SN 2013ai: a link between hydrogen-rich and hydrogen-poor core-collapse supernovae

Figure 3 shows the optical spectra of SN 2013ai. In the optical, at all epochs, we see narrow optical emission in H$\alpha$, H$\beta$, [O ii] $\lambda$3727, [O iii] $\lambda$5007 and [S ii] $\lambda\lambda$6717, 6731 at the redshift of NGC 2207. The earliest spectra, taken $11$ days before $V$ maximum, show only a broad H$\alpha$ P Cygni profile with narrow Na I D $\lambda\lambda$5890 and 5896 likely from the interstellar medium (ISM). H$\beta$ is not present until later times, first seen around $3.2$ days before $V$ maximum. A montage of the optical spectra is presented in Figure 3.

Starting with the spectrum taken $3$ days after $V$ maximum, the H$\alpha$ absorption becomes split, likely due to Si ii $\lambda$6355 at these early times (Gutiérrez et al., 2017). Fe-group lines are also present during this epoch, primarily Fe ii $\lambda\lambda$5169, 5267, and 5363. We also see the emergence of the Ca ii NIR triplet as a broad and blended P Cygni line around 8500 Å.

From $11-20$ days after maximum, SN 2013ai became even redder, as more flux is lost to absorption by Cr, Sc, Ba and Fe in the blue. The usual Fe-group lines seen in SNe II are present between 3500-5500 Å; however, these lines are weaker than what are seen in a normal SN II (see Gutiérrez et al. 2017; de Jaeger et al. 2019). H$\alpha$ has a strong absorption component in its P Cygni profile, with a minimum at a velocity of 11,000 km s^-1, while the second absorption, which was tentatively associated with Si ii, is still present as a “notch” in the feature. There is a lack of absorption from Sc ii and Ba ii on the blue side of the H$\alpha$ absorption line.

By $30$ days after maximum, the notch in H$\alpha$ is no longer present in the spectrum of SN 2013ai, suggesting that the notch was not due to high velocity (HV) H i (see Gutiérrez et al. 2017 for more information on this feature). The spectrum shows relatively little evolution over the following period, from $30$ to $73$ days post explosion, the continuum continues to become redder, while the most notable change in the lines is an increase in the strength of the Na I D absorption.

The late time optical spectrum obtained $230$ days after $V$ maximum with GTC+OSIRIS is shown in Figure 6. This spectrum was taken 250 days from the estimated explosion epoch and is shown in comparison to the SN IIP SNe 1999em and the SN IIb SN 1993J at similar times. The spectrum of SN 2013ai contains very little flux from the SN. Nonetheless, several lines characteristic of core-collapse SNe in their nebular phase are detected, namely broad H$\alpha$ emission, [Ca ii] $\lambda$7291, and possibly [O i] $\lambda$6300. The reddening corrected [Ca ii] to [O i] line ratio appears approximately similar to that in SN 2004et (see Jerkstrand et al., 2012), suggesting that the core mass of the star which exploded was probably quite similar and not exceptionally massive. However, the weakness of these lines makes it difficult to accurately measure a flux ratio and suggest that the SN is not sufficiently nebular for an accurate progenitor to be assumed using the method of Jerkstrand et al. (2012). The H$\alpha$ profile at this epoch is dominated by emission and shows a profile with a wide base and narrow top with no absorption. Together with the early X-ray observations, as noted in Section 2, this suggests interaction with circumstellar material (CSM).

The NIR spectrum of SN 2013ai is shown in Figure 7 and exhibits mostly hydrogen and helium features, such as $P_{\delta}$ and $B_{\gamma}$ and the He i/$P_{\gamma}$ blend. In the NIR, SN 2013ai does not exhibit any lack of features like the absence Sc ii and Ba ii in the optical.

Figure 8 shows optical spectra of SN 2013ai compared to other SNe at similar phases. Included are examples of well-studied SNe IIP, IIL, and IIb. This comparison highlights the uniqueness of SN 2013ai in the features seen, their strengths, and expansion velocities. As previously noted, there is a lack of features on the blue side of H$\alpha$ and the Na/He absorption around 5990 Å is significantly shallower than that seen in a normal SN II/IIb.

Figure 7 shows the 14.5 day NIR spectrum of SN 2013ai compared to other SNe II at similar epochs. The SN does not show any signs of being a NIR weak SN II (Davis et al., 2019); however, at such early times it is not always possible to determine a NIR spectroscopic subclass. The He i $1.083\,\mu m$ line is stronger than that of SN 2013hj at a similar phase. Both SNe show a boxy $P_{\beta}$ emission profile. Overall, the NIR features of SN 2013ai are much more normal than in the optical. However, similar to the optical, the features in the NIR have much higher velocities than other SNe IIP/L at similar times.

The top row of Figure 9 compares the pEW of all optical features present in the spectra of SN 2013ai with the mean from Gutiérrez et al. (2017) and all data from the de Jaeger et al. (2019) and Liu et al. (2016) samples. The metal lines, from 4500-5500 Å, of SN 2013ai evolve like those of a typical SN II. The 5990 Å He/Na blend is particularly interesting due to its weakness at later times. The pEWs of all features seen in the optical lie amongst normal SNe II values. Similarly, the top row of Figure 10 compares the pEW of all NIR spectral features of SN 2013ai with the sample from Davis et al. (2019). The NIR pEWs, like the optical, lie within the large comparative samples.

The bottom row of Figure 9 shows the absorption velocities of SN 2013ai compared to the samples of Liu et al. (2016), Gutiérrez et al. (2017), and de Jaeger et al. (2019). SN 2013ai exhibits high velocities in all features observed, for a SN II. The 5990 Å He/Na blend is particularly interesting due to its high velocities if assumed to be Na i $\lambda 5993$, giving velocities over 13,000 km s^-1 at early times. While the Fe ii velocities are closer to normal SN II values, they are still noticeably higher than the comparative SN II samples, with the velocities more similar to SESNe. Despite the high velocities, the evolution is decreasing with time, as that of a normal SN II. Figure 10 shows the NIR absorption velocities of SN 2013ai compared to that of the data from Davis et al. (2019). Much like the optical, the velocities are significantly higher than the sample, $\sim 2\sigma$, with the P$\beta$ and P$\delta$ features around 12,000-13,000 km s^-1 and He i $\lambda 1.083\,\mu m$ around 11,500 km s^-1at 22 days past maximum. The offset of velocities from typical SN II values are similar in the optical and NIR. The normal pEWs and high velocities of SN 2013ai suggest that the features are not broad, but are fast, like those of a SN IIb. However, the spectral features present are typical of a SN II.

The optical absorption velocities at 50 days past explosion, the middle of the plateau for a normal SN II, were also compared to the Gutiérrez et al. (2017) and de Jaeger et al. (2019) samples in order to see if any other SNe II have been found with velocities as high as SN 2013ai. No SNe with velocities as high as SN 2013ai were found, except for SN 2007ld from the Gutiérrez et al. (2017) sample, which has a 50 day H$\alpha$ velocity over 11,000 km s^-1. SN 2007ld has a normal photometric and spectroscopic evolution; however, it shows some contamination in the H$\alpha$ emission profile which could be due to He i, a common sign of a SN IIb. Unfortunately, SN 2007ld has no spectra towards the end of its plateau, when a SN IIb can be easily identified. SN 2007ld does not exhibit a long rising light curve or a quick decline after maximum. Super-luminous SNe (e.g. SN 2013fc, SN 2016gsd Kangas et al., 2016; Reynolds et al., 2020) are not included for comparison as SN 2013ai is not super-luminous. However, see SN 2016gsd (Reynolds et al., 2020) for a super-luminous SNe II that exhibits high velocities similar to SN 2013ai.

The spectra of SNe with similar rise times to SN 2013ai (see Section 4) were also examined. The classification spectrum of ASASSN-14kg is plotted with the earliest spectrum of SN 2013ai in Figure 11. The spectra look similar at these early times. There are no later spectra taken of ASASSN-14kg. ASASSN-14kg has a well defined explosion date, $59672.9\pm 0.5$ MJD (Nicolas et al., 2014). Given the similarity of the light curves and early spectra of ASASSN-14g and SN 2013ai, it is likely that SN 2013ai is around the same phase from explosion in the spectra seen in Figure 11. Afsariardchi et al. (2019) presented the H$\alpha$ velocity of SN 2017it at 93 days past explosion, which is higher than that of a normal SNe II. However, it is still $\sim$1000 km s^-1 slower than SN 2013ai. We were unable to find any SN II with expansion velocities comparable to SN 2013ai during the photospheric phase. However, the similarity of the early-time spectra and light curves of ASASSN-14kg and SN 2013ai suggests that they could be alike.

Pre-explosion HST observations were available for SN 2013ai, consisting of HST+Wide-Field and Planetary Camera 2 (WFPC2) images taken on 1996 May 25, see Figure 12. The SN location lies on the WF2 chip, which has a pixel scale of 0.1″/pixel. Four exposures, consisting of two cr-split pairs, were taken in each of the F336W, F439W, F555W, and F814W filters, with total exposure times of 2000s, 2000s, 660s, and 720s, respectively. The pipeline reduced _c0f files were downloaded from the Mikulski Archive for Space Telescopes (MAST)⁴⁴4http://archive.stsci.edu/. The images were first masked with their associated _c1f files, and the cr-split pairs were combined to reject cosmic rays using the crmask task within IRAF. Finally, the dithered images in each filter were aligned and coadded.

Unfortunately, the SN position lies very close to the charge transfer trap 2-337 on the WF2 chip in all images (Whitmore & Wiggs, 1995), which gives rise to a bad column artefact. While it is possible to attempt to reconstruct the flux lost due to the trap, as dithered exposures were available, we instead ensured that the bad column was masked in both cr-split pairs for each filter before coadding. As the cr-split pairs in each filter were offset by $\sim$3 pixels in the x direction, the defect (when masked) does not contribute to the final image when they are shifted and combined.

We also attempted to use the drizzle algorithm (Fruchter & Hook, 1997) as implemented in astrodrizzle within the drizzlepac package to improve the spatial resolution of the F814W image. Before drizzling, the bad column close to the position of SN 2013ai was flagged in the data quality image. A series of drizzled images were produced, to test the effect of varying the output pixel scale and the size of the drop. However, in all cases a faint artefact was still visible at the position of SN 2013ai in the final image. It is likely that the artefact in the output image results from the interpolation over the masked bad column. Thus the drizzled images were not considered any further.

To identify the position of SN 2013ai in our pre-explosion images, we obtained Ks filter imaging of SN 2013ai with the Nasmyth Adaptive Optics System and Near-Infrared Imager and Spectrograph (NAOS-CONICA; NaCo) on the Very Large Telescope UT4 over 2013 March 3-4. SN 2013ai itself was used as a guide star for NAOS, while the S54 camera was employed on CONICA, yielding a pixel scale of 0.0543″/ pixel, over a field of view of 56″$\times$56″. Daytime calibration data consisting of flat fields and long and short exposure darks were reduced with the NaCo pipeline (version 4.3.1), to give a master flat field and a map of aberrant pixels on the detector. Using these, the individual science frames were masked and flat-fielded. All images of SN 2013ai were dithered on-source, facilitating the subtraction of the sky background using the IRAF xdimsum package. After sky-subtraction, the individual frames were aligned and combined to yield a single, deep image with a total exposure time of 5040s.

The NaCo post-explosion image was aligned to the pre-explosion WFPC2 F814W image using a geometric transformation derived from the pixel coordinates of sources common to both images. Two separate sets of sources were used for the alignment: a “good” set of 21 sources which were clearly point-like, detected with good S/N, and which appear to lie within NGC 2207, and a larger superset of 31 sources which we term the “complete” set, where the signal to noise (S/N) of sources were lower, and which also included foreground and slightly extended objects. For both the “good” and “complete” sets, the pixel coordinates of each source were measured in the WFPC2 and NaCo frames, and three separate geometric transformations were then derived using the lists of matched coordinates, one allowing for translation, rotation, and scaling in x and y, and two which also included second and third order polynomial terms, respectively.

The pixel coordinates of SN 2013ai were measured in the NaCo image using three different algorithms (centroid, gauss and ofilter within IRAF), which had a standard deviation of only 0.05 pixels, or 3 mas. The average of the three algorithms was taken as the position of SN 2013ai in the NaCo image. This position was then transformed to the pixel coordinates of the pre-explosion image using each of the transformations derived using both the “good” and “complete” sources. The standard deviation of the transformed positions was 16 mas, and we take this to be indicative of the uncertainty in SN position depending on the geometry or sources used for the alignment. We add this in quadrature with the uncertainty in the SN position in the NaCo observation (3 mas), and the average of the RMS error found when fitting each transformation (58 mas), to obtain the total uncertainty of 60 mas on the location of SN 2013ai in the WFPC2 image.

The SN falls close to, but is not coincident, with a source in the WFPC2 image, which we designate “Source A”. The centre of Source A (see Fig. 12), as measured using three centering algorithms within IRAF phot, lies 193 mas (1.93 pixels) from the transformed SN position. This offset is a factor $\sim$3 greater than the uncertainty in the transformed position of SN 2013ai (60 mas) and so we discount the possibility that this is a single stellar progenitor of SN 2013ai.

By eye, Source A appears to be somewhat extended. To test this, 14 isolated, point-like sources were fit with Moffat profiles to determine their full-width at half maximum (FWHM). The average FWHM of these sources was 1.6$\pm$0.2 pixels. Source A is significantly broader, with a FWHM of 2.5 pixels. On these grounds, it appears that the source may be an unresolved cluster or complex (2.5 WF pixels corresponds to a physical scale of 46 pc at the distance of NGC 2207).

While the position of SN 2013ai lies outside the FWHM of Source A, it still appears to fall within the wings of the PSF. To quantify the degree to which SN 2013ai lies within the position of Source A, the latter was fit with a Moffat profile. Then, we integrated under this profile to determine the flux within an aperture with a radius corresponding to the distance from Source A to SN 2013ai. Assuming that SN 2013ai is associated with a cluster, it was found that 97${}^{+2}_{-6}$ percent of the flux of Source A is in the position of SN 2013ai. In the remainder of this section, we consider the implications for the progenitor of SN 2013ai both if it was associated with Source A, and if it was unrelated.

Photometry was performed on Source A using hstphot, (Dolphin, 2000a) a stand-alone photometry package for use with WFPC2 data. hstphot includes corrections for charge transfer efficiency (CTE) losses, aperture corrections, and zeropoints appropriate to WFPC2 (Dolphin, 2000b). While Source A is somewhat broader than most point sources in the field, it is still relatively well fit by a PSF. We measure PSF-fitted magnitudes for Source A in the HST flight system (Holtzman et al., 1995), with updated zeropoints as per Andrew Dolphin’s webpages⁵⁵5http://americano.dolphinsim.com/wfpc2_calib/ of F336W=22.68$\pm$0.12 mag, F439W=23.90$\pm$0.15 mag, F555W=23.40$\pm$0.12 mag, F814W=22.60$\pm$0.13 mag. Transforming to the standard UBVRI system, these correspond to U=22.68$\pm$0.12 mag, B=23.89$\pm$0.15 mag, V=23.38$\pm$0.12 mag, I=22.56$\pm$0.13 mag. As a test of the PSF-fitting results, aperture photometry was also performed using a small (2 WF pixel) aperture. For all four filters, the difference between PSF-fitting and aperture photometry was $<$0.1 mag, i.e. less than the photometric error.

To estimate an age for Source A, the chorizos SED-fitting package was used (Maíz-Apellániz, 2004). A grid of Starburst99 models was fit (Leitherer et al., 1999) to the measured WFPC2 photometry of the source, constraining the reddening-law parameter R₅₄₉₅ to be 3.1, and the metallicity to the solar value, while allowing the cluster age and extinction to vary. The results of the fitting procedure are shown in Figure 13. The best fitting model (shown in the left panel) has an age of 7.9 Myr and a relatively low extinction of E(4405-5495)=0.12 mag. Comparing to the STARS models⁶⁶6https://people.ast.cam.ac.uk/ stars/ (Eggleton et al., 2011), such a cluster age would imply a progenitor mass of $\sim$21 M_⊙; however, we note that there are also reasonable fits for 11.2 Myr, giving a lower progenitor mass of $\sim$17 M_⊙ and higher extinction. The low mass best fit of $\sim 17$ M_⊙ is at the high end of SN II progenitor masses (Smartt et al., 2009; Sukhbold & Adams, 2020).

The extinction towards the cluster inferred by the fit is considerably lower than that seen towards SN 2013ai itself. There is a tail of solutions in the $\chi^{2}$ plane which accommodates a reddening of E(4405-5495)=0.7 mag, which is consistent with that seen towards the SN. However, these solutions require the cluster to be younger than $\sim$3 Myr, which would in turn lead to a progenitor which is extremely massive, and probably inconsistent with the maximum mass of a star that can maintain its H envelope to the point of core-collapse.

Further analysis was done using the Bayesian fitting technique of Maund (2018), which addresses cluster mass, distances, and filter transformations. Both the WFPC2 photometry and WFPC2 photometry converted to Johnson-Cousins photometry were included as seperate runs. Given the luminosity of the cluster, the mass is constrained to lie in the range $\log(M)=10^{4.5}-10^{5.5}M_{\odot}$. At such masses, individual, luminous stars may have a disproportionate influence on the colour of the cluster that might cause deviations from the expected colour-sequence predicted by STARBURST99 (for much more massive clusters of $10^{6}M_{\odot}$).

The progenitor of SN 2013ai could be a single, isolated red supergiant as found for other Type II SNe. The relatively bright cluster nearby makes the identification of such a progenitor difficult, as the flux in the wings of the cluster may mask that of a fainter progenitor close by. The most convincing demonstration of such a progenitor would rely on image subtraction, where a flux deficit in new HST images taken after SN 2013ai has faded would indicate the absence of progenitor flux which was present in pre-explosion images. As demonstrated by Maund et al. (2014) and Folatelli et al. (2015), difference imaging using late-time data can substantially improve the precision and sensitivity of limits placed on SN progenitors. While such a test will have to wait for new observational data, it is still possible to estimate the sensitivity of pre-explosion data to a single red supergiant progenitor. From photometry of point sources close to the SN position, we estimate a $5\sigma$ limiting magnitude of F814W $\gtrsim$ 24.5 mag for the WFPC2 images. At the distance of NGC 2207, this would imply an absolute magnitude for the progenitor of F814W$>$-9 mag, which unfortunately only allows us to constrain the luminosity of the progenitor to log L/L_⊙$<$5.5. For single star models, this luminosity corresponds to a progenitor zero age main sequence mass of $<$24 M_⊙ (Eldridge & Tout, 2004).

Determining the properties of CCSNe poses a challenge due to the wide range of parameters, e.g. progenitor mass, mass loss pre-explosion, explosion mechanism, mixing, and possible interaction, which affect the observational features. However, using a combination of empirical measurements and modeling we attempt to reconstruct these properties for SN 2013ai and motivate the use of a SN 1993J inspired model.

The chemical and density structures of the outer layers of the progenitor are revealed by the spectral evolution. In general, at early epochs, the spectral features are formed at or above the electron scattering photosphere in the H-rich layers. During this time, the photosphere can be traced with measurements of weak lines that form near the photosphere, e.g. Fe ii. The inner edge of the H-rich layers is determined by the point in time when the Doppler shifts of the Balmer lines becomes constant, around 40 days past explosion in SN 2013ai, whereas Fe ii still receeds. At the photosphere, the ejecta density profile of a SN II can be approximated by a power law, much like that of a stellar atmosphere, $\rho=r^{-n}$, with $n$ the density slope and $r$ the radius at a point in the ejecta (Hoeflich, 1990). Before the recombination phase of H, the differential Doppler shift between Balmer series H-lines can be used to reconstruct the density structure by integration of the density and slope (see Hoeflich, 1988, 1990, for applications to SNe II). The result of applying this technique to SN 2013ai gives a density slope, n, ranging from $\sim$12-25. A typical SN II has a density slope of $\sim$10 at early times (Hoeflich, 1990), significantly flatter than that of SN 2013ai. The steep density profile, H line profiles, and Doppler shifts are more similar to SN 1993J than the other SNe compared to (see Figures 8 and 10) with both SNe having photospheric velocities of $\sim$12,500 km s^-1 at the time of the transition between the H- and He-rich layers, suggesting a similar specific energy in the H-rich layers.

Around a week after maximum, Fe ii is used to trace the photosphere. This is around the same time that a sharp drop in the Fe ii $\lambda 5169$ velocity is observed, as seen in Figure 9. The drop in velocity is likely caused by the photosphere receding quickly through the He layers into the carbon/oxygen (C/O) core. He i has a high ionization potential and requires high energy nonthermal photons to be observed (Graham, 1988). We see little evidence of He i in the optical which supports the notion that the jump in Fe ii velocity is caused by the photosphere quickly passing through the He layer. In the NIR, He i $\lambda 1.083\,\mu m$ is seen, however, this He feature is the basis for many other He transitions. Thus, even an object with little He should show NIR He i $\lambda 1.083\,\mu m$ given its high population.

The light curves are powered by the energy deposition due to the propagating shock wave, the radioactive decay of ${}^{56}Ni\rightarrow$${}^{56}Co\rightarrow$${}^{56}Fe$, and the recombination energy. The rise time is dominated by the diffusion time scales in the optically thick inner layers, and the light curve tail is governed by radioactive decay and, possibly, interaction with the CSM and ISM. The rise time of SN 2013ai is typical of a SN IIb and in particular similar to that of the prototypical SN IIb SN 1993J, which, together with the spectroscopic similarities highlighted above, motivate us to use SN1993J as a starting point to model SN 2013ai. SN 1993J was extensively modeled by Hoeflich et al. (1993) and was found to have a layered He-C/O core of $\sim 10$ M_⊙ with an outer H-rich layer mass of $\sim$ 3 M_⊙ originating from a progenitor with a main sequence mass ($M_{MS}$) of $\sim$ 25 M_⊙. Note that the final H, He, and C/O core masses depend on the phase of stellar evolution during which mass-loss occurs.

For the progenitor evolution of SN 2013ai, we use the stellar evolution code MESA (Paxton et al., 2011). To trigger the explosion, the explosion energy ($E_{exp}$) is put in the inner region as a thermal bomb which causes the ejection of the envelope leaving behind a neutron star or a black hole. For computational efficiency and to cover a sufficiently wide range of parameters, we first used the code developed by Bersten et al. (2011) to model SN 2013ai. To verify and further constrain these models, we then modeled SN 2013ai with the non-LTE hydrodynamical-radiation code HYDRA (Hoeflich, 2003, 2009). A wide range of $M_{MS}$, $E_{exp}$, and mixing, including pure C/O cores, were evaluated. We will omit the “failed” attempts.

The comparison between a frequency independent model from the code of Bersten et al. (2011) with the photometry is shown in Figure 14 using depth dependent mean opacities based on SN 1993J and originating from a 20 M_⊙ star. The resulting models have 0.55 M_⊙ of ⁵⁶Ni. Such a high ⁵⁶Ni mass is problematic for the explosion of a massive star both from observations and theory (see Thielemann et al. 2018 for a review). The bolometric light curve, in principle, is a sensitive measure of the physical parameters (e.g. Suntzeff et al., 1992; Bersten et al., 2011). Despite the advantage of the bolometric light curve, three problem zones may be identified: 1) the reconstruction of the bolometric light curve, 2) distance uncertainties, and 3) asymmetries. Moreover, using the bolometric light curve reduces the information for finding model parameters. Due to these uncertainties, we use monochromatic light curves based on detailed non-LTE models.

In Figures 15 and 16, we show the results of HYDRA starting with the progenitor of SN 1993J but modifying the explosion energy, the mass of the H-rich layers, the explosion energy, and the ⁵⁶Ni mass. The H-rich layer mass was adjusted to 0.2 M_⊙, and the ⁵⁶Ni mass to 0.3 M_⊙. These models are based on the results from the previously mentioned models, hereafter referred to as “LC models”. The results between best fit models of our simulations, presented as (LC, HYDRA), give consistent results within the uncertainties: $M_{MS}=(20,30)$ M_⊙, $E_{exp}=(3.0,2.5)$ foe⁷⁷71 foe = $10^{51}$ ergs, and $M_{Ni}=(0.55,0.3)$ M_⊙.

Given the model best fits, the effect of uncertainties on the $M_{Ni}$, $E_{exp}$, C/O core mass, and distance modulus must be discussed. Linear polarizations give evidence for a symmetry axis and bipolar explosions with $P\approx 1\%$ indicating axis ratios; a measure of how non-spherical an object is, of 1/2-1/4 (Wang et al., 1998; Maund et al., 2007a, b; Stevance et al., 2016; Reilly et al., 2017). To estimate the uncertainty for the light curves, we use SN 1998bw/GRB980425 as a proxy for the explosion of a massive stripped envelope SN (Hoeflich et al., 1999). Detailed analysis of polarization spectra resulted in an axis ratio of 0.7 for SN 1998bw seen from an angle of $30^{o}$ (Hoeflich, 1995) but with significant uncertainties as discussed in Stevance et al. (2020). The possible effect of asymmetry on the $V$-band light curve of SN 2013ai is shown in Figure 17. Asymmetry may change the shape of the light curve by varying the maximum brightness up to $\pm 1$ mag, placing SN 2013ai near superluminous SNe (see Figure 17), and can shift the rise time by about 10 days.

Even for the same model, the light curve rises more slowly and shows a lower maximum brightness that directly translates into uncertainties in $E_{exp}$, and the C/O core mass. The error for the asymmetry and the shape may be regarded as an upper limit because even for purely jet-driven explosions of CCSNe, an extended outer envelope tends to become more spherical with time (Khokhlov et al., 1999; Hoeflich et al., 2004; Couch et al., 2009). The colors are found to be less sensitive as they are a measure of the physical conditions at the photosphere, namely the flux density rather than the total flux. In either case, the tail of the light curve is less effected by polarization because the luminosity becomes isotropic within $\sim 0.2$ mag 1-2 months after the explosion, which translates directly into an error for $M_{Ni}$. For the ⁵⁶Ni mass, both 0.3 and 0.55 M_⊙ fit the data; however, as previously mentioned $0.3-0.55$ M_⊙ of ⁵⁶Ni is higher than expected for a massive star explosion.

Though beyond the physics included in our simulations, note that the theoretical $V$-band light curve at 250 days is fainter by $\sim$ 1 mag than observed. This may indicate a significant contribution by interaction between the SN-ejecta and the ISM or a pre-explosion stellar wind. This theory would also be consistent with the multi-tiered H$\alpha$ emission profile seen in the nebular spectrum of SN 2013ai, which indicates interaction.