Multiepoch VLT-FORS spectro-polarimetric observations of supernova 2012aw reveal an asymmetric explosion
Luc Dessart, Douglas C. Leonard, D. John Hillier, Giuliano Pignata
AAstronomy & Astrophysics manuscript no. ms © ESO 2021January 5, 2021
Multi-epoch VLT − FORS spectro-polarimetric observations ofsupernova 2012aw reveal an asymmetric explosion
Luc Dessart , Douglas C. Leonard , D. John Hillier , and Giuliano Pignata , Institut d’Astrophysique de Paris, CNRS-Sorbonne Université, 98 bis boulevard Arago, F-75014 Paris, France. Department of Astronomy, San Diego State University, San Diego, CA 92182-1221, USA. Department of Physics and Astronomy & Pittsburgh Particle Physics, Astrophysics, and Cosmology Center (PITT PACC), Univer-sity of Pittsburgh, 3941 O’Hara Street, Pittsburgh, PA 15260, USA. Departamento de Ciencias Fisicas - Universidad Andres Bello, Avda. Republica 252, Santiago, 8320000 Chile. Millennium Institute of Astrophysics (MAS), Nuncio Monseñor Sotero Sanz 100, Providencia, Santiago, Chile.Received; accepted
ABSTRACT
We present VLT − FORS spectropolarimetric observations of the type II supernova (SN) 2012aw taken at seven epochs during thephotospheric phase, from 16 to 120 d after explosion. We correct for the interstellar polarization by postulating that the SN polarizationis naught near the rest wavelength of the strongest lines – this is later confirmed by our modeling. SN 2012aw exhibits intrinsicpolarization, with strong variations across lines, and with a magnitude that grows in the 7000 Å line-free region from 0.1 % at 16 d upto 1.2 % at 120 d. This behavior is qualitatively similar to observations gathered for other type II SNe. A suitable rotation of Stokesvectors places the bulk of the polarization in q , suggesting the ejecta of SN 2012aw is predominantly axisymmetric. Using an upgradedversion of our 2D polarized radiative transfer code, we model the wavelength- and time-dependent polarization of SN 2012aw. Thekey observables may be explained by the presence of a confined region of enhanced Ni at ∼ − , which boosts the electrondensity in a cone having an opening angle of ∼
50 deg and an observer’s inclination of ∼
70 deg to the axis of symmetry. With thisfixed asymmetry in time, the observed evolution of the SN 2012aw polarization arises from the evolution of the ejecta optical depth,ionization, and the relative importance of multiple versus single scattering. However, the polarization signatures exhibit numerousdegeneracies. Cancellation e ff ects at early times imply that a low polarization may even occur for ejecta with a large asymmetry. Anaxisymmetric ejecta with a latitudinal dependent explosion energy can also yield similar polarization signatures as an asymmetry inthe Ni distribution. In spite of these uncertainties, SN 2012aw provides additional evidence for the generic asymmetry of type II SNejecta, of which VLT − FORS spectropolarimetric observations are a decisive and exquisite probe.
Key words. radiative transfer – polarization – supernovae: general – supernova: individual: SN 2012aw
1. Introduction
Photometry and spectroscopy provide critical information ontype II supernova (SN). They characterize, for example, the timescale over which the energy stored in the ejecta is released, thecolor of the escaping radiation, the level of ionization of thespectrum formation region, and the composition of the ejecta.Such data do not, however, provide direct constraints on the ge-ometry of the explosion. Lacking spatially resolved observationsfor type II-Plateau (II-P) SNe, spectropolarimetry is the besttechnique to provide constraints on the morphology of SN ejecta,especially at early times (Shapiro & Sutherland 1982). The po-larization is linear, caused by scattering with free electrons, andis typically less than 1% in the continuum (Wang & Wheeler2008). The current spectropolarimetric dataset of type II SNeis limited to 10 −
20 objects, mostly because of the scarcity ofnearby events. Indeed, high signal-to-noise ratio observations arerequired to reveal such a low-level of polarization.SN 1987A is a hydrogen-rich, core collapse SN classified astype II-peculiar. Unlike type II-P SNe, which arise from red-supergiant star explosions, the progenitor of SN 1987A was ablue supergiant star. It is the first SN for which multiepoch multi-band polarimetric observations were obtained (Mendez et al.1988; Vidmachenko et al. 1988; Barrett 1988; Cropper et al.1988; Je ff ery 1991b; Wang et al. 2002). The 0.5% continuum polarization level observed in the optical during the first twomonths is understood to arise from scattering with free electronsin a prolate or oblate ejecta (Hoflich 1991; Je ff ery 1991a). Thepolarization angle shows variations in time and across lines, sug-gesting some departures from axial symmetry. Overall, an asym-metric distribution of Ni may also account for the observedpolarization (Chugai 1992).Subsequently, spectropolarimetric observations have beengathered for a number of nearby type II SNe, but now of theplateau type, starting with SN 1999em (Leonard et al. 2001). ForSN 2004dj, Leonard et al. (2006) collected spectropolarimetricobservations throughout the photospheric and the nebular phase,up to about a year after explosion. The intrinsic polarization ofthe SN was revealed unambiguously by the strong variation inpolarization across lines. Furthermore, the continuum polariza-tion evolved from being small during the photospheric phase tobeing maximum at the onset of the nebular phase, subsequentlydecreasing as the inverse of the time squared. Chugai (2006) pro-posed that the polarization was associated with the excess exci-tation and ionization originating from the presence of high ve-locity Ni fingers or blobs. Although intimately associated withejecta asymmetry, Dessart & Hillier (2011) suggest that the po-larization peak at the onset of the nebular phase is caused by aradiative transfer e ff ect as the ejecta turns optically thin, while Article number, page 1 of 19 a r X i v : . [ a s t r o - ph . H E ] J a n & A proofs: manuscript no. ms its subsequent evolution reflects the drop in ejecta optical depth.Chornock et al. (2010) presented spectropolarimetric observa-tions for a few more type II SNe, confirming the general trendobserved for SN 2004dj. It thus appears that type II SNe typicallyshow a temporal increase in continuum polarization through thephotospheric phase, reaching a maximum at the onset of the neb-ular phase, and subsequently dropping.To supplement this dataset, we initiated a VLT − FORS spec-tropolarimetric program in 2008. Over a number of years, wegathered multiepoch high quality observations for SNe 2008bk,2012aw, and 2013ej, as well as a more sparse dataset for a fewadditional objects. Although this dataset has been presented in aterse format previously (Leonard et al. 2012, 2015), we are nowin a position to present the observations in more detail, and withmodeling results. We start in this paper with SN 2012aw.In the next section, we summarize the results from the polar-ization modeling of Dessart & Hillier (2011). Section 3 presentsthe numerical approach used for the present study, which is alsopresented in detail in Hillier & Dessart (in prep.). As in Dessart& Hillier (2011), we adopt a 2D axisymmetric ejectum but wenow model the polarized spectrum for the whole optical rangerather than focus on isolated lines and the overlapping contin-uum. In Section 4 we present the spectropolarimetric observa-tions of SN 2012aw that we collected with VLT − FORS whilesection 5 presents the results of our modeling of SN 2012awcovering both the photospheric and the nebular phase. In Sec-tion 6, we investigate the degeneracy of the polarization resultsby investigating the impact of our assumptions (magnitude of theasymmetry, adoption of the mirror symmetry with respect to theequatorial plane, and nature of the asymmetry). We present ourconclusions in Section 7.
2. Summary of results from Dessart & Hillier (2011)
Dessart & Hillier (2011) studied the linear polarization in 2Daxisymmetric Type II SN ejecta using a long-characteristic codeas well as a Monte Carlo code. They explored how the intrin-sic continuum and line polarization change with wavelength,bound-bound transition, albedo, SN age, or some properties ofthe asymmetry. Below, we summarize their results since they arerelevant for the present study and the interpretation of polariza-tion signatures in general.Even though electron scattering produces a gray opacity, theabsorption opacity varies between photoionization edges andcauses a variation of the albedo with wavelength. Consequently,the continuum polarization in type II SNe may vary across theoptical range. In the modeling, we take this e ff ect into account byseparating the di ff erent sources of opacity at each wavelength,namely electron scattering on the one hand and on the otherhand the bound-bound and bound-free processes (see also Hillier1994, 1996). Scattering in lines (i.e., bound-bound transitions),and especially resonance lines, is assumed to be unpolarized.In reality, line scattering has a polarizing e ff ect but it is muchweaker than the polarizing e ff ect caused by free electrons (Jef-fery 1991a).The residual polarized flux arises from a non-cancellationof the local polarization integrated on the plane of the sky. Anet polarization can result from an asymmetric distribution ofscatterers, but it can also stem from an asymmetric distribution ofthe flux (for example due to optical-depth e ff ects; Hillier 1994;Dessart & Hillier 2011; Vlasis et al. 2016; Bulla et al. 2015).Even for large asphericity, a large polarization may not resultbecause of strong cancellation e ff ects. This situation holds par-ticularly at early times because of the confinement of the spectral formation region, which favors a low residual polarization. Theintegrated polarized flux is also inhibited by multiple scatteringssince these tend to randomize the scattering directions, makingthe radiation field more isotropic. These e ff ects suggest that SNejecta should produce a small (or at least a smaller) continuumpolarization at earlier times, whatever the level of asymmetry. Atnebular times, when the ejecta optical depth is below unity alongall sight lines, photons typically scatter once with free electrons.Because of the weaker cancellation e ff ects, this situation pro-duces a greater residual polarization for a given asymmetric con-figuration. Under optically thin conditions, the continuum polar-ization scales linearly with continuum optical depth (Brown &McLean 1977). Despite these advantages, the faintness of SNeat nebular times presents a major challenge for spectropolarimet-ric observations (Leonard et al. 2006).Optical depths e ff ects can lead to complex polarization sig-natures, distinct from what obtains under optically thin condi-tions. One such feature is a sign reversal of the polarization(equivalent to a 90 deg change in the polarization angle) in thecontinuum across the optical range (in part driven by the changein albedo between the Balmer edge and the Paschen edge), aswell as sign reversals across line profiles and the adjacent contin-uum. In observations, sign reversals (especially if they are weak)may be hard to diagnose if the interstellar polarization is notknown accurately. Sign reversals have been seen in spectropo-larimetric observations of some type II-P SNe (Chornock et al.2010).In optically thick lines like H α , the polarization is generallyzero somewhere between the location of maximum absorptionin the P-Cygni trough and the location of maximum flux nearline center (see for example Figs. 8 and 16 in Dessart & Hillier2011, in cases where the zero crossing is associated with a signreversal of the polarization). This location may thus be useful forconstraining the interstellar polarization, especially if other linescan also be used for that purpose (for example the Ca ii near-infrared triplet or H β ). In contrast, weak lines may not causemuch reduction to the continuum polarization (for example H δ )and thus only induce an inflection in the continuum polarizationlevel. In the opposite regime of strong line overlap, for examplein regions of line blanketing, both the total flux and the polarizedflux are strongly reduced.For a fixed ejecta asymmetry, the continuum polarization isexpected to reach a maximum when the core is revealed. Thisis understood as stemming from the progressive growth of theextent of the spectrum formation region during the photosphericphase and the drop in envelope optical depth (favoring single in-stead of multiple scatterings). It may occur at fixed asymmetry,or if the asymmetry increases towards the inner ejecta regions.Such a strong increase in continuum polarization has been seenin a few type II SNe (Leonard et al. 2006; Chornock et al. 2010),and this feature is also present in the SN that we discuss in thispaper. At nebular times, when the total electron scattering opticaldepth of the ejecta is below unity, the continuum polarization isexpected to follow a 1 / t evolution for a fixed ionization (sincethe asymmetry and viewing angle do not change with time; seealso Brown & McLean 1977), following the drop in ejecta opti-cal depth. In SN ejecta, however, deviations from this could ariseif the asymmetry varies with depth and if the ionization evolvesin a complicated fashion – for example following the growth ofthe γ -ray photon mean free path with time.All of these conclusions apply to the new models presentedhere using the upgraded long-characteristic code (Section 3.3;see also Hillier & Dessart, in prep.). However, they are nowmuch more directly visible since the upgraded code delivers Article number, page 2 of 19uc Dessart et al.: Modeling of the SN 2012aw spectropolarimetric obsvervations λ obs [ ˚A] S c a l e d F λ + C o n s t . ∆ M V =-0.1445.5d ∆ M V = 0.1462.5d ∆ M V = 0.1471.5d ∆ M V = 0.0991.5d ∆ M V = 0.04107.4d ∆ M V =-0.09120.4d ∆ M V =-0.11 Fig. 1.
Multiepoch spectral comparison between SN 2012aw and ourreference 1D
CMFGEN model 1D-X1 (see Section 3.1). The data cor-respond to the observed total flux for SN 2012aw obtained withVLT − FORS in spectropolarimetric mode. The model is redshifted, red-dened, and then normalized to the observed flux at 7000 Å. The V -bandbrightness o ff set, calculated from the observed photometry (Bose et al.2013; corrected for distance and extinction) and the model photometry,is given for each epoch and is typically of the order of 0.1 mag. Thismodel is named x1p5 in Hillier & Dessart (2019) and is our best matchmodel for SN 2012aw. the full polarized spectrum (across the optical or near-infraredrange) rather than just the polarization for a single isolated lineand its adjacent continuum, as previously done in Dessart &Hillier (2011).
3. Modeling approach
The spectro-polarimetric modeling we perform in this study re-quires a few preparatory steps. All simulations are based on 1Dnonlocal thermodynamic equilibrium (nonLTE) time-dependentradiative transfer simulations of RSG explosion models per-formed with
CMFGEN , computed recently for a separate studyon the diversity of Type II SNe (Hillier & Dessart 2019). These
CMFGEN simulations are used as initial conditions for the 2D po-larized radiative transfer modeling. We use an updated versionof the long-characteristic code described in Dessart & Hillier(2011). The main improvements are the computation of the fulloptical polarized flux as well as extra flexibility for setting up the2D axisymmetric ejecta (see Hillier & Dessart, in prep.).Here, a 2D axially-symmetric ejecta is built using various ap-proaches. The first approach is to introduce a latitudinal scalingto the density (i.e., mass density, ion density, atom density, andelectron density), opacity, and emissivity computed from a 1D
CMFGEN model. The second approach is to “stretch" these quan-tities in radial space, by an amount that depends on latitude. Onelimitation of these two approaches is that the adopted scaling ofradial o ff set may not reflect well the properties of axisymmetricand asymmetric ejecta. In the third approach, we assign distinct1D CMFGEN models to a range of latitudes so all latitudes are λ obs [ ˚A] S c a l e d F λ + C o n s t . ∆ M V = 0.1145.5d ∆ M V = 0.1562.5d ∆ M V = 0.1071.5d ∆ M V = 0.0291.5d ∆ M V =-0.15107.4d ∆ M V =-0.55120.4d ∆ M V =-1.28 Fig. 2.
Same as for Fig. 1, but now showing the results for model 1D-X2b. Compared to model 1D-X1, model 1D-X2b is designed to be morestrongly mixed, and to possess a Ni shell around 4000 km s − . Withthese modifications it deviates more strongly from the observations ofSN2012aw. characterized by a physical, albeit 1D, model. One may use twomodels (defining the properties of the 2D ejecta along the equa-tor and the pole) or a larger set of models to introduce smallerscale asymmetries covering a smaller opening angle.In this work, we used a combination of approaches. For thefirst epoch of spectropolarimetric observations of SN 2012aw,we explored the influence of a latitudinal scaling or a latitudinalradial displacement of a 1D model (options 1 and 2 above). Thissomewhat artificial approach is not followed further. Instead, thebulk of the modeling is based on a mapping in latitude of two dis-tinct 1D CMFGEN models to produce a large scale asymmetry ofthe ejecta. This approach is also applied from early to late timesand thus allows one to see the evolution of the SN polarizationfor a given asymmetric ejecta. Below, we review the propertiesof the 1D
CMFGEN models and the various 2D asymmetric (butaxisymmetric) ejecta that we build from these, before describingin detail how the polarization calculations are performed.
CMFGEN model
SN 2012aw was studied by Hillier & Dessart (2019) as part of alarge investigation on the origin of the diversity of type II SNe,in particular in terms of V -band decline rate and spectral diver-sity during the photospheric phase. The reference model we usefor SN 2012aw is named here 1D-X1 (originally called x1p5 inHillier & Dessart 2019) and corresponds to a 15 M (cid:12) zero agemain sequence star, exploded to yield an ejecta kinetic energyof 1 . × erg, an ejecta mass of 12.12 M (cid:12) , and a Ni massof 0.056 M (cid:12) . Figure 1 illustrates the agreement in the opticalrange (the data are from our spectropolarimetric VLT − FORSprogram) between model 1D-X1 and SN 2012aw, and also thefact that the model is within about 0.1 mag of the observed V band magnitude throughout the high brightness phase (there is asimilar o ff set at nebular times; see Hillier & Dessart 2019). This Article number, page 3 of 19 & A proofs: manuscript no. ms l o g ( L b o l / e r g s − ) − − − − − l o g ( ρ )[ g c m − ) − − − − − − X ( N i ) l o g ( N e / c m − ) T [ K ] V [10 km s − ]0.00.20.40.60.81.0 N ( H + ) / N ( H ) Fig. 3.
Top: Bolometric light curve for the 1D
CMFGEN models 1D-X1and 1D-X2b. Bottom: Mass density, initial Ni mass fraction, electrondensity, gas temperature, and hydrogen ionization fraction vs. velocityfor models 1D-X1 (solid) and 1D-X2b (dashed) at 44 d after explosion.These two models vary in the outer ejecta structure, the inner core struc-ture, and the Ni mixing adopted (strong or weak, presence or absenceof a Ni-rich shell at a Lagrangian mass of 12 M (cid:12) , corresponding to avelocity of about 4000 km s − in the resulting ejecta). The two modelshave the same mass-density profile outside of the inner ejecta (with theexception of the regions beyond 10000 km s − ), but di ff erent electron-density profiles because of the di ff erent Ni profiles. agreement suggests that a spherical ejecta model yields a satis-factory match to the photometric and spectroscopic observationsof SN 2012aw. The deviations from spherical symmetry that we introduce must therefore remain small enough not to deterioratethe agreement between observations and the results for model1D-X1.
In this approach, the density along the polar angle β is ρ ( r , β ) = ρ ( r ) B (1 + A cos β ) . (1)with B set (in all cases) to 1 / (1 + A /
3) (chosen so that (cid:82) B (1 + A cos β ) sin β d β = − A value. Other choices are possible. For example, byvarying the exponent on the cosine one can modulate the latitu-dinal confinement of the asymmetry. This option is not exploredhere. With this adopted density scaling, and at a given radius r ,opacities and emissivities scale with B (1 + A cos β ) , while theelectron-scattering opacity scales with B (1 + A cos β ). The second option is to apply a radial scaling of the 1D
CMFGEN model and use the same density, opacity, and emissivities as the1D
CMFGEN model on that scaled grid. For example, the den-sity (as well as opacities and emissivities) at r along β corre-sponds to its counterpart in the original 1D CMFGEN model at r / B (1 + A cos β ). If the density declines outwards, positive (neg-ative) A values correspond to prolate (oblate) density configura-tions. For an ejecta characterized by a power-law density withexponent n ρ , the radial scaling causes a maximum density con-trast between pole and equator of (1 + A ) n ρ . For n ρ =
12 (which isrepresentative of the outer layers of the ejecta in model 1D-X1),this maximum density contrast corresponds to 14.6 for A = . A = . CMFGEN model with another model
In this approach, we build the axisymmetric ejecta assigning dis-tinct 1D
CMFGEN models to the equatorial and to the polar direc-tions, and interpolate between the models at intermediate lati-tudes. The mapping of a given model over a range of latitudes isflexible. Multiple models could also be used but here we use onlytwo to facilitate the interpretation. The angle β / corresponds tothe maximum polar angle over which the “polar” model applies.Hence, besides the reference model 1D-X1, we used an al-ternate model 1D-X2b. This model deviates from model 1D-X1both for the progenitor and for the ejecta. The core materialwithin the inner 5 M (cid:12) is made completely homogeneous (thisimplies strong mixing) and is also given a fixed density (con-serving total mass and yields), thereby erasing the jump in den-sity present in model 1D-X1 at the He-core edge. The progenitor Article number, page 4 of 19uc Dessart et al.: Modeling of the SN 2012aw spectropolarimetric obsvervations − − − − − [ N e ( V , θ ) / N e ( V , 0 )] − − − − − − − [ X ( Ni )( V , θ ) / X ( Ni )( V , 0 )] Fig. 4.
2D distribution of the free-electron density N e (left) and of the Ni mass fraction (right; the Ni mass fraction is in practice alwayslower than about 0.01 beyond 1000 km s − along all latitudes – see Fig. 3), normalized to their corresponding values along the pole, for model2D-X1-X2b at 44 d after explosion. We use model 1D-X2b up to 25 −
30 deg, and model 1D-X1 beyond. The vertical corresponds to the axis ofsymmetry, so polar angle β of zero and 180 deg, and the horizontal corresponds to the equatorial direction. This axisymmetric ejecta has mirrorsymmetry with respect to the equatorial plane. The black label along the one o’clock direction gives the radial velocity in units of 1000 km s − .The ejecta asymmetry stems from the asymmetric distribution of Ni, as well as di ff erences in the outer ejecta (through the presence of a denseRSG progenitor atmosphere or not) and di ff erences in the inner ejecta (complete mixing and smoothing of the He core material) – see Fig. 3 andSection 3.2.3 for details. is enshrouded by a dense extended atmosphere prior to explo-sion, which produces a lower maximum ejecta velocity as wellas the presence of a dense shell in the outer ejecta. The most im-portant feature is however the presence of a Ni rich shell (in1D), which we add “by hand” at a Lagrangian mass of 12 M (cid:12) , sothat it is located around 4000 km s − in the resulting ejecta oncehomologous expansion is established. In practice, this is donein the radiation-hydrodynamics calculation with V1D (see Livne1993 and Hillier & Dessart 2019 for discussion), at 1000 s afterthe explosion trigger (and therefore in mass space since the ve-locity profile has not settled at that time). Because there is verylittle mass between 12 M (cid:12) (or the ejecta location where the ve-locity is about 4000 km s − ) and the outermost ejecta layers, anexcess in Ni (relative to model 1D-X1) is present throughoutthe outer ejecta. However, the Ni mass fraction remains verysmall, typically below 0.01, and the Ni is mixed beyond about2000 km s − with a material rich in H and He (the material thatused to be in the H-rich envelope of the progenitor star). Model1D-X2b corresponds to a (spherically-symmetric) ejecta with akinetic energy of 1 . × erg, a total mass of 12.43 M (cid:12) , anda Ni mass of 0.047 M (cid:12) .Figure 2 compares the spectral properties of model 1D-X2bwith the observations of SN 2012aw in the optical range. Be-cause of the adjustments made, the resulting SN radiation di ff erssizably from that obtained for model 1D-X1, and thus deviatesfrom SN 2012aw. This was the goal, and is in line with the ear-lier findings of Chugai (2006). In model 1D-X2b, the strongermixing and the presence of a Ni shell at 4000 km s − (addedto give a source of asymmetry in a 2D ejecta; see below) bothlead to a stronger and broader H α line in the second half of theplateau phase. The H α mismatch suggests that there is no strong Ni mixing along all ejecta-centered directions, but that strong Ni mixing, if present, must be limited to a modest solid an-gle. The lower Ni mass also causes an underestimate of the optical brightness compared to SN 2012aw. As long as model1D-X2b covers a small fraction of the whole 2D ejecta volume,the global ejecta and radiation properties should primarily reflectthose of model 1D-X1. We compare the ejecta properties and thebolometric light curves of the 1D models 1D-X1 and 1D-X2b inFig. 3.In most of the simulations we adopt mirror symmetry withrespect to the equatorial plane. Model 1D-X2b is assigned thepolar directions and model 1D-X1 is assigned the equatorial di-rection. The properties at intermediate latitudes is flexible. In thereference setup described in Section 5, model 1D-X2b representsthe ejecta for all polar angles between zero and 22.5 deg. Be-yond 33.75 deg, we use model 1D-X1 instead. In between thesetwo angles, we linearly interpolate in cos β . This interpolationapplies to all relevant quantities for the 2D radiative transfer andin particular the opacities and emissivities. We thus interpolatebetween two physically consistent 1D CMFGEN models (also de-rived from physically consistent radiation-hydrodynamics sim-ulations of the explosion). This is superior to using one 1D
CMFGEN model and applying some density scaling or radial scal-ing on its properties (see previous two sections). Figure 4 illus-trates some of the properties of the resulting 2D ejecta. The leftpanel shows the normalized electron density (in the log base 10)at 44 d after explosion. With this setup, we can explore the con-figuration for an enhanced Ni abundance confined to a narrowrange of latitudes. Such a configuration is known to yield a resid-ual polarization (Chugai 2006).This setup is not an exact representation of what may occurin nature. Detailed 3D simulations of neutrino-driven explosionssuggest that the Ni material may be asymmetrically distributedin a complicated structure of blobs, shells, fingers, elongated to-wards large velocities, and made of essentially pure Ni (Wong-wathanarat et al. 2015). This material is macroscopically mixedin space during the dynamical phase of the explosion, so that
Article number, page 5 of 19 & A proofs: manuscript no. ms it may coexist at a given velocity with H-rich material but it isnot microscopically mixed with it. In our present approach, weinstead apply a macroscopic and a microscopic mixing so thatboth Ni and H-rich material coexist at a given velocity. Thisapproach is perhaps not as problematic as it seems since, withrespect to the polarization of light, it is the influence of Ni onthe electron density that matters. Being non local, this influenceshould create an extended cocoon of enhanced ionization (andthus electron density) around any pure Ni blob. This configu-ration is well captured by our setup.
The simulations presented in this study were performed withthe long characteristic code
LONG POL first implemented to treatcontinuum (Hillier 1994) and line (Hillier 1996) polarization inthe context of multi-scattering axisymmetric envelopes of hotstar winds. This code was subsequently extended to treat thecase of Type II SN ejecta (Dessart & Hillier 2011) but limitedto solving for the polarized flux for an individual line and itsoverlapping continuum.The code has since been improved in two important ways(Hillier & Dessart 2020, in prep.). First, the code now computesthe full optical polarized spectrum and is thus no longer lim-ited to a single line. This implies that line overlap is automat-ically treated, irrespective of the number of overlapping lines.Second, the code is no longer limited to 2D ejecta producedby the latitudinal density scaling or radial stretching of a pre-computed 1D
CMFGEN model. We can now also combine a setof pre-computed 1D
CMFGEN models, assigning a given modelto a specific range of latitudes. The benefit from this latter ap-proach is the improved physical consistency since the opacitiesand emissivities imported into
LONG POL are those from the pre-computed nonLTE 1D
CMFGEN models. This is less artificial thanprescribing an asymmetry through parametrized scalings. A re-maining weakness is that we still interpolate between 1D modelsat a given latitude. Further, the opacities and emissivities (withthe exception of the electron-scattering emissivity) assigned tothis 2D ejecta are not computed from a fully consistent 2D non-LTE model. Practically, the 2D radiative transfer is performed asa formal solution (i.e., with the opacities and emissivities fixed).The other benefit is that we can combine any variety of models,di ff ering in explosion energy, composition such as Ni mass or Ni mixing, or clumping.In
LONG POL , the polarization is produced by electron scat-tering and described by the Stokes parameters I , Q , U , and V (Chandrasekhar 1960). The polarization is thus linear and V isidentically zero. We use the same nomenclature as in Dessart &Hillier (2011) and define the corresponding flux-like polariza-tion quantities F I , F Q , and F U output by the code (the calcula-tions are done in I l , I r , U and V space where I l = . I + Q ), I r = . I − U ); see Hillier 1994 for details) – the definitionsof these various quantities are given in Section 2 of Dessart &Hillier (2011). The geometry of the axisymmetric ejecta is de-scribed with a right-handed set of unit vectors ( ξ X , ξ Y , ξ W ). Theorigin is the center of the ejecta, the axis of symmetry lies along ξ W , ξ Y is in the plane of the sky, and the observer lies in the XW plane (Hillier 1994, 1996). We adopt the same sign conventionas in Dessart & Hillier (2011). The flux F Q is positive (negative)when the polarization is parallel (perpendicular) to the symme-try axis. Because of the imposed axial symmetry and the adoptedorientation, F U is zero. Hence, the polarization fluxes that wepresent from our simulations with LONG POL will be limited to F Q and to P = × | F Q | / F I (and thus corresponding to a per- centage), where F I is the total flux. To keep with the conventionin spectropolarimetric observations, the normalized Stokes pa-rameters q and u are defined as q = F Q / F I and u = F U / F I . Inthis study, we will only discuss the flux-like polarized quanti-ties obtained with LONG POL . We refer the reader to Dessart &Hillier (2011) for a discussion of the maps of the polarizationintensities on the plane of the sky for various asymmetric con-figurations and wavelengths. In general, because of the very lowlevel of polarization from asymmetric SN ejecta, these polariza-tion intensity maps are di ffi cult to analyze.For a LONG POL calculation, we first construct the 2D ejecta(see Section 3.2). The initial information in this 2D model con-sists of radius, density, velocity, electron scattering opacity, ab-sorptive opacity as well as emissivity for the selected range ofwavelengths. The 2D ejecta is mapped with about one hundredradial points and nine polar angles between pole and equator(equally-spaced; mirror symmetry with respect to the equatorialplane is by default assumed). The radiation field I at each ejectalocation ( r , β ) is discretized in azimuth φ and latitude θ . We usethirteen φ angles while the latitudinal rays are set according tothe impact parameter. Fourteen points (or rays) are used to coveruniformly from the ejecta center to the innermost ejecta radius,and the radial grid is used beyond that to define the impact pa-rameter.In LONG POL , most of the computing time is taken by anested β and φ angle loop (in which the 2D radiative transferequation is solved for). This section is parallelized with MPI and LONG POL is run with N β N φ processors (hence typically of ordera few hundred processors). Convergence is reached after 20 −
4. Observations of SN 2012aw − FORS Dataset
We obtained seven epochs of spectropolarimetry of the Type II-P supernova SN 2012aw using the European Southern Observa-tory’s Very Large Telescope (VLT) at Cerro Paranal in Chile(Appenzeller et al. 1998). All observations were made withthe FOcal Reducer and low dispersion Spectrograph (FORS2),mounted at the Cassegrain focus of the Antu (UT1) telescope.The data sample days 16 – 120 following the date of explosion(16.09 March, 2012 UT) estimated by Bose et al. (2013). Detailsof the observational epochs, instrumental setup, exposure times,and observational conditions are given in Table 1. The data arepresented in Fig. 5.Each complete observational sequence consisted of four sep-arate exposures with the half-wave retarder plate positioned ateach of 0, 45, 22.5, and 67.5 deg; multiple complete sequenceswere executed at all epochs except for epoch 2, for which onlyone complete sequence was obtained (see Table 1). All data werebias subtracted and flat-field corrected using dome flats obtainedthrough the polarimetry optics at a waveplate position angle of This grid is by default the same as in the 1D
CMFGEN model andthus equally spaced in optical depth scale. However, the two modelsused may present a recombination front at a di ff erent radius. Becausethe same radial grid is used along all latitudes, it must be augmented toresolve any recombination front along any latitude. VLT observing program 089.D-0515(A): PI G. Pignata.Article number, page 6 of 19uc Dessart et al.: Modeling of the SN 2012aw spectropolarimetric obsvervations
Table 1.
Journal of Spectropolarimetric Observations of SN 2012aw.
Epoch Day a UT Date b MJD c Exposure d Air Mass e Seeing f − − − − − − − / mm grism (“GRIS_300V”) along with the “GG435”order-sorting filter to prevent second-order contamination. This resulted in a useable spectral range of 4350 Å − (cid:48)(cid:48) . ∼
12 Å, as derived from the FWHM of night-skylines. The flux standard used for all observations was BD + ◦ ff eredsignificantly from the parallactic angle (Filippenko 1982); when coupled with the relatively narrow slit used, the potential fordi ff erential light loss – and, hence, inaccurate relative flux calibration – is significant. a Day since estimated explosion date of 2012-03-16.09 UT = MJD 56,002.09 = JD 2,456,002.09; Bose et al. 2013). b yyyy-mm-dd, calculated as the midpoint of all of the individual exposures. c Modified Julian Date (Julian Date - 2400000.5), at the midpoint of all of the individual exposures. d Total exposure time in seconds, with the number of complete sets of four waveplate positions obtained shown in parenthesis. e Air mass range for each set of observations. f Average value of the FWHM of the spatial profile for each epoch, rounded to the nearest 0 (cid:48)(cid:48) . ; reductions carriedout with unflatfielded data were found to be virtually identical tothose made with the flatfielded data, as expected due to the re-dundant number of half-wave positions obtained (see, e.g., Patat& Romaniello 2006).One-dimensional sky-subtracted spectra were extracted us-ing both the optimal (Horne 1986) and standard extraction tech-niques, with an extraction width set to encompass at least threetimes the e ff ective “seeing” of the particular observations (seeTable 1). The wide extraction was used to minimize the e ff ect ofspurious polarization features being introduced by interpolatingthe counts in fractional pixels at the edges of narrow extractionapertures (see Leonard et al. 2001). We subtracted a linear inter-polation of the median values of background windows on eitherside of the object from the object’s spectrum; for consistency,the “sky” background region was set for all extractions to be ± (cid:48)(cid:48) . (cid:48)(cid:48) . ± (cid:48)(cid:48) .
75 : 3 (cid:48)(cid:48) . − linear scale.We formed the normalized Stokes q and u parameters andtheir statistical uncertainties for each complete sequence of ob-servations according to the methods outlined by Miller et al.(1988) and Cohen et al. (1997). For each observational sequence,we derived results using both the optimal and standard extrac-tions separately, and then compared them to each other. In gen-eral, the two extraction algorithms yielded virtually identical re-sults, and so the optimal extractions were preferred due to their(slightly) higher signal-to-noise ratio and ability to discount mi-nor cosmic ray strikes. However, in certain spectral regions thatcontained particularly sharp spectral features that also a ff ectedthe “sky” region (e.g., regions of strong telluric absorption or, For epoch 1, the flat fields were acquired on the night after the scienceobservations. more rarely, severe cosmic-ray strikes across the spatial profileof the object on the CCD), the optimal extraction algorithm pro-duced spurious features in the polarimetry that were not foundin the standard extraction results (and, were not consistent fromone observational set to the next on the same night). Presum-ably, such features occur due to a breakdown of the assumptionof a “smoothly varying” spatial profile for the object, which un-derlies the successful application of the optimal weighting algo-rithm (Horne 1986). This occurred quite frequently at the telluric“A-band” feature near 7600 Å; in one instance, it also occurredin the vicinity of a particularly bad cosmic-ray strike. In theselimited regions (typically ∼ <
40 Å), the standard extraction’s po-larimetric results were inserted; otherwise, the results derivedfrom the optimal algorithm were used.We corrected all data for the small ( ∼ < . ff set between the half-wave plate and thesky coordinate system provided by the VLT website . In addi-tion, we checked this angle on each night by observing polarizedstandard stars from the lists of Clemens et al. (1990; HD 127769)and Cikota et al. (2017; Hiltner652; BD-144922; HDE316232;and Vela1). With the exception of the final (seventh) epoch, allpolarized standard stars’ derived polarization angles (in the V -band; nominal wavelength range of 5050 − ff ered from thevalue reported by Clemens et al. (1990) by 2.4 deg; the causefor this discrepancy is not known, although we do note the un-usually poor seeing (Table 1) and short exposure time for thestar (one second per waveplate position; more typical of othernights was 3 − ff set,and did not apply any corrections based on our observations ofpolarized standard stars. To check for instrumental polarization Article number, page 7 of 19 & A proofs: manuscript no. ms F I [ s c a l e d ] SN 2012aw θ P [ d e g ] λ max,ISP ; q max,ISP ; u max,ISP = (5535 ˚A;-0.0012;-0.0043) P [ % ] Fig. 5.
Multiepoch VLT spectropolarimetric observations of SN 2012aw showing the total flux F I at the top, the observed polarization angle θ P (we show the quantity 180 + arctan( F U , F Q ), in degrees), and the observed percentage polarization P (we show here the traditional polarizationdefined as 100 (cid:112) q + u ; see, e.g., Leonard et al. 2001). The quantities θ P and P are displayed with a binning of 15 Å / bin. Each epoch is colorcoded (see labels and Table 1). We overplot the adopted interstellar polarization based on a Serkowski law (Serkowski et al. 1975; Cikota et al.2017) derived by assuming that the intrinsic polarization is zero at the regions of the strong, consistent, depolarization seen near the flux emissionpeaks of H β , Na i D, H α , and the Ca ii near-infrared triplet at epoch 6 (thick black line; see Section 4.2 for details). beyond that already accounted for through application of the lin-ear relations provided by Cikota et al. (2017), we also observedthe null polarization standard HD 109055 (Clemens et al. 1990;Berdyugin et al. 1995) at every epoch, and always found it to benull to within 0 . q and u parametersderived from all complete observational sequences were inter-compared for consistency and then combined according to theirstatistical weights. We then removed the interstellar polarizationderived in Section 4.2, removed a redshift of 778 km s − (de Vau-couleurs et al. 1991, via NED), rebinned to 5 Å bin − , and cre-ated a final, rest-frame dataset with a wavelength range of 4350Å – 9150 Å. We also computed results binned to 15, 25, 50 and100 Å bin − for presentation purposes. To generate the total fluxspectrum for each epoch, a final correction for telluric absorptionbands (Wade & Horne 1988) was made.To illustrate the continuum polarization as a function of timewe show in the left panel of Fig. 6 the normalized Stokes q and u measurements for all seven epochs in three spectral re-gions devoid of strong lines around 6000, 7000, and 8000 Å. Allmeasurements lie in a single quadrant of the ( q , u ) plane, and tofirst order lie along a straight line. The biggest departure froma straight line occurs for the data at 120 d. The o ff set is signif- icant, although the data at this epoch also has the worst signal-to-noise ratio. If SN 2012aw were axisymmetric all polarizationdata would lie along a straight line in the ( q , u ) plane, and, inthe absence of interstellar polarization, would pass through theorigin. The presence of a dominant axis in the data sets indicatesthat SN 2012aw must possess a strong degree of axial symmetry.However, the scatter (and in some cases one could argue for anepoch dependent o ff set) of the polarization observations aboutthe straight line indicate the structure cannot be simply axisym-metric.A study of the polarization behavior for H α (right panel ofFigure 6) yields a similar conclusion to that reached using thecontinuum polarization. Again, all measurements lie in a singlequadrant of the ( q , u ) plane, and to first order lie along a straightline. As for the continuum, it is the observations at 120 d thatshow the largest o ff set. Because of the presence of a dominantaxis we will rotate the observations so that the bulk of the po-larization lies in q (Section 4.3). This has the consequence thatsign reversals in q can easily be seen in the same plot, whereassuch reversals are not seen in P and imply a 90-deg change inthe angle of polarization. Article number, page 8 of 19uc Dessart et al.: Modeling of the SN 2012aw spectropolarimetric obsvervations − − − q [%] − − − u [ % ] SN 2012aw observationsContinuum regions − − q [%] − − − − − u [ % ] SN 2012aw observationsH α Fig. 6.
Normalized Stokes parameters for SN 2012aw in three spectral regions devoid of strong lines covering 6000 − − − q and u were derived by taking the mean of the binned (by 15 Å), de-redshifted dataover the stated ranges; uncertainties shown are the 1-sigma spread of the data values over the same ranges. Right: Same as left, but now zooming-inon the H α region extending ± − away from line center. In both panels, a black dot gives the average interstellar polarization inferredat 6500 Å and the dashed line indicates the dominant axis adopted for the rotation of the Stokes parameters in subsequent sections. Correcting for interstellar polarization is a delicate problem. Inthis study, we have tried three di ff erent approaches. The first ap-proach is to assume that the intrinsic polarization of the SN iszero at the first epoch. Theoretically, this may be motivated bythe fact that instabilities, born in the progenitor core during theexplosion or at the H / He interface after shock passage, influencelittle the outer ejecta. Unfortunately, there is clearly some varia-tion in polarization across lines even at the earliest epoch, incom-patible with the expected smooth featureless polarization fromthe interstellar medium. As this option also conflicts with the ex-pected depolarization in strong lines at later epochs (in particularat epochs 5, 6, and 7; see Section 5) it was abandoned.The second option is to use the strong, consistent depolariza-tion seen in H α just blueward of the emission peak (see Fig. 5)as a location of assumed zero intrinsic polarization. Indeed, atlater epochs, all observations yield the same polarization in thisregion, while the polarization in the surrounding spectral regionsvaries considerably with each epoch. This choice seems suitable.Our final choice is similar to option 2 above, except that wenow fit a Serkowski law (Serkowski et al. 1975) through thestrong depolarizations seen in H β , Na i D, and the Ca ii near-infrared triplet, as well as H α , at epoch 6 (similar results arefound using epochs 3 to 7). Using these four spectral locationsas a reference for the interstellar polarization, we searched forthe set of values λ max , ISP , q max , ISP , and u max , ISP that produced thebest fitting curve given by a Serkowski law. For a given q max , ISP and u max , ISP correspond θ ISP = . u max , ISP / q max , ISP ) and p max , ISP = q max , ISP cos(2 θ ISP ) + u max , ISP sin(2 θ ISP ) . (2)The wavelength dependence of the interstellar polarization isthen given by p ISP ( λ ) = p max , ISP exp (cid:0) − K ISP ln ( λ max , ISP /λ ) (cid:1) , (3) where K ISP = − . + . λ max , ISP (Cikota et al. 2017). Wethen obtain q ISP ( λ ) = p ISP ( λ ) cos(2 θ ISP ) ; u ISP ( λ ) = p ISP ( λ ) sin(2 θ ISP ) . (4)With this approach, the best fitting values for SN 2012aware λ max , ISP = q max , ISP = − u max , ISP = − q ISP and u ISP . As a consequence,there is a small but a non negligible intrinsic polarization ofSN 2012aw at early epochs (SN ages of 16.1 and 46.0 d) includ-ing prior to the recombination phase. The multiepoch observa-tional dataset, uncorrected for interstellar polarization, togetherwith the adopted interstellar polarization, is shown in Fig. 5.Adopting this insterstellar polarization and focusing on thespectral range between 6900 and 7200 Å (Fig. 5), SN 2012awyields a non-zero polarization (about 0.2 % level) even at the ear-liest times. It also shows a strong increase in polarization duringthe second half of the plateau phase, reaching a maximum ofabout 1.2 % as the SN starts plunging in brightness and transi-tioning to the nebular phase. This has been seen in SN 2004dj(Leonard et al. 2006), SNe 2006ov and 2007aa (Chornock et al.2010) , SN 2008bk (Leonard et al. 2012), and 2013ej (Leonardet al. 2015). SN 2012aw could not be observed at nebular timesas it went behind the sun. It was too faint when it later emerged. For an aspherical but axially symmetric ejecta, the polarization(at all times and at all wavelengths) lies on a diagonal line inthe ( q , u ) plane, and this line passes through the origin. Sincethe slope of this line simply reflects the orientation of the ejecta SN 2006my shows a high polarization of about 1% at the onset of thenebular phase but only one epoch was observed.Article number, page 9 of 19 & A proofs: manuscript no. ms F I [ S c a l e d ] Model: [email protected] − q r o t [ % ] − u r o t [ % ] P [ % ] F I [ S c a l e d ] Model: [email protected] − q r o t [ % ] − u r o t [ % ] P [ % ] Fig. 7.
Left: Comparison between the model 2D-X1-SCL0p5 and observations of SN 2012aw for epoch 1 (i.e., 16.1 d after explosion) using adensity scaling with A of 0.5 (left) or 4.0 (right). All inclinations between polar and equatorial directions are shown. For SN 2012aw, we correctfor the instrumental polarization, for the interstellar polarization and redshift (as well as reddening for F I ). The observed normalized Stokesparameters have been rotated by 28 deg (see section 4.3). A binning of 25 Å is used for q rot and u rot , and only 5 Å for F I . Note that the polarization P displayed in the bottom panels for the SN 2012aw data is formally the rotated Stokes parameter (RSP), as defined by Trammell et al. (1993) anddiscussed extensively by Leonard et al. (2001). RSP is similar to q rot , except that (1) it is positive definite; and (2) it is formed by using a smoothedversion of the polarization angle rather than setting it equal to a constant value across the spectrum. Due to the second reason, it tends to capture agreater amount of the polarization, leaving little residual in the rotated U Stokes parameter (URSP). Note that for all models u rot is null across thespectrum, and so is not plotted. symmetry axis on the plane of the sky, one can rotate it so thatall the polarization lies in q . This then allows a direct compari-son with our model calculations in which all the polarization liesin q . In observations, noise and departures from strict axial sym-metry leave a residual signal in u even after such a rotation hasbeen applied.Hence, when we compare to our polarization results for ax-isymmetric models, we apply a rotation of the observed Stokesvectors by an angle θ rot = . δ u /δ q ), where δ u and δ q are measured in the ( q , u ) plane. This is done to put most of theobserved polarization in q . The rotated fluxes are then given by q rot = q cos(2 θ rot ) + u sin(2 θ rot ) (5)and u rot = − q sin(2 θ rot ) + u cos(2 θ rot ) . (6)The data of SN 2012aw, corrected for interstellar polariza-tion, yields an observed polarization angle of 118 deg in theplane of the sky. However, we choose to rotate the Stokes vec-tors by the angle orthogonal to this – i.e., of 28 deg – in order toproduce a sign of the rotated Stokes vectors that agrees with themodels, for ease of comparison (rotating through an angle 90 degdi ff erent simply changes the sign of the polarization). With this rotation, the bulk of the polarization is now in q , and u is gener-ally close to zero. This works with only modest success at earlytimes when the polarization level is low, but it seems suitable forepochs four and later, as shown in the next section.
5. Modeling results and comparisons toobservations of SN 2012aw
As discussed in Section 3.2, we used three di ff erent ways to in-troduce the asymmetry in our 2D ejecta. In the first two sectionsbelow, we describe the results for the first epoch adopting a lati-tudinal density scaling (Section 3.2.1) or a latitudinal stretching(Section 3.2.2) of a 1D CMFGEN model. In the subsequent sec-tion, we use the combination of two models (Section 3.2.3) andcompare to all observed epochs of SN 2012aw (Section 4). Forclarity, we will call these models the scaled model, the stretchedmodel, and the hybrid model.Throughout this section, we present multi-panel figures thathave the same structure. The top panel shows the total flux F I for both the model and the observations (normalized to unityat 7100 Å). The next two panels show the normalized, rotatedStokes parameters q rot and u rot (we plot the percentage value),also corrected for interstellar polarization (see previous section).Finally, the bottom panel shows the percentage polarization P Article number, page 10 of 19uc Dessart et al.: Modeling of the SN 2012aw spectropolarimetric obsvervations F I [ S c a l e d ] Model: [email protected] − q r o t [ % ] − u r o t [ % ] P [ % ] F I [ S c a l e d ] Model: [email protected] − q r o t [ % ] − u r o t [ % ] P [ % ] Fig. 8.
Same as Fig. 7 but now for the 2D ejecta models built using a stretching of model 1D-X1, with a parameter A of 0.25 (left; model2D-X1-STR0p25) and 0.50 (right; model 2D-X1-STR0p50). for both the observation (black) and the model (colors other thanblack). The model results are sometimes shown for all inclina-tions. The inclination of zero degree yields zero polarization.The other inclinations are spaced uniformly every 11.25 deg, col-ored from blue to red (colorblind rainbow colormap) as the in-clination is increased.Compared with the simulations presented in Dessart &Hillier (2011), the present simulations provide the total and po-larized flux spectrum over the entire optical range. We explicitlyaccount for line overlap or blanketing and can directly compareto the optical VLT − FORS spectropolarimetric data that we col-lected for SN 2012aw.
Figure 7 compares the observations of SN 2012aw at the firstepoch (16.1 d after explosion) with models in which a latitudinaldensity scaling has been applied to the 1D
CMFGEN model 1D-X1. The left panel shows the results for A = . A = . F I (top panel). The polarized flux F Q (orequivalently q ) is positive throughout most of the optical range,which means that the electric vector is parallel to the symmetryaxis. This is an optical depth e ff ect because in the optically-thinregime, F Q would be negative in this case. Switching to an oblateconfiguration would result in a sign change of the polarization(rotation of the polarization angle by 90 deg). In absolute terms, the polarized flux is here greater where the flux is greater. Thepercentage polarization is also greater at shorter wavelengths,albeit modestly. Theoretically, one may expect a greater polar-ization at shorter wavelength because of the greater albedo to-wards the Balmer edge (Dessart & Hillier 2011), especially atthis young SN age when the material in the spectrum formationregion is at least partially ionized. The polarization changes be-tween lines and continuum, as well as within the lines. We see asign reversal of the polarization in H α , and for many lines in thecase of a stronger asymmetry (right panel of Fig. 7). For a largescaling ( A = . P rises to a maximum of 0.6% inthe largely line-free region between Fe ii α .The observed polarization of SN 2012aw at the correspond-ing epoch di ff ers in a number of ways from the model predic-tions. It is greater at redder wavelengths, although at this epochthe polarization is quite low, and only about 0.2% at 7000 Å,thus significantly lower that the maximum value obtained withthe models discussed above (about 0.6%). Observations also re-veal a sizable change in polarization across H β and H α , but notother lines (perhaps because the data are too noisy). While H β and H α in the models also show polarization, the structure doesnot match the observations. In q the model shows an enhancedpolarization in the P Cygni absorption component (not seen inthe observations), and the polarization absorption trough is o ff -set from that observed. In P , the models show a very compli-cated structure which contrasts with the relatively smooth struc-ture seen in the observations. The di ffi culty in interpreting thisepoch is exacerbated by the uncertainties in the interstellar po-larization, the low level of polarization, and the significant signalthat remains in the rotated u parameter. Article number, page 11 of 19 & A proofs: manuscript no. ms km s − ]0.250.300.350.400.450.500.550.600.650.70 S h a p e f a c t o r γ r Fig. 9.
Evolution of the shape factor γ ( r ) in the hybrid model versusvelocity for the eight epochs modeled (the last epoch at 160 d has noobservational counterpart). The symbols correspond to the photosphericradii in the equatorial direction (square) and the polar direction (dot). For an asymmetric ejecta resulting from a radial stretching of the1D
CMFGEN model 1D-X1 (see section 3.2.2 for explanations),the results are quite di ff erent from models with a latitudinal den-sity scaling (Fig. 8). The changes to the total flux are greater thanfor the models with the density scaling. The flux F I at 7100 Åincreases by 40% (90%) as we vary the inclination from 0 to90 deg in model 2D-X1-STR0p25 (2D-X1-STR0p50; see Sec-tion 3.2.2 for details on the nomenclature). Once we renormal-ize the spectra at 7100 Å, the variation with inclination appearsmostly in lines in the form of a wavelength shift. This shift ismore noticeable in model 2D-X1-STR0p50. The polarization isalso overall lower. Varying the magnitude of the radial stretchingchanges little the resulting polarization, despite the large den-sity contrast between pole and equator (corresponding to val-ues of 14.6 and 130.0 for models 2D-X1-STR0p25 and 2D-X1-STR0p50 since the ejecta follow a power-law density with ex-ponent twelve in the outer regions). In both cases, we see strongsign reversals of the continuum polarization when the inclina-tion is changed. There are also strong reversals of the polariza-tion across line profiles for inclinations corresponding to low lat-itudes (near equator-on views). Such signatures are likely dueto optical depth e ff ects with a complicated cancellation of thelocal polarization on the plane of the sky. These can occur atearly times because asymmetry impacts the distribution on theplane of the sky of both the scatterers and the flux. Hence, forthe case of a latitudinal stretching, the model fails to reproducethe greater polarization at longer wavelengths, is unable to re-produce the level of polarization observed, and fails to match thestructure across H α . CMFGEN models
In this section, we use two distinct 1D
CMFGEN models to build anasymmetric ejecta. The ejecta are symmetric with respect to theequatorial plane so it corresponds to a bipolar explosion in whicha cone of opening angle 50 deg along the symmetry axis is char-acterized by distinct properties relative to lower latitudes. In oursetup, the di ff erence is the steeper density fall-o ff beyond about10,000 km s − and the larger Ni content within the H-rich lay- ers of the ejecta (see Fig. 3). In the context of polarization, themain impact of the Ni enhancement is not the change in opac-ity or abundance (both have a minor influence) but instead theboost it induces to the density of free electrons (the mass densitybelow about 10,000 km s − hardly changes with angle, so the im-pact is on the ionization). Hence, there is at least one source ofasymmetry at any given velocity, which implies that this setupmay produce a net polarization at any epoch. The initial condi-tions used in this case are presented in detail in Section 3.2.3,to which we refer the reader for additional information (see alsoFig. 3 − Ni enhancement a ff ects mostly the innermostejecta layers, so the asymmetry should be weak early on andgrow as the inner aspherical ejecta layers are revealed by the re-ceding photosphere. Consequently, in our hybrid model, the ratioof polar and equatorial photospheric radii evolves from 1.02, to1.08, 1.22, 1.36, 1.59, 1.52 and 1.26 as the SN ages from 16.1,to 46.0, 63.0, 71.9, 91.9, 107.9, and 120.0 d.Another method for illustrating the structure is with theshape factor defined by Brown & McLean (1977) which (to-gether with the optical depth and the inclination) fully describesthe continuum polarization properties of an axially-symmetricejecta illuminated by a point source. In this paper, we introducea depth-dependent shape factor γ ( r ) and define it as γ ( r ) = (cid:82) ∞ r (cid:82) − N e ( r , µ ) µ d µ dr (cid:82) ∞ r (cid:82) − N e ( r , µ ) d µ dr , (7)where N e ( r , µ ) is the free-electron density at ( r , µ ) ( µ being thecosine of the polar angle β ). When r = R min we obtain the expres-sion of the shape factor as in Brown & McLean (1977). Becauseof the complicated dependence of the emissivities and opacitieson depth, no single parameter can fully describe the asymmetricnature of the ejecta. The shape factor (defined as in Eq. 7) forour hybrid model is shown for all epochs in Fig. 9.In Figs. 10 −
11, we compare the multi-epoch spectro-polarimetric observations of SN 2012aw with the results of the2D polarized radiative transfer code based on this new axisym-metric ejecta configuration (the figure layout is the same as thatused in Fig. 7). Figure 10 shows the results for the first fourepochs (at 16.1, 46.0, 63.0, and 71.9 d after explosion), andFig. 11 shows the results for the last three epochs (91.9, 107.9,and 120.0 d after explosion) together with the predictions of themodel at 160 d (for which there is no data). The later epochs arebetter suited for analyzing the SN polarization since the polariza-tion level is higher and the data has a higher signal to noise ratio.The variations across lines are better resolved. For this compar-ison, we selected the inclination of 67.5 deg since it yielded agood match to the observations at all epochs except the first one.At the first two epochs, the observed polarization is low,about 0 . − . ff ect due to the competing Article number, page 12 of 19uc Dessart et al.: Modeling of the SN 2012aw spectropolarimetric obsvervations F I [ S c a l e d ] Model: [email protected] − q r o t [ % ] − u r o t [ % ] P [ % ] F I [ S c a l e d ] Model: [email protected] − q r o t [ % ] − u r o t [ % ] P [ % ] F I [ S c a l e d ] Model: [email protected] − q r o t [ % ] − u r o t [ % ] P [ % ] F I [ S c a l e d ] Model: [email protected] − − q r o t [ % ] − u r o t [ % ] P [ % ] Fig. 10.
Same as Fig. 7, but now for the hybrid model 2D-X1-X2b (for an inclination of 67.5 deg) and the observations of SN 2012aw for epochs1, 2, 3, and 4. influence of flux and optical depth (i.e., between the asymmetryof the emission and of the scatterers as seen on the plane of thesky).At later epochs (63 to 107.9 d), the hybrid model cap-tures the qualitative and the quantitative evolution observed in SN 2012aw, both for the total flux (and thus photometry) and forthe polarized flux. This is obtained without any adjustment to theejecta structure so that the change in polarization arises from theintrinsic evolution of the 2D ejecta as they expand and cool etc.
Article number, page 13 of 19 & A proofs: manuscript no. ms F I [ S c a l e d ] Model: [email protected] − − − q r o t [ % ] − − u r o t [ % ] P [ % ] F I [ S c a l e d ] Model: [email protected] − − − − q r o t [ % ] − − − − u r o t [ % ] P [ % ] F I [ S c a l e d ] Model: [email protected] − − − q r o t [ % ] − − u r o t [ % ] P [ % ] F I [ S c a l e d ] Model: 2D-X1-X2bInc=67.5degAge = 160d − − − q r o t [ % ] P [ % ] Fig. 11.
Same as Fig. 10, but now for epochs 5, 6, and 7. For the epoch 8 at 160 d, there is no observational data for SN 2012aw.
Across the profile of a strong line, the polarization is ex-pected to show a jump in the blue part of the P-Cygni profiletrough. This is observed in Na i D and the Ca ii near-infraredtriplet, but rarely in other lines. In contrast, the model shows thisas a narrow feature in most lines, including H α . The di ffi culty ofobserving this feature is aggravated by the rebinning of the data, which was used to enhance the signal-to-noise ratio per bin. Thefeature originates from the deficit of unscattered (direct and thusunpolarized) flux from the SN while the residual flux arises fromphotons scattered back into the line of sight. In an asymmetricejecta, such scattered photons yield a net polarization. In the Ca ii near-infrared triplet, the model shows the polarization jump fur- Article number, page 14 of 19uc Dessart et al.: Modeling of the SN 2012aw spectropolarimetric obsvervations < P c o n t > [ % ] P cont (SN 2012aw) P cont (Model) m V m V (SN 2012aw) m V (Model) Fig. 12.
Evolution of the observed continuum polarization of SN 2012aw, corrected for interstellar polarization (filled symbols; averaged valueover the observed range 6900 to 7200 Å) and its V -band magnitude (open circles; values shown on right axis, from Bose et al. 2013), along withmodel predictions (closed diamonds for polarization, open diamonds for photometry); note that the model photometry accounts for extinction anddistance dilution as in Hillier & Dessart (2019). The symbols at 160 d are for the model only since the last observation is at 120 d. In this hybrid2D axisymmetric ejecta model, the ratio of polar and equatorial photospheric radii evolves from 1.02, to 1.08, 1.22, 1.36, 1.59, 1.52 and 1.26 asthe SN ages from 16.1, to 46.0, 63.0, 71.9, 91.9, 107.9, and 120.0 d. ther to the blue than in the observations. Close inspection of thetotal flux F I shows that the model overestimates the width of theCa ii near-infrared triplet line at late epochs (more so at epochs5, 6 and 7), so the mismatch in q or P is probably rooted in theslight overestimate by the model of the ejecta expansion rate.In strong lines, the polarization is zero or close to zero some-where between the P-Cygni trough and the location of maxi-mum flux. This feature of line depolarization was used earlierto constrain the interstellar polarization towards the line of sightto SN 2012aw. It is clearly observed at all epochs (more so forepochs three to seven) in H β , Na i D, H α , and the Ca ii near-infrared triplet. It is also reproduced by the model. This resultsfrom the depolarization of line photons. Recombination linesemit photons isotropically and thus cannot produce a net polar-ization. The large line optical depth also leads to absorption ofpossibly polarized photons, thereby destroying the polarizationthey carried. The line photons that scatter with free electronswithin the ejecta can however produce a residual polarization.However, this is associated with a redshift (because of the largebulk velocity of the scatterers) so this polarized flux appears inthe red wing of each line. As we move to the red wing of lines, the polarized flux islarge, and larger for stronger lines. This arises from line photonsscattered by free electrons. The e ff ect yields an excess total fluxin the red wing of all lines. Because of the asymmetry of the dis-tribution of free electrons, it also yields a clear polarized flux F Q , If the scattering was done by an external scatterer at rest, such as adistant dust sheet, line photons could be associated with a residual polar-ization even at line center since the scattering process would introduceno velocity shift. whose strength scales with F I . However, P is at the same levelin the red wing of lines and in the adjacent continuum. This sug-gests that line photons and continuum photons pick up the samelevel of polarization when they last scatter in the ejecta (and thatstatistically they bear the same level of polarization before thatlast scattering). The observed polarization behavior across linesis quite distinct in type Ia SNe, which show a maximum polar-ization within the absorption trough of the strongest lines and alow level of polarization elsewhere (see, e.g., Wang et al. 2003).In general there is good agreement between the observed andtheoretical line profiles, especially after the first two epochs. Onediscrepancy is that the model profiles exhibit a larger variationin polarization at the blue edge of the P Cygni absorption thando the observations. Particularly striking is the agreement for thepolarization across the O i i α .In the continuum, the level of polarization increases withwavelength in the optical range, although this e ff ect, which ismatched by the model, is quite weak. At late times (epochs fourto seven), in the region of line blanketing shortwards of 5000 Å,the model predicts no clean (line free) continuum region sostrong line depolarization occurs. In the red part of the spectrum,and in particular between H α and O i ii Article number, page 15 of 19 & A proofs: manuscript no. ms
Other discrepancies convey some important information onthe source of polarization and departures from what is adoptedin the hybrid model. Clearly visible at epochs 5 and 6 is the over-estimate of the polarized flux associated with the scattered andred-shifted H α photons, appearing as a bump in q or P . In con-trast, the observations suggest the same level of polarization forthese photons and the adjacent continuum-only photons furtherto the red. This mismatch is, however, only so obvious in H α andnot, for example, in Na i D. The origin of the discrepancy mightbe an inadequate Ni distribution which causes a too extendedboost to the electron density (for example, a more localized en-hancement in Ni at large velocity may be more suitable).Another important discrepancy, which could be related to thepoint discussed in the previous paragraph, is the lack of modelpolarization in the blue part (say < ff ected by line blanketing due to Fe ii and Ti ii ). Onesimple way of resolving this issue is to invoke a more distant,i.e. a more external scattering source, for example in the forma higher velocity Ni enhancement. In this context, this exter-nal scatterer would yield a polarization that would scale with thenumber of incoming photons. The polarization in the blue couldthen be slightly smaller than at long wavelength because of theresidual influence of line opacity at large velocities in the ejecta.In the comparisons shown in Figs. 10 and 11, the observedpolarization (second and third panels from top) was rotated sothat the bulk of the polarization lies in the flux F Q . Doing this,the flux F U is close to zero and shows only modest variationswith wavelength, in contrast to what is observed in F Q (this ismore easily seen at late epochs when the polarization in line-free regions reaches 0.5 to 1.0%). At all epochs except the firstone, the flux F Q is negative at all wavelengths in both the modeland the observations. Together with the fact that F U is close tozero and much smaller (in magnitude) than F Q , this propertysuggests that the ejecta is primarily axisymmetric. This propertyis also evident from the data distribution in the ( q , u ) plane shownin Fig. 6.Figure 12 summarizes the evolution of the continuum polar-ization (taken from the averaged value over the range 6900 to7200 Å) as well as the V -band brightness for both SN 2012awand for the hybrid model discussed in this section. The modelmatches well the light curve for the plateau duration and bright-ness, and for the tail brightness. This occurs because the asym-metry has little e ff ect on the total observed flux – the hybridmodel displays the same light curve and spectral evolution asthe spherically symmetric model 1D-X1 (with the exception ofa slight overestimate of the H α line strength and width). The hy-brid model also matches the low level of polarization at earlytimes and its steep rise in the second half of the plateau. Thehybrid model then predicts that the continuum polarization willdrop at late times, essentially following the drop in optical depth.Under optically-thin conditions, the polarization scales linearlywith τ (Brown & McLean 1977) and in SN τ drops as 1 / t atlate times (assuming constant ionization). Such an evolution hasbeen observed over a sizeable timeline (Leonard et al. 2006) andexplained by polarization modeling (Dessart & Hillier 2011). F I [ S c a l e d ] − − F Q [ n o r m a li z e d ] P [ % ] β = β = β = Fig. 13.
Influence of the half opening angle β / on the total flux F I (top), the normalized polarized flux F Q (middle) and the percentagepolarization P (bottom) for the hybrid model 2D-X1-X2b at 107 d andusing an inclination of 67.5 deg. [See Section 6.1 for discussion.]
6. Uniqueness of the solution
In the preceding section, we have explored various configura-tions for the aspherical (2D axisymmetric) ejecta. Here, we studythe sensitivity of our results to changes in the 2D ejecta structurein the hybrid model. The hybrid model used model 1D-X2b forthe polar latitudes (corresponding to a cone with an opening an-gle of 50 −
60 deg), and model 1D-X1 for other latitudes. In thissection, we describe the results when the half opening angle ofthe cone is increased (Section 6.1), when we drop the assump-tion of mirror symmetry (equivalent to comparing a bipolar and aunipolar explosion; Section 6.2), and when the explosion energyof the ejecta in the polar regions is increased (Section 6.3).
Figure 13 shows the influence of the half opening angle β / on the radiative signatures for model 2D-X1-X2b at 107 d. Inthis particular setup, model X2b extends from the pole up to β / of 11.25, 22.5, or 33.75 deg. For an inclination of 67.5 deg,we find that the normalized polarized flux (middle panel ofFig. 13) is qualitatively independent of β / , while quantitatively,the percentage polarization increases with β / (bottom panel ofFig. 13). For lower inclinations, optical depth e ff ects cause a signflip (rotation by 90 deg) of the polarization F Q for di ff erent val-ues of β / . Such optical depth e ff ects complicate the analysisof the observed polarization during the photospheric phase. Fur-thermore, the polarization increase with increasing β / occursup to a half opening angle of about 60 deg, beyond which theconfiguration is somewhat equivalent to swapping model X1 andX2b for a half opening angle of π/ − β / . Article number, page 16 of 19uc Dessart et al.: Modeling of the SN 2012aw spectropolarimetric obsvervations F I [ S c a l e d ] − − F Q [ n o r m a li z e d ] P [ % ] Mirror sym.No mirror sym.
Fig. 14.
Same as Fig. 13, but now showing the impact of the assumptionof mirror symmetry. With (without) mirror symmetry, the ejecta (or theexplosion) is bipolar (unipolar). [See Section 6.2 for discussion.]
Figure 14 shows the results for the hybrid model 2D-X1-X2bwith and without the assumption of mirror symmetry. The an-gle β / is 22.5 deg and the inclination is 67.5 deg with respectto the symmetry axis. With mirror symmetry, the ejecta has abipolar morphology, with the properties of model X2b along thepoles. Without mirror symmetry, the ejecta is unipolar so thatmodel X2b only covers polar angles smaller than 22.5 deg. InFig. 14, we adopt an inclination of 67.5 deg. In this case, themodel with mirror symmetry (i.e., bipolar explosion) shows apercentage polarization in the 7000 Å region that is more thantwice greater than when mirror symmetry is ignored (i.e. unipo-lar explosion). The o ff set di ff ers from a factor of two because ofoptical depth e ff ects. For example, for an inclination of 10 deg,the percentage polarization is the same in both cases becausethe asymmetry from the receding part of the 2D ejecta is ob-scured. For an inclination of 45 deg, the level of polarization isvery di ff erent (sign flip, di ff erent absolute polarization level) be-tween the two cases, again because of optical depth e ff ects. Foran inclination of 90 deg (edge on), the percentage polarizationin the 7000 Å region is exactly twice larger if mirror symme-try is assumed rather than ignored. In the mirror-symmetry case,each hemisphere contributes the same residual polarization andthe same flux at any wavelength. Without mirror symmetry, onlyone hemisphere contributes, yielding exactly half the continuumpolarization level obtained with mirror symmetry. Finally, what-ever the inclination, the total flux F I is essentially identical inboth cases. Figure 15 compares the ejecta and radiative properties of the hy-brid models 2D-X1-X2b and 2D-X1-X2B, which di ff er in thatthe polar regions have an enhanced Ni mass fraction (for-mer model) or an enhanced explosion energy (latter model). Al- though these two asymmetric ejecta are produced by very dif-ferent means, they yield very similar total and polarized fluxes.The reason is that in the polar direction, the higher explosion en-ergy leads to a greater density at large velocities. Although theionization is the same along all latitudes (because of the simi-lar influence of the deeply embedded Ni), the electron densityvaries with latitude, reflecting the variation in mass density (seeleft panel of Fig. 15). This polar enhancement in electron den-sity (not caused by an excess in Ni), produces the continuumpolarization in model 2D-X1-X2B. In contrast, the mass densityin model 2D-X1-X2b is essentially constant with latitude but thegreater Ni mass fraction along the pole yields a greater ioniza-tion and thus a greater density of free electrons along the polesthan at lower latitudes. The results in Fig. 15 indicate that suchdistinct physical processes (polar enhanced Ni mixing or polarenhanced explosion energy) yield a very similar polarization sig-nal. This emphasizes the degeneracy of polarization signatures.
7. Conclusion
We have presented VLT − FORS spectropolarimetric observa-tions of the type II SN 2012aw spanning seven epochs of the pho-tospheric phase until the onset of the drop-o ff from the plateau.Unfortunately, no spectropolarimetric observation of SN 2012awwas made during the nebular phase. SN 2012aw presents po-larization characteristics that are qualitatively and quantitativelysimilar to what has been observed in type II SNe so far, such asthe emblematic SN 2004dj (Leonard et al. 2006). A generic prop-erty of Type II SNe, corroborated here with 2012aw, is the rel-atively low polarization early in the plateau phase (although seethe counter-example of SN 2013ej; Leonard et al. 2015; Mauer-han et al. 2017, or SN 2017gmr; Nagao et al. 2019) and theprogressive rise to a maximum polarization of about 1 % as theejecta turns optically thin at the end of the high-brightness phase.By rotation of the Stokes vectors, it is possible to place mostof the SN 2012aw polarization along a single, fixed polarizationaxis, which implies that the ejecta of SN 2012aw has a dominantaxis and that the polarization arises from ejecta with an oblate orprolate morphology.We have modeled the SN 2012aw linear polarization usinga long characteristic code that assumes axial symmetry (Hillier1994, 1996; Dessart & Hillier 2011; Hillier & Dessart 2021).The code provides the full optical total and polarized flux F Q (for symmetry and geometry reasons, F U is zero) at any epochduring the photospheric or nebular phase. At present, the favoredoperation mode for the code is to build a 2D axially-symmetricejecta using two 1D CMFGEN models and assigning them a spe-cific range of latitudes. Such a hybrid 2D model can thus mimican explosion with a higher energy or a higher Ni mass fractionalong specified latitudes.Our modeling results support the notion that the intrinsic po-larization is negligible within strong lines somewhere betweenthe location of maximum absorption and the location of maxi-mum line flux. We use this property to correct for the interstel-lar polarization. With this choice, the intrinsic polarization ofSN 2012aw is small but non zero at the earliest epochs.The parameter space for producing such 2D hybrid modelsis extended since we may use any 1D
CMFGEN from the largeset of models that we have calculated for type II SNe over theyears. We also have freedom when assigning a given model and agiven latitude. Having selected two models and a specific model-latitude assignment to produce a 2D ejecta, the same setup isused for all epochs. No further adjustment is made to the setupduring a sequence of polarization calculations. In the present pa-
Article number, page 17 of 19 & A proofs: manuscript no. ms − − − l o g ( ρ )[ g c m − ) − − − − − X ( N i ) V [10 km s − ]6810 l o g ( N e / c m − ) F I [ S c a l e d ] Asymmetric explosion energy − − F Q [ n o r m a li z e d ] P [ % ] Fig. 15.
Left: Comparison of the mass density, Ni mass fraction, and electron density for the 1D models used to build the 2D ejecta 2D-X1-X2b and 2D-X1-X2B. Right: Same as Fig. 13, but now comparing the results for the hybrid model 2D-X1-X2b (bipolar ejecta with an enhanced Ni mass fraction along the polar regions) and the hybrid model 2D-X1-X2B (bipolar ejecta with an enhanced explosion energy along the polarregions). In both models the angle β / is 22.5 deg. [See Section 6.3 for discussion.] per, we focused on one hybrid model composed of model 1D-X1 (also named x1p5 in Hillier & Dessart 2019) and of model1D-X2b, which deviates from model 1D-X1 by having enhanced Ni mixing at large velocities. For an inclination of ∼
70 deg,we find that this hybrid model can reproduce the SN 2012aw po-larization at all epochs except the first two, without any adjust-ment. The model predicts the polarization is maximum in the linefree region between H α and O i ff ery 1991a), we also find that the po-larization signatures are degenerate so other configurations canalso reproduce the observations. Acknowledgements.
D.C.L. acknowledges support from NSF grants AST-1009571, AST-1210311, and AST-2010001, under which part of this researchwas carried out. D.J.H. thanks NASA for partial support through the astrophysi-cal theory grant 80NSSC20K0524. Support for G.P. is provided by the Ministryof Economy, Development, and Tourism’s Millennium Science Initiative throughgrant IC120009, awarded to The Millennium Institute of Astrophysics, MAS.This work was granted access to the HPC resources of CINES under the alloca-tion 2018 – A0050410554 and 2019 – A0070410554 made by GENCI, France.This research has made use of NASA’s Astrophysics Data System BibliographicServices. Based on observations collected at the European Southern Observatory,Chile, under programme 089.D-0515(A).
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