Boundary Shear Acceleration in the Jet of MKN501
aa r X i v : . [ a s t r o - ph . H E ] A ug Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 8 June 2018 (MN L A TEX style file v2.2)
BOUNDARY SHEAR ACCELERATION IN THE JETOF MKN501
S. Sahayanathan ⋆ Astrophysical Sciences Division, Bhabha Atomic Research Centre, Mumbai - 400085, India
ABSTRACT
The high resolution image of the jet of the BL Lac object MKN501 in radio, show alimb-brightened feature. An explanation of this feature as an outcome of differentialDoppler boosting of jet spine and jet boundary due to transverse velocity structureof the jet requires large viewing angle. However this inference contradicts with theconstraints derived from the high energy γ -ray studies unless the jets bends over alarge angle immediately after the γ -ray zone (close to the central engine). In thisletter we propose an alternate explanation to the limb-brightened feature of MKN501by considering the diffusion of electrons accelerated at the boundary shear layer intothe jet medium and this consideration does not require large viewing angle. Also theobserved difference in the spectral index at the jet boundary and jet spine can beunderstood within the frame work of shear acceleration. Key words: galaxies: active - galaxies: jets - BL Lacertae objects: individ-ual(MKN501) - acceleration of particles - diffusion
BL Lac objects are the extreme class of active galactic nu-clei(AGN) with weak or no emission lines and are cate-gorized along with flat spectrum radio quasars(FSRQ) asblazars. Their spectra cover a broad range of photon ener-gies starting from radio to gamma rays with a few of themdetected in TeV energies by ground based Air Cerenkovexperiments(Krawczynski (2004); Katarzy´nski, Sol, & Kus(2001); Sambruna (2000),Costamante & Ghisellini (2002)).These sources are found to be strongly variable withflare time scales ranging from days to less than anhour (Gaidos et al. (1996); Coppi & Aharonian (1999);Sambruna (2000); Krawczynski et al. (2000)). The shorttime variability and their detection at very high energiesdemand that the emission region should be moving down ajet at relativistic velocities close to the line of sight of the ob-server (Ghisellini et al. (1993); Dondi & Ghisellini (1995)).The strong polarization detected in radio/optical energiesand the non-thermal photon spectra indicates the radio tox-ray spectra is due to synchrotron radiation from a non-thermal electron distribution cooling in a magnetic field.However the gamma ray emission from these sources isstill not well understood. Leptonic models explain the highemission as inverse Compton scattered synchrotron photonsby the electron population responsible for the synchrotron ⋆ E-mail: [email protected] process itself(SSC) (Maraschi, Ghisellini, & Celotti (1992);Bloom & Marscher (1996); B¨ottcher (2000)) where as inhadronic models it is due to the synchrotron proton emissionand proton-photon interactions involving an external photonfield (synchrotron proton blazar model(SPB)) (Mannheim(1998); M¨ucke et al. (2003)). Under unification hypothesis ofradio-loud AGN, BL Lac objects are considered to be alignedjet version of Fanaroff-Riley type I (FRI) radio galaxies(Urry & Padovani (1995)).MKN501 is a nearby BL Lac object (z=0.034) and alsothe second extra galactic source detected in TeV photon en-ergies by ground based Cherenkov Telescopes(Quinn et al.(1996)). It was later detected in MeV photon energies by thesatellite based experiment EGRET (Kataoka et al. (1999)).The radio images of MKN501 show a jet emerging froma bright nucleus (Edwards et al. (2000); Giovannini et al.(1999); Aaron (1999); Giroletti et al. (2004)). The highresolution (milli arc second) radio images show a trans-verse jet structure with the edges being brighter thanthe central spine commonly referred as ”limb-brightened”structure (Edwards et al. (2000); Giovannini et al. (1999);Giroletti et al. (2004)). This feature is usually explainedby the ”spine-sheath” model where the velocity at the jetspine is larger compared to the velocity at the boundary.Such a radial stratification of velocity across the jet ariseswhen jet moves through the ambient medium and the vis-cosity involved will cause a shear at the boundary. Three-dimensional hydrodynamic simulations of relativistic jets c (cid:13) S. Sahayanathan (Aloy et al. (2000)) and two-dimensional simulations of rel-ativistic magnetized jets (Leismann et al. (2005)) also sup-ports the presence of jet velocity stratification due to itsinteraction with the ambient medium. The existence of ve-locity shear at the jet boundary was first suggested byOwen, Hardee, & Cornwell (1989) to explain the morphol-ogy of M87 jet. Perlman et al. (1999) later confirmed itthrough the polarisation studies of M87 jet. If the jet is mis-aligned towards the observer, it may happen for a propercombination of velocities we see a Doppler boosted imageof the boundary compared to the less boosted spine giv-ing rise to a limb-brightened structure(Komissarov (1990);Laing (1996)). A possible consequence of the velocity shearis the alignment of the magnetic field at the boundary par-allel to the flow velocity due to stretching of the frozen-infield lines of the plasma(Kahn (1983)). The polarisation an-gle observed at the jet boundary of MKN501 is perpendic-ular to the jet axis(Pushkarev et al. (2005); Aaron (1999))indicating a parallel magnetic field. However it should benoted here that the polarisation angle at the jet spine in-dicates a perpendicular magnetic field and this along withthe parallel magnetic field at the jet boundary can bean outcome of a dynamically dominant toroidal magneticfield structure (Pushkarev et al. (2005); Gabuzda (1999);Gabuzda, Murray, & Cronin (2005)). The radial velocitystratification of the jet can introduce Kelvin-Helmholtz in-stability and the stability of jets against this instabilitywas studied by many authors(Turland & Scheuer (1976);Blandford & Pringle (1976); Ferrari, Trussoni, & Zaninetti(1978); Hardee (1979); Birkinshaw (1991)).Giroletti et al. (2004) have studied the limb-brightenedstructure of MKN501 jet considering the differential Dopplerboosting at the jet spine and the boundary (Laing (1996);Komissarov (1990)) and concluded the viewing angle (an-gle between the jet and the line of sight of the ob-server) of the radio jet should be more than 15 o . How-ever high energy studies of MKN501 demands the view-ing angle of the jet should be ≈ o in order to ex-plain the observed rapid variability and the high energyemission(Katarzy´nski, Sol, & Kus (2001); Tavecchio et al.(2001)). Considering the fact that the gamma ray emissionis originated from the inner part of the jet close to nucleus,Giroletti et al. (2004) suggested a bending of the jet mayhappen immediately after the gamma-ray zone to explainthe required large viewing angle of the radio jet. However themechanism required to bend the jets are still not well under-stood (jets deflected due to the pressure gradient in externalmedium is studied by Canto & Raga (1996); Raga & Canto(1996); Mendoza & Longair (2001)) and moreover the ob-served large bending of the jet in the radio maps can beapparent one because of projection effects. This projectioneffects are even amplified when the jet is close to the line ofsight. Though it needs to be noted here that jets with largebending angle are indeed observed(Savolainen et al. (2006)).The limb-brightened structure can also be explainedif we consider the synchrotron emission from the par-ticles accelerated at the boundary and this inferencedoes not require large viewing angle. Eilek (1979, 1982)considered the acceleration of particles due to tur-bulence initiated by Kelvin-Helmholtz and Rayleigh-Taylor instabilities at the jet boundary. Particles at theboundary can also be accelerated via shear accelera- tion (Berezhko (1981); Berezhko & Krymskii (1981)) andthis case is considered in the present work. The accel-eration of particles in a shear flow or by turbulenceis well studied by various authors for both relativisticand non relativistic case (Earl, Jokipii, & Morfill (1988);Webb (1989); Ostrowski (1990); Stawarz & Ostrowski(2002); Rieger & Duffy (2006); Stawarz & Petrosian (2008);Virtanen & Vainio (2005)).In this letter we explain the observed limb-brightenedfeature of MKN501 by considering the diffusion of electronsaccelerated at the jet boundary via shear acceleration. In thenext section we show the required condition for the shear ac-celeration to be dominant over turbulent acceleration and in § § H o = 75km s − Mpc − and q = 0 . The particle acceleration process at the jet boundary can bedescribed by the diffusion equation in momentum space. Theevolution of an isotropic phase space distribution is givenby(Melrose (1968)) ∂f ( p ) ∂t = 1 p ∂∂p „ p D ( p ) ∂f ( p ) ∂p « (1)where D ( p ) is the momentum diffusion coefficient. The char-acteristic acceleration timescale can be written as t acc = p » ∂∂p ` p D ( p ) ´– − (2)If we consider a sheared flow, the electrons are scatteredacross different velocity layers by turbulent structures whichare embedded in the shear flow. Berezhko (1981) showed insuch case there will be a net gain of energy in the elec-trons getting scattered and this process is referred as shearacceleration. The momentum diffusion coefficient in caseof a shear flow can be written as(Rieger & Duffy (2006);Rieger, Bosch-Ramon, & Duffy (2007)) D s ( p ) = χp τ (3)where τ is the mean scattering time given by τ ≃ λ/c with λ the mean free path and χ is the shear coefficient given fora relativistic flow as(Rieger & Duffy (2004)) χ = c r ) − „ ∂ Γ ∂r « (4)where Γ( r ) is the bulk Lorentz factor of the flow and r isthe radial coordinate of the jet cross section. Using (3), theshear acceleration timescale( t ( s ) acc ) for τ = τ o p ξ will be t ( s ) acc = 1(4 + ξ ) χτ (5)In case of turbulent acceleration(stochastic), the particlesare scattered off by randomly moving scattering centres and c (cid:13) , 000–000 gets energized by second order Fermi acceleration. The mo-mentum diffusion coefficient in this case can be approxi-mated as(Rieger, Bosch-Ramon, & Duffy (2007)) D t ( p ) ≃ p τ „ V A c « (6)where the Alfven velocity( V A ) is given by V A = B √ πρ (7)here B is the magnetic field and ρ the mass density of thejet. Hence the turbulent acceleration timescale( t ( t ) acc ) will be t ( t ) acc = 3 τ (4 − ξ ) „ cV A « (8)For shear acceleration to be dominant over turbulent acceler-ation t ( s ) acc < t ( t ) acc . If we consider Bohm diffusion ( ξ = 1) thenthe mean free path of the electron aligned to the magneticfield ( λ k ) scales as the gyro radius ( r g )(Achterberg & Ball(1994)), λ k ≃ η γm e c eB , where η is a numerical factor ( η > γ ( ≫
1) is the Lorentz fac-tor of the electron scattered. Since the magnetic field atthe jet boundary of MKN501 is parallel to the jet axis (ortoroidal)(Aaron (1999); Pushkarev et al. (2005); Gabuzda(1999)), we consider τ ≃ λ k /c . Also if we consider ∂ Γ ∂r ≃ ∆Γ∆ r (9)where ∆Γ is the difference between the bulk Lorentz factorat the jet spine and the jet boundary and ∆ r is the thicknessof the shear layer, then the condition for shear accelerationto be dominant over turbulent acceleration will be∆ r < ηγm e c (∆Γ) eB » πρ r ) − – (10)If we consider the mass density of the jet is dominated bycold protons and if the number of protons are equal to thenumber of non-thermal electrons, then the jet mass densitycan be written in terms of equipartition magnetic field( B eq )as ρ ≃ m p B eq (2 α − πm e c αγ min (11)and (10) will be∆ r < . ηγc (∆Γ) eB eq » m e m p (2 α − αγ min (Γ( r ) − – (12)where α is the observed photon spectral index, m p is theproton mass and γ min is the Lorentz factor of electron re-sponsible for the minimum observed photon frequency ν min .The equipartition magnetic field can be expressed in termsof observed quantities as B eq ≃ .
62 1Γ( r ) ( m e ceν min ) » d L F ( ν min ) V σ T (2 α − – G (13)where F ( ν min ) is the flux at the minimum observed fre-quency ν min , d L is the luminosity distance, V is the vol-ume of the emission region and σ T is Thomson cross sec-tion. Hence, for Γ( r ) ≫ α ≃ .
7, shear acceleration will dominate the particle spectrum at the jet boundary ofMKN501 if the thickness of the shear layer∆ r < . × − × “ η ” „ ∆Γ10 « “ ν obs . ” “ ν min ” − ×× „ F (10MHz)910mJy « − „ R . « parsec (14)Where R is the radius of the spherical region considered.(We assume 10 MHz as minimum observed frequency andthe flux at 10
MHz is obtained from the flux at 1 . GHz considering the same spectral index. The flux at 1 . GHz and R in (14) are obtained from a region around R.A 10 mas anddeclination − mas from Fig.7 of Giroletti et al. (2004)).The corresponding equipartition magnetic field B eq for Γ =5 is 1 . × − G .The electrons accelerated by shear acceleration cool viasynchrotron radiation. The cooling time for synchrotron lossis given by t cool = 6 πm e cγσ T B eq (15)Using (5) and (15), we find t ( s ) acc t cool ≃ . × − „ B . × − G « “ η ” − „ Γ( r )5 « ×× „ ∆ r − parsec « „ ∆Γ10 « − (16)and since t ( s ) acc ≪ t cool , shear acceleration dominates oversynchrotron cooling. It can be noted that (16) is independentof the electron energy and hence the maximum energy ofthe electron will be decided by the loss processes other thansynchrotron loss (which are not considered in this simplistictreatment).If we maintain the general form of mean scattering time τ = τ p ξ , then for shear acceleration to dominate over turbu-lent acceleration the thickness of the shear layer (∆ r ) shouldbe ∆ r < . × τ p ξ (∆Γ)Γ( r ) » (4 + ξ )(2 α − α (4 − ξ ) γ min – cm (17)It can be noted that (10) is equal to (17) if we set in thelatter ξ = 1 and τ p ξ = ηr g /c . Particles accelerated at the shear layer of the jet bound-ary, diffuse into the jet medium before getting cooled offvia synchrotron radiation. As the magnetic field at the jetboundary is parallel to the jet axis (or toroidal)(Aaron(1999); Pushkarev et al. (2005); Gabuzda (1999)), the ra-dial diffusion of the electron into the jet medium is deter-mined by cross field diffusion. The cross field diffusion coeffi-cient can be approximated as (Axford (1965); Jokipii (1987);Achterberg & Ball (1994)) κ ⊥ ≈ η r g c (18)Where η ( >
1) is the scaling factor determining the fieldaligned mean free path (see ( § c (cid:13) , 000–000 S. Sahayanathan
The radial distance R diff that the electron diffuse be-fore getting cooled can then be approximated as R diff ≈ √ κ ⊥ t cool (19)Using (15) and (18) and considering the equipartition mag-netic field we get R diff ≃ . × − “ η ” − „ B . × − G « − parsec(20)Since the thickness of the shear layer ∆ r ≪ R diff (refer(14) and (20)), the thickness of the limb brightened structurewill be ≈ R diff . This corresponds to an angular distance of4 . × − mas which is beyond the resolution of present daytelescopes.For τ = τ p ξ , the cross field diffusion coefficient will be κ ⊥ ≃ τ r g p − ξ (21)Using (15) and (19) we get R diff ≃ . × B − τ − p − ξ cm (22)and hence the thickness of the limb brightened structure willbe energy dependent for ξ = 1. If we add mono-energetic particle injection term ( δ ( p − p o ))and particle escape term ( − /t esc ) in (1), then the steadystate equation in case of shear acceleration for p > p o and ξ = 1 can be written as p d f s dp + 5 p df s dp − f s χτ o t esc = 0 (23)and in case of turbulent acceleration it will be p d f t dp + 3 df t dp − f t ψt esc = 0 (24)where ψ = V A c τ o . If we substitute p = 1 /x in (23) we get x d f s dx − df s dx − f s χτ o t esc = 0 (25)Equations (24) and (25) can be solved analytically (Kepinski(1905)) and the solutions are complex and are given by f s = „ χτ o pt esc « ×× » a s J „ i r χτ o pt esc « + b s Y „ i r χτ o pt esc «– (26)and f t = „ ψt esc p « » a t J „ i r pψt esc « + b t Y „ i r pψt esc «– (27)Where J n ( z ) and Y n ( z ) are the Bessel functions of first andsecond kind and a s , b s , a t and b t are constants. For negli-gible escape ( t esc → ∞ ), using the limiting forms of Besselfunctions (Abramowitz & Stegun (1972)), the solutions (26)and (27) approaches a power law f s ∝ p − and f t ∝ p − .The shear accelerated particle number density will then be n s ( p ) ∝ p − and the corresponding synchrotron photon flux will be S ν,shear ∝ ν − / . For turbulent acceleration thenumber density will be independent of p ( n t ( p ) ∝ p ) andhence the observed synchrotron photon flux will be a flatone S ν,turb ∝ ν / . The spectral index map of MKN501 jetindicates a steep photon spectra at the boundary and a flatspectra at the spine(Giroletti et al. (2004)). Hence it can beargued that the shear acceleration may be dominant at thejet boundary of MKN501 and turbulent acceleration at thejet spine. However ξ is usually related to the turbulent spec-tral index (Biermann & Strittmatter (1987)) which may bedifferent at the jet boundary and jet spine. As the AGN jet moves through the ambient medium theviscosity involved will cause a shear at the jet boundaryand hence acceleration of particles in these shear layer isunavoidable. If the shear gradient ∂ Γ /∂r is very steep or ifthe shear layer is very thin (14), then shear acceleration candominate over the turbulent acceleration initiated by theinstabilities at the jet boundary(Eilek (1982)). Turbulentacceleration may play an important role at the interior re-gions of the jet (Virtanen & Vainio (2005)) and can providean alternative to explain the emission from the inter knotregions of AGN jets (Macchetto (1996); Jester et al. (2001)).The observed hard spectra at the jet spine (Giroletti et al.(2004)) also supports this inference since turbulent accelera-tion can produce a hard particle spectra(Virtanen & Vainio(2005)) (also shown in section § (4). The electrons acceler-ated by the turbulence can be reaccelerated by shocks andcan form a broken power law electron spectrum. This canpossibly explain the break in the radio-to-x-ray spectra ofthe knots of FRI jets (Sahayanathan (2008)).Giroletti et al. (2004) calculated the jet viewing angle( θ ) using the correlation between the core power and thetotal power (Giovannini et al. (2001)). They estimated thejet viewing angle to be within 10 o < θ < o by compar-ing the observed core radio power and the expected intrin-sic core power derived from the correlation. However thisestimation may vary if the core flux density variability ismore than factor 2. Also considering the variation of theparameter values in the correlation with increase numberof samples, this may not provide a strong constrain on thejet viewing angle. The estimate of θ based on the adiabat-ically expanding relativistic jet model (Baum et al. (1997))may not be a strong constraint as it considers a simplifiedsituation. Also the constrain is less severe in case of per-pendicular magnetic fields and observed polarisation stud-ies have indicated the presence of perpendicular magneticfields at jet spine (Pushkarev et al. (2005); Aaron (1999)).Stawarz & Ostrowski (2002) proposed a model similar to thepresent one, however their aim was to show the observa-tional implications of the two-component particle spectrum(power law distribution with high energy pile-up) formed atthe boundary shear layer and the complex beaming pattern. The observed limb brightened structure seen in the radiomaps of MKN501 jet can be explained if we consider the c (cid:13) , 000–000 shear acceleration of particles at the boundary due to ve-locity stratification and their diffusion into the jet medium.This inference does not demand large viewing angle whichis required otherwise for the explanation via differentialDoppler boosting of the jet spine and boundary. We haveshown that shear acceleration dominates over turbulent ac-celeration at the boundary if we consider thin shear layer ora sharp velocity gradient. Also for the estimated set of pa-rameters, shear acceleration timescale is much smaller thansynchrotron cooling timescale allowing acceleration of elec-trons to be possible. The thickness of the limb brightenedstructure will be decided by the distance electrons have dif-fused into the jet medium before loosing its energy via syn-chrotron radiation. However the estimated thickness is be-yond the resolution of present day telescopes. Simple ana-lytical solution of the steady state diffusion equation consid-ering mono-energetic injection and particle escape, indicatesa steep particle spectra for the electrons accelerated at theshear layer in comparison with turbulent acceleration. Theradio spectral index map of MKN501 jet is also observed tohave steep spectra at the boundary supporting the presenceof shear acceleration.The author is grateful to the anonymous referee forhis useful comments which helped in clearing many of theignorances and a better understanding. The author ac-knowledges the useful discussions with S. Bhattacharyya, N.Bhatt, M.Choudhury and A.Mitra. The author is grateful toL. Stawarz and F. M. Rieger for enlightening information onvarious topics related to shear acceleration. REFERENCES
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