A synchrotron self-Compton model with low energy electron cut-off for the blazar S5 0716+714
aa r X i v : . [ a s t r o - ph ] O c t Astronomy & Astrophysics manuscript no. 7962 c (cid:13)
ESO 2018November 16, 2018
A synchrotron self-Compton model with low energy electroncut-off for the blazar S5 0716+714
Olivia Tsang and J. G. Kirk
Max-Planck-Institut-f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, GermanyReceived . . . / Accepted . . .
ABSTRACT
Context.
In a self-absorbed synchrotron source with power-law electrons, rapid inverse Compton cooling sets in when thebrightness temperature of the source reaches T B ∼ K. However, brightness temperatures inferred from observations ofintra-day variable sources (IDV) are well above the ”Compton catastrophe” limit. This can be understood if the underlyingelectron distribution cuts off at low energy.
Aims.
We examine the compatibility of the synchrotron and inverse Compton emission of an electron distribution with low-energy cut-off with that of IDV sources, using the observed spectral energy distribution of S5 0716+714 as an example.
Methods.
We compute the synchrotron self-Compton (SSC) spectrum of monoenergetic electrons and compare it to the observedspectral energy distribution (SED) of S5 0716+714. The hard radio spectrum is well-fitted by this model, and the optical datacan be accommodated by a power-law extension to the electron spectrum. We therefore examine the scenario of an injection ofelectrons, which is a double power law in energy, with a hard low-energy component that does not contribute to the synchrotronopacity.
Results.
We show that the double power-law injection model is in good agreement with the observed SED of S5 0716+714.For intrinsic variability, we find that a Doppler factor of
D ≥
30 can explain the observed SED provided that low-frequency( <
32 GHz) emission originates from a larger region than the higher-frequency emission. To fit the entire spectrum,
D ≥
65 isneeded. We find the constraint imposed by induced Compton scattering at high T B is insignificant in our model. Conclusions.
We confirm that electron distribution with a low-energy cut-off can explain the high brightness temperature incompact radio sources. We show that synchrotron spectrum from such distributions naturally accounts for the observed hardradio continuum with a softer optical component, without the need for an inhomogeneous source. The required low energyelectron distribution is compatible with a relativistic Maxwellian.
Key words. galaxies: active – galaxies: high redshift – galaxies: jets – BL Lacertae objects: individual: S5 0716+714
1. Introduction
Observations of many extra-galactic radio sources havefound rapid flux variations at radio frequency (e.g.Kedziora-Chudczer et al. 2001), some of which fluctu-ate over a time scale of a day or less. They are re-ferred to as intra-day variable sources (IDV). The vari-ability time scale is often used to constrain the size ofthe source based on causality arguments. Using this con-straint, one can derive a variability brightness temperature(Wagner & Witzel 1995) T var = 4 . × F ν (cid:18) λd L t obs (1 + z ) (cid:19) K (1)where the flux density F ν , wavelength λ , luminosity dis-tant d L , and observed variability time scale t obs are mea-sured in Jy, cm, Mpc, and days, respectively. Send offprint requests to : O. Tsang, e-mail: [email protected]
The high radio flux frequently measured in IDVsources implies an extremely high brightness temper-ature, often many orders of magnitude above 10 K.Kellermann & Pauliny-Toth (1969) have shown that, as-suming the electron distribution follows a single powerlaw, the luminosity of the inverse Compton scattered pho-tons exceeds that of the synchrotron photons when thebrightness temperature of the source reaches ∼ K.Above this threshold, rapid cooling of the relativistic elec-trons due to inverse Compton scattering — the “Comptoncatastrophe” — forbids a further increase in the bright-ness temperature (see e.g. Kellermann 2002, for a recentreview of the brightness temperature problem). The limit-ing value is even lower, T B < K, if the magnetic fieldand particle energy density of the source is driven towardsequipartition (Readhead 1994). The observed variabilityin some sources can be interpreted as the result of extrin-sic effects, which, at first sight, relaxes the size constraint.
O. Tsang and J. G. Kirk: SSC model for S5 0716+714
For example, the flux variations of PKS 1519 −
273 andPKS 0405 −
385 are convincingly identified as interstellarscintillation. Nevertheless, all realistic models of the scin-tillation mechanism impose a new constraint on the sizeand require a brightness temperature of T B > K insome cases (Macquart et al. 2000; Rickett et al. 2002), farexceeding the limit imposed by the Compton catastrophe.A prevalent feature associated with IDV sources isa flat or inverted spectrum ( α ≤
0, with flux F ν ∝ ν − α ) at radio-millimeter wavelengths (e.g., Gear et al.1994; Kedziora-Chudczer et al. 2001). Optically thick syn-chrotron emission from power-law electrons rises as ν / ,too fast to account for the observed spectra. Opticallythin synchrotron emission in the scope of the conven-tional interpretation of the synchrotron theory has a flux F ν ∝ ν − ( s − / , where s is the power-law index of theelectrons (d N e / d γ ∝ γ − s ). If α = ( s − / ≤
0, the num-ber density of electrons diverges towards high γ . Imposinga high-energy cut-off in the electron spectrum avoids thedivergence and may account for the commonly observedspectral steepening at optical frequencies, but Marscher(1977) showed that electron spectra with s ≤ T B ∼ K at GHz frequencies is possiblewith only a moderate Doppler boosting factor of ∼
10. InTsang & Kirk (2007), we discussed the parameters of themonoenergetic model and showed that the assumption ofequipartition of energy in the source does not prevent theCompton catastrophe. We also showed that an injectionof highly relativistic electrons or strong acceleration in thesource cannot produce temperatures much higher than ourlimit due to copious electron-positron pair production.In this paper, we examine the spectral properties ofsynchrotron emission from monoenergetic electrons andfrom an electron distribution that is a double power lawin energy, by comparing the model spectra with the ob-servations of S5 0716+714, a BL Lac object that is one ofthe brightest known IDV sources, as well as a gamma-ray blazar (Hartman et al. 1999). In doing so, we as-sume that the dominant targets for inverse Comptonscattering are produced within the source (SSC model).The emission from gamma-ray blazars can also be in-terpreted in the context of models in which the tar-get photons are created externally (EC model), for ex-ample in the broad line region, the accretion disk, ora molecular torus (Sokolov & Marscher 2005). However,in many sources there is no observational evidence of a significant external photon source. This is the case forS5 0716+714, where, despite much effort over the pastthree decades, no emission lines have been detected (e.g.,Bychkova et al. 2006). Furthermore, XMM-Newton obser-vations of S5 0716+714 in 2004 analysed by Ferrero et al.(2006) and Foschini et al. (2006) show two spectral com-ponents in the 0 . −
10 keV band, whose variability prop-erties appear to favour the SSC interpretation. The re-cent extensive simultaneous observations of this objectfrom radio to optical frequencies by Ostorero et al. (2006),together with INTEGRAL pointings at GeV γ -ray ener-gies during the same period, provide the best test for ourmodel.In the following, we present the computation of the sta-tionary electron distribution and the resulting synchrotronand inverse Compton spectra. The model spectra com-puted using the monoenergetic electron approximation,as described in Tsang & Kirk (2007), are presented first.Although adequate for the radio emission, the monoen-ergetic model cannot reproduce the entire spectrum ofS5 0716+714. We therefore investigate an electron dis-tribution that is a double power law in energy — a hardlow-energy part that softens to a high-energy tail abovea characteristic energy. In this way, the inverted opticallythin radio emission is retained and complemented by non-thermal synchrotron emission from the high energy tail. Insection 2, we briefly describe these injection models. Theresulting stationary electron distribution is calculated insection 3 and used for the computation of the synchrotronand inverse Compton spectra. In section 4, we comparethe predictions of these models with the observed spectralenergy distribution (SED) of the source to S5 0716+714.Our findings and some limitations of our approach arediscussed in section 5 and our conclusions presented insection 6.
2. The model
The homogeneous monoenergetic model discussed previ-ously (Kirk & Tsang 2006; Tsang & Kirk 2007) can becompletely characterised by the Doppler boosting factor D = 1 / [Γ(1 − β cos ϑ )] ( cβ is the source speed with re-spect to the rest frame of the host galaxy, ϑ the an-gle between the velocity and the line of sight, and Γ =(1 − β ) − / ), the redshift of the host galaxy z , and foursource parameters, the electron number density N e , themagnetic field strength B , the linear size of the source R , and the electron Lorentz factor measured in the restframe of the source γ . For the purpose of comparisonwith observations, these can be transformed into a dif-ferent set of parameters. Details of the transformationcan be found in Kirk & Tsang (2006), in which N e , B ,and γ are replaced by the characteristic frequency of syn-chrotron emission, ν s = γ ν , where ν = 3 eB/ (4 πmc ),the Comptonisation parameter ξ , which is the ratio ofthe luminosity of each successive generation of inverseCompton scattered photons to the luminosity of the pre-vious generation: ξ = 4 γ τ T /
3, (where τ T = N e Rσ T is the . Tsang and J. G. Kirk: SSC model for S5 0716+714 3 Thomson optical depth), and the optical depth τ s to syn-chrotron self-absorption at the observing frequency. Thesize of the source, R , can be constrained, for example, byapplying causality arguments to the variation time, ∆ t , ofthe source: R < c ∆ t D / (1 + z ).We present in Section 4 the model spectra from mo-noenergetic electrons that show good agreement with theobservations of S5 0716+714 at radio frequencies. The op-tical data can be fitted by this model if a high-energypower-law “tail” is added. To do this, we consider aninjection spectrum of the form Q ( γ ) ∝ ( γ/γ p ) − s for γ min < γ < γ max , where the power-law index s equals s for γ < γ p , and s for γ > γ p (Fig. 1). The elec-tron number density at a given time is proportional to γ − s max for s < ∝ γ − s min for s >
1, and ∝ ln γ max for s = 1. In the high-energy branch of the injection spec-trum, for γ > γ p , we require that s >
1, so that electronnumber density congregates towards γ p . In the low-energybranch, γ < γ p , the electrons congregate at γ p if s < γ = γ p , whichis achieved by demanding s < /
3. Under the conditions s < / s >
1, the low-frequency synchrotron spec-trum is well-approximated by that of monoenergetic elec-trons with Lorentz factor γ p .The electron injection spectrum cuts off at γ min to-wards low energy and at γ max towards high energy. Theexact value of γ min is unimportant, since, as explainedabove, synchrotron emission and opacity are dominated byelectrons with γ = γ p in the low-energy part of the injec-tion spectrum, where γ max determines the high frequencycut-off in the synchrotron spectrum, at ν max = γ ν ,and the highest photon energy achievable through inverseCompton scattering in the Klein-Nishina limit, whichequals γ max mc .To summarise, the injection spectrum has the form Q ( γ ) = Q ( γ/γ p ) − s , γ min ≤ γ < γ p ( γ/γ p ) − s , γ p ≤ γ < γ max (2)where Q is the electron injection rate per unit volumeper unit γ at γ = γ p .
3. Stationary solution
The shape of the synchrotron spectrum is determined bythe stationary electron-energy distribution. Electrons in-jected into the source according to Eq. (2) are subject toradiative cooling while in the source and evacuate thiszone on a time-scale close to the light crossing time, t esc ∼ R/c . The evolution of the electron spectrum is gov-erned by the kinetic equation (Kardashev 1962): ∂n e ∂t = Q (cid:18) γγ p (cid:19) − s − ∂∂γ ( n e ˙ γ total ) − n e t esc (3)where, for simplicity, we denote the differential electronnumber density (d N e / d γ ) by n e . The second term on the Γ cool H B L Γ p Γ cool H R L Log Γ Log Q H Γ L(cid:144)
Log n H Γ L A B C D
Fig. 1.
Schematic representation of the electron injectionspectrum and the stationary differential number densityas a function of γ . The heights of the spectra have beenadjusted for easy comparison and are not to scale. Thesolid line shows the double power-law injection spectrumwith power-law index s for γ < γ p , and s for γ > γ p . Thedashed line shows the case where γ cool = γ (R)cool > γ p . Thedifferential electron number density n e ∝ γ − s in regionsA and B, n e ∝ γ − s in region C, and n e ∝ γ − ( s +1) inregion D. The dotted line shows the case where γ cool = γ (B)cool < γ , with n e ∝ γ − s in region A, n e ∝ γ − inregion B, and n e ∝ γ − ( s +1) in regions C and D.right hand side of Eq. (3) is the rate of change of theelectron Lorentz factor due to radiative losses. This termis the sum of the rates for synchrotron losses and for lossesfrom inverse Compton scattering:˙ γ total = ˙ γ s + ˙ γ IC (4)where˙ γ s = 4 σ T U B mc γ . (5)The third term is the rate at which electrons escape fromthe source.In the stationary state, Eq. (3) can be solved analyti-cally: n e ( γ ) = 1 f I ( γ ) Z γ Q ( γ ′′ )˙ γ ′′ total f I ( γ ′′ ) d γ ′′ (6)with the integrating factor f I ( γ ) = ˙ γ exp (cid:20) − Z γ (cid:0) ˙ γ ′ total t esc (cid:1) − d γ ′ (cid:21) . (7)However, Eq. (3) is only a rough description of asource, for example, because of the crude treatment ofparticle escape involved in setting t esc = R/c . Therefore,rather than use Eq. (6), we choose to use an approximatesolution that more clearly demonstrates the effects that cooling and the evacuation of electrons from the emissionregion have on the electron energy distribution.We first identify the Lorentz factor, γ cool , which de-termines the electron energy at which radiative cooling O. Tsang and J. G. Kirk: SSC model for S5 0716+714 dominates losses due to particles escaping the emissionregion:˙ γ total γ (cid:12)(cid:12)(cid:12)(cid:12) γ = γ cool = 1 t esc . (8)In principle, γ cool can be evaluated only if the entire elec-tron distribution is already known, since ˙ γ IC depends onthe spectrum and intensity of emitted radiation. However,in practise, a simple iterative scheme enables it to be foundrapidly in all the cases we have computed. Assuming it isknown, solutions of Eq. (3) that are valid in the limits γ ≪ γ cool and γ ≫ γ cool are easily found. In the first case,cooling is unimportant, and it immediately follows that n e = t esc Q ( γ/γ p ) − s for γ ≪ γ cool . (9)In the second, escape is unimportant, and the appropriatesolution is found by integrating the kinetic equation once: n e = ˙ γ − Z ∞ γ Q ( γ ′ /γ p ) − s d γ ′ for γ ≫ γ cool . (10)These solutions intersect close to the point γ = γ cool . Ourapproximation consists in adopting the solution withoutcooling given in Eq. (9) for all Lorentz factors below theintersection point and the solution without escape givenin Eq. (10) for all Lorentz factors above the intersectionpoint.In addition, we assume and verify a posteriori (seeSection 5) that ˙ γ IC can be approximated by the expressionfor inverse Compton scattering of the synchrotron photonsin the Thomson regime:˙ γ IC = 4 σ T U s mc γ (11)where U s is the energy density of synchrotron photons inthe source. In this case, ˙ γ total = γ / ( γ cool t esc ), and ourapproximate solution is n e = t esc Q ( γ ) γ < aγ cool t esc γ cool γ − R ∞ γ d γ ′ Q ( γ ′ ) aγ cool ≤ γ (12)where a ( ∼
1) is determined by requiring the solution (butnot its first derivative) to be continuous.The Lorentz factors γ cool and γ p give rise to breaksin n e , which correspond to the breaks in the synchrotronspectrum at ν p = γ ν and ν cool = γ ν . Notice that, if s < γ p ), n e is approx-imately proportional to ˙ γ − ∝ γ − , whereas if s > γ p ), n e is approximately ∝ γ − ( s +1) .Two types of stationary spectra result from Eq. (12),depending on whether the peak of the injection spec-trum, γ p , is below or above γ cool . Figure 1 shows the in-jection spectrum as a solid line, the stationary spectrawhere γ p > γ cool as a dotted line and where γ p < γ cool as a dashed line. When electrons are predominantly re-moved from a certain energy range by leaving the source( t esc < t cool ), the spectrum retains its original shape, n e ∝ γ − s , since t esc is independent of particle energy.On the other hand, when synchrotron losses dominates,such that t esc > t cool , the stationary solution is n e ∝ γ − for γ < γ p , and n e ∝ γ − ( s +1) for γ > γ p . For the com-putation of the low frequency synchrotron emission, thedistribution can be approximated by a monoenergetic oneat γ cool in the first case and γ p in the second.The iterative procedure used to find γ cool is as follows:The loss rate is defined as˙ γ total = 4 σ T U B (1 + δ )3 mc γ. (13)Then, starting with δ = 0, γ cool is evaluated from Eq. (8)and, using the electron distribution given by (12), U s isevaluated as described in Sect. 3.1. The value of δ is read-justed to δ = U s /U B and the cycle repeated until succes-sive values differ by less than 1%. In the examples dis-cussed in this paper, convergence was achieved after twoiterations. Because the change in γ cool between iterationswas only roughly a factor of 2, the final emission spectrumwas close to that found using δ = 0. The synchrotron specific intensity, following straightfor-wardly from the radiative transport equation, is I (S) ν = S ν [1 − exp( − τ s )] (14)where the optical depth to synchrotron radiation is τ s = α ν · R , and α ν is the absorption coefficient (e.g., Longair1992, Chapter 18) α ν = − √ σ T α f mc hν ν L sin φν × Z γ max γ min γ F ( x ) dd γ (cid:18) n e ( γ ) γ (cid:19) d γ (15)where α f is the fine structure constant, φ the angle be-tween the magnetic field and the direction of the emit-ted radiation, x = ν/ ( γ ν ), F ( x ) = x R ∞ x K / ( t )d t , and K / is the modified Bessel function of order 5 /
3. Thesource function S ν is S ν = − mν R γ max γ min F ( x ) n e ( γ )d γ R γ max γ min γ F ( x ) dd γ (cid:16) n e ( γ ) γ (cid:17) d γ . (16)In the monoenergetic approximation, the source functionsimplifies to S ν = mν γ F ( x ) K / ( x ) . (17)Equation (14) is integrated over frequency and angle togive the energy density of synchrotron photons in thesource U s = 4 π c ζ Z ∞ I ν d ν (18)where ζ is a geometrical factor that is shown inTsang & Kirk (2007) to be ζ = 2 / . Tsang and J. G. Kirk: SSC model for S5 0716+714 5 The synchrotron photons are repeatedly scattered bythe energetic electrons to higher energies. Denoting by i the number of times a photon is scattered, the rateof scattering the ( i − th generation of photons intothe frequency interval d ν i by a single electron (see e.g.,Georganopoulos et al. 2001, Eq. (4)) is (cid:18) d n ph d t d ν i (cid:19) sp = 3 σ T c ν i − γ f ( y ) N ν i − (19)where N ν i − = 4 πc ζI ν i − hν i − (20)is the number density of the target photons, and I ν i − thespecific intensity of the ( i − th generation of photons.The first generation of scattered photons is produced di-rectly from the synchrotron photons: i = 1, I ν = I (S) ν .Rybicki & Lightman (1979, Chapter 7) assumed that scat-tering in the Thomson regime is isotropic in the restframe of the electron, and obtained f ( y ) ≈ f iso ( y ) =2(1 − y ) /
3. Here, we include the Klein-Nishina effects (e.g.,Georganopoulos et al. 2001), in which case f ( y ) = (cid:20) y ln y + y + 1 − y + (4 ǫ i − γ y ) (1 − y )2(1 + 4 ǫ i − γ y ) (cid:21) × P (1 / γ , , y ) , (21) y = ǫ i ǫ i − γ (1 − ǫ i /γ ) (22)where ǫ i − and ǫ i are the energy of the target photonsand scattered photon, respectively, in units of mc , and P (1 / γ , , y ) = 1 for 1 / γ ≤ y ≤
1, and zero otherwise.Assuming a spherical source, the rate of scatteringphotons with energy hν i − to energy hν i , in the observer’sframe, from a homogeneous distribution of electrons withdifferential number density n e can be found by integratingover the electron energy distribution, (cid:18) d n ph d t d ν i (cid:19) = 4 π (cid:18) R (cid:19) Z ∞ d γn e (cid:18) d n ph d t d ν i (cid:19) sp . (23)Note that R (the linear size of the source) is divided by 2to obtain the source radius.The specific intensity of the i th generation photonsis then the scattering rate of the electron distribution inEq. (23) integrated over all target photon frequency, I (C) ν i = (cid:18) d E d t d ν i d r dΩ (cid:19) = Z ∞ d ν i − (cid:18) d n ph d t d ν i (cid:19) hν i π ( R/ , (24)and for a general electron distribution n e , it can be writtenas I (C) ν i = π Rσ T ν i Z ∞ d γγ n e Z ∞ d ν i − ν i − I ν i − f ( y ) . (25)For a monoenergetic electron distribution, Eq. (25) can besimplified to I (C) ν i = π τ T ν i γ Z ∞ d ν i − ν i − I ν i − f ( y ) . (26) Equations (26) and (25) are integrated numerically formonoenergetic electrons and for an electron distributiongiven by Eq. (12).
4. The BL Lac object S5 0716+714
Observations of S5 0716+714 have shown that thesource exhibits intra-day variability in the radio andoptical bands (e.g. Ghisellini et al. 1997; Raiteri et al.2003). Correlation between radio (at 5 GHz) and opti-cal (at 650 nm) variability suggest that scintillation, aprocess that is not effective at high radio and opti-cal frequencies, does not play a large part in the ob-served variability (Quirrenbach et al. 1991; Wagner 2001).More recent multi-frequency studies of S5 0716+714 (e.g.Ostorero et al. 2006; Agudo et al. 2006) have obtainedsimultaneous measurements from radio to optical fre-quencies during the INTEGRAL pointing period, andthe non-detection of the source by INTEGRAL has pro-vided upper limits at X-ray frequencies. Flux variationswere detected at 32 and 37 GHz over a period of ∆ t =4 . H =70 km sec − Mpc − , with Ω λ = 0 .
7, Ω M = 0 .
3, and Ω k = 0,and a redshift z > . T var > (2 . ± . × K.Bach et al. (2005) analysed the data set of VLBI im-ages of 11 jet components of S5 0716+714 at 4.9 GHz,8.4 GHz, 15.3 GHz, and 22.2 GHz, observed between 1992and 2001. Assuming that all the jet components move withthe same speed along the jet (i.e. all components have thesame bulk Lorentz factor), they propose that the observedwide range (from 5.5 c to 16.1 c ) of apparent componentspeeds is due to variations in the viewing angle and limitthe Lorentz factor and the viewing angle of the VLBI jet toΓ >
15 and θ < ◦ , respectively. Under these conditions,the range of Doppler factors would be D ≈ − α − of − . − . ν ≈ Hz would correspond to the self-absorption fre-quency ν abs . In the near-infrared to optical band, obser-vations from 2001 − ν K = 1 . × Hz and ν B =6 . × Hz can be fitted by the power law F ν ∝ ν − . .Here, we apply the two homogeneous models describedin section 2: the monoenergetic one, which successfullymodels the hard radio spectrum with relatively few freeparameters, and the one with double power-law injection(and, consequently more free parameters), which also en-ables the high-energy emission to be modelled. We adopt O. Tsang and J. G. Kirk: SSC model for S5 0716+714 a value of z = 0 . t = 4 . R = c ∆ t D / (1 + z ). The values wefind for the free parameters of the models and for severalparameters derived from these are given in Table 1. Thespectra predicted by the two models are shown in Figs. 2and 3, and are discussed separately in the next two sec-tions.Identifying the creation of our homogeneous sourcewith an event that leads to the ejection of an individualblob observed with VLBI, we estimate the minimum jetpower implied by each set of parameters by multiplyingthe total energy content of the source by the average rateat which blobs are ejected from the core. In the co-movingframe of the source, for monoenergetic electrons, the totalenergy content is E ′ blob = (cid:18) B π + N ′ e γmc (cid:19) R ′ (27)and, for power-law electrons, E ′ blob = (cid:18) B π + Z γ max γ min n ′ e ( γ ) γmc d γ (cid:19) R ′ . (28)Transforming to the rest frame of the galaxy, E blob =Γ E ′ blob , so that the jet power P jet = E blob (cid:18) t ej (cid:19) , (29)where ∆ t ej is the average time difference between theejection of successive blobs. Linear fits of the change inthe position of the 11 jet components (Bach et al. 2005)suggest that a new component is ejected from the coreevery 0 . − . t ej = 1 . In the monoenergetic model, the spectrum is specified byfour parameters, ν abs , ν p , D , and ξ , as well as z and ∆ t (which are kept fixed for all models). The self-absorptionfrequency ν abs is determined by the first spectral break at ∼ ν p corresponds to the spectral cut-off. InFig. 2, we compare two models in which one has a cut-offat the spectral break at ∼ . Hz, and the other cuts offjust before reaching the optical point. The Doppler factor D affects the level of the observed flux both by determiningthe linear size of the source and determining the amountof boosting the flux receives. The Comptonisation param-eter ξ determines the ratio of the synchrotron flux to theinverse Compton flux, as well as the value of γ p . Therefore,after specifying ν abs and ν p , ξ must be adjusted to com-pensate for its effect on the level of the observed flux andto ensure the inverse Compton spectra do not exceed theINTEGRAL upper limits, while minimising D .Figure 2 shows the measurements and variable rangesobtained from the simultaneous multi-frequency obser-vation of S5 0716+714 from the study conducted by Ν @ Hz D -13-12-11-10-9-8 Log Ν F Ν @ erg (cid:144) cm (cid:144) s D Fig. 2.
Spectral energy distribution of S5 0716 +714.Multi-frequency simultaneous data from Ostorero et al.(2006) are shown as black symbols. Black dots show datapoints, variation ranges are shown by a vertical bar be-tween two symbols, and downward arrows show upperlimits. Values of the parameters are shown in Table 1.The model spectra are computed from a distribution ofmonoenergetic electrons and are shown with solid anddashed lines. The dashed line shows the strong-magnetic-field model where the parameters are chosen such that itgoes through the data points at optical frequency, whereasthe solid line shows weak-magnetic-field model where theparameters are chosen to mimic the spectral turning at10 . Hz. The values of the parameters are shown inTable 1.Ostorero et al. (2006). Also shown are the spectra pre-dicted by the model assuming electrons are monoener-getic. The Doppler boosting factor is D = 55 in bothmodels. The weak-magnetic-field model has a synchrotronself-absorption frequency of ν abs = 3 . ν p = 300 GHz (the val-ues of other parameters are shown in Table 1). This modelgives a brightness temperature of T B = 3 . × K at ν obs = 32 GHz ( T B = c F ν / (2 k B ν θ ), where k B is theBoltzmann constant and θ d the angular diameter of thesource). The synchrotron spectrum shows good agreementwith the data points at radio frequencies. The first-orderinverse-Compton spectrum gives emission from opticalto soft X-ray frequencies and the second-order spectrumgives gamma-ray emission up to ∼
40 MeV, while emis-sion from higher orders scattering is negligible due to theKlein-Nishina effect.The strong-magnetic-field model has ν abs = 3 . ν p = 55 × Hz (the valuesof other parameters can be found in Table 1) and gives abrightness temperature of T B = 3 . × K at the ob-serving frequency ν obs = 32 GHz. The synchrotron spec-trum extends up to optical frequencies, and gives a reason-able fit at radio frequencies up to ∼ . Hz. The first-order inverse-Compton spectrum gives X-ray emission,the second-order inverse-Compton spectrum is greatly re-duced by the Klein-Nishina effects, and very little gamma-ray emission is produced. . Tsang and J. G. Kirk: SSC model for S5 0716+714 7
The spectral break at 10 . Hz is well-fitted by theweak-magnetic-field model. We are unable to obtain a setof parameters to allow the first inverse Compton spectrumto reproduce the optical data. Simple qualitative analysisshows that mimicking the optical data points with thefirst inverse Compton spectrum is inconsistent with ob-servation. The level of flux that the first inverse-Comptonspectrum would require in order to account for the opticaldata is much higher than the synchrotron flux (i.e. ξ ≫ γ would therefore be required, resulting in thespectrum extending to frequencies far beyond the opticalband. The first inverse-Compton spectrum would there-fore exceed the INTEGRAL upper limits, and the veryhigh X- and γ -ray flux would result in copious electron-positron pair production as the γ -ray photons interactwith the synchrotron photons.Alternatively, one can attempt to include the opticaldata in the synchrotron spectrum, as shown by the strong-magnetic-field model. The Lorentz factor of this model ishigher than γ cool , which implies that the particles lose asignificant portion of their energy by synchrotron radia-tion before they vacate the source, so the electron spec-trum will evolve into one that is proportional to γ − . Thisset of parameters therefore violates the monoenergetic as-sumption. Furthermore, the predicted spectrum fails toaccount for the spectral break at ν ∼ . Hz, and theoptical flux is very sensitively to the electron Lorentz fac-tor. This model is, therefore, inconsistent. It is apparentthat, in order to reproduce the observed optical emission,a power-law component in the electron spectrum at γ > γ p must be incorporated, which emits synchrotron radiationat a frequency above ν p . The Comptonisation parameter ξ in this model must ac-count for the γ dependence of the electron density. It istherefore redefined as ξ = 43 Rσ T Z ∞ γ n e d γ, (30)which is consistent with the definition used in the monoen-ergetic model. However, for the purpose of fitting the fluxin the INTEGRAL frequency range, ∼ − Hz,we introduce a parameter r p , that determines the ratioof flux between ν p and γ ν p . Inspection of the solid linein Fig. 2 shows that synchrotron photons at the spectralbreak ν p ≈
300 GHz are scattered to γ ν p ≈ . Hz.Therefore, r p is a parameter that allows us to specifythe spectral break ν p and then to adjust the flux in theINTEGRAL frequency range. The normalisation constantin the double power-law injection spectrum, Q is elimi-nated in favour of the parameter r p r p = 43 γ Rσ T Z ∞ n e ( γ )d γ, (31)with n e ( γ ) given by Eq. (12). In the monoenergetic ap-proximation, r p is equivalent to ξ . Therefore, using the monoenergetic model and specifying ν abs and ν p , the ra-dio and the X-ray flux can be adjusted with ξ ≡ r p and D , and ν cool can be calculated.In the low-Doppler-factor model (Figs. 3 and 4), weattempt to minimise the Doppler factor of the source.According to Wagner et al. (1996) and Ostorero et al.(2006), the variability displayed by S5 0716+714 is in-trinsic, and the variation time ∆ t = 4 . D = 30.The power-law indices of the injection spectrum usedto generate the spectrum of the low-Doppler-factor modelare s = − γ < γ p do not contribute significantly to the syn-chrotron opacity, and s = 2 .
60, chosen for constructingthe spectral shape in the infrared to optical band. The restof the parameters are varied while keeping the Dopplerboosting factor fixed. To find the limiting case, we havechosen the self absorption frequency to be ν abs = 32 GHzand find that the minimum Doppler factor that can gen-erate a high enough level of flux at 32 GHz and beyondis D = 30. The values of the other parameters can befound in Table 1. At the observing frequency of 32 GHz,the brightness temperature in the frame of the observer is T B = 1 . × K. The frequency at which the synchrotronspectrum cuts off does not affect the spectral shape at lowfrequencies. However, ν max = D / (1 + z ) × ν γ is con-strained by the optical data, which imposes a lower limiton ν max , and the INTEGRAL upper limits, which imposean upper limit on ν max . The maximum value is shown in this model , where ν max = 10 Hz. This equates to γ max = 5 . × with D = 30 and z = 0 . ν ≥
32 GHz.In the high-Doppler-factor model (Figs. 3 and 4) weassume the emission at all frequencies — including thelow frequency ( <
32 GHz) radio — originates from a sin-gle homogeneous source. This is suggested by the correla-tion between the variability at 5 GHz and 650 nm observedin February 1990 (Quirrenbach et al. 1991; Wagner et al.1996). The parameter r p must be kept small, so that theinverse Compton spectra are below the INTEGRAL upperlimits. This is achieved at the expense of a relatively highDoppler factor, at D = 65. The brightness temperature at32 GHz is T B = 2 . × K. This model also shows theminimum value of ν max , found to be ν max = 1 . × Hz.
O. Tsang and J. G. Kirk: SSC model for S5 0716+714
Table 1. Model parameters corresponding to
Figs. 2 and 3.
Mono (dashed) Mono (solid) Power-law (dashed) Power-law (solid)Primary Parameters z t (days) 4.1 4.1 4.1 4.1 D
55 55 30 65 ν p (Hz) 5 . × . × . × . × ν cool (Hz) 1 . × . × . × . × ν max (Hz) . . . . . . . × . × ξ − . − . . . . . . . Secondary Parameters ξ . . . . . . .
92 0 . R (pc) 0.15 0.15 0.08 0.17 θ d ( µ as) 32.5 32.5 17.7 38.4 γ p
691 800 244 696 γ cool
123 4 . × . × . × γ max . . . . . . . × . × N e (cm − ) 0.02 0.70 3.15 0.32 B (mG) 648 3.43 34.7 2.65 δ . . . . . . U B /U par . × . × − . × − T B (K) 3 . × . × . × . × τ T × − × − × − × − P jet (ergs/s) 1 × × × × Note: θ d is the angular diameter of the source, U par the energy density of the particles, and P jet is the jet power in the restframe of the host galaxy, as predicted by each model. The compactness of all four models are negligibly small so not includedin the discussion. The high-Doppler-factor model shows that if the emis-sion at all frequencies originate from a single source region,it must be boosted by a much higher Doppler factor thanproposed for the jet components by Bach et al. (2005).Even with a viewing angle ϑ ≈
0, in which case
D ≈ >
33 is required, suggesting eitherthat the source was travelling at a much higher speed thanduring the observations analysed by Bach et al. (2005) orthat an interpretation of the superluminal motion in theVLBI jet that infers a much higher bulk Lorentz factorshould be applied. One such suggestion is a conical jet(Gopal-Krishna et al. 2006). Alternatively, the jet compo-nents may decelerate as they travel down the jet (Marscher1999; Georganopoulos & Kazanas 2003; Ghisellini et al.2005). The wide range of superluminal velocities shown in Bach et al. (2005) would then be a combined result ofthe variations in speed, as well as of the viewing angle.
5. Discussion
The spectra of the four models — two monoenergetic andtwo with double power-law electron injection — shown inFigs. 2 and 3, show brightness temperatures at 32 GHzthat are well above the conventional Compton limit.However, due to the lack of low-energy electrons, thesebrightness temperatures in fact lie below the threshold ofthe Compton catastrophe (i.e., ξ <
1) and are, therefore,sustainable by the source.An independent limit on T B is provided by inducedCompton scattering, when low-energy electrons couplewith high-frequency photons. The photon occupation . Tsang and J. G. Kirk: SSC model for S5 0716+714 9 Ν @ Hz D -14-13-12-11-10-9 Log Ν F Ν @ erg (cid:144) cm (cid:144) s D Fig. 3.
The spectral energy distribution of S5 0716 +714, as represented in Fig. 2. The model spectra, shown as solid anddashed lines, are computed from a quasi-monoenergetic electron distribution in the form of Eq. (12). The dashed linerepresents the low-Doppler-factor model where the Doppler boosting factor is minimised, whereas the solid line showsthe high-Doppler-factor model where the values of the parameters are chosen to account for all radio and optical datapoints. The dashed gridline shows the position of 32 GHz. The values of the parameters are shown in Table 1. Historicaldata, as compiled by Ostorero et al. (2006) and shown as grey symbols, at radio-to-optical frequencies are fromKuehr et al. (1981), Waltman et al. (1981), Eckart et al. (1982), Perley (1982), Perley et al. (1982), Lawrence et al.(1985), Saikia et al. (1987), Kuehr & Schmidt (1990), Moshir et al. (1990), Hales et al. (1991), Krichbaum et al.(1993), Gear et al. (1994), Hales et al. (1995), Douglas et al. (1996), Rengelink et al. (1997), Zhang et al. (1997),Riley et al. (1999), Cohen et al. (2002), Raiteri et al. (2003); UV data from Pian & Treves (1993), Ghisellini et al.(1997); X-ray data from Biermann et al. (1992), Comastri et al. (1997), Kubo et al. (1998), Giommi et al. (1999),Tagliaferri et al. (2003), Pian et al. (2005); and γ -ray data from McNaron-Brown et al. (1995), Hartman et al. (1999),Collmar (2006).number n ph ( ν ) ∝ I ν /ν , which implies that the pho-ton occupation number is high at and below the peak(where ν = ν abs ) of the synchrotron spectrum. In thepresence of low-energy electrons, induced Compton scat-tering of photons at frequency ν abs to frequencies ν <ν abs becomes increasingly significant as the synchrotronflux grows. Sunyaev (1971) showed that this process lim-its the brightness temperature at a certain frequency to T B < mc / ( k B τ T ) = 5 × K for τ T ∼
1. Sincell & Krolik(1994) demonstrated by numerical simulations that rel-ativistic induced Compton scattering limits the bright-ness temperature of a self-absorbed synchrotron sourceto T B < × ν − / ( s +3)GHz γ ( s +2) / ( s +5)min K, where ν GHz isthe observing frequency in units of GHz, γ min is the lowenergy cut-off in the electron spectrum ∝ γ − s . For a con-ventional power-law electron spectrum in which γ min = 1,this gives a limit of T B < × K at 1 GHz.One might suspect that, at such high brightness tem-peratures as are predicted by the models shown here, theeffect of induced Compton scattering should be significant.Qualitative arguments reveal the contrary in our model,since the low-energy cut-off in the electron spectrum iseffectively γ p , and the occupation number ( ∝ F ν /ν ) ofphotons at frequencies that would couple with electronsat γ p is negligibly small. Alternatively, when using the re- sult of Sincell & Krolik (1994), one finds that the limitcorresponds to T B < . × ν − / ( γ p / ) / K for s = s ≈ . γ min = γ p .The spectral break at ∼ . Hz was interpreted byOstorero et al. (2006) as the result of the change in opacityof the source. In our interpretation, the break is a result ofa corresponding spectral break in the electron spectrum;below this frequency, the synchrotron spectrum remainsoptically thin. The self-absorption frequency ν abs lies ata much lower frequency ( ∼ γ p lies below or above γ cool , since thisaffects the final shape of the electron spectrum, as ex-plained in Section 3. If the number of electrons leavingthe energy γ p mc is dominated by cooling by radiation,the synchrotron spectrum continues from F ν ∝ ν / be-tween ν abs and ν cool , to F ν ∝ ν − / between ν cool and ν p , then F ν ∝ ν − s / between ν p and ν max , and it iscut off exponentially beyond ν max . In this case, the lowradio-frequency spectrum resembles that of a monoener-getic electron distribution of energy γ cool mc . If, on the Ν @ Hz D -15-14-13-12-11-10 Log Ν F Ν @ erg (cid:144) cm (cid:144) s D Ν @ Hz D -15-14-13-12-11-10 Log Ν F Ν @ erg (cid:144) cm (cid:144) s D Fig. 4.
The spectral energy distribution of S5 0716 +714and the model spectra, as represented in Fig. 2, inthe radio-to-optical band. Top panel shows the low-Doppler-factor model where the Doppler boosting factoris minimised. Bottom panel shows the high-Doppler-factormodel where the values of the parameters are chosen toaccount for all radio and optical data points.other hand, losses are dominated by electrons evacuatingthe emission zone on a time scale of t esc = R/c , the syn-chrotron spectrum then continues from F ν ∝ ν / between ν abs and ν p , to F ν ∝ ν − ( s − / between ν p and ν cool , then F ν ∝ ν − s / between ν cool and ν max , and it is again cutsoff exponentially beyond ν max . The low radio-frequencysynchrotron spectrum can be approximated in the sameway as the one from a monoenergetic distribution of elec-tron of energy γ p mc .Observations of S5 0716+714 from infrared to opti-cal frequencies suggest that the spectral energy distribu-tion between the frequencies ν K = 1 . × Hz and ν B = 6 . × Hz can be well-fitted by the power law F ν ∝ ν − . (Hagen-Thorn et al. 2006). Clearly, the toppanel in Fig. 4 is much too hard at these frequencies.However, the spectrum can be softened by lowering thecut-off frequency of the synchrotron spectrum ν max (i.e.,bottom panel in Fig. 4). By decreasing ν max to approxi-mately ν K , the spectrum begins an exponential drop at orjust before reaching the relevant frequency range and, asa result, softens the spectrum, without altering the levelof flux or the spectral shape at frequencies < ν max .Figure 2 and the bottom panel in Fig. 4 demonstratethat, if the variability of S5 0716+714 is intrinsic and ex-tends down to <
32 GHz, the Doppler-boosting factor of Γ Γ (cid:144) Γ s Γ Γ (cid:144) Γ s Fig. 5.
Cooling rates normalised to the synchrotron lossrates (dashed lines at ˙ γ = ˙ γ s ). The rate for inverseCompton scattering is shown as a dotted line, the totalloss rate as a thin solid line (blue online). The approximaterate used to compute the electron distribution is shown asa thick solid line (black online). The top (bottom) panelshows the results of the low(high)-Doppler-factor modelshown as dashed (solid) lines in Fig. 3. The vertical dot-ted lines correspond to γ = aγ cool (see Eq. (12)) on theleft, and to γ = γ max on the right.the emission region must be much higher than the 20 − ν abs to increase, as shown by the toppanel in Fig. 4, therefore requiring the assumption that theemission at frequency below ν abs originates from a largerregion than inferred from the variability at 32 GHz and37 GHz. Requiring the value of ν abs ≤
32 GHz, we findthat, to remain below the INTEGRAL upper limits, werequire a minimum Doppler factor of D = 30 and a min-imum self-absorption frequency of ν abs = 32 GHz. Belowthese minima, the model spectrum would either have alevel of flux below the measured flux at radio frequenciesor the subsequent first inverse Compton spectrum wouldlie above the INTEGRAL upper limits.The models shown here depart from the equipartitionof magnetic and particle energy, and are dominated by theenergy of the relativistic electrons (except for the rejectedmodel). Estimating the energy required by the source fromthe host galaxy, P jet , we find that the power required from . Tsang and J. G. Kirk: SSC model for S5 0716+714 11 the source by our models is consistent with what is ex-pected from a low-energy peaked BL Lac object (see e.g.,Nieppola et al. 2006).An approximation inherent in our method is that, asfar as their effect on the electron distribution is concerned,inverse Compton losses are dominated by single scatter-ings off the synchrotron photons and may be treated inthe Thomson approximation. For each of the models pre-sented in Fig. 4, we show in Fig. 5 the total rate of radia-tive cooling of an electron (thin solid lines, blue online)computed using the full Klein-Nishina loss rate for scatter-ing on the full output spectrum. This should be comparedwith the thick solid lines (black online) that give the lossrates used in computing the electron density according toEq. (3). These lines are horizontal and give the quantity˙ γ total / ˙ γ s = 1 + δ for the converged model solutions. Therange over which cooling is important in our approximatesolution lies between the vertical lines at γ = aγ cool and γ max in this figure. (In the lower panel, these lines lie closetogether.) The maximum deviation in this range is roughly20% and occurs at γ = γ max for the model with a relativelylow Doppler factor (plotted as a dashed line in Fig. 4). Thedeviation for the higher Doppler factor model is less than3%. We conclude that this approximation has a negligibleeffect on the electron distribution. The computation of theoutput spectrum from the electron distribution is not af-fected by this approximation, since the full Klein-Nishinaexpression for the emissivity is used.A testable prediction of our models is correlated vari-ability. The hard α ≈ − . ≈ x = ν/ν s ≪ x = 1, and tends to 100% at x ≫
1. The effect of a tan-gled field within the source reduces these values, but isindependent of observing frequency. Thus, as in conven-tional inhomogeneous models, the degree of linear polar-isation is predicted to increase with frequency. However,more quantitative predictions would require considerationof effects, such as internal Faraday rotation, and lie out-side the scope of the present paper.Finally, we note that the hard electron injection spec-trum that we have adopted ( n e ∝ γ ) corresponds tothe low-energy ( γmc < k B T ) part of a relativistic ther-mal distribution. According to Table 1, the correspondingelectron temperature would lie at around 10 K. In anelectron-ion plasma, this corresponds to the temperatureof shocked gas behind a mildly relativistic shock front, ifone assumes that the electrons and ions equilibrate to acommon temperature. Recent P.I.C. simulations suggestthat this assumption may indeed be justified (Spitkovsky2007).
6. Conclusion
Using the specific case of S5 0716+714 as an example,we confirm that it is possible to produce high brightnesstemperatures at GHz frequencies in compact radio sourceswithout the onset of catastrophic cooling, provided thatthe radiating particles have a distribution that is suffi-ciently hard below a characteristic. In addition, we showqualitatively that induced Compton scattering is insignif-icant in sources with a low-energy electron cut-off despitethe high brightness temperature, the underlying reasonbeing the low occupation number of the photons that cancouple with the electrons at the cut-off energy.The model where an electron distribution that is adouble power law in energy, peaking at γ p , is injected intothe source offers more flexibility at higher frequencies inthe synchrotron spectrum (from infrared to optical) at theexpense of more free parameters, compared to either mo-noenergetic or single power-law distributions. These pa-rameters should be constrained by simultaneous observa-tions due to the highly variable nature of IDV sources. Inthe case of S5 0716+714 where such data is available, the spectral break at about 230 GHz determines the value of γ p , the optical data at 5 × Hz gives the lower limit of ν max , as well as constraining the spectral index s , and theINTEGRAL upper limits give the upper limit of ν max andalso constrain the value of r p , which in turn determinesthe electron density.The example of S5 0716+714 illustrates several impor-tant spectral properties of an electron distribution with alow-energy cut-off, as described in the previous sections.The most noticeable feature is the hard, inverted, op-tically thin synchrotron spectrum, spanning a wide fre-quency range, which is a prevalent feature in compactradio sources at radio frequencies (e.g., Gear et al. 1994;Kedziora-Chudczer et al. 2001). Other features are thespectral breaks at ν p = γ ν , ν cool = γ ν , and theexponential cut-off at ν max = γ ν . This model, there-fore, allows a simple homogeneous source to reproduce thecommon features shown by many IDV sources. Acknowledgements.
We thank Luisa Ostorero and StefanWagner for helpful discussions and for providing us with easyaccess to the observational data. We would also like to thankthe anonymous referee for constructive comments and sugges-tions that we feel have led to a significant improvement in thispaper.
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