Abundance stratification in type Ia supernovae -- VI: the peculiar slow decliner SN\,1999aa

The abundance distribution in the ejecta of the peculiar slowly declining Type Ia supernova (SN\,Ia) SN\,1999aa is obtained by modelling a time series of optical spectra. Similar to SN\,1991T, SN\,1999aa was characterised by early-time spectra dominated by \FeIII\ features and a weak \SiII\,6355\,\AA\ line, but it exhibited a high-velocity \CaII\,H\&K line and morphed into a spectroscopically normal SN\,Ia earlier. Three explosion models are investigated, yielding comparable fits. The innermost layers are dominated by $\sim 0.3$\,\Msun\ of neutron-rich stable Fe-group elements, mostly stable iron. Above that central region lies a \Nifs-dominated shell, extending to $v \approx 11,000$ -- $12,000$\,\kms, with mass $\sim 0.65$\,\Msun. These inner layers are therefore similar to those of normal SNe\,Ia. However, the outer layers exhibit composition peculiarities similar to those of SN\,1991T: the intermediate-mass elements shell is very thin, containing only $\sim 0.2$\,\Msun, and is sharply separated from an outer oxygen-dominated shell, which includes $\sim 0.22$\,\Msun. These results imply that burning suddenly stopped in SN\,1999aa. This is a feature SN\,1999aa shares with SN\,1991T, and explain the peculiarities of both SNe, which are quite similar in nature apart from the different luminosities. The spectroscopic path from normal to SN\,1991T-like SNe\,Ia cannot be explained solely by a temperature sequence. It also involves composition layering differences, suggesting variations in the progenitor density structure or in the explosion parameters.


INTRODUCTION
Type Ia supernovae (SNe Ia) are among the most luminous transients in the Universe. They are thought to be the thermonuclear explosions of carbon-oxygen (CO) white dwarfs close to the Chandrasekhar limit (Hillebrandt & Niemeyer 2000;Mazzali et al. 2007;Livio & Mazzali 2018). A relation between the peak luminosity with the width of the light curve (Phillips 1993) makes SNe Ia standardisable candles and has led to their practical use as distance indicators and for the discovery of dark energy (Riess et al. 1998;Perlmutter et al. 1998).
The rise of the SN Ia light curve is caused by the deposition of the gamma-rays and positrons emitted in the decay of the 56 Ni synthesised during the explosion (Arnett 1982;Kuchner et al. 1994; 1 E-mail:charlesaouad@aascid.ae Mazzali et al. 1998Mazzali et al. , 2001. The optical photons created in this process remain trapped until the ejecta become optically thin as they expand , allowing their diffusion. Therefore, the peak of the light curve is directly proportional to the mass of 56 Ni synthesised, while its width is related to the photon diffusion time, which is a function of ejected mass, kinetic energy, the radial distribution of 56 Ni, and of the effective opacity, which is itself a function of temperature, density, and composition . Even though the majority of SNe Ia constitute a nearly equivalent group of intrinsically bright events and their spectroscopic features are fairly similar, observations indicate a scatter in their spectroscopic properties (Branch 2001;Silverman et al. 2012a;Siebert et al. 2019;Jha et al. 2019); see Filippenko (1997) for a review.
A question that arises is how distinct these events are. A clear separation could mean that they are of intrinsically different nature, while a continuity of properties would suggest quasisimilar events, with the observed diversity being caused by smoothly changing parameters. Important factors are the physical mechanism through which the white dwarf reaches ignition densities, the mass at explosion, and the explosion mechanism.
Different regimes under which the burning flame propagates lead to different nucleosynthetic yields, different composition structures, and therefore different spectral features. Simulations of pure deflagration models were unable to reproduce 56 Ni masses of ∼ 0.5 M and kinetic energies ∼ 10 51 ergs, as derived from observations (Mazzali et al. 2007). In contrast, a pure detonation, in which the burning front propagates supersonically and ignites the fuel by compressive heating, incinerates the whole star to iron-group nuclei and cannot explain the presence of intermediate-mass elements (IMEs) in the ejecta outer layers. Alternative successful models have been proposed in which the deflagration front transits to a detonation at some specific density (deflagration to detonation transition, or DDT) (Khokhlov 1991b). One-dimensional simulations of delayeddetonation models have proven successful in reproducing many of the observed spectral features of SNe Ia, in particular, the presence of a layer of IMEs, the product of partial burning of carbon and oxygen. These models can also account for the energy budget of the most energetic events and for the observed photospheric velocities. However, the exact physics of how this transition occurs is still a subject of extensive research (Woosley 2007).
The early-time spectra of normal SNe Ia are characterised by lines of singly-ionized IMEs such as Mg, Si, S, Ca, and iron-group elements (hereafter, Fe-gp). As time progresses, Fe lines increase in strength until they dominate the appearance of the spectrum a few weeks after maximum light (Filippenko 1997;Parrent et al. 2014). However, in some "peculiar" events characterised by high luminosity (SN 1991T, and SNe of its subgroup; Filippenko et al. e.g., 1992), singly-ionised IMEs only start to appear near maximum light, never reaching the same intensity as in normal SNe Ia. Their early-time spectra are instead dominated by doubly-ionised species such as Fe III and Si III. The presence of these lines requires high temperatures in the outer ejecta.
SNe Ia with properties intermediate between those of SN 1991T and normal SNe Ia have been discovered. One case in particular is that of SN 1999aa, the subject of this study (Garavini et al. 2004;Jha et al. 2006;Matheson et al. 2008). Similar to SN 1991T, SN 1999aa was a slow decliner, with ∆m15(B) measurements ranging from 0.75 mag (Krisciunas et al. 2000) to 0.85 mag (Jha et al. 2006). The earliest spectra of SN 1999aa resemble those of SN 1991T in being dominated by Fe III lines and by the weakness of singly-ionised IME lines, in particular Si II 6355Å. However, unlike SN 1991T, they showed a high-velocity Ca II H&K feature. SN 1999aa morphed to looking like a normal SN Ia earlier than did SN 1991T. In fact, one week before B maximum, S II 5468, 5654Å and Si II 6355Å were already visible in SN 1999aa. Figures 1 and 2 show optical spectra of SN 1999aa compared to SN 1991T and the spectroscopically normal SN 2003du, respectively ∼ 10 days before and near B maximum.
A theoretical understanding of SN 1999aa should help clarify the spectroscopic sequence from normal to SN 1991T-like events. A first step toward this is to derive the composition and stratification of the ejecta. This can be done using the so-called "abundance tomography" technique (Stehle et al. 2005  a temporal series of spectra to reproduce their features consistently. At early times, the spectra are characterised by a pseudocontinuum on which P Cygni profiles of the lines that dominate near the momentary photosphere are superimposed. As the ejecta expand, the photosphere recedes inward and reveals progressively deeper layers. This approach was successfully used to model several SNe Ia: SN 2002bo (Stehle et al. 2005), SN 2004eo (Mazzali et al. 2008), SN 2003du (Tanaka et al. 2011), SN 1991T (Sasdelli et al. 2014), and SN 1986G (Ashall et al. 2016).
Here, we use the abundance tomography technique to investi-gate the properties of SN 1999aa. In Section 2 we describe the data used, and in Section 3 we explain the modelling methods. We present our modelling results in Sections 4 and 5. In Section 6 we discuss the abundance tomography results. We use the derived abundances to compute a bolometric light curve in Section 7, and Section 8 discusses our results. Our conclusions are drawn in Section 9.

DATA
SN 1999aa was discovered independently by Nakano et al. (1999), Armstrong & Schwartz (1999), and Qiao et al. (1999). The host galaxy is NGC 2595, a barred spiral of morphological classification SAB(rs)c with redshift z = 0.0144 (Epinat et al. 2008;van Driel et al. 2016). Distance moduli to the galaxy based on the Tully-Fisher relation range from µ = 32.30 ± 0.53 mag (Bottinelli et al. 1985) to µ = 34.44 ± 0.47 mag (Theureau et al. 2007). Distances using the light curve of SN 1999aa vary from µ = 33.43 ± 0.16 mag (Amanullah et al. 2010) to µ = 34.58 ± 0.24 mag (Riess et al. 2004). Photometric data are taken from Jha et al. (2006), Krisciunas et al. (2000), Qiao et al. (1999), Armstrong & Schwartz (1999), Yoshida et al. (1999), and Altavilla et al. (2004). Late-time unpublished data are based on observations collected with the Optical Imager Galileo (OIG) at Telescopio Nazionale Galileo (TNG) -La Palma. The TNG + OIG U BV RI frames were reduced following standard procedures and made use of the ECSNOOPY package (Cappellaro 2014) using the point spread function (PSF) fitting technique for the SN measurement. The BV RI SN magnitudes were then calibrated with reference to the magnitudes of field stars retrieved from Krisciunas et al. (2000), while for the U band, we converted the SDSS catalog magnitudes of the local sequence into Johnson U following Chonis & Gaskell (2008). The final TNG + OIG magnitudes are shown in Table 1, where the mean photometric errors, estimated with artificial-star experiments, are given in parentheses.
The spectra used in this study are available at the Weizmann Interactive Supernova Data Repository (WISeREP) (Yaron & Gal-Yam 2012); they are listed in Table 2. The spectra were calibrated against photometric observations. Calibration was performed in the U , B, V , and R bands by multiplying the spectra with a line of constant gradient or with a low-order smoothed spline. We ensured that the flux in the spectra in any passband did not vary by more than ∼ 10% from the observed flux in that filter passband.

MODELLING TECHNIQUES
Spectra in the photospheric phase have been modelled using a Monte Carlo spectrum synthesis code Lucy 1999a,b). The code assumes a sharp photosphere. As the ejecta expand homologously, the photosphere recedes in velocity space and consequently in mass coordinate. Thermal equilibrium is assumed. Photons emitted from the photosphere propagate through the expanding ejecta and interact with the gas through line absorption, including line branching (Mazzali 2000), or electron scattering. The required input parameters are the density structure of the SN ejecta, the emergent bolometric luminosity L Bol , the photospheric velocity v ph , the time from the explosion t0, and the abundances of the elements as a function of depth above the photosphere.
The distance and the extinction to the SN are needed in order to scale the flux. Since the distance to NGC 2595 is not known accurately, we treat it as a free parameter, within the range allowed by the literature. We tested several values of the distance modulus (µ) for three different spectra. For larger distances the high luminosity +16.8 +13.8 +7.8 +1.8 -0.2 -1.9 -2.9 -3.9 -5.0 -6.2 -7.0 -7.9 -8.9 -10.2 -11.2 +HV shell W7 at t=100s DD3 at t=100s DD2 at t=100s days from B max Figure 3. The three density profiles used in the modelling: W7 (Nomoto et al. 1984), DD2, and DD3 (Iwamoto et al. 1999). Vertical dashed lines mark the photospheres of the synthetic spectra.
causes a high temperature, which in turn leads to unrealistic ionisation. The opposite happens for distances that are too small. The best models are obtained with µ = 34.00 mag, which is very close to the mean value calculated using the various distance moduli reported in the literature,μ=33.975±0.34 mag. We adopt an extinction value E(B − V ) = 0.04 mag (Schlegel et al. 1998) for the Milky Way and assume E(B − V ) = 0.00 mag for the host galaxy (Krisciunas et al. 2000). We also tested different rise times, between 19 and 21 days; the best results are obtained with a value of 20 days. We use three different density-velocity distributions: the classical fast deflagration model, W7 (Nomoto et al. 1984), and two moreenergetic delayed-detonation models, DD2 and DD3 from Iwamoto et al. (1999). These density profiles are shown in Fig. 3.
Having fixed µ, t0, and E(B − V ), the modelling starts with the earliest spectrum. Different values of L Bol are tried until the synthetic spectrum matches the observed one in flux. After that, v ph is iterated to match the position of the spectral features and the overall temperature. In parallel, the abundances are modified until the model matches the observation.
For the following spectrum in the sequence, a new, smaller v ph is defined. This will introduce a new shell where new abundances can be determined. This process is repeated for each spectrum. As the spectra evolve, deeper layers are revealed and the abundance stratification is gradually built.

THE PHOTOSPHERIC PHASE
We modelled 15 spectra, from day −11 to day +14 from B maximum. The input parameters are shown in Table 2. The synthetic spectra corresponding to the three explosion models we use are shown in Figs. 4,8,9,10,and 11, overlaid on the observed spectra.

The early-time spectra
In Fig. 4 we show models for the earliest spectra, ranging from −11 to −9 days from B maximum. The synthetic spectra reproduce the observed features well. In particular, they exhibit deep absorption lines of Fe III and Si III, and the overall flux matches the observed  Fe-group elements: A small amount of iron is needed at the outer shells to reproduce the deep observed features near 4250 and 5000Å. This is stable Fe; at these early stages, 56 Ni would not have had time to decay significantly into 56 Fe. The mass fraction of Fe at v > 12, 600 km s −1 needed to reproduce the observed features is ∼ 0.015-0.018 (Fig.5). The presence of stable Fe in the outer shells has been reported in other SNe Ia. Sasdelli et al. (2014) give Small amounts of 56 Ni, Ti, and Cr are needed at these epochs to block the ultraviolet (UV) flux and redistribute it redward. The abundance of 56 Ni is not constrained at these epochs, as no visible line in the spectrum is reproduced by 56 Ni or Co alone. Unfortunately, the spectra of SN 1999aa do not extend bluer than ∼ 3400Å, where a prominent feature dominated by Co III should be expected at ∼ 3200Å (Mazzali et al. 1995;Stehle et al. 2005;Sasdelli et al. 2014).
Calcium: The early-time spectra of SN 1999aa show a deep ab-  (Garavini et al. 2004). This feature is due to high-velocity Ca II H&K (Mazzali et al. 2005b).
We are able to produce it with X(Ca) ≈ 0.0035 at v > 21, 000 km s −1 with the W7 model. The DD2 and DD3 models have more mass at high velocity, and therefore X(Ca) ≈ 0.00025 is sufficient (see Tanaka et al. 2011). This is much less than the Ca abundance reported in some spectroscopically normal SNe Ia (Tanaka et al. 2008). However, those SNe exhibit a much stronger Ca II H&K lines and a much earlier appearance of the Ca II NIR feature than in both SN 1999aa and SN 1991T. On the other hand, the abundance we obtained is similar to that obtained for SN 1991T, for which Sasdelli et al. (2014) estimate a Ca abundance < 0.0003 at v > 17000 km s −1 using the DD3 density profile. Regardless of the density profile used, our results suggest that the abundance of Ca at high velocities is supersolar (X(Ca ) = 0.00006, Asplund et al. 2009).
The strength of the Ca II H&K feature is very sensitive not only to the Ca abundance at high velocity, but also to the parameters that directly affect ionisation, in particular the electron density. The presence of free electrons decreases the ionisation and favours singly-ionised species (Mazzali et al. 2005a,b). Adding H results in a higher electron density. Following Tanaka et al. (2011), in Fig. 6 we show how the Ca II H&K feature can be reproduced with different Ca abundances coupled with different amounts of H at the outermost shells (v > 21, 000 km s −1 ). However, because of the degeneracy between the Ca abundance and the electron density, it is not possible to determine the Ca mass fraction. Hydrogen may result from the interaction of the ejecta with the circumstellar medium (Mazzali et al. 2005b) or may be a remnant of accretion on the surface of the WD (Livio & Mazzali 2018) . Even though small amounts of H are sufficient to reduce the ionization at the dilute outermost layers , and therefore create the HVFs ubiquitously observed in SNe Ia spectra (Mazzali et al. 2005b), larger amounts (X(H) 0.3), will give rise to an Hα feature that is not seen in the observed spectrum. The lack of H signatures can be taken as an argument against the single degenerate scenario (Marietta et al. 2000;Panagia et al. 2006) but it is not enough to rule it out. (Justham 2011;Hachisu et al. 2012)(for a review, see Livio & Mazzali 2018).
Silicon, Sulphur, Magnesium: The Si II 6355Å line is much weaker in the earliest spectra of SN 1999aa than in normal SNe Ia. It grows in strength as the spectra evolve. However, the feature near 4400Å, which is due to Si III 4553, 4568, 4575Å, is prominent in the earliest spectra, as the high temperature favours doubly-ionised species. These Si lines are well reproduced in the synthetic spectra. The Si mass fraction is 0.025 at v > 12, 600 km s −1 , but it rapidly increases to 0.73 at v > 12, 300 km s −1 .
The two S II features at 5468 and 5654Å are not present at these early epochs, and only start to show at day ∼ −6. The Mg II 4481Å line is never visible in the spectra of SN 1999aa, as that region is dominated by Fe III lines.
Carbon, Oxygen: A C II 6578, 6583Å line has been detected in some SNe Ia (Mazzali 2001;Parrent et al. 2011). It can be observed on top of the Si II 6355Å P Cygni emission, but it is not a common feature. This line is not visible in SN 1999aa. An upper limit to the carbon abundance of ∼ 0.0005 by mass at v > 12, 600 km s −1 can be determined (Fig. 7). The absence of C in the outermost layers favours delayed-detonations models (Khokhlov 1991b;Marion et al. 2006). It is also possible that carbon is present, but most of it is in a doubly ionized state and therefore does not produce a visible feature in the spectrum. It is difficult to excite any lower level of any optical line of C III at the temperatures of even a luminous SN Ia. Carbon may also be present at much higher velocities, but in order to investigate this, we need earlier observations which unfortunately are not available for SN 1999aa.
The synthetic spectra using W7 show a shallow O I 7744Å line (see Fig. 4). This feature is not seen in spectra of SN 1999aa. The DD2 and DD3 profiles have less mass in the region between 13,000 and 16,000 km s −1 , where the line is formed (see Fig. 3), and produce a much shallower feature that matches better the observed spectra (days −11 and −10). The oxygen abundance can be constrained by considering the abundances of other elements that are present at high velocity (Si, S, Ca, Fe, and 56 Ni  at v > 12, 600 km s −1 , but is absent already at v ≈ 12, 300 km s −1 . This behaviour is remarkably similar to that of SN 1991T (Sasdelli et al. 2014). In contrast, Tanaka et al. (2011) report the presence of oxygen down to a velocity of 9400 km s −1 for the normal SN Ia 2003du.

Pre-maximum-light spectra
Figs. 8 and 9 show spectra ranging from −8 to −3 days from B maximum. The photospheric velocity evolves from 11,500 to 10,000 km s −1 .
Fe-group elements: The Fe III lines observed near 4300 and 5000Å increase in strength. Our synthetic spectra reproduce their evolution. At these epochs, the fraction of Fe originating from 56 Ni decay becomes significant. At day −7, it already constitutes 20% of the total Fe mass fraction, increasing to 30% at day −3. We obtain good fits for X(Fe stable ) ≈ 0.1 in the shells that are probed. The abundance of 56 Ni increases from 0.05 at v = 11, 500 km s −1 to 0.53 at v = 10, 950 km s −1 .
Calcium: The synthetic spectra reproduce well the Ca II H&K feature. The Ca II near-infrared (NIR) triplet is still not seen in the observed spectra at day −6, and this is confirmed in our synthetic spectra. The near-photospheric Ca abundance at these epochs is ∼ 0.0015 for all models.
Silicon, Sulphur: The Si II 6355Å line gets deeper with time. This is well replicated in our synthetic spectra, as are the Si III feature near 4400Å and the Si II 4130Å line. The Si abundance is 0.72 at v ≈ 11, 500 km s −1 , and it decreases to 0.26 at v ≈ 10, 000 km s −1 . The two S II lines at 5468 and 5654Å start to show at day −6 and grow stronger with time. The sulphur abundance is 0.15 at v ≈ 11, 500 km s −1 , decreasing slightly to lower velocities.
Carbon, Oxygen: C and O are not needed at these epochs. Any abundance of C would produce a line that is not seen in the observed spectra. Oxygen is not needed because the Fe-group elements and IMEs are sufficient to complete the composition at these velocities. Fig. 10 shows spectra ranging from −2 to +2 days from B maximum. The photospheric velocity evolves from 9600 to 8900 km s −1 . The synthetic spectra predict excess flux redward of ∼ 6000Å. At these epochs, as the photosphere recedes inside the 56 Ni-dominated shell, energy is partially deposited above the photosphere, and the assumption of blackbody emission at the photosphere is not entirely correct.

Spectra at maximum brightness
Fe-group elements: At these epochs, Fe lines are quite deep. The Fe abundance is high, because iron from 56 Ni decay is now a significant contribution (∼ 30%) to the Fe abundance.  near 5000Å becomes broader because of the contribution of Fe II lines. This is reproduced reasonably well in our synthetic spectra.
Calcium: The synthetic spectra still reproduce well both the depth and the shape of the Ca II H&K feature. At these epochs, it becomes contaminated by Si II and Co III lines in its bluer part and by Si II and Fe III lines in its redder part (see Silverman et al. 2015). The Ca II NIR triplet begins to appear two days after B maximum, and this is reproduced in the synthetic spectra. This feature is seen much earlier in spectroscopically normal SNe Ia, where it is much stronger than in SN 1999aa even ∼ 12 days before maximum light (Stehle et al. 2005;Mazzali et al. 2008;Tanaka et al. 2011). Instead, in SNe 1991T and 1999aa it only appears a few days after B maximum. Calcium extends down to v ≈ 9600 km s −1 .
Silicon, Sulphur: The shape and depth of the prominent Si II 6355Å line are well replicated in the synthetic spectra. The silicon abundance is 0.25 at v ≈ 9600 km s −1 , decreasing to 0.1 at v ≈ 8900 km s −1 . S II 5468, 5654Å are now prominent, and increase in strength with time. Our synthetic spectra reproduce their evolution and the ratio of their depths reasonably well. The S abundance is 0.12 by mass at v ≈ 9600 km s −1 , decreasing to 0.05 at v ≈ 8900 km s −1 . Fig. 11 shows spectra ranging from 8 to 17 days after B maximum. The photospheric velocity evolves from 7150 to 4250 km s −1 . At these epochs, the quality of the fits starts degrading, as the photosphere resides deep in the 56 Ni-dominated region. Therefore, we do not use these epochs to infer abundances, but rather employ the nebular-phase models. Nevertheless, the synthetic spectra reproduce the observed ones sufficiently well.

Post-maximum-light spectra
Fe-group elements: At these epochs (∼ 30-40 days after the explosion), more than about 70% of all Fe originates from the decay of 56 Ni. The Fe II feature near 5000Å splits into three components that are fairly reproduced in the synthetic spectra. This is the consequence of the lower degree of line blending at slower velocities.
Calcium: The strong Ca II H&K line is still reproduced fairly well. The Ca II NIR triplet is now clearly visible, and it shows two distinct features, which are well reproduced in shape.
Silicon, Sulphur: The synthetic spectra still reproduce well the Si II 6355Å line. Si II 4130Å is now contaminated by Fe II 4173, 4233, 4351Å and Co II 4160, 4145Å. The S II 5468, 5654Å lines are also contaminated by a contribution from Fe III. The feature near 5700Å may be due to Na I D absorption   and +17 without major modifications to the Na ionisation structure .

SPECTRA IN THE NEBULAR PHASE
Two epochs of nebular spectroscopy are available for modelling. Both were obtained with the Low-Resolution Imaging Spectrometer (Oke et al. 1995) on the Keck-I 10 m telescope. One spectrum was taken on 1999 Nov. 9 (exposure time 600 s; airmass 1.01), the other on 1999 Dec. 5 (exposure time 300 s; airmass 1.01), corresponding to 275 and 301 rest-frame days after explosion, respectively; see Silverman et al. (2012a) for details regarding data acquisition and reduction.
The spectra were modelled using our non-local thermodynamic equilibrium (NLTE) code, which is based on the assumptions set out by Axelrod (1980). The emission of gamma rays and positrons from a distribution of 56 Ni is computed, and the propagation and deposition of these particles is determined using a Monte Carlo scheme as outlined first by Cappellaro et al. (1997). Opacities κγ = 0.027 cm 2 g −1 and κ e + = 7 cm 2 g −1 are assumed in order to compute the deposition of energy. The energy that is deposited is used partly for impact-ionisation, while the rest heats the ejecta via collisional excitation. The population of excited levels is computed in NLTE. Heating is then balanced by cooling via line emission. Most emission is in forbidden lines, in particular of the elements that dominate the inner ejecta (i.e., Fe, Co, and Ni), but some is also via permitted transitions, in particular of Ca II. The ejecta are assumed to be transparent to optical radiation, so no transport is performed. As discussed by Mazzali et al. (2007) and others, the code can use a one-dimensional stratified density and composition, and it can be employed to explore the inner layers of an SN Ia and thus to complete the tomography experiment in regions that are not accessible during the early photospheric phase.
The same three explosion models used for the early-time data are tested in the nebular phase, at both available epochs. Using the density distribution of the original models and the composition for the outer regions derived from the early-time models, we now modify the abundances in the inner layers (v < 8000 km s −1 ) in order to optimise the fits. A best fit is defined empirically, as it is impossible to match every line and not all lines carry the same weight of information, but basically we need to match both the intensity of the lines (which depends on the amount of 56 Ni as well as of the emitting element) and their width (which traces the radial distribution of  the emitting elements as well as indirectly that of 56 Ni, since heating from radioactive decay must reach the emitting region). Collisional data are not perfectly known for many of the Fe lines in the optical region, so we cannot expect that all emission lines will be simultaneously reproduced. We focus therefore on reproducing the strongest emission lines. Fortunately, these include emission from both Fe III (the emission near 4800Å) and Fe II (the emission near 5200Å), so we can control the ionisation of Fe, which is the dominant species in the inner ejecta at the time of the nebular spectra.
Figs. 12 and 13 show the fits to the two nebular spectra. We used the same composition at both epochs, which confirms that radioactive decay is the sole powering mechanism of the SN luminosity.
The mass of 56 Ni synthesised is ∼ 0.65 M for all three models. The stable Fe mass is highest when using DD2, but it is still within the expected range of values ). The extra Fe seems to be located at 9000-12,000 km s −1 . Stable Fe is necessary to reduce the ionisation degree and obtain a reasonable ratio of the [Fe III] and [Fe II]-dominated features. The mass of stable Ni is quite low, and this is reflected by the weakness of the only visible Ni line, [Ni II] 7380Å. This is common to all SNe Ia we have studied, and suggests that little stable Ni is synthesised even in the dense innermost regions of SNe Ia. A moderate degree of clumping   Table 3, where the expected EK is also shown and compared to that of the original models.

ABUNDANCE TOMOGRAPHY
The mass fractions of different elements as a function of mass and velocity for the three density profiles are shown in Figures 14, 15, and 16, compared to the original abundance distributions in the hydrodynamical models (Nomoto et al. 1984;Iwamoto et al. 1999). The inner core, up to v ≈ 2500 km s −1 , is dominated by stable Fe with a small amount of 56 Ni. Stable Fe-group elements are synthesised by electron capture in the high-density/temperature core (ρ ≥ 10 8 g cm −3 ; T ≥ 5 × 10 9 K) during the explosion, when nuclear statistical equilibrium (NSE) is attained (Arnett 1982;Iwamoto et al. 1999;Woosley et al. 2007). The distribution of these elements that we derive is in general agreement with the various explosion models.
Moving outward, a 56 Ni-dominated shell extends over ∼ 0.8-1 M , out to v ≈ 11, 000 km s −1 . Practically no stable Ni is present in this region, in contrast to all explosion models, while a significant amount of stable iron is present, similar to the model prediction in the inner regions of this shell but significantly above it in regions between 3000 and 8000-9000 km s −1 . This results in a larger production of stable Fe, at the expense of stable Ni, when our results are compared to the original models.
A narrow, IME-dominated shell characterises velocities ∼ 11, 000-12,000 km s −1 . The abundance of IMEs decreases sharply above this velocity. In the hydrodynamic models, this shell extends to higher velocities. The confinement of the IMEs in a narrow shell was also suggested by Garavini et al. (2004) based on the velocity evolution of Si II 6355Å. IMEs are the result of incomplete burning, when the densities drop to ∼ 10 7 g cm −3 . Their sudden depletion suggests a sudden drop in burning, which may be a key element to understand the structure of the progenitor and the explosion mechanism. The weakness of the IME lines in the early-time spectra of SN 1999aa and other SN 1991T-like SNe Ia is therefore an abundance effect (see Jeffery et al. 1992;Filippenko et al. 1992), and not only a temperature effect. The abundance of 56 Ni is still significant in this region.
Above the IME shell, an O-rich outer layer is present. We could not conclusively determine the C abundance as no strong C features  . Nebular-phase spectra obtained on 1999 Dec. 5, corresponding to 281 rest-frame days after B maximum (black). Line identification and colour codes are similar to those in Fig. 12 are observed. These outermost layers determine the appearance of the earliest spectra (see Fig. 3). Small amounts of Ca are necessary to form the Ca II high-velocity features (HVFs). A small abundance of stable Fe, roughly a few per cent, is necessary in order to form Fe lines at the earliest epochs. This is larger than the solar abundance.
The host-galaxy metallicity at the location of SN 1999aa is 12 + log(O/H) = 8.45 (Galbany et al. 2016), about a factor of two below solar, suggesting that Fe at these shells is probably the result of explosive nucleosynthesis (see also Hachinger et al. 2013). In general, the presence of Fe at these shells is more consistent with DD2 and DD3 than with W7. Only a very small amount of 56 Ni is present in the O-rich layer, as also previously reported in other SNe Ia (Stehle et al. 2005;Tanaka et al. 2011). The distribution of 56 Ni is in general consistent with the explosion models.

Building the bolometric light curve
We constructed a bolometric light curve of SN 1999aa in the range 3000-10,000Å. The U BV RI light curves were splined with a time resolution of 1 day, dereddened with the extinction curve of Cardelli et al. (1989) using E(B − V ) = 0.04 mag (Schlegel et al. 1998) and reduced to the rest frame. Daily spectral energy distributions in the above wavelength interval were constructed using the flux zeropoints of Fukugita et al. (1995). For each epoch, we integrated the U − to I-band flux after interpolating the flux between the central wavelengths of the filters, and added at the blue and red boundaries of the interval the fluxes obtained extrapolating the spectrum with a flat power law to 3000Å and 10,000Å, respectively. The final bolometric light curve was resampled to the epochs of the actual optical observations. Since the first four measurements (i.e., prior to 1999 Feb. 13.5) are unfiltered, they have been assimilated to V -band fluxes and a bolometric correction was applied to them equivalent to the difference between the early bolometric magnitude and the simultaneous V -band magnitude. Bolometric luminosities were obtained using the luminosity distance of NGC 2595 (63.1 Mpc); they are shown in Fig. 17 as black circles.
We evaluated the contribution of the NIR flux to the bolometric light curve. NIR photometry in the J and K bands is available at four epochs after maximum brightness (Krisciunas et al. 2000). The NIR luminosity in the range 10,000-24,000Å was constructed following a procedure analogous to the one adopted in the optical. Flat power laws were used to estimate the flux shortward of the J band and longward of the K band. Luminosities over the range 3000-24,000Å at the four epochs when NIR observations are available are shown in Fig. 17 as red circles.
No UV observations are available for SN 1999aa, so we cannot account for flux at wavelengths shorter than the Bessell U filter (λ < 3000Å ). The UV should make a significant contribution only at the earliest epochs (see below).

Modelling the bolometric light curve
Having studied the abundance distribution for a few possible explosion models in SN 1999aa, one way to verify the results is to test them against another observable. The light curve is one such observable. As is customary in our work, we computed synthetic bolometric light curves using the density and abundance distributions of the three models we tested. We used a Monte Carlo code that initially follows the emission and deposition of gamma rays and positrons, exactly as in the nebular spectrum calculations. The energy that is deposited is then assumed to be converted to optical photons, which  300 Figure 17. The U BV RI bolometric light curve of SN 1999aa (black dots), compared to the synthetic light curves computed using the density and abundance profiles of the three explosion models: W7 (red), DD2 (green), and DD3 (blue). Red points represent luminosities at the epochs when NIR observations are available from Krisciunas et al. (2000).
are in turn transported through the ejecta using a time-dependent Monte Carlo scheme as outlined by Mazzali et al. (2001). The propagation of the optical photons is subject to an opacity. In the case of SNe Ia (and of all H-poor SNe), line opacity is the dominant opacity (Pauldrach et al. 1996). Line opacity can be parameterised based on the number of active lines in different species and the relative abundance of that species in the ejecta . Photon diffusion also depends on the mass in the ejecta and their expansion (i.e., their E k ).
We computed synthetic bolometric light curves for our three explosion models. These are compared to the bolometric light curve of SN 1999aa in Fig. 17. All three synthetic light curves match the observed one reasonably well. While this suggests that the models we used and the abundances we derived are credible, it is difficult to choose a best-fitting model. Although DD2 yields the closest 56 Ni mass to the value we obtained for SN 1999aa, the correspondence between the values we derived for the masses of the various elements and those in the original hydrodynamic calculation is not always perfect. Also, owing to the lack of early UV data, it is hard to constrain the densities in the outer layers. We can only conclude that DD2 is a reasonable model, but some modification is required. Most likely, a specific model would have to be derived for SN 1999aa, which may be similar to DD2 but may differ in some areas, as was the case for SN 2011fe (Mazzali et al. , 2015.

DISCUSSION
Our synthetic spectra show reasonably good fits to the observed ones for the three density profiles used, with only small differences between them. For example, the Si II 6355Å feature (Fig. 4), the O I 7744Å line (Fig. 8), and the Fe II lines near 5000Å (Fig. 10) are better reproduced with the DD2 and DD3 density profiles than with W7. However, these differences are marginal, and based on this criterion alone it is difficult to select a best-fit model.
The yields of the most important elements or groups of elements are recapped in Table 3. From these yields we computed the expected kinetic energy yield for each model using the formula × 10 51 erg (1) , where E bind = 0.46 10 51 erg is the binding energy of the white dwarf. Results are given in Table 3. The values we obtain are slightly smaller than the original models. The difference may be explained by the weak burning at the outer shells. A similar behaviour was seen in SN 1991T (Sasdelli et al. 2014).
Most significantly, although IMEs reach a high abundance in a shell at ∼ 11, 000 km s −1 , the IME-dominated shell is very narrow, and therefore has little mass. At the outer edge, unlike in normal SNe Ia, the abundance of IMEs drops very sharply at v ≈ 12, 000 km s −1 , above which oxygen dominates.
This suggests that the weakness or absence of Si II and S II features in the earliest spectra of SN 1991T-like SNe Ia is not only an ionisation effect but also the result of a low abundance (Mazzali et al. 1995). In these peculiar SNe the IME abundance in the outermost layers is very small, and therefore the spectra start looking like those of normal SNe Ia at a later time.
In order to check the effect of the Si abundance on the spectra, we computed synthetic spectra using increasing quantities of Si at v ≈ 12, 600 km s −1 , at the expense of O, while keeping L Bol and v ph unchanged (Fig. 19). The Si II 6355Å line in SN 1999aa is well reproduced with a Si abundance of ∼ 0.025. As the Si abundance increases the line gets stronger, and it matches the spectrum of SN 2003du when the abundance is ∼ 0.1 at high velocities, which is comparable to the abundances reported in SN 2002bo (Stehle et al. 2005) and SN 2004eo (Mazzali et al. 2008). The abundance derived by Tanaka et al. (2011) for SN 2003du is even higher (∼ 0.3 at v ≈ 10,500-15,000 km s −1 ).
Although the spectroscopic properties of SN 1999aa suggest that it is physically intermediate between SN 1991T and normal SNe Ia, its photometric properties do not. Our modelling shows that the amount of 56 Ni synthesised in SN 1999aa (∼ 0.65 M ) is less than in SN 1991T (∼ 0.78 Msun; Sasdelli et al. 2014), suggesting that SN 1999aa should be less luminous than SN 1991T (see Fig.  20). However, SN 1999aa was a slower decliner than SN 1991T. SN 1999aa has estimated ∆m15(B) values ranging from 0.75 mag (Krisciunas et al. 2000) to 0.85 mag (Jha et al. 2006), which may be taken to imply that it was actually more luminous than SN 1991T (∆m15(B)= 0.94 mag).
However, a comparison of the bolometric light curves of the two SNe shows that relying on a ∆m15(B) alone would be misleading. The light curve of SN 1991T is brighter throughout, as it should be based on the 56 Ni mass. However, it peaks much earlier than that of SN 1999aa. This is because the 56 Ni abundance in the outer layers of SN 1991T is larger than in SN 1999aa, causing a faster rise to a very luminous maximum (see Fig.18). The luminous phase is then sustained by the larger 56 Ni mass, but the contrast between the nominal luminosity at peak and that 15 days later is larger than in SN 1999aa, which reaches maximum brightness later. This may mean that ∆m15(B)is not valid for the SN 1991T class (see also Pinto & Eastman 2000;Woosley et al. 2007;Scalzo et al. 2012), and it was also suggested for objects at the faint end of the luminosity-width relation (Ashall et al. 2018). On the other hand, SN 2003du and SN 1999aa, reach peak luminosities that differ by only log (L) 0.05, even though they have different decline rates and different spectroscopic properties. Despite the distance uncertainties, this result can be taken to confirm that both of these events synthesize a similar mass of 56 Ni as suggested from our spectral modeling (∼ 0.62-0.65 M , see Tab 3) .
Even though the abundance distributions in SNe 1999aa and 1991T are similar, their spectroscopic evolution shows differences. These can be explained by the difference in luminosity between the two SNe. We computed synthetic spectra at day −11 starting from the model that matches SN 1999aa and progressively increased the luminosity (Fig.21). As the luminosity increases, the spectrum changes, until it finally starts resembling that of SN 1991T: the Si II 6355Å line becomes weaker, and so does Ca II H&K. The same is true for the Fe III features observed near 3200, 3500, 4200, and 4900Å.
SNe Ia exhibit very similar spectroscopic properties beyond maximum brightness. Therefore, an explosion-progenitor scenario that can explain the complete spectroscopic sequence should be one that allows variations only in the outer layers. The sudden depletion of the IMEs in the outer shells of SN 1999aa is not easy to explain within the framework of conventional delayed-detonation explosion models (Khokhlov 1991b;Iwamoto et al. 1999). One possible explanation may be an explosion that initially proceeds very efficiently but then suddenly stops, leaving an only weakly burned outer layer. One such class of models is pulsation-driven detonations (Ivanova et al. 1974;Khokhlov 1991a;Hoeflich et al. 1995). In these configurations, the progenitor is characterised by an outer layer with low density, which could be the result of the pre-expansion of a white dwarf that has gone through an initial failed burning phase, or to a binary merger. This results in a steep density gradient and may cause IMEs to be confined in a relatively narrow velocity range. However, these models predict no burning in the outermost layers, and therefore the presence of a copious amount of C (Baron et al. 2008), which is not observed in SN 1999aa or SN 1991T. Additionally, simulations of these models show IME lines at very early times, and do not resemble the spectra of SN 1991T-like SNe Ia (Dessart et al. 2014). Furthermore, three-dimensional versions of these models exhibit a large degree of mixing and cannot explain the stratification seen in SN 1999aa (Plewa et al. 2004;Kasen & Plewa 2005;).
In general, none of the currently available models can explain the entire spectroscopic properties of SNe Ia over a large range of luminosities. Nevertheless, the pulsation-driven scenario remains interesting for SN 1991T-like SNe because it only affects the outer ejecta. Based on our current knowledge, this particular scenario should only kick in when 56 Ni production is very high.

CONCLUSIONS
We have modelled a series of optical spectra of the peculiar slow decliner SN 1999aa, from −12 to + ∼ 300 days from B maximum to infer the composition layering of its ejecta. Three different density profiles were used -the fast deflagration W7 and two delayed detonation models, DD2 and DD3. We have compared our results with spectroscopically normal events as well as with SN 1991T.
Our main results can be summarised as follows.
• All three density profiles yield synthetic spectra similar to the observed ones and follow their evolution. In particular, an Fe IIIdominated early-time spectrum with shallow IME lines, typical of the SN 1991T class, is reproduced.
• The internal composition of SN 1999aa is dominated by neutron-rich iron-peak elements, as in normal SNe Ia. This is followed by a 56 Ni shell (mass ≈ 0.65 M ). Above this lies a narrow IME shell which is sharply separated from the outer, O-dominated shell.
• The confinement of IMEs to a narrow velocity range and their  Figure 18. The distribution of the most important elements in SN 99aa, SN 1991T, and some spectroscopically normal SNe Ia. Left-hand side, top to bottom: Si, S, and Ca. Right-hand side, top to bottom: stable Fe, 56 Ni, O. SNe 1999aa and 1991T have similar stratification properties: a more complete dominance of 56 Ni in the inner layers (2000-10,000 km s −1 ), a narrow IME shell peaking near 11,000 km s −1 but terminating sharply above ∼ 12, 000 km s −1 , a larger prevalence of oxygen in the outer layers, suggesting less burning in these regions. The dashed lines in the first panel show v ph at days −11.2 and at B maximum light. The continuous lines show the position of v ph at the epochs when Fe II lines start to appear in SN 1999aa (green) and SN 1991T (blue). depletion in the outermost layers indicates a sudden shift from a regime of strong burning to one of weak incomplete burning. This behaviour is remarkably similar to that of SN 1991T, but is not observed in normal SNe Ia. Therefore, it is reasonable to conclude that SNe 1999aa and 1991T share a similar explosion mechanism, despite their somewhat different luminosities.
• The observed stratification may be the result of sharp density gradients in the outer shells of the progenitor.
• The spectroscopic path from normal SNe Ia to the brightest peculiar events cannot be explained solely by a luminosity/temperature sequence. It should involve composition layering differences suggesting variations either in the density structure of the progenitor white dwarf at the outer layers or in details of the explosion.
• Within the SN 1991T class, IME confinement coupled with dif-ferences in luminosity (i.e., 56 Ni production) may explain the observed spectra.  Figure 21. Early-time spectra of SNe 1999aa and 1991T compared to synthetic spectra computed for increasing luminosity but the same composition.
As the luminosity increases the spectra morph from looking like SN 1999aa to looking like SN 1991T. ifornia, and NASA; the observatory was made possible by the generous financial support of the W. M. Keck Foundation. The Kast spectrograph on the Shane 3 m telescope at Lick Observatory was made possible through a gift from William and Marina Kast. Research at Lick Observatory is partially supported by a generous gift from Google. We thank the staffs at the various observatories where data were obtained.

DATA AVAILABILITY
The spectroscopic data used in this article are available at the Weizmann Interactive Supernova Data Repository (WISeREP) (Yaron & Gal-Yam 2012).