Polarimetric modeling of corotating interaction regions (CIRs) threading massive-star winds

0.29 MB·English

by  R. Ignace

Astronomy&Astrophysicsmanuscriptno.final (cid:13)c ESO2015
January30,2015
Polarimetric modeling of corotating interaction regions (CIRs)
threading massive-star winds
RichardIgnace1,NicoleSt-Louis2,andFe´lixProulx-Giraldeau2
1 DepartmentofPhysics&Astronomy,EastTennesseeStateUniversity,JohnsonCity,TN37614,USA
2 De´partementdePhysique,PavillonRogerGaudry,Universite´Montre´al,C.P.6128,Succ.Centre-Ville,Montre´al,Quebec,H3C3J7
5 ABSTRACT
1
0 Context. Massivestarwindsarecomplexradiation-hydrodynamic (sometimesmagnetohydrodynamic) outflowsthatarepropelled
2 bytheirenormously strongluminosities. Thewindsareoftenfound tobestructuredandvariable, butcanalsodisplayperiodicor
quasi-periodicbehaviorinavarietyofwinddiagnostics.
n Aims. Theregularvariationsobservedinputativelysinglestars,especiallyinUVwindlines,haveoftenbeenattributedtocorotating
a interactionregions(CIRs)likethoseseeninthesolarwind.WepresentlightcurvesforvariablepolarizationfromwindswithCIR
J
structures.

9 Methods. Wedevelop a model for a time-independent CIR based on a kinematical description. Assuming optically thin electron
2 scattering,weexploretherangeofpolarimetriclightcurvesthatresultasthecurvature,latitude,andnumberofCIRsarevaried.
Results. We find that a diverse array of variable polarizations result from an exploration of cases. The net polarization from an

] unresolved source is weighted more toward the inner radii of the wind. Given that most massive stars have relatively fast winds
R
comparedtotheirrotationspeeds,CIRstendtobeconicalatinnerradii,transitioningtoaspiralshapeatafewtoseveralstellarradii
S inthewind.
. Conclusions. WindswithasingleCIRstructureleadtoeasilyidentifiablepolarizationsignatures.Bycontrastallowingformultiple
h
CIRs,allemergingfromarangeofazimuthandlatitudepositionsatthestar,canyieldcomplexpolarimetricbehavior.Althoughour
p
modelisbasedonsomesimplifyingassumptions,itproducesqualitativebehaviorthatweexpecttoberobust,andthishasallowedus
-
o toexploreawiderangeofCIRconfigurationsthatwillproveusefulforinterpretingpolarimetricdata.
r
t Keywords. Polarization–Stars:early-typestars–Stars:massive–Stars:rotation–Stars:winds,outflows–Stars:Wolf-Rayet
s
a
[
1. Introduction only 7 % of O stars (Wadeetal. 2013), and these are thought
1
to have a fossil origin. However, it has been suggested that
v Acorotatinginteractionregion(CIR)arisesfromtheinteraction smaller scale, localized magnetic structures could be responsi-
3 ofwindflowsthathavedifferentspeeds:rotationofthestarul- ble for the onset of DACs (Kaper&Henrichs 1994). More re-
6 timatelyleadstoacollisionbetweenthedifferentspeedflowsto cently, Cantiello&Braithwaite (2011) have presented a model
5
7 produceaspiralpatterninthewind.CIRshavebeenwell-studied thatsuggeststhatsuch structurescouldbe widespreadforstars
0 inthesolarwind(e.g.,Rouillardetal.2008;Burlagaetal.1983, thatharborasubsurfaceconvectionzone.Suchzoneshavebeen
. 1997; Gosling&Pizzo 1999). They have also been invoked to predictedtoexistinhotmassivestarsbyCantielloetal.(2009)
1
explainavarietyofphenomenaobservedinmassivestarwinds. owingtoanopacitypeakcausedbythepartialionizationofiron-
0
Evidence of large-scale structure in the winds of massive stars groupelements.Detectingsuchsmall-scalemagneticstructures
5
1 is manifold. For example, the ubiquitousness of discrete ab- atthesurfaceofmassivestarsisanextremelydifficulttaskwhen
: sorptioncomponents(DACs) in the ultraviolet(UV) spectra of dealingwithOstars(Kochukhov&Sudnik2013)andevenmore
v OB stars (Howarth&Prinja 1989; Fullertonetal. 1997), some difficultforWRstarswherethephotosphereishiddenbytheop-
Xi showntohavearecurrencetimescalerelatedtotherotationrate ticallythickwind.
(Kaperetal.1999),arethoughttobeaconsequenceofthepres-
r
a enceinthewindofCIRs.Cranmer&Owocki(1996)predicted
thatsuchregionsshoulddevelopfollowingaperturbation,such
as a bright or dark spot, on the stellar photosphere that propa-
gates in the wind, thus generating a complex spiral-like struc- Studyingthewindstructuresthemselvescanprovideuseful
ture of low and high density and velocity regions as the star constraintsthatwillshednewlightontheirorigin.Polarimatry
rotates.CIRs havealso beenassociated with the periodicspec- is a powerful means of studying asymmetries in stellar winds.
troscopic,photometric,andpolarimetricvariabilityobservedin Indeed, in view of their high temperatures, massive-star out-
someWolf-Rayet(WR) stars (St-Louisetal. 1995;Moreletal. flows contain a copious number of free electrons that scatter
1997,1999;Chene´&St-Louis2010). light,therebygeneratinglinearpolarization.Forunresolvedand
Nevertheless, the origin of the perturbationsgenerating the spherically symmetric envelopes, there is total cancellation of
departure from spherical symmetry in massive-star winds re- the polarization as integrated across the source. Consequently,
mains unclear. Possible physical processes include non-radial anetpolarizationintrinsictothesourcedemonstratesdeviation
pulsationsandmagneticfields.Thelatterisparticularlypromis- by the source geometry from the spherical. Brown&McLean
ing. Large-scale,organizedmagnetic fields have been foundin (1977)showedthatforopticallythinscatteringandanaxisym-
1
Ignace,St-Louis,Proulx-Giraldeau:PolarimetricmodelingofCIRsinwinds
metricenvelope,thenetpolarizationdependsonthreefactors1: equatorialCIRforarotatingstar,andfinallythepolarizationfor
theviewinginclination,anaverageopticaldepthforthescatter- CIRsisexploredatarbitrarylatitudes.
ingenvelope,andtheshapeoftheenvelope.
Since electron scattering is gray, broadbandcontinuum po-
2.1.Polarizationfromopticallythinelectronscattering
larization measurementscan be used to characterize the asym-
metry at any given time. If time-dependent measurements are Foropticallythinscattering,theamountofscatteredlightfroma
available,moreaccurateconstraintscanbeobtainedbecausethe circumstellarenvelopeisdeterminedbyintegratingthe volume
structure will then be observed from different viewing angles. emissivity over the extent of the envelope. The volume emis-
Spectropolarimetry can also be used to study the structure in sivity j for electron scattering dependson the electron number
theline-formingregionandtoobtainkinematicinformation.In density n , the Thomson scattering cross-section σ , and the
e T
thecaseofCIRs,thisisparticularlyimportantbecausetheyare amountofincidentradiation,aswellasgeometricalfactorsrelat-
thoughtto originatein the photosphereand propagatethrough- ingtodipolescatteringtowardtheobserver.Itisassumedthatthe
outtheline-formingregionofthewind. staristhesolesourceofilluminationforcircumstellarelectrons,
To understand the phenomena of CIR and use their occur- meaningthatthecircumstellarenvelopeissufficientlyoptically
renceasadiagnosticofwindandatmosphericstructure,models thin so that the amountof scattered light is small compared to
areneededtointerpretobservations.Mullan(1986)wasanearly thestarlight.
proponent of CIRs observed in the solar wind as a viable ex- WeconsideracoordinatesystemwiththeEarthalongthez-
planationforcertaintypesofvariability(notedabove)observed axis. The star has an axis z∗ that is its rotation axis. The angle
in massive star winds. An important breakthrough in the field of inclinationbetweenthese axesissignified byzˆ·zˆ∗ = cosi0.
camewith2DhydrodynamicsimulationsbyCranmer&Owocki Itisfurtherassumedthaty = y ,andthereforexˆ·xˆ = cosi as
∗ ∗ 0
(1996),whomodeledanequatorialCIRforasteady-state,line- well. FollowingBrown&McLean(1977) for a pointsourceof
driven wind. Following on this, Dessart (2004) developed 3D illumination,thescatteringanglebetweenarayofstarlightand
simulations for CIRs off the equator. Both of those works ex- theobserverisχ.Ascattererisorientedabouttheobserver’saxis
plore the consequences for wind line-profile effects, such as withazimuthψfromthex-axis.
DACs and other periodic signatures. Brownetal. (2004) con- In the Stokes vector prescription, the polarization proper-
sideredinversetechniquesforextractingkinematicinformation ties of intensity are described in terms of I, Q, U, and V (e.g.,
aboutCIRsfromwindemissionlines. Chandrasekhar 1960). The latter is for circular polarization,
Pertinenttothiscontribution,therehasalsobeenworkonthe which will not be considered further. Stokes I is a measure of
polarization that can arise from CIR structures. Harries (2000) thetotalintensity.Stokes-QandU describethe linearpolariza-
explored a number of effects that give rise to variable polar- tionofthelight.Forexample,therelativelinearpolarizationis
ization for rotating stars, including a model for an equatorial
CIRstructure.HarriesexploredtheseeffectsusingMonteCarlo Q2+U2
p= , (1)
radiative transfer simulations. In contrast, Ignaceetal. (2009) p I
adopteda similar kinematicprescriptionforthe structureof an
equatorialCIRinanapplicationforthevariablelinearpolariza- andthepolarizationpositionangleis
tionseeninthebluesupergiantstarHD92207.Inthispaperwe
U
extendtheseinitialresultsinaparameterstudyofvariablepolar- tan2ψ = . (2)
p
izationfromawindwithCIRs.Neitheroftheprecedingpapers Q
allowed for multiple CIRs, and both were restricted to a CIR
Again, we assume that the total flux of light will be domi-
intheequatorialplane.Hereresultsarepresentedthatallowfor
natedbydirectstarlight,meaning(a)thattheamountofscattered
multipleCIRsandatarbitrarylatitudes.
starlight is small and (b) that the amount of absorbed starlight
The structure of the paper is as follows. Section 2 presents
by the circumstellar envelope is miniscule. Consequently, the
the base modelfor the CIR structureand the resulting variable
emissivityfortheStokes-Icomponentisnotaconcernhere.The
polarization, following but expanding on Ignaceetal. (2009).
emissivitiesforStokes-QandUaregivenby
The results of our parameter study are presented in Section 3.
ConcludingremarksaregiveninSection4,withparticularcon-
σ 3
sideration of applicationsfor observations.Our results are pre- j = T L n × sin2χ cos2ψ, (3)
sentedforawindmodelthatissetbyacommonlyusedvelocity Q (cid:18)4πr2(cid:19) ν e 4
structure.AnAppendixprovidesasolutionforadifferentveloc-
and
itystructureasacomparisonexample.
σ 3
j = T L n × sin2χ sin2ψ. (4)
U (cid:18)4πr2(cid:19) ν e 4
2. PolarizationfromasimpleCIR
Hereristheradiusforascatteringelectronfromthecenterofthe
In the next section, the approach of Brown&McLean (1977), star,andL isthemonochromaticstellarluminosityatfrequency
ν
modified by Cassinellietal. (1987), for optically thin electron ν.Aspreviouslynoted,theangleχandψpertaintothegeometry
scattering (i.e., single scattering) in an axisymmetric geometry ofthescattering.
isreviewed.Then,anapplicationisconsideredforaconestruc- The star has spherical coordinates(r,ϑ,ϕ). The observer is
ture in an otherwise sphericalwind. Thenwe show how to use located at ϕ = 0◦ and ϑ = i (see Fig. 1). Using µ = cosϑ,
0
theconeasabasisforcomputingvariablepolarizationfroman Brown&McLean(1977)showedthatforanaxisymmetricdis-
tributionofelectronscatterers,integrationaboutψgives
1 The analysis of Brown&McLean (1977) does not include limb
2π
darkening. Brownetal. (1989) considered limbdarkening, which can
introducechromaticeffects,sinceitisλ-dependent. sin2χ cos2ψdψ=−πsin2i0(1−3µ2), (5)
Z
0
2
Ignace,St-Louis,Proulx-Giraldeau:PolarimetricmodelingofCIRsinwinds
Thiscorrectivemodifiesthepolarizationbybeinginsertedas a
“weighting”factorintheintegrandofequation(7):
L 3 n
q= P = τ sin2i e D(r˜)(1−3µ2)dr˜dµ. (10)
0 0
Lν 16 Z Z n0!
Thenetrelativepolarizationisdeterminedbyqwithu=0,ow-
ing to the axisymmetryof the envelope.We note that q can be
positiveor negative,signifyingthe orientationof the netpolar-
izationagainstthe sky.Theprojectedsymmetryaxisoftheen-
velopeistakenliealongasthenorth-southdirectioninthesky.
Consequently, a structure like a bipolar jet would have q < 0,
whereasonelikeanequatorialdiskwouldhaveq>0.Theresult
showsthatthepolarizationscalesinrelationtoviewinginclina-
tionassin2i.Additionally,thepolarizationislinearintheoptical
depthscalingparameterfortheenvelopewithq ∝ τ .Theratio
0
q/τ isthusafunctionpurelyfromtheenvelopegeometry.
0
2.2.Thecaseofasimplecone
Our treatment for CIRs is in terms of segments of a cone that
arephase-laggedfromeachotherinaradiallydependentway.It
thereforeisusefultofirstreviewthetreatmentofthepolarization
fromasimplecone.
Fig.1. Scattering geometryfor Thomsonscattering. Here zˆ , zˆ,
∗ We considera cone with z the symmetryaxis of the cone:
∗
andrˆareunitvectorsforthestellarrotationaxis,observeraxis,
the cone lies directly over the stellar rotation axis with open-
and local radial. The scatterer hascoordinates(ϑ,ϕ). The scat-
ingangleβ .Thestarhasasphericallysymmetricwind,andthe
teringanglesare(χ,ψ),andiistheviewinginclination.Theap- 0
conerepresentsanaxisymmetricsectorofthewindthathasei-
pearance of π/2 ensures that a net polarization aligned with zˆ
∗ therhigherorlowerdensity,therebybreakingthesphericalsym-
hasq>0,versusq<0ifperpendicular.
metryofthesystemandleadingtoanetpolarizationfromelec-
tronscatteringofstarlightasdescribedintheprevioussection.In
relationtotheresultspresentedinthatsection,themainconsid-
and
erationshereare(a)torelatethepolarizationtothepropertiesof
thewindand(b)tousetheresultasabasisforourconstruction
2π
sin2χ sin2ψdψ=0. (6) ofaCIR.
Z0 Thestellarwindistakentohavemass-lossrate M˙ andwind
terminal speed v . The gas is ionized with a mean molecular
∞
TheseimplythatonlytheU-polarizationvanishesuponintegra-
weightperfreeelectronµ .Thenumberdensityofelectronsin
tion aboutthe scattering volume for an unresolvedsource, and e
thesphericalwindisgivenby
onlytheQ-polarizationsurvives.Thetotalpolarizationpisthen
givenby M˙/µ m 1
n = e H ≡n r˜−2w−1, (11)
w 4πR2v "r˜2w(r˜)# 0
L 3 n (r˜,µ) ∗ ∞
q= P = τ sin2i e (1−3µ2)dr˜dµ, (7)
0 0
Lν 16 Z Z " n0 # wheren0 isaparameterforthescaleofthenumberdensitythat
isgivenbythefactorsforwindandstarparametersatthebegin-
wnuhmerbeeLrQdeinsstihtyeloufmscinaottseirteyros,fr˜po=larr/izRe∗dthlieghrat,dniu0ssoinmteherewfeirnedncaes vneinlogcoitfy.thWeeeuqsueataiosnta,nadnadrdww=indv(vr˜e)l/ovc∞ityislathwegniovermnablyized wind
normalizedtothestellarradius,andτ anopticaldepthparame-
0
ter.Thefactorτ0isgivenby b γ
w(r˜)= 1− , (12)
r˜!
τ =n σ R . (8)
0 0 T ∗
with γ ≥ 0 the velocity law exponent, and b a dimensionless
Cassinellietal.(1987)introducedacorrectiontermtorelax
parameterthatsetstheinitialwindspeedatthewindbase,with
the assumption of a point source star. The correctionis known w =(1−b)γ.Examplesusedinthispaperadoptγ=1(although
0
as the finite depolarization factor, D(r). The factor assumes a theAppendixpresentsasolutionwithγ=2).
star of uniform brightness (i.e., no limb darkening). Although
Theelectrondensityintheconeisparametrizedby
Brown&McLean (1977) considered more general representa-
tionsforthiscorrection,weassumeastarofuniformbrightness. n =(1+η)n (r˜), (13)
c w
The factor D accountsfor the varyingdegreeof anisotropy
oftheradiationfieldasperceivedbyascatteringelectronatdif- whereηisadimensionlessparametertakingvaluesintheinter-
ferentdistancesfromthestar.Thefactorisgivenby valη∈[−1,∞).Ineffect,ηrepresentsanexcessordecrementof
densityintheconerelativetotheotherwisesphericalwind.The
1 conehaszerodensityifη= −1.Ifη =0,theconehasthesame
D(r˜)= r1− r˜2. (9) density as the wind, in which case sphericity is preserved, and
3
Ignace,St-Louis,Proulx-Giraldeau:PolarimetricmodelingofCIRsinwinds
oneexpectszeronetpolarization.Inthisformwearetakingthe
runofdensitywithradiusintheconetohavethesamefunctional
formas in the wind. Thisneed notbe the case, and certainlya
differentradial dependencefor the density in the coneas com-
paredtothewindcouldbetreated(e.g.,perhapsγisdifferentfor
theflowintheconeascomparedtothewindflow).
TheresultsofBrown&McLeanandCassinellietal.aregen-
eral to any axisymmetric distribution of scatterers. However, a
coneisanexampleofaspecialclassofaxisymmetricenvelopes
inwhichthedensityismathematicallyseparableintermsofra-
dialandangularfunctions.With µ = cosθ, we definethefunc-
tion f andgas
n /n = f(r˜)g(µ). (14)
c 0
In the application here, f(r˜) = r˜−2w−1 is the same both in the
cone and in the wind. For a cone the function g = 1 for µ ≥
cosβ andg=0forµ<cos(β ).
0 0
Consider now the polarization from the cone. It is deter-
minedbyavolumeintegralovertheentireenvelope,inboththe
coneandthenon-coneregions.Thepolarizationisgivenby
3 n (r˜,µ)
q= τ sin2i e D(r˜)(1−3µ2)dµdr˜, (15)
0 0
16 Z Z n0
Fig.2. Plot of the normalized cumulative polarization in terms
wheren istheelectrondensity,whichiseithern orn .Since
e w c ofdistancefromthe star. Thefigureapplieswhenthepolariza-
theconedensityscaleswiththewinddensity,theintegrationcan
tioncalculationisseperable(seetext).Inthecaseγ = 1,refer-
berecastas
enceslinesshowthathalfofthepolarizationwillbedetermined
within half a stellar radius above the atmosphere;90% will be
3 determinedbetweenoneand5.6stellarradii.
q = τ sin2i ×
0 0
16
∞ +1 n
e D(r˜)(1−3µ2)dµdr˜
"Z1 Z−1 n0 Inrelationtotheradialintegralfactor,Γ,itisusefultoconsider
∞ +1 n the sensitivity of the polarization to structure at different radii
+η Z1 Zµ0 n0e D(r˜)(1−3µ2)dµdr˜#, (16) ξintoth1e.wInintdh.isFfiogrumret2heshpolowtsditshpelavyasluΓeaosftΓheacsuimntuelgartaivteedpforloamr-
where µ = cosβ . The first integral is an integration over the ization from the stellar surface outto the radiusr˜ = ξ−1 (for a
0 0
sphericalwind,whichofcoursevanishes.Thesecondintegralis givenvalueofΛ),hencethelabelfortheverticalaxis.Theplot
overthecone,whichscaleswiththefactorη. is normalized to have a unit value when the integral is carried
Theseparabilitymeansthattheintegrationandradiusandin out to infinite distance. The reference lines show the locations
anglecanbewrittenasaproductoffactors,with where 50%and 90% of the polarizationare obtained.In terms
of the radial weighting, half of the polarization is set between
q= 3 τ sin2i ΓΛ, (17) 1.0andabout1.5stellarradii;90%issetbetween1.0andabout
16 0 0 5.6 stellar radii. The remaining 10% arises between 5.6R and
∗
beyond. As a result, when discussing the polarization due to a
where
curvedCIR, the bulk of the polarizationwill be determinedby
∞ thestructureoftheCIRattheinnerfewstellarradii.
Γ= f(r˜)D(r˜)dr˜, (18)
A wordaboutconventionisusefulto mentionat thispoint.
Z
1
The observersystem being used assumes the orientationof the
and cone as projected in the sky lies along a north-south direction
(i.e.,alonga meridian).Fora conewith η > 0,andnotingthat
+1
Λ= g(µ)(1−3µ2)dµ. (19) Λ0 < 0,thenetpolarizationwouldbenegative.Thissimplyin-
Z−1 dicatesthattheelectricvectorofthenetpolarizationisparallel
to the east-west line, which is orthogonalto the orientation of
For the problem at hand, and using a change of variable with
theconeprojectedontothesky.Ifη<0,thenq>0becausethe
ξ =r˜−1,thesefunctionsbecome
polarization is dominated by scatterers lying outside the cone,
∞ 1 D(ξ) andthepolarizationpositionangleliesparalleltothecone.
Γ= r˜−2w−1D(r˜)dr˜= dξ, (20) Finally,thattheconeorientationisnorth-southcorresponds
Z Z w(ξ)
1 0 to an orientation of the projected axis of symmetry on the sky
and asψ=90◦asmeasuredfromwestthroughnorth.Obviously,an
arbitrarysourcecouldhaveanorientationonthesky,andthenet
+1 polarizationwouldnotchange.Itisstraightforwardtorepresent
Λ=Zµ0 (1−3µ2)dµ=−µ0(1−µ0)2 ≡Λ0. (21) thispolarizationandpositionangleasfollows.Let p0 = |q(ψ =
4
Ignace,St-Louis,Proulx-Giraldeau:PolarimetricmodelingofCIRsinwinds
90◦)|.Thenforarbitraryorientationasperceivedbyanobserver, (1998) to model the magnetic field topology in the context of
thenormalizedStokesqanduparametersbecome wind-compressiontheory.)
In the rotating frame of the star, we imagine a generalized
“spot”ofopeningangleβ andsolidangleΩ attheequatorofa
0 0
q = p cos(2ψ), and (22) starthatrotateswithperiodPandangularvelocityω.Thisspot
0
isthesourceofadensityperturbationonthewind(i.e.,thecause
u = p sin(2ψ). (23)
0 ofη , 0).Intherotatingframe,thespotisfixed.Ifweassume
thattheflowfromthespotonlyhasaradialcomponent,thenin
IndevelopingasimplifiedmodelforthepolarizationfromCIRs,
the rotatingframe,a fluid elementemergingfromthe spotwill
weusetheseconstructionsinwhatfollows.
progressivelydisplacethespotazimuthallyinproportiontoωt,
wheretistheintervaloftimeafterthefluidparcelwasinjected
intotheflow.
2.3.CIRsintheequatorialplane
Generally, the fluid element should of course emerge from
A CIR amounts to a spiral-shaped zone of flow that threads a thespotwithanazimuthalspeedofωR .However,withthecom-
∗
stellar wind. Its structure can indeed be complicated, as seen plexityofnon-radialforceconsiderations,wesimplyignorethe
inthehydrodynamicsimulationsofCranmer&Owocki(1996) details of the azimuthal componentof speed v that a fluid el-
ϕ
and Dessart (2004). Our goal is to construct an approximation ementwouldhave.Certainly,aparametrizationforv couldbe
ϕ
for the structure of a CIR to rapidly explore model parameter included.Infact,forv theradius-dependentazimuthalvelocity
ϕ
spaceandtrendsinthevariablepolarizationwithrotationphase. intheframeofthestar,theequationofmotionforthecenterof
To this end, we approximate a cross-section of a CIR with a thefluidelementintherotatingframeofthestarwouldbe
sphereas a “sphericalcap”.This caphas circularcross-section
andisidenticaltotheintersectionofasphereofradiusrwitha dϕ′ = 1 vϕ(r)−ωr , (24)
cone.Our constructionfora CIR is simply these cap segments dr r " v #
r
foracone,butwhereeachsuccessivesegmentsinradiusareaz-
imuthallyphase-lagged.Infactwerequireanequationofmotion where ϕ′ is the azimuth about the star in the rotating frame as
forthecentersofthesegments. measuredfromthespot(i.e.,ϕ′ =0◦isthelocationofthespot).
Initially,weconsideraCIRthatisspiralinshapewherethe OurmodelforthespiralCIRassumesv (r)=0.
ϕ
center of the spirallies in the equatorialplane.The calculation Thesolutiontotheprecedingequationthenbecomes
forthepolarizationcantakeanarbitraryfunctionofdensityinto
accountwithradiusinsidetheconeandevenavaryingsolidan- R 1 1
gle of the segments with radius. As long as the CIR segments ϕ′ =− ∗ dξ, (25)
are individually axisymmetric and scattering is optically thin, r0 Zξ ξ2w(ξ)
theapproachdescribedin precedingsectionscanbeemployed. wherer = v /ωisaconvenientparameterizationrepresenting
0 ∞
However,to reduce the number of free parameters, we assume
the “winding radius” of the CIR spiral. Fast radial flow means
thatallsegmentsoftheCIRhaveaconstantsolidangle,andwe
thatthespiralismostlyalinearconeatradiiclosetothestarbe-
adoptthe same parameterizationfor the CIR density as for the
causetherotationisrelativelyslow,andsor /R islarge,yield-
0 ∗
coneoftheprecedingsection:thescalingwithηsuchthatηisa ing ϕ′ is roughly constant for r < r . In contrast, fast rotation
0
constantthroughouttheCIR,andtheγvelocityexponentisalso
relativetotheradialflowspeedindicatesarapidwindingupof
thesameinboththeconeandthewind.
thespiral,whichisrepresentedbyarelativelylowvalueofr .
0
Thequestionnowiswhattousefortheequationofmotion In the observer’s frame, the equation for the center of the
ofthesegmentcenters.Whatisthegeometricformforthespi- spiralis
ral? A physicallymotivated approachcould be based on wind-
compressiontheory (Bjorkman&Cassinelli 1993; Ignaceetal. ϕ=ϕ +ϕ′(r)+ωt, (26)
0
1996).Thatmodelisforarotatingwindinwhichstreamlineflow
issolvedviakinematicrelations,namelyavelocitylawandcon- whereϕ allowsfortheazimuthallocationofthespottobede-
0
servationofangularmomentum.Thismodelforrotatingwinds finedintermsofanarbitraryreference.
leads to density enhancements toward equatorial latitudes at Forγ=1,theintegralequationinξisanalytic.Thesolution
theexpenseofpolarlatitudes.However,wind-compressionthe- is
ory assumes a centralforce for the wind driving.Owockietal.
(1996)haveshownthatthisdoesnotholdforline-drivenwinds ϕ′(r=ξ−1)=−R∗ 1−ξ +b ln w . (27)
formassivestars.Indeed,Owockietal.findthatincludingnon- r " ξ uw !#
0 0
radialforcesleadstodensity-enhancedpolarflows.
Forourpurposesweadoptpurelyradialstreamlinesforwind Analyticsolutionscanalsobefoundforotherintegervaluesof
flow in the observer’s frame. This means that a parcel of fluid γ(seetheAppendix).Forourcaseofγ=1,theinfluenceofthe
injected into the wind from the star travels along a radial line, windingradius,r /R ,forthe shapeoftheCIRspiralisshown
0 ∗
butsince thestar rotates,subsequentparcelsofgasalso follow in Figure 3. Two reference circles are shown for radii of 2R
∗
a radial trajectory, but are successively displaced in azimuth. and4R . Thefourcurvesareforthesolutiontoϕ′ fordifferent
∗
Consequently,the CIR amounts to a spiral pattern of flow that valuesof r /R = 2,4,6,and 8, as labeled.Lowervaluesof r
0 ∗ 0
spins as a whole with the rotation period of the star. It is basi- indicate relatively high equatorial rotation speeds as compared
callythesameproblemasforperceivingaspiralpatternofwater to the wind terminal speed. Curves with lower r have higher
0
flowfromarotatinghose.Theshapeisfoundfromthestream- valuesofϕ′atagivenradius.
line flow for a CIR in the rotatingframe of the star and recog- Toevaluatethepolarization,akeyresultistorecognizethe
nizing that the shape is equivalent to the spiral pattern as seen separability of the spatial integrations for our adopted model.
by an observer. (A similar approach was used by Ignaceetal. Ourassumptionisthatthecross-sectionfortheCIRisthesame
5
Ignace,St-Louis,Proulx-Giraldeau:PolarimetricmodelingofCIRsinwinds
1 D(ξ)
Γ = sin2i sin2ψdξ, (33)
u Z w(ξ)
0
with
sin2i cos2ψ = sin2ϕ(t,ξ)−cos2i cos2ϕ(t,ξ), and (34)
0
sin2i sin2ψ = −cosi sin2ϕ(t,ξ). (35)
0
Thetotalpolarizationandthepolarizationpositionanglebecome
3
p= ητ Λ Γ2+Γ2, (36)
16 0 0 q q u
and
u Γ
tan2ψ = = u, (37)
p q Γ
q
wherethelastequalityisspecificallytrueforthemanyassump-
tions that we have adopted, such as every cross-sectional seg-
mentoftheCIRhavingthesamesolidangle.Intheopticallythin
limit,thepolarizationpositionangleisindependentofthewind
opticaldepth,thevalueofη,andeventheCIRcross-sectionas
encapsulatedinΛ .Theonlyrelevantfactoristheshapeofthe
0
spiralasafunctionofradiusasweightedbythegeometricalfac-
torsassociatedwithThomsonscattering.
Fig.3.ExamplesofthespiralshapefortheCIRsasafunctionof
It is straightforward to include multiple CIR structures by
the windingradius, r . The figure displaysthe centroidof four
0 computingtheqanduvaluesforeachoneseparately.Thisholds
differentCIRswithdistancefromthestar.Thestaristheshaded
aslongaseachoneisopticallythin.ForN CIRsattheequator,
magentacircle.TheCIRsareintheequatorialplane,shownhere
eachonehavingitsownvalueofη,ϕ ,andβ fortheith CIR,
asseen fromabove.Thetwo largerdottedcirclesare reference i 0,i i
theStokesqanduparametersarelinearcombinationswith
circlesat2R and4R .Thedifferentspiralsaredistinguishedby
∗ ∗
differentwindingradiiaslabeled.Lowervaluesof r place the
0
transition froma cone-shapedCIR to a spiral one closer to the N
star;highervaluesofr0indicatethattheCIRhassmallcurvature qtot = qi, and (38)
untilachievinglargedistancesfromthestar. Xi=1
N
u = u. (39)
asforacone;therefore,Λ fortheconeholdsfortheCIR.The tot i
polarizationforaCIRredu0cestoevaluatingΓ.However,theso- Xi=1
lutionforΓ,whichisanintegrationinradius,musttakeaccount However,itisusefultoformulatethetotalmasslossoftheflow.
of the radius-dependentphase-laggingofsegmentsas givenby Theangle-averagedelectronopticaldepthoftheentireenvelope
thesolutionofequation(27). becomes
The phase-lagging must be related to the observer system N Ω
of coordinates. The result from Brown & McLean reveals that τ¯ =τ 1+ η i . (40)
tFahnoedrpapooClsaIirtRiizo,anetiaaocnnhgflsreeo.gmImnaeengfftievicsetn,eescsaoecnnhitcisaaellglsymeagetmnatehndatisffsecira=elenist(ϕwin(icrt,lhitn)s)aintai2noidn. IfeonewassociaXit=e1s aic4eπrtain M˙ with the wind component,then a
ψ=ψ(ϕ(r,t)).Usingsphericaltrigonometry,onecanshowthat totalmass-lossratecanbedeterminedfromtheprecedingequa-
tionafterinclusionofCIRs.
ItisworthmentioningthatCIRsarisefromredistributionof
sin2icos2ψ = sin2ϕ−cos2i cos2ϕ, and (28) windflow,ascontrastedtoamodelofsectorswithenhancedor
0
sin2isin2ψ = −cosi sin2ϕ. (29) diminished mass loss relative to an average spherical wind. In
0 redistribution one envisions CIRs as perturbations of the wind
ThenthepolarizationforaCIRatagivenatimeintermsofthe density, and the summation term in equation (40) must vanish.
normalizedStokesparametersqanduisgivenby In this framework a “CIR” would actually consist of two CIR
structures,onewith positiveηand the otherwith negativeη so
astoconservethemass-lossrate.Inotherwordsredistributionof
q(t) = 3 ητ0Λ0Γq, and (30) densityimpliesthatτ¯e = τw,andourmodelwouldthensuggest
16 pairsofCIRswithhigh(η )andlow(η)densities,suchthat
h l
3
u(t) = 16ητ0Λ0Γu, (31) ηhΩh+ηlΩl =0 (41)
foreverypair.Modelresultstobepresentedinsection3adopta
where
frameworkofhighandlowdensitystreamsinsteadofaconsid-
erationofredistribution.Thepointisthatredistributiveapproach
1 D(ξ) couldbemodeledinacrudewaywithpairsofCIRsasjustde-
Γ = sin2i cos2ψdξ, and (32)
q Z w(ξ) scribed.
0
6
Ignace,St-Louis,Proulx-Giraldeau:PolarimetricmodelingofCIRsinwinds
2.4.CIRsfromarbitrarylatitudes
Offtheequator,theprocessesforformulatingtheCIRgeometry
andthepolarizationislargelythesame.AttheequatortheCIR
remains centered in the equatorialplane. At other latitudes the
centerofaCIRmovesalongaconicalsurfaceunderourscheme
(since the flow is radial). The equation of motion for ϕ(t,r,θ )
0
forastreamoriginatingatlatitudeθ isalmostunchangedfrom
0
the equatorialcase. With ̟ = rsinθ , thedifferentialequation
0
forthecentersofCIRsegmentsis
dϕ′ ω
=− . (42)
d̟ v
r
Giventhatforourapproximation,θ = θ isfixedatallradiifor
0
aCIRoriginatingatθ ,thesolutionforϕbecomes
0
R 1 dξ
ϕ(t,r˜,θ )=ωt+ϕ − ∗ sinθ . (43)
0 0 r0 0 Zξ ξ2w
The main difference is the appearanceof the factor sinθ . The
0
interpretationis thatthereis nowan“effectivewindingradius”
withreff =r0/sinθ0,andthisbecomeslargerforlatitudescloser
tothepole.Atthepole,reff →∞,andaCIRemergingfromthe
polewouldbeaconicalstream.
Fig.4. Periodicvariationsofpolarizationwithstellarrotational
A major difference from the equatorial case comes in the phaseforCIRswithdifferentwindingradiusvalues.Leftpanels
formofthemorecomplexsphericaltrigonometricrelationsthat
areforpolarizationlightcurveswithphase;rightpanelsarefor
transformbetweenthestarcoordinatesystemandthatoftheob-
variationsintheq-uplane.Toppanelsareforaviewinginclina-
server.TheexpressionsneededforΓqandΓu become tion of i = 30◦ and lower for i = 60◦. Differentcurvesare for
differentwindingradii,withvaluesr /R =1(black),3(red),9
0 ∗
sin2i cos2ψ = sin2θ sin2ϕ−(cosθ sini (green),and27(blue).
0 0 0
−cosi sinθ cosϕ)2, and (44)
0 0
sin2i sin2ψ = sini0 sin2θ0 sinϕ Second,andmostimportantforourpurposes,theinterstellar
−cosi sin2θ sin2ϕ. (45) contribution is constant. Any variable polarization is therefore
0 0
necessarily intrinsic to the stellar source.One cannotgenerally
With these relations in hand, polarization light curves can be saythattheinterstellarpolarizationisgivenbytheobservedmin-
computed for a CIR emerging from an arbitrary location on imum q and u values. In other words, even if the star did not
the star. Indeed,using the approachof the precedingsection, a displayvariablepolarization,itmaystillhaveanon-zerointrin-
modelwithmultipleCIRscanalsobecalculated. sic polarization. Nonetheless, variable polarization is taken as
The next section details an exploration of parameters as a evidenceforintrinsicpolarization.Iftheamplitudeofthatpolar-
samplingofthekindsofvariablepolarizationsthatcanresultfor izationislargerthantheminimumvalue,thenonemaycertainly
anumberofselectscenarios. conclude that the stellar polarization is large compared to the
interstellarcomponent.
With these points in mind, what follows presents idealized
3. Results
modelresultsforq (t)andu (t)thatignoretheISMcomponent,
∗ ∗
Wehaveconductedabroadparameterstudyforthevariablepo- as well as considerations of sources of random or systematic
larizationresulting fromour modelof CIRs. Beforediscussing (e.g., instrumental) errors present in the data. Our assumption
these results, we wish to emphasize two points. First, these is thatour modelswould be applied to processeddata after in-
models represent the intrinsic source polarization. Real mea- terstellarandsystematiceffectshavebeencorrected.
surements are contaminated by an additional component aris- The parameter space for CIR simulations is large owing to
ing from interstellar polarization. An unresolved star that is ourrathergeneralmodel.OurmodelforthestructureofCIRsis
spherically symmetric will generally have a measurable polar- admittedly parameteric. We have strived to minimize the num-
izationofthepolarizinginfluenceoftheinterstellargasanddust beroffreeparametersinourapproach.Nonetheless,simulation
throughwhichthestarlightmusttraveltoreachEarth.Thisinflu- variablesincludeopeningangleβ,viewinginclinationi,latitude
ence dependson wavelength. The effect is frequentlymodeled of the CIR ϑ, azimuthal location ϕ, the winding radius r0/R∗,
or removed using a “Serkowski Law” (Serkowskietal. 1975). and the density enhancementfactor η. In addition, simulations
Supposethe star hasan intrinsic normalizedpolarizationgiven allowforanarbitrarynumberofCIRstructuresN.
by[q (λ),u (λ)]attheEarth.Theinterstellarcontributioncanbe As a sampling of this enormous range of possible models,
∗ ∗
modeled as adding an additional [q (λ),u (λ)]. The mea- we focusonfourbasic sets:the effectofchangingthewinding
ISM ISM
suredpolarizationwillthenbecome(q +q ,u +u ).The radius for one equatorial CIR, the effect of multiple CIRs all
∗ ISM ∗ ISM
ISM contributionessentially producesa wavelength-dependent emerging from the equator of the star, the effect of CIRs from
translation of the intrinsic stellar polarization in the q−u dia- different latitudes as seen from different viewing inclinations,
gram. andtheeffectofdifferentnumbersofrandomlydistributedCIRs
7
Ignace,St-Louis,Proulx-Giraldeau:PolarimetricmodelingofCIRsinwinds
Fig.5. Periodic variable polarization for two equatorial CIRs. Fig.6. Periodic variations in polarization are shown here for
TheformatisthesameasinFig.4withlightcurvestotheleft, CIRsthatemergefromdifferentlatitudesofthestar.Thelayout
q-uplanevariationstotheright,aviewingperspectiveofi=30◦ ofthefigureisthesameasinFig.5.HerethecolorsareforCIRs
at the top, and i = 60◦ along the bottom. The two CIRs are at latitudes of ϑ = 20◦ (black), 40◦ (red),60◦ (green),and 80◦
shiftedfromeachotherbyanangle∆ϕof0◦ (black),30◦ (red), (blue).
60◦ (green),90◦ (darkblue),120◦ (lightblue),150◦ (magenta),
and180◦(orange).
3.2.EquatorialCIRs
WeconsideramodelwithtwoequatorialCIRs.ThefirstCIRhas
η = 1 and emergesfrom an azimuthof ϕ = 0◦. A second also
as seen from a typical viewingperspective.Exceptfor the first hasη = 1 butemergesfromϕ = 0◦ to 180◦ in30◦ increments.
set of models, the winding radius will be fixed at r0/R∗ = 5. (Modelsfrom 180◦ to 360◦ are degeneratewith our set, within
Thisamountstoanequatorialrotationspeedthatis20%ofthe a phaseshift.)Thetwoare co-addedwithq = q +q andu =
wind terminalspeed.In addition,the openingangle β = 15◦ is u +u . 1 2
fixed.Thusatagivenradius,theCIRcoversasolidangleofjust 1 2
Figure 5 displays polarimetric light curves for this set of
under 2% of the “sky” as seen from the star. Since our model models.ThedisplayisthesameasinFigure4.Thedifferentcol-
isforopticallythinscattering,thepolarizationparametersq,u, orsforthe differentazimuthaloffsetsbetweenthe pairof CIRs
and pallscalelinearlywiththeopticaldepthτ ,soresultswill
0 is indicated in the caption. All of the light curves are double
bepresentedasnormalizedtoτ aspercentagevalues.
0 peaked,but(a)the shapeofthepeaks,(b)theintervalbetween
thepeaksinphase,and(c)theamplitudeofthepeaksdependon
therelativepositioningofthetwoCIRsinazimuth.Ofparticu-
3.1.Changingthewindingradius
lar interest is that for the parametersshown, the two peaks are
Figure4showsmodelresultsforasingleequatorialCIRasthe closest when the CIRs emerge at 90◦ from one another around
wind radius r /R is varied. Left panels are polarization light theequator.
0 ∗
curves; right are q-u planes. The top panels are for a viewing
inclinationofi=30◦,whereasthebottompanelsarefori=90◦.
3.3.CIRsatdifferentlatitudes
Thecurvesareforr /R = 1,3,9,and27.Highervaluesforthe
0 ∗
windingradiusimplythatCIRsstarttocurveatlargerradii. Inthissetofmodels,asingleCIRstructurethreadsthewind,but
Notsurprisingly,smallwindingradiitendto havelowrela- withtheCIRemergingfromdifferentlatitudesandviewedfrom
tivepolarization(i.e.,p/τ )amplitudes.Thisisbecausematerial differentinclinations.Figure6showsresultsinaformatsimilar
0
is more spread out in azimuth about the star, and so a greater to Figure 4. Like that figure, the upper panels are for i = 30◦
range of scattering angles are sampled, leading to polarimetric andthelowerfori=60◦:leftisforpolarizationlightcurvesand
cancellation.Higher valuesof r /R lead to higher relative po- right for the q-u plane. The different curves are for CIRs from
0 ∗
larizationamplitudes.Inallcases,twopeaksseparatedbyhalfa differentlatitudes,withϑ=20◦,40◦,60◦,and80◦.
rotationalphaseareseeninthepolarizationlightcurvebecause Changingthe latitude has distinctiveeffects on the variable
thereisa singleCIR structure.Inthe q-uplane,these leadtoa polarizationascomparedtotheprevioustwosectionsthatcon-
double loop. The two troughs of the light curves (and the two sideredonlyequatorialCIRs.Changingthelatitude,allelsebe-
loopsintheq-uplane)arenotidenticalbecauseofstellaroccul- ingequal,hasconsequencesforwhetherthelightcurveissingle
tationinconjunctionwiththeCIRbeingasymmetric. ordoublepeaked.Intheq-uplane,theeffectsareeitherasingle
8
Ignace,St-Louis,Proulx-Giraldeau:PolarimetricmodelingofCIRsinwinds
or a doubleloop.The reasonis thata CIR thatis moreclosely
locatednearthepoletendstobemorenearlystationaryinpro-
jection on the sky, despite the rotationof the star. The limiting
caseisaCIRatthepole,whichisjustaconicalflowthatmain-
tainsafixedprojectedpositionangle.
3.4.RandomdistributionsofCIRs
The case that may perhaps most closely represent real stars is
one with multiple CIR structures emerging from different az-
imuthandlatitudelocations.Inthisexamplesevenmodelshave
beencalculatedwithindividualCIRs.Arandomnumbergenera-
torwasusedtoplacetheindividualCIRsaroundthestar.Values
ofη were randomlyassigned in the rangeof −1 to +4.Table 1
summarizestheassignedparametersforthesevencases.
Figures7and8showpolarizationlightcurvesandq-uplanes
using the following construction. The viewing perspective is
fixedati = 60◦.AlltheCIRshaver /R = 5.Thentheresults
0 ∗
aresuperimposedandnormalizedaccordingto
n
1
q = q (46)
net n i
Xi=1
and
Fig.7. Periodicpolarizationlightcurvesforarandomizeddistri-
n butionofCIRsacrossthestar.Eachcurverepresentstheaddition
1
u = u (47) of onemorerandomlyplacedCIR. Thecurvescorrespondto a
net i
n
Xi=1 number of CIRs of 1 (black), 2 (red), 3 (green), 4 (dark blue),
5 (lightblue),6 (magenta),and 7 (orange).Table 1 lists model
where n runs from 1 to 7. Figures 7 and 8 thus show results
parametersoftheindividualCIRs.
for one CIR, two CIRs, three CIRs, and so on, up to all seven
CIRstogether.Thisisanincrementalapproachforseeingwhat
happensasadditionalrandomlyplacedCIRstructuresareadded
tothemodel.
spherical wind. There is no denying that we took some liber-
Thenormalizationensuresthatthecurvesarecomparingthe
tiesinpresentingasimplifiedstructureforaCIR:circularcross-
samethings.Forexample,ifweweretoco-addsevenmodelsof
section, constant solid angle, uniform density within the CIR,
exactly the same CIR, the resulting polarization would simply
and a simple prescriptionfor its spiral curvature.However,the
be the same curve as one CIR, only with a polarization that is
advantages of the approach allow, for the first time, a broad
seventimesgreater.Normalizationensuresthatas morecurves
exploration of consequences for variable polarization signals.
areadded,thecurvesarebeingcomparedonthesamebasis.
The goals were to adopt a flexible, semi-analytic description
The result is a diverse set of possible outcomes. The black
of a CIR structure, basically motivated by hydrodynamic sim-
curve shows the case of just one CIR. Adding additionalCIRs
ulations, in which to explore the range of polarimetric be-
moves through the curves in color in the order of red, green,
havior easily and rapidly. These results may then be used in
darkblue, lightblue,magenta,and finally orange.A varietyof
conjunction with other methods, such as UV line variability
shapeswithmoreorfewerpeaksresult.Addtothisthefactthat
(for a recent example, see Prinjaetal. 2012), optical variabil-
forthisexample,allCIRshavethesameopeningangleandthat
ity(e.g.,Chene´&St-Louis2010),andperhapsX-rayvariability
onlyoneviewinginclinationcaseisbeingshown,itseemsclear
(Ignaceetal.2013;Massaetal.2014).
thatawidevarietyofvariablepolarizationscouldresult.
AnimportantresultofourstudyisallowanceforCIRsfrom
latitudesotherthantheequator.Themajorityofpreviousworks
4. Conclusion have focused on equatorial CIRs (one exception being Dessart
2004). The motivation is often one of simplicity. Frequently,
In this paper we have described a parametric representation of
CIRs are considered in relation to variable blueshifted abso-
a corotating interaction region (CIR) that threads an otherwise
prtion for UV line data involving the intersection of the CIR
with the absorption column between the observer and the star.
Table1.ModelparametersforrandomlydistributedCIRs Consideration of an equatorial CIR makes the modeling eas-
ier. In contrast, polarization is sensitive to the fully three-
dimensionalstructureof the asymmetricdistributionof density
Case η ϕ(degs) cosϑ
a −0.64 220 +0.18 aboutthestar.(Thetrade-offinrelationtoline-profilestudiesis
b +1.83 250 −0.89 thelossofvelocityshiftinformation.)A majorconsequenceof
c +0.20 219 −0.22 havingCIRsfromdifferentlatitudesratherthanjusttheequator
d +0.48 290 +0.98 is a far more diverse range of polarimetric light curves and of
e +0.63 14 +0.22 behavior in the q-u plane. The potential-added complexity can
f −1.94 326 −0.63 be a strength when used in conjunctionwith other diagnostics,
g +0.52 154 −0.74 suchaslinevariability.
9
Ignace,St-Louis,Proulx-Giraldeau:PolarimetricmodelingofCIRsinwinds
Fig.8. Polarization modelresults from Fig. 7 displayed in the Fig.A.1. Comparison of the azimuth location of the CIR cen-
q-uplane.Thecolorscorrespondtotheidentificationsindicated troidwithradiusfromthestar.Hereblueisforγ=1,andredis
inFig.7. for γ = 2. Thisexampleuses a wind radiusof r = 10R . The
0 ∗
valueofφ′ isgenerallybetween50%to100%largerforγ = 2
thanforγ = 1indicatingthataspiralCIRisconsiderablymore
ThiscontributionfocusedonthetheoreticalaspectsofCIRs
curvedatagivenradiusforthehigherγcase.
and variable polarization. There are several data sets to which
these results couldbe applied.Ina follow-uppaper,our model
willbeemployedtointerpretvariablepolarizationfortheWolf-
Rayet (WR) star EZ CMa (Drissenetal. 1989). WR stars are
knowntoformpseudo-photospheres,whereopticaldepthunity
occursinthewinditself(e.g.,Hillier1987).Althoughourmodel
isforopticallythinscattering,usefulresultscanstillbeobtained
through adoption of the “last scattering approximation” (e.g.,
Stenflo1982).Themeritforsuchanapproachisempiricallysug-
gestedbySchulte-Ladbecketal.(1992)forEZCMa.However,
the approach will require modifications to the prescription for
CIRcurvature,whichwillbeexplainedinthefollow-uppaper.
Acknowledgements. The authors gratefully acknowledge helpful comments
from an anonymous referee. RI wishes to acknowledge support for this re-
searchthroughagrantfromtheNationalScienceFoundation(AST-0807664).
NSLacknowledgesfinancialsupportfromtheNaturalSciencesandEngineering
Research(NSERC)ofCanada.
AppendixA: CIRsfortheCaseγ = 2
Our results have focused exclusively on the case of γ = 1 for
thewindvelocitylaw.Ofcourse,othervaluesofγmaybecon-
sidered.Ingeneral,thesolutionforthespiralshapeofthespiral
willnotbeanalyticbutmustbeevaluatednumerically.However,
analytic solutions for ϕ(ξ) can be found for integer values of
γ. Expected values of γ for early-type star winds range from
γ = 0.5 from the original Castoretal. (1975) paper for line- Fig.A.2.Comparisonofpolarizationlightcurvesfortheγ = 1
andγ = 2casesshowninthe previousfigure.Thepolarization
driven winds to γ ≈ 0.8 from Pauldrachetal. (1986) that aug-
mentedtheinitialresultsofCastoretal.totheγ=1thatistyp- isplottedasnormalizedtothewindopticaldepthalongaradial.
The case of γ = 1 produces a relatively stronger polarization
ically used forthe innerwind of WR stars (e.g.,Schmutzetal.
signal (per unit optical depth) than in the case of γ = 2. The
1989) and then to γ ≈ 3, which has been suggested for some
polarization extrema are slightly shifted in phase between the
supergiantwinds(e.g.,Prinjaetal.1995).
Asanexample,thecaseofγ =2isprovidedhereasacom- twocases.
parisoncasetoγ = 1.Ingeneral,highervaluesofγtendtoin-
creasetheradialwidthoverwhichthebulkofwindacceleration
10