Spectral analysis of the 91bg-like Type Ia SN 2005bl: Low luminosity, low velocities, incomplete burning
Stephan Hachinger, Paolo A. Mazzali, Stefan Taubenberger, Ruediger Pakmor, Wolfgang Hillebrandt
aa r X i v : . [ a s t r o - ph . H E ] D ec Mon. Not. R. Astron. Soc. (2009) (MN L A TEX style file v2.2)
Spectral analysis of the 91bg-like Type Ia SN 2005bl:Low luminosity, low velocities, incomplete burning.
Stephan Hachinger , Paolo A. Mazzali , , , Stefan Taubenberger ,R ¨udiger Pakmor , Wolfgang Hillebrandt Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild-Str. 1, 85748 Garching, Germany Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy Istituto Nazionale di Astrofisica-OAPd, vicolo dell’Osservatorio 5, 35122 Padova, Italy
ABSTRACT
The properties of underluminous type Ia supernovae (SNe Ia) of the 91bg subclass have yetto be theoretically understood. Here, we take a closer look at the structure of the dim SN Ia2005bl. We infer the abundance and density profiles needed to reproduce the observed spec-tral evolution between − d and +12 . d with respect to B maximum. Initially, we assumethe density structure of the standard explosion model W7; then we test whether better fitsto the observed spectra can be obtained using modified density profiles with different totalmasses and kinetic energies. Compared to normal SNe Ia, we find a lack of burning prod-ucts especially in the rapidly-expanding outer layers ( v & km s − ). The zone between ∼ and km s − is dominated by oxygen and includes some amount of intermediatemass elements. At lower velocities, intermediate mass elements dominate. This holds downto the lowest zones investigated in this work. This fact, together with negligible-to-moderateabundances of Fe-group elements, indicates large-scale incomplete Si burning or explosive Oburning, possibly in a detonation at low densities. Consistently with the reduced nucleosyn-thesis, we find hints of a kinetic energy lower than that of a canonical SN Ia: The spectrastrongly favour reduced densities at & km s − compared to W7, and are very well fittedusing a rescaled W7 model with original mass ( . M ⊙ ), but a kinetic energy reduced by ∼ (i.e. from . · erg to . · erg). Key words: supernovae: general – techniques: spectroscopic – radiative transfer
Type Ia supernovae (SNe Ia) play a key role in modern astro-physics. They are invaluable as distance indicators for cosmology(e.g. Perlmutter et al. 1997, 1999; Riess et al. 1998; Astier et al.2006; Wood-Vasey et al. 2007) because of the high accuracy withwhich the absolute luminosity of most SNe Ia can be inferred.The luminosity varies among different objects, but the variationscorrelate with distance-independent light-curve parameters suchas the decline in magnitudes in the B -band within 15 days af-ter B maximum (Phillips 1993). Luminosity calibration tech-niques exploiting this fact are mostly applied to “normal” SNeIa (Branch, Fisher & Nugent 1993) not showing poorly-understoodpeculiarities. These SNe supposedly emerge from a homogeneoussample of progenitors, which are thought to be C-O white dwarfs(WDs) accreting matter from a non-degenerate companion star(single-degenerate scenario).In the single-degenerate paradigm, the smooth varia-tions among “normal” SNe Ia (Branch, Fisher & Nugent 1993;Nugent et al. 1995) can be explained within a delayed-detonationscenario (Khokhlov 1991): an initially subsonic explosion (defla- gration) undergoes a deflagration-detonation transition (DDT) andproceeds as a supersonic detonation afterwards. The efficiency andextent of burning in the initial deflagration may then vary from ob-ject to object, which affects the nucleosynthesis and causes the ob-served variability (Mazzali et al. 2007).Extremely sub- or superluminous SNe Ia (e.g.Filippenko et al. 1992; Phillips et al. 1992; Leibundgut et al.1993; Howell et al. 2006), on the other hand, are more difficultto explain. Here, progenitors deviating from the Chandrasekharmass may play a role, or some explosions might result from amerger of two WDs (double-degenerate scenario). Progenitorsystems producing peculiar SNe Ia might also produce some rather“normal” explosions, contaminating the sample of homogeneousexplosions used for distance determination. Clarifying whichexplosion scenarios lead to SNe Ia at which rates is thereforeimportant for supernova cosmology, but it will also be of valuefor other fields. Studies concerned with the binary progenitorsand population synthesis (e.g. Ruiter, Belczynski & Fryer 2009),observed supernova rates (Greggio, Renzini & Daddi 2008) or theimpact of supernovae on their surroundings (e.g. Sato et al. 2007)will profit from understanding the origin of peculiar supernovae. c (cid:13) (cid:13) Hachinger et al.
Thus motivated, we analyse the 91bg-like SN 2005bl(Taubenberger et al. 2008). SNe of the 91bg subclass are dim anddecline rapidly (e.g. Filippenko et al. 1992; Leibundgut et al. 1993;Turatto et al. 1996; Garnavich et al. 2004). They were used, withother SNe, to infer the slope of the relation between luminosityand decline rate of SNe Ia (Phillips 1993), but later it became clearthat dim SNe decline even more rapidly than expected from a linearluminosity − decline-rate relation among normal SNe (Phillips et al.1999; Taubenberger et al. 2008). Spectroscopically, 91bg-like SNeshow characteristic peculiarities, such as low line velocities around B maximum (e.g. Filippenko et al. 1992) and clear spectral signa-tures of Ti II , indicating lower ionisation (Mazzali et al. 1997). Allthese properties together are consistent with a low mass of newly-synthesised Ni. To date, no elaborate explosion models have con-vincingly reproduced 91bg-like SNe Ia. Pure deflagration modelsshow even lower expansion velocities than observed in these ob-jects, especially when little Ni is produced (cf. Sahu et al. 2008).Delayed-detonation models might explain 91bg-like objects withina unified scenario for SNe Ia (Mazzali et al. 2007). Yet, there arehints of qualitative differences. One example are the improved fitsto spectra of SN 1991bg of Mazzali et al. (1997), enabled by a re-duction in ejecta mass and kinetic energy with respect to canonicalvalues. Ultimately, only refined analyses of photometric and spec-troscopic properties can constrain explosion models.We use a spectral synthesis code to analyse the structureand abundance stratification of SN 2005bl, reproducing its ob-served spectral evolution. The “abundance tomography“ method(Stehle et al. 2005), which we use, exploits the fact that the opti-cally thick region of the ejecta becomes smaller as time progresses.Thus, deeper and deeper layers contribute to spectrum formation.Modelling a time series of spectra, we infer the abundance profilefrom the outer envelope to as deep a layer as possible. We thentest whether variations in mass or explosion energy are needed toexplain the differences between spectra of normal and dim SNeIa. This is done performing abundance tomography with variousdensity profiles, and assessing the quality of the resulting spectralfits. The range in masses and energies sampled by the modifiedmodels starts at . · M Ch / ∼ · erg and extends to . · M Ch / ∼ · erg ( M Ch : Chandrasekhar mass, . M ⊙ ). This choice hasbeen motivated by parameters inferred for observed extreme SNeIa of all kinds (Mazzali et al. 1997; Howell et al. 2006).The paper is structured as follows: First, we give a short in-troduction to the methods employed (Sec. 2). We then present themodels for SN 2005bl (Sec. 3), discuss and assess them (Sec. 4),and finally draw conclusions (Sec. 5). The radiative transfer code we use and the abundance tomog-raphy method have already been described (Stehle et al. 2005;Mazzali et al. 2008). Thus, we focus on aspects necessary for anunderstanding of the present study.
We use a 1D Monte Carlo (MC) radiative transfer code(Abbott & Lucy 1985, Mazzali & Lucy 1993, Lucy 1999, Mazzali2000 and Stehle et al. 2005) to compute SN spectra from a givendensity and abundance profile. The aim is to infer the chemicalstructure adjusting the abundances within the envelope until an op-timal fit to the observed spectra is obtained. The code computes the radiative transfer through the SN ejectaabove an assumed photosphere. The densities within the envelopeare calculated from an initial density profile describing the state ofthe ejecta after homologous expansion has set in, which is a fewseconds after the explosion (e.g. R¨opke & Hillebrandt 2005). Theejecta expand radially with r = v · t , where r is the distance fromthe centre, t the time from explosion (see beginning of Sec. 3), and v the velocity. Radius and velocity can be used interchangeably ascoordinates.From the photosphere, which is located at an adjustable v ph ,thermal radiation [ I + ν = B ν ( T ph ) ] is assumed to be emittedinto the atmosphere. This is of course quite a crude approxima-tion to the pseudo-continuous radiation field deep in the ejecta(Sauer, Hoffmann & Pauldrach 2006). Notable deviations mainlyappear in the red and infrared, where a departure of the flux levelfrom that of the observed spectra sometimes cannot be avoided.The radiation is simulated as ”photon packets“, which un-dergo Thomson scattering as well as line excitation-deexcitationprocesses, treated in the Sobolev approximation. The process ofphoton branching is included, which implies that the transitions forexcitation and deexcitation can be different. In a branching event,the photon packet is not split up. Instead, it is emitted as a wholewith a new frequency corresponding to a possible downward tran-sition. This ”indivisible packet“ approach (Lucy 1999) enforces ra-diative equilibrium. The downward transition is randomly selected,taking into account effective emission probabilities. Thus, if a largenumber of packets are simulated, the distribution of decays reflectsthe actual one.A modified nebular approximation, which mimics effects ofnon-local thermodynamic equilibrium (NLTE), is used to calculatethe action of the radiation field onto the gas. For each of the zones into which the envelope is discretised here, a radiation tem-perature T R and an equivalent dilution factor W are calculated.These quantities mostly determine the excitation and ionisationstate (Abbott & Lucy 1985). Only the variables describing the stateof the gas are discretised; for the paths and redshifts of photons, andfor the positions of interaction surfaces of lines, continuous valuesare allowed.The code iterates the radiation field and the gas conditions.Furthermore, T ph is automatically modified so as to match a givenoutput luminosity L , taking backscattering into account. After con-vergence, the emerging spectrum is obtained from a formal integralsolution of the transfer equation (Lucy 1999). As a first step, we adopt the density structure of the standard ex-plosion model W7 (Nomoto, Thielemann & Yokoi 1984) as a ba-sis for our calculations. We then repeat the abundance tomographywith modified density profiles, changing the total mass and kineticenergy of the explosion. To achieve this, the values for each gridpoint in the W7 velocity-density structure are scaled uniformly (i.e.all velocities by one scaling factor, and all densities by another one)according to: ρ ′ = ρ W · (cid:18) E ′ k E k ,W (cid:19) − / · (cid:18) M ′ M W (cid:19) / (1) v ′ = v W · (cid:18) E ′ k E k ,W (cid:19) / · (cid:18) M ′ M W (cid:19) − / . (2)Here, ρ ′ and v ′ are the density and velocity coordinates of each c (cid:13) (cid:13) im SN Ia 2005bl: abundances and density profile. Table 1.
Density models used in this work, and their total kinetic energy E ′ k and mass M ′ . The models are named according to the scaling factorsfor kinetic energy and mass with respect to W7 (cid:16) E ′ k E k ,W and M ′ M W (cid:17) , andsorted according to their kinetic energy E ′ k . Model E ′ k /E k ,W M ′ /M W E ′ k [ erg] M ′ [ M ⊙ ] E ′ k M ′ ,(cid:16) E k M (cid:17) W w7e0.35 0.35 1.00 0.47 1.38 0.35w7e0.5m0.5 0.50 0.50 0.66 0.69 1.0w7e0.5m0.7 0.50 0.70 0.66 0.97 0.7w7e0.5 0.50 1.00 0.66 1.38 0.5w7e0.5m1.25 0.50 1.25 0.66 1.73 0.4w7e0.7m0.7 0.70 0.70 0.93 0.97 1.0w7e0.7 0.70 1.00 0.93 1.38 0.7w7e0.7m1.25 0.70 1.25 0.93 1.73 0.6w7e0.7m1.45 0.70 1.45 0.93 2.00 0.5w7m0.7 1.00 0.70 1.33 0.97 1.4w7 1.00 1.00 1.33 1.38 1.0w7m1.25 1.00 1.25 1.33 1.73 0.8w7m1.45 1.00 1.45 1.33 2.00 0.7w7e1.45m1.45 1.45 1.45 1.93 2.00 1.0 grid point after the scaling. E ′ k and M ′ are the new total kineticenergy and mass.The scaled density models used in this work are listed in Ta-ble 1, which gives an overview of the respective E ′ k E k and M ′ M ratios.We have not implemented every possible energy-mass combinationwithin the limits given in Section 1. Instead, we first constrainedourselves to a few test cases. Then, we sampled the E ′ k − M ′ planemore densely in the region where models of acceptable qualityemerged (see Sec. 4.1).W7 naturally shows some differences with respect to more re-cent and realistic hydrodynamical simulations, and the scaled den-sity profiles can also be expected to do so. However, it is possibleto obtain good fits to spectra of “normal“ SNe Ia like SN 2002bo(Stehle et al. 2005) using the W7 density structure. The results forthe scaled profiles should therefore bring out possible differencesbetween dim SNe Ia and normal ones. The abundance tomography method (Stehle et al. 2005) uses a se-ries of photospheric spectra to establish the abundance distributionwithin a supernova. The idea is that the opaque core of the expand-ing ejecta shrinks with time. Thus, a time series of spectra carriesinformation about the abundances in the ejecta at different depths.In the picture adopted in our code, involving an approximate pho-tosphere, the photosphere recedes to lower velocities with time.Deeper and deeper layers become visible, leaving their imprint onthe spectra.The earliest spectrum available can be used to obtain the pho-tospheric velocity at that time and the abundances in the outer enve-lope. To this aim, we optimise the code input parameters to matchthat spectrum, as in a one-zone spectral model (e.g. Mazzali et al.1997). The subsequent spectrum will carry the imprint of the mate-rial in the outer envelope and additionally that of the layers insidewhich the photosphere has receded. Because the abundances in theouter zone are already known, the abundances of the layers whichhave become visible can now be inferred, together with the newvelocity of the photosphere. This procedure is then continued withlater spectra.The optimum parameters inferred from a spectral modelare subject to some uncertainty (see also the discussion inMazzali et al. 2008). One important reason for this can be degener- acy, which makes the spectra appear similar for different parametersets. The composition adopted for an outer layer in an early-epochmodel may therefore be in conflict with a later spectrum, if the laterspectrum is still influenced by the outer layers. In such cases, werevised the parameters for the outer layers so as to optimise theearlier and later spectra at the same time.
We analyse five spectra of SN 2005bl, taken at − d, − d, − d, +4 . d and +12 . d with respect to B maximum. Observationaldata and one-zone spectral models have already been presented inTaubenberger et al. (2008). As in that paper, we assume a total red-dening of E ( B − V )=0 . and a B -band rise time of 17d to cal-culate the time t from the onset of the explosion.Later spectra were not modelled, as the photosphere hasalready receded to v ph < km s − at +12 . d [for compari-son, Mazzali et al. (2008) found v ph =4700 km s − at +14 d and v ph =2800 km s − at +21 d in SN 2004eo]. As the photospherereaches the Ni-rich zone, some energy deposition should real-istically take place above the photosphere itself. This is is not takeninto account in our code. Thus, to explore the innermost layers onewould need to model nebular spectra (which are not available forSN 2005bl) at least as a consistency check.The outermost ejecta of SNe Ia may partly consist of un-burned material (cf. Mazzali et al. 2007). As the one-zone mod-els for SN 2005bl (Taubenberger et al. 2008) showed too muchabsorption by burned material at high velocities, we introduced azone with strongly reduced abundances of burning products above v & km s − . This was done limiting the mass fractions ofburning products in this zone to . their value at the photosphereat − d. The unburned material at v & km s − , which thenconstitutes & by mass at these velocities, is assumed to consistof carbon and oxygen in a ∼ : ratio. In a preliminary stratified-abundance model, this was found clearly to improve the syntheticspectra, also with respect to the one-zone models (cf. Sec. 3.1.6).Consequently, we implemented such a zone in all our stratified-abundance models (except when using the w7e0.35 density profile,which has negligible densities in the outer layers).Below, we first discuss an abundance tomography experimentbased on the original W7 density structure. We compare our syn-thetic spectra with the observed ones and with the one-zone modelspectra of Taubenberger et al. (2008). After discussing the abun-dance profile, we then present models with different total massand kinetic energy. Parameters (abundances, photospheric veloci-ties, etc.) of all models are compiled in Appendix A. The spectral models discussed here are shown in Fig. 1, where themost important spectral features are marked. v ph =8400km s − At this epoch, the supernova shows a spectrum dominated bysingly-ionised species. In normal SNe, usually also doubly-ionisedspecies are detected at such early epochs (e.g. Mazzali et al. 2008).The zone between and km s − is dominated by oxygen.The absence of the Si II λ emission peak suggests absorptionby C II λ . However, the mass fraction of C between and km s − must be < ; otherwise the C II feature would c (cid:13) (cid:13) Hachinger et al. C a II, S i II ( S i II ) ( M g II ) T i II C r II S i II ( ) F e II S II S i II S i II C II O I M g II M g II C a II M g II SN 2005bl: −6d05bl−w7 stratified modelHomogeneous model (Taubenberger ’08)SN 2005bl: −5d05bl−w7 stratified modelHomogeneous model (Taubenberger ’08) F l ( a r b i t r a r y un i t s ) SN 2005bl: −3d05bl−w7 stratified modelHomogeneous model (Taubenberger ’08)SN 2005bl: +4.8d05bl−w7 stratified modelHomogeneous model (T ’08) 3000 4000 5000 6000 7000 8000 9000 l [ Å ] C a II, T i II, S i II m o s t l y T i II m o s t l y C r II F e II N a I S i II S i II m o s t l y T i II F e II O I S i II ( ) C a II M g II S i II SN 2005bl: +12.9d05bl−w7 strat. model
Figure 1.
Model sequence based on the W7 density profile (05bl-w7 sequence, green lines). Observed spectra (black lines) and one-zone model spectra(magenta lines) from Taubenberger et al. (2008), which only extend to +4 . d, are shown for comparison. Identifications of the most prominent features aregiven at the beginning and the end of the sequence. Especially in the +12 . d spectrum, many blends appear, so that the identifications are only approximate.c (cid:13) (cid:13) im SN Ia 2005bl: abundances and density profile. become too deep. Burned material (oxygen as a burning productexcluded) makes up for no more than ∼ in mass according tothe observed line depths.While numerous lines of intermediate-mass elements (IME)are visible, there are no absorptions that can unambiguously beattributed to Fe. We determined an upper limit to the Fe abun-dance of . , avoiding the appearance of a spurious Fe II fea-ture at ∼ ˚A. Yet, some burning products heavier than Si andS are seen in the spectra: some per mille of Ti and Cr are neces-sary to model the absorption trough at ∼ ˚A and the feature at ∼ ˚A, respectively. These elements also contribute significantlyto line blocking in the UV (Sauer et al. 2008). v ph =8100km s − The April 17 spectrum is very similar to the previous one. As thematerial directly above the photosphere is highly ionised, many fea-tures in this spectrum depend strongly on the abundances above km s − .At ∼ ˚A, the stratified model has an absorption trough toodeep. This is mostly due to the Si II λ line, whose strengthlargely depends on the Si abundance above v =8400 km s − . Wechose this abundance so as to match the Si II λ line of this andthe previous spectrum, and a simultaneous match of the Si II λ line was not possible. Apart from this and some flux mismatch inthe red, the observations are fitted well. v ph =7500km s − This model again matches the observed spectrum nicely in mostregions. The Ti-dominated trough at ∼ ˚A is now deeper than inthe earlier spectra, and relatively hard to fit. A good model requiresTi abundances of the order of a few percent at the photosphere, andrelatively large Ti abundances in the zones above. Thus, we set theTi mass fraction to between and km s − . At largervelocities, the abundances are sharply constrained to some per milleby the features in the − d spectrum.There is still no evidence for significant amounts of Fe in thespectrum. Fe mass fractions of a few per cent in the layers between and km s − are compatible with the observations, butnot strictly required. v ph =6600km s − In order to fit this spectrum with its low flux in the UV and blue, themodel atmosphere must contain sufficient amounts of Ti, Cr and Fe.The layers at v> km s − contain relatively small amounts ofthese elements, as dictated by the pre-maximum spectral featuresand UV flux. To compensate for this, large amounts are neededclose to the photosphere. While the flux-blocking in the UV is quitesensitive to the abundances close to the photosphere, the depth ofindividual features (such as Si II λ ) is still more strongly in-fluenced by the composition at > km s − .The most notable deviation the model from the observed spec-trum occurs in the blue wing of O I λ , where there is too muchabsorption. In the outermost zone, O could only be replaced by C,but we already have a ∼ : I absorption strength by a factor of 2 in these layers, wewould have to postulate a ∼ : and km s − , theamount of oxygen cannot be reduced (cf. Sec. 3.1.1). In Sec. 3.2.2, we will show that a reduction of the density in the outer layers cancure this problem.There is some mismatch around ˚A, which seems to becaused by a low pseudo-continuum. This impression is howeveralso due to Na I D absorption at the peak between the S II troughand the Si II λ feature. We introduced a small amount of Naabove km s − to obtain at least some Na I D absorption at +12 . d. The spurious absorption appearing at +4 . d then indicatesinaccuracies in the Na ionisation profile and its evolution with time,a common issue with synthetic spectra (Mazzali et al. 1997). +12 . , v ph =3250km s − This model carries some conceptual uncertainty, as a possible en-ergy deposition by Ni above the photosphere is not simulated inour code. Yet, the overall fit is satisfactory. Some incompatibilitieswith the abundances inferred for the outer layers could not be re-solved. It was, for example, impossible to get rid of the absorptionsat ∼ and ∼ ˚A, which are due to Ti II λλ , , and Si II λ , respectively. These lines were not visible in theearlier spectra.The photosphere is now deep inside the Si-dominated zone.The extended red wing of the observed feature at ∼ ˚A, causedmostly by Si II λ , indicates a large Si mass fraction. On theother hand, the small flux in the blue and UV already demands alarger fraction of Fe-group elements. While the exact amounts ofFe, Co and Ni are somewhat uncertain, their sum can be estimatedto be ∼ . The exact number depends on the abundances of otherelements blocking UV flux (mostly Ti and Cr) between and km s − , which are somewhat uncertain. Compared to one-zone models [Taubenberger et al. (2008), shownin Fig. 1 as the magenta line], the stratified model sequenceclearly constitutes an improvement in fitting the observations. Themain reason for this is the C/O-dominated shell introduced at v> km s − , which makes spectral lines of burned materialnarrower. The changes with respect to the one-zone model are es-pecially apparent in the pre-maximum spectra: the Ca II H&K andSi II λ lines absorb less at high velocities, so that in the bluewings of the features only small mismatches are left. The Ti II -dominated trough around ˚A now has more structure. Somedeviations, even a bit more apparent then in the one-zone models,remain in the red wing of Si II λ in the earliest spectra. Thisis largely due to re-emission in this wavelength range, caused byelements such as Ti and Cr which block and redistribute UV flux.These elements are, however, necessary to model the spectral fea-tures (see Sec. 3.1.1). In Fig. 2, we compare the abundance profile derived in our to-mography experiment to the nucleosynthesis in W7 (Iwamoto et al.1999), which approximately represents a normally-luminous SNIa (Nomoto, Thielemann & Yokoi 1984). In our models, unburnedmaterial (counting in all of the oxygen) constitutes a much largerfraction of the ejecta, almost the outer ∼ . M ⊙ . Our analysis of theouter layers is still a bit coarse. A better-resolved analysis, yieldingmore exact results e.g. for the amount of IME between and c (cid:13) (cid:13) Hachinger et al. m a ss f r a c t i on [ % ] enclosed mass [M sun ]velocity [km/s]W73000 5000 8000 11000 15000COMgSiSCaTi+CrFe Ni m a ss f r a c t i on [ % ] enclosed mass [M sun ]velocity [km/s]05bl−w73000 5000 8000 11000 15000COMgSiSCaTi+CrFe Ni Figure 2.
Abundances of W7 nucleosynthesis calculations (Iwamoto et al.1999, top panel) vs. abundance tomography of SN 2005bl, based on theoriginal W7 density profile (05bl-w7 model, bottom panel). km s − , would be possible if spectra at earlier epochs wereavailable (see Sec. 4.4).Below the outer ∼ . M ⊙ , the ejecta of SN 2005bl are dom-inated by IME. The transition happens in the zone between and km s − . The exact transition velocity is difficult to infer,as the post-maximum spectra show only a limited sensitivity to theSi abundances below km s − . Mazzali et al. (1997) have con-ducted a fine analysis of the O I λ line profile in SN 1991bg,and found a lower cut-off velocity of km s − for O. We thusimplemented a relatively sharp decrease of the O abundance infavour of Si below the − d photosphere.The layers between and km s − already consist of ∼ Ni in W7. We, in contrast, find (besides IME) compar-atively large abundances of Ti and Cr as products of incompleteburning at these velocities (peak values in the order of some percent). These elements contribute to the formation of the observedtrough around ∼ ˚A which is characteristic of 91bg-like ob-jects past maximum, but also to the line blocking in the UV. Tosome extent, their effects can also be mimicked by Fe, Co and Ni.With overly large amounts of Fe, however, individual lines in theoptical may show up, and the flux distribution in the UV and bluemay deviate from what is observed. Large abundances of Ni orits decay product Co outside the centre would be in conflict withthe nebular spectra of dim SNe Ia, which show very narrow lines(Mazzali et al. 1997).The deepest zones that we reach with our analysis are still dominated by IME. However, there are signs of a transition to theNSE-burning zone: the large amount of line blocking and flux re-distribution needed to fit the +12 . d spectrum clearly points to-wards Fe-group abundances of several .Compared to the one-zone models of Taubenberger et al.(2008) epoch by epoch, the abundances of burning products at therespective photospheres are larger. In a model with homogeneouscomposition, the inferred abundances will always be some averagebetween those at the photosphere and those further outwards, whereless burning products are present. We now show some representative spectral models based on mod-ified density profiles (Sec. 2.2). The reader interested in the abun-dances is referred to Section 4.3 and Appendix A. Here, we fo-cus on the differences in the spectra with respect to the W7-basedmodels. To facilitate the understanding of these differences, we firstdiscuss the properties of the scaled density models.Our scaled density models span a range of masses and kineticenergies (see Table 1). Scaling the total mass and energy, the am-plitude and/or form of the W7 density structure is changed. Whatexactly happens depends on the E ′ k M ′ ratio of the final profile withrespect to E W M W . Here, we distinguish the following three cases,which result in three classes of scaled density profiles: E ′ k M ′ = E W M W : In this case, the scaled velocity-density profile isobtained from W7 by reducing the density at each velocity by auniform factor. The form of the density profile in velocity space isthus left unchanged. E ′ k M ′ < E W M W : Here, the energy per unit mass is reduced. Thismeans that mass elements are ”shifted“ towards lower velocities.The density profile becomes steeper in velocity space, and the rel-ative amount of mass at high velocities is smaller. E ′ k M ′ > E W M W : Increasing the energy per unit mass ”shifts“ materialoutwards, opposite to the case before. As the spectra of 91bg-likeSNe Ia lack absorption at high velocities in all lines, this is gen-erally disfavoured. Thus, we calculated only one model sequencewith such a density profile (w7m0.7).The models we discuss below are exemplary for these threescaling types. They are named after the underlying density models(e.g. 05bl-w7m0.7 is based on w7m0.7). E ′ k M ′ = E W M W : 05bl-w7e0.7m0.7 In these models, the density is decreased at all radii. This leads toa slight improvement of the spectra (Fig. 3), as the photospheresare deeper inside the ejecta, and the absorption velocities tend to belower. Owing to the lower densities, larger mass fractions of burnedmaterial are necessary to fit the line depths. At high velocities, how-ever, oxygen still dominates and the high-velocity absorption in theO I λ line only becomes a bit weaker. E ′ k M ′ < E W M W : 05bl-w7e0.7 In the w7e0.7 density profile (Fig. 4), the densities are sig-nificantly increased below ∼ km s − and decreased above ∼ km s − . Thus, the spectral features become narrower com-pared to the W7-based sequence. Line widths and positions nowgenerally fit the structure of the observed spectra better. c (cid:13) (cid:13)(cid:13)
Abundances of W7 nucleosynthesis calculations (Iwamoto et al.1999, top panel) vs. abundance tomography of SN 2005bl, based on theoriginal W7 density profile (05bl-w7 model, bottom panel). km s − , would be possible if spectra at earlier epochs wereavailable (see Sec. 4.4).Below the outer ∼ . M ⊙ , the ejecta of SN 2005bl are dom-inated by IME. The transition happens in the zone between and km s − . The exact transition velocity is difficult to infer,as the post-maximum spectra show only a limited sensitivity to theSi abundances below km s − . Mazzali et al. (1997) have con-ducted a fine analysis of the O I λ line profile in SN 1991bg,and found a lower cut-off velocity of km s − for O. We thusimplemented a relatively sharp decrease of the O abundance infavour of Si below the − d photosphere.The layers between and km s − already consist of ∼ Ni in W7. We, in contrast, find (besides IME) compar-atively large abundances of Ti and Cr as products of incompleteburning at these velocities (peak values in the order of some percent). These elements contribute to the formation of the observedtrough around ∼ ˚A which is characteristic of 91bg-like ob-jects past maximum, but also to the line blocking in the UV. Tosome extent, their effects can also be mimicked by Fe, Co and Ni.With overly large amounts of Fe, however, individual lines in theoptical may show up, and the flux distribution in the UV and bluemay deviate from what is observed. Large abundances of Ni orits decay product Co outside the centre would be in conflict withthe nebular spectra of dim SNe Ia, which show very narrow lines(Mazzali et al. 1997).The deepest zones that we reach with our analysis are still dominated by IME. However, there are signs of a transition to theNSE-burning zone: the large amount of line blocking and flux re-distribution needed to fit the +12 . d spectrum clearly points to-wards Fe-group abundances of several .Compared to the one-zone models of Taubenberger et al.(2008) epoch by epoch, the abundances of burning products at therespective photospheres are larger. In a model with homogeneouscomposition, the inferred abundances will always be some averagebetween those at the photosphere and those further outwards, whereless burning products are present. We now show some representative spectral models based on mod-ified density profiles (Sec. 2.2). The reader interested in the abun-dances is referred to Section 4.3 and Appendix A. Here, we fo-cus on the differences in the spectra with respect to the W7-basedmodels. To facilitate the understanding of these differences, we firstdiscuss the properties of the scaled density models.Our scaled density models span a range of masses and kineticenergies (see Table 1). Scaling the total mass and energy, the am-plitude and/or form of the W7 density structure is changed. Whatexactly happens depends on the E ′ k M ′ ratio of the final profile withrespect to E W M W . Here, we distinguish the following three cases,which result in three classes of scaled density profiles: E ′ k M ′ = E W M W : In this case, the scaled velocity-density profile isobtained from W7 by reducing the density at each velocity by auniform factor. The form of the density profile in velocity space isthus left unchanged. E ′ k M ′ < E W M W : Here, the energy per unit mass is reduced. Thismeans that mass elements are ”shifted“ towards lower velocities.The density profile becomes steeper in velocity space, and the rel-ative amount of mass at high velocities is smaller. E ′ k M ′ > E W M W : Increasing the energy per unit mass ”shifts“ materialoutwards, opposite to the case before. As the spectra of 91bg-likeSNe Ia lack absorption at high velocities in all lines, this is gen-erally disfavoured. Thus, we calculated only one model sequencewith such a density profile (w7m0.7).The models we discuss below are exemplary for these threescaling types. They are named after the underlying density models(e.g. 05bl-w7m0.7 is based on w7m0.7). E ′ k M ′ = E W M W : 05bl-w7e0.7m0.7 In these models, the density is decreased at all radii. This leads toa slight improvement of the spectra (Fig. 3), as the photospheresare deeper inside the ejecta, and the absorption velocities tend to belower. Owing to the lower densities, larger mass fractions of burnedmaterial are necessary to fit the line depths. At high velocities, how-ever, oxygen still dominates and the high-velocity absorption in theO I λ line only becomes a bit weaker. E ′ k M ′ < E W M W : 05bl-w7e0.7 In the w7e0.7 density profile (Fig. 4), the densities are sig-nificantly increased below ∼ km s − and decreased above ∼ km s − . Thus, the spectral features become narrower com-pared to the W7-based sequence. Line widths and positions nowgenerally fit the structure of the observed spectra better. c (cid:13) (cid:13)(cid:13) im SN Ia 2005bl: abundances and density profile. Si II l l l l F l ( a r b i t r a r y un i t s ) Si II l l l l l [ Å ]SN 05bl: +12.9dModel05bl−w7 model Figure 3. II λ and O I λ features in detail. Changes in thedensity profile usually manifest themselves most clearly in these two features (see also Sec. 4.1).c (cid:13) (cid:13) Hachinger et al.
Si II l l l l F l ( a r b i t r a r y un i t s ) Si II l l l l l [ Å ]SN 05bl: +12.9dModel05bl−w7 model Figure 4. (cid:13) (cid:13) im SN Ia 2005bl: abundances and density profile. Owing to the lower densities in the outer part, there is less lineblocking by heavy elements. This decreases the flux redistribution,so that the flux level in the red wing of Si II λ and redwards ismatched better, especially at early times.At the same time, the spurious high-velocity absorption ispractically gone in Ca H&K and Si II λ , but even more im-portantly in O I λ . The reason for this is again the decreaseddensity in the outer layers. As the abundances in the outer layers,especially of oxygen, are not fundamentally changed with the den-sity modification, the decrease in density translates into weaker ab-sorption at high velocities. E ′ k M ′ < E W M W : 05bl-w7m1.25 Despite the larger mass, the 05bl-w7m1.25 spectra show a some-what improved quality compared to W7 (Fig. 5, see also line ve-locity measurements in Sec. 4.1). This illustrates that a super-Chandrasekhar total mass is not necessarily incompatible with thespectra of SN 2005bl. Remarkably, the improvement over the W7-based sequence is due to decreased densities in the outermost layers( v & km s − ) of the warped density profile. E ′ k M ′ > E W M W : 05bl-w7m0.7 Here, the densities in the outermost layers are increased with re-spect to W7. This can directly be seen in the spectra (Fig. 6): allthe problems which are reduced in 05bl-w7e0.7 (compared to theoriginal W7-based model sequence) are now exacerbated.
Having discussed some representative cases in Section 3.2, we nowsystematically compare all models calculated on the basis of differ-ent density profiles. Our aim is to judge the quality of each modelsequence in a simple and meaningful manner. To achieve this, weintroduce three quality criteria:(i)
Consistence of spectra.
The main motivation to test modi-fications of the density were mismatches in the line velocities orwidths remaining in the W7-based model sequence, especially inO I λ and Si II λ . Other lines of the spectrum did notshow deviations as apparent, apart from Ca II H&K, which behavesquite similar to Si II λ . To assess if the lines are better fittedusing different density profiles, we measured the velocities of O I λ and Si II λ in each synthetic and observed spectrumat − d, − d and +4 . d. Then we calculated, for each model se-quence and line, the velocity difference between the observed andthe synthetic spectra, averaged over the epochs.(ii) Consistence of kinetic energy.
We calculated a hypotheti-cal kinetic energy ( E k,hyp ) for each of the abundance profiles in-ferred. This is the nuclear energy release (assuming a pre-explosion Ca II H&K, Si II λ and O I λ are usually the strongest linesin our spectra, which therefore have the highest probability of developinghigh-velocity absorptions. composition of equal amounts C and O) minus the binding en-ergy | E bind | of the WD (gravitational energy minus thermal and,in case of rotation, rotational energy). To judge the quality of amodel sequence, we then compared the kinetic energy assumed inthe density scaling ( E ′ k ) to E k,hyp . The calculation of E k,hyp dependson some assumptions, the first of which is that the mass fractionof IME in the obscured core below the +12 . d photosphere is of that above the +12 . d photosphere. Actually, this mass frac-tion may be between zero and the IME mass fraction above the +12 . d photosphere. The possible error due to this is given be-low. The binding energy | E bind | of the progenitors (except for the . M ⊙ ones) was calculated following Yoon & Langer (2005),who assume a white dwarf rotation profile resulting from binaryevolution. We used their ” BE ( M ; ρ c ) “ relation (eq. 33), assum-ing a central density ρ c of . · g / cm (which is typical for WDignition) for M ′ > M Ch . For sub-Chandrasekhar WDs, the centraldensities are lower even in the absence of rotation. We assumednegligible rotation for these cases, and obtained the central densityfor a given mass inverting formula (22) of Yoon & Langer (2005) .(iii) Expected light-curve width.
For models with good consis-tence based on the first two criteria, we additionally can checkwhether the density and abundance structure implies a width of thebolometric light curve ( τ LC ) compatible with that of dim SNe Ia. Wecalculated an expected light curve width for each model sequence,following Mazzali et al. (2007), from the respective kinetic energy E k , ejecta mass M ′ , and total masses of IME and NSE material M IME , M
NSE as: τ LC = N · ˜ κ E − k M ′ . Here, ˜ κ = (0 . M IME + M NSE ) /M ′ is proportional to the opacityestimate of Mazzali et al. (2007), and N is a normalisation factorchosen so as to agree with their estimates of light-curve widths. Inorder to calculate ˜ κ , we can assume different burning efficienciesin the core, as above; additionally, we may adopt as E k either thehypothetical value E k,hyp or the value E ′ k from the density scaling.We thus calculated again an average τ LC and an estimate of theerror introduced by these degrees of freedom. In order to judge themodels, the values τ LC were compared to τ LC,dim =13 . d, which isthe average expected light curve width for the similarly dim SNe1991bg and 1999by (Mazzali et al. 2007).We now discuss the quality of the models in terms of the threecriteria. The differences in Doppler velocity of the O I λ and Si II λ lines between observed and synthetic spectra are shown inTable 2. In this table, the models are ranked according to the abso-lute value of the ”mean velocity difference“, which is the averageover both lines and all epochs.A decent match of line velocity is obtained especially forthe 05bl-w7e0.7 model, but also, for example, for some super-Chandrasekhar mass models with E ′ k /M ′ lower than W7. This By ”binding energy“ and ”gravitational energy“ we always mean the ab-solute values here, i.e. we treat them as positive numbers. This formula cannot be applied for our lowest-mass models. There-fore, we inferred the binding energy of a . M ⊙ progenitor from a WDmodel with constant temperature, which uses the Timmes equation of state(Timmes & Arnett 1999). This equation of state takes into account a vari-able degree of electron degeneracy.c (cid:13) (cid:13) Hachinger et al.
Si II l l l l F l ( a r b i t r a r y un i t s ) Si II l l l l l [ Å ]SN 05bl: +12.9dModel05bl−w7 model Figure 5. (cid:13) (cid:13)(cid:13)
Si II l l l l F l ( a r b i t r a r y un i t s ) Si II l l l l l [ Å ]SN 05bl: +12.9dModel05bl−w7 model Figure 5. (cid:13) (cid:13)(cid:13) im SN Ia 2005bl: abundances and density profile. Si II l l l l F l ( a r b i t r a r y un i t s ) Si II l l l l l [ Å ]SN 05bl: +12.9dModel05bl−w7 model Figure 6. (cid:13) (cid:13) Hachinger et al.
Table 2.
Time-averaged line velocity differences in Si II λ and O I λ from models to observed spectra (denoted by h ∆ v ( Si II λ i and h ∆ v ( O I λ i , respectively). Positive differences mean that the lines are too fast (blue) in the synthetic spectra. The models are sorted according to themean velocity difference averaged among both lines (ascending in absolute value). Model E ′ k /E k ,W M ′ /M W h ∆ v ( Si II λ i h ∆ v ( O I λ i h ∆ v i shows that a reduced density in the outer layers is the key to a betterfit in the lines. To fit the observed lines well, models near the Chan-drasekhar mass need a E ′ k /M ′ smaller by ∼ − with respectto W7. With too large a reduction in energy, line velocities becometoo low (see e.g. negative velocity differences for the 05bl-w7e0.5model). At low masses, generally a smaller reduction in E ′ k /M ′ suffices: 05bl-w7e0.5m0.5 ( M ′ =0 . M ⊙ ) as an extreme modelstill gives a satisfactory fit with E ′ k /M ′ =( E k /M ) W .Remarkably, for all mass values probed in this work, a reason-ably good model can be obtained (judged by the line velocities).The kinetic energy E ′ k of all well-fitting models, however, is lowerthan E k ,W . In Table 3 we show our hypothetical kinetic energy values, as wellas the quantities from which they were calculated. We then judgethe models by the ratio of E k,hyp to the kinetic energy assumed inthe density scaling ( E ′ k ). Ideally, this ratio should be equal to one;the larger the deviation, the lower the rank of a model.For density profiles with the same mass, but different kineticenergy E ′ k , the hypothetical kinetic energy E k,hyp usually varies sys-tematically. In density models with smaller E ′ k , densities are re-duced in the high-velocity layers (see Sec. 3.2.2), which containmostly unburned material. At the same time, densities are increasedin lower layers, where the material is mostly burned. The velocity atwhich the transition (between unburned and burned material) hap-pens does not vary much from model to model as it is constrainedby spectral features. Therefore, the change in the density profile re-sults in a larger ratio of burned to unburned material and a larger E k,hyp . Similarly, when E ′ k is increased, E k,hyp decreases. Equality,i.e. consistence, between E k,hyp and E ′ k is usually reached at a re-duced value of E ′ k /M ′ with respect to W7. The required reductionvaries with the mass of the models (see Sec. 4.2).In Table 3, two supermassive models (05bl-w7e0.7m1.25,05bl-w7e0.7m1.45) rank top. However, it should be noted that theenergetic quality criterion again does not single out a certain mass,but sets a point of energetic consistence for each mass. All modelswith larger E ′ k will then feature too little nucleosynthesis to explainthe assumed kinetic energy. The opposite holds for models withlower E ′ k . We calculated estimates of the width of the bolometric light curvefor the models ranking best in spectroscopic and energetic consis-tence at each mass M ′ . The resulting values, and those of the quan-tities needed for the calculation, are given in Table 4.The deviation of τ LC from τ LC,dim =13 . d strongly dependson the mass M ′ . Models with larger mass clearly tend to have alarger light-curve width, although they often have lower values of ˜ κ , as relatively small abundances of burning products are needed tomatch the observed line strengths with the synthetic spectra.Although our expected light-curve widths are quite rough es-timates, one can clearly state that the criterion disfavours masseslargely deviating from the Chandrasekhar mass. The least massivemodel, with a mass of M ′ =0 . M Ch , presumably will not producea broad enough light curve. Likewise, the models at M ′ =1 . M Ch will probably exhibit too broad a light curve, although these mod-els are not strictly incompatible with τ LC,dim , as a large inaccuracyin τ LC,dim results from the large mass in the obscured core. E ′ k − M ′ plane Fig. 7 gives an overview of all models in an E ′ k − M ′ plane. Accord-ing to their quality in spectroscopic terms, the models are markedwith different colours; the energetic consistence is indicated byhatches.In the figure, we also indicate where spectroscopically andenergetically consistent models can generally be expected in theplane: a black line is drawn approximately where the transition be-tween too large and too small line velocities occurs. This line isstraight and runs from massive models with low E ′ k /M ′ to sub-massive models with E ′ k /M ′ ≈ ( E k /M ) W . A green line approx-imately divides the regions of too large and too small nucleosyn-thetic energy yields. It lies in the same region as the line of spec-troscopic consistence, but is curved because the WD binding en-ergy shows a disproportionately strong increase with WD mass.For models up to M ′ ≈ . M Ch , the binding energy is negligiblecompared to the nuclear energy release, whereas at higher massesit is considerable, forcing the line of consistence towards smaller E ′ k and larger nucleosynthesis yields.The two lines of consistence are especially close to one an-other for masses M ′ . M Ch . Models at M ′ =1 . M Ch are eitherspectroscopically or energetically inconsistent, at least under theassumptions we made in this work. Additionally, the light-curve c (cid:13) (cid:13) im SN Ia 2005bl: abundances and density profile. Table 3.
Energetic balance of the models (see text for a description of the quantities). The models are ordered according to the deviation of their E k,hyp E ′ k ratiofrom 1 (cf. last column). Model E ′ k /E k ,W M ′ /M W E ′ k E nucl E bind E k,hypa E k,hyp E ′ k − [ erg] [ erg] [ erg] [ erg]05bl-w7e0.7m1.45 0.70 1.45 0.93 2.10 1.15 0.96 ± ± ± ± ± ± ± ± ± ± ± ± ± ± a The error estimate only reflects the error due to the unknown composition below the photosphere at +12 . d. Table 4.
Light-curve width estimates for the spectroscopically and energetically most consistent models at each mass M ′ . The models are ordered accordingto the deviation of the ratio τ LC τ LC,dim from 1; we assume τ LC,dim =13 . d (see text). Model E ′ k /E k ,W M ′ /M W E ′ k M ′ E k,hyp ˜ κ a τ LCa τ LC τ LC,dim − [ erg] [ M ⊙ ] [ erg] [d]05bl-w7e0.7 0.70 1.00 0.93 1.31 0.77 0.31 ± ± ± ± ± ± ± ± ± ± ± ± a The error estimate reflects the errors due to the unknown composition below the photosphere at +12 . d, and due to the uncertainties in E k . Figure 7.
Overview of all models evaluated in this work. Colour and hatches mark the energetic and spectroscopic consistence of the models, as indicated bythe quantities E k,hyp E ′ k − and h ∆ v i , respectively (darker colours / denser hatches meaning better consistence; numerical values see Tables 2 and 3). The greenline divides the regions where the models have too large and too small a hypothetical kinetic energy yield, compared to the kinetic energy assumed in thedensity scaling. To the left of the black line models show too low line velocities; to the right, the opposite holds.c (cid:13) (cid:13) Hachinger et al. r [ g / c m ] a t t = ( a ft e r e x p l o s i on ) velocity [km/s] w7e0.7 profilew7e0.7m1.25 profileW7 profile Figure 8. w7e0.7 compared to the standard W7 and the w7m1.25e0.7 den-sity profiles. criterion indicates that the width of the light curve is too largefor these most massive models. Our least massive models with M ′ =0 . M Ch are also disfavoured in this respect, as they wouldprobably show too rapid a light-curve evolution.Criteria like those used here could give stricter limits still, ifthe chemical composition in the inner layers was known. This re-quires studies of nebular spectra of dim SNe Ia. As discussed above, our models give no clear indication for a de-viation from the Chandrasekhar mass. The simplest modificationleading to better spectral fits and roughly consistent energetics issimply a moderate downscaling of the energy, as in the 05bl-w7e0.7model. Thus, we consider the 05bl-w7e0.7 model a “reference”. InFigures 8 and 9, we show the density and abundance profiles ofthe 05bl-w7e0.7 model. Other spectroscopically consistent mod-els show similar densities in the outer layers, and thus also similarabundances in that zone. This can be verified in Figures 8 and 9,where the 05bl-w7e0.7m1.25 model is also plotted for comparison.The 05bl-w7e0.7 model features . M ⊙ of unburned mate-rial (including all oxygen; C constitutes . M ⊙ ). IME are dom-inant, with a total abundance of . M ⊙ above km s − , thevelocity of the photosphere at +12 . d. Stable Fe is present in sig-nificant amounts ( . M ⊙ ). The mass of Ni (including decayproducts) above km s − is . M ⊙ , which is a bit higherthan the . − . M ⊙ found in Mazzali et al. (1997) above km s − for SN 1991bg. However, some of the Ni couldbe replaced by other UV-blocking elements without changing thequality of the fit. Some . M ⊙ of material are still hidden belowthe +12 . d photosphere, where the IME abundances may still besignificant (Si of the order of several ).Alternative spectroscopically consistent models show similarpatterns in the abundance profile in velocity space, but the exactdensities and abundances below ∼ km s − are somewhat dif-ferent. In 05bl-w7e0.7m1.25, as an example, the densities in the in-ner zones are larger. Thus, the abundances of Fe, Ti and Cr must belower too keep UV opacities reasonable. For Si, moderate changesin the number density do not cause big changes in the spectra.Therefore, the smaller Fe, Ti and Cr abundances can be balancedby slightly larger Si abundances. The analysis presented here could still be refined for the outer-most and innermost layers. The exact abundance stratification inthe outer envelope cannot be inferred from the spectrum at − d,whose photospheric velocity is already quite low. For the inner lay-ers, especially the density structure and thus the abundance of Si issomewhat uncertain (see Sec. 4.3). In order to make a more precisestudy of dim SNe Ia possible, additional spectra in the very earlyand in the nebular phase are needed.The potential of an analysis of the nebular spectra has alreadybeen shown in Mazzali et al. (1997). Here, we would like to illus-trate the benefit of early time spectra, showing their sensitivity tothe abundances in the outer envelope. We checked the influence ofthe abundances between v ≈ km s − and km s − on the − d spectrum, and found that these abundances have some effectsdifficult to distinguish from those of the chemical composition atlower velocities. Moreover, the (small) abundances of burned mate-rial at & km s − cannot be exactly determined, as these onlyaffect the extreme blue wings of the spectral features.To explore the effect of the abundances in the outer envelopeon early-time spectra, we calculated model spectra at − d and − d (Fig. 10). The luminosities at these epochs were crudelyestimated from the luminosity at − d under the assumption of aquadratic light curve rise (cf. Riess et al. 1999). We first calculatedspectra assuming the 05bl-w7e0.7 density and abundance structure.For each of the two epochs, the photospheric position was shiftedfrom its value at − d to higher velocities, until the backscatteringwas reasonably reduced. This resulted in photospheric velocities of and km s − , respectively.After calculating these initial models, we explored the effectof changes in the chemical composition, performing three addi-tional code runs for each epoch. In the first two runs, we reducedIME and heavier elements to of their original abundances,respectively. For the − d model, these changes were applied tothe whole atmosphere. In the − d model, we kept the originalcomposition at velocities > km s − constant, in order to showthe sensitivity to the abundances in the zone not probed by the − d spectrum. In the third code run, finally, we removed oxy-gen in favour of carbon (so that the mass fraction X ( C )=80% ).This change was applied to the whole atmosphere, also at -10d,as otherwise an inverted composition (larger C abundances furtherinwards) would have resulted.In Fig. 10, we show the resulting spectra and give line iden-tifications to clarify the effect of the modified abundances. More-over, it is indicated which lines do and which do not change sig-nificantly with the modifications. At − d, the synthetic spectrumlooks vastly different from the spectrum of a normal SN Ia. We il-lustrate this in the upper panel of Fig. 10 by additionally plottingthe earliest SN Ia spectrum ever observed (SN 1990N at − d,Leibundgut et al. 1991). Compared to spectra of normal SNe Ia,but also to the − d spectrum, lines of less ionised species appearowing to the low temperatures, which result from the low luminos-ity. Si II and S II lines, which normally characterise SNe Ia, areabsent. The spectrum is especially sensitive to the abundances ofNa, Ca and Fe-group elements (with Na, uncertainties in the ioni-sation remain a caveat, see Sec. 3.1.4). Furthermore, C I features arepresent around ˚A and ˚A. As the C abundance is alreadyquite large in the outermost layers of 05bl-w7e0.7, these featuresdo not react strongly to a further increase of X ( C ) . However, ifmuch less carbon was present, they should gradually disappear. c (cid:13) (cid:13) im SN Ia 2005bl: abundances and density profile. m a ss f r a c t i on [ % ] enclosed mass [M sun ]velocity [km/s]05bl−w7e0.7a) 3000 5000 8000 11000 15000COMgSiSCaTi+CrFe Ni m a ss f r a c t i on [ % ] velocity [km/s]05bl−w7e0.7b) COMgSiSCaTi+CrFe Ni m a ss f r a c t i on [ % ] enclosed mass [M sun ]velocity [km/s]05bl−w7e0.7m1.25c) 3000 5000 8000 11000 15000COMgSiSCaTi+CrFe Ni m a ss f r a c t i on [ % ] velocity [km/s]05bl−w7e0.7m1.25d) COMgSiSCaTi+CrFe Ni Figure 9.
Abundance tomography of SN 2005bl based on w7e0.7. The abundances are plotted versus enclosed mass (panel a) and velocity (panel b). Forcomparison, we also show the 05bl-w7m1.25e0.7 abundances (panels c and d). In velocity space, the patterns of the abundance profiles are very similar. At − d, the structure of the spectrum resembles somewhatmore that at − d. Yet, the spectrum has little in common withthat of the moderately subluminous, spectroscopically rather nor-mal SN 2004eo at − d (Pastorello et al. 2007; plotted in the lowerpanel of Fig. 10). Compared to − d, the − d spectrum still showshints of lower temperatures: because of the scarce population ofexcited levels, the S II “W-trough” does not show up. For the samereason, the C II λ feature is weak. In the model with a larger Cmass fraction, however, some of the strongest lines of C I begin toabsorb at ∼ ˚A. Furthermore, there are absorptions due to O I ,Na I , Si II , Ti II , Cr II and Fe II , which should allow for an analysisof the abundances in the outer layers as soon as observations areavailable.The amount of extra information which can be inferred fromearly-time spectra will, of course, also depend on the actual lumi-nosity of the SN at these epochs. Larger luminosities mean highertemperatures, making lines of different ions appear. However, ourresults already suggest that there are interesting possibilities to in-fer the chemical composition of the outermost ejecta. We conducted an abundance tomography of SN 2005bl and con-firmed that nuclear burning in dim, 91bg-like SNe Ia stops at less advanced stages compared to normal SNe Ia. The spectra indicatethat the abundance of burned material above ∼ km s − is muchlower than even in moderately-luminous objects (Mazzali et al.2008). From ∼ km s − down to ∼ km s − , IME dom-inate the ejecta. This points towards large-scale incomplete Si-burning or explosive O burning (e.g. Woosley 1973). A detona-tion at low densities, as it proceeds in the outer layers of delayed-detonation models (Khokhlov 1991), may be responsible for theabundance pattern we find. Assuming this, we need to under-stand how low densities could prevail in such a large fraction ofthe envelope at the onset of the detonation. Up to now, all ex-plosion models which pre-expand the star by a deflagration andthen detonate (e.g. Hillebrandt & Niemeyer 2000, Badenes et al.2003, Gamezo, Khokhlov & Oran 2004, R¨opke & Niemeyer 2007,Bravo et al. 2009) produce larger amounts of Ni. This indicatesthat either the pre-expansion is too weak or the amount of Ni pro-duced in the deflagration stage is already too large. As it is uncer-tain whether a suitable single-degenerate model can be found, thepossibility of a double-degenerate origin of dim SNe Ia deservesattention.Besides the abundances, we have obtained information aboutthe density profile of SN 2005bl. We showed that the spectra areincompatible with the presence of significant amounts of oxygenat v & km s − . Together with the low abundances of burningproducts, this indicates a general lack of material at high veloci- c (cid:13) (cid:13) Hachinger et al. F l ( a r b i t r a r y un i t s ) l [ Å ] w7e0.7 synthetic spectra at −15d,v ph =15200kms −1 , log(L bol / L )=7.04 T i II C a II T i II e t c . N a I C I ( p ) O I ( M g II, C I ) C a II C I (r , b ) O I ( b ) M g II ( b ) SN 90N,−14d(scaled) original compositionX(C) = 0.80X(IME) * 0.2X(Sc..Ni) * 0.2 3000 4000 5000 6000 7000 8000 9000 10000 F l ( a r b i t r a r y un i t s ) l [ Å ] w7e0.7 synthetic spectra at −10d,v ph =11750kms −1 , log(L bol / L )=8.13 ( M g II ( b )) C r II, F e II ( S i II ( b )) T i II, C r II ( F e II ) N a I S i II S i II ( s ) C II ( b ) e t c . O I M g II ( b ) O I ( b ) C a II ( s , b , o ) C I M g II ( b ) O I ( b ) M g II ( b ) O I ( b ) SN 04eo,−11d(scaled,smoothened) original compositionX(C) = 0.80 everywhereX(IME) *0.2 at v<15200kms −1 X(Sc..Ni)*0.2 −− " −− T i II C a II ( o ) ( S i II ( b )) T i II Figure 10.
Early time synthetic spectra for 15d and 10d before B maximum, based on 05bl-w7e0.7. The black lines are spectra calculated with the originalcomposition. The other lines illustrate the most notable changes which occur when setting X ( C ) to (red, dashed line) and when reducing IME (Mg upto Ca, blue, solid line) or heavier elements (Sc to Ni, magenta, dotted line) to 1/5 of their original abundances, respectively (in the velocity ranges indicated).For comparison, we plotted spectra of SN 1990N at -14d and of SN 2004eo at -11d as grey thick lines below the synthetic spectra.Approximate identifications are given for prominent features. Weak lines are given in parentheses; lines not reacting to changes in the abundances are shownwith dotted marks, with the reasons for the insensitivity indicated as follows (based on a rough analysis): (b) – line is heavily blended; (o) – line is formedmainly at > km s − ; (r) – relative change of abundance is too small; (s) – line is partially saturated. ties, albeit less extreme than in objects like SN 2005hk (Sahu et al.2008). We tested whether a good fit to the observed spectra ispossible using a modified W7 model, scaled to a different totalmass and/or energy. Indeed, a reduction of ∼ in total kineticenergy yielded a spectroscopically, and also energetically consis-tent Chandrasekhar-mass model (05bl-w7e0.7). Such consistencecan also be reached with somewhat super- or sub-Chandrasekharmass density profiles, provided that they are similar to w7e0.7 at v & km s − . Deviations of & from the Chandrasekharmass seem disfavoured. With our most massive models ( . M Ch ),it proved impossible to obtain spectroscopic and energetic consis-tence at the same time. In addition, these models as well as theleast massive ones ( . M Ch ) most likely would yield a light curvenot matching that of a dim SN Ia.Sharper constraints on density models, as well as on the abun-dance structure in the innermost and outermost layers may be ob- tained from very-early-epoch and nebular spectra of dim SNe Ia.More extensive observations are needed in order to complete ourpicture of these objects, and of SNe Ia in general. ACKNOWLEDGEMENTS
This work was supported in part by the European Community’s Hu-man Potential Programme under contract HPRN-CT-2002-00303,‘The Physics of Type Ia Supernovae’, and by the DFG-TCRC 33“The Dark Universe”. We would like to thank the anonymous ref-eree for constructive comments. c (cid:13) (cid:13) im SN Ia 2005bl: abundances and density profile. References
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APPENDIX A: PARAMETERS OF THE MODELS
Table A1 shows the code input parameters of all spectral modelsmentioned in the main paper. Apart from the abundances, the codetakes as input the photospheric velocity v ph , the time from explo-sion t (see main text) and the bolometric luminosity L bol . For differ-ent models of a given spectrum, these luminosities can differ a bit,depending on the model spectral energy distribution. In addition tothe input values, Table A1 also gives the calculated temperature ofthe photospheric black body emission, T BB , for each model. c (cid:13) (cid:13) Hachinger et al.
Table A1.
Parameters of the models. Abundances are given only for more significant elements.
Model epochs lg (cid:18) L bol L ⊙ (cid:19) v ph T BB Element abundances (mass fractions)[d] [km s − ] [K] X (C) X (O) X (Na) X (Mg) X (Al) X (Si) X (S) X (Ca) X (Ti) X (Cr) X (Fe) a) X ( Ni) a) a) The abundances of Fe, Co and Ni in our models are assumed to be the sum of Ni and its decay chain products ( Co and Fe) on the one hand, and directly synthesised / progenitor Fe on the other hand.Thus, they are conveniently given in terms of the Ni mass fraction at t = 0 [ X ( Ni ) ], the Fe abundance at t = 0 [ X ( Fe ) ], and the time from explosion onset t . c (cid:13) (cid:13)(cid:13)