Three-dimensional modeling of Type Ia supernovae - The power of late time spectra

a r X i v :a s t r o -p h /0504317v 1 14 A p r 2005

Astronomy &Astrophysics manuscript no.3D˙mod February 2,2008

(DOI:will be inserted by hand later)

Three-dimensional modeling of Type Ia supernovae -The power

of late time spectra ?

Cecilia Kozma 1,Claes Fransson 1,Wolfgang Hillebrandt 2,Claudia Travaglio 3,2,Jesper Sollerman 1,

Martin Reinecke 2,Friedrich Konrad R¨o pke 2,Jason Spyromilio 4

1

Stockholm Observatory,AlbaNova,Department of Astronomy,10691Stockholm,Sweden

2

Max-Planck-Institut f¨u r Astrophysik,Karl-Schwarzschild-Strasse 1,D-85741Garching,Germany 3

Istituto Nazionale di Astro?sica (INAF)-Osservatorio Astronomico di Torino,via Osservatorio 20,10025Pino Torinese (Torino),Italy 4

European Southern Observatory,Karl-Schwarzschild-Strasse 2,D-85748Garching,Germany

Received —?,Accepted —?

https://www.wendangku.net/doc/4410025761.html,te time synthetic spectra of Type Ia supernovae,based on three-dimensional de?agration models,are presented.We mainly focus on one model,“c3256

Send o?print requests to :cecilia@astro.su.se ?

Based on observations collected at the European Southern Observatory,Paranal,Chile (ESO Programmes 67.D-0134and 69.D-0193).

rapidly increasing number of observed SNe Ia at interme-diate and high redshifts,attempts are made to constrain

also the equation of state of the “dark energy”that is thought to be the cause of the acceleration.

However,in any such studies it is important to under-stand the systematic errors,which are here likely to dom-inate.The intrinsic dispersion,i.e.originating from prop-erties of the explosions cannot be controlled by observing a large number of SNe Ia.We therefore here compare re-alistic models to detailed observations to get a better un-derstanding of the physics leading to the homogeneity of the observed properties,and the inherent uncertainties.The presently favored model for a Type Ia super-nova is the thermonuclear explosion and disruption of an accreting carbon-oxygen white dwarf with a mass close to the Chandrasekhar limit.Guided by the presence of intermediate-mass elements in the observed spectra these models assume that thermonuclear burning begins as a

2Kozma et al.:3D modeling

(subsonic)de?agration,which may or may not change into a(supersonic)detonation at later times.Recently,there has been great progress in the modeling of such explo-sions by means of combustion-hydrodynamics codes.In particular,three-dimensional de?agration calculations of increasing complexity have been performed by a number of groups,Reinecke et al.(2002a,2002b),Gamezo et al. (2003)and Garcia-Senz&Bravo(2005).Although there are rather large di?erences in the ways turbulent ther-monuclear?ames are modeled by the groups,they arrive at similar conclusions,namely that the chemical structure of the supernova should be highly inhomogeneous.In par-ticular,they?nd substantial amounts of unburned carbon and oxygen close to the center.Whether this is a result of inadequate modelling of the nuclear burning,or is a real feature can best be decided by comparing model spectra with observations.

In this paper we demonstrate the possibility of testing the multidimensional hydrodynamics and nucleosynthesis calculations by studying the late time spectra.The emis-sion at late times,https://www.wendangku.net/doc/4410025761.html,ter than～100days,is predomi-nantly emerging from the central parts of the supernova, where the e?ects of the explosion are most pronounced. Therefore,by comparing model spectra,based on di?er-ent explosion models,to observations one can draw conclu-sions about the explosive nucleosynthesis,hydrodynamic mixing,the amount of56Ni formed,and the total energy of the explosion.In contrast,early spectra are more sen-sitive to the chemical structure of the high velocity outer regions.The two phases therefore nicely complement each other.The input models used in this study come from the calculations by Travaglio et al.(2004),who have made a detailed post processing nucleosynthesis of the hydrody-namic models by Reinecke et al.(2002a,2002b)and R¨o pke (2005).The hydrodynamic code only had a minimal nu-cleosynthesis network included,su?cient for a good ap-proximation of the thermonuclear energy release.

In section2we?rst summarize the three-dimensional hydrodynamical models and the nucleosynthesis we use as input to our spectral code.Then we discuss the late-time spectral modeling.Our results are given in section3.In section4we discuss our?ndings,and what we can learn from late time spectra.Finally in section5we summarize our results.

2.Modeling

2.1.3D hydrodynamical modeling

Our input models for the spectral-synthesis code come from the multidimensional hydrodynamic calculations of Reinecke et al.(2002a,2002b).In order to save computer time with still su?cient numerical resolution,only an oc-tant of the star was simulated and mirror symmetry was assumed at the inner boundaries.This arti?cial symmetry causes some problems with the shapes of the emission lines of the nebular spectra,as will be discussed later.Recently R¨o pke(2005)has evolved one of their three-dimensional explosion models up to10seconds,at which time the as-sumption of homologous expansion is valid.

The simulations of Reinecke et al.(2002a,2002b)at-tempt to model the nuclear burning and hydrodynam-ics from?rst principles,as far as possible.The nuclear burning is modeled as a subsonic,turbulent de?agration. Because the length scales vary from sub-mm scales to di-mensions comparable to the radius of the white dwarf, some simplifying assumptions about the physics of the ?ame front on small length scales have to be invoked.The ?rst assumption is that since the?amefront cannot be numerically resolved it is replaced by a sharp disconti-nuity between“fuel”and“ashes”.This discontinuity is then modeled by a level-set function.The remaining un-known is the normal velocity of the level set,i.e.the tur-bulent?ame speed.Since in the?amelet regime of tur-bulent combustion(valid for Type Ia supernovae down to densities of around107g cm?3)the?ame velocity is independent of the physics on microscopic scales,it can be determined from a sub-grid scale model of the unre-solved turbulent velocity?uctuations.These,in turn,re-sult from a Kolmogorov-like turbulent cascade with energy fed in by large-scale hydrodynamic instabilities,predomi-nantly the Rayleigh-Taylor(buoyancy-driven)instability. The important parameters determining the evolution of the explosion are the central density and the composition of the white dwarf.All simulations used in the present study started with a central density of2.9×109g cm?3 and equal carbon and oxygen mass fractions.

The numerical experiments in Reinecke et al.(2002a, 2002b)show that the way the explosion is initiated can have large e?ects on the outcome,especially on the ex-plosion energy and the56Ni mass.This is likely to be determined by the evolution leading up to the ignition. According to Garcia-Senz&Woosley(1995)possible ig-nition conditions could be a couple of?oating“blobs”of burning material accelerated to a fraction of the sound ve-locity near the center of the star.Alternatively,large scale convective motions could lead to ignition in extended re-gions at the center(H¨o?ich&Stein2002),or a multi-point ignition with an exponentially increasing and asymmetric number of ignition points out to approximately150–200 km o?-center(Woosley,Wunsch&Kuhlen2004).

In the simulations by Reinecke et al.the e?ects of the ignition were explored by assuming di?erent ignition topologies.In particular,in one set of models the explosion is initiated at the center,while in a second set of calcula-tions the explosion starts in a?nite number of randomly distributed,?oating bubbles.The result of this is that the more complex the initial conditions are,the more56Ni seems to form and the stronger the explosion energy.This is not too surprising because,in general,a more complex geometry has a larger surface area,increasing the burning rate and leading to a higher energy.Also,the amount of unburned carbon and oxygen that is dredged-down to the center between the rising Rayleigh-Taylor blobs depends on the geometry of the ignition region,but this e?ect has not yet been fully explored.

Kozma et al.:3D modeling3

Finally,the numerical resolution is important,mostly for the nucleosynthesis but,to a lesser extent also for the kinetic energy of the explosion.This is especially impor-tant for the mass of56Ni produced and also the amount of unburned carbon and oxygen near the center of the star,and both are crucial parameters for the calculations of synthetic spectra.The reason can easily be understood. For instance,the number of initial burning blobs is limited by the number of grid points.Low resolution means few blobs with rather large sizes,in contrast to the few kilome-ters expected from analytical models(Woosley,Wunsch& Kuhlen2004).These large blobs rise fast and burn little fuel,leaving behind lots of unburned gas.In contrast,high resolution allows for more realistic initial conditions and thus for more burning early on in the explosion.Similar arguments hold for other initial conditions.The further evolution,however,is much less a?ected because once tur-bulence has developed the subgrid-scale model takes care of resolution e?ects.Resolution studies underway(R¨o pke &Hillebrandt2005c)seem to indicate that even compu-tations with(static)grids of5123mesh points may not describe the early evolution correctly as far as the nucle-osynthesis yields are concerned.We will come back to this question later.

In the present paper we will use three di?erent 3D models,referred to as b30768,b5256,and c3256.In Table1we summarize the main charac-teristics of the di?erent hydrodynamical models studied here.The label c3refers to central burning,while b30and b5refer to models,where the burning starts in30and 5blobs,respectively.The last number refers to the res-olution of the model,in theses cases7683or2563grid points,respectively.While the b5256,and c3256 are from Reinecke et al.(2002b),the high resolution model b30768is discussed in Travaglio et al.(2004).From what we discussed in the previous paragraph it is obvious that the nuclear abundances obtained from the models with low numerical resolution need to be considered with caution.

The calculations by Reinecke et al.(2002b)follow the explosion up to1.2–1.5seconds.After that time the outermost layers are’escaping’their?xed Eulerian grid. However,a problem for any spectral modeling is that the ejecta have at that time not yet reached homologous ex-pansion,but the velocities and densities are still a?ected by pressure and gravity.This problem has recently been solved by R¨o pke(2005),who calculated the hydrodynam-ics for the c3256model by Reinecke et al.(2002b)up to10seconds after the explosion,using a moving grid. We will in this paper refer to this model as c3256

3d10s model,and just brie?y discuss the models b30768, c3256and b5256with respect to the initial condi-tions and resolution.

The kinetic energies for the models are given in Table1.These energies all refer to the end of the sim-ulations,i.e.at1.2–1.5s.However,as shown by R¨o pke

(2005),the kinetic energy does not change much between 1.5s and10s.The maximum velocity for the tracer parti-

cles(and the supernova ejecta)in the c3256

3d3d 3d

3d10s.The nuclear reac-

tion network contains383nuclear species,and is based on Thielemann et al.(1996)and Iwamoto et al.(1999).Weak

interaction rates are updated from Langanke&Martinez-

Pinedo(2000)and Martinez-Pinedo et al.(2000).

Initially,the tracer particles contain only12C,16O,

and22Ne.In all the models used in the present study for ～30to40%of the particles the temperatures are suf-?ciently high for nucleosynthesis to transform the initial

composition all the way up to iron group elements.In the remaining part,on the other hand,no or only incom-plete burning occurs,depending on the temperatures and

4Kozma et al.:3D modeling

densities.As the nucleosynthesis proceeds up to10sec-onds,the?nal abundances we use in our nebular calcula-tions are somewhat di?erent from those in the c3256 model at1.5seconds.This is mainly a consequence of the expanding computational grid in the model followed to 10seconds,which reduces the?ame resolution.The mass of carbon in unburned particles is0.31M⊙and of oxy-gen0.35M⊙.Including also the partially processed par-ticles,the total mass of carbon and oxygen in the model is0.34and0.42M⊙,respectively.The remaining parti-cles have undergone nuclear processing,resulting in inter-mediate mass elements and iron peak elements.We refer to these as burned particles.The total mass of56Ni is in the c3256

3d10s model.

https://www.wendangku.net/doc/4410025761.html,te time spectral modeling

The modeling of late time spectra is based on the code described in detail in Kozma&Fransson(1998a).The code was originally used to model Type II supernovae and has been applied to SN1987A(Kozma&Fransson1998a; Kozma&Fransson1998b).Since then the code has been improved in a number of ways,and has also been extended to apply to Type Ia supernovae(Sollerman et al.2004).We here therefore summarize the main features and changes of the model.

The energy input to the ejecta at these late epochs is due to decays of radioactive elements formed in the ex-plosion.In our calculations we include decays of56Ni to 56Co,and56Fe,as well as decays of57Ni and44Ti.We cal-

culate the amount of gamma-ray and positron energy de-posited into heating,ionization,and excitation,by solving the Boltzmann equation,as formulated by Spencer&Fano (1954).The relative fractions going into these channels de-pend on the composition and degree of ionization.In our model the gamma-ray and positron deposition is therefore calculated for each tracer particle separately.Our treat-ment of the nonthermal deposition is described in detail in Kozma&Fransson(1992).

The temperature,ionization and level populations in each tracer particle are calculated in steady state for a speci?c time by solving the statistical equations,together with the energy balance.We?nd that steady state is a good approximation up to500days.At later times de-partures from steady state become increasingly important.

The following ions are included:H I-II,He I-III,C I-III,N I-II,O I-III,Ne I-V,Na I-II,Mg I-III,Si I-III,S I-V, Ar I-V,Ca I-III,Fe I-V,Co I-V,Ni I-II.Among these the following are treated as multilevel atoms,with the number of levels within brackets:H I(20),He I(16),O I(13),Mg I(10),Si I(21),Ca II(6),Fe I(121),Fe II(191),Fe III (112),Fe IV(45),Ni I(14),Ni II(18),Co II(108),Co III(87).To calculate the cooling and line emission from the remaining ions,the level populations are calculated in steady state,using two-,?ve-,or six-level atoms.

Details and references for most of the atomic data used are given in Kozma&Fransson(1998a).In addition to the atomic data given in this paper,the code has been updated speci?cally to model Type Ia’s(Sollerman et al.2004).In particular,higher ionization stages of especially the iron group elements have been included.The following charge transfer reactions important for the modeling of Type Ia’s have been added:Fe I+Fe II→Fe II+Fe I,Fe I+Fe III→Fe II+Fe II,and Fe I+Fe IV→Fe II+Fe III. Rate coe?cients for these three reactions are from Krstic, Stancil,&Schultz(1997),as given by Liu et al.(1998).

We have improved the modeling of iron by updating the total recombination rates and including rates to indi-vidual levels for Fe I(Nahar,Bautista,Pradhan1997),Fe II(Nahar1997),Fe III(Nahar1996).

Both Co II and Co III are included as multilevel atoms. For Co II,with108levels,A-values are from Kurucz(1988) and Quinet(1998),and the energies are from Pickering et al.(1998).For Co III,with87levels,all data are from Kurucz(1988).Recombination rates to Co II and Co III are unknown.To estimate recombination rates for these ions,we use the same values as for Fe II and Fe III.We set the total recombination rates to Co II and Co III equal to the total recombination rates to Fe II and Fe III,and assume that the distribution to individual levels are simi-lar to the individual levels in Fe II and Fe III.

Also collision strengths for the Co II and Co III tran-sitions are unknown.As a rough approximation we set the collision strengths equal to?=0.1for forbidden transitions,and for the allowed transitions we use Van Regemorter’s formula(van Regemorter1962).As we show below,the contribution of Co II-III is negligible at the epochs of interest,and the lack of atomic data for these ions is therefore for this application not serious.

In all calculations presented in this paper we assume photoionization to be unimportant.As was argued in Kozma&Fransson(1998b),the reason for this is that the high energy photons responsible for the photoionization are absorbed and scattered by lines in the ejecta.Because the number of resonance transitions at these energies are extremely large multiple resonance scattering,followed by branching into the optical,will e?ciently reduce the UV-?eld,responsible for the photoionization.In the optical range we do take?uoresence by the Ca II H and K lines ex-plicitly into account in our calculation of the spectra.This is done by assuming that all emission within6000km s?1, on the blue side of the H and K lines,to be absorbed and re-emitted in the Ca II IR-triplet and theλλ7291,7324 lines.This is the dominant mechanism for the formation of these lines(Kozma&Fransson1998b).Other permit-ted lines are included,using the Sobolev approximation to treat the scattering.We do,however,not include?uores-ence and scattering from line to line.Branch et al.(2005) compare spectra modeled by the synthetic spectrum code Synow,which include multiple scattering,but not any cal-culation of the level populations,to observations of

SN

Kozma et al.:3D modeling5 1994D at nebular phases.From this they suggest that the

e?ects of resonance scattering are seen at～300days in

Ca II H,K and IR-triplet,and Na I.This is in agreement

with the fact that these lines,as well as several Fe II lines,

are optically thick also in our calculations.

3.Results

In this section we discuss the late-time spectra based on

the c3256

3d3d3d

3d10s model.As we will show,the physical con-

ditions in the ejecta,and therefore the spectrum,are sen-sitive to the density structure.Therefore,it is not possi-ble to draw any?rm conclusions from late time modeling based on the1.2s and1.5s models.They might,however, indicate the e?ects of di?erent numerical resolution and initial burning conditions on the late time spectra.

Because of the importance of the density on the physi-cal properties and the emission,we have plotted the distri-bution of the number densities at300days for the Fe-rich, intermediate mass elements,and

unburned tracer parti-cles in model c3256

3d10s model. The spectrum at300days is analyzed in more detail in Fig.4,where we show the contributions from the lines of Fe I,Fe II,Fe III,Ni II,Si I,O I,and C I.

The main di?erence between the spectra at the two epochs is the decrease in ionization from day300to day 500,as can be seen in the abundances of the iron ions.At day500the emission from Fe I has increased,and the emis-sion from Fe II and III has decreased somewhat compared to the emission at300days.At both epochs,however,the dominant ion is Fe II.

The most interesting results of these calculations are the very strong lines from carbon and oxygen.For both epochs[C I]λλ9824,9850and[O I]λλ6300,6364are dominating the spectrum.At300days also C II]λ2326, C Iλλ2966,2968and[C I]λ8727are strong,but they decrease in strength at day500.With regard to the UV lines,it should be noted that the spectrum below～3500?A is very uncertain due to the e?ects of multiple scattering.

In the lower panel of Fig.3we show the IR spectrum between1and2.5μm.At these wavelengths there are no strong oxygen or carbon lines.The spectrum is dominated by Fe II,but also shows strong Si I and Ni II lines.The strongest lines are[Fe II]1.257μ,1.644μand[Si I]1.099μ. The only intermediate mass element giving rise to a sig-ni?cant emission feature in the models is silicon.As can be seen in Fig.3,the[Si I]1.099μline is strong in the 300day spectrum,while it has almost disappeared at500 days.Cobalt does not contribute signi?cantly to the spec-tra at300and500days,and the strength of the cobalt

6Kozma et al.:3D modeling

lines decreases with time.At 300days only ～7%of the radioactive cobalt remains.Uncertainties in especially the cobalt collisional excitation and recombination data are therefore of minor importance.The strongest cobalt line in the models is [Co II]1.019μ.

While most line features in both the optical and IR are severely blended,the IR lines are the ones least a?ected.For studies of line pro?les these are therefore the most suitable.In particular the Ni II line from stable 58Ni at 1.940μis from this point of

view especially interesting.In Fig.5we compare a model spectrum at 350days to observations of SN 1998bu on day 398,SN 2000cx on day 378,and SN 2001el on day 336after explosion.We have renormalized the ?ux of the three spectra to 350days,using a decline rate of 0.0138mags per day in the V band (Sollerman et al.2004).We have also scaled the observed spectra to a common distance of 10Mpc.

The spectrum of SN 1998bu,taken from Spyromilio et al.(2004),was obtained at 398days after explosion.This is one of the best S/N spectra available at this epoch,and one of the few covering the region above 8000?A .A problem is,however,that the spectrum is contaminated by a light echo (Cappellaro et al.2001),which especially in the blue gives a substan-tial contribution to the ?ux.We have removed this as discussed in Spyromilio et al.(2004).For the reddening of SN 1998bu we adopt E B ?V =0.30and a distance of 11Mpc (Suntze?et al.1999).SN 1998bu has a normal luminosity and decline rate.

SN 2000cx was observed using VLT/FORS 360days past maximum light.These observations were performed with the 300V grism,an order-sorting ?lter GG375and a 1.′′3wide slit.The data were reduced in a standard way,including ?ux-calibration using spectrophotmetric stan-dard stars,as outlined by Sollerman et al.(2004).For the smoothed spectrum of SN 2000cx shown in Fig.5we use a reddening E B ?V =0.08(Schlegel et al.1998)and a distance of 33Mpc (Candia et al.2003).SN 2000cx was a rather peculiar SN Ia around maximum light,peculiari-ties including an unusual light-curve and an unusual color evolution (Li et al.2001).However,as shown in Sollerman et al.(2004),at later phases the light curve for this super-nova actually behaved quite normally.In Fig.5we also compare the late spectrum from SN 2000cx to the two other SN Ia and we ?nd that,at these epochs,the three spectra are quite similar.

The observations of SN 2001el were obtained with the same instrumental set-up as for SN 2000cx at an epoch of 318days past maximum.For the reddening we adopt E B ?V =0.25and a distance of 17.9Mpc (Krisciunas et al.2003).SN 2001el was a rather normal SN Ia as far as its luminosity and decline rate are concerned (Krisciunas et al.2003),but it was the ?rst “normal”SN Ia that showed strong intrinsic polarization and a high velocity detached Ca II IR triplet,interpreted as a clumpy shell (Kasen et al.2003).

The ?rst thing to note from the comparison of the model with the observed spectra is the general similarity

of most of the strong features in the spectrum.In particu-lar,this applies to the Fe II-III features,which are all seen,albeit with somewhat di?erent relative ?uxes (see below).The absolute level of the model spectrum is lower than the observed spectra,which is explained by the low 56Ni mass

in the c3

2563d 10s

model is not able to satisfactorily reproduce the charac-teristic three-bump feature between 4500?A and 5500?A .In particular,the observed strong bump at ～4700?A ,which is mainly due to Fe III (Fig.4),is too weak in the model,as can be seen in Fig.5.This indicates that the model has a too low degree of ionization.

Another slight discrepancy is the feature seen in the observations between 5700and 6200?A ,but which is ab-sent in the model.This region is in the model dominated by Fe II emission.In addition Si I,Fe I,Fe IV,Co II,and Ni II have transitions at these wavelengths.

In Fig.6we compare the modeled IR spectrum at 350days to observations of SN 1998bu at 344days from Spyromilio et al.(2004).We ?nd that the model in this wavelength region is able to reproduce most of the fea-tures seen in the observations.In particular,the relative strengths of the di?erent Fe II lines agree very nicely with the observations.There is an indication that the [Si I]1.099μline is also present.Unfortunately,the observed spectrum only covers half of the line.

The densities in the other three explosion models,which were only calculated up to 1.2–1.5seconds,are signi?cantly higher than for the 10second model,if scaled to the same epoch.If run to the homologous stage they would,however,most likely end up with a lower density.In particular,the b30768model has the highest ki-netic energy and may therefore end up as the lowest den-sity model.Nevertheless,these models demonstrate that higher densities result in lower temperatures and lower de-grees of ionization.Therefore,for these models the Fe III bump is completely absent.

Kozma et al.:3D modeling7 In the near-IR,the low degree of ionization in these

models result in a strong[Fe I]1.443μline,which is very

weak in the c3256

3d3d3d

3d10s model.This depends

mainly on the di?erences in densities.The higher the den-

sity is,the smaller the spread in temperature will be.

Also for particles not burned all the way to NSE,rich

in intermediate mass elements,we?nd that particles with

similar compositions gather along certain boundaries.The

temperatures for some of these particles are even higher

than for the Fe-rich particles rich in56Ni,due to the less

e?cient cooling in these particles.

3.3.Ionization

In the right panels of Fig.7we show the electron fraction

as function of density within each Fe-rich mass element at

300and500days for the c3256

8Kozma et al.:3D modeling

>～0.1,in the high density regions.Conversely,Fe III

becomes increasingly important as energy input increases.

Similar to the temperature,the spread in electron frac-

tion for the Fe-rich particles in the b30768,b5256,

and the c3256models are smaller than for the

c3256

3d10s model with varying amounts of

unburned matter.The results are shown in Fig.8,where

we show the spectra in the wavelength regions contain-

ing the[O I]λλ6300,6364and[C I]λ8727lines,to-

gether with observations of SN1998bu,SN2000cx,and

SN2001el.The absolute?ux calibration of the spectra of

SNe1998bu and2000cx is performed by comparison to

contemporary broad band photometry.For SN2001el we

use a late time magnitude based on the assumption that

this supernova has the same time-evolution(?m350

)as

V

the well observed SN1996X.We integrate the spectra un-

der the V band?lter function and scale it to match the

extrapolated late V band magnitude.Finally,the spec-

tra are corrected for Galactic extinction,transferred to a

common distance and scaled to a common epoch.

The dotted curve in Fig.8shows the original model.

The model contains in total0.42M⊙of oxygen and0.34

M⊙of carbon,residing both in the unburned particles and

in the partially burned particles.Note that the observed

spectra are calibrated on an absolute scale.We can there-

fore directly compare the?uxes in the O I and C I lines

with the observations,and not only relative to the’con-

tinuum’.

As an extreme we have arti?cially removed all carbon

and oxygen emission in one model,shown as the lowest

short-long dashed line in Fig.8.We also show two models

where we have varied the number of unburned particles.In

the?rst we removed all unburned particles,including only

the carbon and oxygen residing in the partially burned

particles(0.07M⊙of oxygen and0.03M⊙of carbon).In

the second we reduced the number of unburned particles

by a factor of10,resulting in oxygen and carbon masses of

0.11M⊙and0.06M⊙,respectively.From this?gure it is

seen that the original model is in clear disagreement with

the observations for both the C I and O I lines.The model

with only the carbon and oxygen in the partially burned

particles is just compatible with the observations.As an

upper limit to the unburned mass we estimate this to be

close to the0.03M⊙of carbon and0.04M⊙of oxygen

in the dashed line model in Fig.8.This is in addition to

the carbon and oxygen in the partially burned gas.We do

stress that these limits only apply to the low velocity gas

in the core.

Although we do not?nd any signs of oxygen in our

observations there are early claims in the literature of de-

Kozma et al.:3D modeling9 tection of[O I]λλ6300,6364in SN1937C(Minkowski,

1939).This has,however,not been con?rmed by modern

observations.

In the wavelength range4500–5500?A the observa-

tions show three characteristic bumps(Fig.5).These can

all be seen in the model,and are explained as a mixture

of Fe II and Fe III emission.The relative strengths of the

Fe II and Fe III features are sensitive to the densities in

the emitting gas.In Fig.9we show the contributions from

the di?erent ionization stages of iron to this wavelength

region for particles of di?erent density ranges.In the up-per panel we show the contribution from the Fe-rich tracer particles in the range(1?3)×105cm?3,and in the two lower panels for the density ranges(3?10)×105cm?3 and(1?3)×106cm?3.The contributions to the total?ux in this wavelength region from the above density ranges are17%,34%and49%,respectively.

Fig.9clearly shows the e?ects of varying the density. Especially the strength of the Fe III feature around4600–4800?A is sensitive to the density of the emitting gas. An increase in density reduces the degree of ionization, due to increased recombination rates,and consequently the Fe II abundance increases relative to Fe III.In the c3256

3d10s model is therefore indicated from these observations.Also the too strong Fe II complex at7000–8000?A,and the blend of Fe II and C I at8500–9500?A indicate a too low ionization.In addition,the large dip in the model spec-trum at～6000?A would be reduced by a lower density.

In the other three non-homologous models the densi-ties are even higher,and Fe II is totally dominating the spectrum in this wavelength region.From this we see the importance of following the hydrodynamical calculations until homologous expansion is reached.

In addition to decreasing the densities,an alternative way to increase the ionization is to increase the amount of 56Ni formed in the explosion.The amount of56Ni in the

c3256

3d

3d10s model is the combination of a lower mean density and a higher56Ni mass.

If we compare the synthetic spectra based on the non-homologous c3256and b30768models at300and 500days(not shown here),we?nd that both models show too strong O I and C I lines.The12C and16O masses for the di?erent models are given in Table1.While there is a considerable di?erence in these masses depending on the resolution and initiation of the burning,this is not enough to solve the problem with the strong O I and C I features. This could be due to the fact that model b30768was not evolved far enough in time.But it could also re?ect a deeper problem of the models.

There are several assumptions and uncertainties in the 3D hydro calculations that might a?ect the still existing di?erences between the computed late time spectra and our observations.First of all,low-resolution models such as the c3256

3d

3d

10Kozma et al.:3D modeling 1992;Niemeyer&Woosley1997).Light curve modeling

by H¨o?ich&Khokhlov(1996),based on1D simulations,

has shown that delayed detonations give good?ts to the

light curves and early spectra,provided the transition den-

sity from the de?agration to the detonation phase is prop-

erly adjusted,introducing a new free parameter.Recently

Gamezo et al.(2004a,2004b)found in a3D model that a

transition from de?agration to detonation can remove ef-

?ciently low velocity carbon and oxygen.In addition,the

mass of56Ni increased considerably.

Plewa et al.(2004)present a gravitationally con?ned

detonation mechanism in which a single rising de?agra-

tion bubble triggers a detonation near the stellar surface.

These models are thought to result in mildely asymmetric explosions,and with energetics and chemical composition

similar to those of the delayed detonation models.

It is,however,in our view premature to draw any

strong conclusions of a preferred explosion mechanism

in either direction.In particular,it is not at all clear

what kind of physics might cause a transition to a det-

onation.On the contrary,all investigations carried out

so far seem to rule out the mechanisms responsible for

de?agration-to-detonation transitions in laboratory com-

bustion for SNe Ia(Niemeyer1999;Lisewski et al.2000;

R¨o pke et al.2004a,2004b;Bell et al.2004).Moreover,it

is unclear even whether or not a detonation ignited in one

pocket of unburned C-O fuel can incinerate the rest of

the already fast expanding star.Because the detonation

front in a SN Ia is weak and propagates with sonic veloc-

ity pressure waves may not be able to penetrate through

regions consisting of burned material.It is quite possible

that numerical di?usion has caused the almost complete

burning found in the simulation of Gamezo et al.(2004a,

2004b)(Niemeyer,Livne,private communication).

3d 5.Summary

Our main aim in this paper is to show the potential of test-

ing multidimensional explosion models by detailed mod-

eling of the emission later than～100days after the ex-

plosion.To do this we have calculated late time spectra

based on3D hydrodynamical and nucleosynthesis simula-

tions.We have mainly studied one model which has been

calculated to the homologous stage,c3256

Kozma et al.:3D modeling11

Supernova Explosions”under contract HPRN-CT-2002-00303.

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12Kozma et al.:3D modeling

Model Mode of ignition Time M(unburned O)Kinetic energy

(s)(M⊙)(ergs) 3d76830.420.28

3d25630.330.24

3d25630.310.35

3d10s central100.35～4.4×1050 Table1.The explosion models studied in this paper from Travaglio et al.(2004),and from R¨o pke(2005).The time in this table is the time for which the hydrodynamics and nucleosynthesis has been calculated.

Kozma et al.:3D modeling13

Fig.1.The positions of the Fe-rich(red),unburned(blue)and intermediate(green)tracer particles at10s in the c3256

3d10s model at300days.

14Kozma et al.:3D modeling

Fig.3.Model spectra at300days(thick lines)and500days(thin lines)for the c3256

Kozma et al.:3D modeling15

Fig.4.The dominating contributions to the model spectra from c3256

16Kozma et al.:3D modeling

Fig.5.Model spectrum at350days,for the c3256

Kozma et al.:3D modeling17

Fig.6.Model IR spectrum at350days,for the c3256

18Kozma et al.:3D modeling

Fig.7.The crosses mark density and temperature(left hand panels)and density and electron fraction(right hand panels)for each of the Fe-rich mass elements the c3256

Kozma et al.:3D modeling19

Fig.8.The region around the[O I]λλ6300,6364and[C I]λ8727lines based on the c3256

20Kozma et al.:3D modeling

Fig.9.Spectra based on the c3256