COMPARISON OF DIVERSITY OF TYPE IIB SUPERNOVAE WITH ASYMMETRY IN CASSIOPEIA A USING LIGHT ECHOES

Three LEs of Cas A were discovered, and their spectra were collected and processed by Rest et al. (2008b) and R11. While we refer to those publications for a detailed description of the data, here we discuss relevant characteristics of the LE spectra.

The data were taken with the Low Resolution Imaging Spectrometer (Oke et al. 1995) on the 10 m Keck I telescope and with the Blue Channel spectrograph (Schmidt et al. 1989) on the 6.5 m MMT. The LEs were originally labeled LE2116, LE2521, and LE3923 by R11; however, hereafter we refer to them by the direction of the line of sight of Cas A that each one probes: SW, NW, and NE, respectively. The LEs are separated by: SW-NW ∼ 70°, SW-NE ∼ 855, and NE-NW ∼ 109°, thus truly probing different directions of the explosion. The native resolution of these spectra is around 7 Å and the dispersion is 1 Å for the NE and SW LEs and 2 Å for the NW LE.

For the NW LE, two spectra were taken by R11 at epochs separated by one day, and are analyzed separately in their work. In order to improve the signal-to-noise ratio (S/N) of our measurements, we combine these two spectra into a single spectrum in the wavelength range covered by both, namely $3192\,\mathring{\rm{A}} \leqslant \lambda \leqslant 8400\,\mathring{\rm{A}}$. We smooth (Section 3.2) and renormalize the spectra and use their average, and combine the errors in quadrature.

R11 measured and compared the velocity of the He i λ5876, Hα, and Ca ii near-infrared (NIR) triplet absorption features in the three LE spectra and in the spectra of SN 1993J and SN 2003bg. R11 chose these lines as a metric to measure asymmetry because they are strong, isolated lines in the spectra of SNe IIb, and we expect their absorption velocity (measured as the location of the feature minimum) to approximately track the velocity of the outer envelopes of the star. By looking at this velocity in different directions, we can build a picture of the asymmetry of the SN.

However, since the dust sheets are not perfect reflectors, one cannot compare the light echo spectra directly to each other or to spectra from SNe in the literature, nor can one invert the effect of the dust to recover the emitted spectra. Instead, one must first consider what effects the dust had on the Cas A spectrum and then reproduce those effects in a chosen reference spectrum so that they are comparable. These modeling techniques are described in R11 and by several earlier works (Couderc 1939; Chevalier 1986; Emmering & Chevalier 1989; Sugerman 2003; Patat 2005). The modeling of the effect of reflection on spectra is described in Rest et al. (2011b) and we will use the same LE models, which are also used in Rest et al. (2011a). An in-depth description of the method can be found in these works to which we refer the reader for details, but provide a brief summary in Section 3.1.

In order to simulate the effect of dust scattering, one must take both the spectral and photometric time evolutions into account (Section 3.1). Therefore, R11 chose the spectra of SN 1993J and SN 2003bg as the reference spectra since these SNe IIb are both exceptionally well observed. However, an analysis by L16 revealed that when it comes to our chosen metric, the absorption velocities of the three lines we are focusing on, these SNe are actually some of the most extreme, and have atypical evolutions (e.g., Hα for SN 1993J, see Figure 1). We instead use an SN IIb template that describes the average behavior of SNe IIb as our reference spectra when evaluating the asymmetry of Cas A (Figure 2). The SN IIb spectral template was produced by L16 using 227 spectra of 14 SNe IIb taken from the SuperNova IDentification code spectra library (Blondin & Tonry 2007), the sample presented in Modjaz et al. (2014), and relevant spectra from the literature available before September 2014. Template spectra, i.e., mean spectra, were created at phases ranging from ${t}_{V\max }=-20$ to +45 in five-day intervals and also at phases 55, 65, 70, 90, 95, and 105 by taking one spectrum from all SNe IIb that had been observed within ±2 days of the target phase and averaging. In addition, in order to retain a measure of the diversity of the SNe spectra that were used for constructing the mean spectra, we adopt the standard deviation spectral arrays provided by L16 and plot them as contours in our plots of the template spectra (Figure 2). Full details of how the template was generated can be found in L16.

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We can use these template spectra to confirm the asymmetry of Cas A; however, we cannot use them to determine whether the asymmetry can explain the diversity of absorption velocities seen in SNe IIb. To do this, we look at a selection of the individual SNe that make up the template. Our selection is chosen such that the complete range of absorption velocities is represented, as identified in L16, and in particular, those SNe that have the most extreme velocities are included in our sample. Figure 1 shows the velocity evolution of all SNe studied by L16 and was used to inform our choice. The SNe used, along with the references for their spectra and photometry are: SN 1993J (Jeffery et al. 1994; Barbon et al. 1995; Richmond et al. 1996; Matheson et al. 2000; Fransson et al. 2005; Modjaz et al. 2014), SN 1996cb (Qiu et al. 1999; Matheson et al. 2001a; Modjaz et al. 2014), SN 2011dh (Marion et al. 2014), SN 2003bg (Hamuy et al. 2009), SN 2008ax (Pastorello et al. 2008; Milisavljevic et al. 2010; Bianco et al. 2014; Modjaz et al. 2014), SN 2011dh (Marion et al. 2014), and SN 2011ei (Milisavljevic et al. 2013). Since the Ca triplet feature falls in NIR wavelengths, and NIR spectra of stripped SN are rare, the Ca line comparison cannot be conducted for all SNe in our sample, and we restrict ourselves to only the He i λ5876 and Hα lines for this part of the analysis.

The dust sheets that cause the LEs do not act as perfect mirrors, and thus the spectra we observe will not be the same as those emitted by the SN. One reason for this is that the dust sheets have a finite size and thickness, which means that light from multiple epochs in the event timeline will reach the observer at the same time by scattering off different parts of the dust. Therefore, the spectrum we observe in an LE is really the sum of spectra from all epochs weighted appropriately with weights that depend on properties of the dust, of the observation, and on the photometric evolution of the transient, as opposed to that of a single epoch, which would be observed if viewing the SN directly.

Since the effect of this integration cannot be removed from the observed Cas A spectra, we shall process a reference object in such a way as to simulate how it would have appeared had it exploded at the same position as Cas A and been observed in the same LEs. It will then be possible to directly compare the spectra of the reference object and the Cas A LE spectra. For this purpose, a window function (Tylenda 2004; Rest et al. 2011b), w(t), is defined for each LE such that the observed spectrum, $S(\lambda )$, is related to the emitted spectra, $s(\lambda ,t)$, by

$$\begin{eqnarray}&&S(\lambda )={\int }_{-\infty }^{\infty }s(t,\lambda )w(t){dt}.\end{eqnarray} \tag{ 1 }$$

It should be noted that the relative normalization of the different epochs that contribute to $s(t,\lambda )$ is important and thus knowledge of the photometry of the reference object is essential.

Rest et al. (2011b) calculated the window functions of the three LEs of Cas A under study using properties of the dust sheet and the observing slit, the distance from the source and the photometric evolution of the transient. The window functions we use in this work are identical in all respects except where photometry is incorporated and are shown in Figure 3. To apply the window functions correctly, we need flux as a function of time and wavelength, and spectra from each epoch must be weighted appropriately so that they have the correct relative normalization. However, since the spectra we use only have good relative calibration, but not absolute calibration, we must use the corresponding light curves to infer the correct normalization of each spectrum. While R11 used light curves of individual SNe (SN 1993J and SN 2003bg) we must use a light curve relevant to the template spectra of L16 that we use. We must therefore use template SNe IIb light curves derived from the photometric observation of a large sample of SNe IIb. Such light curves have been created by F. B. Bianco et al. (2016, in preparation). Later, when we use individual SN for the comparison, the spectra and light curves of the specific SN will be used.

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SN 1993J, which was used as a reference by R11, has a prominent double peak in its early time photometric data. A double-peaked rise structure is present in several SNe IIb, yet is not universal, and its details are diverse. The double peak is not present in our template for epochs ${t}_{V\max }\gt -15$ days. There is therefore a noticeable difference in the window functions shown in Figure 3 and those in R11 (shown as dashed lines), despite the minimal difference in construction.

Furthermore, since it is hard to quantify the effect of reddening due to dust in the SN's host galaxy, we cannot trust the shape of the spectra on large wavelength scales. The templates have therefore been constructed using continuum-subtracted spectra (L16) and assumptions must be made in order to account for the missing continuum. Since we will be focusing only on individual features in the spectra, the assumption that we choose to make is that the continuum varies only on very large wavelength scales and hence is flat throughout the range of wavelengths covered by each feature, which is on the order of 200 Å. With this assumption reinserting the continuum is trivial and each epoch is simply weighted by a single band light curve, observed in the appropriate photometric band for a given feature. Figure 2 shows the template spectra as they would appear in the three LEs of Cas A with each panel focusing on a single feature in the spectrum. For the Ca ii NIR triplet and Hα, the choice of which band to use for normalization is obvious; however, the He i λ5876 line lies directly in between the V and r bands. We therefore use both bands for this line and compare the results. As is evident from this plot, the choice of band is not significant and we therefore use the V band for the remaining analysis.

For some SNe (SN 1993J, SN 2008ax, SN 2011dh, and SN 2011ei) the light curves in the literature do not cover the full range of phases for which spectra have been obtained. For these SNe, we extrapolate the light curve to late phases using a linear extrapolation in log space (magnitudes) by fitting a linear decline to the SN brightness for ${t}_{V\max }\gt 50\,\mathrm{days}$, so that we can appropriately normalize all the spectra. A linear extrapolation is consistent with our light-curve templates as well as with the expectation from ⁵⁶Co decline. We are therefore able to include all observed spectra in our calculations. The window functions only have power for ${t}_{V\max }\lesssim 100$ days, as is evident from Figure 3, and therefore spectra taken at later phases than this contribute negligibly to the final spectrum. SN 2011dh and SN 1993J have light curves measured to 80 and 94 days after V-max, respectively, so this extrapolation is of little consequence for these SNe. However, the light curves of SN 2011ei and SN 2008ax are only measured to 33 and 50 days, respectively, and so the extrapolation is more important. For these SNe, we use only the last two points for the extrapolation, since the slope at earlier phases may not be indicative of the SN's decline at late phases. We notice that the last V-band measurement in the light curve of SN 2011ei (Milisavljevic et al. 2013, Figure 3) indicates a very shallow decay rate, much slower than would be extrapolated from the earlier datapoints (the last of which is at ${t}_{V\max }=19$ days). While we have no reason to doubt this measurement, and some SN IIb light curves do show flattening after 20 days (Prentice et al. 2016), we felt it was important to test the effect that this datapoint has on our results. We therefore generated two sets of LE spectra for SN 2011ei using two extrapolations of its light curve, one including the measurement at 33 days (a very shallow decline) and one ignoring it (a rather steep decline). The only result affected by this choice is the velocity of the He i λ5876 line seen through the NW LE (whose window function has more power at late times), which changes from −9100 to −9600 km s⁻¹, a small change that leaves the conclusions of this paper intact.

The metric we choose for our comparison is the absorption velocities of three features in the spectrum: He i λ5876, Hα, and the Ca ii NIR triplet, as was done by R11. We employ a Monte Carlo (MC) method, adapted from the one used in L16, to calculate these velocities, the details of which follow. We work separately on each feature by choosing an ∼400 Å region around the expected location of the feature and generate synthetic spectra within that region from the observed flux values and their uncertainties. We then fit a quadratic polynomial to an 80 Å range around the flux minimum for each realization, and take the location of the minimum of the quadratic fit as our velocity measurement, using the relativistic Doppler formula to convert between wavelength and velocity. For the Cas A spectra, we smooth each synthetic spectrum using the Gaussian process described below before fitting the quadratic since the low S/N of these spectra make direct fits unreliable.

The distribution from which our synthetic observations are drawn should be the same as the one our actual observation was drawn from, i.e., the distribution of observations we would expect were we to return to our telescope on several different nights. For the template and the directly observed spectra, it is a good enough approximation to assume that the observed values are close enough to the true spectral flux to draw synthetic observations from a distribution centered on the observed flux. However, due to their low S/N at native spectral resolution, we cannot generally assume that this holds for the LE spectra. Instead of binning the spectra to suppress the noise, we can calculate the correct distribution using Gaussian processes in the following way. We first place a prior on the expected spectrum driven by our expectation that SN spectra vary smoothly. We then condition this prior on the measured spectrum to obtain the final distribution from which our synthetic observations, f_i at wavelength ${\lambda }_{i}$, are drawn. We choose our prior to be an exponential with a mean of zero and a covariance of

$$\begin{eqnarray}&&k({\lambda }_{1},{\lambda }_{2})=N\exp \left(-\displaystyle \frac{{(v({\lambda }_{1})-v({\lambda }_{2}))}^{2}}{{\sigma }^{2}}\right)\end{eqnarray} \tag{ 2 }$$

between any two fluxes measured at wavelengths ${\lambda }_{1}$ and ${\lambda }_{2}$, where $v(\lambda )$ is the velocity of wavelength λ as measured by the relativistic Doppler formula. N and σ are parameters of our model, which represent the S/N and the minimum expected line width respectively. We choose N = 1 and $\sigma =3\times {10}^{3}$ km s⁻¹, in order to be consistent with the smoothing algorithm used by R11. The mean vector, $\bar{f}$, and covariance matrix, cov f, of the final distribution from which our synthetic observations are drawn are then

$$\begin{eqnarray}&&\bar{f}=K{(K+{{ \mathcal E }}^{2})}^{-1}{f}_{* }\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\mathrm{cov}\,f=K+{{ \mathcal E }}^{2}-K{(K+{{ \mathcal E }}^{2})}^{-1}K,\end{eqnarray} \tag{ 4 }$$

where K is the covariance matrix of our prior, with elements ${K}_{{ij}}=k({\lambda }_{i},{\lambda }_{j});$ ${ \mathcal E }$, with elements ${{ \mathcal E }}_{{ij}}={\epsilon }_{i}{\delta }_{{ij}}$, is a diagonal matrix of the observational errors of our measurements and f_* is the observed flux. Full details of this calculation can be found in, e.g., Rasmussen & Williams (2006).

We then generate ${N}_{\mathrm{MC}}={N}_{\lambda }{(\mathrm{ln}{N}_{\lambda })}^{2}$ simulations where N_λ is the number of wavelength bins within the range of interest (Babu & Singh 1983).⁶ We smooth each simulated spectrum by the matrix $K{(K+{{ \mathcal E }}^{2})}^{-1}$, treat it as an independent observation and process it as we did for the directly observed spectra.

Figure 4 shows the range of velocities found superimposed on the Cas A LE spectra and the template spectra processed through the appropriate window functions.

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By comparing the velocity of our three absorption features in the LE spectra of Cas A to those of the template spectra convolved with the relevant window functions of the dust sheets, we can isolate effects due to the asymmetry of Cas A from effects due to differences in the dust sheets. If Cas A were symmetric, we would expect the velocity of Cas A to be the same in all three directions with respect to the template, i.e., we would expect either the line velocities from all three directions to be consistent with the templates or for them all to be faster or all slower by the same amount. Any deviation from this is an indication of asymmetry.

As can be seen in Figure 4 for all three lines the highest velocity, relative to the template velocity, is observed in the NW direction. For the Hα line, we see that Cas A is consistent with the template when seen from the NW direction but has a lower velocity than the templates, with significance $2\sigma$, when seen from the NE direction and $1\sigma$ when seen from the SW direction. Similarly, for the Ca ii NIR triplet, Cas A is perfectly consistent with the template when viewed from the NW direction, whereas it is just on the slow end of 1σ consistency in the SW direction and is slower by $2\sigma$ in the NE. The He i λ5876 line has the largest uncertainty of the three so care must be taken when considering this line. Nevertheless, we see that Cas A has higher velocity than the template, with a significance of $\sim 2\sigma$ when viewed from the NW direction whereas it is consistent at $1\sigma$ when viewed from the SW and NE directions. The uncertainty in the NE LE in particular is very large for this line, thus, although the velocity is consistent with the templates at $1\sigma$, this has little impact on the significance of our conclusions.

Although we do not find a statistically significant difference in velocity for any individual line by itself, the fact that all three lines show NW as the fastest moving direction, as well as the other evidence for asymmetry of Cas A mentioned in Section 1 leads us to the same conclusion as R11: that Cas A is indeed asymmetric and, in particular, the ejecta is moving more quickly in the NW direction. Furthermore, our calculations strengthen this result since we use mean SN IIb spectra instead of individual SNe and employ MC methods to quantify the differences in velocity more robustly.

In order to see if this asymmetry, as seen in the Cas A LE spectra, is enough to explain the diversity of absorption velocities seen in our sample of SNe IIb, we repeat the same steps as before, but use the spectra of the individual SNe in our sample as references instead of the template. Figure 5 shows the absorption velocities, as measured by our MC technique, of our selection of SNe IIb processed through the window functions of the three dust sheets, as well as the LEs of Cas A. These measurements are also tabulated in Table 1. The sample of directly observed SNe IIb includes SNe with the most extreme velocities as identified by L16 (see Figure 1), and for the Hα and He i λ5876 lines, measurements of the fastest and slowest well observed SNe in the sample are also plotted separately in Figure 6.

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For both the He i λ5876 and Hα lines, the velocity of Cas A is consistent to within $2\sigma$ with the velocity of either the fastest, or slowest SNe in our sample in at least one LE. Because of this, we cannot rule out that the diversity observed in SNe IIb, as measured from the blueshift velocity of the lines, is entirely due to asymmetry in the explosion, while the kinetic energy per unit mass could be identical for all objects in the sample. Meanwhile, diversity in the kinetic energy per unit mass may of course manifest itself in other observables, for example, in luminosity, line ratios, or temperatures, which our analysis does not probe.

Again we notice that the velocity difference between the different LEs of Cas A is greater than that of any other SN in our sample, especially in Hα, again highlighting that the asymmetry of Cas A is real, and not just an artifact of the differences in the dust sheets (while in our sample of SNe IIb each object is, of course, seen only from a single line of sight, and all differences in the different panels arise from the convolution with different LE window functions).

In Figure 6, only the SNe with the most extreme velocities are plotted, and the velocities are grouped by SN. Data points with the same symbol, representing the same LE, should be compared, and we see the following.

• 1.  
  While the velocities of the reference SNe are only slightly affected by the different window functions, a big difference between the LEs of Cas A is seen, which we attribute to a direct detection of asymmetry in Cas A. This effect is even more apparent when one considers that, for most of the reference SNe, the window function of the NW dust sheet causes the feature to appear slower, whereas for Cas A, the velocity in this direction is consistently the highest of the three (Figure 6).
• 2.  
  Some of the velocities of Cas A are consistent with the fastest SNe IIb (e.g., NW He i λ5876), whereas others are consistent with the slowest (e.g., NE Hα). Thus the range of velocities seen in the LEs of Cas A is of the same order as the spread of velocities in our sample of SNe IIb, indicating that the asymmetry of Cas A, if common among SNe IIb, would manifest itself as diversity in a sample of directly observed SNe IIb consistent with that which we observe.
• 3.  
  For all lines and all directions there is at least one SN IIb whose velocity difference from Cas A is consistent with 0 within $2\sigma$ (see the right panel of Figure 5). This shows that Cas A is not an outlier of the distribution of velocities in SNe IIb and hence that treating the asymmetry of Cas A as typical is a justified assumption.
• 4.  
  The SNe IIb displaying the most extreme velocities in He i λ5876 are not the same objects that show extreme velocities in Hα. In fact, for all LEs, Cas A has high He i λ5876 velocities, but low Hα compared to our sample and our template.