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Mon.Not.R.Astron.Soc.000,000-000(1999) Printed1February2008 (MNplainTEXmacrosv1.6) Radiation force on relativistic jets in active galactic nuclei Qinghuan Luo and R. J. Protheroe Department of Physicsand Mathematical Physics, The Universityof Adelaide, Adelaide, SA 5005, Australia Accepted date;Receiveddate 9 ABSTRACT 9 Radiative deceleration of relativistic jets in active galactic nuclei as the result of 9 inverse Compton scattering of soft photons from accretion discs is discussed. The 1 Klein-Nishina (KN) cross section is used in the calculation of the radiation force due n to inverse Compton scattering. Our result shows that deceleration due to scattering a in the KN regime is important only for jets starting with a bulk Lorentz factor larger J than 103. When the bulk Lorentz factor satisfies this condition, particles scattering 8 in the Thomson regime contribute positively to the radiation force (acceleration), 1 but those particles scattering in the KN regime are dominant and the overall effect is deceleration. In the KN limit, the drag due to Compton scattering, though less 1 v severe than in the Thomson limit, strongly constrains the bulk Lorentz factor. Most 3 of the power from the deceleration goes into radiation and hence the ability of the 2 jet to transport significant power (in particle kinetic energy) out of the subparsec 2 region is severely limited. The deceleration efficiency decreases significantly if the jet 1 containsprotonsandtheprotontoelectronnumberdensityratiosatisfiesthecondition 0 np/ne0 >2γmin/µp where γmin is the minimum Lorentz factor of relativistic electrons 9 (or positrons) in the jet frame and µ is the proton to electron mass ratio. p 9 / Key words: Scattering – plasmas – relativistic jets – AGN – radiation: nonthermal h p - o r 1 INTRODUCTION Compton scattering can be more effective in slowing down t s ratherthan accelerating thejet, andcan constrain thebulk a High energy observations of blazars strongly suggest that flow of thejet in that region. : v the emission comes from relativistic jets of active galactic There are AGN emission models in which protons are i nuclei (AGN) (e.g. von Montigny et al. 1995; Thompson et accelerated to ultra high energies (e.g. Königl 1994). One X al. 1995; Gaidos et al. 1996; Quinnet al. 1996; Schubnellet of the attractive features of this type of model is that pro- r al. 1996). Other evidence for relativistic AGN jets includes tons can be accelerated to high energies without significant a theobservation of variability in radio, optical, and high en- energy loss. These high energy protons can interact with ergy gamma ray emission from blazars. Although there is photon fields producing electron-positron pairs. It is possi- no consensus on the details of AGN, it is widely accepted blethatthroughpaircascades,anultrarelativisticjetispro- thatajetmayformnearablackholewithanaccretiondisc duced.Alternatively,anultrarelativisticjetcanbeproduced (e.g. a review by Blandford 1990). Since a substantial frac- through rapid acceleration of e± such as by magnetic re- tionofthebindingenergyofaccretingmaterialisdissipated connection (e.g. Haswell et al. 1992) orbyrotation-induced and converted to radiation, the disc is a strong source of electrostatic potential drop as in pulsars (Michel 1987). In softphotons(e.g.asblackbodyradiationorreprocessedra- bothcases,thebulkflowofthejetmayinitiallyhavealarge diation). Interaction of relativistic particles in the jet with Lorentzfactor,andplasmasinthejetcanbehighlyrelativis- thesesurroundingphotonfieldscan beimportant,and may tic. Therefore, scattering in theKlein-Nishina (KN) regime contributetotheobservedgammarayemission,e.g.through isimportant,andshouldbeincludedincalculationofradia- inverseComptonscattering(Reynolds1982;Melia&Königl tion force duetoinverseCompton scattering. Thereis then 1989; Dermer & Schlickeiser 1993). Disk emission can also thequestionofwhetherthejetwithinitiallylargebulkflow affectthejetdynamicsneartheblackholethroughradiative speed is subject to the same (severe) radiation deceleration acceleration or deceleration. The effect of Compton scat- as in theThomson regime, and thiswill beexplored in this tering on the jet flow was discussed by O’Dell (1981), and paper. Phinney (1982, 1987), and it was suggested that undercer- TheconstraintonthebulkflowbyComptonscattering tain (quite stringent) conditions, the jet can be accelerated should strongly depend on the soft photon distribution of through Compton scattering. However, as pointed out by the disc. In our discussion, disc emission is assumed to be several authors (e.g. Phinney 1987; Melia & Königl 1989; axisymmetric, and so theonly effect on the jet flow consid- Sikora et al. 1996), in the region close to the black hole, ered is in the jet direction (it is usually assumed that the 2 Luo and Protheroe jet is normal tothediscplane). However,therearecases in theaccretiondiscemission,andisaxisymmetricwithrespect whichaccretionmaynotbecircular,e.g.eccentricaccretion to the jet axis (which is in the z direction). A schematic canoccuriftheblackholeisinabinarysystem(e.g.Begel- diagram for such a disc-jet system is shown in Figure 1. man, Blandford & Rees 1980; Eracleous et al. 1995), and Emission from the disc is modeled as the sum of emission this possibility will be considered in a forthcoming paper from seriesofringscenteredat theblackhole;each ofthem (Luo1998). emitsphotonswithacharacteristicenergyε˜ whereRisthe R The deceleration efficiency strongly depends on the radiusofthering.Thefluxnumberdensityofphotonsfrom composition of the jet. An AGN jet that contains protons the ring R R+dR is given by F(ε˜,R) = F(R)δ(ε˜ ε˜ ) wouldbesubjecttolessseveredecelerationbecauseprotons where ε˜= Γ∼(1+β cosθ)ε is the photon energy (in m− cR2) b e scatter photons with much smaller cross section than that seeninK,εisthecorrespondingenergyseeninK ,θisthe j forelectrons(orpositrons)andthiseffectivelyincreasesthe anglebetweenthephotonpropagation directionandthejet jet inertia. axis, and F(R)is given by (Blandford 1990) In this paper, we extend calculations of the radiation forcebyincludingtheKlein-Nishinaeffectastheearliercal- F(R) 4.4 1030 1eV M˙ culations were done only in the Thomson scattering regime ≈ × (cid:18) ε˜R (cid:19)(cid:18)LEdd/c2(cid:19) (2.2) (e.g.O’Dell1981; Phinney1982; Sikoraetal.1996). InSec. −1 −3 M R 2,theaverageradiationforceonthebulkplasmaflowisde- Icm−2s−1, rived in both the Thomson and Klein-Nishina limits. Con- ×(cid:18)108M⊙(cid:19) (cid:18)GM/c2(cid:19) straints by radiative deceleration on the Lorentz factor of whereI =(1 R /R)1/2,R istheradiusof theinner- min min thejet are discussed in Sec. 3. − most part of the disc, M is the mass of the black hole, G isNewton’sconstant,L istheEddingtonluminosity.We Edd assume R to bethemaximum radius of thedisc, within 2 RADIATIVE EFFECT ON JET FLOW max which emission can be adequately described by (2.2). ThroughComptonscatteringofexternalphotons,individual A detailed model for emission from the accretion disc particles in plasmas lose energy and at thesame time there shouldincludeeffectssuchasreprocessingofradiation,disc is momemtum transfer to the plasma. For plasmas in a jet, corona,etc.Asweareconcernedwiththeradiativeeffecton the momemtum transfer can affect the jet dynamics, i.e. thejet flow, all thesedetails will beignored and weassume the bulk flow can be either accelerated or decelerated (i.e. thatdiscemissionisblackbodywithε˜ 2.7k T(R)/m c2, R ≈ B e radiative drag). where T(R) is the effective temperature of emission from Consider a comoving cell with energy E˜ (in m c2 per the ring with R. Then, for an optically thick disc, T(R) e unit volume) in a relativistic jet with the bulk Lorentz fac- is identified as the surface temperature, and it is given by tor Γ = 1/(1 β2)1/2 where β is the bulk velocity in c. T(R)=(3GMM˙I/8πR3σ )1/4, that is, − b b SB The tilded quantities correspond to the values in the lab frame(K).Weassumethatthejetcontainsmainlyelectron- M˙ 1/4 M −1/4 T(R) 5 105 positron pairs (there may be protons as well but in small ≈ × LEdd/c2 108M⊙ number)andthatwithinthecellelectron-positronpairpro- (cid:18) (cid:19) (cid:18) (cid:19) (2.3) −3/4 R duction is not important. The latter assumption may not I1/4K, be accurate since pairs can be injected continuously along × GM/c2 (cid:18) (cid:19) the jet. Nonetheless, it allows us to single out the radiative where R R , σ is the Stefan-Boltzmann constant. effectduetoCompton scattering, andtheresultshould not ≥ min SB Figure2showsplotsofF(R)forR /R =5and6,where change qualitatively when the pair production effect is in- min g cluded. Let f be theradiative force (in mec per second per tRiogn=alGraMdi/ucs2. ≈Th1e.4fl8u×xe1s0a1r3ecmpe(aMke/d10a8tMR⊙)=is(t7h/e6)gRravita- unitvolume)onthecellseeninthejetcomovingframe(Kj). 5.8R and 7R , respectively. 0 min ≈ Then, the rate of change of Γ due to radiation force on the g g Let n (ε,Ω,R)dR be the number density of photons cell is given by (e.g. Phinney1982; Sikora et al. 1996) ph emitted from thering between R and R+dR,where ε and dΓ = βbfz, (2.1) Ω=(φ,cosθ)aretheenergyanddirectionofincomingpho- dt E tons, respectively. Using (2.2), the photon number density whereE isthecellenergyinKj,andweassumethatthejet (perunitR),nph(ε,Ω,R),inKj canbewrittenas(e.g.Der- is along the z direction. Then, for f >0, thejet is acceler- mer & Schlickeiser 1993) z ated away from thesource, and for fz <0, it is decelerated RF(ε˜,R) cosθ˜ β tfoorwcaerfdztdheepseonudrsceo,ni.teh.erapdairattiicolne ddirsatgr.ibIunti(o2n.1w),htichheiasvienrflague- nph(ε,Ω,R)= 2πc(R2+z2)δ(cid:18)cosθ−1−βbRc−osθ˜bR(cid:19),(2.4) encedbythespecificaccelerationorenergylossmechanisms. where cosθ˜ = z/(R2+z2)1/2, z is the distance from the Inthefollowingdiscussion,weonlyconsiderradiativeeffects discsurfaceR, θ˜ istheangleofincoming photonsrelativeto dueto Compton scattering. R thethejet axis in K . j 2.1 Radiation from the disc 2.2 Radiation force Tostudytheradiativeeffect onjetdynamicsduetoinverse Comptonscattering,aspecificmodelfortheexternalphoton The radiation force f (on the jet bulk flow) can be derived fieldisrequired.Weassumethattheexternalphotonfieldis by calculating the rate of average momemtum transfer to Relativistic jets 3 z(cid:13) scatteredphotons.Asweuseaxisymmetricdiskemission as given by (2.2), the only relevant component of the force is along z axis. Jet(cid:13) WefirstderivetheradiationforceintheThomsonlimit. g(cid:13) In the Thomson scattering regime, scattering is elastic and -rays(cid:13) wehaveε′ ε′whereε′andε′ arerespectivelytheenergies s≈ s of the incoming and scattered photon in the electron rest e(cid:13) frame (K′). Further, we assume a beaming approximation in which the scattered photons propagate approximately in the electron’s direction in K provided that γ 1. Based j ≫ e(cid:13)~(cid:13) ontheseconsiderations,anapproximationforthedifferential R(cid:13) cross section in K can be obtained as (A1), and the force j is calculated as R(cid:13) R(cid:13) BH(cid:13) min(cid:13) max(cid:13) Rmax RF(R) disc(cid:13) fz =σT dRR2+z2 Γ2ε˜R(1−βbcosθ˜R) ZRmin (2.6) Figure 1.AschematicdiagramofanAGNjet. ×(cosθ˜R −βb) 32hγ2βi+1 ne0, (cid:16) (cid:17) logF(R)(cm- 2 s - 1 ) whereweassumethatthediscemission isimportantwithin 26.5 R R R , σ is the Thomson cross section, the min ≤ ≤ max T averageismadeovertheelectron(orpositron)distribution, ... = dΩ dγ(...)n (γ,Ω )/n , and n is the average 26 h i e e e e0 e0 numberdensity.Thedistributionn (γ,Ω )isassumedtobe e e R isotropic in K with a power law: j 25.5 C n (γ,Ω )= 0n γ−p, (2.7) e e 4π e0 25 where p is the electron spectral index with values between 2 and 3, γ γ γ , the constant C is chosen min max 0 ≤ ≤ 24.5 such that (2.7) is normalized to ne0. We assume the cyclotron timescaleismuchlessthantherelevanttimescales of acceleration or deceleration, electron (positron) energy loss. Therefore, all quantities in (2.7) can be regarded as 0.6 0.8 1 1.2 1.4 1.6 average over the cyclotron time. For γ γ and log(R/Rg) p 2, we have C (p 1)γp−1. Usinmgax(2≫.7), wmienhave γ2≥ [(p 1)/(3 0p≈)]γp−1−(γ3−pmin γ3−p) which reducesto Figure 2. Photon flux density from the ring as a function of hγ2i≈2γ2− ln(γ −/γ m)informpax=−3. min R. We assume M = 108M⊙, M˙ = 0.1LEdd/c2. The upper and h iI≈t follmoiwns frommax(2.m6i)nthat acceleration (f > 0) occurs lower curves are obtained by assuming Rmin = 5Rg and 6Rg, for cosθ˜ > β corresponding to when electrzons see most respectively. R b incoming photons with θ 0, and deceleration (f < 0) for cosθ˜ < β corresponRd≈ing to when electrons seez most plasmasinthejetasaresultofinverseComptonscattering, R b photons with θ π in the jet frame. Therefore, the key which is given by R ≈ role in radiative acceleration or deceleration is played by f = dΩ dγ dpn (γ,Ω ) anisotropic scattering, which can be understood as follows. e dt e e Assume a cell moving along the jet containing relativistic Z plasma with an isotropic distribution. The particles in the = c dR dΩ dγn (γ,Ω ) dε dΩn (ε,Ω,R)D − e e e ph (2.5) jet frame see anisotropic photon fields, i.e. the radiation Z Z Z Z mainlycomesfrombehindorinfrontdependingonΓofthe dε dΩ dσ (ε kˆ εkˆ), bulk motion. When cosθ˜R <βb, in the jet frame, although × s s dεsdΩs s s− eachelectronwouldscatterphotonsintoitsdirectionofmo- Z Z (cid:19) tion, on average there is more momemtum beamed along where D = 1 β kˆ, γ = 1/(1 β2)1/2 is the particle the jet direction. Thus, the cell is subject to a force in the − · − Lorentzfactorinthejetframe(K ),n (ε,Ω)isthephoton oppositedirection.ThecriticalΓ,abovewhichtheradiative j ph density, p = γβ is the momemtum (in mec), ne(γ,Ωe) is deceleration (fz < 0) occurs, can be derived from (2.6) by the particle distribution, Ωe = (φe,cosθe) is the direction setting fz = 0 (e.g. Sikora et al. 1996). Since the critical Γ of electron motion, dσ/dε dΩ is the differential cross in isgenerally small (e.g.Phiney1987; Sikoraetal.1996), our s s K whose approximations in theThomson and KNregimes discussion concentrates on deceleration of jets with a large j are given respectively by (A1) and (A2). The direction of initial Γ. incoming andscattered photonsin K are representedbykˆ Inthecaseofdeceleration,sincetheflux(2.2)ispeaked j and kˆ , respectively. All quantities with subscript s are for atR=R =(9/7)R ,anapproximationforf atz R s 0 min z ≫ 0 4 Luo and Protheroe m is given by c R c R 4 1 f 1µ ηΓ2 g 0 γ2 n z ≈− 3 p z2 z h i e0 , (2.8) 0.75 R c(cid:16)R (cid:17)4 = 2µ ηΓ2 g 0 γ2 n ln(γ /γ ), − 3 p z2 z min e0 max min 0.5 (cid:16) (cid:17) where µ = m /m is the proton to electron mass ratio, p p e 0.25 R = 7R /6 is the radius at which the flux is peaked, 0 min η = Ld/LEdd is the radiation efficiency of disc emission, 0 L =2πm c2 dRRF(R)ε˜ istheluminosityofthesource, d e R and the equality is for p = 3. The radiation force increases -0.25 rapidly with ΓR2. -0.5 Although(2.6)isderivedusingthebeamingapproxima- tionasdescribedbyδ(Ωs Ωe)inEq.(A1),apartfromβ 1 -0.75 − ≈ intheaverage γ2β ,itisingoodagreementwiththeresult h i derived by Sikora et al. (1996) using covariant formalism. 0.8 1 1.2 1.4 1.6 1.8 2 One may also reproduce the result given by O’Dell (1981) wherethephotonsarefromapointsource,forwhichwemay log( R / R g ) assume θ˜ = 0 and that 2π dRRF(R)ε˜ L /m c2 is R R → d e the luminosity of the source. In this special case, the parti- Figure 3. Plots µc as afunction of R/Rg for different z and γ. cles are accelerated away fromR thesource. The solidand dashed curves correspond respectively to γ =102 In calculating (2.6) and (2.8), we assume ε′ < 1. In and103.Ineachcase,thecurvesfrombottom totopcorrespond the electron rest frame, the photon energy is ε′ = ε′ to z/Rg = 5, 10, 30, 50. We assume the bulk Lorentz factor R ≡ Γ=104. γ(1 βcosΘ )Γ(1 β cosθ˜ )ε˜ , where Θ is the angle betw−eentheinRcomin−gphbotonRdireRctionandelRectronmotion logG (in K ). Then, the condition for all particles to scatter in 4 j theThomson regime is 3.8 γε <1/2. (2.9) R with ε = Γ(1 β cosθ˜ )ε˜ . This condition requires that 3.6 R − b R R photons of the highest energy, seen by particles with Θ = R 3.4 π, satisfy ε′ < 1, which may not be the case if Γ, or the R average Lorentz factor ( γ ) of the plasma in K ,are large. h i j 3.2 3 2.3 The Klein-Nishina regime For an ultrarelativistic jet with a large bulk Lorentz factor 2.8 or when particles are highly relativistic in the jet comoving frame, Compton scattering should be treated in the Klein- 2.6 Nishina scattering regime. The minimum Γ such that all 0.8 1 1.2 1.4 1.6 particles of energy γ are in the KN regime can be derived as follows. In the jet frame, since the soft photon direction log(R/Rg) is θ π, electrons with θ = π see the lowest energy R e ≈ Figure 4.TheminimumΓforKNscatteringtobedominantin photons. Since cosΘ =cosθ cosθ +sinθ sinθ cos(φ R R e R e φ ) cosθ cosθ (where φ and φ are the azimutha−l deceleration are plotted as functions of R/Rg. The curves from anegle≈so−fthepRhotoneandelectron),andefromε′ 1,wecan top to bottom correspond to z/Rg = 20, 10, 5, 2. We assume derive the condition for all particles to scattRer≥in the KN γwmitihn =M˙8=0,0R.1mLiEnd=d/5cR2.g, and usethe temperature function (2.3) regime, Γ 2 in radiative drag is less strict than (2.10), (2.11), and is > , (2.10) γ ε˜ (1 β cosθ˜ ) generally satisfied for moderate Γ. R − b R In general, given a particle distribution, the Thom- or son and KN approximations apply respectively to parti- Γγ < 21ε˜R(1+cosθ˜R), (2.11) cΓlγes(1witβhµcocsoθseθ<)(1µc aβndcocsoθ˜sθ)eε˜≥=µ1c,, twhhaetries,µc satisfies c R b R R − − where we assume that the plasma is relativistic in the jet 1 1 Γγ(1 β cosθ˜ )ε˜ frame. For ε˜ = 6 10−5, we have Γ/γ > 5 104/(1 − − b R R, for γε 1/2; caonsdθ˜(R2).1o1r)Γar/Reγt<he3K××N1c0o−n5d(i1tio+ncfoosrθ˜pRa)r.tiIcnleesquoafl×iγtiwesit(h2.a1n0−y) µc =1β, Γγ(βb−cosθ˜R)ε˜R for γεRR ≥<1/2. (2.12) θ ,whichrequiresanextremeratio(verylargeorverysmall) Thus,theconditions(2.10),(2.11)correspondtoanextreme e  Γ/γ. However, in the following discussion we show that the case; if one of these two conditions is satisfied, then µ = c condition for theKNscattering tobethedominantprocess 1, and in this case the KN approximation applies to all − Relativistic jets 5 particles. Figure 3 shows plots of µc as functions of R/Rg. - fz,KN/<g> ne0 Forgivenzandγ,theKNscatteringregimeislocatedabove - fz,T/<g> ne0 thecurve(i.e. for electrons or positrons with cosθ µ ). e c ≥ To calculate f , particles of energy γ in thecell are di- z videdintotwopartsaccordingto(2.12).Letfz,T andfz,KN 103 be the force due to scattering by particles with cosθe ≤µc fz,T/<g> ne0>0 (the Thomson regime) and cosθ > µ (the KN regime), e c respectively. Then, theradiation force can be expressed as 102 fz =fz,T+fz,KN. (2.13) fz,T/<g> ne0>0 wheref andf canbecalculatedapproximatelyusing z,T z,KN the cross sections (A1), (A2) as given in the Appendix (cf. 10 Eq. A4 and A5). Assuming that incident photons in the jet frame are approximately beamed, cosθ 1, which is R ≈− valid for Γ 1, we have 2 2.5 3 3.5 4 4.5 5 ≫ logG Rmax RFε˜ Γ2 f = 1σ C n dR R (1 β cosθ˜ )2 z,T −2 T 0 e0 2π(R2+z2) − b R ZRmin Figure 5. Variation of fz,T/hγine0 (dashed), fz,KN/hγine0 γmax (solid) with Γ for z = 5Rg (upper) and z = 10Rg (lower). The × dγγ−p γ2 12(µ2c−1)+ 32(µ3c+1)+ 14(µ4c−1) dottedcurve(forz=5Rg)iscalculatedbyignoringscatteringin Zγmin n h i theKNregime. +(µ +1)+ 1(µ2 1) , (2.14) c 2 c− with α=2/(p 1)+ln(64e)/p ln2/(p+1), which ranges o − − and from4.4(p=2)to2.6(p=3).ForlargeΓsatisfying(2.16), Rmax RF 1 fz,KN overtakesfz,T andbecomeimportant.Forevenlarger fz,KN ≈−83σTC0ne0 dR2π(R2+z2) 4ε˜ Γ, we find ×Zγmγminaxdγγ−pn(cid:0)Z1R−minµ2c(cid:1)ln2+(1−µc) R (2.15) fz ≈fz,KN =−138σTne02ZεRR1(mpaxd1R) 2π(RR2F+z2) (2.18) +2εγR (1−µc)ln(2/√e)+2ln(2εRγ) H(γεR−12), ×2ε˜R 1+ Rγmin−p ln(2εRγmin) , h i h io whereR istheradiusoftheringwhichemitsphotonswith whereH(γε 1/2)=1forγε 1/2andH(γε 1/2)= 1 R− R ≥ R− energy ε = 1/(2γ ). If ε > 1/(2γ ) for R R 0 for γεR < 1/2. In the Thomson approximation, µc = 1, R , RRis replacemdinby R R . Figurem5inshows fmin/≤γ n≤ wehavefz =fz,T,whichreducesto(2.6)withcosθR ≈−1. (Emqa.x14)1, f / γ n (Eqm.in15) as functions ofzΓ,Tfohr iz e=0 For µc < 1, fz,T can be positive even though we assume z,KN h i e0 5R , 10R . The disc parameters are the same as in Figure cosθ = 1. This is because f includes particles with g g R − z,T 2. We assume p = 2.8, R = 5R , R = 50R . The 1 cosθe < µc, and if most of them have θe > π/2 we min g max g − ≤ value of f is significantly lower than that calculated us- would have f > 0. In contrast, f is always negative z z,T z,KN ing the Thomson approximation (dotted curve), indicating since it includes only those particles with µ < cosθ 1, c e ≤ the importance of the KN scattering. The KN scattering mostofwhichmoveforward alongthejetandcontributeto startstoovertaketheThomsonscatteringwhenΓ>300for drag.From(2.14)and(2.15),weseethatwhenµ <1scat- c z = 5R and Γ > 200 for z = 10R , and it is described by tering in theKNregime becomes important while theforce g g Eq. (2.17). Scattering in the KN regime becomes dominant duetoscatteringintheThomsonregimestartstodecrease. in decelerating the jet at Γ > 103 and the approximation From (2.12), this condition becomes (2.18)applies.Theforcecomponentduetoscatteringinthe 1 Γ . (2.16) Thomson regime becomes positive for large Γ since most ≥ 2γ(1 β cosθ˜ )ε˜ particles that scatter in the Thomson regime move toward − b R R Figure 4shows thecondition (2.16) as afunction of R with the black hole and hence contribute to acceleration of the γ being replaced by γ . We assume γ = 80, p = 3. Thus, cell.However,thenetforceisstillnegativebecausethecom- the average Lorentzhfacitor in the jet frame is γ = 160. ponent due to forward moving particles that scatter in the This gives, for example, Γ>400 for z =2R , anhdiΓ>700 KN regime dominates. For extremely large Γ, we may have g for z = 5Rg. These lower limits can be further reduced as 2εR >pγmin/(p−1)ln(2εRγmin),andthen,thesecondterm scattering in the KNregime extendsto higher γ. on the right hand side of (2.18) becomes important, and Integration overγ in Eq.(2.14) and(2.15) canbedone fz fz,KN increases logarithmically with Γ. ≈ analytically using (2.7), and is given by Eq. (A8), (A9) in the Appendix. For 2ε < 1/γ , (2.15) has the following R min 3 RADIATIVE DECELERATION approximation f A relativistic jet can form in the vicinity of a black hole z,KN ≈ through either (1) a hydrodynamic process, in which ra- ασ n RmaxdR RF(R) 3(εRγmin)p−1, (2.17) diation from an accretion disc plays an important role in − T e0ZRmin 2π(R2+z2) 16ε˜R producing and collimating the jet, or (2) a MHD process, 6 Luo and Protheroe in which magnetic fields play major role in the jet flow. In 15 eithercase,regardlessofthedetailsofjetproduction,which _ 1100 arepoorlyunderstood,thejetmustpassthroughintensera- z diationfieldsfrom discemission andbesubjecttoradiative 55 deceleration.Inthefollowingdiscussion,weassumethatthe jet has initially a large Γ and then undergoes deceleration 3 due to the radiation force discussed in earlier sections, and weexaminetheconstraintonthebulkflowwhenscattering 2 in theKN regime is included. log(-d G / G d_z) 1 3.1 Constraint on Γ 0 22 The bulk Lorentz factor Γ as a function of z can be cal- 33 culated by integrating (2.1) along the jet direction. For de- logG 44 celeration from the KN to Thomson regime, we use (2.13) 5 together with (2.14) and (2.15) for which integration over R can be done numerically. We first consider an electron positron jet with an energy density (in m c) given by Figure6.Aplotof−dlnΓ/dz¯vs.z¯(withz¯=z/Rg)andΓ.The e discparametersarethesameasinFigure5. E = n γ . (A relativistic jet containing protons will be e0 h i considered in Sec.3.2.) From thedistribution (2.7), we have -dlnG /dz_ p−2 103 p 1 γ E ≈ne0γminp−2 1− γmin , (3.1) − " max! # for p > 2 and γmax γmin. Figure 6 shows dlnΓ/dz 102 ≫ as a function of z and γ. Radiative drag is severe, i.e. dlnΓ/dz 1, for small z and moderate Γ, but decreases ≪− as z increases. Plots of dlnΓ/dz as a function of z for an electron-positron jet are given in the solid curves in Figure 101 7. For a very large bulk Lorentz factor, the deceleration efficiency decreases significantly because scattering is in the KN regime. For example, we havea much smaller dlnΓ/dz 100 for Γ=105, as shown in Figure 7. In the Thomson limit, we can solve (2.1) with (2.8) to find Γ Γ(z) 0 , (3.2) 10 20 _ 30 40 50 ≈ 1+ηΓ0ξT z where Γ is the initial valueat z=z , and 0 0 Figure7.Plotsof−dlnΓ/dz¯vs.z¯(z¯=z/Rg).Thesolidcurves, ξT(z0,z)= 115µpRz0g Rz00 5 1− zz055 hγγ2i. (3.3) ffroormapbuortetoemlecttorotno-ppocsoirtrreosnpojentd(iin.eg.tnop/Γn=e01=050,,5c0f.,E10q2.,3.110)3.,Tahree (cid:16) (cid:17) (cid:16) (cid:17) h i dotted anddashed curvesareforthejetcontaining coldprotons Forp=3andthedistributiongivenby(2.7),ξ reducesto T with np/ne0 = 0.1 and 0.5, respectively (cf. Sec. 3.2). In each R R 5 z5 ofthese two cases, the curves frombottom totop correspond to ξ 102 g 0 1 0 γ ln(γ /γ ). (3.4) Γ=50,102,103.ThediscparametersarethesameasinFigure T ≈ z0 z0 − z5 min max min 5. (cid:16) (cid:17) (cid:16) (cid:17) Assuming that theinitial value of Γ is large and to particles scattering in the KN regime. From (2.17), and ηΓ0ξT ≫1, using (2.1), the bulk Lorentzfactor is calculated to be one has theasymptotic valuegiven by Γ(z) Γ0 , (3.6) Γ( ) 1 . (3.5) ≈ [1+ηΓ0(p−2)ξKN]1/(p−2) ∞ ≈ ηξT(z0,∞) with For, p = 3, γ /γ = 5 103, γ = 50, R = 2R , ξ = max min × min min g KN ηan=d0z.01,=an1d0RΓg(, e)ven1w0iftohrΓη0=≫0.011,.wTehhuas,vienΓt(h∞eT)h≈om1sfoonr 3α(p−2)2µpε˜Rp−3Rg R0 2p−1 1 z0 2p−1 γp−2, (3.7) limit, radiation∞dra≈g is indeed a severe constraint on Γ as 16π2p−1(2p−1) R0(cid:16)z0(cid:17) h −(cid:16) z (cid:17) i min alreadyshowninearlierwork,e.g.byMelia&Köngl(1989), where p > 2, α = 4.4 2.6, ε˜ = 5 10−5. Deceleration − R × Sikora et al. (1996). issignificantly slowerthanthatcalculated fromtheforcein Deceleration describedby(3.2)appliesonlyforsmallΓ theThomsonapproximationsinceξ issmallerthanξ by KN T or γ . When Γ satisfies (2.16), deceleration is mainly due afactorof 5γ ln(γ /γ )(for p=3).Eq.(3.6) applies max min max min Relativistic jets 7 only within a narrow range of Γ (i.e. 1/ γ ε 1/γ ) G( )z h i≤ R ≤ min as shown in Figure 5. 5 10 15 ForevenlargerΓ ,thedragismainlyduetoscattering 105 0 in the KN regime (2.18). Retaining only the first term on theright hand side of (2.18), we have 104 3ηµ R z Γ(z)=Γ p g 1 0 , (3.8) 0− 16πhγiε˜2R z0 (cid:16) − z (cid:17) 3 for z z z , where z is the distance at which Γ(z ) is 10 0 1 1 1 ≤ ≤ sosmallthat(3.8) doesnotapply.Figure8showstherapid dropof Γ(z)withthedistancez,whereweassume thatthe 102 jetstartswithΓ =105 atz =5R and10R ,respectively. 0 0 g g In each case, the first curve from the left corresponds to the deceleration of an e± jet. The solid curves correspond 10 to deceleration in the Thomson regime, and are calculated using(2.6).ThebulkLorentzfactordecreasesfromΓ=105 to Γ = 103 within a distance ∆z z . Radiation drag ≪ 0 5 10 15 is effective except for Γ0 ≥(3ηµp/16πhγiε˜2R)(Rg/z0)≈5× z/Rg 106(η/0.1)(R /z )(for γ =100).However,forsuchalarge g 0 h i Γ , the second term becomes important, and Γ decreases 0 Figure 8. Deceleration of jets. Deceleration in the KN regime exponentially according to and Thomson regime are indicated respectively by dashed and ln(Γ/Γ ) solidpartofthecurves.ThejetisassumedtostartwithΓ=105 0 3ηµ (p≈ 1) z R 1 cosθ˜ (3.9) at z =5Rg and z =10Rg. In each case, the curves from left to p − dz′ g − R ln(2ε γ ). right correspond to np/ne0 = 0 (a pure electron-positron jet), − 4πp Zz0 z′2 ε˜Rhγiγmin R min np/ne0=0.1,andnp/ne0=0.5.Theefficiency ofdiscemission, η=0.05,isused.Protonsareassumedtobecold. For p=3, we have η R R2 z3 ln(2ε γ ) ln(Γ/Γ ) 5.3 g 0 1 0 R min , (3.10) 0 ≈− 0.1 z3 − z3 ε˜ γ2 0 R min (cid:16) (cid:17) (cid:16) (cid:17) 3.3 Energy loss due to inverse Compton with ε = Γ ε˜ . The drag strongly depends on γ , and R 0 R min scattering can bereduced for large γ . min In the calculation of the radiation force, we assume that the plasma in the jet is highly relativistic. In practice, the 3.2 A jet containing protons particledistributionisdeterminedbythevariousenergyloss Ifajetisheavilyloadedwithprotons,theeffectofradiation processesandtheinjectionrateofhighenergyparticles.The drag is reduced significantly, since protons scatter photons energy loss rate due to inverse Compton scattering can be with much a smaller cross section compared to electrons, derivedfrom γ˙ = β dp/dtwithdp/dtgivenby(2.5).In − − · andthepresenceofprotonseffectivelyincreases theinertia. theThomsonlimit,thelossrateisobtainedinthejetframe When the jet contains cold protons, the energy density is (K ) as j E = 2n γ +n µ where n is the number density of protonsea0ndminthe elpectpron spectrpal index is assumed to be γ˙ = RmaxdR RF(R) Γ2(1 β cosθ˜ )24σT γ2ε˜ p = 3 (cf. Eq. 2.7). Suppose that n is so large that the −h ia R2+z2 − b R 3 R p ZRmin (3.12) condition 4 cR R δ µpnp >1 (3.11) ≈ 23µpη z2g z0 Γ2γ2, ≡ 2γminne0 (cid:18) (cid:19) is satisfied. The value δ in Eq. 3.11, corresponding to the where ... a represents an average overΩe, and theapprox- h i ratioofthecoldprotonenergydensitytotheelectronenergy imation is valid only for Γ z/R0. In the large angle ap- density, can increase significantly if protons are relativistic proximation, we have γ˙ a≫ γ2Γ2. In the small angle ap- in the jet frame. Then, f /E is reduced by a factor δ. The proximation θ˜ Γ−1−,hi.ei. t∝hephotonfield isfrom apoint z R ≪ asymptotic Γ( ) given by (3.5) would increase by a factor source,wehave γ˙ a γ2/Γ2.From(3.12),thetimescale ∞ −h i ∝ δ. For example, for γmin =50, one must include the inertia inK for energylossforΓγ <1/ε˜R andz >R0 isestimated of protons if n /n > 0.05, and the drag force would be to be p e0 significantly reduced. z2 z 4 1 The effect of protons on the deceleration efficiency is t 3 shown in Figure 7 by dotted curves for np/ne0 = 0.1 and IC ≈ 2cRg(cid:18)R0(cid:19) Γγη (3.13) dashed curves for np/ne0 = 0.5. We have smaller dlnΓ/dz 0.1 ε˜ 1/ε˜ z 6 than that for the e± jet, indicating a decrease in the de- =0.017s η 6 1R0−5 ΓγR 10R , g celeration efficiency. A relatively weaker radiation drag on (cid:16) (cid:17)(cid:18) × (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) jet containing protons can also be seen in Figure 8. The where ε˜ = 6 10−5 corresponds to 30 eV photons. For R × jets with n /n = 0.1 and 0.5 have much higher terminal Γγ 1/ε˜ , the time scale should be calculated in the KN Lorentz facptorse0than thepuree± jet. limi≥t. R 8 Luo and Protheroe In the KN scattering regime, their energy loss rate is thecharacteristictransversesizeofthejet,R R =1.5 j g ≈ × much slower, i.e. 1013m(M/108M⊙), the condition ρg <Rj should be easily satisfied.Itcanalsobeshownthattheapproximationmade Rmax RF(R) 3σ γ˙ dR T ln(2√eγε ) earlier (Eq. 2.5), i.e. neglecting electron gyration, is valid. −h ia ≈ZRmin R2+z2 8ε˜R R (3.14) Thecyclotrontimescaletc ≈ρg/c≈10−3s(γ/104)(1G/B), 3µpη cRg ln(2√eγΓε˜ ). is much shorter than tIC, tKN, ts, and even much shorter ≈4 ε˜2 z2 R than the characteristic time of deceleration, which is typi- R cally >1s. Unlike (3.12) in the Thomson limit, the energy loss rate increases only slowly with increasing Γ and γ. In K, the characteristic time for theenergy loss is 4 CONCLUSIONS AND DISCUSSION 0.1 ε˜ z 2 Γγε˜ Radiative deceleration of relativistic jets due to inverse t 0.03s R R , (3.15) Compton scattering is discussed in both the Thomson and KN ≈ (cid:16) η (cid:17)(cid:18)6×10−5(cid:19)(cid:16)10Rg(cid:17) ln(3.3Γγε˜R) KN scattering regimes. We show that KN scattering is im- where Γγε˜ 1. portant in deceleration of ultrarelativistic jets with initial R ≥ Plasmas in the jet are likely to be magnetized. Then, bulkLorentzΓ>103.ForΓ 1,neartheblackholeandin ≫ particles would also radiate synchrotron emission. The en- thejetframe,theincomingphotonsarebeamedtowardsthe ergy loss rate dueto synchrotron emission is given by hole, and so the main contribution to the drag of the jet is fromhead-onscatteringbyforwardmovingparticles.These dγ 1 σ B2γ2 T , (3.16) particlesaremorelikely tobeintheKNregimeastheysee dt ≈ 4π mec photonsofthehighestenergy(ε′>1).Particlesscatteringin where B is the magnetic field in the jet frame. Thus, the theThomsonregimemaycontributepositivelytotheradia- synchrotron time scale is tionforcebutoverallKNscatteringisdominantandresults 10G 2 102 indeceleration.Thus,scatteringintheKNregimeshouldbe ts≈3×103s B γ . (3.17) includedincalculationsoftheradiationforcewhenΓ>103. OurresultshowsthatintheKNregime,radiationdragisre- (cid:16) (cid:17) (cid:16) (cid:17) Theenergylossduetosynchrotronemission canbecompa- duced,butstillseverelyconstrainsthespeedofthejetbulk rable to (3.13) and (3.15) only for strong magnetic fields or flow.Thus,Compton dragcan decelerate ane± jet starting large γ. with any Γ>10 in theregion sufficiently close totheblack Because of the short time scales indicated by (3.13) hole down to the value Γ < 10. The efficiency of decelera- and (3.15) near the black hole, for plasmas in the cell to tionissignificantlyreducedifthejetcontainsprotonssince behighlyrelativistic oneneedsanefficientinjection ofelec- protons have much smaller scattering cross section and the tron positron pairs, e.g. through cascades or acceleration. effect of protons is to increase theinertia of thejet. The relevant processes, either pair cascades or direct ac- Accordingtotheunifiedscheme(e.g.Urry&Padovani celeration of electrons (positrons), must be fast enough to 1995), blazars are radio-loud quasars, and if this is true, overcome energy loss due to inverse Compton scattering or a relativistic jet with sufficient particle kinetic luminosity synchrotronradiation.Althoughdetailsofaccelerationnear is required to power radio lobes at a larger distance. The ablackholearenotwellunderstood,theexistenceofarapid results given here show that because of radiation drag the acceleration mechanism that is able to accelerate particles ability of an electron-positron jet topower large scale radio toultrarelativisticenergieswithinashorttimescalecannot lobesisseverelyconstrained.Forexample,considerane±jet be ruled out. One possibility is acceleration by a rotation- with an initial luminosity N˙ Γ L , whereN˙ and Γ 1 i i d i i ≈ ≫ induced potential drop, similar to that occuring in pulsars, are the initial particle flux and Lorentz factor. As a result whereparticlesextractedfrom theneutronstarareacceler- of Compton drag, the flow slows down to Γ < 10 with f atedtoultrarelativisticenergieswithinamuchshortertime most of the energy going into radiation. Thus, even with scale than the rotation period (e.g. Michel 1987). Particles pair cascades N˙ N˙ , we may still have N˙ Γ N˙ Γ . f i f f i i injected at radial distance Rinj to the black hole are accler- The constraint can≫be overcome if e± are re-accele≪rated at ated to energy γ σ2/3∆R/Rinj after travelling a distance a distance further away from the central region or if there ≈ ∆R,whereσ=e∆Φ/mec2,∆Φisthepotentialdropwhich is outflow of accelerated protons as suggested in a two-flow can be estimated from the total power output, L0, of the modelbySol,Pelletier&Asseo(1989).Intheirmodel,radio AGN, and ∆R/Rinj 1 (e.g. Michel 1987). Then, the ac- lobesare powered byoutflowof p-eplasma jetsand e± jets ≪ celeration time tacc =∆R/c is estimated tobe are dominant only in the subparsec region. Inourdiscussion,theexternalsoftphotonsareassumed γR γ R tacc = cσ2i/n3j ≈10−3s 104 2Ringj , (3.18) to come only from the disc emission which is the sum of (cid:16) (cid:17)(cid:16) (cid:17) blackbodyemissionfrom seriesofringscentredattheblack where σ2/3 1010 for L = 1046ergs−1. Since t < t , hole, and is similar to the model discussed by Demer & ≈ 0 acc IC t ,electrons(positrons)canbeaccelerated totheenergies Schlickeiser (1993). Generalization of the calculation to in- KN in theKN regime. cludeotherradiation fields,suchasthatduetoadisctorus For magnetized plasmas, the condition for relativistic (Protheroe & Biermann 1997), and reprocessed radiation, plasmas to be contained in the jet requires that the gy- shouldbestraightforward. Theeffectonthejetduetoscat- roradius being less than the characteristic transverse size, tering of photons from the broad-line region may also be R , of the jet. Since the gyroradius is given by ρ = important,andisnotconsideredhere.Incalculatingtheav- j g 2 107cm(γ/104)(1G/B)whereBisthemagneticfield,for erage force, we assumed an isotropic (angular) distribution × Relativistic jets 9 of electrons (or positrons) with a power law. Some mecha- beaming approximation. nism, e.g. pitch angle scattering, is required for isotropiza- To evaluate (2.5), particles with γ can be broadly sep- tion since inverse Compton scattering itself tends to make arated intotwogroups:onegroup ofparticleswith cosθ < e thedistribution anisotropic. µ )scatterintheThomson regime with(A1)andtheother c group with cosθ µ scatterin theKNregime with (A2). e c ≥ This approximation is similar to that was used Dermer & ACKNOWLEDGEMENTS Schlickeiser(1993)incalculatingscatteredphotonspectrum. Thus,using(A1) and(A2),theradiation force (2.5) can be QLacknowledgesthereceiptofAustralianResearchCouncil written approximately into theform (ARC) Postdoctoral Fellowship. dp f dΩ dγ zn (γ,Ω ) REFERENCES z ≡ e dt e e Z RF(R) Begelman, M. C.,Blandford,R. D.& Rees, M. J.1980, Nature, σ dR 287,307. ≈− T 2π(R2+z2) Z Blandford,R.D.1990,inActiveGalacticNuclei,eds.R.D.Bland- ford,H.Netzer&L.Woltjer,(Springer-Verlag,Berlin)p.161. ×ε˜RΓ2(1−βbcosθ˜R)2 dΩedγne(γ,Ωe) Blumenthal, G. R. & Gould, R. J. 1970, Rev. Mod. Phys., 42, Z 237. (1 βcosΘ ) γ2(1 βcosΘ)cosθ cosθ H(1 ε′ ) Dermer,C.D.&Schlickeiser,R.1993,ApJ,416,458. × − R − e− R − R Eracleous,M.,Livio,M.,Halpern,J.P.&Storchi-Bergmann,T. (cid:26) h i 1995,ApJ,438,610. +3ln(2√eε′R) γcosθ ε cosθ )H(ε′ 1) , (A3) Gaidos,J.A.etal.1996, Nature,383,319. 8 Γγε˜ e− R R R− R (cid:27) Haswell,C.A.,Tajima,T.&Sakai,J.I.1992.401,495. (cid:0) Ko¨nigl,A.1994,inTheFirstStromloSymposium:ThePhysicsof whereH(ε′ 1)=1forε′ 1andH(ε′ 1)=0forε′ < ActiveGalaxies,ed.G.V.Bicknell,M.A.Dopita&P.J.Quinn, R− R ≥ R− R 1,n (γ,Ω )istheparticledistributionasdefinedearlier(Eq. ASPConf.SeriesVol.54,p.33. 2.7)e. Theeterms with H(1 ε′ ) and H(ε′ 1) correspond Luo,Q.1998,inpreparation. − R R − respectively to scattering in theThomson and KN regimes. Melia,F.&Ko¨nigl,A.1989,ApJ,340,162. Michel,F.C.1987,ApJ,321,714. Accordingly,fz iswrittenasasumoftwocomponents, O’Dell,S.L.1981,ApJ,243,L147. fz = fz,T+fz,KN where fz,T, fz,KN describe the contribu- Phinney,E.S.1982, MNRAS,198,1109. tions to the force from scattering in the Thomson and KN Phinney, E. S. 1987, in Zensus, J. A. & Pearson, T. J. eds. Su- regime, respectively,and they are perluminalRadioSources,(CambridgeUniv.Press),p.301. Protheroe, R. J., Biermann, P. L. 1997, Astroparticle Phys., 6, RF(R) 293. fz,T=−σT dR2π(R2+z2)ε˜RΓ2(1−βbcosθ˜R)2 Quinn,J.etal.1996,456,L83. Z Reynolds,S.P.1982, ApJ,256,38. dγ dφe µcdcosθe n (γ,Ω )(1 βcosΘ ) (A4) Schubnell,M.S.etal.1996,ApJ,460,644. × 2π 2 e e − R Sikora,M.etal.1996, MNRAS,280,781. Z Z Z−1 Sol,H.,Pelletier,G.&Asséo,E.1989,MNRAS,237,411. γ2(1 βcosΘ)cosθ cosθ , × − e− R Thompson,D.J.etal.1995,ApJS,101,259. h i Urry,C.M.&Padovani,P.1995,PASP,107,803. and vonMontigny,C.etal.1995,ApJ,440,525. RF(R) f = σ dR ε˜ Γ2(1 β cosθ˜ )2 z,KN − T 2π(R2+z2) R − b R APPENDIX A: MOMEMTUM TRANSFER DUE Z TO COMPTON SCATTERING dγ dφe 1dcosθe n (γ,Ω )3ln(2√eε′R) (A5) × 2π 2 e e 8 Γγε˜ Calculation of momemtum transfer (2.5) requires the full Z Z Zµc R tKaNilsdbiffyeBrelnutmiaelnctrhoasls&dσG/doεuslddΩ(s19w7h0i)c.hHweares,dwisecuussseedthienfdoel-- × γcosθe−εRcosθR), lowing approximation (in K ) (Reynolds 1982; Dermer & (cid:0) j where µ , n (γ,Ω ) are defined by (2.12), (2.7). Integra- c e e Schlickeiser 1993) tion over cosθ can be easily done using approximation e dσ sinθ cosθ 1, cosθ 1. In this approxima- σ δ ε γ2ε(1 βcosΘ) δ(Ω Ω ), (A1) | R| ≪ | R| ≈ R ≈ − dεsdΩs ≈ T s− − s− e tion, the integrands in (A4) and (A5) can be regarded as independentof φ . Then,integrating overcosθ ,we obtain for the Thomso(cid:2)n regime (ε′<1) and(cid:3) e e dσ 3σT ln(2√eε′)δ(ε γ)δ(Ω Ω ), (A2) fz,T= dεsdΩs ≈ 8ε′ s− s− e RFε˜ Γ2 1σ C n dR R (1 β cosθ˜ )2 dγγ−p for theKN regime (ε′ 1), where σ is theThomson cross −2 T 0 e0 2π(R2+z2) − b R section, ε′ is the inco≥ming photonTenergy in the electron Z Z rest frame, cosΘ = cosθcosθe+sinθsinθecos(φ−φe), Θ × γ2 21(µ2c−1)− 32βcosθR(µ3c+1)+ 41(µ4c−1) istheanglebetween theincomingphotondirection andthe n h i electronmotion,thedeltafunctionδ(Ωs−Ωe)describesthe −cosθR(µc+1)+ 12(µ2c−1) , (A6) o 10 Luo and Protheroe RF 1 f 3σ C n dR dγγ−p z,KN ≈−8 T 0 e0 2π(R2+z2)4ε˜ Z R Z 1 µ 1 µ2 ln2 − c × − c − βcosθ R (A7) n2ε(cid:0) cosθ(cid:1) R R (1 µ )ln(2/√e) − γ − c 1h + 1 ln(2ε γ) H(γε 1). − βcosθ R R− 2 R (cid:16) (cid:17) io Eq. (A6) reduces to Eq. (2.6) or (2.8) by assuming µ =1, c i.e. all particles are in the Thomson scattering regime. Let I andI representγ-integrationin(A6)and(A7);I and T KN T I are calculated to be KN 1 1 I = 1+ γ−p−1 γ−p−1 T 2(p+1)ε2 2ε2 c − max R R (cid:16) (cid:17)(cid:16) (cid:17) 1 4 + γ−p γ−p + γ3−p γ3−p (A8) 3pε3 max − c 3(3 p) c − min R(cid:16) (cid:17) − (cid:16) (cid:17) 2 + γ1−p γ1−p , p 1 min − c − (cid:16) (cid:17) 2 I = γ1−p γ1−p KN p 1 c − max − (cid:16) (cid:17) 1 ln(4e) + +2ε ln(4/e)+4ε /p γ−p γ−p p εR R R c − max (A9) h1 ln2 i(cid:16) (cid:17) +ln(4/e) γ−p−1 γ−p−1 −p+1 ε2 c − max R h i(cid:16) (cid:17) 4ε + R γ−pln(2ε γ ) γ−p ln(2ε γ ) , p c R c − max R max h i where γ = max 1/ε ,γ . Then, (A6) and (A7) can be c { R min} furtherwritten into theform f = z,T Rmax RFε˜ Γ2(1 β cosθ˜ )2 (A10) − 21σTC0ne0 dR R2π(R−2+bz2) R IT, ZRmin Rmax RFI 1 f 3σ C n dR KN . (A11) z,KN ≈−8 T 0 e0 2π(R2+z2)4ε˜ ZRmin R ThispaperhasbeenproducedusingtheRoyalAstronomical Society/Blackwell Science TEX macros.