Analysis of the Flux and Polarization Spectra of the Type Ia Supernova SN 2001el: Exploring the Geometry of the High-velocity Ejecta

34 Pages·1.6 MB·English

by  Daniel Kasen

DraftversionFebruary2,2008
PreprinttypesetusingLATEXstyleemulateapjv.11/12/01
ANALYSIS OF THE FLUX AND POLARIZATION SPECTRA OF THE
TYPE IA SUPERNOVA SN 2001EL:
EXPLORING THE GEOMETRY OF THE HIGH-VELOCITY EJECTA
Daniel Kasen1, Peter Nugent1, Lifan Wang1, D.A. Howell1, J. Craig Wheeler2,
Peter Ho¨flich2, Dietrich Baade3, E. Baron4, P.H. Hauschildt5
[email protected]
Draft version February 2, 2008
ABSTRACT
SN 2001el is the first normal Type Ia supernova to show a strong, intrinsic polarization signal. In
3 addition,duringtheepochspriortomaximumlight,the CaIIIRtripletabsorptionisseendistinctly and
0 separately at both normal photospheric velocities and at very high velocities. The high-velocity triplet
0 absorption is highly polarized, with a different polarization angle than the rest of the spectrum. The
2
uniqueobservationallowsustoconstructarelativelydetailedpictureofthelayeredgeometricalstructure

n of the supernova ejecta: in our interpretation, the ejecta layers near the photosphere (v ≈ 10,000
a km s−1) obey a near axial symmetry, while a detached, high-velocity structure (v ≈ 18,000−25,000
J kms−1)withhighCaIIlineopacitydeviatesfromthephotosphericaxisymmetry. Bypartiallyobscuring

6 the underlying photosphere, the high-velocity structure causes a more incomplete cancellation of the
1 polarizationofthe photosphericlight,andsogivesrisetothe polarizationpeakandrotatedpolarization
angle of the high-velocity IR triplet feature. In an effort to constrain the ejecta geometry, we develop a
1
techniqueforcalculating3-Dsyntheticpolarizationspectraanduseittogeneratepolarizationprofilesfor
v
several parameterized configurations. In particular, we examine the case where the inner ejecta layers
2
are ellipsoidal and the outer, high-velocity structure is one of four possibilities: a spherical shell, an
1
ellipsoidal shell, a clumped shell, or a toroid. The synthetic spectra rule out the spherical shell model,
3
1 disfavor a toroid, and find a best fit with the clumped shell. We show further that different geometries
0 canbe moreclearlydiscriminatedifobservationsareobtainedfromseveraldifferentlinesofsight. Thus,
3 assuming the high velocity structure observed for SN 2001el is a consistent feature of at least a known
0 subsetoftypeIasupernovae,futureobservationsandanalysessuchasthesemayallowonetoputstrong
/ constraints on the ejecta geometry and hence on supernova progenitors and explosion mechanisms.
h
p
- 1. introduction showedan intrinsic continuum polarizationof about 0.7%
o
(Howell et al. 2001). Chemical inhomogeneities were also
r 1.1. Spectropolarimetry of Supernova
t suggested to explain the rather noisy polarization data of
s
a The geometricalstructureofsupernovaejecta,asdeter- SN1996x(Wangetal.1997). Inaddition,strongintrinsic
: mined empirically from observations, can give important polarizationhasbeenmeasuredinalltypesofcorecollapse
v
clues as to the nature of the supernova progenitor system supernovae (Wang et al. 1996).
i
X andexplosionphysics. Spectropolarimetryisacrucialtool Anon-zerointrinsicpolarizationmeasurementindicates
r in constraining the shape of unresolved supernovae. The that a supernova is aspherical, but using the spectropo-
a scattering atmospheres found in supernovae can linearly larimetry to constrainthe supernova geometry usually re-
polarize light. For an unresolved, spherically symmetric quirestheoreticalmodeling. Thedetailedtheoreticalstud-
system the differently aligned polarization vectors around ies so far have been confined to axisymmetric configura-
the disk will cancel, resulting in zero net polarization. If tions. Shapiro & Sutherland (1982) first estimated the
thesymmetryaroundthelineofsightisbroken,however,a continuum polarizationexpected froman ellipsoidal, elec-
net polarization can result due to incomplete cancellation tronscatteringsupernovaatmosphere. Ho¨flich(1991)used
of polarization vectors (Shapiro & Sutherland 1982). a Monte Carlo code to calculate the continuum polariza-
The polarizationobservationsofSN2001elpresentedin tion from several axisymmetric configurations, including
Wang et al. (2002) (hereafter Paper I) are the first obser- an off-center energy source embedded in a spherical elec-
vations of a spectroscopically normal Type Ia supernova tron scattering envelope. Calculations of synthetic super-
(SN Ia)whichshowa significantintrinsic polarizationsig- nova polarization spectra have also been performed, but
nal. Most previous observations of SN Ia showed no ob- usually only for the ellipsoidal geometries (see however
servable polarization, given the signal-to-noise of the ob- Chugai (1992)). In the past, such ellipsoidal models have
servations(Wangetal.1996). Theonlyotherindicationof done a fair job in fitting gross characteristics of the avail-
a clear non-zero polarization in a SN Ia was the sublumi- able spectropolametric observations, for example those of
nous and spectroscopically peculiar SN Ia 1999by, which SNe 1987A(Jeffrey 1991),1993J (H¨oflich et al. 1996)and
1 LawrenceBerkeleyNationalLaboratory, Berkeley,CA94720
2 DepartmentofAstronomy,UniversityofTexasatAustin,Austin,TX78712
3 EuropeanSouthernObservatory,Karl-Schwarzschild-Strasse2,D-85748Garching,Germany
4 DepartmentofPhysicsandAstronomy,UniversityofOklahoma,Norman,OK73019
5 DepartmentofPhysicsandAstronomyandCenterforSimulationalPhysics,UniversityofGeorgia,Athens,GA30602
1
2 D. Kasen et al.
SN 1999by (Howell et al. 2001). P-Cygni features due to Si II, S II, Ca II and Fe II (see
SN 2001el presents an exciting development in that no e.g. Branch et al. (1993)). The blueshifts of the minima
axially symmetric geometry is able to account entirely for ofthese featurescanbe usedto estimate the photospheric
the spectropolametricobservations. Inparticular,we sug- velocities of SN 2001el,which for all features are found to
gest that the supernova ejecta consists of nearly axially- be v ≈ 10,000 km s−1. The only truly unusual feature
ph
symmetric inner layers (v . 15,000 km s−1), surrounded of the flux spectrum is a strong absorption near 8000 ˚A,
by a detached, high-velocity structure (v ≈ 20,000 − which is discussed in detail below.
25,000 km s−1) with a different orientation. The anal- We concentrate our analysis on the earliest spectrum
ysis of the system therefore requires that we consider the (Sept.25),ofSN2001el. Afulldescriptionofthe flux and
synthesis of polarization spectra for 3-D configurations. polarization spectra at all epochs is given in Paper I.
In this paper we take an empirical approach, and use
a parameterized model to try to extract as much model 1.3. High Velocity Material in SNe Ia
independent informationaboutthehighvelocitystructure
The most interesting feature of SN 2001el is the strong
in SN 2001el as the observations will permit. A unique
absorption feature near 8000 ˚A. The absorption has
3-D reconstructionof the geometryis not possible, as this
a “double-dipped” profile, consisting of two partially
constitutes a kind of ill-posed inverse problem. However,
blendedminima separatedby about150˚A.It seems to be
by restricting our attention to various parameterized sys-
a pure absorption feature with no obvious emission com-
tems, we can draw some rather general conclusions about
ponent to the red. The feature is still strong on Sept 30,
the viability of different geometries. In particular, we ex-
but has weakened considerably by Oct 9. By the Nov 9
aminethecasewheretheinnerejectalayersareellipsoidal
observations,the8000˚Afeaturehasvirtuallydisappeared
and the outer, high-velocity structure is one of four pos-
(see Paper I).
sibilities: a spherical shell, an ellipsoidal shell, a clumped
Hatano et al. (1999) identified a much weaker 8000 ˚A
shell, or a toroid. We develop a technique for calculat-
feature in SN 1994D as a highly blueshifted Ca II IR
ing 3-D synthetic polarization spectra of the high veloc-
triplet. The double-dipped profile now visible in the Sept
ity material. The synthetic spectra rule out the spherical
25SN2001elspectrumsupportsthis conclusion. Thered-
shell model, disfavor a toroid, and find a best fit with the
mostlineofthetriplet(λ8662)producesthered-sidemin-
clumped shell.
imum while the two other triplet lines (λ8542 & λ8498)
Geometrical information extracted empirically from
blend to produce the blue-side minimum. The synthetic
spectropolarimetry must eventually be compared to de-
spectra to be presented in §4 confirm that the IR triplet
tailedmulti-dimensionalexplosionmodels. Asofyet,none
can reproduce the shape of the double minimum. Unfor-
of the computed explosion models appear directly appli-
tunately,theearlyspectradonotextendfarenoughtothe
cable to SN 2001el. 3-D deflagrationmodels of a SN Ia in
blue to observe a corresponding high velocity component
the earlyphases havebeen computed by Khokhlov(2000)
to the Ca II H&K lines. We have investigated all other
and Reinecke et al. (2002). These models show a quite
potentiallines thatmighthavecausedthe 8000˚Afeature,
inhomogeneous chemical structure, with large plumes of
but none were able to reproduce the feature without pro-
burned materialextending into unburnedmaterial. So far
ducing another unobserved line signature somewhere else
the calculations only cover the early stages of the explo-
in the spectrum.
sion, before free expansion is reached. It is possible that
AdoptingtheIRtripletidentificationforthe8000˚Afea-
at some point the deflagration transitions into a detona-
ture, the implied calcium line of sight velocities span the
tion wave (Khokhlov 1991). The detonation may smooth
range18,000−25,000km s−1. This should be contrasted
out the inhomogeneities in the chemical composition by
with the photospheric velocity of 10,000 km s−1 as mea-
burning away the unburnt material between the plumes
sured from the normal SN Ia features in SN 2001el. We
(H¨oflich et al. 2002; Khokhlov 2000). It could also intro-
therefore make the distinction between the photospheric
duce aglobalasymmetryifitoccursatanoff-centerpoint
material, which gives rise to a seemingly normal SN Ia
(Livne 1999). Other possible sources of asymmetry in-
spectrum (hereafter, the “photospheric spectrum”), and
clude rapid rotationof a white dwarfprogenitor(Mahaffy
thehigh velocity material (HVM),whichproducestheun-
& Hansen 1975), and the binary nature of the progenitor
usual8000˚AIRtripletfeature. Inthefluxspectrum,there
system (Marietta et al. 2000).
is a clear separation between the photospheric triplet ab-
sorption at 8300 ˚A and the HVM feature at 8000 ˚A. In
1.2. Supernova SN 2001el
the polarization spectrum, the angle and degree of polar-
Monard(2001)discoveredSN2001elinthegalaxyNGC ization of the 8000 ˚A feature each differ from the photo-
1448. The brightness of this nearby supernova (m ≈ 12 spheric spectrum. Both of these imply a rather sudden
B
at peak) made it an ideal candidate for spectropolarime- change of the atmospheric conditions in the HVM.
try. Spectropolametric observations were taken on Sept AhighvelocityCaIIIRtripletfeaturehasbeenobserved
25, Sept 30, Oct 9 and Nov 9 of 2001. Details on the ob- inotherSNeIa,albeitrarelyandneverasstrong. Thepre-
servations and the data reduction of the spectra analyzed max spectra of SN 1994D (Patatet al. 1996;Meikle et al.
in this paper can be found in Paper I. 1996), show a similar, but much weaker absorption. The
In Figure 1a we show the flux spectrum of SN 2001el SiIIandFeIIlinesofthesespectraalsosuggestsomema-
for the first epoch (we have removed the redshift due to terial is moving faster than 25,000 km s−1 (Hatano et al.
the peculiar velocity of the host galaxy). The flux spec- 1999)The earliest spectrum of SN 1990Nat day -14 (Lei-
trum of SN 2001el resembles the normal SN Ia SN 1994d bundgut et al. 1991) has a deep, rounded 8000 ˚A feature,
at about 7 days before maximum light, with the expected and the spectrum also showed evidence of high velocity
Spectropolarimetry of SN 2001el 3
silicon or carbon (Fisher et al. 1997). The 8000 ˚A feature of the intensity; in this case:
has also been observed in the maximum light spectrum
(I −I )
of SN 2000cx (Li et al. 2001). In this case, however, the P = 0 90 (2)
I +I
line widths are narrower and the two minima are almost 0 90
completely resolved. The polarization position angle (labeled χ) is defined as
In SN 2001el, the only clear-cut evidence for high ve- the angle at which the transmitted intensity is maximum.
locity material seems to be the 8000 ˚A feature. There is It is tempting to think of the polarization as a (two di-
no strong Si II 6150 absorption at v > 20,000, although mensional) vector, since it has both a magnitude and a
a weak absorption cannot be ruled out because at this direction. Actually the polarization is a percent differ-
wavelength (5880 ˚A) it would blend completely with the ence in intensity, and intensity is the square of a vector
neighboring Si II λλ5958,5979 feature. There is also no (the electric field). The polarization is actually a quasi-
clear indication of high velocity Fe II or S II. The blue vector, i.e. polarization directions 180◦ (not 360◦) apart
edge of the CaII H&K feature on Oct. 9 – the first avail- are considered identical. The additive properties of the
able spectrum to go far enough to the blue – is at 27,000 polarization thus differ slightly from the vector case, as
km s−1. The likelihood of this being HVM is suspect be- evidenced by the fact that the polarization is canceled by
cause of the strong possibility of line blending. Since the another equal beam oriented 90◦ to it, rather than one at
8000˚Afeatureistheonlyunambiguousdetectionofahigh 180◦ as in vector addition.
velocity material in SN 2001el, we hereafter refer to it as In this case, a useful conventionfor describing polariza-
the HVM feature. tion is through the Stokes Parameters, I,Q and U, which
Our analysis will focus almost entirely on the 8000 ˚A measure the difference of intensities oriented 90◦ to each
HVM feature. In §2 we give an introduction to polariza- other. A Stokes “Vector” can be defined and illustrated
tion in supernova atmospheres; §3 describes a parameter- pictorially as:
ized model that allows us to generate synthetic polariza-
tionspectra,andin§4weusethemodeltoexplorevarious I I0◦ +I90◦ l+↔
geometries for SN 2001el. In §5 we consider the signature I= Q = I0◦ −I90◦ = l−↔ (3)
of each geometry when viewed from alternative lines of U! I45◦ −I−45◦! ց−ր!
sight. The implication of these constraints on the progen- whereI90◦,forinstance,designatestheintensitymeasured
◦
itors and explosion mechanisms of SNe Ia’s is discussed with the polarizing filter oriented 90 to a specified direc-
briefly in the conclusion. tion called the polarization reference direction. To deter-
minethesuperpositionoftwopolarizedbeams,onesimply
2. supernova spectropolarimetry adds their Stokes vectors. A fourth Stokes parameter V
2.1. Polarization Basics measures the excess of circular polarization in the beam.
Non-zero circular polarization has not been measured in
The polarization state of light describes an anisotropy supernova, and no circular polarization observations were
in the time-averaged vibration of the electric field vector. taken for SN 2001el; therefore we will not discuss Stokes
Abeamofradiationwheretheelectricfieldvectorvibrates V inthispaper. Forscatteringatmosphereswithoutmag-
inonespecificplaneiscompletely(orfully)linearlypolar- neticfields,theradiativetransferequationforcircularpo-
ized. A beam of radiation where the electric field vector larizationseparatesfromthelinearpolarizationequations,
vibrates with no preferreddirection is unpolarized. Imag- allowingustoignoreV inourcalculations(Chandrasekhar
ine holding a polarization filter in front of a completely 1960).
linearly polarized light beam of intensity I0. The filter We further define the fractional polarizations: q =Q/I
only transmits the component of electric field parallel to and u = U/I. The degree of polarization, P, and the po-
the filter axis. Thus as the filter is rotated, the transmit- sition angle χ can then be written in terms of the Stokes
ted intensity, which is proportional to the square of the Parameters:
electric field, varies as I(θ)=I cos2θ.
0
Q2+U2
Thelightmeasuredfromastrophysicalobjectsisthesu- P = = q2+u2
perposition of many individual waves of varying polariza- p I (4)
tion. Imaginealightbeamconsistingofthesuper-position χ= 1tan−1(U/Q)= 1tapn−1(u/q)
oftwo completelylinearlypolarizedbeamsofintensityI0, 2 2
◦
and I , whose electric field vectors are oriented 90 to
90 A single plot that captures both the change of polar-
each other. If the beams add incoherently, the transmit-
ization degree and position angle over a spectrum is the
ted intensity is the sum of each separate beam intensity:
q-u plot of Figure 2. Each point in this figure is a wave-
I(θ)=I cos2θ+I cos2(θ+90◦) lengthelementofthe spectrum,andforeachpointwecan
0 90
(1) read off P and χ at that wavelength much as we would
=I cos2θ+I sin2θ
0 90 read a polar plot. According to Equation 4, the degree of
Ifthebeamsareofequalintensity,I0 =I90,thenthetrans- polarization P is given by the distance of the point from
mitted intensity shows no directional dependence upon θ the origin, while the position angle χ is half that of the
– i.e. the light is unpolarized. In this sense, we say that plot’s polarangle. Inthis senseq andu canbe thoughtof
the polarizationof a light beam is “canceled” by an equal as the two components of a two dimensional polarization
intensity beam of orthogonal – or “opposite” – polariza- quasi-vector.
tion. If I 6= I the cancellation is incomplete, and the
0 90
beam is said to be partially polarized. The degree of po-
2.2. Polarization in Supernova Atmospheres
larizationP isdefinedasthemaximumpercentagechange
4 D. Kasen et al.
The majoropacities in a supernovaatmosphere are due direction. (2)Thelightfromthephotospherelimbismore
to electron scattering and bound-bound line transitions. highlypolarizedthanthatfromthecenter. Thisisbecause
The continuum polarization of supernova spectra is at- the radiation field at the limb is highly anisotropic – i.e
tributed to electron scattering. The line opacity can cre- highly peaked in the outward (radial) direction. In ad-
ate features (either peaks or troughs) in the polarization dition, photons scattered into the line of sight from the
spectra. supernova limb, have generally scattered at angles closer
◦
To understand the polarizing effect of an electron scat- to 90 .
tering, note that an electron scatters a fully polarized Ifthe projectionofthe supernovaalongthe line ofsight
beam of radiation according to dipole sin2ψ angular dis- is circularly symmetric, as in Figure 3a, the polarization
tribution,whereψ istheanglemeasuredfromtheincident of each emergent specific intensity beam will be exactly
polarization direction. Now unpolarized light can be rep- canceled by an orthogonal beam one quadrant away. The
resented by a super-position of two equal intensity, fully- integrated light from the supernova will therefore be un-
polarizedorthogonalbeams. Uponelectronscattering,the polarized. A non-zeropolarizationmeasurementdemands
twodifferently orientedbeams getredistributedaccording some degree of asphericity; for example in the ellipsoidal
to differently oriented dipole patterns; thus in certain di- photosphere of Figure 3b, vertically polarized light from
rections they are no longer equal and do not cancel. The the long edge of the photosphere dominates the horizon-
scattered light is therefore polarized with the percent po- tally polarized light from the short edge. The integrated
larizationdepending upon the scattering angleΘ between specific intensity of Figure 3b is then partially polarized
incident and scattered rays: with q > 0. Because an axisymmetric system has only
1−cos2Θ one preferred direction, symmetry demands that the po-
P = (5) larization angle is aligned either parallel or perpendicular
1+cos2Θ
to the axis of symmetry, thus u = 0 for the geometry of
◦
Light scattered at 90 is fully polarized, while that which Figure 3b.
◦
is forward scattered at 180 remains unpolarized. The The effect of line opacity on the polarization spectrum
direction of the polarization is perpendicular to the scat- can be complicated. In general, light resonantly scattered
teringplanedefinedbytheincomingandoutgoingphoton in a line can become polarized in much the same way as
directions. described above for electrons. However because random-
Deepenoughwithinthesupernovaatmosphere,thelight izing collisions tend to destroy the polarization state of
becomes unpolarized for two reasons: (1) Below a certain an atom during an atomic transition, the light scattered
radius, known as the thermalization depth, the absorptive fromlinesinsupernovaatmospheresisoftenassumedtobe
opacity dominates the scattering opacity and photons are completelyunpolarized(e.g. Ho¨flichetal.(1996)–wedis-
destroyed into the thermal pool. The energy is subse- cuss this assumptionin more detailin §3.4). In ellipsoidal
quently re-emitted as blackbody radiation which, being models, it has been shown that the effect of depolarizing
the result of random collision processes, is necessarily un- lineopacityisprimarilytocreateadecreaseinthelevelof
polarized. (2) Deep within the atmosphere, the radiation polarizationinthespectrum(H¨oflichetal.1996). Because
field becomes isotropic. Because the radiationincident on SNIahavemorelines inthe blue,the polarizationinsuch
a scattereristhen equalinalldirections,the netpolariza- models typically rises from blue to red.
tion of scattered light will cancel. In general, however, the fact that a line is depolarizing
Thepolarizationoftheradiationoccursabovetheinner doesnotmeanitnecessarilyproducesadecreaseinthede-
unpolarized depth, where the election scattering opacity greeofpolarizationinthespectrum. Theactualeffectwill
dominatesandtheradiationfieldbecomesanisotropicdue depend sensitively upon the geometry of the line opacity
to the escapeofphotonsoutofthe supernovasurface. We andtheelectronscatteringmedium. Forexample,suppose
call this region the electron-scattering zone. The surface theelectron-scatteringregimeisspherical,butinanouter,
above the electron scattering zone at which point pho- detachedlayerthereisanasymmetricclumpoflineoptical
tons have a high probability of escaping the atmosphere, depth, as shown in Figure 3c. Because the line obscures
is the supernova photosphere. Formationof the well-know light of a particular polarization, the cancellation of the
P-Cygni line profiles in supernovae is due to line opac- polarization of the photospheric specific intensity beams
ity from material primarily above the photosphere. This willnotbecomplete. Thelinethusproducesapeak inthe
region is called the line-forming region. polarization spectrum and a corresponding absorption in
Figure 3 illustrates how the polarization of specific in- the flux spectrum. We call this effect of generating polar-
tensity beams emergent from an spherical, pure electron ization features the partial obscuration line opacity effect
scattering photosphere mightlook. The double-arrowsin- or just partial obscuration. In the case of Figure 3c, the
dicate the polarization direction of a beam, with the size clump primarily absorbs diagonally polarized light, so we
of the arrowindicating the degreeof polarization(not the expect the polarization peak to have a dominant compo-
intensity). Note the following two facts: (1) The polar- nent in the u-direction.
ization is oriented perpendicular to the radial direction. A non-axially symmetric supernova is shown in Fig-
This follows from nature of the anisotropy of the radia- ure 3d. The electron scattering medium is ellipsoidal, so
tion field. At all points in the atmosphere (except the the continuum spectrum will be polarized in the q direc-
center) more radiation is traveling in the radial direction tion. The clump of line opacity, which breaks the ax-
than perpendicular to it. Because the polarization from ial symmetry, preferentially obscures diagonally polarized
electronscatteringisperpendiculartothescatteringplane, light so the line absorption feature will be polarized pri-
thedominantscatteringofradiallytravelinglightwillpro- marily in the u direction. As we see in the next section,
duce an excess of polarization perpendicular to the radial
Spectropolarimetry of SN 2001el 5
this type of two-axis configuration is a relevant one for material,itdoesnotgreatlyaffectouranalysisoftheHVM
SN 2001el. feature.
2.3. The Polarization of SN 2001el 2.3.2. Polarization of The HVM Feature
2.3.1. Polarization of The Photospheric Spectrum The HVM flux absorption feature is associated with a
polarization peak in the spectrum (Figure 1b). Unlike
Theq-uplotofSN2001elisshowninFigure2. Inorder
the flux absorptionprofile, the polarizationpeak does not
to interpretthe intrinsicsupernovapolarization,onemust
showacleardoublefeature. Althoughthenoiseofthe po-
firstsubtractofftheinterstellarpolarization(ISP),caused
larization spectrum makes it difficult to analyze the line
by the scattering ofthe radiationoffasphericaldust grain
profile, it appears that a peak due to the red triplet line
along the way to the observer. The ISP has a very weak
(λ8662)isabsentorsuppressedcomparedtothebluelines
wavelengthdependence,(Serkowskietal.1975)andthere-
(λ8498 & λ8542).
forechoosingthemagnitudeanddirectionoftheISPisba-
InFigure2,thewavelengthscorrespondingtotheHVM
sicallyequivalenttochoosingthezeropointoftheintrinsic
feature are shown with closed circles. The HVM polar-
supernova polarization in the q-u plane of Figure 2. The
ization angle deviates from the photospheric one, point-
particular choice of ISP can dramatically affect the theo-
ing instead mostly in the u-direction. The HVM fea-
reticalinterpretationofthepolarizationdata(seeLeonard
ture also shows an interesting looping structure – as the
et al. (2000); Howell et al. (2001)).
wavelength is increased, the polarization moves counter-
ThechoiceoftheISPthatleadstothesimplesttheoret-
clockwisein the q-uplane. “q-uloops”suchas these have
ical description is shown as the green square in Figure 2.
been observed before, for example in the H-alpha feature
In this case the photospheric part of the spectrum (open
of SN 1987A (Cropper et al. 1988).
circles),apartfromsomescatter,drawsoutastraightline
The different polarization angle of the HVM feature
in the q-u plane – i.e. the degree of polarization changes
means that the geometry of SN 2001el cannot be com-
acrossthe photosphericspectrumbutthe polarizationan-
pletely axially symmetric. The Stokes U parameter
gle remains fairly constant. This would be the case if all
changes sign upon reflecting the system about the polar-
of the photospheric materialfollowed the same axial sym-
izationreference axis (see Equation3) andtherefore must
metry. The intrinsic polarization spectrum (i.e. percent
bezeroforanysystemwithareflectivesymmetry,suchas
polarization versus wavelength) of SN 2001el using this
the axially-symmetric system of Figure 3b. The non-zero
choice of ISP is shown in Figure 1b. The degree of po-
u-polarization can not solely be a kinematic effect either,
larizationrises from blue to red, as expected in ellipsoidal
foralthoughtheSNejectaisexpanding,thevelocitylawis
modelsduetothehigherlineopacityintheblue. Thelevel
supposedto be a spherical,homologousone(v ∝r) which
of continuum polarization in the red is about 0.4%, and
preserves the reflective symmetry. As the supernova ex-
the SiII6150linerepresentsadepolarizationbyaboutthe
pands andevolvesthe density contoursofthe systemmay
sameamount. Modelsofellipsoidalelectronscatteringat-
changeasouterlayersthinoutandrevealdifferentpartsof
mospheres indicate that level of polarization may roughly
the underlying material; however unless the velocity law
correspond to an deviation from spherical symmetry of
deviates from homology and shows some preferential di-
about 10% (H¨oflich 1991).
rection, the reflective symmetry will always be preserved
Althoughthe squareinFigure2isfavoredbysimplicity
and we must have u = 0 at all times. In order to get a
arguments, it is preferable to make a direct measurement
non-zero u component, we must break the reflective sym-
oftheISP,ifpossible. Atlateepochsitisbelievedthatthe
metry of the geometry with an off-axis component, such
supernova ejecta becomes optically thin to electron scat-
as the clump of Figure 3d.
tering. The intrinsic supernova continuum polarization
A natural explanation of the relatively large degree of
wouldthenbezero,andtheobservedpolarizationdueonly
polarizationand change of polarizationangle of the HVM
to the ISP. Paper I estimated the ISP in this way, using
feature is partial obscuration of polarized photospheric
observations taken on Nov 9. Assuming the intrinsic su-
light, somewhat like Figure 3d. We find in §4 that this
pernova polarization is zero at this time, the determined
interpretation can also account for the q-u loop. In the
ISP (with an estimated error contour) is shown as the
nextsectionwedescribeatechniqueforcalculatingpartial
green triangle in Figure 2 . Although the ISP thus deter-
obscuration that allows us to directly compare synthetic
mined is not grosslyinconsistentwith the simplest choice,
polarizationspectrato the data. Othermechanisms could
itseemstoindicatethatthepolarizationzeropointliesoff
presumably be invoked to explain the HVM polarization
of the main q-u line. If this is true, the angle across the
peak,butinthispaperweonlyconsiderthe effectsofpar-
photospheric spectrum is no longer constant. The pho-
tial obscuration.
tospheric material approximates an axial symmetry, but
an off-axis, sub-dominantcomponent (e.g. a photospheric
3. the two-component polarization model
clump) must existto accountfor the offsetofthe q-uline.
Becausethemainpurposeofthispaperistoexplorethe To compute polarization in multi-dimensions most in-
geometry of the HVM, not the photosphere, we will sim- vestigatorshaveemployedMonte Carlomethods (Code &
plify our discussion by ignoring any off-axis photospheric Whitney 1995; Wood et al. 1996; Ho¨flich 1991). This ap-
components. We will assume the polarization zero point proach has the benefits of generality and ease of coding,
oftheaxially-symmetriccomponentisgivenbythesquare butwiththedrawbackofextremecomputationalexpense.
and that the photosphere can be approximately modeled Averylargenumberofphotonsmustbefollowedtoescape
as anellipsoid. Although the paricularISP choicehas im- along each line of sight in order to overcome the random
portant implications for the geometry of the photospheric Poisson noise. This noise must be kept much less than
6 D. Kasen et al.
a fraction of a percent in order to confront the small ob- proximation is that the photon will only interact with a
served polarization levels. It is therefore cumbersome to line in the small region of the atmosphere where the pho-
use Monte Carlo codes in a parameterized way to explore ton is Doppler-shifted in resonance with the line. The
the huge parameter space available with 3-D geometries. radiative transfer problem then becomes localized to such
Inthecaseofthe HVM,asimplificationispossiblethat “resonance regions”. Free expansion is established in su-
allowsfor a much faster andmore insightfulcomputation. pernova atmospheres shortly after the explosion; the ve-
Assuming that the electron densities in the HVM regime locity vector at a point in the atmosphere is in the radial
are around 107cm−3, the optical depth to electron scat- direction and is given by ~v = (r−r )/trˆ, where r is the
0
tering through the HVM shell is τ = n σ R ≈ 10−3. radius at time t since explosion, and r is the initial ra-
es e t sh 0
Therefore one can ignore electron scattering in the HVM dius which is usually small and can be ignored. Consider
andtheradiativetransferproblemseparatesnaturallyinto a beam of radiation emanating from the photosphere and
the two regimes of photosphere and HVM. The photo- propagating through this atmosphere in the z direction,
sphere acts as a source of polarized light illuminating a at an impact parameter p and azimuthal angle φ. Such a
region of basically pure line optical depth in the HVM. beam was illustrated pictorially as a double-arrow in Fig-
Assuming the lines are depolarizing,the only effect of the ure 3; here we quantify it with a Stokes specific intensity
HVMistoobscuresomeofthepolarizedphotosphericlight vector I0(λ,p,φ). If the wavelength of the beam in the
andre-emitsomeunpolarizedlightintotheobserver’sline observer frame is λ, then the wavelength in the local co-
of sight. movingatmosphereframeisgivenbythe(non-relativistic)
Because the model makes a sharp distinction between Doppler formula:
an inner polarized source (the photosphere) and an outer ~v·zˆ z
λ =λ 1+ =λ 1+ (6)
line-forming region (the HVM), we call this approach the loc
c ct
two-componentmodel. Themodelisbasicallyawaytofor- (cid:18) (cid:19) (cid:18) (cid:19)
Suppose the only opacity in the atmosphere is due to one
malizethesimplepicturesofFigure3. Thetwo-component
line with rest wavelength λ . A beam of radiation will
0
model is constructed to apply to the detached layers of
come into resonance with the line when λ = λ , which
loc 0
the HVM. For line forming materialnear the photosphere
by Equation 6 is at a point:
a sharp separation of the two regimes would be artificial
z =ct(λ /λ−1) (7)
sinceelectronscatteringisnotentirelynegligibleintheline r 0
For each wavelength λ in an observed spectrum there is
forming region. Because the two-component model does
thus a unique point in the z-direction at which the beam
not account for the multiple scattering between lines and
comesinresonancewiththeline. AccordingtotheSobolev
electrons,photosphericspectrasynthesizedwithitmaybe
approximation, the emergent Stokes specific intensity I
incorrect. On the other hand because the model captures
that reaches the observerat infinity after passing through
some of the essential features of various geometries, some
the line forming region is given by:
qualitative insight may still be gained with respect to the
lines formed near the photosphere. As we are only con- I(λ,p,φ)=I0(λ,p,φ)e−τ +(1−eτ)S(λ,p,φ,zr) (8)
cerned with the HVM in this paper, this is not relevant where τ is the Sobolev line optical depth at the point
for the present work. (p,φ,zr) and S is the Stokes source-function of the line
at this point. Both quantities will be explained further in
the sections to come. The first term in Equation 8 rep-
3.1. The Sobolev Approximation
resents photospheric light attenuated by the line optical
The Sobolev approximation is a method for computing depth; the second term represents light scattered or cre-
line formation in atmospheres with large velocity gradi- atedtoemergeintothelineofsightbytheline. Equation8
ents. Sobolev models (under the assumption of a sharp is identical to the usual, unpolarized expression for the
photospherepluslineformingregion)havefrequentlybeen Sobolev approximation (see Rybicki & Hummer (1978)),
used to analyze supernova flux spectra. Typically spheri- except now the terms in boldface are all Stokes vectors.
calsymmetryisassumed(e.g. (Branchetal.1983;Hatano Togeneratetheobservedspectrumofanunresolvedob-
et al. 1999)) but the method has also been applied in 3D ject,thespecificintensityofEquation8mustbeintegrated
(Thomas et al. 2002). Derivations of the Sobolev method overthe projectedsurface ofthe atmosphere,i.e. overthe
and justification of the approximation in the modeling of p−φ plane. A wavelength λ in the observed spectrum
supernova atmospheres can be found in (Rybicki & Hum- thusgivesusinformationaboutthe lineopticaldepthand
mer 1978; Castor 1970; Jeffery & Branch 1990); here we sourcefunctionintegratedoveraplaneatz . Suchaplane,
r
only quote the important results. which is perpendicular to the observer’s line of sight, is
The geometry used in the models is shown in Figure 4. called a constant-velocity (CV) surface.
Weuseacylindricalcoordinatesystem,(p,φ,z)oralterna- In the case of an monotonically expanding atmosphere
tively aCartesianone(x,y,z). Ineithercasethe observer withmorethanoneline,abeamofradiationwillcomeinto
lineofsightischosenasthezaxiswithzdecreasing toward resonance with each line one at a time, starting with the
theobserver(i.e. theobserverisatnegativeinfinity). The bluestline andmovingto the red. In this case Equation8
polarizationreference axis is chosento lie along the φ=0 is readily generalized:
(or y) direction, which is also the photosphere symmetry N
axis. I(λ,p,φ)=I0(λ,p,φ)exp − τi
For atmospheres in general expansion, such as super- (cid:18) i=1 (cid:19)
X (9)
novae, the wavelength of a propagating photon is con- N i−1
stantly redshifting with respect to the local comoving + Si(λ,p,φ)[1−eτi]exp − τj
frame of reference. The insight behind the Sobolev ap-
i=1 (cid:18) j=1 (cid:19)
X X
Spectropolarimetry of SN 2001el 7
wheretheindicesiandjrunoverthelinesfromredtoblue. of additional opacities in the photospheric regime will de-
Before considering the integration of Equation 9 over the crease the polarization from the pure electron scattering
CV planes, we discuss in more detail the terms I0, S, and results presented here.
τ. Using a Monte Carlo code, we computed the functions
I (p) and P (p) for the above scenario. Unpolarized pho-
z z
3.2. The Photospheric Intensity tons were emitted isotropically from the inner boundary
surface. Thepolarizationofthesephotonsweretrackedas
In this section we calculate the intensity and polariza-
they scattered multiple times through the electron scat-
tion of specific intensity beams emergentfroman electron
scattering photosphere. We first consider I0(p,φ) in the tering zone. Photons that were back-scattered onto the
inner boundary surface were assumed to be re-absorbed
casethatphotosphericregimeisspherical(asinFigure3a)
and were omitted from the calculation. The Monte Carlo
and later show how to adapt the result to the ellipsoidal
codeusedinthiscalculationisanewonedevelopedtofur-
case. From the circular symmetry, the intensity and de-
ther study supernova polarizationin cases where the two-
gree of polarization of a specific intensity beam can only
componentmodelisnotapplicable. Adetaileddescription
depend upon the impact parameter p and not on φ. Let
of the Monte Carlo code will be presented in a future pa-
I (p) represent the specific intensity in the zˆ direction at
z
per. Wenotethattheoutputhasbeencheckedagainstthe
p, and P (p) the degree of polarization of this beam. The
z
resultsofChandrasekhar(1960)andCassinelli&Hummer
polarized specific intensity is I (p)P (p) which will be di-
z z
(1971),andseveralothercasesincludingHillier(1994)and
vided between the Q and U Stokes parameters.
the analytic results of Brown & McLean (1977).
For φ = 0, the polarization points in the horizontal, or
The computed functions I (p) and P (p). are shown in
negative Q direction – i.e. Q(p,φ = 0) = −I (p)P (p) z z
z z
Figure 5. Here p is given in units of the photosphere ra-
while U(p,φ=0)=0. The Q and U components at arbi-
dius, defined as the radius at which the optical depth to
trary φ are derived by rotating this expression by φ. The
electron scattering equals 1. The intensity and polariza-
resulting Stokes vector is:
tion for p < 1 do not differ much from the plane-parallel
I I (p)
0 z case, with P = 13% at p = 1. The photospheric spe-
I0 = Q0 = −Pz(p)Iz(p)cos(2φ) (10) cific intensityzdoes not, however,terminate sharply at the
U ! −P (p)I (p)sin(2φ)!
0 z z photosphericradiusasisusuallyassumedinSobolevmod-
The fact that the trigonometric rotationterms depend on els; rather a significant amount of light is scattered into
2φ rather than φ reflects the fact that the polarization is the line of sight out to p ≈ 1.4. Since this limb light is
actually a quasi-vector (Chandrasekhar 1960). highly polarized (up to 40%) it is important to include it
Inthe two-componentmodelone mustpre-computethe in our calculations. Actually most of the polarized flux
functions Iz(p) and Pz(p). Chandrasekhar first obtained comes from an annulus at the edge of the photosphere.
the result for a pure electron scattering, plane-parallelat- I (p) has become negligible out at the HVM distances of
z
mosphere(Chandrasekhar1960);inthatcaseIz(p)follows p≈2,whichconfirmsthatwecanmakeaclearseparation
closely the linear limb darkening law, while the degree of between the photospheric and HVM regimes.
polarization Pz(p) rises from zero in the center to 11.2% In Figure 5 we also compare the n = 7,τes = 3 results
at the limb; however, the plane-parallel approximation is to other models with differing density laws and optical
notagoodoneforsupernovae,whichhaveextendedatmo- depths. From the similarity of the n=7 and n =5 mod-
spheres (i.e. the thickness of the electron scattering zone elsinFigure5aand5bitisclearthatthecalculationswill
isasizablepercentageofitsradius). Inanextendedatmo- not depend sensitively on our choice of power law index.
sphere the radiationfield tends towardamore anisotropic Eveniftheindexwereaslowasn=3,(orworse,noteven
distribution, peaking in the outward direction. This in- describedby a strictpowerlaw)the behaviorofI (p) and
z
creasedanisotropyofthe radiationfield leads to generally P (p) should still show the same qualitative trends. From
z
higher limb polarizations. Cassinelli & Hummer (1971) Figure 5c and 5d we see the results also do not depend
solved the polarized radiative transfer Equation for ex- much on τ as long as τ &3.
es es
tended, spherical electron scattering spheres with density The results givensofarhavenottakenintoaccountthe
power laws of index n=2.5 and n=3. They find the polar- asphericity of the photosphere in SN 2001el. One could
ization can become higher than 50% at the limb. redo the Monte Carlo calculations for various axisymmet-
We modelthe photosphericregimeas aninner unpolar- ric configurations, but the small degree of polarization in
izedboundarysurface,surroundedbyapureelectronscat- SN2001elsuggestsa rathersmall(∼10%)deviationfrom
teringenvelopewithapowerlawelectrondensityρ∝r−n. spherical symmetry, so it is not a bad approximation to
We choose n = 7, a density law motivated by SN Ia ex- apply the spherically symmetric specific intensities to a
plosion models and one that has been often used in di- slightly distorted photosphere. This technique of using
rect spectral analysis (Nomoto et al. 1984; Branch et al. spherical results to calculate the polarization from dis-
1983). The optical depth (in the radial direction) from tortedatmosphereshas been used, invarious manners,by
the inner boundary surface to infinity is set at τes = 3. many other authors (Shapiro & Sutherland 1982; McCall
The assumption of a pure electron scattering atmosphere 1984; Jeffrey 1991; Cassinelli & Haisch 1974).
should be a good one for the wavelength range we are in- Inourmodelswewillonlyconsiderthecaseofanoblate
terestedin. Thephotonsthatredshiftintoresonancewith ellipsoidalatmosphere with axis ratio E and viewededge-
the high velocity IR triplet are those with wavelengths on. We define an ellipsoidal coordinate:
from 8000-8500 ˚A, and there are no strong lines or ab-
η = x2+E2y2 (11)
sorptiveopacitiesinthisregionofthespectrum(seePinto
& Eastman (2000)). At other wavelengths the presence Our approximation is that the emergent Stokes inten-
p
8 D. Kasen et al.
sity from a position η,φ is given by Equation 10 with the effect of collisions. After a photon has excited the
I (p=η,φ=φ)andP (p=η,φ=φ). Inthiscasewefind atom,the atomisinapolarizedstatewithaspecificmag-
z z
an axis ratio of E ≈ 0.9 is necessary to produce the 0.4% neticsublevelM.Ifthecollisionaltimescaleisshorterthan
polarization observed in the red continuum of SN 2001el. thelifetimeofthetransition,collisionswilldestroythepo-
This result agreeswith previous,2-D calculations (Jeffrey larizationstateoftheatombyredistributingtheatomover
1991; Ho¨flich 1991). all the nearly degenerate magnetic sublevels, thereby pro-
While the above photospheric model provides a sim- ducing an spherically symmetric configuration. The scat-
ple and rather general description of an axially symmet- tered light will thus be isotropic and unpolarized. This is
ric photosphere, there is no easy way to assure ourselves theassumptionmadeinthemodelsofHowelletal.(2001)
that this photospheric model is unique. The actual spe- (and references therein).
cificintensityemergentfromanellipsoidalatmospherecan In this paper we use exclusively an isotropic, unpolar-
dependonthedepthandshapeoftheinnerboundarysur- ized line source function. In addition to the depolarizing
face, as well as the inclination of the system. Moreover, effectofcollisions,wesuggesttwofurtherreasonswhythe
thepolarizationofthephotosphericspectrumofSN2001el effect of intrinsic line polarization is likely a small effect
could arise from a different kind of asphericity altogether, in the case of the HVM feature. (1) If we evaluate the
for instance an off-center Ni56 source, or a clumpy atmo- polarizability factor for the lines of the IR triplet we find
sphere. In the absence of a single preferred photospheric that W is almost zero for λ8542 (W = 0.02) and ex-
2 2
model, we proceed with the above model, but reiterate actly zero for λ8662. Accordingto the Hamilton prescrip-
thatitremainsjustoneofmanypossiblescenarios. Other tion, only the λ8498 line has a moderate polarizing effect
choices of I (p,φ) and P (p,φ) must be investigated on a (W =0.32),butthislineisbyfartheweakestofthethree.
z z 2
case by case basis. Note however that since the IR triplet lines are not res-
onance lines, the Hamilton prescription does not strictly
3.3. The Line Optical Depth apply and complicated NLTE polarizing effects could be
operative (Trujillo Bueno & Manso Sainz 1999). (2) For
In our synthetic spectra fits, we take the optical depth
opticallythicklines,photonswillmultiplescatterwithina
of the λ8542 line, as a free parameter τ . The optical
1
resonance regionbefore escaping. On averagethe number
depths of the other two lines (λ8662, λ8498) are derived
of scatters in the resonance regionis givenby N =1/P
from τ . All three triplet lines come from nearly degen- esc
1 where the escape probability P is given by the Sobolev
erate lower levels, so in LTE the relative strength of each esc
formalism:
linedependsonlyupontheweightedoscillatorstrengthgf 1−e−τ
of the atomic transition. Even if the level populations de- P = (12)
esc
τ
viatefromLTE,oneexpectsthedeviationtoaffecteachof
the nearly degenerate levels in the same way. The λ8542 This multiple scattering has two depolarizing effects: (1)
line has the largest gf value; λ8662 is 1.8 times weaker, theradiationfieldinthelinetendstowardanisotropicdis-
and λ8498 10 times weaker. tribution(2)theprobabilityofthedestructionofaphoton
intothethermalpoolwillbeincreased. Foropticallythick
3.4. The Line Source Function lines the line-scattered light will then tend to be unpolar-
ized. On the basis of the spectral fits of §4, we will argue
The line source function represents light scattered by
that the lines of the IR triplet are saturated (τ & 5) for
1
the line, created from the thermal pool or from NLTE ef-
the HVM in front ofthe photosphere andthus largelyun-
fects. Scatteringinalinecanpolarizelight–asinthecase
polarized.
of electron scattering, the effect is due to the anisotropic
Foranisotropic,unpolarizedsourcefunctionthe Stokes
redistribution ofthe different polarizationdirections. The
vector is:
angular redistribution depends in general on the angular S S
I 0
momentum J of the upper and lower levels of the atomic S= S = 0 (13)
Q
transition. S ! 0!
U
Hamilton(Hamilton1947)hasconsideredthelinearpo-
where S is the unpolarized source function. The actual
larization from a resonance line, free from collisions. He 0
valueofS requiresafullNLTEcomputationoftheatomic
showedthatthe angularredistributionfunctionfromsuch 0
levels. For our purposes a useful parameterizationis:
alinecouldbewrittenasthesumofanisotropicanddipole
S =(1−ǫ′)J¯+ǫ′B(T) (14)
term,therelativecontributionsdependingupontheangu- 0
lar momentum of the transition levels. The dipole contri- The first term represents impinging light scattered by the
butionhasexactlythesamepolarizingeffectasanelectron line,andsodependsuponthemeanlocalradiationfieldin
scattering,while the isotropiccontributionis unpolarized. the line J¯; the second term represents light created from
Thefinalpolarizingeffectisthusgenerallydilutedascom- the thermal pool and so depends upon the Planck func-
paredtotheelectronscatteringcase,andcanbedescribed tionB andthetemperatureT. Therelativeimportanceof
′
by apolarizabiltyfactor W , whichvariesfrom0 for a de- the twofactorsis governedby ǫ, the probabilitya photon
2
polarizinglineto1foralinethatpolarizeslikeanelectron is destroyed into the thermal pool on traversing the reso-
′
(Stenflo1994). BecausetheHamiltonapproachprovidesa nance region of a line. In the Sobolev approximation ǫ is
simple prescription for estimating the intrinsic polarizing given by:
ǫ
effects of line scattered light, it has often been used out- ǫ′ = (15)
P +ǫ(1−P )
side its scope to calculate polarized line profiles for non- esc esc
resonance lines (Jeffrey 1991). where ǫ is the usual static atmosphere destruction prob-
The Hamilton prescription does not take into account ability. In NLTE models of supernova atmospheres one
Spectropolarimetry of SN 2001el 9
finds ǫ between 0.05 and 0.1 (Nugent 1997). Note as quantities we measure: F (λ),F (λ), and F (λ). What
I Q U
the probability of a photon’s escape (P ) decreases, the we do measure can be thought of as certain “moments”
esc
′
chances that it gets thermalized (ǫ) increases. of the τ distribution over each CV plane. F is a type of
I
For the value of J¯ in the HVM, we use the radiation “zeroth moment”, which depends mostly upon how much
incident from the photosphere, ignoring multiple scatter- material is covering the photosphere, with little depen-
ingofphotonsbetweenthetripletlines(foradiscussionof dence on its geometrical distribution. On the other hand
this approximation, see Thomas et al. (2002)). The pho- the F and F , because of the cos2φ and sin2φ factors,
Q U
tospheric radiation in the HVM is geometrically diluted behavesomewhatlike“firstmoments”,andaresensitiveto
by a factor of roughly πr2 /4πr2 ≈ 1/16. Thus for howτ isdistributedoverthephotosphere. Becausethean-
ph HVM
a pure scattering line (ǫ′ = 0), the intensity of the line gle factors cos2φ,sin2φ are rather low-frequency, smaller
sourcefunction isabout16times weakerthanthe average scale structures will be averaged out over the integrals,
photospheric intensity. At the other extreme, for a ther- andthe polarizationmeasurementswillonlyconstrainthe
malizedline (ǫ′ =1)andanHVM temperatureof5500K, large scale structures in the HVM.
the line source function is about 4 times weaker than the Before proceeding with the spectral synthesis calcula-
average photospheric intensity. tions let us summarize the assumptions that go into the
Becausethelinesourcefunctionlightisunpolarizedand two-component model. (1) The electron scattering opac-
relatively weak, we find in the end that it has little affect ity in the HVM is negligible. (2) the photospheric regime
on the synthetic line profiles. The exact value of ǫ is thus is reasonably well described by a pure electronscattering,
not of great importance. In our models, we use ǫ=0.01. power law atmosphere, surrounding a finite, unpolarized
source at τ ≈ 3. (3) For small (∼10 percent) devia-
es
3.5. The Integrated Spectrum tions from sphericity in the photosphere, the angular de-
pendence of the polarized radiation field does not deviate
To obtain the observed Stokes fluxes at a certain wave-
significantly from the spherical results (4) The line source
length one must integrate the specific intensity over the
functionlightisunpolarized(5)Multiplescatteringamong
CV planes of each line. For those CV planes behind the
the triplet lines and between the HVM and photospheric
photosphere, we must also account for the attenuation of
regime can be ignored.
the line source function light due to scattering off elec-
tronsasthebeampassesthroughthephotosphericregion.
4. the geometry of the high velocity material
If we define τ (p,φ,z) as the electron scattering optical
es
depth along the z-direction from the point (p,φ,z) to the The speed of the two-component model allows us to
observer, then a fraction (1 − e−τes) of photons will be explore many different configurations for the HVM. We
scattered out of the line of sight on their way to the ob- report on four possibilities here, each of which may ap-
server. Weassumethesephotonsaresimplyremovedfrom proximatea structure thatis the resultofsome particular
the beam and are not subsequently re-scattered into the physicalmechanism: (1)Asphericallysymmetricshell(2)
line of sight. Anellipsoidalshellwithanaxisofsymmetryrotatedfrom
Forasinglelineatmosphere,theintegratedStokesfluxes the photosphere axis of symmetry. (3) A clumped spher-
atwavelengthλcorrespondtomaterialfromtheCVplane ical shell (4) A toroidal structure with a symmetry axis
zr and are given by: rotated with respect to the photospheric axis. The geom-
etry used in the models is shown in Figure 4.
FI(λ)= Iz(p,φ)e−τ+ The photosphere is modeled as discussed in §3.2, as an
Z Z (cid:20) oblate ellipsoid with axis ratio E = 0.91, viewed edge-
(1−e−τ)S (p,φ,z )e−τes pdpdφ on. It is not the purpose of this paper to explore the
0 r
detailed geometry of the photosphere, therefore the ellip-
(cid:21) (16)
soidal model was chosen as the simplest possibility that
FQ(λ)= Pz(p,φ)Iz(p,φ)cos(2φ)e−τpdpdφ captures the essential features of the axisymmetric pho-
Z Z tosphere. The photosphere symmetry axis is the y-axis,
F (λ)= P (p,φ)I (p,φ)sin(2φ)e−τpdpdφ which is also the polarization reference direction. The
U z z
photospheric intensity is assumed to follow a blackbody
Z Z
◦
Theintegralscanbeeasilygeneralizedforthecaseofmul- distribution with a temperature T = 9000 K chosen to
bb
tiple lines by applying Equation 9. fit the slope of the red continuum. We do not attach any
Given our scenario of how the high velocity CaII po- physical significance to the value of T , but consider it
bb
larization is formed by partial obscuration, Equations 16 only a convenient fit parameter.
give us some insight into what extent the HVM geometry The parameterizationofthe variousHVMgeometriesis
isconstrainedbythepolarizationmeasurements. Forsim- keptsimple andgeneral. The HVM is chosenaxiallysym-
plicity, consider the formation of a single, unblended line, metric, with the orientation of the HVM axis defined by
above a spherical photosphere, and suppose we are try- the two angles γ and δ. The velocities v and v denote
1 2
ing to reconstruct the distribution of Sobolev line optical theinnerandouterradialboundariesoftheHVM,whileψ
depth τ(p,φ,z) over the entire ejecta volume. The Stokes is the opening angle (see Figure 4). The reference optical
flux at a certain wavelength gives us information about τ depth τ of the λ8542 line is assumed constant through-
1
over the corresponding CV plane at z . As Equations 16 out the defined structure boundaries. Although this is an
r
demonstrate we obviously will not be able to uniquely re- idealization of the real HVM, it allows us to isolate the
construct the distributionof τ overthis plane, because all defining geometrical features of each structure individu-
of the information gets integrated over to give the three ally. Table 1 summarizes the fitted parameters of each
10 D. Kasen et al.
HVM geometry considered in the sections to follow. Be- redtripletline (with agf value 1.8times larger),the blue
fore considering the specific models, we first discuss the minima of the IR triplet feature will be about twice as
generalconstraintsthatmustbe metbyanyHVM model. deep the red one unless both lines are saturated. Because
the minima inthe HVM feature are ofabout equaldepth,
4.1. General Constraints we conclude that the two lines are indeed saturated (i.e.
τ &5) and the z-plane covering factor is in fact the min-
Figure 6 is a diagram of the formation of the CaII IR 1
imal one, f =43%.
tripletfeatureinSN2001el. TheHVMhasforillustration min
(3) The shape of the flux profile may also constrain the
beenshownasasphericalshell. Theatmospherecanbedi-
value τ . Note that two minima in the flux profile have
videdintothreeregions,thehigh-velocitymaterialineach 1
roughlyequalwidths. Ontheotherhandifallthreetriplet
region having a different affect on the spectrum. (1) The
lines are saturated the blue minima will tend to be wider
absorption region: Material in the tube directly in front
than the red, due to the blending of the λ8498 with the
of the photosphere absorbs photospheric light and emits
λ8542line. Thissuggeststhattheλ8498lineisweakwhile
line source function light into the line of sight. Since the
theothertwolinesarestrong,asituationthatoccurswhen
line source function intensity is usually weaker than the
τ ≈5.
photospheric intensity, this effect produces an absorption 1
(4) Finally, the HVM polarization feature points pri-
featureinthespectrum. (2)Theemissionregion: material
marily in the u-direction. This means the distribution of
in the outer lobes does not obscure the photosphere but
◦
theHVMisweightedalongthe45 linetothephotosphere
onlyaddslinesourcefunctionlight;thisproducesanemis-
symmetry axis.
sionfeaturetothe redofthe absorption. (2)Theoccluded
region: Material in the tube behind the photosphere is
occluded by the photosphere and is not visible. 4.1.2. Constraints on the Emission Region Material
Because in our models it is the partial obscuration of
The material in the emission region may be observable
polarized photospheric light that gives rise to the HVM
as a flux emission feature to the red of the HVM absorp-
polarizationfeature, all of our geometricalinformationon
tion. If, for example, the HVM was a spherical shell, this
the HVM will be about the distribution ofCaII in the ab-
emission feature would extend from about z = −20,000
sorption region. Whether there is any HVM CaII in the to z = 20,000, or over 1000 ˚A. The emission from a shell
emission region,and if so, what its geometry may be, will
is then verybroad,but because the line sourcefunction is
be very difficult to say. In addition we will have abso-
much less than the photospheric intensity, the feature is
lutely no information about the material in the occluded
typically weak and difficult to detect in the spectrum. A
region. In the spherical HVM shell of Figure 6, about 5%
serious problem, evident from Figure 6, is that the HVM
of the material is in the absorption region, 5% is in the
emission feature overlaps with the photospheric triplet
occluded region, and 90% is in the emission region. Thus
absorption and emission, making it difficult to separate
we only probe a small portion of the potential HVM. We
the two contributions. Only for the HVM material with
now consider the general constraints of these regions in z & 15,000 (i.e. λ > 8700 ˚A) is the HVM emission fea-
more detail.
turenotblendedwiththephotospheric. Unfortunatelythe
available spectra of SN 2001el do not extend that far to
4.1.1. Constraints on the Absorption Region Material
the red.
We can list 4 general constraints on the HVM absorp- The emission region material also affects the polariza-
tion region material that are directly deducible from the tion levelby diluting the photospheric light with unpolar-
Sept. 25 spectra: ized line source function light, thus creating a depolariza-
(1) The width of the HVM flux absorption feature con- tionfeatureinthespectrum. Ofcoursethisdepolarization
strainsτ tobenon-zeroonlyovertheline-of-sightvelocity feature gives no additional clue as to the orientation of
1
range18,000−25,000kms−1. τ isthusconfinedtoarel- the emission region material, as the unpolarized line light
1
atively thin region that is significantly detached from the carries no directional information. The polarization spec-
photosphere. The edges ofthe flux feature are sharp,sug- trum of SN 2001el does have a significant depolarization
gesting that the boundaries of the HVM are well-defined. to the red of the HVM peak, but since the overlapping
(2)AttheminimumoftheHVMabsorptionthefluxhas photospheric triplet feature may also depolarize at these
decreased by 43% from the continuum level. For geome- wavelengths, it is again not easy to use this to directly
trieswheretheHVMcoverstheentirephotosphere,theop- constrainthe HVM emissionregionmaterial. Inourmod-
ticaldepthimpliedisτ ≈.8. Onthe otherhand,somege- els,wedonotattempttofittheregionredwardof8200˚A,
ometriesmayhavehigheropticaldepthsandsmallercover- where the HVM feature is blended with the photospheric
ingfactors,theminimalcoveringfactorbeingf =43% feature.
min
forwhenthelinesarecompletelyopaque. Notethatinthis We find that the red emission/depolarization feature is
context the term “covering factor” denotes the percent of not a very sensitive diagnostic of emission region mate-
the photospheric area obscured by the slice of HVM on a rial. The effectonthe spectrumis showninFigure7 fora
plane perpendicular to the line of sight, corresponding to sphericalshellwithvariousvaluesofthedestructionprob-
the resonance surface of a certain wavelength. Since this ability ǫ. For a pure scattering line (ǫ=0) the emissionis
differsfromthetraditionalusageoftheterm,wehereafter hardlyvisible. Forthethermalizedline(ǫ=1)andatem-
call this the z-plane covering factor. perature T = 5500K, the emission would be substantial
We can use the double-dipped flux profile to constrain butdifficulttoseparatefromthephotosphericcomponent.
thez-planecoveringfactoroftheHVM.Becausetheλ8542 A value ǫ = 1 is also unlikely for supernova atmospheres;
blue triplet line is intrinsically stronger than the λ8662 NLTE models find ǫ≈0.05.