED, without taking into account the evolution due to injection and cooling effects.

Moreover, if the electrons in the Thomson regime, the peak frequency of the SSC component is

In the case of the EC process, soft photons are fed externally, the peak frequency is given by

where, $\nu_{\rm ext}$ is the frequency of external photons, $\nu_{\rm ext}=2.46\times 10^{15}\,{\rm Hz}$ for the case of external photons coming from the broad line region (BLR) and $\nu_{\rm ext}=7.7\times 10^{13}\,{\rm Hz}$ for the case of external photons coming from the dusty torus (DT) (Tavecchio & Ghisellini, 2008; Ghisellini & Tavecchio, 2015).

The magnetic field and the electron energy are crucial to the radiation model of blazar jets. In a leptonic model, one can estimate these two parameters, $B$ and $\gamma_{\rm p}$, through equations 1, 4 and 5 in both SSC and EC processes. We calculated $B$ and $\gamma_{\rm p}$ for the 1900 blazars, including 1141 BL Lacs and 759 FSRQs, the SSC process is assumed to be the emission case of BL Lacs and the EC process is considered for the FSRQs. An average replacement is used for those sources without available redshift or Doppler factors, $\langle z_{\rm F}\rangle=1.21$ and $\langle\delta_{\rm F}\rangle=17.47$ for FSRQs, $\langle z_{\rm B}\rangle=0.528$ and $\langle\delta_{\rm B}\rangle=11.26$ for BL Lacs. In addition, for the EC process in FSRQs, we consider the external photons from the DT for those HSPs, while the external photons with an origination of the BLR for those LSPs and ISP. In this case, we have 754 FSRQs using the model of external soft photons coming from the BLR, this is consistent with the assumption of soft photon origin for FSRQs in literature (e.g., Tan et al., 2020), and also consistent with the fact that the FSRQs show significant broad emission lines and the emission lines contribute to the EC component significantly (Xiao et al., 2022a). The rest of the 5 FSRQs are considered as HSPs, and the HSPs are naturally considered as TeV candidates (Zhu et al., 2023). The TeV emission could be severely absorbed by interacting with BLR soft photons, thus we assume these 5 FSRQs with soft photons from the DT. The results of $\log B$ and $\log\gamma_{\rm p}$ are listed in columns (9) and (10) of Table 1, and are displayed in Figure 1. The Gaussian fit is applied to the distributions, and the fitting results give a mean value $\log B^{\rm B}=-0.51$ with a standard deviation of 1.66 for BL Lacs and a mean value $\log B^{\rm F}=1.76$ with a standard deviation of 0.85 for FSRQs; a mean value $\log\gamma_{\rm p}^{\rm B}=4.17$ with a standard deviation of 0.32 for BL Lacs and a mean value $\log\gamma_{\rm p}^{\rm F}=2.01$ with a standard deviation of 0.43 for FSRQs. A two-sample Kolmogorov–Smirnov (K–S) test, is employed to test whether each parameter for these two subclasses is from the same parent distribution. In the K-S test, the probability smaller than the critical value ($p=0.05$) would be used to reject the null hypothesis, which is that the two distributions are coming from the same parent distribution. Our results of K-S tests show $p\sim 0$ for both $\log B$ and $\log\gamma_{\rm p}$ distributions, suggest the BL Lac $\log B^{\rm B}$ distribution and the FSRQ $\log B^{\rm F}$ distribution come from different parent distributions, as well as for the BL Lac $\log\gamma_{\rm p}^{\rm B}$ distribution and the FSRQ $\log\gamma_{\rm p}^{\rm F}$ distribution.

Besides, we calculate the range of both $\log B$ and $\log\gamma_{\rm p}$ for the 808 BCUs in our sample, the limits are obtained by assuming the seed photons come from either the SSC or from the EC. During the calculation, the average replacement is used for those BCUs without available redshift and Doppler factors. The results are also listed in columns (9) and (10) of Table 1.

The jet emission region size ($R$) is usually constrained by a causality reason, which is the variability time scale, and expressed as

$R$ is easily obtained if we assume a variability timescale $\Delta t$ for those blazars with available $z$ and $\delta$. Throughout this paper, we assume $\Delta t=1\,{\rm day}$ (Nalewajko, 2013; Fan et al., 2013) to estimate $R$. Consider a constant and symmetric jet geometry, if the full jet cross section is responsible for the emission region diameter, then the distance from the central SMBH to the emission region $d_{\rm em}$ and the emission region size $R$ are correlated as

where $\phi$ is the semi-aperture opening angle of the jet (Acharyya et al., 2021). $\phi$ is an undetectable quantity, the value of $\tan\phi=0.1$ is fixed to study the blazar jet power in Ghisellini & Tavecchio (2009) and $\tan\phi\lesssim 0.25$ is suggested in Dermer et al. (2009). In this work, we can assume this opening angle $\phi$ to be close to the viewing angle $\theta$ for blazars ($\theta\simeq\phi$) due to the fact that the blazar jet is pointing at observers. Actually, $\theta$ can be estimated by using the apparent velocity of the resolved jet components and expressed as

where $\beta_{\rm app}$ is the apparent velocity of the jet component (Kellermann et al., 2004; Liodakis et al., 2018; Xiao et al., 2019), which is usually observed via the Very Long Baseline Interferometry (VLBI) technique (Lister et al., 2009, 2019, 2021). We manage to collect $\theta$ from Liodakis et al. (2018) for 182 blazars (35 BL Lacs and 147 FSRQs) of our sample and list them in column (5) of Table 2, and show them in Figure 2. We notice that there are 142 of 147 FSRQs with $\tan\!\theta<0.25$, taking 96.9%, there are 30 of 35 BL Lacs with $\tan\!\theta<0.25$, taking 85.7%, . Then we calculate the $d_{\rm em}$ for these 182 sources and list the results in column (6) of Table 2, and show them in Figure 3. The distribution of the emission region distance gives a mean value $\log d_{\rm em}^{\rm B}=17.37$ with a standard deviation of 0.91 for BL Lacs and $\log d_{\rm em}^{\rm F}=17.49$ with a standard deviation of 0.88 for FSRQs. A K-S test result of $0.31$ suggests that the $\log d_{\rm em}^{\rm B}$ and $\log d_{\rm em}^{\rm F}$ could come from the same distribution.

In the system of a blackhole-based jet, the particle-field relations depend on uncertain jet formation, particle acceleration, and radiation mechanisms (Dermer et al., 2014). It is natural that systems with interacting components often tend to equipartition. As a synchrotron source, blazar jet contains relativistic electrons with some energy density $U_{\rm e}$ and a magnetic field whose energy density is $U_{\rm B}=B^{2}/(8\pi)$. In order to avoid involving those poorly understood microphysics processes in the jets, Dermer et al. (2014) assumed a condition that the equipartition between magnetic-field and non-thermal electron energy densities holds for blazar jets, as it was used in the analysis of a wide variety of astrophysical systems; see Burbidge (1959) for the radio lobes, Pacholczyk (1970) for a study of radio sources, and Beck & Krause (2005) for a theoretical equipartition study of synchrotron observations. On large scales, the equipartition also corresponds to the minimum jet power condition as suggested by Ghisellini & Celotti (2001), in which the minimum jet power is required to produce the radiation we observe at all wavelengths. The simplest equipartition relation is $U_{\rm e}=\xi U_{\rm B}$, $\xi$ is close to the unity (Ghisellini & Celotti, 2001; Dermer et al., 2014). It is believed that the jet is in global equipartition between $U_{\rm e}$ and $U_{\rm B}$, however, local out-of equipartition is also allowed to explain peculiar observation phenomenon. The very high energy (VHE) detection of 3C 279 (MAGIC Collaboration et al., 2008) required either emission from leptons far out of equilibrium accompanied by poor fits to the X-ray or synchrotron data in the leptonic frame. In 2018, Fermi observed a characteristic peak-in-peak variability pattern on time scales in minutes of 3C 279 resulting from magnetic reconnection (Shukla & Mannheim, 2020).

In Figure 1, we notice that the FSRQs show a stronger magnetic field than the BL Lacs, while the latter shows a higher energy of electrons than the former. This is consistent with the typical blazar radiation mechanism paradigm, in which FSRQ contains a photon-rich environment so that the energy of relativistic electrons in the emission blob could efficiently dissipate via the IC process. And, BL Lacs is believed to be located in a photon-starving environment, thus the accelerated relativistic electrons can be well preserved. Moreover, the distribution of $\log\gamma_{\rm p}$ shows a clear separation between the FSRQs and the BL Lacs. This result reveals the energy of the relativistic electrons is distributed in a wide range, which contains large variance for different types of blazars, the FSRQs taking the side of lower energy while the BL Lacs taking the side of higher energy. We suggest using the intersection point value, $\log\gamma_{\rm p}=3.20\pm 0.01\,(\gamma_{\rm p}\simeq 1.6\times 10^{3})$, of the two Gaussian profiles to divide FSRQs and BL Lacs.

With known $B$, $\gamma_{\rm p}$, and $N(\gamma)$, one can obtain the magnetic energy density $U_{\rm B}$ and the electron energy density

where the minimum Lorentz $\gamma_{\rm min}=10$ and the maximum Lorentz factor $\gamma_{\rm max}=1\times 10^{6}$ of electrons is assumed, $m_{\rm e}$ is the rest mass of electron. Figure 4 shows the distribution of $\log U_{\rm e}$ and the ratio $\log(U_{\rm e}/U_{\rm B})$. The Gaussian fit gives results of a mean value $\log U_{\rm e}^{\rm B}=-1.47$ with a standard deviation of 3.12 for BL Lacs and $\log U_{\rm e}^{\rm F}=-4.40$ with a standard deviation of -2.14 for FSRQs; a mean value $\log(U_{\rm e}^{\rm B}/U_{\rm B}^{\rm B})=0.98$ with a standard deviation of 6.23 for BL Lacs and a mean value $\log(U_{\rm e}^{\rm F}/U_{\rm B}^{\rm F})=-6.45$ with standard deviation of 3.68 for FSRQs. K–S tests of $\log U_{\rm e}$ and of $\log(U_{\rm e}/U_{\rm B})$ for BL Lacs and FSRQs, and give both results of $p\sim 0$. The results suggest $\log U_{\rm e}^{\rm B}$ and $\log U_{\rm e}^{\rm F}$ are from different distributions, as well as for the distributions of $\log(U_{\rm e}^{\rm B}/U_{\rm B}^{\rm B})$ and $\log(U_{\rm e}^{\rm B}/U_{\rm B}^{\rm F})$. Based on our results shown in Figure 1 and 4, we suggest BL Lacs have an averagely larger energy and energy density of electron distribution, meanwhile, it also shows a larger electron-to-magnetic energy ratio than FSRQs.

Based on the distribution of $\log(U_{\rm e}/U_{\rm B})$ in Figure 4. It is clear that most of the FSRQs are away from the value $\log(U_{\rm e}/U_{\rm B})=0$, which is the condition of equipartition between magnetic-field and non-thermal electron energy densities, while the BL Lacs have a mean value near $\log(U_{\rm e}/U_{\rm B})=0$. Thus, our results suggest that the BL Lacs stay in a ‘quasi-equipartition’ state, while the FSRQs do not.

Variation, one of the characterizing properties of blazars, has been observed across all frequencies and time scales (eg., Urry, 1996; Dermer, 1999; Fan, 1999; Singh & Meintjes, 2020; Webb et al., 2021; Otero-Santos et al., 2022; Amaya-Almazán et al., 2022). Blazar variability time scales are observed from years to months, to days, and even to minutes (Wagner & Witzel, 1995; Fan et al., 1998, 2018; Aharonian et al., 2007; Albert et al., 2007). The basic idea of the mechanism for these different time scales of variabilities is that a disturbance created near the black hole travels outward with a Lorentz factor $\Gamma$ and radiates energy at a distance $\gtrsim\Gamma^{2}r_{\rm g}$, where $r_{\rm g}$ is the gravitational radius. This scenario works for most of the cases of variability in optical, X-ray, and even $\gamma$-ray bands, together with those models referring to the jet spiral structure, precession, or geometric effects in the jet (Gopal-Krishna & Wiita, 1992; Camenzind & Krockenberger, 1992), all the blazar variability seems mostly well explained. Minutes time scale variability observed in the TeV band (Aharonian et al., 2007; Albert et al., 2007) demands every efficient particle acceleration/dissipation mechanism (Shukla & Mannheim, 2020; Wang et al., 2022a) or extremely small emission zone. Although the variability mechanisms have been explored and discussed by many people, the initial seed instability of arising blazar variabilities, in different time scales, was few discussed.

A two-fluid model, which was proposed by (Sol et al., 1989), coupled with the Kelvin–Helmholtz instability could have a chance. In this scenario, two fluids (one fluid is a nonrelativistic jet with electron-proton plasma, and another one is a relativistic jet with an electron-positron plasma) with different speeds and compositions, the Kelvin–Helmholtz instability occurs at the junction of the two jet components and generate significant disturbances (Romero, 1995; Cai et al., 2022). But, this instability only happens when the magnetic field is weaker than the critical magnetic field strength

where $N$ is the particle number density that we obtained via equation 3, $\Gamma$ is assumed to be equal to $\delta$. We calculate the $B_{\rm c}$ for FSRQs and BL Lacs in our sample, and an average value replacement is applied to those sources without available $\delta$ during the calculation, the results are listed in column (12) of Table 1. Based on our calculation, there are 682 sources (taking 36% sources of the FSRQs and BL Lacs) that have a magnetic field weaker than its critical one. The result suggests that there is a significant number of Fermi blazars could show the Kelvin–Helmholtz instability in the emission zone. There are possible consequences of these instabilities. The instability could be amplified due to accumulation, thus showing detectable variability and the variability time scale is dependent on the energy dissipation efficiency. Or the instability could be generated in a manner of kink, and thus show detectable violent variability. For instance, these kink instabilities can disrupt the jet (Porth & Komissarov, 2015; Barniol Duran et al., 2017) and trigger magnetic reconnection (Giannios & Spruit, 2006; Shukla & Mannheim, 2020). The magnetic reconnection accelerates particles, and these particles dissipate rapidly in a very short timescale, e.g., the 3C 279 gamma-ray flare (Shukla & Mannheim, 2020).

With the assumption of two-fluid model Kelvin–Helmholtz instability, our results suggest that more than one-third of the blazars are able to show this instability. While the result is still modest if we consider observed variabilities are initially arising from the seed instabilities because almost all the observed blazars show strong variabilities although these variabilities happen in different bands and at different epochs. Possible reasons are the following (1) the SED fitting results of particle number density, that we applied to calculate $B_{\rm c}$, are biased by high states of blazars; (2) other mechanisms of seed instability, that were not considered in this work, may involve. However, one should keep in mind that even if all the blazars can generate instabilities, the strong and violent variability only happens occasionally because most of the instabilities fade before they are amplified to generate significant variabilities.

The location of the emission region is one of the essential properties in blazars. In the framework of the one-zone leptonic model, emissions in different bands are believed to be radiated in the same region. However, the location of the blazar emission region is controversial. The most common method to determine the location of the emission region is by assuming a constant jet geometry and taking the full jet cross section as the emission region diameter, then obtaining the distance between the emission region and the SMBH through the basic trigonometric relations. Foschini et al. (2011) used the $\sim$2 yr of Fermi-LAT observations to study the locations of several blazars and found the location should be within the BLR. While the expected spectral cut-off, arising from the photon-photon pair annihilation of $\gamma$-rays with the helium Lyman recombination continuum within the BLR (Poutanen & Stern, 2010), at the GeV band is not always observed. Not even the cut-off could also be explained by the other consequence, e.g., a break on the EED suggested by Dermer et al. (2015). Furthermore, the detection of TeV emission from blazars suggests the TeV emission region should locate outside the BLR because of the severe attenuation, the interactions between TeV photons and the photons in the BLR, would not allow us to observe the TeV emission from blazars.

In this work, we made our research on the location of the blazar emission region. We applied a timescale of 1 day, which was suggested by Nalewajko (2013) for as a typical variability timescale in the source frame in the Fermi $\gamma$-ray band, this timescale has been used in many works (e.g., Ghisellini et al., 1998; Nalewajko, 2013; Fan et al., 2013; Chen, 2018; Pei et al., 2022). One can set an upper limit on the emission region size via inequality 6, thus its distance from the SMBH can be estimated via equation 7. An approximation, which is assuming the viewing angle to be equal to the jet semi-opening angle, is proposed and employed in this work. We notice that most of our sources with semi-opening angles of $\tan\!\phi<0.25$, see the upper panel of Fig. 3. Our result is partly consistent with the range for the semi-opening angle, which is suggested as $0.1<\tan\!\phi<0.25$ (Ghisellini & Tavecchio, 2009; Dermer et al., 2009). Meanwhile, there are 116 of 147 FSRQs (taking 78.9%) and 20 of 35 BL Lacs (taking 57.1%) with $\tan\!\phi\leq 0.1$. In this case, our result suggests that maybe the viewing angle is smaller than the actual semi-opening angle and the line of sight should lie within the jet cone.

As we can see from the bottom panel of Fig. 3, the distributions show that the emission region is located at a distance of $1.68\times 10^{14}\,{\rm cm}$ to $6.61\times 10^{19}\,{\rm cm}$, corresponds to $5.4\times 10^{-5}\,{\rm pc}$ to $21.3\,{\rm pc}$ for all blazars. This large variance of 6 orders of magnitude directly arises from the variance of the viewing angle, from 0.04 for 4FGL J2035.4+1056 to 84.12 for 4FGL J0113.7+0225. The mean values of $\log d_{\rm em}^{\rm B}=17.37$ (0.076 pc) for BL Lacs and $\log d_{\rm em}^{\rm F}=17.49$ (0.1 pc) for FSRQs are very close to each other, and the result of K-S test confirmed that they are from the same distribution. Figure 5 shows a comparison of $\log d_{\rm em}$ from the present work and from Ghisellini & Tavecchio (2015), in which they estimated the $\log d_{\rm em}$ for 221 sources and gave an average value of $\sim 1\times 10^{17}\,{\rm cm}$ (0.03 pc), the comparison also shows that the $\log d_{\rm em}$ from the present work is larger than those from their work. The comparison result could arise from an underestimated semi-opening angle and thus an overestimated distance from the SMBH. In addition, different methods we employed to estimate the $\log d_{\rm em}$ could also arise the discrepancy.

Nevertheless, we check the emission region’s relative location with the BLR and with the DT. The distances from the BLR and from the DT to the SMBH are assumed to scale with the square root of the accretion disk luminosity,

and

where the $L_{\rm disk,\,45}$ is the accretion disk luminosity $L_{\rm disk}$ in units of $10^{45}\,{\rm erg\cdot s^{-1}}$ (Ghisellini & Tavecchio, 2008). The accretion disk luminosity can be estimated by the BLR luminosity $L_{\rm disk}\simeq 10L_{\rm BLR}$ (Calderone et al., 2013). We manage to collect the $L_{\rm BLR}$ from Xiao et al. (2022b) for 143 of the 182 blazars and list them in Table 2, we find that there are 66 (taking 46.2%) sources with emission regions located within the BLR, 63 (taking 44.0%) sources with emission regions located between the BLR and the DT, and 14 (9.8%) sources with emission regions located beyond the DT. Locating the emission region much further out of the DT is not appropriate because there are no important sources of external photons. The reason for the 14 sources with overestimated $d_{\rm em}$ is that small semi-opening angles, less than 0.1, are employed.

To sum up, our results suggest about half of the blazars with the emitting regions $d_{\rm em}<d_{\rm BLR}$ and another half with the emitting regions $d_{\rm BLR}<d_{\rm em}<d_{\rm DT}$. This is clearly inconsistent with the result suggested in Ghisellini et al. (2014), in which they suggested 85% sources with emission region located within the BLR and 15% sources with emission region located between the BLR and the DT. This discrepancy could be caused mainly by two reasons, namely, the sample size and the methods of estimating the $d_{\rm em}$. While the method influence can not be the main reason because the deviation caused by one particular method should evenly affect $d_{\rm em}$ for all sources in the sample and should not significantly change the portion of $d_{\rm em}$ within the BLR or within the DT. The best way to eliminate this discrepancy is to compile a larger/complete sample of blazars to estimate the distance from the SMBH to the emission region and determine the emission region’s relative location to the BLR or the DT.