External Compton and Internal Shock Model

# Seed Photon Fields of Blazars in the Internal Shock Scenario

## Abstract

We extend our approach of modeling spectral energy distribution (SED) and lightcurves of blazars to include external Compton (EC) emission due to inverse Compton scattering of an external anisotropic target radiation field. We describe the time-dependent impact of such seed photon fields on the evolution of multifrequency emission and spectral variability of blazars using a multi-zone time-dependent leptonic jet model, with radiation feedback, in the internal shock model scenario.

We calculate accurate EC-scattered high-energy spectra produced by relativistic electrons throughout the Thomson and Klein-Nishina regimes. We explore the effects of varying the contribution of (1) a thermal Shakura-Sunyaev accretion disk, (2) a spherically symmetric shell of broad-line clouds, the broad line region (BLR), and (3) a hot infrared emitting dusty torus (DT), on the resultant seed photon fields. We let the system evolve to beyond the BLR and within the DT and study the manifestation of the varying target photon fields on the simulated SED and lightcurves of a typical blazar. The calculations of broadband spectra include effects of absorption as -rays propagate through the photon pool present inside the jet due to synchrotron and inverse Compton processes, but neglect absorption by the BLR and DT photon fields outside the jet. Thus, our account of absorption is a lower limit to this effect. Here, we focus on studying the impact of parameters relevant for EC processes on high-energy (HE) emission of blazars.

BL Lacertae objects: general — Galaxies: jets — Hydrodynamics: — Radiation mechanism: non-thermal — Radiative transfer: — Relativistic processes
5

## 1 Introduction

Blazars are known for their highly variable broadband emission. They are characterized by a doubly humped spectral energy distribution (SED), attributed to non-thermal emission, and spectral variability. The SED and variability patterns can be used as key observational features to place constraints on the nature of the particle population, acceleration of particles, and the environment around the jet that is responsible for the observed emission. Conversely, incorporating the nature of the particle population and the jet environment, as accurately as possible, in modeling such observational features can enable us to reach a better agreement between theoretical and observational results. Thus, exploring the environment of a blazar jet in an anisotropic and time-dependent manner is important for connecting the pieces together and putting tighter constraints on the origin of -ray emission.

Blazars, a combination of BL Lacertae (BL Lac) objects and flat spectrum radio-loud quasars (FSRQs), are divided into various subclasses depending on the location of the peak of the low-energy (synchrotron) SED component. The synchrotron peak lies in the infrared regime, with Hz, in low-synchrotron-peaked (LSP) blazars comprising FSRQs and low-frequency peaked BL Lac objects (LBLs). In the case of intermediate-synchrotron-peaked (ISP) blazars, consisting of LBLs and intermediate-frequency peaked BL Lacs (IBLs), the synchrotron peak lies in the optical - near-UV region with Hz. The synchrotron component of high-synchrotron-peaked (HSP) blazars, which include essentially all high-frequency-peaked BL Lac objects (HBLs), peaks in the X-rays at Hz (Abdo et al., 2010; Böttcher, 2012). The high-energy (HE) component of blazars can be a result of inverse Compton (IC) scattering of synchrotron photons internal to the jet resulting in synchrotron self-Compton (SSC) emission (Bloom & Marscher, 1996). It could also be due to upscattering of accretion-disk photons (Dermer & Schlickeiser, 1993), and/or photons initially from the accretion disk being scattered by the broad-line region (BLR) (Sikora et al., 1994; Dermer, Sturner, & Schlickeiser, 1997), and/or seed photons from a surrounding dusty torus (DT) (Kataoka et al., 1999; Błażejowski et al., 2000). In the case of HBLs, the HE component is usually well reproduced with a synchrotron/SSC leptonic jet model (e.g., Finke, Dermer, & Böttcher, 2008; Aleksić et al., 2012), whereas an additional external Compton (EC) component is almost always required to fit the high-energy spectra of FSRQs, LBLs, and IBLs (e.g., Chiaberge & Ghisellini, 1999; Collmar et al., 2010).

Detailed numerical calculations for Compton scattering processes have been carried out for many specific models of blazar jet emission that involve their environment. Dermer & Schlickeiser (1993, 2002) have calculated Compton scattering of target photons in the Thomson regime from an optically thick and geometrically thin, thermal accretion disk based on the model of Shakura & Sunyaev (1973). Quasi-isotropic seed photon fields due to BLR or DT have also been considered to obtain Compton-scattered high-energy spectra in the Thomson limit by several authors (Sikora et al., 1994; Dermer, Sturner, & Schlickeiser, 1997; Błażejowski et al., 2000). On the other hand, extensive calculations involving anisotropic accretion-disk and BLR seed photon fields have been considered as well (Böttcher, Mause, & Schlickeiser, 1997; Böttcher & Bloom, 2000; Böttcher & Reimer, 2004; Kusunose & Takahara, 2005). Anisotropic radiation fields of the disk, the BLR, and the DT have been studied previously by Donea & Protheroe (2003), but primarily in the context of interaction of these photons with the GeV and TeV photons produced in the jet. Anisotropic treatment of BLR and DT photons, focussing on jet emission and rapid non-thermal flares, was carried out by Sokolov & Marscher (2005). These authors studied parameters describing the properties of BLR and DT that govern the interplay between the dominance of SSC and EC emission and their subsequent impact on SEDs, as well as relative time delays between light curves at different frequencies. For the purposes of their study, they used an integrated intensity - instead of considering line and continuum intensities separately - of the incident emission from the BLR. The emitting plasma was assumed to be located at parsec scales and the evolution of HE emission at sub-pc distances was ignored.

Recently, anisotropic treatment of disk and BLR target radiation fields has been considered by Dermer et al. (2009). The authors have calculated accurate -ray spectra due to inverse Comptonization of such seed photon fields throughout the Thomson and Klein-Nishina (KN) regimes to model FSRQ blazars, although in a one-zone scenario. Also, these authors evolve the system to only sub-pc distances along the jet axis, limiting themselves to locations within the BLR. In addition to this, one-zone leptonic jet models were recently shown (Böttcher, Reimer, & Marscher, 2009) to have severe limitations in attempts to reproduce very high energy (VHE) flares, such as that of 3C 279 detected in 2006 (Albert et al., 2008).

In a more recent approach, Marscher (2013) has considered an anisotropic seed photon field of the DT to calculate the resultant EC component of HE emission from blazar jets, in a turbulent extreme multi-zone scenario. While the -ray spectra are calculated throughout the Thomson and KN regimes, the energy loss rates are limited to only the Thomson regime. For the problem that work addresses, the system is located beyond the BLR, at parsec-scale distances from the central engine.

Here, we extend the previous approach of Joshi & Böttcher (2011, hereafter Paper 1), which calculated the synchrotron and SSC emission from blazar jets, to address some of the limitations of the models mentioned above. We use a fully time-dependent, 1-D multi-zone with radiation feedback, leptonic jet model in the internal shock scenario, shortened to the MUlti ZOne Radiation Feedback, MUZORF, model. We evolve the system from sub-pc to pc scale distances along the jet axis. We consider anisotropic target radiation fields to calculate the HE spectra resulting from EC scattering processes. The entire spectrum is calculated throughout the Thomson and KN regimes, thereby making it applicable to all classes of blazars. We include the attenuation of jet -rays through absorption (described in Paper 1) due to the presence of target radiation fields inside the jet, in a self-consistent manner. The generalized approach of our model lets us account for the constantly changing contribution of each of the seed photon field sources in producing HE emission in a self-consistent and time-dependent manner. This is especially relevant for understanding the origin of -ray emission from blazar jets.

In a number of previous analyses, the region within the BLR has been considered the most favorable location for -ray emission, with a range limited to between 0.01 and 0.3 pc Dermer & Schlickeiser (1994); Blandford & Levinson (1995); Ghisellini & Madau (1996). The reason behind this is the short intra-day variability timescales observed in some -ray flares, which indicated on the basis of light crossing timescales that the emission region is small and hence not be too far away from the central engine Ghisellini & Madau (1996); Ghisellini & Tavecchio (2009). At the same time, the emission region cannot be too close to the central engine without violating constraints placed by the absorption process Ghisellini & Madau (1996); Liu & Bai (2006). As a result, an emission region location closer to the BLR was considered the most favorable position due to the strong dependence of the scattered flux on the level of boosting and the energy of incoming photons (Sikora et al., 1994). Contrary to the above scenario, recent observations have shown coincidences of -ray outbursts with radio events on pc scales (e.g., León-Tavares et al., 2012; Jorstad et al., 2013). This seems to suggest a cospatial origin of radio and -ray events located at such distances. As a result, some authors conclude that the -ray emitting region could also lie outside of the BLR Sokolov & Marscher (2005); Lindfors, Valtaoja, & Türler (2005).

Thus, in order to understand the origin of -ray emission, it is important to let the system being modeled evolve to beyond the BLR and into the DT, and to include its contribution to the production of -ray emission. Here, we focus our attention toward understanding the dependence of -ray emission on the combination of various intrinsic physical parameters. We explore this aspect by including various components of seed photon fields in order to obtain a complete picture of their contribution in producing -ray emission and understand their effects on the dynamic evolution of SEDs and spectral variability patterns.

In §2, we describe our EC framework of including anisotropic seed photon fields from the accretion-disk, the BLR, and the DT. We lay out the expressions used to calculate accurate Compton-scattered -ray spectra resulting from the seed photon fields and the corresponding electron energy loss rates throughout the Thomson and KN regimes. In §3, we describe our baseline model, its simulated results, and the relevant physical input parameters that we use in the study. In §3.2, we present our results of the parameter study and discuss the effects of varying the input parameters on the simulated SED and lightcurves. We discuss and summarize our findings in §4. Throughout this paper, we refer to as the energy spectral index such that flux density, ; the unprimed quantities refer to the rest frame of the AGN (lab frame), primed quantities to the comoving frame of the emitting plasma, and starred quantities to the observer’s frame; the dimensionless photon energy is denoted by .

Appendix A delineates the details of line-of-sight calculations for the BLR line and diffuse continuum emission used in obtaining the intensity of incoming BLR photons.

## 2 Methodology

We consider a multi-zone time-dependent leptonic jet model with radiation-feedback scheme as described in Paper 1. We extend our previous model of synchrotron/SSC emission to include the EC component in order to simulate the SED and spectral variability patterns of blazars. We consider three sources of external seed photon fields, namely the accretion disk, the BLR, and the DT. We evolve the emission region in the jet from sub-pc to pc scales (within the DT) and follow the evolution of the SED and spectral variability patterns over a period of 1 day, corresponding to the timescale of a rapid nonthermal flare. Such a comprehensive approach can be used as an important tool for connecting the origin of -ray emission of a flare to its multiwavelength properties.

As in paper 1, we consider a cylindrical emission region for our current study. We assume the emitting volume to be well collimated out to pc distances, which is a safe assumption to make based on the work of Jorstad et al. (2005), and hence do not consider the effects of adiabatic expansion on the evolution of the strength of the magnetic field or the electron population in the emission region. The size of the emission region is assumed to be small in comparison to the sizes of and distances to the external seed photon field sources. This way, the external radiation can be safely assumed to be homogeneous throughout the emitting plasma, although it is still highly anisotropic in the comoving frame of the plasma. In our current framework, we do not simulate radio emission as the calculated flux is well below the actual radio value. This is because we follow the early phase of -ray production corresponding to a shock position upto 1 pc in the lab frame. During this phase, the emission region is highly optically thick to GHz radio frequencies.

The angular dependence of the incoming radiation and the amount to which it contributes toward EC emission is determined by the geometry of all three seed photon field sources and the location of the emission region along the jet axis. In addition, the anisotropy is further enhanced due to relativistic aberration and Doppler boosting or deboosting in the plasma frame. We assume the external radiation to be constant in time over the period of our simulation. Figure 1 depicts the geometry of all three external sources under consideration. The jet is oriented along the z-axis in a plane perpendicular to the plane of the central engine, which is composed of the black hole and the accretion disk surronding it. The central engine is enveloped by a BLR, considered to be a geometrically thick spherical shell, and is situated inside the cavity of the BLR. These sources are, in turn, encased by a puffed up torus containing hot dust.

In the following subsections, we discuss the sources of seed photons for EC scattering and delineate the expressions that we use to calculate the corresponding emissivities and energy loss rates throughout the Thomson and KN regimes.

### 2.1 The Accretion Disk

In order to calculate the EC scattering of photons coming directly from a central source, we consider an optically thick accretion disk that radiates with a blackbody spectrum, based on the model of Shakura & Sunyaev (1973). The blackbody spectrum is calculated according to a temperature distribution T(R) given by Eq. (4) of Böttcher, Mause, & Schlickeiser (1997). where R is the radius of the disk.

Figure 2 shows a schematic of the disk geometry and the angular dependence of the disk spectral intensity on the position of the emission region in the jet. We assume a multi-color disk and calculate the radius dependent quantity, (where  erg/K is the Boltzmann constant), in order to obtain the EC emissivity and the corresponding electron energy loss rate. The disk is assumed to emit in the energy range from optical to hard X-rays (10 keV), with a characteristic peak frequency of  Hz.

For sake of brevity, the subscript has been dropped from for the rest of this section. Now, the spectral surface energy flux at radius R is given by , where describes the spectrum of a blackbody radiation at radius R with temperature T(R) (Dermer & Schlickeiser, 1993). The differential number of photons produced per second between and and emitted from disk radius R and R + dR, , is given by

 dNdRdtdϵ=2π2RBν[T(R)]hϵ , (1)

where .

The differential spectral photon number density, , is then given by

 dnphdϵdR=˙N(ϵ,R)4πx2c=π2RBν[T(R)]x2chϵ , (2)

where and . Here and in the rest of the paper, a quantity is differential in the variables that are listed in parentheses. If the variables are preceded by a semicolon or only one variable is listed in the parentheses, then the quantity is parametrically dependent on such variable(s).

Converting into by realizing that and assuming azimuthal symmetry of the photon source, is given by

 dnphdϵdΩph=12ππBν[T(R)]2chϵηph . (3)

Using Eq. (3) and the invariance of (Rybicki & Lightman, 1979; Dermer & Schlickeiser, 1993; Böttcher, Mause, & Schlickeiser, 1997),

 n′ph(ϵ′,Ω′ph)=ϵ′2ϵ2nph(ϵ,Ωph) , (4)

we can obtain the anisotropic differential spectral photon number density, , in the plasma frame (Böttcher, Mause, & Schlickeiser, 1997)

 n′ph(ϵ′,Ω′ph)=12c3(mec2h)3ϵ′2⎛⎝eϵ′Γsh(1+βΓshη′ph)Θ(R)−1⎞⎠−11+βΓshη′phη′ph+βΓsh , (5)

Here, is the cosine of the angle that the incoming photon, emitted at radius , makes with respect to the jet axis at height z. The relevant Lorentz transformations are given by (Dermer & Schlickeiser, 1993)

 ϵ=ϵ′Γsh(1+βΓshη′ph) ηph=η′ph+βΓsh1+βΓshη′ph . (6)

The electron energy loss rate and photon production rate per unit volume due to inverse-Compton scattering of disk photons (ECD) can be calculated using Eq. (5). We use the approximation given in Böttcher, Mause, & Schlickeiser (1997) to calculate the energy loss rate of an electron with energy throughout the Thomson and KN regimes:

 −˙γ′ECD=π5r2e30c2(mec2h)3γ′2Γ2shRmax∫RmindR Θ4(R) R(x−βΓshz)2x4I(⟨ϵ′⟩,γ′,η′ph) , (7)

where, is given by either equation (15) or (16) of Böttcher, Mause, & Schlickeiser (1997) according to the regime it is being calculated in.

On the other hand, if all scattering occurs in the Thomson regime, , then the electron energy loss rate can be directly calculated using Eq. (12) of Böttcher, Mause, & Schlickeiser (1997). Figure 3 shows a comparison between the electron energy loss rate obtained using the full expression given in Eq. (7) and the Thomson expression using Eq. (12) of Böttcher, Mause, & Schlickeiser (1997) for ECD. The transition from the KN to Thomson regime is governed by the temperature of the accretion disk such that the transition electron Lorentz factor is given by , where is the maximum temperature of the disk for a non-rotating black hole (BH). The quantity is the gravitational radius corresponding to a BH of mass M (in units of ). The accretion rate of the BH is given by such that the total disk luminosity, , and the accretion efficiency, , are related to as . The Stefan-Boltzmann constant, . In the case of our baseline model, , implying .

As can be seen from the figure, the lines for ECD and ECD-Th do not overlap each other in the Thomson regime. This is due to the fact that Eq. (13) of Böttcher, Mause, & Schlickeiser (1997) was used to calculate the electron energy loss rate due to external Comptonization of disk photons. The expression employs an approximation for all electron energies in calculating the electron energy loss rate. According to the approximation, the exact value of the angle between the electron and the jet axis does not play an important role and could be taken to be perpendicular to the jet axis for all electron energies. Furthermore, the thermal spectrum emitted by each radius of the disk could be approximated by a delta function in energy. Hence, the resulting electron energy loss rate is slightly different from that obtained using the full KN cross-section for an extended source.

The Compton photon production rate per unit volume, in the head-on approximation with , can be calculated from Eqs. (23) and (25) of Dermer et al. (2009). Using the following relationship between spectral luminosity, emissivity, and photon production rate per unit volume (Dermer & Menon, 2009), we obtain

 ϵL(ϵ,Ω)=Vbϵj(ϵ,Ω) ϵj(ϵ,Ω)=mec2ϵ2˙n(ϵ,Ω) . (8)

After substituting the expression from Eq. (5), and converting in terms of R as mentioned above, we can obtain the ECD photon production rate per unit volume, in the plasma frame, under the head-on approximation as

 ˙n′ECD(ϵ′s,Ω′s)=3σT64πc21Γ2sh2π∫0dϕ′phRout∫RindRR(x−βΓshz)2 ϵ′max∫0dϵ′ϵ′eϵ′xΘ(R)Γsh(x−βΓshz)−1∞∫γ′lowdγ′n′e(γ′)γ′2ΞC , (9)

where the subscript stands for Compton scattered quantities, is the Thomson cross-section for an electron, and is the electron number density. The quantity is the solid-angle integrated KN Compton cross-section, under the head-on approximation (Dermer et al., 2009; Dermer & Menon, 2009) given by

 Ξc=γ′−ϵ′sγ′+γ′γ′−ϵ′s−2ϵ′sγ′ϵ′(1−cosΨ′)(γ′−ϵ′s)+ϵ′2sγ′2ϵ′2(1−cosΨ′)2(γ′−ϵ′s)2 (10)

The quantities and are given by

 ϵ′max=2ϵ′s(1−cosΨ′) and γ′low=ϵ′S2⎡⎣1+√1+2ϵ′ϵ′S(1−cosΨ′)⎤⎦ , (11)

where , given by Eq. (6) of Böttcher, Mause, & Schlickeiser (1997), is the cosine of the scattering angle between the electron and target photon directions. We take without loss of generality, based on the assumed azimuthal symmetry of electrons in the emission region. Eqs. (2.1), (21) (see §2.2), and (32) (see §2.3) are evaluated such that in the case of those scatterings for which , we use Eq. (44) of Dermer et al. (2009) to calculate the Compton cross-section in the Thomson regime under the head-on approximation.

For cases where all scattering occurs in the Thomson regime, we can substitute the following differential cross-section, in the head-on approximation, (Dermer & Menon, 2009):

 d2σCdϵ′SdΩ′S=σTδ(Ω′S−Ω′e)δ(ϵ′S−γ′ϵ′(1−βγ′cosψ′)) , (12)

where the subscript corresponds to electron related quantities, and Eq. (5) in the expression

 ˙n′EC(ϵ′S,Ω′S)=c4π∫4πdΩ′ph∞∫1dγ′n′e(γ′)∞∫0dϵ′n′ph(ϵ′,Ω′ph)(1−βγ′cosψ′)∫4πdΩ′ed2σCdϵ′SdΩ′S (13)

to obtain the Thomson regime photon production rate per unit volume in the plasma frame:

 ˙n′Th(ϵ′S,Ω′S)=σT8πc2(mec2h)3ϵ′2SΓ2sh∞∫1dγ′n′e(γ′)γ′62π∫0dϕ′ph11−βγ′cosψ′ Rout∫RindRR(x−βΓshz)2⎛⎜ ⎜⎝eϵ′SxΓshγ′2Θ(R)(x−βΓshz)(1−βγ′cosψ′)−1⎞⎟ ⎟⎠−1 . (14)

### 2.2 The Broad Line Region

Here we model the BLR as an optically thin and geometrically thick spherical shell, extending from radius to , with an optical depth (Donea & Protheroe, 2003) and a covering factor of the central UV radiation (Liu & Bai, 2006). We assume the BLR to consist of dense clouds, which reprocess a fraction of the central UV radiation to produce the broad emission lines (Liu & Bai, 2006; Dermer et al., 2009). For our purposes, we assume that the radial dependence of line emissivity is based on the best fit parameters (s = 1 and p = 1.5) of Kaspi & Netzer (1999), such that the number density of clouds and the radius of clouds , at distance from the BH. In addition, the BLR clouds Thomson scatter a portion of the central UV radiation into a diffuse continuum (Liu & Bai, 2006). The line emission and diffuse continuum can provide important sources of target photons that jet electrons scatter to produce -ray energies (Sikora et al., 1997; Dermer et al., 2009).

In order to obtain the EC component due to the seed photon field of the BLR, we need to calculate the anisotropic distribution of the BLR line and continuum emission. This can be achieved by integrating the line and continuum emissivities along the lines of sight through the BLR to obtain the corresponding intensities (Donea & Protheroe, 2003; Liu & Bai, 2006). We use Eqs. (12) and (13) of Liu & Bai (2006) to calculate the anisotropic intensity of radiation of line emission, , and diffuse continuum, (in units of ) as a function of distance from the central source and angle that the incoming photons make with the jet axis. Figure 4 represents the geometry of the BLR under consideration and the angular dependence of the intensity of radiation from the BLR at the position of the emission region in the jet. We consider three possible positions of the emission region (Donea & Protheroe, 2003) to calculate emissivities and corresponding intensities, , along the jet axis, where is the cosine of the angle that the incoming BLR photon makes with the jet axis. The calculations of these path lengths are described in Appendix A.

The anisotropic profile of emission line intensity obtained using the path length calculations, as described in Appendix A, at the three locations (marked in Figure 4) is shown in Figure 5. The anisotropic intensity due to diffuse BLR emission has a similar profile as that of emission line intensity, and is not shown here for the sake of brevity.

For the purposes of our study, we consider both the broad line emission and the diffuse continuum radiation to calculate the total radiation field of the BLR. The combined field provides the source of target photons for EC scattering (ECBLR) by jet electrons. The BLR is assumed to emit in the energy range from infrared (IR) to soft X-rays (3 keV), with a characteristic peak frequency of   Hz. For the sake of brevity, we drop the subscript from the equations for the rest of this section.

We consider 35 emission lines (34 components from Francis et al. (1991) and the H component from Gaskell, Shields, & Wampler (1981)) to estimate the total flux of broad emission lines. Using Eq. (19) of Liu & Bai (2006) and Eq. (4), we can obtain the differential line emission photon number density (in ) in the plasma frame:

 n′lineph(ϵ′line,Ω′ph;z)=C1Iline(z,μ)Γ4sh(1+βΓshμ′)4NV δ(ϵ′−ϵ′line)ϵ′ , (15)

where , is the dimensionless energy of the incoming photon corresponding to one of the 35 emission line components and is the line strength of each of those 35 components, with that of Ly arbitrarily set at 100 (Francis et al., 1991). We define from to in the plasma frame and use Eq. (2.1) to obtain for the lab frame.

Similarly, using Eqs. (19), (20), and (21) of Liu & Bai (2006) and Eq. (4), we can obtain the differential diffuse continuum photon number density (in ) in the plasma frame as

 n′contph(ϵ′cont,Ω′ph;z)=C2Icont(z,μ)ϵ′2(eϵ′Γsh(1+βΓshμ′)Θ−1)I , (16)

where and , corresponding to a blackbody temperature of K, which has been assumed for the inner region of the accretion disk (Liu & Bai, 2006). The quantity is the total blackbody spectrum, given by

 I=ϵmax∫ϵminϵ3dϵeϵΘ−1 , (17)

with corresponding to the photon frequency Hz, and corresponding to the frequency Hz (Liu & Bai, 2006). Thus, the total anisotropic differential spectral photon number density entering the jet from the BLR is

 n′ph(ϵ′,Ω′ph)=n′lineph(ϵ′,Ω′ph)+n′contph(ϵ′,Ω′ph) . (18)

The electron energy loss rate and photon production rate per unit volume due to ECBLR can be calculated using Eqs. (15), (16) and (18). We use Eq. (6.46) of Dermer & Menon (2009) to obtain the electron energy loss rate in the plasma frame. Substituting Eqs. (6.39) and (6.40) in Eq. (6.46) of Dermer & Menon (2009) yields

 ˙γ′=−3cσT8∫4πdΩ′ph∞∫0dϵ′n′ph(ϵ′,Ω′ph){ln(D)γ′ϵ′M2[(γ′−ϵ′)(M(M−2)−2)−γ′]+ 13ϵ′D3[1−D3+6ϵ′D(γ′−ϵ′)(1+M)(1−βγ′μ′)+ 6D2γ′M(2(γ′−ϵ′)D−γ′(M(M−1)−1))]} , (19)

where and . As can be seen from the above equation, the entire integral is independent of and can be solved analytically. Similarly, after substituting Eqs. (15) and (16) in Eq. (2.2), the integral can be solved analytically for due to the presence of the function in its expression (see Eq. 15). After having carried out these simplifications, Eq. (2.2) is solved numerically to obtain the final electron energy loss rate due to the ECBLR process.

In the case that all scattering occurs in the Thomson regime, , the electron energy loss rate can be directly calculated by substituting and into Eq. (6.46) of Dermer & Menon (2009). After carrying out integrations over and analytically for both and , the Thomson-regime electron energy loss rate expression for the ECBLR process is given by

 ˙γ′=−2πcσTC21∫−1dμ′(1−βγ′μ′)(γ′2(1−βγ′μ′)−1)Γ4sh(1+βΓshμ′)4[Iline(z,μ)+(πΘ)415IIcont(z,μ)] . (20)

Here, we have used the result . As can be seen from Fig. 3, the Thomson approximation for ECBLR deviates from the corresponding full expression at .

The ECBLR photon production rate per unit volume, under the head-on approximation, is calculated using Eq. (6.32) of Dermer & Menon (2009). We write it in terms of the differential photon production rate using Eq. (2.1), and substitute the expression for the differential photon number density of BLR photons from Eq. (18) to obtain

 ˙n′(ϵ′s,Ω′s)=3cσT32π2π∫0dϕ′ph1∫−1dμ′ϵ′max∫0dϵ′n′ph(ϵ′,Ω′ph)ϵ′∞∫γ′lowdγ′n′e(γ′)γ′2ΞC , (21)

where the quantities used in the above equation have been explained in §2.1. As mentioned in §2.1, the Compton cross-section in the above expression is evaluated such that in the case of scatterings for which , we use Eq. (44) of Dermer et al. (2009) to calculate it in the Thomson regime, under head-on approximation.

For cases where all scattering occurs in the Thomson regime, Eqs. (12) and (13) yield

 ˙n′Th(ϵ′S,Ω′S)=cσT4π∫4πdΩ′ph∞∫1dγ′n′e(γ′)(1−βγ′cosψ′)∞∫0dϵ′n′ph(ϵ′,Ω′ph) δ(ϵ′S−γ′2ϵ′[1−βγ′cosψ′]) . (22)

Solving for analytically, we obtain the Thomson regime photon production rate per unit volume as

 ˙n′Th(ϵ′S,Ω′S)=cσT4π∫4πdΩ′ph∞∫1dγ′n′e(γ′)γ′2n′ph(ϵ′Sγ′2(1−βγ′cosψ′),Ω′ph) . (23)

Now substituting Eqs. (15) and (16) in Eq. (23) and solving for the delta function (present in Eq. (15)) in the integral for the line emission part, we can obtain the final expression for the ECBLR Thomson-regime photon production rate per unit volume as

 ˙n′Th(ϵ′S,Ω′S)=cσT4π∫4πdΩ′ph[C12Iline(z,μ)Γ4sh(1+βΓshμ′)4√1−cosψ′ϵ′S∑ϵ′lineNVϵ′3/2linene⎛⎜⎝ ⎷ϵ′Sϵ′line(1−cosψ′)⎞⎟⎠ +C2Icont(z,μ)ϵ′2SI∞∫1dγ′n′e(γ′)γ′6(1−βγ′cosψ′)2⎛⎜ ⎜⎝eϵ′SΓsh(1+βΓshμ′)γ′2(1−βγ′cosψ′)Θ−1⎞⎟ ⎟⎠−1] . (24)

Where we evaluate the line emission term of the above equation for cases where so that and the integral can be solved analytically using the delta function that is present in the expression for the differential line emission photon number density.

### 2.3 The Dusty Torus

We consider a clumpy molecular torus (Sokolov & Marscher, 2005; Marscher, 2013) whose emission is dominated by dust and which radiates as a blackbody at IR frequencies at a temperature T = 1200 K (Malmrose et al., 2011) in the lab frame. The torus lies in the plane of the accretion disk and extends from to . As shown in Fig. 6, the central circle of the torus is located at a distance of from the central source and the cross-sectional radius of the torus is given by . We assume that the incident radiation comes from a portion of the inner surface of the torus. This portion, which is the covering factor, is dependent on the size of the torus.

The DT is assumed to emit IR photons with a characteristic peak frequency of the radiation field,   Hz. For the sake of brevity, we drop the subscript from all quantities listed in this section. The minimum, , and maximun, , incident angles constraining the incident emission from the torus are given by

 θmin=sin−1r√z2+r2−sin−1RT√z2+r2 θmax=sin−1r√z2+r2+sin−1RT√z2+r2 . (25)

These angles are subsequently transformed into the plasma frame according to:

 η′=cosθ′=cosθ−βΓsh1−βΓshcosθ . (26)

The covering factor of the dusty torus, , can be obtained in terms of the fraction, , of the disk luminosity, , that illuminates the torus such that, . Here we take as found for PKS 1222+216 by Malmrose et al. (2011). Also, the following relationship holds between , , and the illuminated area of the torus, :

 LDust=AobsσSBT4fcov , (27)

The illuminated area of the torus visible from a position in the jet is given by

 Aobs≈π24(R2out−R2in) , (28)

where the factor of 1/4 appears because only the front side of the inner torus is illuminated and only half of this is visible to the emitting region in the lab frame. Thus, for given values of , , and , the covering factor of the torus can be obtained from Eqs. (27) and (28). Conversely, for given values of and , we can also obtain the extent of the torus in terms of and the corresponding values of and .

Since the torus emits as a blackbody, the differential spectral photon number density in the plasma frame is simply given by

 n′ph(ϵ′,Ω′ph)=2(mech)3fcovϵ′2eϵ′Γsh(1+βΓshη′)Θ−1 , (29)

where we have used Eq. (4) to convert the differential photon density from the lab to the plasma frame. Fig. 7 shows the anisotropic intensity profile of the DT as a function of incident angle, , in the plasma frame.

We substitute Eq. (29) in Eq. (2.2) to calculate the electron energy loss rate due to ECDT, which yields

 ˙γ′=−3cπσT2(mech)3fcovη′max∫η′mindη′∞∫0dϵ′ϵ′2eϵ′Γsh(1+βΓshη′)Θ−1{ln(D)γ′ϵ′M2[(γ′−ϵ′)(M(M−2)−2)−γ′] +13ϵ′D3[1−D3+6ϵ′D(γ′−ϵ′)(1+M)(1−βγ′η′)+ 6D2γ′M(2(γ′−ϵ′)D−γ′(M(M−1)−1))]}. (30)

For cases where all scattering occurs in the Thomson regime, we follow the steps described in §2.2 to obtain Eq. (20), which yields the Thomson-regime electron energy loss rate for the ECDT process as

 ˙γ′=−4π5cσTΘ415Γ4sh(mech)3fcovη′max∫η′mindη′(1−βγ′η′)[γ′2(1−βγ′η′)−1](1+βΓshη′)4 . (31)

As shown in Fig. 3, the Thomson approximation for ECDT deviates from the corresponding full expression above . We substitute Eq. (29) in Eq. (21) to obtain the ECDT photon production rate per unit volume, under the head-on approximation, as

 ˙n′(ϵ′s,Ω′s)=3cσT16π(mech)3fcov2π∫0dϕ′phη′max∫η′mindη′ϵ′max∫0dϵ′ϵ′eϵ′Γsh(1+βΓshη′)Θ−1∞∫γ′lowdγ′n′e(γ′)γ′2ΞC . (32)

For scatterings occuring entirely in the Thomson regime, we substitute Eq. (29) in Eq. (2.2) to obtain the ECDT Thomson-regime photon production rate per unit volume,

 ˙n′Th(ϵ′S,Ω′S)=cσT2π(mech)3fcov2π∫0dϕ′η′max∫η′mindη′∞∫1dγ′n′e(γ′)γ′6(1−βγ′cosψ′)2⎡⎢ ⎢⎣eϵ′SΓsh(1+βΓshη′)Θγ′2(1−βγ′cosψ′)−1⎤⎥ ⎥⎦−1 .