Polarization and structure of relativistic parsec-scale AGN jets

23 Pages·2005·0.84 MB·English

Mon.Not.R.Astron.Soc.360,869–891(2005) doi:10.1111/j.1365-2966.2005.08954.x
Polarization and structure of relativistic parsec-scale AGN jets
(cid:1)
M. Lyutikov,1,2,3 V. I. Pariev4,5† and D. C. Gabuzda6
1PhysicsDepartment,McGillUniversity,3600rueUniversity,Montreal,QC,CanadaH3A2T8
2CanadianInstituteforTheoreticalAstrophysics,60StGeorge,Toronto,Ont,M5S3H8,Canada
3KavliInstituteforParticleAstrophysicsandCosmology,2575SandhillRoad,MenloPark,CA94305,USA
4DepartmentofPhysicsandAstronomy,UniversityofRochester,Rochester,NY14627,USA
5LebedevPhysicalInstitute,LeninskyProspect53,Moscow119991,Russia
6DepartmentofPhysics,UniversityCollegeCork,Cork,Ireland
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Accepted2005February22.Received2005February11;inoriginalform2004June6 n
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ABSTRACT fro
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We consider the polarization properties of optically thin synchrotron radiation emitted by h
relativistically moving electron–positron jets carrying large-scale helical magnetic fields. In ttp
s
our model, the jet is cylindrical and the emitting plasma moves parallel to the jet axis with ://a
c
a characteristic Lorentz factor (cid:2). Wedraw attention to the strong influence that the bulk ad
e
relativisticmotionoftheemittingrelativisticparticleshasontheobservedpolarization. m
ic
Ourcomputationspredictandexplainthefollowingbehaviour.(i)Forjetsunresolvedin .o
u
thedirectionperpendiculartotheirdirectionofpropagation,thepositionangleoftheelectric p.c
o
vector of the linear polarization has a bimodal distribution, being oriented either parallel or m
perpendiculartothejet.(ii)Ifanultra-relativisticjetwith(cid:2) (cid:1)1whoseaxismakesasmall /m
n
angletothelineofsight,θ ∼1/(cid:2),experiencesarelativelysmallchangeinthedirectionof ras
/a
propagation,velocityorpitchangleofthemagneticfields,thepolarizationislikelytoremain rtic
parallelorperpendicular;ontheotherhand,insomecases,thedegreeofpolarizationcanexhibit le
largevariationsandthepolarizationpositionanglecanexperienceabrupt90◦ changes.This -ab
s
changeismorelikelytooccurinjetswithflatterspectra.(iii)Inorderforthejetpolarization tra
c
tobeorientedalongthejetaxis,theintrinsictoroidalmagneticfield(intheframeofthejet) t/3
6
shouldbeoftheorderoforstrongerthantheintrinsicpoloidalfield;inthiscase,thehighly 0/3
relativisticmotionofthejetimpliesthat,intheobserver’sframe,thejetisstronglydominated /8
6
by the toroidal magnetic field Bφ/Bz (cid:1) (cid:2). (iv) The emission-weighted average pitch angle 9/99
oftheintrinsichelicalfieldinthejetmustnotbetoosmalltoproducepolarizationalongthe 96
2
jet axis. In force-free jets with a smooth distribution of emissivities, the emission should be 4
b
generatedinalimitedrangeofradiinottooclosetothejetcore.(v)Formildlyrelativisticjets, y g
u
whenacounter-jetcanbeseen,thepolarizationofthecounter-jetispreferentiallyorthogonal e
s
totheaxis,unlessthejetisstronglydominatedbythetoroidalmagneticfieldinitsrestframe. t o
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(vi)Forresolvedjets,thepolarizationpatternisnotsymmetricwithrespecttojetaxis.Under 0
4
certain conditions, this can be used to deduce the direction of the spin of the central object A
p
(black hole or disc), whether it is aligned or anti-aligned with the jet axis. (vii) In resolved ril 2
0
‘cylindricalshell’typejets,thecentralpartsofthejetarepolarizedalongtheaxis,whilethe 1
9
outerpartsarepolarizedorthogonaltoit,inaccordancewithobservations.
Weconcludethatlarge-scalemagneticfieldscanexplainthesalientpolarizationproperties
of parsec-scale AGN jets. Since the typical degrees of polarization are (cid:2)15 per cent, the
emittingpartsofthejetsmusthavecomparablerest-frametoroidalandpoloidalfields.Inthis
case, most relativistic jets are strongly dominated by the toroidal magnetic field component
intheobserver’sframe, Bφ/Bz∼(cid:2).WealsodiscussthepossibilitythatrelativisticAGNjets
(cid:1)
E-mail:[email protected]
†Currentlyat:PhysicsDepartment,UniversityofWisconsin-Madison,1150UniversityAve.,Madison,WI53706,USA.
(cid:4)C 2005RAS
870 M.Lyutikov,V.I.ParievandD.C.Gabuzda
maybeelectromagnetically(Poyntingflux)dominated.Inthiscase,dissipationofthetoroidal
magneticfield(andnotfluidshocks)mayberesponsibleforparticleacceleration.
Keywords: MHD–radiationmechanisms:non-thermal–galaxies:active–galaxies:jets–
galaxies:nuclei.
1 INTRODUCTION
VLBIlinearpolarizationstudiesofcompactparsec-scalejetsinradio-loudAGNshaverevealedanumberofgeneraltendencies(e.g.Cawthorne
etal.1993;Gabuzda,Pushkarev&Cawthorne2000;Gabuzda2003,andreferencestherein).Thesetendenciesinclude:(i)aroughlybimodal
distributionoftheelectricvectorpositionangles(EVPAs)forthejetlinearpolarizationwithrespecttojetdirection,withquasarstendingto
havepolarizationorthogonaltothejetandBLLacobjectstendingtohavepolarizationalongthejet(Marscheretal.2002);(ii)theEVPAs
D
sometimes ‘follow’ the jet as it bends, keeping its relative orientation with respect to the jet direction (Fig. 1; see also Owen, Hardee & ow
n
Cornwell1989forthekiloparsecjetinM87);(iii)thejetEVPAssometimesexperienceorthogonaljumpsalongthejet,changingfromaligned lo
a
toorthogonalorviceversa(Gabuzda&Go´mez2001;Pushkarevetal.2005;Fig.2);(iv)insomeresolvedjets,theEVPAinthecentralpart d
e
d
ofthejetliesalongthejet,whiletheedgesarepolarizedorthogonallytothejetdirection(Attridge,Roberts&Wardle1999;Moellenbrock, fro
Roberts&Wardle2000;Pushkarevetal.2005);(v)Faraday-rotationgradientsarefrequentlyobservedacrossthejetsofBLLacobjects m
(Gabuzda,Murray&Cronin2004).Allofthesepropertiesarestatistical,eachhavinganumberofknownexceptions. http
1977T),haecclaerlgeera-sticoanle(em.ga.gnBeltaincdfifoelrdds&arePaaylmneos1t9u8n2i;veCrosanltloypaocucloepst&edLtoovpelalaycean19im94p;orCtaanmternozleinidn1th9e95p)roadnudc,tieosnpe(cei.agl.lyB,lcaonldlfimoradti&onZonfajtehke s://ac
a
jets(e.g.Heyvaerts&Norman1989;Sauty,Tsinganos&Trussoni2002;Heyvaerts&Norman2003);seealsothereviewbyBegelman, de
m
Blandford&Rees(1984).Yettheirimportanceforjetpropagationandemissiongenerationisunderestimated.Mostcommonly,polarization ic
structureisattributedtotransverseorobliquerelativisticshockspropagatinginthejetandcompressingtheupstreamturbulentmagneticfield, .ou
p
sothatthecompresseddownstreammagneticfieldbecomesanisotropicallydistributed(e.g.Laing1980;Hughes,Aller&Aller1989a,b). .c
o
Thismechanismproducesfieldscompressedintheplaneoftheshockandisexpectedtoproducepolarizationalongthejetforthecaseof m
/m
atransverseshock.Numerousobservationsofpolarizationorthogonaltothejetareattributedtoashearedcomponentofmagneticfieldina n
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sheathsurroundingthejet.Thisoverallscenarioisopentocriticism.First,itishardtoseehowweakerbridgeswiththesamecharacteristic s
/a
polarizationconnectingbrightknotscouldbeproduced:thiswouldrequireaquasi-steadysequenceofweakshocks.Secondly,sinceinternal rtic
shockscanbeoblique,witharangeoforientations,itisnotnaturaltohaveabimodaldistributionoftherelativeEVPAs(theEVPAsinBLLac le
-a
seemtobeindisagreementwiththeshockmodelDenn,Mutel&Marscher2000).Thirdly,somefeaturesshowinglongitudinalpolarization b
s
arefarfrombeingcompact(e.g.theresolvedVLBIjetsof1219+285,seeGabuzdaetal.1994,and1803+784,seeGabuzda&Chernetskii tra
c
2003). t/3
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Analternativeinterpretationofthejetpolarization,whichwefavour,isthattheflowcarrieslarge-scalehelicalmagneticfields.Inthis 0
/3
paper,wecalculatethepolarizationpropertiesofarelativisticjetcarryinghelicalmagneticfieldsandcomparethemwithobservations.The /8
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1 0 -1 -2 -3 -4 -5 -6
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Figure1. Observed4-cmtotal-intensitycontoursandsuperposedlinear-polarizationvectorsfortheBLLacobject1749+701:thepolarizationremainsaligned
withthejetdirectionasthejetbends(dataprocessingisdescribedinGabuzda,Pushkarev&Garnich2001a).
(cid:4)C 2005RAS,MNRAS360,869–891
PolarizationofrelativisticAGNjets 871
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Figure2. Observed6-cmtotal-intensitycontoursandsuperposedlinear-polarizationvectorsfortheBLLacobject1418+546:theorientationofthepolarization em
relativetothejetdisplaysjumpsbetweenbeingalignedwithandorthogonaltothejet(dataprocessingandfurtheranalysisarediscussedinPushkarevetal. ic
(2005). .ou
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.c
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resultsareencouraging.Wecanbothreproducetheaveragepropertiesofthejetpolarization,suchasthebimodaldistributionoftheobserved /m
EVPAs,aswellasallowforpossibleexceptions,suchasorthogonaljumpsintheEVPAalongthejet. n
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TherehavebeentwoapproachestostudiesofthesynchrotronpolarizationofAGNjets.Ontheonehand,Laing(1980,1981)developed s/a
detailedmodelsforopticallythinsynchrotronemissionfromnon-relativisticjets.Recently,suchnon-relativisticmodelsweregeneralized rtic
toincludeplasmaeffectsonthepropagationoftheradiationinsidethejet,FaradayrotationandtheCotton–Moutoneffect(Zheleznyakov le-a
&Koryagin2002;Beckert&Falcke2002;Beckert2003;Ruszkowski2003).Intheseworks,althoughplasmaisrelativistic,theradiation bs
andpolarizationtransportareconsideredinastaticmedium.Inactualjets,theplasmaisinastateofbulkmotionwithrelativisticspeeds. tra
c
Ontheotherhand,thepolarizationpropertiesofthesynchrotronemissionfromrelativisticAGNjetshavebeendiscussedbyBlandford& t/3
6
0
Ko¨nigl(1979),Bjornsson(1982)andKonigl&Choudhuri(1985),butnotinthecontextoflarge-scalespiralmagneticfields.Pariev,Istomin /3
&Beresnyak(2003)haveinvestigatedthepolarizationpropertiesofaspecificconfigurationofaspiralmagneticfieldwithauniformaxial /86
9
field,zerototalpoloidalcurrentinthejet,andwithshearedrelativisticspiralmotionoftheplasmaperpendiculartothemagneticfieldlines. /9
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Inthispaper,wepresenttheresultsoffullyrelativisticcalculationsforvariousjetinternalstructuresandtheresultingpolarizationstructures. 9
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WetakeastepbackandtrytoanswerthequestionofwhetherwecanreproducethemostsalientpolarizationfeaturesofAGNjetsunder 4
b
genericassumptionsabouttheinternalhelicalmagneticfieldinthejet(basicallygeneralizingtheworkofLaing1981,torelativisticjets).In y
g
ordertoanswerthisquestion,onemustknowtheinternalstructureofthejet(distributionofpoloidalandtoroidalmagneticfields)andthe ue
s
distributionofsynchrotronemissivities.Thesearehighlyuncertainintermsoftheoreticalprinciples,andarealsosubjecttoaveragingover t o
n
thewholejet(forunresolvedjets)andalongthelineofsight(forresolvedjets). 0
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Conventionally (and erroneously for a relativistically moving plasma!), the direction of the observed polarization for optically thin A
p
regionsandtheassociatedmagneticfieldsareassumedtobeinone-to-onecorrespondence,beingorthogonaltoeachother,sothatsome ril 2
observerschosetoplotthedirectionoftheelectricvectorofthewave,whileothersplotvectorsorthogonaltotheelectricvectorsandcall 0
1
themthedirectionofthemagneticfield.Thisiscorrectonlyfornon-relativisticallymovingsources,andthuscannotbeappliedtoAGNjets. 9
First,relativisticboostingchangestherelativestrengthofthemagneticfieldcomponentsalongandorthogonaltothelineofsight,which
transformdifferentlyundertheLorentzboost.Thus,ifwecouldseethemagneticfieldsdirectly,thestrengthofthejetpoloidalandtoroidal
fieldsmeasuredinthelaboratoryframewouldbedifferentfromthosemeasuredinthejetframe.Secondly,sincetheemissionisboostedby
therelativisticmotionofthejetmaterial,thepolarizationEVPArotatesparalleltotheplanecontainingthevelocityvectoroftheemitting
volumevandtheunitvectorinthedirectiontotheobservernˆ,sothat theobservedelectricfieldofthewaveisnot,ingeneral,orthogonal
totheobservedmagneticfield(Blandford&Ko¨nigl1979;Lyutikov,Pariev&Blandford2003).Thus,toreconstructtheinternalstructureof
relativisticjetsfrompolarizationobservationsoneneedstoknowboththepolarizationandthevelocityfieldofthejet.Overall,plottingthe
directionoftheelectricfield(EVPA)shouldbeconsideredtheonlyacceptablewaytorepresentpolarizationdataforsourceswhererelativistic
motionmaybeinvolved.
Becauseofthefiniteresolutionofrealpolarimetricobservations,theobservedintensitiesofthepolarizedandunpolarizedcomponents
areaveragesovertheeffectivebeamsize.Whentheresolutionisrelativelylow,thetransversedistributionoftheintensityisunresolved.Inthis
case,onlytheintensityandpolarizationaveragedoverthejetsizeisobserved.Ifthejetiscircularincrosssectionandthedistributionsofthe
(cid:4)C 2005RAS,MNRAS360,869–891
872 M.Lyutikov,V.I.ParievandD.C.Gabuzda
magneticfields,emittingparticlesandvelocityfieldsareaxisymmetricacrossthejet,thenthewholejetcanbedecomposedintoacollection
ofinfinitesimallythinshells.Theseshellshaveconstantcomponentsofthespiralmagneticfieldandconstantemissivityalongtheirperimeter,
andaremovinguniformlywithaconstantvelocity.Thesummationofthepolarizationfromthefrontandbackcrossingpointsofalineofsight
withsuchashellresultsintheobservedEVPAofeachshelleitherbeingparallelorperpendiculartothejetaxis.Consequently,theintegrated
polarizationfromacylindricalaxisymmetricjetiseitherparallelorperpendiculartothejetaxis.Thisistrueregardlessofrelativisticeffects
andisalsotrueinthepresenceofrotationofthejet(Parievetal.2003).Realjetsarenotcylindrical,anddivergewithdistancefromthecore,
andtheymayhaveanellipticalorotherwisenon-axisymmetriccrosssection.Wequalitativelyexpectthattheseeffectswillleadtodeviations
oftheEVPAfromastrictlybimodaldistribution,andeventotheappearanceofπ/4EVPAswithrespecttothejetaxis.
Polarization along or orthogonal to the jet is often interpreted in terms of toroidally or poloidally dominated jets. Since the ratio of
thetoroidaltopoloidalfieldsdependsonthereferenceframe,oneshouldbecarefulindefiningwhatismeantby,forexample,atoroidally
dominatedjet.Astronglyrelativisticjetwithcomparabletoroidalandpoloidalfieldsinitsrestframe(defined,forexample,asaframein
whichtheparticledistributionisisotropic)willbestronglytoroidallydominatedintheobserverframe. D
o
Thechoiceofreferenceframeinwhichthestructureofthejetisconsidereddependsonthequestionstobeasked.Forexample,the w
n
intrinsicstabilityofthejet(e.g.withrespecttokinkmodes)dependsmostlyonthepropertiesofthejetinitsrestframe.Itisconjecturedthat lo
a
stronglytoroidallydominatedjets(intheirrestframe)shouldbecomeunstable(seeworksonnon-relativisticMHDstabilityofjets(Torricelli- de
d
Ciamponi&Pietrini1990;Appl&Camenzind1992;Appl,Lery&Baty2000).Thoughtherearestablelaboratoryfieldconfigurationswith fro
dominanttoroidalfields(reverse-fieldpinches),theyrequireacarefularrangementofcurrentsthatisnotlikelytooccurunderastrophysical m
h
conditions.Inaddition,relativisticforce-free,differentiallyrotatingpinches1 arelikelytobemorestablethanthenon-relativisticanalogue ttp
owingtothestabilizingeffectsofdifferentialrotation(Istomin&Pariev1994,1996).However,westressthatbecauseofthespecificchoiceof s://a
theuniformaxialmagneticfieldusedintheseworks,thedifferentialrotationmightnothaveastabilizingeffectformoregeneralequilibrium c
a
magneticfieldconfigurations(seefurtherconsiderationsinLyubarskii1999;Tomimatsu,Matsuoka&Takahashi2001).Thesetwoarguments de
m
showthatthefieldconfigurationintherestframemaybesomewhatdominatedbythetoroidalfield,butisunlikelytobestronglytoroidally ic
.o
dominated.Theobservationalappearanceofjets,suchastheirintensityandlinearpolarization,alsodependsmostlyontheinternalstructure u
p
ofthejetandtheemissivitydistribution,butapropertransformationtotheobserverframeisneeded.Ontheotherhand,interactionsofajet .c
o
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withthesurroundingmediumandthecorrespondinginstabilities(e.g.Kelvin–Helmholtz)dependonthejetpropertiesintheobserverframe
/m
(Blandford&Pringle1976;Turland&Scheuer1976;Hardee1979;Bodo,Mignone&Rosner2004). n
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Our polarization calculations indicate that, in the jet rest frame, the toroidal and poloidal fields are generally comparable, so that in s
/a
theobserverframe,therelativisticjetsaredominatedbythetoroidalmagneticfield:thehighertheDopplerfactorofthejet,thelargerthe rtic
observedtoroidalmagneticfield.Notethat,inthisspecificcase,theobservedpolarizationelectricvectorsaveragedoverthejetsizewould le
-a
bealignedwiththejetdirection,andtheconclusionthatthecorrespondingmagneticfieldinthejetframeisatleastslightlydominatedby b
s
the‘transverse’(toroidal)B-fieldcomponentwould,infact,becorrect.Thisillustratesthefactthatcertainofthegeneralconclusionsthat tra
c
havebeendrawninearlierworkbasedonobservationswithrelativelylittleresolutionacrossthejetsmaywellbecorrect,eventhoughitis t/3
6
truethattheobservedelectricvectorsarenot,ingeneral,orthogonaltothemagneticfield.Notealsothattheconclusioninitalicsaboveis 0/3
alsoconsistentwiththedynamicalevolutionofthelarge-scalepoloidalandtoroidalfields:itisexpectedthattheratioofthepoloidaltothe /8
6
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toroidalmagneticfieldintheobserverframedecreaseslinearlywiththecylindricalradiusofthejetasitexpandsfromthecentraldisc(sizes /9
oftheorderofhundredsofau)toathicknessof∼0.1pc;(seeSection7). 99
6
2
Wealsoassumethattheemissionisopticallythinandneglectpossibleplasmapropagationeffects,suchasFaradayandCotton–Mouton 4
b
effects. This approximation is usually well justified (Pariev et al. 2003). Faraday rotation is absent in a pair plasma, as well as intrinsic y
g
depolarizationinsidethejet.VLBIpolarizationmeasurementsaredoneatfrequenciesintherange1.6to43GHz.Synchrotronself-absorption u
e
isverysmallatGHzandhigherfrequencies.For B ∼10−2 G,n ∼0.1cm−3,and p ≈2.5,anapproximateestimateforthesynchrotron st o
self-absorptioncoefficientisκs =(5×103 pc)−1(ν/1GHz)−2−p/2 (e.g.Zhelezniakov1996).Thecorrespondingopticaldepththroughthe n 0
1-pc jet is only 2 × 10−4. The typical energy of particles emitting in this frequency range in a 10−2 G magnetic field is 100 MeV to 1 4 A
p
GreelaVt.ivTishteiccpoonwveerrs-liaownopfarltiinceleardpisotlraibriuzteiodnisnwtoitchirpcu≈la2r.p5oalnadrizaemdirnaidmioumwaLvoersen(tthzefaCcotottro∼n–1M0,othuetoensteimffeactet)foisrtahlseocosmnvaelrl.siFoonrcbooetfhficthieenrtmisalκacn=d ril 201
(500pc)−1(ν/1GHz)−3(Sazonov1969).Thisistoosmalltoinfluencethetransferoflinearpolarizationinsidethejet,butcouldbethesource 9
forasmallcircularpolarization,atthelevelofafractionofapercent.Wefocusonlinearpolarizationinthiswork.
Asafirststep,weconsiderthesynchrotronemissionofanunresolved,thin,circularcylindricalshellpopulatedbyrelativisticelectrons
withapower-lawdistributionandmovinguniformlywithconstantvelocityβ =v/c.Thepropertiesofthesynchrotronemissionarethen
determinedbythreeparameters:theinternalpitchangleofthemagneticfieldψ(cid:6),theLorenzfactoroftheshellinthelaboratoryframe(cid:2)and
theviewingangleθ thatthelineofsighttotheobservermakeswiththejetaxisintheobserver’sreferenceframe.Notethat,inthecaseof
unresolvedjets,thisisequivalenttocalculatingthepolarizationfromajetwithapitchangleproperlyaveragedbytheemissivity.
ByperformingthecorrectLorentztransformationsofthejetandradiationelectromagneticfields,weconstructpolarizationmapsfor
variousobservationanglesθandjetparameters.Wefindthat,inordertoproducepolarizationalongthejetaxis,thepitchangleofthehelical
1Here,byarelativisticforce-freepinch,wemeanthattheelectricfieldsareoftheorderofthemagneticfields,andthechargedensitiesareoftheorderof
j/nec,seebelow.
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PolarizationofrelativisticAGNjets 873
magneticfieldinthejetframemustbeoftheorderofunityorgreater,B(cid:6)/B(cid:6) ∼1.ByvirtueoftheLorentztransformation,thisimpliesthat
φ z
therelativisticjetsareobservedtobedominatedbythetoroidalmagneticfield Bφ/Bz(cid:1)(cid:2).Inaddition,wefindthatachangeoftheEVPA
inthejetcanoccuronlyforanarrowrangeofrest-framepitchangles,π/4(cid:2)ψ(cid:6)(cid:2)π/3,whichallowsonetomeasure(andnotonlyderivea
lowerlimitfor)theratioofthetoroidalandpoloidalmagneticfields:Bφ/Bz∼(cid:2)(cid:1)1.
Further,weconsidertheemissionfrom‘filled’jets,i.e.whenmostofthejetvolumecontributestotheemissivity.Tofindtheemission
propertiesofsuchjets,onealsoneedstointegratethejetemissivityoverthejetvolume.Thisrequiresknowledgeoftheinternalstructure
ofthejetandthedistributionofsynchrotronemissivitythroughoutthejet,whichintroduceslargeuncertaintiesintothemodel.Totestthe
sensitivityoftheresultstoaparticularjetmodel,weinvestigatethreepossibleforce-freemodelsforthejetstructure(diffusepinch,pinch
withzerototalpoloidalmagneticfluxandhigh-orderreverse-fieldpinch)andtwoprescriptionsforthenumberdensityofrelativisticparticles
(proportionaltothesecondpowerofthecurrentdensityandtothemagneticenergydensity).Weconcludethattheemission-weightedaverage
pitchangleinthejetmustnotbetoosmallifwewishtoproducelongitudinalpolarization.Thisimpliesthataquasi-homogeneousemissivity
distributioncannotreproducethevarietyofpositionanglebehaviourobserved–theaveragepolarizationremainsprimarilyorthogonaltothe D
o
jet,dominatedbythecentralpartsofthejetwherethemagneticfieldinthecomovingframeisprimarilypoloidal(sincethetoroidalfieldmust w
n
vanishontheaxis).Sincethisisnotgenerallythecase,thesynchrotronemissioninsuchjetsmustbegeneratedinanarrowrangeofradii, lo
a
wherethetoroidalandpoloidalmagneticfieldsareofthesameorderofmagnitudeinthejetframe.ThoughLaing(1981)arguedagainstthis de
d
possibilityonthegroundsthatitshouldproducestrongly‘edge-brightened’profilesinresolvedjets,subsequentobservationsoftheresolved fro
jetsofCenA(Clarke,Burns&Feigelson1986)andM87(Biretta,Owen&Hardee1983)did,infact,showsuchanintensitydistribution. m
h
ttp
s
2 SYNCHROTRON EMISSION FROM STEADY RELATIVISTIC FLOWS ://a
c
a
Consider a cylindrical jet (Fig. 3) viewed by an observer. Below we denote all quantities measured in the local frame comoving with an d
e
elementaryemittingvolumewithprimes,whileunprimedquantitiesarethosemeasuredintheobserver’sframe.Letr,φandzbecylindrical m
ic
coordinatescentredonthejetaxisandx,yandzbethecorrespondingrectangularcoordinates.Themagneticfieldisintheφ–zdirection. .o
u
Thecomponentsofallvectorswrittenbelowarecomponentswithrespecttotherectangular(cid:1)coordinatesx,yandz.Thevelocityβ=v/cof p.c
thebulkmotionisdirectedalongthez-direction(thecorrespondingLorentzfactoris(cid:2)=1/ 1−β2).Wedonotconsiderbulkrotationof om
/m
thejetinthiswork.Anobserverislocatedinthex–zplane,sothattheunitvectoralongthedirectionoftheemittedphotonscanbewritten
n
nˆ ={sinθ,0,cosθ}.Wedenotethemagneticfieldinthejetby B,themagneticfieldintheelectromagneticwaveemittedbyasmallvolume ra
s
ohfatthoevejertabvyebct,othreweilllecmtreiacnfiaeludniintvtheectjoertibnythEa,tadnidretchteioenl,eec.tgri.cBˆfieisldaiunntihteveelcetcotrroinmtahgendeitrieccwtioavneoefmthiettemdabgyneatiscmfiaellldvoinlutmheeoobfstehrevejer’tsbfyraem.Ae, /article
eˆisaunitvectorinthedirectionoftheelectricfieldintheelectromagneticfieldintheobserver’sframe,bˆ(cid:6)isaunitvectorinthedirectionof -ab
s
themagneticfieldintheelectromagneticwaveinthecomovingframe,andsoon.Weassumethatthedistributionfunctionfortheemitting tra
c
particlesintheframecomovingwithanelementofthejetisisotropicinmomentumandisapowerlawinenergy: t/3
6
dn= Ke(cid:9)−pd(cid:9)dVd(cid:10)p. (1) 0/3
/8
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9
/9
9
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z 4
b
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j n t o
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y 9
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φ
Figure3. Geometryofthemodel.
(cid:4)C 2005RAS,MNRAS360,869–891
874 M.Lyutikov,V.I.ParievandD.C.Gabuzda
Here,dnisthenumberofparticlesintheenergyinterval(cid:9),(cid:9)+d(cid:9),dV istheelementaryvolume,d(cid:10) istheelementarysolidangleinthe
p
directionoftheparticlemomentum p,K =K (r)and p=constant.
e e
Since the emitting particles are ultra-relativistic and we neglect the generation of circular polarization, we do not include circular
polarizationinourmodel(StokesV =0).Wealsoneglectapossibletangledcomponentofthemagneticfieldintheemissionregion.Under
theseassumptions,ourestimatesprovideanupperlimitonthepossiblepolarization.
TheStokesparametersperunitjetlengthforastationaryflowarethengivenby(seealsoLyutikovetal.2003;Parievetal.2003)
(cid:2) 
I = pp++71/κ3(Dν)2(1+κz(ν)(cid:2)2)+(p−d1S)/2 sdinSθ Ke D2+(p−1)/2|B(cid:6)sinχ(cid:6)|(p+1)/2,
Q = K D2+(p−1)/2|B(cid:6)sinχ(cid:6)|(p+1)/2cos2χ˜,
U = DD22((11++κzz(ν))22)++((pp−−11))//22 (cid:2) ssidinnSθθ Kee D2+(p−1)/2|B(cid:6)sinχ(cid:6)|(p+1)/2sin2χ˜,  (2) Dow
V =0. nlo
a
d
Thefunc√ti3onκ((cid:7)ν)3ips−1(cid:8) (cid:7)3p+7(cid:8) e3 (cid:7) 3e (cid:8)(p−1)/2 ed fro
κ(ν)= (cid:2) (cid:2) ν−(p−1)/2, (3) m
4 E 12 E 12 m c2 2πm3c5 h
e e ttp
whereeandm arethechargeandmassofanelectron,cisthespeedoflight,(cid:2) istheEulergamma-function,zisthecosmologicalredshift s
andDisthelume inositydistancetothejet.Theintegrationinequations(2)isoveErdS=rdrdφ,0<φ<2 π,0< r<Rjforunresolvedjets ://aca
(R isthejetradius),while,forresolvedjetsdS=hdhdφ/sin2φ,arcsinh/R<φ<πsgnh−arcsinh/R,wherehistheprojecteddistance d
j e
ofthelineofsightfromthejetaxis(Parievetal.2003).Inequation(2),DistheDopplerboostingfactor mic
D= 1 , (4) .oup
(cid:2)(1−βcosθ) .c
o
χ(cid:6)istheanglebetweenthelineofsightandthemagneticfieldintherestframeofaplasmaelement,andχ˜ istheobservedEVPAintheplane m
/m
oftheskyseenbytheobserver,measuredclockwisefromsomereferencedirection. n
Wenextintroduceaunitvectorlˆnormaltotheplanecontainingnˆ andthereferencedirectionintheplaneofthesky.Wechoosethe ras
/a
directionoftheprojectionofthejetaxistotheplaneoftheskyasareferencedirection,sothatlˆ={0,1,0}.Then, rtic
cosχ˜ =eˆ×(nˆ×lˆ), sinχ˜ =eˆ×lˆ, (5) le-a
b
whereeˆisaunitvectorinthedirectionoftheelectricfieldofthewave. stra
CalculationoftheStokesparametersusingequations(2)isnotasstraightforwardasinthecaseofaplasmaatrest.Severalkeyingredients c
t/3
needtobetakenintoaccountinthecaseofrelativisticallymovingsynchrotronsources(Cocke&Holm1972;Blandford&Ko¨nigl1979; 6
0
Bjornsson1982;Ginzburg1989).First,thesynchrotronemissivitydependsontheanglebetweenthedirectionofanemittedphotonandthe /3
/8
magneticfieldintheplasmarestframe.Secondly,sinceemissionisboostedbythebulkrelativisticmotion,thepositionangleofthelinear 69
polarizationrotatesparalleltothenˆ–vplane.Thefractionalpolarizationemittedbyeachelementremainsthesame,butthedirectionsofthe /99
9
polarizationvectoroftheradiationemittedbydifferentelementsarerotatedbydifferentamounts.Thisleadstotheeffectivedepolarizationof 6
2
4
thetotalemission.Thetheoreticalmaximumfractionalpolarizationforahomogeneousfieldcanbeachievedonlyforauniformplane-parallel b
y
velocityfield.Thirdly,integrationalongthelineofsightandovertheimageofthejetintheplaneoftheskyforunresolvedjetsisbestcarried g
u
outinthelaboratoryframe(theframewherethesourceisatrestasawhole). es
Letnˆ(cid:6) beaunitvectorinthedirectionofthewavevectorintheplasmarestframe,and Bˆ(cid:6) beaunitvectoralongthemagneticfieldin t o
n
theplasmarestframe.Theelectricfieldofalinearlypolarizedelectromagneticwavee(cid:6)isdirectedalongtheunitvectoreˆ(cid:6)=nˆ(cid:6)×Bˆ(cid:6)andthe 04
magneticfieldofthewaveb(cid:6)isdirectedalongtheunitvectorbˆ(cid:6)=nˆ(cid:6)×eˆ(cid:6).MakingtheLorentzboosttotheobserverframe,wefind(Lyutikov Ap
etal.2003) (cid:9) (cid:10)  ril 2
0
nˆ(cid:6) = nˆ+(cid:2)(cid:2)v[1(cid:2)−(cid:2)+1((nˆnˆ××vv))]−1 ,  19
nˆ×q(cid:6)
eˆ = (cid:1) , (6)
q(cid:6) = Bˆ(cid:6)+q(cid:6)2nˆ−×(nˆ(v××qB(cid:6)ˆ)2(cid:6))− (cid:2) (Bˆ(cid:6)×v)v,
1+(cid:2)
wherehereandinallfurtherexpressionswesetc=1.Wecanexpresstherest-frameunitvector Bˆ(cid:6)intermsoftheunitvector Bˆ alongthe
magneticfieldinthelaboratoryframe.AssumingidealMHD,thereisnoelectricfieldintheplasmarestframe,E(cid:6)=0.Wethenobtain
(cid:11) (cid:12) 
Bˆ = (cid:1)1−(1Bˆ(cid:6)×v)2 Bˆ(cid:6)− 1+(cid:2)(cid:2)(Bˆ(cid:6)×v)v ,
(7)
Bˆ(cid:6) = (1+(cid:2)(cid:1))Bˆ +(cid:2)2(Bˆ ×v)v , 
(1+(cid:2)) 1+(cid:2)2(Bˆ ×v)2
(cid:4)C 2005RAS,MNRAS360,869–891
PolarizationofrelativisticAGNjets 875
toget

eˆ = (cid:1) nˆ×q ,
q2−(nˆ×q)2 (8)

q = Bˆ +nˆ×(v×Bˆ).
ThisisageneralexpressiongivingthepolarizationvectorintermsoftheobservedquantitiesBˆ,nˆandv.Quitecompactformofexpression(8)
wasderivedin(Lyutikovetal.2003).
Relativisticaberrationmakestheobservedelectricfieldinthewavenon-orthogonaltotheobservedBfieldandtoitsprojectiononthe
sky.Owingtotheconservationoftherelativisticinvariante(cid:6)×(b(cid:6)+B(cid:6)),wefindthat(seealsoBlandford&Ko¨nigl1979)
e×B=(v×B)×(nˆ×e). (9)
Forexample,theanglebetweeneˆ(whichliesintheplaneofthesky)andBˆ is
(cid:9) (cid:10)
cosζ =eˆ×Bˆ = (Bˆ ×(cid:1)nˆ) Bˆ ×(nˆ×v) , (10) Dow
q2−(nˆ×q)2 nlo
a
whichisingeneralnotequaltozero.TheobservedelectricfieldisorthogonaltotheobservedBfieldifeither Bˆ liesinthenˆ–vplaneor de
d
(Bˆ ×nˆ)=0. fro
m
Next,weapplytheabovegeneralrelationstothecaseofsynchrotronemissionbyathincylindricalshellcarryingtoroidalmagneticfield
h
andmovinguniformly(cid:13)withvelocityβparalleltoitsaxisinthez-direction.Theshellisviewedatanangleθ withrespecttoitsaxis,sothat ttp
s
v=β{0,0,1} ://a
(11) ca
nˆ ={sinθ,0,cosθ}. d
e
m
LetthemagneticfieldB(cid:6)intheemittingshellinthejetframebehelical: (cid:13) ic.ou
BB(cid:6)(cid:6) == BBφ(cid:6)(cid:6)B{ˆ−(cid:6),siBnˆ(cid:6)φ=,csoisnφψ,(cid:6)0{−}+sinBφz(cid:6){,0c,o0s,φ1},0=}+Bc(cid:6)osisnψψ(cid:6)(cid:6){{0−,0si,n1φ},,cosφ,0}+B(cid:6)cosψ(cid:6){0,0,1} (12) p.com/m
n
whereψ(cid:6) isthemagneticfieldpitchangleintheshellrestframe, Bˆ(cid:6) isaunitvectoralongthemagneticfieldintheshellframe,andB(cid:6) is ra
s
tdheepemnadgennicteudoeftohfethmeafigneeldti.cAfitetlhdissapnodinetm,wisseivairteieisn.tTerheissteisdcionntshideeermedisisnioSnecotfioans4in.gleshellanddonotspecify,forthemoment,theradial /article
Forthisparticularchoiceofgeometry,wefind -ab
s
nˆ(cid:6)=D[sinθ,0,(cid:2)(cosθ−β)], (13) tra
1 (cid:7) cosψ(cid:6)(cid:8) ct/36
Bˆ = (cid:1)1−β2cos2ψ(cid:6) −sinψ(cid:6)sinφ,cosφsinψ(cid:6), (cid:2) , (14) 0/3/8
(cid:9) (cid:10) 69
cosχ(cid:6)=(nˆ(cid:6)×Bˆ(cid:6))=D (cid:2)cosψ(cid:6)(cosθ−β)−sinθsinφsinψ(cid:6) , (15) /99
9
whereBˆ istheunitvectoralongthemagneticfieldinthelaboratory(observer)frame.Notethat 62
 4
Bˆ ×nˆ = cosθco(cid:2)s(cid:1)ψ(cid:6)1−−(cid:2)βs2incoθss2iψnφ(cid:6) sinψ(cid:6), by gues
Bˆ ×v= (cid:1) βcosψ(cid:6) , (16) t on
Bˆ ×nˆ×v=−(cid:2)β(cid:1)s1i1n−−θβsβi2n2cψcoos(cid:6)s2c2oψψs(cid:6)φ(cid:6) .  04 April 2
0
1
9
Thepitchangleinthelaboratoryframe,ψ,islargerthanintherestframe:
tanψ =(cid:2)tanψ(cid:6). (17)
Aswehavealreadypointedout,oneofthemaineffectsoftherelativisticLorentztransformationisthattheobservedmagneticfield
andpolarizationvectorsarenot,ingeneral,orthogonal(Blandford&Ko¨nigl1979).Forcylindricaljets,theexpressionfortheangleζ istoo
lengthytobereproducedhere(seealsoAppendixA).Thedependenceofζ onφisplottedinFig.4.InFig.5,weplottheobservedrelative
orientationsoftheelectricfieldinthewaveandthemagneticfieldprojectedontotheplaneofthesky.Notethat,forastationaryjet,the
electricfieldisalwaysperpendiculartothemagneticfield,whilethisisnolongertrueforarelativisticallymovingjet.Infact,theelectric
fieldofthewavemaybecomealmostparalleltothemagneticfieldforso(cid:1)meviewingangles.
Thedegreeoflinearpolarizationoftheobservedradiationis(cid:13)= Q2+U2/I.TheresultantEVPAmeasuredbytheobserver,χ˜ ,
res
isobtainedfrom
Q U
cos2χ˜ = (cid:1) , sin2χ˜ = (cid:1) , 0(cid:2)χ˜ <π. (18)
res Q2+U2 res Q2+U2 res
(cid:4)C 2005RAS,MNRAS360,869–891
876 M.Lyutikov,V.I.ParievandD.C.Gabuzda
D
o
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fro
m
h
ttp
s
://a
c
a
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e
Figure4. Angleζ betweentheobserveddirectionofthemagneticfieldBˆ andthepolarizationvectorinthewaveasafunctionofφforacylindricalshell m
with(cid:2)=10,θ=0.1andsixvaluesofψ(cid:6).Inaccordancewithformula(10), ζ =90◦wheneitherthemagneticfieldispurelyaxial(ψ(cid:6)=0)oratthevisible ic.o
edgesofthejet(φ=90◦andφ=270◦). u
p
.c
o
m
/m
n
ra
s
/a
rtic
le
-a
b
s
tra
c
t/3
6
0
/3
/8
6
9
/9
9
9
6
2
4
b
y
g
u
e
s
t o
n
0
4
A
p
ril 2
0
1
9
Figure5. Examplesofobservedelectricfieldinthewaveandmagneticfieldforθ=π/3.Projectionsofthemagneticfieldontheplaneoftheskyaredenoted
byarrowswithlargehead,electricfieldsaredenotedbyarrowswithsmallheads.(a)Purelypoloidalmagneticfieldψ(cid:6)=0,(b)toroidalmagneticfieldψ(cid:6)=
π/2,(c)helicalfieldwithψ(cid:6)=π/4viewedintherestframe,(d)helicalfieldwithψ(cid:6)=π/4movingwith(cid:2)=2.Forstationaryjets(a–c)theelectricvector
ofthewaveisalwaysorthogonaltothemagneticfieldinthejetintheobserverframe;formovingjets(cased)thisisnottrueanymore.
Itcanbeverifiedthat,foracylindricallysymmetricflowandmagneticfield,thechangeofφtoπ−φintheintegrals(2)doesnotchange
thevalueofQandthesignofUisreversed.Therefore,theStokesparameterUintegratestozero.Consequently,ifQ>0thenχ˜ =0,and
res
ifQ<0thenχ˜ =π/2.Thus,forcircularjets,theobservedEVPAcanbeeitherparallelorperpendiculartotheprojectionofthejetaxis
res
ontotheplaneofthesky.Thisistruefordifferentiallyrotatingjetsaswell(Parievetal.2003).Thisnaturallyexplainstheobservedbimodal
distributionofthejetEVPAs.
(cid:4)C 2005RAS,MNRAS360,869–891
PolarizationofrelativisticAGNjets 877
SinceU =0foracircularjet,wedefinethefractionalpolarizationas(cid:13)=Q/I,retainingthesignofQ,sothat (cid:13)canbesmallerthan
zero.For(cid:13)>0,thepolarizationisalongthejet,whilethepolarizationisorthogonaltothejetfor (cid:13)<0.
GiventhattheanglebetweentheobservedEVPAandobservedmagneticfieldforindividualradiatingplasmaelementsarenot,ingeneral,
orthogonal,dependingontheviewingconditions,wecanaskwhyitisthecasethatreasonablyclearpatternshaveappearedintherelationship
betweentheobservedEVPAsandthelocaljetdirection.Thiscanbeunderstood,inpart,asaconsequenceofthefactthattheresolution
providedbyVLBIobservationsisofteninsufficientfullytoresolvethejetsinthetransversedirection,sothattheEVPAintegratedoverthe
jetcrosssectionisobserved.ThisisessentiallyequivalenttothestatementabovethattheobservedEVPAintegratedoveracylindricaljetis
expectedtobeeitherparallelorperpendiculartotheprojectionofthejetaxisontotheplaneofthesky.TheobservedEVPAwillessentially
bedeterminedbytheratio B(cid:6)/B(cid:6);ifthejetsareapproximatelycylindrical,theobservedEVPAcanbeusedtodrawconclusionsaboutthe
φ z
ratioofthetoroidalandpoloidalmagneticfieldcomponentsintherestframe,butitisnotalwayspossibletotranslatetheseintounambiguous
conclusionsaboutthisratiointheobserver’sframe.(Note,however,thatmostconclusionsabouttheB-fielddirectionthathavebeeninferred
fromthedirectionoftheEVPAhave,infact,implicitlytakenthistomeantheB-fieldinthesourcerestframe.) D
o
w
n
3 HOLLOW CYLINDRICAL JETS loa
d
e
d
3.1 Unresolvedhollowjets(cylindricalshell) fro
m
Wefirstconsidercylindricaljetswiththeiremissivityconfinedtoanarrowcylindricalshell.Inordertointerpretobservationsthatdonot h
resolvethetransversedependenceofthepolarizationacrossthejet,weintegratetheemissivityoftheshellovertheazimuthalangleinthe ttps
observer’sframe.Foraconstantvelocityfieldoftheshell,allrelativisticeffectscanbeaccountedforbytheLorentztransformation:the ://a
c
degreeofpolarizationcanbecalculatedinthejetrestframefortheviewingangleinthejetrestframeθ(cid:6)obtainedfromexpression(13) ad
e
sinθ cosθ−β m
sinθ(cid:6)= , cosθ(cid:6)= , (19) ic
(cid:2)(1−βcosθ) 1−βcosθ .o
u
where0<θ <πand0<θ(cid:6)<π.Inaddition,theLorentztransformationofthejetmagneticfieldis Bz=B(cid:6)z,Bφ =(cid:2) B(cid:6)φ andresultsinthe p.co
changeofthepitchangledescribedbyexpression(17).Thoughcorrespondingnon-relativisticcalculationshavebeendonebyLaing(1981), m
/m
theinterpretationofthesecalculationsforarelativisticallymovingshellisdifferent. n
ThegeneralexpressionsforpolarizationgiveninSection2canbesimplifiedconsiderablyfornon-relativisticjets.Settingβ=0,wefind ras
 /a
sin2χ(cid:6) =cos2ψ(cid:6)sin2θ+ 1sin2θsin2ψ(cid:6)sinφ+(cos2θ+cos2φsin2θ)sin2ψ(cid:6) rtic
2 le
cos2χ˜ = cos2φ1si−n2(ψco(cid:6)s−θ(ccoossψψ(cid:6)(cid:6)−sinsiθn+θscinosφθssininψφ(cid:6))s2inψ(cid:6))2.  (20) -abstra
c
Thedegreeofpolarization(cid:13)fromacylindricalshellpopulatedbyparticleswithvariouspower-lawindices p=1,2.4,3areplottedinFig.6 t/3
6
asafunctionoftheviewingangleandpitchangle(seealsoLaing1981).(Atypicalpower-lawindexforpc-scalejetsis p =2.4,which 0/3
correspondstoaspectralindexofα=−0.7,Sν ∝να;see,forexample,Gabuzda&Go´mez2001;Gabuzda,Go´mez&Aguda2001b.)2As /86
9
canbeseenfromFig.6(positivevaluesindicatepolarizationalongthejet,andnegativevaluespolarizationorthogonaltothejet),thereisa /9
9
weakdependenceofthepolarizationonthespectralindex.Inparticular,flatterspectra(smallerp)producemorecomplicatedpolarization 9
6
2
structure,sothatsourceswithflatterspectraaremorelikelytoproduceachangeofpolarizationfromparalleltoperpendicular. 4
b
Forrelativisticjets,weadoptp=2.4asafiducialnumberandplotthefractionalpolarizationforvariousviewinganglesandpitchangles y
g
intherestframeandthelaboratoryframe(Fig.7).Inaddition,weplotinFig.8thefractionalpolarizationasafunctionofthebulkLorentz ue
factor(cid:2)forθ =10◦. st o
n
Severalconclusionscanbedrawnuponanalysingtheseplots.Considerfirstthepolarizationasafunctionoftheviewingangleforvarious 0
rest-framepitchangles(Fig.7a).Forextremevaluesofthepitchangleintherestframe(purelytoroidalmagneticfield,ψ(cid:6)=π/2,andpurely 4 A
pπo/l3oi(cid:2)daψlm(cid:6)(cid:2)agnπe/t2ic,tfiheeldpoψla(cid:6)r=iza0ti)o,nthaelwcoaryrsesrpemonadininsgaploonlagritzhaetiaoxnisi,spaelaoknigng((cid:13)at>θ0∼)a1n/d(cid:2)oarnthdodgeocnraela(s(cid:13)ing<s0h)artopltyhfeojreθt.(cid:2)Fo(cid:1)rl1ar.gFeopriitnctheramngedleisa,te pril 20
1
pitchangles,ψ(cid:6)(cid:2)π/3,thepolarizationcanchangefromparalleltoorthogonaldependingontheviewingangle,withsuchachangebeing 9
morelikelytooccurforlowervaluesofp.Forsmallerpitchangles,ψ(cid:6) (cid:2)π/4,thepolarizationishigh,weaklydependentontheviewing
angle,andorientedorthogonaltothejet.
Thus,theaveragepolarizationisaverysensitivefunctionofthejetparameters.Forsomejetparameters,thepolarizationpropertiesare
fairlyconstant,while,insomecases,theycanchangeappreciablyduetosmallvariationsinthejetparameters.
Mostimportantly,theobservationofpolarizationalongthejetrequiresthatthetoroidalmagneticfieldinthejetframebeatleastofthe
orderofthepoloidalfield.Forrelativisticjets,thismeansthatsuchjetsarestronglydominatedbythetoroidalfieldintheobserver’sframe,
Bφ/Bz ∼(cid:2) B(cid:6)φ/B(cid:6)z (cid:1)1.Incaseswhenachangeofsignisseen,wecanplacestrongconstraintsontheinternalpitchangle,sincesuch
2Aparticularlysimplecaseis p=3,whentheintegrationsinequations(2)canbedoneexactly(seealsoParievetal.2003,foranotherexactlyintegrable
case).Wefindfortheaveragepolarizationofanunresolvedshell:
(cid:13)=−(3/2)(1+3cos2ψ(cid:6))sin2θ(5−cos2θ−cos2ψ(cid:6)−3cos2θcos2ψ(cid:6))−1 (21)
(cid:4)C 2005RAS,MNRAS360,869–891
878 M.Lyutikov,V.I.ParievandD.C.Gabuzda
0.5 0.5 0.5
0 0 0
-0.5 -0.5 -0.5
0 50 100 150 0 50 100 150 0 50 100 150
D
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(a) (b) (c) nlo
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ttp
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0.5 0.5 0.5 ://a
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0 0 0 ic
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-0.5 -0.5 -0.5 /m
n
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0 20 40 60 80 0 20 40 60 80 0 20 40 60 80 /a
rtic
(d) (e) (f) le
-a
b
s
Figure6. Polarizationfraction(cid:13)=Q/I fornon-relativisticcylindricalshells.Firstrowofplotsare(cid:13)asafunctionoftheobserverangleθ for p=1(a), tra
2.4(b),3(c);secondrowofplotsare(cid:13)asafunctionofthepitchangleψ(cid:6)for p=1(d),2.4(e),3(f).Positivevaluesindicatepolarizationalongthejet; ct/3
negative–polarizationorthogonallytothejet. 6
0
/3
/8
6
9
/9
transitionscanbeseenonlyforalimitedrangeofpitchangles,π/3(cid:2)ψ(cid:6) (cid:2)π/4.A90◦ changeoftheEVPAcanbeinitiatedbyalreadya 9
9
smalldeviationofthejetdirectioncaused,forexample,bytheprecessionofthejetbase(ifθ ∼1/(cid:2),inordertochangetheobservedEVPA 62
4
byanangleoftheorderofunity,therealdirectionmaychangeby(cid:15)θ(cid:9)1/(cid:2)).Similarly,aslightaccelerationofthejetorinjectionofasmall b
y
amountoftheadditionaltoroidalmagneticflux(asinthe‘sweepingmagnetictwist’modelofNakamura,Uchida&Hirose2001andKigure gu
etal.2004).Ontheotherhand,arelativisticjetviewedatalargeangle,θ(cid:1)1/(cid:2),willbeeitherweaklypolarizedalongthejet(forB(cid:6) (cid:1)B(cid:6)) es
orstronglypolarizedorthogonallytothejet(forB(cid:6)φ (cid:2)B(cid:6)z);seeFig.7a. φ z t on 0
4
A
p
3.2 Resolvedhollowjets(cylindricalshells) ril 2
0
Wenextturntoresolvedjets.Ifemissionisconfinedtoanarrowcylindricalshell,Ke∝δ(r −Rj)thenintegrationoverdSgivesamultiplier 19
h/arcsinh/R,sothatpolarizationbecomes
j
p+1
(cid:13)= cos2χ˜. (22)
p+7/3
InFig.9weplotthefractionalpolarizationforshellsmovingwith(cid:2)=10,forvariouspitchanglesψ(cid:6)andviewinganglesθ.
First,notethatthepolarizationclosetotheedgeoftheobservedjetisalwaysorthogonaltothejet.Thiscaneasilybeunderstoodfrom
simplegeometricalconsiderations:atthevisibleedgeofthecylindricalshell,theprojectionofthetoroidalmagneticfieldontotheplaneofthe
skybecomeszero,sothatonlythepoloidalmagneticfieldcontributestotheobservedsynchrotronemission,andthispartofthejetisobserved
tobepolarizedorthogonaltothejetaxis.Forθ (cid:9)1/(cid:2) andθ (cid:1)1/(cid:2),thepolarizationinthecentralpartofthejetisalongthesymmetry
axis.Also,inbothcasesofverylarge(θ (cid:1)1/(cid:2))andverysmall(θ (cid:9)1/(cid:2))viewingangles,theprofileofthepolarizationbecomescloseto
symmetricwithrespecttothejetaxis.Forintermediatevaluesofθ ∼1/(cid:2),thepolarizationinthecentralpartsofthejetislongitudinalifthe
pitchangleψ(cid:6) >45◦ andorthogonalifψ(cid:6) <45◦.Thiscanexplaininanaturalwaythefactthatresolvedjetssometimeshavepolarization
alignedwiththejetinthecentralregionandorthogonaltothejetattheedge.
(cid:4)C 2005RAS,MNRAS360,869–891