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The common superluminal nature of the first identified [MATH] -ray blazars, namely 3C 273, 3C 279, and PKS 0528+134, led Dermer et al. ( 1992 to propose a Compton-scattering origin for the [MATH] rays. In this model, jet electrons Compton-scatter accretion-disk photons that intercept the jet plasma. The nonthermal jet ... |
[MATH] -ray energies (Sikora et al., 1994 , as will radiation from a surrounding dusty torus (Kataoka et al., 1999 ; Błażejowski et al., 2000 . The accretion-disk and scattered radiation will attenuate jet [MATH] -rays through |
[MATH] pair-production attenuation (Becker & Kafatos, 1995 ; Blandford & Levinson, 1995 Expressions for the [MATH] -ray spectral energy distributions (SEDs) of blazars produced by Compton scattering processes have been derived and calculated for many specific models of the black-hole/blazar jet environment. In the case... |
and Błażejowski et al. ( 2000 . Detailed numerical calculations including both accretion disk and scattered radiation fields, have been made by, e.g., |
Kusunose & Takahara ( 2005 ); Böttcher & Bloom ( 2000 and Böttcher & Reimer ( 2004 Compton scattering in the Klein-Nishina regime is not so simple to treat compared to analyses restricted to the Thomson regime, but is unavoidable for blazar analysis in the era of the Fermi Gamma-Ray Space Telescope (formerly known as G... |
Georganopoulos et al. ( 2001 suggested to transform the comoving electron distribution to the stationary frame and then scatter the target photons to [MATH] -ray energies, using the formula first derived by Jones (Jones, 1968 ; Blumenthal & Gould, 1970 . This approach is generalized in this paper to surrounding anisotr... |
The usual spectral modeling approach proceeds by injecting power-law electrons and evolving these particles while they produce the output synchrotron and Compton-scattered radiation (e.g., Dermer & Schlickeiser, 1993 ; Böttcher et al., 1997 For example, Moderski et al. ( 2005 |
calculate electron energy evolution and spectral formation throughout the Thomson and Klein-Nishina regimes for different ratios of synchrotron and isotropic radiation field energy densities. They show that reduced Compton losses in the Klein-Nishina regime compared to synchrotron losses can lead to spectral hardening ... |
Here we extend a method of blazar analysis recently proposed for TeV blazars (Finke et al., 2008 that avoids these difficulties. For a standard |
[MATH] -ray blazar model, where isotropically distributed electrons spiral in a randomly oriented magnetic field with mean magnetic field strength [MATH] in the fluid frame, the measured synchrotron flux directly reveals the electron spectrum responsible for the synchrotron radiation. The only uncertainties are the mea... |
Georganopoulos et al. ( 2001 to surrounding anisotropic radiation fields. The temporally-evolving electron spectrum in blazars can be derived in this approach from simultaneous multiwavelength blazar data. Values of |
[MATH] [MATH] , and jet power can then be deduced. The related treatment for XBLs applied to PKS 2155-304, including more details about the derivation of the electron spectrum from the synchrotron component, the derivation and calculations of the SSC component and internal |
[MATH] -ray opacity by the synchrotron photons, is given by Finke et al. ( 2008 Analysis of blazar SEDs using this approach is presented in Section 2, where formulas to calculate Compton-scattered internal and external radiation and a [MATH] -function approximation for [MATH] |
opacity from the internal radiation field are given. Derivations of the Compton-scattered spectrum for specific examples of external radiation fields consisting of a monochromatic point source of radiation radially behind the jet, a Shakura-Sunyaev disk model, and a model BLR radiation field are derived in Section 3. D... |
2. One-Zone Synchrotron/Synchrotron Self-Compton Model with [MATH] Opacity We consider a one-zone model for blazar flares. Multiple zones could still be allowed, but the product of the duty cycle and number of zones would have to be small enough that interference of emissions from the different regions would still perm... |
A radiative event from the source emission region that varies on a comoving timescale [MATH] is related to the observed variability timescale through the relation [MATH] , where [MATH] is redshift; thus the comoving blob radius is |
[MATH] . The inequality allows us to neglect light-travel time effects from different parts of the emitting volume and avoid integrations over source volume. Within this zone, the nonthermal electrons with isotropic pitch angle distribution are described by the total comoving electron number spectrum [MATH] in terms of... |
2.1. Synchrotron and Self-Compton Components The [MATH] synchrotron radiation spectrum can be approximated by the expression [EQUATION] |
where [EQUATION] [MATH] is the luminosity distance, [MATH] is the speed of light, [MATH] is the Thomson cross-section, [MATH] is the source redshift, and the comoving magnetic-field energy density of the randomly-oriented comoving field with comoving mean intensity |
[MATH] is [MATH] We use [MATH] and [MATH] to refer to the dimensionless photon energy in the observer and comoving frame, respectively. Here and throughout this paper, unprimed quantities refer to the observer’s frame, and primed quantities refer to the frame comoving with the AGN’s jet, with the exception being [MATH]... |
[EQUATION] where [EQUATION] [MATH] is the ratio of [MATH] and the critical magnetic field [MATH] (Dermer & Schlickeiser, 2002 and [MATH] is the comoving volume of the blob. Note that [MATH] . Eq. ( ) gives a good representation to the source electron distribution when the [MATH] spectral index |
[MATH] (i.e., for spectra softer than [MATH] , adopting the convention [MATH] ) and away from the high-energy exponential cutoff of the spectrum (see Finke et al., 2008 , for comparison) |
The SSC [MATH] flux is given by [EQUATION] The formula of Jones ( 1968 (see also Blumenthal & Gould ( 1970 ) gives the SSC [MATH] flux, |
[EQUATION] where [EQUATION] [EQUATION] The synchrotron photons provide a target radiation field with spectral energy density [EQUATION] |
using eq. ( ). The scattered photon energy in the comoving frame is related to the observed photon energy [MATH] by the relation |
[EQUATION] From the limits on the integration over [MATH] implied by the limits on [MATH] we find [EQUATION] and [EQUATION] (see Finke et al., 2008 , for a detailed derivation of synchrotron/SSC models and application to blazars) Here the maximum lepton Lorentz factor injected into the radiating fluid is [MATH] , and t... |
[MATH] otherwise; the Heaviside function with one entry is defined such that [MATH] when [MATH] , and [MATH] otherwise. The [MATH] SSC spectrum is therefore given by |
[EQUATION] where [MATH] and [MATH] The maximum [MATH] SSC flux at photon energy [MATH] can be approximated in the Thomson limit by the expression |
[EQUATION] where the peak frequencies are related by [EQUATION] (Tavecchio et al., 1998 ; Finke et al., 2008 Here [MATH] is the [MATH] peak of the synchrotron component, which reaches its maximum at [MATH] |
Second-order SSC takes place when the SSC photons are again Compton scattered by electrons in the same blob, and may account for superquadratic variability of the [MATH] -ray flux with respect to the synchrotron flux (Perlman et al., 2008 . In principle, these photons can again be Compton scattered to arbitrarily highe... |
[MATH] with [MATH] in eq. ( 13 ), so that [EQUATION] 2.2. [MATH] Opacity Gamma-ray photons are subject to [MATH] attenuation by synchrotron photons produced in the radiating plasma, by ambient photons in the environment of the black hole (starred frame), and by photons of the intergalactic radiation field. The [MATH] p... |
[EQUATION] (Jauch & Rohrlich, 1976 ; Nikishov, 1961 ; Gould & Schréder, 1967 ; Brown, Mikaelian, & Gould, 1973 where [MATH] is the center-of-momentum frame Lorentz factor of the produced electron and positron, |
[MATH] [EQUATION] and [MATH] cm is the classical electron radius. The interaction angle [MATH] , given by the relation [EQUATION] |
is the angle between the directions of the photon detected with energy [MATH] and the target photon with energy [MATH] The absorption probability per unit pathlength is |
[EQUATION] For absorption by synchrotron photons within the radiating volume, [MATH] and [MATH] , and the target synchrotron radiation field is given by eq. ( ). In this case, the optically-thin [MATH] -ray emission spectrum is modified by the factor [MATH] for a spherical geometry, where |
[EQUATION] Here [MATH] is the total [MATH] optical depth integrated over pathlength. For absorption by ambient photons in the vicinity of the AGN, [MATH] is the photon energy in the AGN rest frame. For cosmic [MATH] absorption, the target photons are given by the spectrum of the intergalactic background light, which ev... |
3. Compton-Scattered External Radiation Fields In the one-zone model, the [MATH] spectrum of Compton-scattered external radiation fields is given by the Compton spectral luminosity [MATH] according to the relation |
[EQUATION] where [MATH] , from eq. ( 10 ), and [MATH] . The latter equality means that the photon direction is not deflected in transit to the observer. The Compton spectral luminosity is given by |
[EQUATION] [EQUATION] having already introduced the approximation that the scattered photon travels in the same direction as the relativistic scattering electron, i.e., |
[MATH] . Because of this approximation, the cosine of the angle [MATH] is given by eq. ( 19 ). The invariant collision energy [EQUATION] |
because [MATH] . The relation [MATH] gives the specific spectral number density of target photons with energy [MATH] , the starred quantities referring to the frame stationary with respect to the black hole. |
The Compton cross section in the head-on approximation is given by [EQUATION] (Dermer & Schlickeiser, 1993 ; Dermer & Böttcher, 2006 where |
[EQUATION] [EQUATION] [MATH] is given by eq. ( 24 ). The Compton spectral luminosity in the head-on approximation becomes [EQUATION] |
The lower limit on the electron Lorentz factor [MATH] and the upper limit [MATH] implied by the kinematic limits on [MATH] are [EQUATION] |
and [EQUATION] Eq. ( 28 ) is the starting point to calculate accurate Compton-scattered spectra involving relativistic electrons and external photon fields with arbitrary anisotropies. In contrast to the comoving electron spectrum used in the SSC calculation, the calculation of Compton-scattered radiation uses the elec... |
[MATH] for relativistic particles is given by [EQUATION] implying that [EQUATION] noting that [MATH] , and [MATH] when [MATH] , required for the head-on approximation. For an isotropic comoving distribution of electrons, |
[MATH] . Hence [EQUATION] or [EQUATION] In terms of the measured synchrotron [MATH] spectrum, eq. ( ), the source Compton spectrum for external Compton (EC) scattering in a standard one-zone model for blazars is, in general, given by the four-fold integral |
[EQUATION] with [EQUATION] using eq. ( ). The number of integrations can obviously be reduced by choosing symmetrical target photon geometries. |
3.1. Point Source Radially Behind Jet First we consider the flux when nonthermal electrons in a relativistic jet Compton-scatter photons from a point source of radiation, isotropically emitting and located radially behind the outflowing plasma jet. For a monochromatic point source with luminosity [MATH] |
and energy [MATH] the spectral luminosity can be expressed as [EQUATION] The spectral energy distribution of the target photon source at distance [MATH] from the point source is therefore given by |
[EQUATION] Substituting eq. ( 38 ) into eq. ( 33 and solving gives [EQUATION] Using eq. ( 22 ), eq. ( 39 ) becomes [EQUATION] or with eq. ( ), |
[EQUATION] where [MATH] is defined in eq. ( 36 ), [MATH] is defined by eq. ( 26 ) with [MATH] replaced by [MATH] and [EQUATION] Eqs. ( 40 ) and ( 41 ) give the Compton-scattered spectrum from a point source of radiation located radially behind the jet, generalizing the Thomson-regime result (Dermer et al., 1992 to incl... |
(Dermer & Schlickeiser, 2002 3.1.1 Reduction to the Thomson Regime We now derive the Thomson limit for the [MATH] spectrum, eq. ( 40 ). Because we consider relativistic electrons [MATH] , we are restricted to the condition |
[MATH] , which occurs according to eq. ( 42 when either [MATH] or [MATH] . The Thomson condition can be expressed as [MATH] , which is guaranteed when [MATH] , in which case [MATH] . Another statement of the Thomson condition is that [MATH] which, with [MATH] , again implies that |
[MATH] . Thus [EQUATION] For the scattering kernel, eq. ( 26 ), [MATH] and [MATH] in the Thomson regime, so [EQUATION] Away from the endpoints of the spectrum, [MATH] and |
[MATH] . Hence [EQUATION] defining [EQUATION] For the comoving electron distribution, eq. ( ), in the power-law form [EQUATION] eq. ( 45 ) becomes |
[EQUATION] [EQUATION] where the final expression applies in the regime [MATH] This can be written as [EQUATION] which can be compared with the Thomson-regime expression |
[EQUATION] (Dermer et al., 1992 ; Dermer & Schlickeiser, 1993 2002 , where [EQUATION] 3.1.2 Accurate Thomson Regime Spectrum Eq. ( 49 ) for the Thomson-scattered spectrum was derived assuming [MATH] away from the endpoints of the spectrum. Using the full expression for [MATH] |
and a power-law electron distribution gives an accurate expression for the spectrum of a localized jet of isotropically entrained electrons Thomson scattering a point source, monochromatic radiation field that enters the jet from behind. The result is |
[EQUATION] where [EQUATION] For the power-law electron distribution [MATH] , eq. ( 47 ), the accurate analytic Thomson-regime [MATH] flux from an isotropic monochromatic point source radiation field located behind the jet is |
[EQUATION] [EQUATION] where [MATH] and [MATH] In the asymptotic limit [MATH] eq. ( 54 ) approaches [EQUATION] where [EQUATION] The values of [MATH] , and [MATH] for [MATH] , and [MATH] respectively. For the Thomson approximation away from the endpoints of the spectrum, given by eq. ( 49 ), the corresponding coefficient... |
3.1.3 Solution with Compton Cross Section From eqs. ( 39 ) and ( 22 ), [EQUATION] Introducing [MATH] and [MATH] and changing variables to [MATH] gives, for the power-law electron distribution, eq. ( 47 ), |
[EQUATION] where [EQUATION] and [MATH] [MATH] The series solution of eq. ( 59 ) is given by [EQUATION] Eq. ( 59 ) can be solved analytically for integral [MATH] |
3.2. Shakura-Sunyaev Accretion Disk Field For the emission spectrum of an accretion disk surrounding the supermassive black hole, we consider the cool, optically-thick blackbody solution of Shakura and Sunyaev ( 1973 . The disk emission is approximated by a surface radiating at the blackbody temperature associated with... |
[EQUATION] where [MATH] is the efficiency to transform accreted matter to escaping radiant energy. The Eddington luminosity [MATH] ergs s -1 , where the mass of the central supermassive black hole is [MATH] and the black hole is accreting mass at the rate [MATH] (gm s -1 ). |
For steady flows where the energy is derived from the viscous dissipation of the gravitational potential energy of the accreting matter, the radiant surface-energy flux |
[EQUATION] (Shakura and Sunyaev, 1973 , where [EQUATION] [MATH] , and [MATH] for the Schwarzschild metric. Integrating equation ( 62 ) over a two-sided disk gives |
[MATH] . Assuming that the disk is an optically-thick blackbody, the effective temperature of the disk can be determined by equating equation ( 62 ) with the surface energy flux [MATH] A monochromatic approximation for the mean photon energy [MATH] at radius |
[MATH] of the accretion disk with mean temperature [MATH] is given by [EQUATION] so [EQUATION] where [EQUATION] Here [MATH] is the angle between the directions of the jet and the photon that intercepts the jet, |
[EQUATION] where [MATH] Lengths marked with a tilde are normalized to [MATH] . The final expression in eq. ( 64 ) is valid when [MATH] , though we assume it is reasonably accurate to [MATH] |
The intensity of the Shakura-Sunyaev accretion disk model along the jet axis is given by [EQUATION] (Shapiro & Teukolsky, 1983 . Thus |
[EQUATION] (Dermer & Schlickeiser, 2002 Substituting eq. ( 69 ) into eq. ( 33 ), using the relation [MATH] , gives [EQUATION] [EQUATION] |
The integration over angle in eq. ( 70 is limited by the inner radius of the accretion disk, so that [MATH] for a Schwarzschild black hole, and |
[EQUATION] and [EQUATION] The other limit on the angular integration arises because of the restriction given by eq. ( 30 ), so that |
[EQUATION] which restricts the integral to a maximum value of [MATH] and therefore [MATH] In the calculation of [MATH] , eq. ( 19 ), we take [MATH] without loss of generality because of the assumed azimuthal symmetry of the accretion-disk emission. |
The result for the accretion-disk radiation field scattered by isotropic, relativistic jet electrons is a 3-fold integral—reduced from a 4-fold integral by approximating the disk blackbody spectrum by its mean thermal energy at different radii. When expressed in terms of the measured synchrotron [MATH] |
spectrum using eq. ( ), the result for the accretion-disk radiation field is [EQUATION] [EQUATION] recalling the definitions of [MATH] and [MATH] from eqs. ( ) and ( 36 ), respectively. This can be reduced to a 2-fold integral by approximating a typical scattering as occurring at [MATH] so that [MATH] Because it is fea... |
3.2.1 Regimes in Compton-Scattered Accretion-Disk Spectra The limiting behaviors of the Compton scattered spectra can be understood based on simple [MATH] -approximations in the Thomson regime. External Compton scattering of the Shakura-Sunyaev disk with radiant luminosity [MATH] |
in the near field, i.e., [MATH] , can be approximated as [EQUATION] where [EQUATION] In the far field ( [MATH] ), [EQUATION] where |
[EQUATION] (Dermer & Schlickeiser, 2002 3.2.2 [MATH] Opacity from Accretion Disk Photons Photons from the accretion disk will interact with high energy |
[MATH] -rays to produce electron-positron pairs, modifying the very-high energy (VHE; multi-GeV – TeV) [MATH] -ray spectrum by a factor of [MATH] The absorption opacity [MATH] can be calculated by inserting the photon density, [MATH] for an accretion disk into eq. ( 20 ) and integrating [MATH] from [MATH] to [MATH] For... |
[EQUATION] where [MATH] is given by eq. ( 69 ). For an azimuthally symmetric accretion disk, the optical depth to [MATH] pair production attenuation for a photon with observed energy [MATH] traveling outward along the jet axis starting at height [MATH] |
is given by [EQUATION] where [MATH] and [MATH] For the Shakura-Sunyaev disk extending to the innermost stable orbit of a Schwarzschild black hole, one sees from eq. ( 80 and the definition of [MATH] , eq. ( 65 ), and [MATH] eq. ( 66 ), that |
[EQUATION] A first-order correction to [MATH] opacity for a photon traveling at a small angle angle [MATH] along the jet axis is obtained by replacing [MATH] with [MATH] which implicitly assumes that typical interactions take place at azimuth |
[MATH] A detailed calculation of the [MATH] opacity from accretion disk photons is given by Becker & Kafatos ( 1995 3.2.3 Numerical Results |
The electron distribution is assumed to be well-represented by the Band et al. ( 1993 -type function [EQUATION] [EQUATION] This distribution is essentially two smoothly joined power laws with power law number indices [MATH] and [MATH] , and low- and high-energy cutoffs, [MATH] and [MATH] , respectively, in the electron... |
The total jet power in the stationary frame of the host galaxy is given by [EQUATION] (Celotti & Fabian, 1993 ; Celotti et al., 2007 ; Finke et al., 2008 , where [MATH] is the total energy density in the jet, [MATH] is the jet power from particles, and [MATH] is the jet power from the magnetic field. Here the factor of... |
We performed a parameter study by varying model parameters with respect to a baseline model, with baseline parameters given in Table 1. We consider a [MATH] supermassive black-hole jet source located at redshift [MATH] . The jet is radiating at [MATH] from the black hole and has Doppler factor [MATH] and bulk Lorentz f... |
[MATH] the light crossing time for the Schwarzschild radius of a [MATH] black hole. The jet opening angle [MATH] for standard parameters. While varying the parameters, the synchrotron spectrum was kept relatively constant by varying |
[MATH] and [MATH] following the relations given in § 3.2.1 (except when changing angle). This was done with a [MATH] fitting technique to the baseline synchrotron spectrum |
(see Finke et al., 2008 . The transition between the near field and far field takes place at [MATH] , so that the baseline height of the jet is in the near field. |
The Compton-scattered accretion disk spectra are calculated from eq. ( 70 ). Fig. shows the effects of changing the magnetic field. With increasing [MATH] , fewer electrons are required to make the same synchrotron flux, so that both the first and second-order SSC flux, and the flux of the Compton-scattered accretion-d... |
With increasing disk power [MATH] , the accretion-disk radiation and also the Compton-scattered accretion-disk component becomes progressively larger, as shown in Fig. Note that the temperature of the disk photons at a given radius increases [MATH] . Fig. displays the dependence of the blazar SED on height [MATH] of th... |
[MATH] and [MATH] the SSC components are unaffected; eventually, the Compton-scattered disk flux falls below the SSC fluxes. The second-order SSC is visible when the external Compton flux decreases enough. Fig. shows how variations in [MATH] |
affect the low energy part of the synchrotron, SSC, and external Compton components. In Fig. , the Doppler factor is varied from the baseline value of 25, with the same approximate synchrotron flux. As [MATH] |
increases, the SSC component decreases, while the Compton-scattered accretion disk component increases. This behavior follows from eqs. ( 14 ) and ( 75 ), for the following reasons: For larger Doppler factors and fixed variability times, the radius becomes larger and fewer electrons are required to make the synchrotron... |
Fig. illustrates how the blazar SED is affected by changes in viewing angle. Contrary to Figs. ( ) – ( ), we let the synchrotron flux vary. In this calculation, [MATH] , and the observation angle and therefore [MATH] changes. It is interesting to note that the SSC flux varies least with changes in viewing angle, while ... |
[MATH] in the flat portion of the [MATH] SED, with the convention that flux density [MATH] . The relative magnitudes of the SSC components are explained from eq. ( 14 ), noting that |
[MATH] , so that [MATH] . The peak flux of the scattered disk component in the near-field regime, [MATH] , from eq. ( 75 ). A larger viewing angle or smaller Doppler factor implies a smaller size scale of the emitting region for a given variability time [MATH] The characteristic Thomson scattering depth is [MATH] for a... |
Fig. shows a calculation of the accretion-disk opacity [MATH] for photons traveling along the jet axis, using eq. ( 80 ) and the parameters given in Table 1. Here we assume that the Shakura-Sunyaev disk reaches to the innermost stable orbit ( [MATH] ) of a Schwarzschild black hole. At larger jet heights, the accretion-... |
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