A Method for Locating a High-energy Dissipation Region in a Blazar

The production site of gamma-rays in a blazar jet is an unresolved problem. We present a method to locate a gamma-ray emission region in the framework of a one-zone emission model. From measurements of the core-shift effect, the relation between the magnetic field strengths (B′) in the radio cores of the jet and the distances (R) of these radio cores from the central supermassive black hole (SMBH) can be inferred. Therefore, once the magnetic field strength in the gamma-ray emission region ( ) is obtained, one can use the relation of B′–R to derive the distance (Rdiss) of the gamma-ray emission region from the SMBH. Here, we evaluate the lower limit of by using the criteria that the optical variability timescale tvar should be longer than or equal to the synchrotron radiation cooling timescale of the electrons that emit optical photons. We test the method with the observations of PSK 1510-089 and BL Lacertae, and derive pc for PSK 1510-089 with tvar ∼ a few hours and pc for BL Lacertae with tvar ∼ a few minutes. Here, δD is the Doppler factor and A is the Compton dominance (i.e., the ratio of the Compton to the synchrotron peak luminosities).

B diss by using the criteria that the optical variability timescale t var should be longer than or equal to the synchrotron radiation cooling timescale of the electrons that emit optical photons. We test the method with the observations of PSK

Introduction
Blazars are one class of radio-loud active galactic nuclei (AGNs) pointing their relativistic jets at us. According to the features of optical emission lines, blazars are usually divided into two classes: BL Lacertae objects (BL Lac objects) with weak or even no observed optical emission lines and flatspectrum radio quasars (FSRQs) with strong optical emission lines (Urry & Padovani 1995). Multiwavelength radiations spanning from radio and optical to TeV gamma-ray energies have been observed from blazars. Blazar emission is generally dominated by nonthermal radiation from the relativistic jet. The broadband spectral energy distribution (SED) of a blazar has two distinct humps. It is generally believed that the first hump is the synchrotron radiation of relativistic electrons in the jet; however, the origin of the γ-ray hump is uncertain.
In blazar jet physics, an open question is the location of the gamma-ray emission region, which controls the radiative cooling processes in both leptonic and hadronic models. The location also means the place where the bulk energy of the jet is converted to an energy distribution of high-energy particles. Because the gamma-ray emission region is usually compact, it cannot be directly resolved by current detectors. Many methods have been proposed to constrain the location of the gamma-ray emission region (e.g., Jorstad et al. 2001Jorstad et al. , 2010Liu & Bai 2006;Tavecchio et al. 2010;Agudo et al. 2011;Dotson et al. 2012;Yan et al. 2012;Nalewajko et al. 2014;Böttcher & Els 2016). One popular method is to model the SEDs of FSRQs, and the location of the gamma-ray emission region (i.e., the distance from central back hole to the gamma-ray emission region) is treated as a model parameter (e.g., . Recently, some methods independent of SED modeling have been proposed to locate the gamma-ray emission region. Dotson et al. (2012) suggested that the energy dependence of the decay times in flare profiles could reflect the property of IC scattering. If the decay times depend on gamma-ray energies, it indicates that IC scattering happens in the Thomson region where the electron cooling time due to IC scattering depends on the energy of the electron. This situation will occur when the gamma-ray emission region locates in dust torus where the seed photons for IC scattering have a mean energy of ∼0.1 eV (Dotson et al. 2012(Dotson et al. , 2015Yan et al. 2016b). Moreover, the variability timescales of gamma-ray emission also provide hints on the location of the gamma-ray emission region. For instance, fast gamma-ray variability indicates that the emission region is very compact, which is usually thought to be close to the central black hole (e.g., Tavecchio et al. 2010).
Currently, there is no consensus on the location of the high energy dissipation region. The results given by the methods mentioned above are very inconsistent, from 0.01 pc to tens of parsecs (Nalewajko et al. 2014).
As suggested by Wu et al. (2018), we also use the relation of B′-R derived in the measurements of the radio core-shift effect (e.g., O'Sullivan & Gabuzda 2009;Sokolovsky et al. 2011;Zamaninasab et al. 2014) to constrain the location of high energy dissipation in blazars. In Wu et al. (2018), the magnetic field strength in the gamma-ray emission region was derived in modeling SED. Here, we use the optical variability timescale to constrain the magnetic field strength in the gamma-ray emission region, and therefore our method is fully independent of SED modeling.

Method
The variabilities of synchrotron radiations (e.g., variability timescale and time delay between emissions in different bands) have been suggested to estimate the comoving magnetic field (e.g., Takahashi et al. 1996;Böttcher et al. 2003).
Optical emission with fast variability from a blazar is believed to be the synchrotron radiation of relativistic electrons. If electron cooling is dominated by synchrotron cooling, the cooling timescale of electron in the comoving frame is given by (e.g., Tavecchio et al. 1998) is the energy density of the magnetic field in the comoving frame, m e is the mass of the electron, σ T is the cross section of Thomson scattering, and γ is the electron energy. Meanwhile, the observational synchrotron frequency is written as 1 is the Larmor frequency and z is the redshift.
The observational variability timescale t var can be taken as the upper limit for the cooling timescale in the observer frame cooling cooling D , i.e., t var t cooling . Then, we can get the lower limit for the magnetic field strength from Equations (1) and(2), i.e., where t var is in units of seconds and ν syn is in units of Hz. If electron cooling is dominated by EC cooling in the Thomson regime, the cooling timescale of the electron in Equation (1) should be modified by a factor of (1+k) −1 (Böttcher et al. 2003), and then the lower limit of ¢ B diss in Equation (3) is modified by a factor of (1+k) −2/3 . Here, k is the ratio of the energy densities between an external photon field and the magnetic field in the comoving frame. k can be replaced with Compton dominance A, i.e., the ratio of IC to synchrotron peak luminosity (Finke 2013).
In analogy to the calculation of A in Nalewajko & Gupta (2017), 5 A can be calculated as A=L 1−100 GeV /L optical , where L 1−100 GeV is the luminosity between 1 and 100 GeV and L optical is the optical luminosity.
Radio telescopes have the capability of resolving the structure of blazar jets on an approximately parsec scale. Very long baseline interferometry (VLBI) observations showed that coreshift effect (the frequency-dependent position of the VLBI cores) is common in AGNs. The core-shift effect is caused by synchrotron self-absorption (e.g., Blandford & Königl 1979). Under the condition of the equipartition between the jet particle and magnetic field energy densities, the core-shift effect can be used to evaluate the magnetic field strength along the jet, and a relation between the magnetic field strength (B′, in units of Gauss) and the distance along the jet (R, in units of pc) was found, i.e., B′∝(R/1 pc) −1 G (e.g., O'Sullivan & Gabuzda 2009; Zamaninasab et al. 2014).
Assuming that this relation still holds on in the subparsec scale of the jet, we then can use it and the lower limit for ¢ B diss derived by using optical variability to constrain the distance of gamma-ray emission region from supermassive black hole (SMBH; R diss ).

Results: Testing the Method with PKS 1510-089 and BL Lacertae
Here, we use observations of two blazars to test the feasibility of our method. During the VHE flare, its optical emission also presented activity. The R-band (the frequency is 4.5×10 14 Hz) flux decreased from 1.4×10 −11 erg cm −2 s −1 to 1.1×10 −11 erg cm −2 s −1 within ∼2 hr (see Figure 2 in Zacharias et al. 2017). We adopt the variability timescale t var ≈2 hr.
In hadronic models, the electron cooling in an FSRQ is dominated by synchrotron cooling (e.g., Böttcher et al. 2013;Diltz et al. 2015). Using Equation (3)   Taking δ D =30 and A∼k=20, we derive R diss <0.5 pc for hadronic models and R diss <3.5 pc for leptonic models. Note that A is sensitive to observing time for FSRQs, hence an A obtained from simultaneous observations should be used in practice.
The sizes of the broad-line region (BLR) and dust torus can be estimated with the disk luminosity L disk (Ghisellini & Tavecchio 2009;Ghisellini et al. 2014 (Castignani et al. 2017), and we obtain r BLR ≈0.1 pc and r DT ≈0.8 pc.
Because BLR photons will attenuate gamma-ray photons above ∼30/(1+z) GeV, 6 the detection of VHE photons from PKS 1510-089 indicates that its gamma-ray emission region should be outside the BLR. Our results support the scenario that the gamma-rays of PKS 1510-089 are produced in the dust torus. This is consistent with the result in Dotson et al. (2015).
This method is also used to constrain the location of the gamma-ray emission region in BL Lac objects. O'Sullivan & Gabuzda (2009) derived a relation of B′≈0.14·(R/1 pc) −1 G for BL Lacertae (2200+420; z=0.069). Covino et al. (2015) reported a very fast optical variability for this source. On 2012 September 1 the R-band flux decayed by a factor of about 3 in 5 minutes. For Fermi-LAT BL Lac objects, A is in the range from 0.1 to 3 (Finke 2013;Nalewajko & Gupta 2017). Abdo et al. (2011) showed that from a low state to a flare state, k varies from ∼0.1 to 3 for BL Lacertae. Using δ D =30 and k=3, we have R diss <0.01 pc for hadronic models and R diss <0.02 pc for leptonic models. Raiteri et al. (2009) estimated that the accretion disk luminosity L disk is 6×10 44 erg s −1 for BL Lacertae. The energy density of the photon field attributed to accretion disk radiation at R diss =0.02 pc is ∼0.4 erg cm −3 , which is much greater than the energy density of the BLR photon field of ∼0.01 erg cm −3 (e.g., Ghisellini & Tavecchio 2009;Hayashida et al. 2012). However, this situation prohibits the production of VHE photons because of the γ-γ absorption by accretion disk photons and BLR photons. Therefore, the detection of VHE photons (e.g., Arlen et al. 2013) cannot be accompanied by a fast optical variability with t var ∼5 minutes.
Here, we aim to present the constraints on R diss given by using various optical variability timescales, i.e., the feasibility of our method. From the above descriptions, one can find that our method is very effective, especially for the source having fast optical variability. In a specific study it is better to choose simultaneous optical flare with respect to gamma-ray emission to obtain variability timescales and Compton dominance, and the definition of the variability timescale should be clarified.

Discussion
Our method for locating the gamma-ray emission region relies on two assumptions: (1) optical and gamma-ray emissions are produced in the same region; and (2) the relation of B′-R obtained from radio core-shift measurements can be extrapolated into the subparsec scale of the jet.
In general, the first assumption still works in the current blazar science, although a class of orphan gamma-ray flares seems challenge the one-zone emission model (e.g., MacDonald et al. 2017). For the second assumption, O'Sullivan & Gabuzda (2009) extended the relation to a distance of 10 −5 pc at the SMBH (very close to the black hole jet-launching distance) and found that the extrapolated magnetic field strengths are in general consistent with that expected from theoretical models of magnetically powered jets (e.g., Blandford & Königl 1979). So far, the above assumptions are reliable.
Besides the two above assumptions, our constraint slightly relies on the value of the Doppler factor, d µ D 1 3 , while it depends on Compton dominance A in leptonic models, ∝(1+A) 2/3 . In addition to the relation of B′-R, our method only requires simultaneous γ-ray and optical observations.
The stringency of our constraint mainly depends on the precision of the measurement for the relation of B′-R. Pushkarev et al. (2012) and Zamaninasab et al. (2014) derived this relation for over 100 blazars by measuring the core-shift effect. Combining the measurement of this relation and the optical variability timescale, one can independently constrain the location of gamma-ray emission region in a blazar.
We use two 1 3 pc for hadronic models. Using a typical value of δ D =30 and a large enough value of A=20, we derive R diss <0.5 pc for hadronic models and R diss <3.5 pc for leptonic models.
For BL Lacertae, we use a very short optical variability timescale of t var ≈5 minutes reported in Covino et al. (2015)  2 3 pc for leptonic models. Using a typical value of δ D =30 and a large enough value of A=3, we have R diss <0.01 pc for hadronic models and R diss <0.02 pc for leptonic models.
One can see that with various optical variability timescales from minutes to a few hours, the high energy emission region can be located within a parsec or subparsec scale from the central black hole in the framework of a one-zone emission model. The lower limit for the distance of the emission region from the SMBH can be estimated by the absorption of GeV-TeV photons from low-energy photons around the jet (e.g., Liu & Bai 2006;Bai et al. 2009;Böttcher & Els 2016).
By modeling blazar SED, one can determine emission mechanisms and physical properties of the relativistic jets (e.g., Ghisellini et al. 2010Ghisellini et al. , 2014Kang et al. 2014;Zhang et al. 2015). In the previous studies, ¢ B diss , R diss , and other model parameters are fitted together. There are degeneracies between model parameters (see Yan et al. 2013 for correlations between model parameters given by the Markov Chain Monte Carlo fitting technique). Our method provides independent constraints for ¢ B diss and R diss , and break degeneracies between model parameters. This will lead to better understanding of emission mechanisms and physical properties of the relativistic jets.
It should be noted that the relation of B′-R is derived under the assumption of the equipartition between electron and magnetic field energy densities (e.g., O'Sullivan & Gabuzda 2009). On the aspect of SED modeling, it is found that the SEDs of FSRQs and low-synchrotron-peaked BL Lac (LBL) objects can be successfully fitted at the condition of (near-)equipartition between electron and magnetic field energy densities in leptonic models (e.g., Abdo et al. 2011;Yan et al. 2016a;Hu et al. 2017), while the leptonic modeling results for the SEDs of high-synchrotron-peaked BL Lac (HBL) objects are far out of equipartition (e.g., Dermer et al. 2015;Zhu et al. 2016). Therefore, for consistency, our method is applicable for FSRQs and LBL objects in the framework of leptonic models. The jet equipartition condition in hadronic models is rather complex. The modeling results are inconsistent (e.g., Böttcher et al. 2013;Diltz et al. 2015). It is unknown whether the (near)-equipartition condition could be achieved in hadronic models.

Conclusion
We presented an effective method for constraining the location of a gamma-ray emission region in a blazar jet in the framework of a one-zone emission model. Our method uses the relation of B′-R derived in the VLBI core-shit effect in the blazar jet. The lower limit for magnetic field strength in the gamma-ray emission region is estimated by utilizing the fact that the optical variability timescale should be longer than or equal to the synchrotron radiation cooling timescale of the electrons that produce optical emission. Then, the upper limit for the location of the gamma-ray emission region is derived with the relation of B′-R. Our method is applicable for LBL objects and FSRQs.