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

0.39 MB·English

by  Rene W. Goosmann

Astronomy&Astrophysicsmanuscriptno.3555pol (cid:13)c ESO2008
February5,2008
Modeling optical and UV polarization of AGNs I.
Imprints of individual scattering regions
Rene´ W.Goosmann1,2 andC.MartinGaskell3
1 AstronomicalInstituteoftheAcademyofSciences,BocˇniII1401,14131Prague,CzechRepublic
e-mail:[email protected]
2 ObservatoiredeParis-Meudon,5placeJulesJanssen,92190Meudon,France
7
3 DepartmentofPhysics&Astronomy,UniversityofNebraska,Lincoln,NE68588-0111,USA
0
e-mail:[email protected]
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2 ReceivedJune2005;acceptedDecember2006

n
a ABSTRACT
J
Context.SpectropolarimetryofAGNsisapowerfultoolforstudyingthestructureandkinematicsoftheinnerregionsofquasars.
6
Aims.WewishtoinvestigatetheeffectsofvariousAGNscatteringregiongeometriesonthepolarizedflux.

Methods.Weintroduceanew,publiclyavailableMonteCarloradiativetransfercode,S,whichmodelspolarizationinducedby
3
scatteringofffreeelectronsanddustgrains.WemodelavarietyofregionsinAGNs.
v
Results.Wefindthat theshape of thefunnel of thedustytorushasasignificant impact onthepolarizationefficiency. Acompact
2
toruswithasteepinnersurfacescattersmorelighttowardtype-2viewinganglesthanalargetorusofthesamehalf-openingangle,
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θ .For θ < 53◦,thescattered light ispolarized perpendicularly tothesymmetryaxis, whilst for θ > 60◦ itispolarizedparallel
0 0 0 0
to the symmetry axis. In between these intervals the orientation of the polarization depends on the viewing angle. The degree of
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polarizationrangesbetween0%and20%andiswavelengthindependentforalargerangeofθ .Observedwavelength-independent
0 0
opticalandnear-UVpolarizationthusdoesnotnecessarilyimplyelectronscattering.Spectropolarimetryatrest-framewavelengths
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lessthan2500 Åmaydistinguishbetweendust andelectronscatteringbutisnotconclusiveinallcases.Forpolar dust,scattering
0
spectraarereddenedfortype-1viewingangles,andmadebluerfortype-2viewingangles.Polarelectron-scatteringconesarevery
/
h efficientpolarizersattype-2viewingangles,whilstthepolarizedfluxofthetorusisweak.
p Conclusions.WepredictthatthenetpolarizationofSeyfert-2galaxiesdecreaseswithluminosity,andconcludethatthedegreeof
- polarization should be correlated with the relative strength of the thermal IR flux. We find that a flattened, equatorial, electron-
o
scattering disk, of relatively low optical depth, reproduces type-1 polarization. This is insensitive to the exact geometry, but the
r
observedpolarizationrequiresalimitedrangeofopticaldepth.
t
s
a Keywords.Galaxies:active–Polarization–Radiativetransfer–Scattering–Dust
:
v
i
X
1. Introduction entation effects (Lawrence & Elvis 1982; De Zotti & Gaskell
r 1985). Since then, the dusty-torusmodel has become the stan-
a One of the foremost problems in AGN research is that the in-
dard unified model (see Antonucci 1993) dividing AGNs into
nermostregionsofAGNscannotberesolvedintheopticaland
twosub-types:“type-1”AGNswhichareseenclosetoface-on,
UVwithcurrenttechnology.However,thelightofAGNsispo-
and“type-2”AGNswhichare seen close toedge-on.Intype-1
larizedoverabroadwavelengthrange,andthisallowsustoput
AGNs the central energysource and its surroundings(e.g., the
importantconstraintsonthegeometryoftheemittingandscat-
BLR) can be seen, whilst in type-2 AGNs the torus blocksour
tering regions. Spectropolarimetric observations giving the de-
direct view of these inner regions. While this obscuration and
tailedwavelengthdependenceofthepolarizedfluxgivefurther the IR emission from the torus are the most obviouseffects of
cluestothenatureofthepolarizingmechanism.
the torus, scattering from the dust will add polarized flux. The
OurinferencesoftheinnermoststructuresofAGNshaveso polarizationspectrumofanopticallythickdustytorushasbeen
farbeenobtainedindirectly.Rowan-Robinson(1977)suggested the subject of several modeling projects (Kartje 1995, Wolf &
thatAGNsaresurroundedbyadustytorusandinthesamepaper Henning1999,Watanabeetal.2003).
he gives a suggestion by M. V. Penston that Seyfert2 galaxies
areseenclosetoedge-onso thattheactivenucleusisobscured
WhenDibai&Shakhovskoy(1966)andWalker(1966)dis-
by the torus. Support for this picture came from the important
covered optical polarization of AGNs, it was initially taken to
discoveryby Keel (1980) that Seyfert1 galaxies(active galax-
be evidenceofopticalsynchrotronemission,since synchrotron
ies showing a broad-line region; BLR) are preferentially seen
radiation has a high intrinsic polarization. However, Angel et
face-on.Keel (1980) also investigatedabsorptioneffects inside
al.(1976)foundtheBalmerlinesinNGC1068tobepolarized
thehostgalaxiesandemphasizedtheneedofadditionalnuclear
similarlytothecontinuum,thusimplyingthatscatteringwasre-
absorption in Seyfert galaxies with respect to normal spirals.
sponsible for the polarization of both the lines and continuum.
Keel’sworkledtofurtherconfirmationoftheimportanceofori- The difference they found in polarization between the narrow-
lineregion(NLR)andBLRplacesthescatteringregionoutside
Sendoffprintrequeststo:Rene´W.Goosmann theBLR,butinsidetheNLR
2 Goosmann&Gaskell:ModelingAGNPolarization
When light is scattered, the angle of polarization depends wedescribeS,anewgeneral-purpose,publicly-available,1
onthedirectionofthelastscattering,sooneexpectstheangleof MonteCarlocodeformodelingwavelength-dependentpolariza-
polarizationtoberelatedtothestructureoftheAGN.Stockman, tioninawidevarietyofscenarios,andwepresentsomeresults
Angel,&Miley(1979)madetheseminaldiscoverythatforlow- ofourstudyofAGNpolarization.
polarization,highopticalluminosity,radio-loudAGNs,the op- InthispaperweconfineourselvestousingS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 polar conesand electron disks. We investigatethe
symmetricscatteringregionnearthesourceofopticalradiation effectsofgeometricalshapeandopticaldepthofgivenregions.
wasanotherpossibility. Noneofourmodelsareintendedtoreproduceobservationalpo-
larimetricdataforanyspecificobject.Rather,wewanttoinves-
Antonucci(1982)pointedoutthatwhilstmanyradiogalax-
tigategeneralconstraintsonthescatteringregionsandthegeom-
ies showed a similar parallel alignmentof the polarizationand
etryofAGNs.Wediscussconsequencesfortheobservedpolar-
radioaxes,therewas,unexpectedly,apopulationshowingaper-
izationdichotomybetweentype-1andtype-2objects.Weleave
pendicularrelationship.Itwas subsequentlyshown(Antonucci
asidethequestionofinteractionsbetweendifferenttypesofscat-
1983)thatrelatively-radio-quietSeyfertgalaxiesshowasimilar
teringregionsforpaperII(Goosmann&Gaskell,inpreparation)
dichotomybetweenthepredominantly,butnotexclusively,par-
wherewealsoconductmoredetailedmodelingofAGNsinthe
allel polarizationin face-ontype-1 Seyfertsand the perpendic-
unifiedscheme.
ular polarizationof type-2 Seyferts(see Antonucci1993, 2002
The present paper is organized as follows: in section 2 we
forreviews).Thesediscoveriesmadeasynchrotronoriginofthe
summarize modeling of optical and UV polarization of AGN
polarizationmuchlesslikely.
and the main results obtained previously. Section 3 describes
Polarization perpendicular to the axis of symmetry is eas- our code S. In section 4 we present modeling results for
ily producedby scattering off material close to the axis. There equatorial,toroidaldust distributions.Section 5 is dedicated to
is good observational evidence for the existence of ionization electronanddustscattering in polardouble-cones.In section 6
cones along the polar axis in numerousobjects (see Kinney et we investigate the polarization signature of equatorial regions
al.1991andreferencetherein).Polarscatteringhasbeenpartic- forelectronscattering.Ourresultsarediscussedinsection7and
ularlywellstudiedintheSeyfert-1galaxyNGC1068.Antonucci wegivesomeconclusionsinsection8.
&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).
tionmodelingcodeswhichwewillcompareourSmodel-
Thedetectionofahiddenbroad-lineregioninNGC1068by ingwith.
Antonucci & Miller (1985) was of great importance for AGN Young et al. (1995, 1996), Packham (1997), and Young
research since it provided strong support for the unified theo- (2000) developed an analytical radiative transfer model, the
ries of AGN activity. More hidden type-1 nuclei have subse- Generic Scattering Model (GSM), for polarization modeling.
quently been found by analysis of their polarized-flux spectra ThemodelisbasedontheunifiedAGNmodel.Extendedemis-
(seee.g.,Miller&Goodrich,1990;Tran,Miller,&Kay,1992; sionregionscanbedefined,andscatteringprocessesaswellas
Hines & Wills, 1993; Kay, 1994; Heisler, Lumsden, & Bailey, dichroicabsorptionareconsidered.Themodeledgeometriesin-
1997; Tran, 2001; Smith et al., 2004). Similar work on the ra- clude toroidal, disk-like, and conical regions of dust and free
dio galaxy 3C 321 was done by Young et al. (1996), and Tran electrons.Forscatteringmaterialinmotion,Dopplereffectsare
et al. (1999) could identify an active nucleus inside an ultra- included.Themodelisfairlyeffectiveinreproducingspectropo-
luminous infra-red galaxy using spectropolarimetry. Recently, larimetric data of Seyfert galaxies (see, for example, Young et
hiddentype-1nucleihavealsobeenfoundinfivetype-2quasar al. 1999forMrk509).Inparticular,it reproducesvariationsof
candidates(Zakamskaetal.2005). thepolarizationacrossbroademissionlines(Smithetal.2005).
Themodelissemi-analyticalandthereforedoesnottakemulti-
The new generation of large telescopes is delivering spec-
plescatteringintoaccount.
tropolarimetryofemissionlineprofileswithgoodvelocityreso-
Wolf&Henning(1999)presentaMonte-Carlocodeusedto
lution (see, for example, the spectropolarimetry of the Seyfert
computethe polarizationobtainedby scattering inside axisym-
1.5 galaxy NGC 4151 presented by Martel 1998, and the at-
metric regions. They consider dust and electron scattering for
lasofspectropolarimetryofSeyfertgalaxiespresentedbySmith
polar double-conesand equatorialtori. In the Monte-Carlo ap-
etal.2002). Examinationofvelocity-dependentpolarizationof
proach two or more of such componentscan be combined and
emissionlinespromisestorevealvaluableinformationaboutthe
theresultingpolarizationspectraaremodeledforvariousincli-
geometryoftheBLR(Smithetal. 2005).Similarly,spectropo-
nations of the system. Aside from spectropolarimetric model-
larimetryofquasarabsorptionlineshelpsconstrainthegeometry
ing, the code by Wolf & Henning (1999) can also producepo-
ofbroadabsorptionlineQSOs(Goodrich&Miller1995,Cohen
larizationimages,whichare providedforvarioustorusgeome-
etal.1995,Hines&Wills1995,Ogleetal.1999).
tries.Animportantelementinthiscodeisthatmultiplescatter-
In order to understandthese manyfacets of AGN polariza- ing,which becomesimportantforopticaldepths> 0.1,is con-
tion, and their implications for the underlying geometry, theo- sidered accurately by including the dependence of the scatter-
reticalmodelingisnecessary.Analyticalapproachestoradiative ingangleandthepolarizationof ascatteredphotononitsinci-
transferthathavebeencarriedoutsofararegenerallylimitedto dentStokes vector. For dustscattering, two differentgrain size
theconsiderationofsingle-scatteringmodels.Computersimula-
tionsareneededtoinvestigatemultiple-scatterings.Inthispaper 1 http://www.stokes-program.info/
Goosmann&Gaskell:ModelingAGNPolarization 3
distributionswereexamined:oneparameterizationrepresenting
Galacticdust,andtheotherfavoringlargergrains.
Kartje(1995)alsodevelopedaMonteCarloCodeandinves- observer
z
tigated quasar schemes with either a torusgeometryor conical
stratifiedwindsalongthepolaraxis.Inadditiontopolarization last scattering event
byscattering,healsoconsiderspolarizationbydichroicextinc-
tion due to magnetically-aligneddust grains. For a simple uni-
fiedtorusmodelhefindsthatthedominantparameterofthepo- Θ
larization, P,isthetorushalf-openingangle:fortype-2objects
creation of
one can find significant polarization (up to 30%) with a posi- a new photon
φ
tion angle directed perpendicular to the axis of symmetry; for
type-1objectsPisnegligible.Kartjeobtainsanimportantresult y
when he investigatesconical stratified-wind regionscontaining
free electrons closer to the central source and dust farther out:
emission region
theamountofpolarizationrangesbetween0%and13%,match-
ing observedvalues, and the direction of the E-vectordepends
scattering region
on the viewing angle in a manner that agrees with the above-
scattering events
mentioned type-1/type-2 dichotomy. The polarization percent-
age can be increased if there is magnetic alignment of dust x
grains, but the general dependence of P on the viewing angle Fig.1.Aphotonworkingitswaythroughthemodelspace.
seemstobeageometricaleffect.
Another Monte-Carlo polarization code is presented by
Watanabe et al. (2003). It is applied to modeling of optical region through various scattering processes until they become
andnear-infraredspectropolarimetricdataofthetype-2Seyfert absorbed or manage to escape from the model region (Fig. 1).
galaxiesMrk463E,Mrk1210,NGC1068,andNGC4388.The The polarization properties of the model photons are given by
codecontainselectronanddustscatteringroutinesquitesimilar theirstoredStokesvectors.
to thoseusedbyWolf & Henning(1999).Itconsidersmultiple Photonsarecreatedinsidethesourceregions,whichcanbe
scatteringanddichroicabsorptionindustytori,spheresaswell realizedbydifferentgeometries.Thecontinuumradiationisnor-
aselectronanddustscatteringindouble-conicalregions.Theab- mallysimplydefinedbytheindexαofanF ∝ ν−α powerlaw.
ν
sorptionandscatteringpropertiesofthedustarecarefullycalcu- TheStokesvectorsofthephotonsareinitiallyset tothevalues
latedbyMietheory.Watanabeetal.(2003)examinewavelength- ofcompletelyunpolarizedlight.
dependentpolarizationpropertiesfordifferentgeometriesovera Various scattering regions can be arranged around the
broad-wavelengthrangeandgiveconstrainsaboutpossiblescat- sources. The programofferse.g. toroidal,cylindrical,spherical
teringcomponentswithin theobjectstheyobserved.Theycon- or conical shapes. These regions can be filled with free elec-
clude that a combination of dust and electron scattering in po- tronsordustconsistingof“astronomicalsilicate” andgraphite.
lar regions can reproducethe optical polarization propertiesof A photon works its way through the model region and gener-
Mrk463EandMrk1210.TheslopeofopticalpolarizationNGC allyundergoesseveralscatterings.Theemissiondirections,path
1068isalmostflatfavoringelectronscatteringasthe dominant lengthsbetweenscatteringevents,andthescatteringanglesare
polarizing process. For the near-infrared range polarization of computedby MonteCarlo routinesbased onclassical intensity
theseobjectscanbemodeledbydichroicabsorptionofaligned distributions. During each scattering event the Stokes vector is
dust grains in a torus. However, scattering off Galactic dust in changedby multiplicationwith the correspondingMueller ma-
atoruscannotsimultaneouslyreproducethenear-infraredpolar- trix.Fordustscattering,absorptionisimportant,andalargefrac-
izationandthetotalflux.Watanabeetal. (2003) hencesuggest tionofthephotonsneverreachesthevirtualobserver.Therele-
thatthegrainsizecompositionofAGNsmightbedifferentfrom vantcross sectionsand matrix elementsfor dust scattering and
ourGalaxy. absorptionare computedon the basis of Mie theory applied to
This list of previous polarization modeling is not exhaus- sizedistributionsofsphericalgraphiteandsilicategrains.
tive. For example, Blaes & Agol 1996 and Agol & Blaes If a photon escapes from the model region, it is registered
1996 have presented modeling of the wavelength-dependent by a web of virtual detectors arranged in a spherical geometry
polarization signature of accretion disks at the Lyman limit, aroundthesource.Thefluxandpolarizationinformationofeach
and Kishimoto (1996) modeled polarization due to electron- detector is obtained by adding up the Stokes parameters of all
scattering off clumpy media in polar regions of AGN. Also, a detectedphotons.Ifthemodeliscompletelyaxiallysymmetric
new Monte-Carlo model, which includes polarization transfer these can be azimuthally integrated and, if there is plane sym-
for the continuum and for broad quasar absorption lines, was metry,thetopandbottomhalvesarecombined.Theobjectcan
recently presented by Wang, Wang, & Wang (2006). We have be analyzed in total flux, in polarized flux, percentage of po-
restrictedthisbriefreviewtorecentmodelingofpolarizationby larization, and the position angle at each viewing angle. The
dustandelectronscattering,sincethisisourprimaryconcernin light travel time of each photon is also recorded, so it is pos-
thepresentpaper. sible to model time-dependent polarization (Gaskell, Shoji, &
Goosmann,inpreparation).
3. Stokes–anoverview
3.1.MonteCarlomethod,photoninitialization,andsampling
ThecomputerprogramSperformssimulationsofradiative
thefreepathlength
transfer,including the treatmentof polarization,for AGNs and
related objects. The code is based on the Monte Carlo method Using the Monte Carlo methodit is possible to generatea ran-
andfollowssinglephotonsfromtheircreationinsidethesource domeventxaccordingtoagivenprobabilitydensitydistribution
4 Goosmann&Gaskell:ModelingAGNPolarization
p(x).Let p(x)be definedon the interval[0,x ]. We canthen scatteringoffasphericalparticle,thefollowingrelationbetween
max
constructtheprobabilitydistributionfunctionP(x)andrelateit theincomingandscatteredelectricfieldsholds:
toarandomnumber,r,between0and1asfollows:
E S (θ) 0 E
1 x k,s = 2 k,i . (6)
r= P(x)= p(x′)dx′. (1) E⊥,s ! 0 S1(θ)! E⊥,i!
C Z
0
The scattering matrix elements, S (θ) and S (θ), are inde-
The constant C is a normalization constant resulting from 1 2
pendentoftheazimuthalangleφ.IncaseofThomsonscattering,
integrationoverthewholedefinitioninterval[0,x ].Giventhe
max thereabsolutevaluesobeytosimpleanalyticexpressions:
randomnumber,thecorrespondingvalueof xforasingleevent
isobtainedbyinvertingequation(1).Agooddescriptionofthe
MonteCarlomethodcanbefoundinCashwell&Everett(1959). |S (θ)|2 = 1, (7)
In the following, we describe the main routines of S and 1
|S (θ)|2 = cos2θ. (8)
denote all random numbers computed from equation (1) by r, 2
i
withi=1,2,3....
For dust scattering, the albedo and the matrix elements of
To generate a model photon, its initial parameters of po-
a standard dust grain are calculated from Mie theory (see sec-
sition, direction of flight, and wavelength all have to be set.
tion 3.3). The albedo at the photon wavelength is compared to
Different geometries for the continuum region, broad-line re-
gion,andnarrow-lineregionareavailableinS.Assuming a random number,r5, in order to decide whether the photon is
absorbedorscattered.Ifthephotonisabsorbeditislost,andthe
a constant density of the emitting material, a random position
cyclestartsoverwiththegenerationofanewphoton.
for the new photonis sampled.The flight directionis givenby
Thepolarizationvectorofeachphotonliesperpendicularto
two angles, θ and φ, defined with respect to a standard polar
itstrajectory,insidetheso-calledpolarizationplane,anddenotes
coordinate system. Assuming isotropic emission, the sampling
thepreferreddirectionoftheE-vector.Itisdefinedwithrespect
equationsfortheanglesareasfollows:
toaco-movingcoordinatesystem.Thepolarizationinformation
ofeachphotoniscodedbyfourStokesparametersI,Q,U,and
θ = arccos(1−2r ), (2) V,representingthe4-dimensionalStokesvector.Weassumethat
1
newlycreatedphotonscomingfromthesourceareunpolarized.
φ = 2πr . (3)
2 Hence,theirStokesvectorshavethesimpleform:
The wavelength of the photon is sampled according to the
I 1
intensityspectrumoverarange[λ ,λ ].Thisleadsto:
min max
Q 0
 = . (9)
λ=Heλhrλemαm,iniαn(cid:16)+λλdmmearinxn3(cid:17)o(cid:16)r3tλe,αmsaxth−eλuαmsuina(cid:17)liα1po,wffooerrrααlaw=,i11n.,dex of the intens(i4ty) uannUVgdlWeergiφtoh,eeos00acacchudrosscuabatrlteoeurrinondtgatteihvoeenn:cttu,hrterheefincrtsoflt-imrgohotavttiidnoignre,ccobtoiyorntdhienoafatztehimseyusptthehoma-l
spectrum. ton. It rotates the E-vector inside the polarization plane to the
Ifweignorescatteringsbackintothebeam,theintensityof position of the new scattering plane (see Fig. 2). Physically, it
aphotonbeamtraversingaslabofscatteringmaterialwithpar- doesnotaffectthepolarizationstate,althoughtheStokesvector
ticlenumberdensityNandcross-sectionσwilldropbyafactor undergoesthefollowingcoordinatetransformation.
of eNσl, with l being the distance traveled inside the scattering
region.Fromthis,onecanderivethesamplingfunctionofl: I∗ 1 0 0 0 Iin
Q∗ 0 cos2φ sin2φ 0 Qin
l= TN1hσelfna(c1to−rr41).is the mean free path length. Depending(o5n) UV∗∗=00 −si0n2φ cos02φ 01UViinn. (10)
Nσ The second rotation occurs in the scattering plane by the
thescatteringmaterial,theprogramuseseitheradustextinction
scatteringangleθ.ThechangeoftheStokesvectorisdetermined
cross-sectionσ computedfromMietheoryor,incaseofelec-
ext bytheMuellermatrix,whichforscatteringoffasphericalparti-
tronscattering,theThomsoncrosssectionσ .
ES clehastheform:
Iout S S 0 0 I∗
3.2.Polarizationformalism,scattering,andphotondetection 11 12
Qout 1 S S 0 0 Q∗
Taonuhdsewtphooelriarkrtidrzaeanstsicofronibrmepdraotepi.oegnr.tidienusrFionifsgcthhsceear,tptHheroeitnnognnisenvgse,anm&tspYlreeodlrykineon(S1p9r9ev4i)-. UVoouutt= k2d2  0012 0022 −SS3334 SS3444UV∗∗. (11)
Thetheoreticalbasisfortheformalismpresentedinthissection The entries of the Mueller matrix are obtained by simple
canbefoundinBohren&Huffman(1983). relations from the elements of the scattering matrix, S (θ) and
1
If we considera photonbeingscattered off a sphericalpar- S (θ).
2
ticle(seeFig.2),theoutgoingelectromagneticwaveassociated The angle-dependent classical intensity distribution of a
with the photon can be resolved into two components, E and scatteredelectromagneticwavemeasurestheprobabilityoffind-
k
E . These components refer to directions of the electric-field ingascatteredphotonatagivendirection.Suchprobabilityden-
⊥
vector parallel and perpendicular to the scattering plane. For sity distributions are derived from equation (6). An important
Goosmann&Gaskell:ModelingAGNPolarization 5
3.3.Computationofdustproperties
y z Mathis,Rumpl,&Nordsieck(1977,MRN)suggesteddustcom-
Ex* positionstoreproduceextinctioncurvesobservedinourGalaxy.
z φ’ x Eyout Exout Tbuhteioynaspsruompoerdtivoanrailoutostaysp,ewsiothf dausbteginrgainthsehagvrainingraadsiizuesdainsdtris-
Exin an arbitrarypower-law index.Our parameterizationof the dust
180° − γ Θ Second turn by Θ propertiesfollowsthatofMRNandgivesagooddescriptionof
i observedGalacticextinctioncurves.Theusercanchoosethear-
y
Ey* Ex* bitraryminimumandmaximumradiiofthegrain-sizedistribu-
φ’ First turn byφbringingEx tion,itspowerlawindex,andtherelativeabundancesofgraphite
into the new scattering plane and“astronomicalsilicate”.
x Exin Eyin extinTchteiornescurlotsssfrsoecmtioMnsie, aslcbaetdteorsin,gantdheeolreym,ei.net.s, socfatthteerisncgatatenrd-
ing matrix, are computed using the code given by Bohren &
Huffman(1983). We importedcomplexdielectric functionsfor
graphite and silicate measured by Draine & Lee (1984). For
Fig.2. Geometry and denotations for a single scattering event. graphite,twodielectricfunctionshavetobeconsideredsincethe
TheinsetshowsthefirstrotationoftheE-vectorbytheangleφ, opticalpropertiesfor lightpolarizedparalleland perpendicular
theviewonthepolarizationplaneisalongthenegativez-axis. to the crystals axis differ from each other. The code therefore
workswith the two differentgraphitetypes having abundances
in a ratio of 1:2. It computes a weighted average for the dust
aspectincludedinSisthatthescatteringdirectionissam- compositionand grain size distribution defined. The procedure
pled dependingon the incident polarizationvector. The degree isdescribed,forexample,inWolf(2003).Thepropertiesofthe
P and the position angle γ of the polarization before scatter- resulting“standarddustgrain”arethenusedbyalldust-related
inigarecomputedfromtheiincidentStokesvectorandenterthe routinesofS.
samplingequationsofθandφ=φ′+180◦−γ: We confine ourselves to using standard Galactic dust such
i
as is seen in the solar neighborhood,even though there is evi-
dence that the tori of AGNs mighthave differentcompositions
θ andgrainsizedistributions(seeCzernyetal.2004,Gaskelletal.
P(θ) = N |S1(θ′)|2+|S2(θ′)|2 sin(θ′)dθ′, (12) 2004,andGaskell&Benker2006).FollowingWolf&Henning
Z
0 (cid:16) (cid:17) 1999,weparameterizeGalacticdustbyamixtureof62.5%car-
P (φ′) = 1 φ′− |S1(θ)|2−|S2(θ)|2P sin2φ′ . (13) bonaceous dust grains and 37.5% “astronomical silicate”. We
θ 2π |S (θ)|2+|S (θ)|2 i 2 ! considergrainradii, a, from 0.005µm to 0.250µm with a dis-
1 2
tributionn(a) ∝ as with s = −3.5.The resultingcross-sections
The number N is a normalizationconstant in orderto have andthealbedoareshowninFig.3asafunctionofwavelength.
(12)rangefrom0to1forscatteringanglesbetween0◦and180◦. Thefigureshowsthatforthisparticulardustmodel,thealbedo
Tosampleφandθ,therighthand-sidesoftheseequationshave isratherflatwithavalueof0.55–0.6overthewavelengthrange
tobesetequaltorandomnumbersr ,r .Theequationsarethen considered.Forwavelengths.2500Åitfallsto0.4.Thecross-
6 7
solvedfortheangles. sectionsalldecreaseregularlywithwavelength,withtheexcep-
Notethatthesamplingisindependentoftheincidentpolar- tionofthewell-knownhumparound2175Å.
ization for θ but notfor φ. In severalMonte-Carlopolarization
transfer codes describedin the literature, the incidentpolariza-
4. Simulationoftorusgeometries
tion does not affect the sampling of the scattering angles. This
does not present a problem if one considers unpolarized inci- Inthissectionweinvestigatehowmuchofthepolarizationprop-
dent radiation and low optical depths. Also for very high op- ertiesoftype-1andtype-2AGNscanbeproducedbyauniform-
tical depths, when multiple-scattering neutralizes the polariza- density torus alone. Kartje (1995) modeledthe polarizationin-
tioninsidethescatteringregion,theincidentpolarizationcanbe ducedbyscatteringoffacylindricallyshapedtorus.Theirtorus
neglected. However, results for intermediate optical depths are model was adopted from a fit to NGC 1068 given by Pier &
sensitive to the sampling method and they should consider the Krolik (1992). This torus is geometrically rather compact and
polarizationstateoftheincidentphoton. islocatedwithina radiusof1pcfromthecentralsource.Such
Whenaphotonescapesfromthemodelregionitisrecorded a cylindrical torus is not necessarily physical, so we examine
byoneofthevirtualdetectors.Itisthennecessarytorotatethe whether the results of Kartje can be confirmed with more gen-
polarizationplanearoundtheflightdirectionuntilitmatchesthe eraltori,andweextendtherangeofparameterspaceexplored.
referenceaxisofthedetector.TheStokesvectorsofallincoming
photonscanfinallybeaddeduptothevaluesIˆ,Qˆ,Uˆ andVˆ.The
4.1.Curvedsurfacesversussharpedges
netpolarizationpropertiesarederivedfrom:
The dusty tori examined by Kartje (1995), Wolf & Henning
(1999), Young (2000), and Watanabe et al. (2003) have rather
Qˆ2+Uˆ2+Vˆ2 sharpedges,and,sincewefindthatpolarizationresultscande-
P = , (14)
p Iˆ pendstronglyongeometricaldetails,wehaveinvestigatedaless
artificialtorusgeometrywith an ellipticalcross-section.To ex-
1 Uˆ
γ = arctan . (15) aminetheinfluenceofsharpedgesofthecylindricaltorusonthe
2 Qˆ polarization,wedefineatoruswithsimilardimensions,andthe
6 Goosmann&Gaskell:ModelingAGNPolarization
1.0
0.9
0.8
0.2
0.7
o 0.6
d
e 0.5
b
al 0.4
0.3 P
0.2 0.1
0.1
0.0
6×10-12
2cm]5×10-12 0
on [4×10-12
secti3×10-12
oss-2×10-12 1
cr
1×10-12
0 1100--11
2000 4000 6000 8000 10000
λ [Angström]
fFoirg.t3h.isCphaapraecrtearsisaticfupnrcotpioenrtioefsowfatvheelednugstthc.oTmoppo:sailtbioendoadvoapltueed. F / Fcent 10-2
Bottom: cross-sections for extinction (black, solid), scattering 1100--33
(red,dashed),andabsorption(blue,dotted).
10-4
2000 3000 4000 5000 6000 7000 8000
λ [Angstrom]
Fig.5. Modeling a cylindrical torus with an elliptical cross-
section and θ = 30◦ (see section 4.1). Top: polarization, P.
0
Bottom: the fraction, F/F , of the centralflux, F , seen at dif-
∗ ∗
ferentviewinginclinations,i.Legend:i = 87◦ (edge-on)(black
crosses), i = 76◦ (orangetriangles with points down), i = 70◦
(intermediate) (maroon stars), i = 57◦ (purple triangles with
points up), i = 41◦ (green diamonds), i = 32◦ (red squares),
andi=18◦(face-on)(bluecircles).
cosi because it givesequal flux per bin for an isotropic source
located at the center of the model space if there is no scatter-
ing.Ourfigureisquitesimilartothecorrespondingdiagramsin
Fig.4. Geometry of the three torus models we consider: (1) Kartje‘spaper(seehisFig.5).
the cylindricaltorus used by Kartje, (2) a compact elliptically- TheonlydifferencebetweenourresultsandthoseofKartje
shapedtorus, and (3) an extendedellipticaltorus. All tori have is that we generally obtain slightly lower polarization degrees
thesamehalf-openingangleΘ0 andaslightlydifferentwavelength-dependentslopeforthescat-
tered flux. This can be explained by the fact that we calculate
ourcross-sectionsfromMie theoryof a specific dustcomposi-
same optical depth in the V band (τ ) ∼ 750 along the radius tionwhilstKartjeusedcross-sectionsgivenbyMezger,Mathis,
V
in the equatorial plane. Thus, practically no photon is able to &Panagia(1982).
penetratethroughthetorusandonlyscatteringoffitssurfaceis Wealsoinvestigatedthepolarizationofacompacttoruswith
relevant.However,ourtorushasan ellipticalcross-section(see an elliptical cross-section for changing θ0. Again we obtained
Fig.4)insteadoftherectangularcross-sectionusedbyKartje. similar results (not shown) to those for Kartje‘s cylindrically-
Our results compare very well to those obtained by Kartje shaped tori. Thus, the differences in polarization between the
(1995). In Fig. 5 we show polarization and flux (normalized ellipticalandcylindricaltoriarenegligible.Havingsharpedges
to the flux of the centralsource)versuswavelengthat different in the cylindrical model rather than the more realistic rounded
viewingdirections.Thetorusconsideredhasahalf-openingan- edgesoftheellipticaltorusdoesnotintroducespuriouseffects.
gleofθ = 30◦.Thepositivevaluesof Pdenotethatthepolar-
0
ization vector is orientedperpendicularlyto the symmetry axis
4.2.Theeffectoftheshapeoftheinneredgeofthetorus
(type-2polarization).In our simulationsthe torus is filled with
standardGalacticdust,parameterizedasdescribedattheendof A real torus is undoubtedlythicker than the geometricallythin
section3.Wesampleatotalof108photonsandrecordspectraat cylindrical torus of Kartje. Direct imaging of NGC 4261 (=
10differentviewinganglesscaledincosi,whereiismeasured 3C 270) shows that the dusty torus in that AGN extends out
fromtheaxisofthetorus.We showourresultsasafunctionof to 230 pc (Ferrarese, Ford, & Jaffe 1996). A similar dust lane
Goosmann&Gaskell:ModelingAGNPolarization 7
acrossthenucleusofM51(=NGC5194)extendsby∼100pc
(Fordetal.1992).Theinnerradiioftoriareobtainedbyinfra-
red reverberationmapping of the hot dust and are in the range
oftenstohundredsoflight-daysforSeyfertgalaxies(seeGlass 0.2
2004andSuganumaetal.2006).
Theouterregionsoftorihaveconsiderableopticaldepth,so
theirpreciseshapeisunimportant,sincenophotonsescapepar-
alleltotheequatorialplaneofthetorus.Theshapeoftheinner P
0.1
regionfacingthecentralenergysourceismorerelevant.Current
torus models commonly consider inner surfaces that are con-
vex towards the central source. We thus model optically-thick,
uniform-density tori with elliptical cross sections, an inner ra-
dius of 0.25 pc, and an outer radius of 100 pc. We compare 0.0
these results to the modelingof a more compacttoruswith the
same half-opening angle, θ = 30◦, as in section 4.1. We de-
0
termine the dust density by fixing τ at ∼ 750. Variability ob- 1
V
servationsimplythatthe size ofthe opticalandUV-continuum
sourceinSeyfertgalaxiesislessthanafewlight-days,asisalso 1100--11
expectedfromsimpleblack-bodyemissivityarguments.Hence,
npwiohtieennts-ilcziokeneoseifdmtehirseisnicgoonsncrtaeintgtueiourinmn.gNsoooffutertchteheaittnotrhouiussr,cwmoenoscdiadenlerananetidgolneacsrsteutmhmeaeifinas- F / Fcent 10-2
validforobjectswithhigherluminositiesbecauseboththesize 1100--33
of the centralemission regionand the inner radiusof the torus
scalewithluminosity.
10-4
The resulting spectra at different inclinations are shown in
Fig.6.Iftheviewingangle,i,islessthanθ0(thuscorresponding 2000 3000 4000 5000 6000 7000 8000
toa type-1object),we onlyobservearegulartype-1spectrum.
λ [Angstroem]
Wefindthatthereisnosignificantpolarizationinthiscase.Ifwe
look at a type-2 objectat a higher inclination angle, only scat- Fig.6.Modelingalargetoruswithanellipticalcross-sectionand
tered (and hencepolarized)lightis detected.This is analogous θ = 30◦ (see section 4.2). Top: polarization, P. Bottom: the
0
to the results obtained for the compact torus shown in Fig. 5. fraction,F/F ,ofthecentralflux, F ,seenatdifferentviewing
∗ ∗
Theoverallshapeofthepolarizationspectrumforbothsizesof inclinations, i. Legend: i = 70◦ (intermediate) (maroon stars),
thetorusisrathersimilaraswell.Withincreasingviewingangle i = 63◦ (pinktriangleswithpointstotheright),i = 57◦ (purple
thelevelofthepolarizationspectrumrises,reachesamaximum, triangleswithpointsup),i=49◦(browntriangleswithpointsto
anddecreasesagaintowardsedge-onlinesofsight.Theshapeof the left), i = 41◦ (green diamonds),i = 32◦ (red squares), and
the P-spectrumdoesnotchangesignificantlybetweendifferent i=18◦(face-on)(bluecircles)
type-2inclinations.
Therearedifferencesbetweenourresultsofmodelingalarge
torus (case 3 in Fig. 4) with half-openingangle θ = 30◦, and
0
the analogous compact torus (case 2 in Fig. 4) with identical
half-openinganglebutsmallerdimensions.Astrikingdifference
occurs in the angular flux distribution: the large torus scatters
considerably fewer photons towards an observer at intermedi-
ateviewinganglesbecausetheyhittheouterpartsofit(seethe
illustration in Fig. 7). Towards edge-on viewing directions the
probability of seeing scattered photons is much lower than for
thesmalltorus.Thespectralslopeofthescatteredradiationalso
differsbetweenthetwotori.Whilethespectrumisflatinthecase
of a compacttorusit rises towardsthe blue for the large torus.
This can be explained by the increasing tendency of forward- Fig.7.Comparisonofthecompactandtheextendedtoruswith
scatteringatshorterwavelengths.Photonsescapingathigherin- θ0 =30◦intheV-band.
clinationshavetoundergoback-scattering;thisismorelikelyto
happenatlongerwavelengths.
There are also differences in the polarization signatures of sufficientstatistics.Ontheotherhand,itclearlyfollowsfromour
both tori. Although the overallspectral dependenceof P is the computationsthatthespectralfluxatanglesi > 76◦ isreduced
same, the level of P is changed. The strongest changes are at by a factor of almost ∼ 2×107 with respect to the flux of the
higher inclinations when the central source is becoming ob- source.Therefore,thepolarizedfluxattheseanglesisverylow.
scured by the torus. As with the total flux (see above), for the WeshowthedifferencesinV-bandtotalfluxandpolarization
largertorus,Pissignificantlylower(comparethecaseofi=70◦ betweenalargeandasmalltorusinFig.8.Thetoppanelshows
betweentheupperpanelsoffFig.5andFig.6).Foralargetorus, thepolarizationasafunctionoftheviewingangle,andthebot-
our current models sampling several 109 photons do not con- tompanelthefractionofthelightreachingtheobserver.Aswas
strainthepolarizationwellatveryhighinclinations.Thenumber shown above,the differencesbetween the two torusshapes are
ofphotonsscatteredintothesedirectionsistoosmalltoallowfor mostimportantat higherinclinations.At i ∼ 70◦ the degreeof
8 Goosmann&Gaskell:ModelingAGNPolarization
0,15
0.2
0,1
P 0,05
0.1
P 0
0.0 -0,05
-0,1
1
1100--11 -0,15
2000 3000 4000 5000 6000 7000 8000
F / Fcent 10-2 λ [Angstrom]
Fig.9. Polarization averaged over type-2 viewing angles (see
1100--33 section4.3).ApositivevalueofpolarizationdenotesanE-vector
orientedperpendiculartothe torussymmetryaxis;fornegative
10-4 valuesthe E-vectoris alignedwith the projectedaxis. Legend:
θ = 10◦ (blackdashedline),θ = 20◦ (solidredline),θ = 30◦
0 0 0
40 45 50 55 60 65 70 (green dot-dashed line), θ = 45◦ (blue dots), θ = 50◦ (long
0 0
yellow dashes), θ = 60◦ (browndouble dots and dashes), and
Inclination [deg] 0
θ =75◦(pinkdouble-dashesanddots).
0
Fig.8.Differencesbetweenlargeandsmalltoriwithanelliptical
cross-sectionintheV-band(seesection4.2).Top:polarization,
P, and bottom, the fraction, F/F , of the central flux, F , as a
∗ ∗
functionofviewinginclinations,i.Thedashedlinesdenotethe Varying the opening angle shows several important things.
thin elliptical torus (case 2), the solid line the extended torus Forθ < 53◦ theabsolutevalueofthepolarizationdecreasesas
0
(case3) theopeningangleincreases(seeFig.9),aswasfoundbyKartje
(1995)forcompacttori.Thepolarizationvectorisorientedper-
pendicularlyto the axis for all viewing directions i > θ , as is
0
observed in type-2 AGN. For θ > 60◦, only parallel polariza-
0
polarizationreachesadifferenceof6%,andthefluxdiffersbya
tionvectorscanbeseenatviewinganglesi>θ .Inthisrangeof
0
factorofalmost100. θ theabsolutedegreeofpolarizationincreaseswiththeopening
0
angle.
The reasonfor the flip of the relative position angle can be
4.3.Theeffectofthetorushalf-openingangle
explained by the scattering phase function, and by the geome-
Kartje (1995)has shownthatthe half-openingangle,θ , of the try of the inner parts of the torus (Kartje 1995). For a distant
0
torusisanimportantparameterfortheobscurationandreflection observerlookingatthetorusalonganoff-axislineofsight,the
properties.Whilemodelinglargetori,weexaminehalf-opening scattered radiationcomesfrom the inner surface walls. In part,
angles ranging from 10◦ to 75◦. Variation of θ is realized by these consist of the inner torus wall facing the observer most
0
changing the vertical half-axis of the elliptical torus cross sec- directly, but they also consist of the two surfaces on the side.
tion.Theothermodelparametersaredefinedasfortheprevious Duetothescatteringgeometry,thephotonsscatteredofftheside
caseofθ =30◦insection4.2. wallsarepolarizedalongtheprojectedsymmetryaxis,whilstthe
0
photonscomingfromthefarwallareperpendicularlypolarized.
The ratio of the solid angle that the far side of the visible in-
4.3.1. Toriwithnarroworwideopenings
nersurfacesubtendstothesolidanglethatthevisibleinnerside
wallssubtendchangeswiththehalf-openingangleofthetorus,
Forlargetori,θ isadominantparameterforboththedegreeof
0 andsodoestheoverallpolarizationvector.
polarizationandthepositionangle,γ.InFig.9weshowthepo-
larizationofthescatteredradiationasafunctionofwavelength
and for variousθ . Due to a similar overallshape of the wave-
0 4.3.2. Transitioncase:intermediatetorushalf-opening
lengthdependenceofP,weaveragethepolarizationovertype-2
angles
viewing angles, i, with i > θ . We thereby exclude the high-
0
est inclinations with an insufficient number of photons, where Forintermediateopeningangleswith53◦ < θ < 60◦ theorien-
0
the statistics of P are too poor.For viewing angleswith i < θ tation of the polarizationpositionangle seen at type-2viewing
0
(correspondingto type-1 objects seen face-on) the polarization anglesdependsontheexactinclination.Weillustratesuchacase
isnegligible. inFig.10,wherewesetθ =57◦.
0
Goosmann&Gaskell:ModelingAGNPolarization 9
0.03 0.15
0.02
0.1
0.01
V-band Peff
P
0
0.05
-0.01
-0.02 0
0 20 40 60 80 0 20 40 60 80
Half-opening angle Θ [deg] Half-opening angle Θ [deg]
0 0
Fig.10.Modelinganexpandedtorusatanintermediateopening Fig.11.Effective polarization, P (see section 4.3), fortype-2
eff
angle of θ = 57◦. The graphshows the polarizationdegree in viewingpositionsasa functionofthe half-openingangle ofan
0
the visual band versus inclination angle i. A positive value of extendedtorus(case3).
polarization denotes an E-vector oriented perpendicular to the
torussymmetryaxis;fornegativevaluesthe E-vectorisaligned
withtheprojectedaxis. θ outsidethisintervalthewavelength-dependenceofPisrather
0
lowanddoesnotexceedafactorof2.Sincethescatteringcross
sectionofinterstellargrainsincreasesstronglyfromtheoptical
For a line of sight passing close enough to the horizon of to the UV, wavelength-independent polarization is commonly
thetorus(i.e.,wheniisonlymoderatelylargerthanθ0)wefind supposed to be the fingerprintof electron scattering. However,
that the polarization vector is parallel, which means that type- scattering in opaque dust clouds produces relatively grey scat-
1polarizationcanbeproducedatobscuredviewinginclinations tering(Kishimoto2001).
(Fig.10).Iftheinclinationincreasesfurtherthepolarizationvec- Ourapparentlycontradictoryresultofrelativelywavelength
torswitchesbacktotype-2polarization.Itisinterestingtonote independentpolarizationwithdustscatteringarisesbecausewe
thatsuchatoruscanproducesignificantpolarizationdegreesup are considering scattering off optically-thick material, and be-
to2%forbothorientationsofthe E-vector. cause of the relatively small variation of the albedo over the
Inordertoillustratetheintegraleffectoftheopeningangle optical and UV spectral regions (see Fig. 3). The approximate
onthepolarization,weplotinFig.11thepolarization,Peff,av- constancyofthealbedoisbecausethescatteringandabsorption
eraged overall type-2 viewing positions, and over wavelength, cross-sectionsvaryin asimilar mannerwithwavelength.Since
asa functionofthehalf-openingangleofthe torus.Thediffer- we assume an optically-thick torus, we see emergent photons
ence between type-1and type-2 polarizationis ignoredin Peff. thathavebeenscatteredatanopticaldepthτ∼1.Thisisregard-
TheabsolutevaluesofPareintegrated. lessofwavelength2.Theincreaseinscatteringcross-sectionwith
The figure shows that the torus polarizes most effectively decreasing wavelength only means that the shorter wavelength
whenhavingeitherasmalloralargehalf-openingangle.Inthe photonswe see havebeenscatteredcloser to thesurfaceof the
transitionregionbetweentype-1andtype-2polarization(i.e.for torus.
53◦ < θ < 60◦)theintegratedpolarizationgoesthroughamini-
A significant change in albedo with wavelength, however,
mum.
will cause a color dependencyin the intensity and polarization
of the scattered light3. Shortwardsof ∼ 2500Å the albedode-
4.4.Wavelengthinsensitivityofpolarizationduetodust creases,butthisrangeisatthelowerlimitofthespectralrange
scattering considered in our modeling. The effect can be seen in the nor-
malized flux spectra of the torus models shown in Figs. 5 and
Wavelength-independentpolarizationiswidelytakentobeasig- 6. FortheGalacticdustcompositionweimplemented,itisless
natureofelectronscattering,butwehaveshownin Figs. 6and visibleinthepolarizationspectra.
9thatdustscatteringcanalsoproducewavelength-independent
Another grain property that needs to be considered is the
scattering. Thusa flat polarizationcurve is not a uniquesigna- degree of asymmetry of the scattering since this is effectively
ture of electron scattering. As Zubko & Laor (2000) pointout,
anangle-dependentalbedochange.Towardshorterwavelengths,
the wavelengthdependenceof polarizationprovidesa probeof
Galactic dust grains are more strongly forward scattering and
the grain scattering properties. Inspection of the wavelength-
dependentpolarizationcurvesforthelargetorusgeometriescon- 2 Thisisthereasonthatthesunlitsidesofcloudsintheearth‘satmo-
sidered above (see Figs. 6 and 9) shows that the polarization sphereareextremelywhite.
for half-opening angles with 30◦ < θ0 < 60◦ is wavelength- 3 Thisisthecauseofcolorationsintheatmosphereofthegiantplan-
independentovertheopticalandmostoftheUV.Forvaluesof ets.
10 Goosmann&Gaskell:ModelingAGNPolarization
the polarization phase function changes (see Draine 2003). As
forThomsonscattering,forward-scatteredlighthasalowerpo- 1.0
larizationthansideways-scatteredlight.Thepolarizationspectra
obtainedforthetorusmodelsdependonthesephasefunctions.
0.8
Theyadditionallyexplainwhya slightwavelength-dependence
of the polarization can be found for very narrow or very wide
openinganglesofalargetorus(seeFig.9). 0.6
P
0.4
5. Polarizationfrompolar-scatteringregions
Scattering in polar regionsof AGNs has allowed the discovery 0.2
of hidden Seyfert-1 nuclei in type-2 objects by radiation being
periscopicallyscatteredaroundtheobscuringtorus.Thecentral
0.0
parts of the polar double cone have to be at least moderately
ionizedduetotheintenseradiationfromtheAGN.Themedium 1.0
couldbeassociatedwiththewarmabsorberseeninmanyAGN
(see Komossa 1999 for a review). The Doppler shift of the X- 0.8
ray absorption lines indicates that the medium is outflowingat
roughly1000km/s.Withincreasingdistancefromthecenter,the
0.6
otdhuuetsfltsuomwbulisvmtebaloeticoointpytriaacdnaidlulyisn,tthdeiunnss,tiatcysootuyflpdteha-el1srooadbbijeaeptcitrosensaerdneetcn.roHetaoosweb.esvBcueerry,eotdhn.ids F / Finc 0.4
5.1.Polarelectronscattering 0.2
Thepolarizationinducedbyscatteringinpolar,conicalelectron-
0.0
scatteringregionshasbeenthesubjectofseveralpreviousstud- 10 20 30 40 50 60 70 80 90
ies.Brown&McLean(1977)developedaformalismtocompute Inclination i [deg]
the polarization expected from scattering inside optically thin,
Fig.12. Modeling double polar cones of various half-opening
axisymmetricscatteringregions.Thisformalismwasappliedby
angles.Top:polarization,P,withpositivevaluesdenotingtype-
Miller&Goodrich(1990)andMilleretal.(1991)tocomputethe
2polarization(perpendiculartothesymmetryaxis).Bottom:the
polarizationforpolarscatteringcones.Wolf&Henning(1999)
fraction,F/F ,ofthecentralflux.Bothvaluesareplottedversus
∗
andWatanabeatal.(2003)extendedthemodelingtooptically-
theinclinationiwithrespecttotheobserver.Thedifferentsym-
thick material using Monte-Carlo techniques that can account
bols denote different half-opening angles of the double-cone.
formultiplescatterings. Legend:θ = 10◦ (black circles), θ = 30◦ (red squares), and
WeconfirmsuchresultsinFig.12usingS.Thefigure θ = 45◦ (Cbluediamonds).TheopticCaldepthbetweentheinner
C
showsthedegreeofpolarizationandthetotalfluxasafunction
andoutershelloftheconesissettoτ =1.
es
of the observer‘s inclination for an electron scattering double-
coneofuniformdensityandwiththeopticaldepthτ =1.This
es
optical depth is measured in the vertical direction between the
law.ComparisonofFig.13withFig.7ofWatanabeetal.(2003,
innerandtheoutershellofonecone.Inordertoisolatetheef-
bottompanel)showsthatthedifferencein Pisverysmallwith
fectsofthescatteringconefromthepolarizationinducedbythe
respecttoauniform-densitytorus.
disk,weuseananisotropicallyemittingcentralsourcewiththe
emission angles being restricted to the solid angle defined by
the scattering cones. The three curves denote the half-opening
5.2.Polardustscattering
angles θ = 10◦, θ = 30◦, and θ = 45◦. The inclination is
C C C
measuredfromthesymmetryaxisofthedouble-cone. Beyond the dust sublimation radius the scattering cone could
As expected, polar electron-scattering cones produce type- containdust.Weinvestigatedthisusingasimilarbi-conicalge-
2 polarization directed perpendicularlyto their symmetry axis. ometry and our Galactic dust prescription – see section 3. In
Thedegreeofpolarizationrises fromface-onto edge-onview- Fig. 14 we show the polarization and total flux resulting spec-
ingangles.Thelattereffectisduetotheangle-dependentpolar- tra for a centrally-illuminateddust cone seen at differentview-
ization phase function of Thomson scattering. For wider open- ing angles i. The half-opening angle of the cone has been set
inganglesofthecones,thenetpolarizationPdecreasesbecause toθ = 30◦,anditsopticaldepthintheV-bandtothemoderate
C
it is the result of integrating a broaderdistribution of polariza- valueofτ =0.3.Thestrongwavelengthdependenceofthedust
V
tion vectors. The breaks of the polarization curves at i = θ extinctionproperties(seeFig.3)isclearlyvisibleinthefigure.
C
in Fig. 12 are due to the impact of multi-scattering inside the It differs for polar viewing angles, which cross the cone, from
cones, the analogous breaks in total flux curves are due to the thosealongequatorialdirections.Theformeronesshowthedust
angle-restrictedcentralemission. extinctionseenintransmission,whilethelatteronesshowdust
InFig.13weplottheinfluenceoftheopticaldepthonthepo- reflection.
larizationforthepolar-coneswithθ = 30◦.Thevariouscurves ThetotalfluxshowninthebottompanelofFig.14issignifi-
C
denote different optical depths. A similar case was considered cantlyreddenedwheni<θ .Inadditiontothat,thewellknown
C
by Watanabe et al. (2003). The density of their electron cones extinctionfeatureat2175µmisseen.Itsdepthdecreaseswithin-
varies with the distance from the center according to a power creasinginclination.Inreflection(i.e.,alongequatorialviewing