Modeling optical and UV polarization of AGNs I. Imprints of individual scattering regions

18 Pages·2013·0.46 MB·English

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Martin Gaskell Publications Research Papers in Physics and Astronomy
1-1-2007
MMooddeelliinngg ooppttiiccaall aanndd UUVV ppoollaarriizzaattiioonn ooff AAGGNNssII.. IImmpprriinnttss ooff
iinnddiivviidduuaall ssccaatttteerriinngg rreeggiioonnss
René W. Goosmann
Astronomical Institute of the Academy of Sciences, Bočni II 1401, 14131 Prague, Czech Republic,
[email protected]
C. Martin Gaskell
University of Nebraska-Lincoln, [email protected]
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Goosmann, René W. and Gaskell, C. Martin, "Modeling optical and UV polarization of AGNsI. Imprints of
individual scattering regions" (2007). Martin Gaskell Publications. 11.
https://digitalcommons.unl.edu/physicsgaskell/11
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A&A465,129–145(2007) Astronomy
DOI:10.1051/0004-6361:20053555 &
(cid:1)c ESO2007 Astrophysics
Modeling optical and UV polarization of AGNs
I. Imprints of individual scattering regions
R.W.Goosmann1,2 andC.M.Gaskell3
1 AstronomicalInstituteoftheAcademyofSciences,BocˇniII1401,14131Prague,CzechRepublic
e-mail:[email protected]
2 ObservatoiredeParis–Meudon,5placeJulesJanssen,92190Meudon,France
3 DepartmentofPhysics&Astronomy,UniversityofNebraska,Lincoln,NE68588-0111,USA
e-mail:[email protected]
Received2June2005/Accepted22December2006
ABSTRACT
Context.SpectropolarimetryofAGNsisapowerfultoolforstudyingthestructureandkinematicsoftheinnerregionsofquasars.
Aims.WewishtoinvestigatetheeffectsofvariousAGNscatteringregiongeometriesonthepolarizedflux.
stokes
Methods.Weintroduceanew,publiclyavailableMonteCarloradiativetransfercode, ,whichmodelspolarizationinduced
byscatteringofffreeelectronsanddustgrains.WemodelavarietyofregionsinAGNs.
Results.Wefindthattheshapeof thefunnel of thedusty torushasasignificant impactonthepolarizationefficiency. A compact
toruswithasteepinnersurfacescattersmorelighttowardtype-2viewinganglesthanalargetorusofthesamehalf-openingangle,
θ . Forθ < 53◦, thescatteredlight ispolarizedperpendicularly tothesymmetry axis, whilstfor θ > 60◦ itispolarized parallel
0 0 0
to the symmetry axis. In between these intervals the orientation of the polarization depends on the viewing angle. The degree of
polarizationrangesbetween0%and20%andiswavelengthindependentforalargerangeofθ .Observedwavelength-independent
0
opticalandnear-UVpolarizationthusdoesnotnecessarilyimplyelectronscattering.Spectropolarimetryatrest-framewavelengths
lessthan2500Åmaydistinguishbetweendustandelectronscatteringbutisnotconclusive inallcases.Forpolardust,scattering
spectraarereddenedfortype-1viewingangles,andmadebluerfortype-2viewingangles.Polarelectron-scatteringconesarevery
efficientpolarizersattype-2viewingangles,whilstthepolarizedfluxofthetorusisweak.
Conclusions.WepredictthatthenetpolarizationofSeyfert-2galaxiesdecreaseswithluminosity,andconclude thatthedegreeof
polarization should be correlated with the relative strength of the thermal IR flux. We find that a flattened, equatorial, electron-
scattering disk, of relatively low optical depth, reproduces type-1 polarization. This is insensitive to the exact geometry, but the
observedpolarizationrequiresalimitedrangeofopticaldepth.
Keywords.galaxies:active–polarization–radiativetransfer–scattering
1. Introduction DeZotti&Gaskell1985).Sincethen,thedusty-torusmodelhas
becomethestandardunifiedmodel(seeAntonucci1993)divid-
One of the foremost problems in AGN research is that the in- ing AGNs into two sub-types: “type-1” AGNs which are seen
nermostregionsofAGNscannotberesolvedintheopticaland close to face-on, and “type-2” AGNs which are seen close to
UVwithcurrenttechnology.However,thelightofAGNsispo- edge-on.Intype-1AGNsthe centralenergysourceanditssur-
larizedoverabroadwavelengthrange,andthisallowsustoput roundings (e.g., the BLR) can be seen, whilst in type-2 AGNs
importantconstraintsonthegeometryoftheemittingandscat- the torus blocks our direct view of these inner regions. While
tering regions. Spectropolarimetric observations giving the de- thisobscurationandtheIRemissionfromthetorusarethemost
tailedwavelengthdependenceofthepolarizedfluxgivefurther obvious effects of the torus, scattering from the dust will add
cluestothenatureofthepolarizingmechanism. polarized flux. The polarization spectrum of an optically thick
Our inferences of the innermost structures of AGNs have dusty torus has been the subject of several modeling projects
so far been obtained indirectly. Rowan-Robinson (1977) sug- (Kartje1995;Wolf&Henning1999;Watanabeetal.2003).
gested that AGNs are surrounded by a dusty torus and in the
same paper he gives a suggestion by Penston that Seyfert 2 WhenDibai&Shakhovskoy(1966)andWalker(1966)dis-
galaxies are seen close to edge-on so that the active nucleus coveredopticalpolarizationofAGNs,itwasinitiallytakentobe
is obscured by the torus. Support for this picture came from evidenceofopticalsynchrotronemission,sincesynchrotronra-
the importantdiscoveryby Keel (1980) that Seyfert 1 galaxies diation has a high intrinsic polarization. However, Angel et al.
(active galaxies showing a broad-line region; BLR) are pref- (1976) found the Balmer lines in NGC 1068 to be polarized
erentially seen face-on. Keel (1980) also investigated absorp- similarlytothecontinuum,thusimplyingthatscatteringwasre-
tion effects inside the host galaxies and emphasized the need sponsible for the polarizationof both the lines and continuum.
ofadditionalnuclearabsorptioninSeyfertgalaxieswithrespect The difference they found in polarization between the narrow-
to normal spirals. Keel’s work led to further confirmation of lineregion(NLR)andBLRplacesthescatteringregionoutside
the importance of orientation effects (Lawrence & Elvis 1982; theBLR,butinsidetheNLR.
Article published by EDP Sciences and available at http://www.aanda.orgor http://dx.doi.org/10.1051/0004-6361:20053555
130 R.W.GoosmannandC.M.Gaskell:ModelingAGNpolarization.I.
When light is scattered, the angle of polarization depends a new general-purpose, publicly-available1, Monte Carlo code
on the direction of the last scattering, so one expects the an- for modeling wavelength-dependentpolarization in a wide va-
gle of polarization to be related to the structure of the AGN. riety of scenarios, and we present some results of our study of
Stockmanetal.(1979)madetheseminaldiscoverythatforlow- AGNpolarization.
stokes
polarization,highopticalluminosity,radio-loudAGNs, theop- Inthispaperweconfineourselvestousing forcalcu-
tical polarization position angles tend to align parallel to the latingthepolarizationimprintsofbasicconstituentsoftheuni-
large-scale radio structure. Although they interpreted this as a fiedscheme.Wecomputethepolarizationspectrumofdustytori
consequence of optical synchrotron emission, they also sug- with various geometries and opening angles, and we consider
gestedthatpolarizationfromanoptically-thin,non-spherically- scattering in polarconesand electrondisks. We investigatethe
symmetricscatteringregionnearthesourceofopticalradiation effectsofgeometricalshapeandopticaldepthofgivenregions.
wasanotherpossibility. None of our models are intended to reproduce observational
polarimetricdataforanyspecificobject.Rather,wewanttoin-
Antonucci(1982)pointedoutthatwhilstmanyradiogalax-
vestigate general constraints on the scattering regions and the
ies showed a similar parallel alignmentof the polarization and
geometryof AGNs. We discuss consequencesfor the observed
radioaxes,therewas,unexpectedly,apopulationshowingaper-
polarization dichotomybetween type-1and type-2 objects. We
pendicularrelationship.Itwassubsequentlyshown(Antonucci
leave aside the question of interactionsbetween differenttypes
1983)thatrelatively-radio-quietSeyfertgalaxiesshowasimilar
of scattering regions for paper II (Goosmann & Gaskell, in
dichotomybetweenthepredominantly,butnotexclusively,par-
preparation)wherewe also conductmoredetailedmodelingof
allel polarizationin face-on type-1Seyferts and the perpendic-
AGNsintheunifiedscheme.
ularpolarizationoftype-2Seyferts(seeAntonucci1993,2002,
Thepresentpaperisorganizedasfollows:inSect.2wesum-
forreviews).Thesediscoveriesmadeasynchrotronoriginofthe
marizemodelingofopticalandUVpolarizationofAGNandthe
polarizationmuchlesslikely.
main results obtained previously. Section 3 describes our code
Polarization perpendicular to the axis of symmetry is eas- stokes. In Sect. 4 we present modeling results for equatorial,
ily produced by scattering off material close to the axis. There toroidaldustdistributions.Section5isdedicatedtoelectronand
is good observational evidence for the existence of ionization dust scattering in polar double-cones.In Sect. 6 we investigate
conesalongthepolaraxisinnumerousobjects(seeKinneyetal. thepolarizationsignatureofequatorialregionsforelectronscat-
1991, andreferencetherein).Polarscattering hasbeen particu- tering. Our results are discussed in Sect. 7 and we give some
larlywellstudiedintheSeyfert-1galaxyNGC1068.Antonucci conclusionsinSect.8.
&Miller(1985)madethekeydiscoverythatthepolarized-flux
spectrum can offer a periscope view of type-2 AGNs because
much of the polarizedflux originatesinside the torus. Detailed 2. Previouscodesandmodeling
HST polarimetry has revealed the polarization structure of the
Inthissection,webrieflysummarizesomerecentAGNpolariza-
ionizationcones(seeCapettietal.1995a,b;Kishimoto1999). stokes
tion modelingcodeswhichwewill compareour mod-
Thedetectionofahiddenbroad-lineregioninNGC1068by elingwith.
Antonucci & Miller (1985) was of great importance for AGN Young et al. (1995, 1996), Packham (1997), and Young
researchsinceitprovidedstrongsupportfortheunifiedtheories (2000) developed an analytical radiative transfer model, the
of AGN activity.More hiddentype-1nucleihavesubsequently Generic Scattering Model (GSM), for polarization modeling.
beenfoundbyanalysisoftheirpolarized-fluxspectra(see e.g., ThemodelisbasedontheunifiedAGNmodel.Extendedemis-
Miller&Goodrich1990;Tranetal.1992;Hines&Wills1993; sionregionscanbedefined,andscatteringprocessesaswellas
Kay 1994; Heisler et al. 1997; Tran 2001; Smith et al. 2004). dichroicabsorptionareconsidered.Themodeledgeometriesin-
Similar work on the radio galaxy 3C 321 was done by Young clude toroidal, disk-like, and conical regions of dust and free
etal.(1996),andTranetal.(1999)couldidentifyanactivenu- electrons.Forscatteringmaterialinmotion,Dopplereffectsare
cleusinsideanultra-luminousinfra-redgalaxyusingspectropo- included.Themodelisfairlyeffectiveinreproducingspectropo-
larimetry. Recently, hidden type-1 nuclei have also been found larimetricdataofSeyfertgalaxies(see,forexample,Youngetal.
infivetype-2quasarcandidates(Zakamskaetal.2005). 1999,forMrk509).Inparticular,itreproducesvariationsofthe
polarizationacrossbroademissionlines(Smithetal.2005).The
The new generation of large telescopes is delivering spec-
model is semi-analytical and therefore does not take multiple
tropolarimetry of emission line profiles with good velocity
scatteringintoaccount.
resolution (see, for example, the spectropolarimetry of the
Wolf&Henning(1999)presentaMonte-Carlocodeusedto
Seyfert 1.5 galaxy NGC 4151 presented by Martel 1998, and
computethe polarizationobtainedby scattering inside axisym-
theatlasofspectropolarimetryofSeyfertgalaxiespresentedby
metric regions. They consider dust and electron scattering for
Smithetal.2002).Examinationofvelocity-dependentpolariza-
polar double-conesand equatorialtori. In the Monte-Carlo ap-
tion of emission lines promises to reveal valuable information
proach two or more of such componentscan be combinedand
about the geometry of the BLR (Smith et al. 2005). Similarly,
the resultingpolarizationspectraaremodeledforvariousincli-
spectropolarimetryofquasarabsorptionlineshelpstoconstrain
nationsofthesystem.Asidefromspectropolarimetricmodeling,
thegeometryofbroadabsorptionlineQSOs(Goodrich&Miller
thecodebyWolf&Henning(1999)canalsoproducepolariza-
1995;Cohenetal.1995;Hines&Wills1995;Ogleetal.1999).
tion images, which are provided for various torus geometries.
In order to understand these many facets of AGN An important element in this code is that multiple scattering,
polarization, and their implications for the underlying ge- which becomes important for optical depths >0.1, is consid-
ometry, theoretical modeling is necessary. Analytical ap- ered accurately by including the dependence of the scattering
proaches to radiative transfer that have been carried out angle and the polarization of a scattered photon on its inci-
so far are generally limited to the consideration of single- dent Stokes vector. For dust scattering, two differentgrain size
scattering models. Computer simulations are needed to inves-
stokes
tigate multiple-scatterings. In this paper we describe , 1 http://www.stokes-program.info/
R.W.GoosmannandC.M.Gaskell:ModelingAGNpolarization.I. 131
distributionswereexamined:oneparameterizationrepresenting
observer
Galacticdust,andtheotherfavoringlargergrains. z
Kartje(1995)alsodevelopedaMonteCarloCodeandinves-
last scattering event
tigated quasar schemeswith either a torusgeometryor conical
stratified windsalongthepolaraxis.Inadditiontopolarization
byscattering,healsoconsiderspolarizationbydichroicextinc-
Θ
tion due to magnetically-aligneddust grains. For a simple uni-
fiedtorusmodelhefindsthatthedominantparameterofthepo- creation of
a new photon
larization, P,isthe torushalf-openingangle:fortype-2objects φ
one can find significant polarization (up to 30%) with a posi-
tion angle directed perpendicular to the axis of symmetry; for y
type-1objectsPisnegligible.Kartjeobtainsanimportantresult
when he investigatesconicalstratified-wind regionscontaining emission region
free electrons closer to the central source and dust farther out:
theamountofpolarizationrangesbetween0%and13%,match- scattering region
ing observedvalues, and the directionof the E-vectordepends scattering events
on the viewing angle in a manner that agrees with the above-
mentioned type-1/type-2 dichotomy. The polarization percent- x
age can be increased if there is magnetic alignment of dust Fig.1.Aphotonworkingitswaythroughthemodelspace.
grains, but the general dependence of P on the viewing angle
seemstobeageometricaleffect.
Another Monte-Carlo polarization code is presented by thesourceregionthroughvariousscatteringprocessesuntilthey
Watanabe et al. (2003). It is applied to modeling of optical become absorbed or manage to escape from the model region
andnear-infraredspectropolarimetricdataofthetype-2Seyfert (Fig. 1). The polarization properties of the model photons are
galaxiesMrk463E,Mrk1210,NGC1068,andNGC4388.The givenbytheirstoredStokesvectors.
codecontainselectronanddustscatteringroutinesquitesimilar Photonsarecreatedinsidethesourceregions,whichcanbe
to those used by Wolf & Henning (1999). It considers multi- realizedbydifferentgeometries.Thecontinuumradiationisnor-
ple scattering and dichroic absorption in dusty tori, spheres as mallysimplydefinedbytheindexαofanFν ∝ ν−α powerlaw.
well as electron and dust scattering in double-conical regions. TheStokesvectorsofthe photonsare initiallysettothe values
The absorption and scattering properties of the dust are care- ofcompletelyunpolarizedlight.
fully calculated by Mie theory. Watanabe et al. (2003) exam- Various scattering regions can be arranged around the
ine wavelength-dependent polarization properties for different sources. The programofferse.g. toroidal,cylindrical,spherical
geometries over a broad-wavelengthrange and give constrains or conical shapes. These regions can be filled with free elec-
aboutpossiblescatteringcomponentswithintheobjectstheyob- tronsordustconsistingof“astronomicalsilicate”andgraphite.
served. They conclude that a combination of dust and electron A photon works its way through the model region and gener-
scattering in polar regions can reproduce the optical polariza- allyundergoesseveralscatterings.Theemissiondirections,path
tionpropertiesofMrk463EandMrk1210.Theslopeofoptical lengthsbetweenscatteringevents,andthescatteringanglesare
polarizationin NGC 1068is almost flat favoringelectron scat- computedby MonteCarlo routinesbasedon classical intensity
teringasthedominantpolarizingprocess.Forthenear-infrared distributions. During each scattering event the Stokes vector is
range polarizationof these objectscan be modeledby dichroic changedby multiplicationwith the correspondingMueller ma-
absorption of aligned dust grains in a torus. However, scatter- trix.Fordustscattering,absorptionisimportant,andalargefrac-
ingoffGalacticdustinatoruscannotsimultaneouslyreproduce tionofthephotonsneverreachesthevirtualobserver.Therele-
thenear-infraredpolarizationandthetotalflux.Watanabeetal. vantcross sections and matrix elementsfor dustscattering and
(2003) hence suggest that the grain size composition of AGNs absorption are computed on the basis of Mie theory applied to
mightbedifferentfromourGalaxy. sizedistributionsofsphericalgraphiteandsilicategrains.
This list of previous polarization modeling is not exhaus- If a photon escapes from the model region, it is registered
tive. For example, Blaes & Agol (1996) and Agol & Blaes by a web of virtual detectors arranged in a spherical geometry
(1996) have presented modeling of the wavelength-dependent aroundthesource.Thefluxandpolarizationinformationofeach
polarization signature of accretion disks at the Lyman limit, detector is obtained by adding up the Stokes parameters of all
and Kishimoto (1996) modeled polarization due to electron- detectedphotons.Ifthemodeliscompletelyaxiallysymmetric
scattering off clumpy media in polar regions of AGN. Also, a these can be azimuthally integrated and, if there is plane sym-
new Monte-Carlo model, which includes polarization transfer metry,thetopandbottomhalvesarecombined.Theobjectcan
for the continuum and for broad quasar absorption lines, was be analyzed in total flux, in polarized flux, percentage of po-
recentlypresentedbyWangetal.(2006).Wehaverestrictedthis larization, and the position angle at each viewing angle. The
briefreviewtorecentmodelingofpolarizationbydustandelec- light travel time of each photon is also recorded, so it is pos-
tron scattering,since this is ourprimaryconcernin the present sible to model time-dependent polarization (Gaskell et al., in
paper. preparation).
3. Stokes–anoverview 3.1.MonteCarlomethod,photoninitialization,andsampling
stokes thefreepathlength
The computer program performssimulations of radia-
tive transfer,includingthe treatmentof polarization,for AGNs Using the MonteCarlo methodit is possible to generatea ran-
and related objects. The code is based on the Monte Carlo dom event x according to a given probability density distribu-
method and follows single photons from their creation inside tion p(x). Let p(x) be defined on the interval[0,x ]. We can
max
132 R.W.GoosmannandC.M.Gaskell:ModelingAGNpolarization.I.
thenconstructtheprobabilitydistributionfunctionP(x)andre- y z
lateittoarand(cid:1)omnumber,r,between0and1asfollows: Ex*
r = P(x)= C1 xp(x(cid:4))dx(cid:4). (1) z φ’ x Eyout Exout
0 Exin
The constant C is a normalization constant resulting from in- 180° − γ Θ Second turn by Θ
tegration overthe whole definitioninterval[0,x ]. Given the i
max y
randomnumber,thecorrespondingvalueof xforasingleevent Ey* Ex*
isobtainedbyinvertingEq.(1).AgooddescriptionoftheMonte φ’ First turn byφbringingEx
Carlo method can be found in Cashwell & Everett (1959). In into the new scattering plane
stokes
thefollowing,wedescribethemainroutinesof andde- x Exin Eyin
noteallrandomnumberscomputedfromequation(1)byr,with
i
i=1,2,3....
To generate a model photon, its initial parameters of po-
sition, direction of flight, and wavelength all have to be set.
Different geometries for the continuum region, broad-line re- Fig.2.Geometryanddenotationsforasinglescatteringevent.Theinset
stokes showsthefirstrotationofthe E-vectorbytheangleφ,theviewonthe
gion,andnarrow-lineregionareavailablein .Assuming
polarizationplaneisalongthenegativez-axis.
a constant density of the emitting material, a random position
for the new photonis sampled. The flightdirectionis givenby
two angles, θ and φ, defined with respect to a standard polar Thescatteringmatrixelements,S1(θ)andS2(θ),areindependent
of the azimuthal angle φ. In case of Thomson scattering, there
coordinate system. Assuming isotropic emission, the sampling
absolutevaluesobeytosimpleanalyticexpressions:
equationsfortheanglesareasfollows:
θ = arccos(1−2r ), (2) |S1(θ)|2 = 1, (7)
1
φ = 2πr . (3) |S2(θ)|2 = cos2θ. (8)
2
Fordustscattering,thealbedoandthematrixelementsofastan-
Thewavelengthofthephotonissampledaccordingtotheinten-
sityspectrumoverarange[λ ,λ ].Thisleadsto: dard dust grain are calculated from Mie theory (see Sect. 3.3).
⎧(cid:6) (cid:7) (cid:8)m(cid:9)in max The albedo at the photonwavelengthis comparedto a random
λ=⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩λλmαminin(cid:7)+λλmarx3(cid:8)r3λ,αmax−λαmin α1 , ffoorrαα=(cid:1)11., (4) soncvuaemtrTtbewhreeeirt,dhpr.5otI,hflaietnrhgieozearnpdtehieororantttooivonednicesotcoafirbdaseoonwfrebwheeaedpcthhhietoriptsothhnlooe.tsoptn,haolntioednsthipseearcbpyseconlredbisectdualroatsrr
min
Here, α denotes the usual power law index of the intensity to itstrajectory,insidetheso-calledpolarizationplane,andde-
notes the preferreddirection of the E-vector. It is defined with
spectrum.
respect to a co-movingcoordinatesystem. The polarizationin-
Ifweignorescatteringsbackintothebeam,theintensityof
formationofeachphotoniscodedbyfourStokesparametersI,
aphotonbeamtraversingaslabofscatteringmaterialwithpar-
ticlenumberdensityN andcross-sectionσwilldropbyafactor Q, U,andV,representingthe4-dimensionalStokesvector.We
of eNσl, with l being the distance traveled inside the scattering assumethatnewlycreatedphotonscomingfromthesourceare
unpolarized.Hence,theirStokesvectorshavethesimpleform:
region.Fromthis,onecanderivethesamplingfunctionofl:
⎛ ⎞ ⎛ ⎞
l= N1σln(1−r4). (5) ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝UQI ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝100⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (9)
The factor 1 is the mean free path length. Depending on the V 0
Nσ
scattering material, the program uses either a dust extinction
Witheachscatteringevent,theco-movingcoordinatesystemun-
cross-sectionσ computedfromMietheoryor,incaseofelec-
ext dergoesadoublerotation:thefirstrotation,bytheazimuthalan-
tronscattering,theThomsoncrosssectionσES. gleφ,occursaroundthecurrentflightdirectionofthephoton.It
rotatesthe E-vectorinsidethepolarizationplanetotheposition
3.2.Polarizationformalism,scattering,andphotondetection ofthenewscatteringplane(seeFig.2).Physically,itdoesnotaf-
fectthepolarizationstate,althoughtheStokesvectorundergoes
stokes
The polarization properties of the photons sampled in thefollowingcoordinatetransformation.
and their transformationduringscattering eventsrely on previ- ⎛ ⎞ ⎛ ⎞⎛ ⎞
obBuaosshiswrefonorr&kthdHeesufcfforrmimbaeandli(es1m.g9.8pi3rne).sFeinsctehderinetthails.s(1ec9t9io4n).cTahnebtehefoourentdicianl ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝UQI∗∗∗⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝100 −cosis0n22φφ csoins022φφ 000⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝UQIiininn⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (10)
If we considera photonbeingscattered off a sphericalpar- V∗ 0 0 0 1 Vin
ticle (see Fig. 2), the outgoing electromagnetic wave associ-
Thesecondrotationoccursinthescatteringplanebythescatter-
ated with the photon can be resolved into two components, E(cid:5) ing angle θ. The change of the Stokes vector is determined by
and E⊥. These components refer to directions of the electric- the Muellermatrix,whichforscatteringoffa sphericalparticle
field vector parallel and perpendicular to the scattering plane.
hastheform:
Forscatteringoffasphericalparticle,thefollowingrelationbe- ⎛ ⎞ ⎛ ⎞⎛ ⎞
t(cid:10)wEEe⊥(cid:5)e,,sns(cid:11)th=e(cid:10)inSco20(mθ)inSg1a0(nθd)s(cid:11)c(cid:10)aEtEte⊥(cid:5),r,iie(cid:11)d.electricfieldsholds: (6) ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝UQVIoooouuuutttt⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝SS001112 SS001222 −SS003334 SS003444⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝UQVI∗∗∗∗⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (11)
R.W.GoosmannandC.M.Gaskell:ModelingAGNpolarization.I. 133
The entries of the Mueller matrix are obtained by simple rela- 1.0
tionsfromtheelementsofthescatteringmatrix,S (θ)andS (θ). 0.9
1 2 0.8
The angle-dependent classical intensity distribution of a 0.7
scatteredelectromagneticwavemeasurestheprobabilityoffind- o 0.6
d
e 0.5
ingascatteredphotonatagivendirection.Suchprobabilityden- b
al 0.4
sity distributionsarederivedfromEq.(6).Animportantaspect 0.3
stokes
included in is that the scattering direction is sampled 0.2
0.1
dependingontheincidentpolarizationvector.ThedegreeP and
i 0.0
the position angle γ of the polarization before scattering are
i 6×10-12
cpoliPmn(gθp)uetq=eudaNtfiroo(cid:1)nmsθot(cid:7)fh|Seθ1ai(nnθc(cid:4)d)i|dφ2e+=nt|SφS(cid:4)2t(+oθk(cid:4)1)e8|s20(cid:8)v◦sei−nct(γoθir(cid:4):)danθ(cid:4)d, enter the s(a1m2-) 2oss-section [cm]2345××××11110000----11112222
1 (cid:10)0 |S (θ)|2−|S (θ)|2 sin2φ(cid:4)(cid:11) cr1×10-12
Pθ(φ(cid:4)) = 2π φ(cid:4)− |S1(θ)|2+|S2(θ)|2Pi 2 · (13) 0 2000 4000 6000 8000 10000
1 2 λ [Angström]
ThenumberN isanormalizationconstantinordertohave(12) Fig.3.Characteristicpropertiesofthedustcompositionadoptedforthis
rangefrom0to1forscatteringanglesbetween0◦ and180◦.To paper as a function of wavelength. Top: albedo value. Bottom: cross-
sample φandθ, the righthand-sidesoftheseequationshaveto sections forextinction(black, solid),scattering(red, dashed), andab-
be set equal to random numbersr , r . The equations are then sorption(blue,dotted).
6 7
solvedfortheangles.
Notethatthesamplingisindependentoftheincidentpolar-
ization for θ butnot for φ. In severalMonte-Carlopolarization
transfer codes described in the literature, the incidentpolariza-
tion does not affect the sampling of the scattering angles. This opticalpropertiesfor light polarizedparallelandperpendicular
does not present a problem if one considers unpolarized inci- to the crystals axis differ from each other. The code therefore
dent radiation and low optical depths. Also for very high op- works with the two differentgraphite types havingabundances
tical depths, when multiple-scattering neutralizes the polariza- in a ratio of 1:2. It computes a weighted average for the dust
tioninsidethescatteringregion,theincidentpolarizationcanbe composition and grain size distribution defined.The procedure
neglected. However, results for intermediate optical depths are isdescribed,forexample,inWolf(2003).Thepropertiesofthe
sensitive to the sampling method and they should consider the resulting“standarddustgrain”arethenusedbyalldust-related
polarizationstateoftheincidentphoton. routinesofstokes.
Whenaphotonescapesfromthemodelregionitisrecorded
We confine ourselves to using standard Galactic dust such
byoneofthevirtualdetectors.Itisthennecessarytorotatethe
as is seen in the solar neighborhood,even though there is evi-
polarizationplanearoundtheflightdirectionuntilitmatchesthe dence that the tori of AGNs might have differentcompositions
referenceaxisofthedetector.TheStokesvectorsofallincoming
andgrainsizedistributions(seeCzernyetal.2004;Gaskelletal.
photonscanfinallybeaddeduptothevaluesIˆ,Qˆ,Uˆ andVˆ.The
2004;andGaskell&Benker2006).FollowingWolf&Henning
netpolarizationpropertiesarederivedfrom:
(1999), we parameterize Galactic dust by a mixture of 62.5%
(cid:18)
carbonaceousdustgrainsand37.5%“astronomicalsilicate”.We
Qˆ2+Uˆ2+Vˆ2
P = , (14) consider grainradii, a, from0.005µm to 0.250µm with a dis-
Iˆ tributionn(a) ∝ as with s = −3.5.Theresultingcross-sections
1 Uˆ andthealbedoareshowninFig.3asafunctionofwavelength.
γ = arctan · (15)
2 Qˆ Thefigureshowsthatforthisparticulardustmodel,the albedo
isratherflatwithavalueof0.55−0.6overthewavelengthrange
<
considered.Forwavelengths∼2500Åitfallsto0.4.Thecross-
3.3.Computationofdustproperties sectionsalldecreaseregularlywithwavelength,withtheexcep-
Mathisetal.(1977,MRN)suggesteddustcompositionstorepro- tionofthewell-knownhumparound2175Å.
duce extinction curves observed in our Galaxy. They assumed
various types of dust grains having a size distribution propor-
tional to as, with a being the grain radius and s an arbitrary
power-law index. Our parameterization of the dust properties 4. Simulationoftorusgeometries
follows thatof MRN and givesa gooddescriptionof observed
Galactic extinction curves. The user can choose the arbitrary Inthissectionweinvestigatehowmuchofthepolarizationprop-
minimum and maximumradii of the grain-size distribution,its ertiesoftype-1andtype-2AGNscanbeproducedbyauniform-
power law index, and the relative abundances of graphite and density torusalone. Kartje (1995) modeled the polarizationin-
“astronomicalsilicate”. ducedbyscatteringoffacylindricallyshapedtorus.Theirtorus
The results from Mie scattering theory, i.e., scattering and model was adopted from a fit to NGC 1068 given by Pier &
extinction cross sections, albedos, and elements of the scatter- Krolik (1992). This torus is geometrically rather compact and
ing matrix, are computed using the code given by Bohren & islocatedwithinaradiusof1pcfromthecentralsource.Such
Huffman(1983). We importedcomplexdielectricfunctionsfor a cylindrical torus is not necessarily physical, so we examine
graphite and silicate measured by Draine & Lee (1984). For whether the results of Kartje can be confirmedwith more gen-
graphite,twodielectricfunctionshavetobeconsideredsincethe eraltori,andweextendtherangeofparameterspaceexplored.
134 R.W.GoosmannandC.M.Gaskell:ModelingAGNpolarization.I.
0.2
P
0.1
0
1
Fig.4.Geometryofthethreetorusmodelsweconsider:(1)thecylin-
drical torus used by Kartje, (2) a compact elliptically-shaped torus, 1100--11
and(3)anextendedellipticaltorus.Alltorihavethesamehalf-opening
angleΘ0 Fcent 10-2
F /
1100--33
4.1.Curvedsurfacesversussharpedges
10-4
The dusty tori examined by Kartje (1995), Wolf & Henning
2000 3000 4000 5000 6000 7000 8000
(1999), Young (2000), and Watanabe et al. (2003) have rather
sharpedges,and,sincewefindthatpolarizationresultscande- λ [Angstrom]
pendstronglyongeometricaldetails,wehaveinvestigatedaless
Fig.5.Modelingacylindricaltoruswithanellipticalcross-sectionand
artificial torusgeometrywith an elliptical cross-section.To ex- θ = 30◦ (see Sect. 4.1). Top: polarization, P. Bottom: the fraction,
0
aminetheinfluenceofsharpedgesofthecylindricaltorusonthe F/F∗, of thecentral flux, F∗,seen at different viewing inclinations, i.
polarization,wedefineatoruswithsimilardimensions,andthe Legend: i = 87◦ (edge-on) (black crosses), i = 76◦ (orange triangles
same optical depth in the V band (τV) ∼ 750 along the radius withpointsdown),i=70◦(intermediate)(maroonstars),i=57◦ (pur-
in the equatorial plane. Thus, practically no photon is able to ple triangleswithpoints up), i = 41◦ (green diamonds), i = 32◦ (red
penetratethroughthetorusandonlyscatteringoffitssurfaceis squares),andi=18◦(face-on)(bluecircles).
relevant.However,ourtorushasanellipticalcross-section(see
Fig.4)insteadoftherectangularcross-sectionusedbyKartje.
Our results compare very well to those obtained by Kartje
4.2.Theeffectoftheshapeoftheinneredgeofthetorus
(1995). In Fig. 5 we show polarization and flux (normalized
to the flux of the centralsource) versuswavelengthatdifferent A real torus is undoubtedly thicker than the geometrically
viewingdirections.Thetorusconsideredhasahalf-openingan- thin cylindrical torus of Kartje. Direct imaging of NGC 4261
gleofθ0 = 30◦. Thepositivevaluesof P denotethatthepolar- (=3C 270) shows that the dusty torusin that AGN extendsout
ization vectoris oriented perpendicularlyto the symmetryaxis to230pc(Ferrareseetal.1996).Asimilardustlaneacrossthe
(type-2polarization).In oursimulationsthe torusis filled with nucleusofM51(=NGC5194)extendsby∼100pc(Fordetal.
standardGalacticdust,parameterizedasdescribedattheendof 1992).Theinnerradiioftoriareobtainedbyinfra-redreverber-
Sect. 3.We samplea totalof108 photonsandrecordspectraat ation mapping of the hot dust and are in the range of tens to
10differentviewinganglesscaledincosi,whereiismeasured hundredsoflight-daysforSeyfertgalaxies(seeGlass2004;and
from the axis of the torus. We show our results as a function Suganumaetal.2006).
ofcosibecauseitgivesequalfluxperbinforanisotropicsource
Theouterregionsoftorihaveconsiderableopticaldepth,so
located at the center of the model space if there is no scatter-
theirpreciseshapeisunimportant,sincenophotonsescapepar-
ing.Ourfigureisquitesimilartothecorrespondingdiagramsin
alleltotheequatorialplaneofthetorus.Theshapeoftheinner
Kartje’spaper(seehisFig.5).
regionfacingthecentralenergysourceismorerelevant.Current
TheonlydifferencebetweenourresultsandthoseofKartje torus models commonly consider inner surfaces that are con-
is that we generally obtain slightly lower polarization degrees vex towards the central source. We thus model optically-thick,
andaslightlydifferentwavelength-dependentslopeforthescat- uniform-density tori with elliptical cross sections, an inner ra-
tered flux. This can be explained by the fact that we calculate diusof0.25pc,andanouterradiusof100pc.Wecomparethese
our cross-sections from Mie theory of a specific dust compo- results to the modelingof a morecompacttoruswith the same
sition whilst Kartje used cross-sections given by Mezger et al. half-openingangle, θ = 30◦, as in Sect 4.1.We determinethe
0
(1982). dustdensitybyfixingτ at∼750.Variabilityobservationsimply
V
Wealsoinvestigatedthepolarizationofacompacttoruswith thatthesizeoftheopticalandUV-continuumsourceinSeyfert
an elliptical cross-section for changing θ . Again we obtained galaxies is less than a few light-days, as is also expected from
0
similar results (not shown) to those for Kartje’s cylindrically- simple black-bodyemissivity arguments. Hence, when consid-
shaped tori. Thus, the differences in polarization between the eringscatteringoffthetorus,wecanneglectthefinitesizeofthe
ellipticalandcylindricaltoriarenegligible.Havingsharpedges continuum source in our model and assume a point-like emis-
in the cylindricalmodel rather than the more realistic rounded sion region. Note that this consideration remains valid for ob-
edgesoftheellipticaltorusdoesnotintroducespuriouseffects. jects with higher luminosities because both the size of the
R.W.GoosmannandC.M.Gaskell:ModelingAGNpolarization.I. 135
0.2
P
0.1
0.0 Fig.7.Comparisonofthecompactandtheextendedtoruswithθ =30◦
0
intheV-band.
1
inclinationshavetoundergoback-scattering;thisismorelikely
1100--11 tohappenatlongerwavelengths.
There are also differences in the polarization signatures of
Fcent 10-2 both tori. Althoughthe overallspectral dependenceof P is the
F / same, the level of P is changed. The strongest changes are at
higher inclinations when the central source is becoming ob-
1100--33 scured by the torus. As with the total flux (see above), for the
largertorus,Pissignificantlylower(comparethecaseofi=70◦
10-4 betweentheupperpanelsoffFigs.5and6).Foralargetorus,our
current models sampling several 109 photons do not constrain
2000 3000 4000 5000 6000 7000 8000
the polarization well at very high inclinations. The number of
λ [Angstroem] photonsscatteredinto thesedirectionsis toosmallto allow for
Fig.6.Modelingalargetoruswithanellipticalcross-sectionandθ = sufficientstatistics.Ontheotherhand,itclearlyfollowsfromour
30◦ (seeSect.4.2).Top:polarization,P.Bottom:thefraction,F/F∗0,of computationsthatthespectralfluxatanglesi > 76◦ isreduced
the central flux, F∗, seen at different viewing inclinations, i. Legend: by a factor of almost ∼2× 107 with respect to the flux of the
i = 70◦ (intermediate) (maroon stars), i = 63◦ (pink triangles with source.Therefore,thepolarizedfluxattheseanglesisverylow.
pointstotheright),i = 57◦ (purpletriangleswithpointsup),i = 49◦ WeshowthedifferencesinV-bandtotalfluxandpolarization
(brown triangles with points to the left), i = 41◦ (green diamonds), betweenalargeandasmalltorusinFig.8.Thetoppanelshows
i=32◦(redsquares),andi=18◦(face-on)(bluecircles) thepolarizationasafunctionoftheviewingangle,andthebot-
tompanelthefractionofthelightreachingtheobserver.Aswas
shown above,the differencesbetween the two torus shapesare
central emission region and the inner radius of the torus scale mostimportantathigherinclinations.Ati ∼ 70◦ the degreeof
withluminosity. polarizationreachesadifferenceof6%,andthefluxdiffersbya
The resulting spectra at different inclinations are shown in factorofalmost100.
Fig.6.Iftheviewingangle,i,islessthanθ (thuscorresponding
0
to a type-1object),we onlyobservea regulartype-1spectrum.
4.3.Theeffectofthetorushalf-openingangle
Wefindthatthereisnosignificantpolarizationinthiscase.Ifwe
look at a type-2 object at a higher inclination angle, only scat- Kartje (1995)hasshown thatthe half-openingangle,θ0, ofthe
tered (and hence polarized)lightis detected. This is analogous torusisanimportantparameterfortheobscurationandreflection
to the results obtained for the compact torus shown in Fig. 5. properties.Whilemodelinglargetori,weexaminehalf-opening
Theoverallshapeofthepolarizationspectrumforbothsizesof angles ranging from 10◦ to 75◦. Variation of θ0 is realized by
thetorusisrathersimilaraswell.Withincreasingviewingangle changing the vertical half-axis of the elliptical torus cross sec-
thelevelofthepolarizationspectrumrises,reachesamaximum, tion.Theothermodelparametersaredefinedasfortheprevious
anddecreasesagaintowardsedge-onlinesofsight.Theshapeof caseofθ0 =30◦inSect.4.2.
the P-spectrumdoesnotchangesignificantlybetweendifferent
type-2inclinations.
4.3.1. Toriwithnarroworwideopenings
Therearedifferencesbetweenourresultsofmodelingalarge
torus (case 3 in Fig. 4) with half-openingangle θ = 30◦, and Forlargetori,θ isadominantparameterforboththedegreeof
0 0
the analogous compact torus (case 2 in Fig. 4) with identical polarizationandthepositionangle,γ.InFig.9weshowthepo-
half-openinganglebutsmallerdimensions.Astrikingdifference larizationofthescatteredradiationasa functionofwavelength
occurs in the angular flux distribution: the large torus scatters and for variousθ . Due to a similar overallshape of the wave-
0
considerably fewer photons towards an observer at intermedi- lengthdependenceofP,weaveragethepolarizationovertype-2
ateviewinganglesbecausetheyhittheouterpartsofit(seethe viewing angles, i, with i > θ . We thereby exclude the high-
0
illustration in Fig. 7). Towards edge-on viewing directions the est inclinations with an insufficient number of photons, where
probability of seeing scattered photons is much lower than for the statistics of P are too poor. For viewingangles with i < θ
0
thesmalltorus.Thespectralslopeofthescatteredradiationalso (correspondingto type-1 objects seen face-on) the polarization
differsbetweenthetwotori.Whilethespectrumisflatinthecase isnegligible.
of a compact torus it rises towards the red for the large torus. Varying the opening angle shows several important things.
This can be explained by the increasing tendency of forward- For θ < 53◦ the absolute value of the polarization decreases
0
scattering at shorter wavelengths. Photons escaping at higher as the opening angle increases (see Fig. 9), as was found by
136 R.W.GoosmannandC.M.Gaskell:ModelingAGNpolarization.I.
0,15
0.2
0,1
P 0,05
0.1
P 0
0.0 -0,05
-0,1
1
-0,15
1100--11
2000 3000 4000 5000 6000 7000 8000
Fcent 10-2 λ [Angstrom]
F / Fig.9.Polarizationaveragedovertype-2viewingangles(seeSect.4.3).
1100--33 A positivevalue ofpolarization denotes an E-vector oriented perpen-
diculartothetorussymmetryaxis;fornegativevaluesthe E-vectoris
alignedwiththeprojectedaxis.Legend:θ = 10◦ (blackdashedline),
10-4 θ = 20◦ (solidredline),θ = 30◦ (green0dot-dashed line),θ = 45◦
0 0 0
(bluedots),θ =50◦(longyellowdashes),θ =60◦(browndoubledots
40 45 50 55 60 65 70 anddashes),0andθ =75◦(pinkdouble-dash0esanddots).
0
Inclination [deg]
Fig.8.Differencesbetweenlargeandsmalltoriwithanellipticalcross- 0.03
sectionintheV-band(seeSect.4.2).Top:polarization,P,andbottom,
thefraction,F/F∗,ofthecentralflux,F∗,asafunctionofviewingin-
clinations,i.Thedashedlinesdenotethethinellipticaltorus(case2),
0.02
thesolidlinetheextendedtorus(case3)
0.01
Kartje (1995) for compact tori. The polarization vector is ori-
eθ0n,teadspiseropbesnedrivceudlairnlytytopet-h2eAaxGisNf.oFroarllθv0ie>wi6n0g◦,doirnelcytiopnarsaille>l PV-band
polarizationvectorscanbeseenatviewinganglesi>θ .Inthis 0
0
rangeofθ theabsolutedegreeofpolarizationincreaseswiththe
0
openingangle.
The reason for the flip of the relative positionangle can be
-0.01
explained by the scattering phase function, and by the geome-
try of the inner parts of the torus (Kartje 1995). For a distant
observerlookingatthetorusalonganoff-axislineofsight,the
-0.02
scattered radiationcomesfromthe inner surface walls. In part, 0 20 40 60 80
these consist of the inner torus wall facing the observer most
Half-opening angle Θ [deg]
directly, but they also consist of the two surfaces on the side. 0
Duetothescatteringgeometry,thephotonsscatteredofftheside Fig.10.Modelinganexpandedtorusatanintermediateopeningangle
wallsarepolarizedalongtheprojectedsymmetryaxis,whilstthe ofθ0=57◦.Thegraphshowsthepolarizationdegreeinthevisualband
photonscomingfromthefarwallareperpendicularlypolarized. versus inclinationangle i.A positivevalue of polarizationdenotes an
E-vectororientedperpendiculartothetorussymmetryaxis;fornegative
The ratio of the solid angle that the far side of the visible in-
valuestheE-vectorisalignedwiththeprojectedaxis.
nersurfacesubtendstothesolidanglethatthevisibleinnerside
wallssubtendchangeswiththehalf-openingangleofthetorus,
andsodoestheoverallpolarizationvector.
thatthepolarizationvectorisparallel,whichmeansthattype-1
polarization can be produced at obscured viewing inclinations
4.3.2. Transitioncase:intermediatetorushalf-opening
(Fig.10).Iftheinclinationincreasesfurtherthepolarizationvec-
angles
torswitchesbacktotype-2polarization.Itisinterestingtonote
Forintermediateopeningangleswith53◦ < θ < 60◦ theorien- thatsuchatoruscanproducesignificantpolarizationdegreesup
0
tation of the polarizationposition angle seen at type-2viewing to2%forbothorientationsoftheE-vector.
anglesdependsontheexactinclination.Weillustratesuchacase Inordertoillustratetheintegraleffectoftheopeningangle
inFig.10,wherewesetθ0 =57◦. onthepolarization,weplotinFig.11thepolarization,Peff,av-
For a line of sight passing close enough to the horizon of eraged over all type-2 viewingpositions, and overwavelength,
thetorus(i.e.,wheniisonlymoderatelylargerthanθ )wefind as a function of the half-opening angle of the torus. The
0
R.W.GoosmannandC.M.Gaskell:ModelingAGNpolarization.I. 137
0.15 regardless of wavelength2. The increase in scattering cross-
sectionwithdecreasingwavelengthonlymeansthattheshorter
wavelengthphotonsweseehavebeenscatteredclosertothesur-
faceofthetorus.
A significant change in albedo with wavelength, however,
will cause a color dependencyin the intensity and polarization
0.1
of the scattered light3. Shortwards of ∼2500 Å the albedo de-
creases,butthisrangeisatthelowerlimitofthespectralrange
Peff considered in our modeling.The effect can be seen in the nor-
malizedfluxspectraofthetorusmodelsshowninFigs.5and6.
FortheGalacticdustcompositionweimplemented,itislessvis-
0.05 ibleinthepolarizationspectra.
Another grain property that needs to be considered is the
degree of asymmetry of the scattering since this is effectively
anangle-dependentalbedochange.Towardshorterwavelengths,
Galactic dust grains are more strongly forward scattering and
the polarization phase function changes (see Draine 2003). As
0
0 20 40 60 80 forThomsonscattering,forward-scatteredlighthasalowerpo-
Half-opening angle Θ [deg] larizationthansideways-scatteredlight.Thepolarizationspectra
0
obtainedforthetorusmodelsdependonthesephasefunctions.
Fig.11. Effectivepolarization, Peff (see Sect.4.3), for type-2 viewing Theyadditionallyexplainwhya slightwavelength-dependence
positionsasafunctionofthehalf-openingangleofanextendedtorus
of the polarization can be found for very narrow or very wide
(case3).
openinganglesofalargetorus(seeFig.9).
5. Polarizationfrompolar-scatteringregions
difference between type-1 and type-2 polarization is ignored
inPeff.TheabsolutevaluesofPareintegrated. Scattering in polar regionsof AGNs has allowed the discovery
The figure shows that the torus polarizes most effectively of hidden Seyfert-1nuclei in type-2 objects by radiation being
when having either a small or a large half-opening angle. In periscopically scattered around the obscuring torus. The cen-
thetransitionregionbetweentype-1andtype-2polarization(i.e. tral parts of the polar double cone have to be at least moder-
for 53◦ < θ < 60◦) the integrated polarization goes through a ately ionized due to the intense radiation from the AGN. The
minimum. medium could be associated with the warm absorber seen in
many AGN (see Komossa 1999, for a review). The Doppler
shift of the X-ray absorption lines indicates that the medium
4.4.Wavelengthinsensitivityofpolarizationdue is outflowingat roughly1000kms−1. With increasingdistance
todustscattering from the center, the outflow velocity and intensity of the radi-
ation decrease. Beyond the sublimation radius, dust could also
Wavelength-independentpolarizationiswidelytakentobeasig-
be present.However,thisdustmustbe opticallythin,astype-1
natureofelectronscattering,butwehaveshowninFigs.6and9
objectsarenotobscured.
that dust scattering can also produce wavelength-independent
scattering. Thus a flat polarization curve is not a unique sig-
nature of electron scattering. As Zubko & Laor (2000) point 5.1.Polarelectronscattering
out,thewavelengthdependenceofpolarizationprovidesaprobe
Thepolarizationinducedbyscatteringinpolar,conicalelectron-
ofthegrainscatteringproperties.Inspectionofthewavelength-
scatteringregionshasbeenthesubjectofseveralpreviousstud-
dependentpolarizationcurvesforthelargetorusgeometriescon-
ies.Brown&McLean(1977)developedaformalismtocompute
sidered above (see Figs. 6 and 9) shows that the polarization
for half-opening angles with 30◦ < θ < 60◦ is wavelength- the polarization expected from scattering inside optically thin,
0 axisymmetricscatteringregions.Thisformalismwasappliedby
independent over the optical and most of the UV. For values
of θ outside this interval the wavelength-dependence of P is Miller&Goodrich(1990)andMilleretal.(1991)tocomputethe
0 polarizationforpolarscatteringcones.Wolf&Henning(1999)
rather low and doesnotexceeda factorof 2. Since the scatter-
andWatanabeetal.(2003)extendedthemodelingtooptically-
ing cross section of interstellar grains increases strongly from
thick material using Monte-Carlo techniques that can account
the optical to the UV, wavelength-independent polarization is
formultiplescatterings.
commonlysupposedtobethefingerprintofelectronscattering. stokes
WeconfirmsuchresultsinFig.12using .Thefigure
However, scattering in opaque dust clouds produces relatively
showsthedegreeofpolarizationandthetotalfluxasafunction
greyscattering(Kishimoto2001).
of the observer’s inclination for an electron scattering double-
Ourapparentlycontradictoryresultofrelativelywavelength coneofuniformdensityandwiththeopticaldepthτ =1.This
independentpolarizationwithdustscatteringarisesbecausewe es
optical depth is measured in the vertical direction between the
are considering scattering off optically-thick material, and be-
innerandtheoutershellofonecone.Inorderto isolatetheef-
cause of the relatively small variation of the albedo over the
fectsofthescatteringconefromthepolarizationinducedbythe
optical and UV spectral regions (see Fig. 3). The approximate
constancyofthealbedoisbecausethescatteringandabsorption 2 Thisisthereasonthatthesunlitsidesofcloudsintheearth’satmo-
cross-sectionsvaryin a similarmannerwith wavelength.Since sphereareextremelywhite.
we assume an optically-thick torus, we see emergent photons 3 This is the cause of colorations in the atmosphere of the giant
that have been scattered at an optical depth τ ∼ 1. This is planets.