Supernova PTF 12glz: A Possible Shock Breakout Driven through an Aspherical Wind

PTF 12glz was discovered on 2012 July 7 by the PTF (Law et al. 2009; Rau et al. 2009) automatic pipeline reviewing potential transients in the data from the PTF camera mounted on the 1.2 m Samuel Oschin telescope (P48, Rahmer et al. 2008). The image processing pipeline is discussed in Laher et al. (2014) and the photometric calibration is described in Ofek et al. (2012). The SN is associated with an r = 18.51 mag galaxy, SDSS¹³ J155452.95+033207.5, shown in Figure 1 and modeled in Section 3.2. The coordinates of the object, measured in the PTF images are α = 15^h54^m5304, δ = +03^d32075 (J2000.0). The redshift z = 0.0799 and the distance modulus μ = 37.77 were obtained from the spectrum and the extinction was deduced from Schlafly & Finkbeiner (2011) and using the extinction curves of Cardelli et al. (1989). All these parameters are summarized in Table 1.

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Table 1.  Summary of PTF 12glz Observational Parameters

Parameter	Value
R.A. α (J2000)	238721000
Decl. δ (J2000)	3535421
redshift z	z = 0.0799
distance modulus μ	37.77 mag
galactic extinction EB−V	0.13 mag

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Previous PTF observations were obtained in the years prior to the SN explosion, but no previous detection of any precursor outburst exists. The most recent nondetection was on 2012 June 25. We present a derivation of the explosion epoch in Section 3.4.

PTF 12glz was observed in multiple bands for almost three years after discovery. The SN was monitored during a rising phase (t < 36 days) and a decay phase (t > 243 days) but not around peak luminosity. All the host-subtracted light curves are shown in Figure 2. The photometry is reported in electronic Table 2 and is available via WISeREP.¹⁴

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GALEX observations of the PTF 12glz field started on 2012 May 26 and 15 observations were obtained with a cadence of ∼3 days. The GALEX NUV camera was operating in scanning mode and observed strips of sky in a drift-scan mode with an effective average integration time of 80 s, to an NUV limiting magnitude of 20.6 mag [AB]. The GALEX data reduction was done using tools¹⁵ by Ofek (2014).

The P48 telescope was used with a 12K × 12K CCD mosaic camera (Rahmer et al. 2008) and a Mould R-band filter. Data were obtained with a cadence of ∼2 days, to a limiting magnitude of R ≈ 21 mag [AB]. For the data reduction of the P48 data, we used a pipeline developed by Mark Sullivan (Sullivan et al. 2006; Firth et al. 2015).

The robotic 1.52 m telescope at Palomar (P60; Cenko et al. 2006) was used with a 2048 × 2048 pixel CCD camera and g', r', i' SDSS filters. Data reduction of the P60 data was performed using the FPipe pipeline (Fremling et al. 2016). We calibrated the P60 data in the following way. The r-band light curve was scaled so that its average value during the time window covered by both telescopes matches the average value of the P48 R-band photometric data. The g-band and i-band data were scaled to match the synthetic photometry of the calibrated spectroscopic data (Section 2.3). The synthetic photometry used for the calibration and for other purposes in this paper was computed with the PyPhot¹⁶ pipeline (M. Fouesneau 2019, in preparation).

Although the photometric data available for PTF 12glz do not cover the peak, the data during the rise and decay allow us to place an upper limit on the absolute magnitude at peak: with M_r ≲ −20, PTF 12glz is at the bright end of the observed SNe IIn, together with, e.g., SN 2006gy (Ofek et al. 2007; Smith & McCray 2007), SNe 2008fq (Thrasher et al. 2008; Taddia et al. 2013), or SN 2003ma (Rest 2009; Rest et al. 2011). In particular, it is brighter than all SNe in the sample by Kiewe et al. (2012), which was designed to be unbiased.

Four optical spectra of PTF 12glz were obtained using the telescopes and spectrographs listed in Table 3. The two first spectra were taken during the light-curve rise, and the two last ones during the decay, at the dates shown in Table 3. The spectra were used to determine the redshift z = 0.0799 from the narrow host lines (Hα and [O iii]). All the observations were corrected for a galactic extinction of E_B−V = 0.13 mag, deduced from Schlafly & Finkbeiner (2011) and using Cardelli et al. (1989) extinction curves, with the parameter R ≡ A(V)/E_B−V (i.e., the ratio of total to selective extinction at V) set to the value R = 3.1. In Section 3.3, we show that our qualitative results are not affected when varying R, e.g., within the interval given in Fitzpatrick (1999).

The spectroscopic observations were calibrated in the following way: the first two and the last spectra, for which we have contemporaneous P48 R-band data, were scaled so that their synthetic photometry matches the P48 R-band value. The third spectrum was scaled in the same way using the overlapping P60 r-band data instead.

The first and last spectra are shown in Figure 3 (the first two spectra are very similar, as well as the last two spectra) and all spectra are available from the Weizmann Interactive Supernova data REPository¹⁷ (WISeREP, Yaron & Gal-Yam 2012).

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The two early spectra, obtained during the rise of the light curve, are characteristic of interacting SNe: a blue continuum with strong and narrow Balmer lines, as well as weak He i (5876, 7065 Å) narrow lines. At short wavelengths, the spectra show absorption from iron, as seen, for example, in SN 2010jl.¹⁸

We fitted a blackbody spectrum to the six-point spectral energy distribution (SED) (corrected for redshift and extinction) obtained by combining (1) the observed photometry in the NUV and P48 R-band (2) the synthetic photometry of the spectra in the g, i and r SDSS bands. The best-fit temperatures and radii are shown in Table 4 and Figure 7 (as stars). In Figure 3, we show the synthetic and observed photometry derived for the earliest spectrum, on which is superimposed the calibrated spectrum and the best blackbody fit.

Table 4.  The Table shows the Best-fit Values of the Continuum and Hα Lines in the Four Spectra of PTF 12glz

Date	Continuum		Hα narrow comp.		Hα intermediate comp.		Hα broad comp.
	TBB	RBB	Δv	FWHM	Δv	FWHM	Δv	FWHM
	(K)	(1015 cm)	(km s−1)	(km s−1)	(km s−1)	(km s−1)	(km s−1)	(km s−1)
2012 Jul 15	${11570}_{-400}^{+350}$	${1.50}_{-0.07}^{+0.11}$	−35	110	−50	680	⋯	⋯
2012 Jul 26	${8400}_{-440}^{+2780}$	${2.91}_{-0.92}^{+0.36}$	−80	240	−100	530	⋯	⋯
2013 Feb 09	${8200}_{-350}^{+340}$	${2.56}_{-0.20}^{+0.24}$	⋯	⋯	−270	1990	−1070	8580
2013 May 09	${8100}_{-390}^{+420}$	${2.15}_{-0.22}^{+0.26}$	⋯	⋯	−240	2310	−830	7470

Note. The early spectra are best fit with a linear combination of a narrow Gaussian component (left column) and an intermediate-width Lorentzian component (central column). The late spectra are best fit with a linear combination of an intermediate-width Gaussian component (central column) and a broad Gaussian component (right column). Δz is the shift of the center of each component compared to the galaxy rest frame (the negative values mean that the component is blueshifted).

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Both early spectra show strong and narrow Balmer lines, which for SNe IIn are interpreted as coming from the slow, unshocked, photoionized CSM. Their broad Lorentzian wings may be the signature of electron scattering, as the Hα photons diffuse ahead of the shock through the dense CSM (e.g., Chugai 2001). After subtracting the best-fit continuum from the spectra, we fitted the narrow Hα lines. We tried several linear combinations of Gaussian and Lorentzian functions: the best fit is a superposition of a narrow Gaussian component with FWHM ≈ 100–200 km s⁻¹ (i.e., unresolved), which we interpret as tracing the slow unshocked CSM and an intermediate Lorentzian component with FWHM ≈ 500–700 km s⁻¹ (if some of the line-broadening comes from electron scattering, this is an upper limit of the CSM speed). The derived speeds and offsets are shown in Table 4. Figure 5 shows the line and the best fit for the latest spectrum.

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No signature of expanding material is visible in the spectrum at this stage, which is consistent with a thick CSM obstructing the view of the SN ejecta at early times. We will show in Section 3.3 how the photometric data are inconsistent with this picture.

The fits mentioned above, as well as all those mentioned in the rest of this paper, were performed using the emcee algorithm (Foreman-Mackey et al. 2013) to sample from the posterior probability distribution. We then used the 10 combinations with the lowest χ² from the Markov chain Monte Carlo as initial conditions for an optimization algorithm to compute the best-fit value (the maximum a posteriori value or "m.a.p.," in the terminology of Hogg et al. 2010). Note that we used MCMC to compute the posterior distribution and deduce the error bars on the one hand and, on the other hand, used a distinct optimization algorithm to solve for the set of parameters maximizing the posterior, i.e., find the m.a.p. (which coincides with the maximum likelihood value and the minimum χ² value because the priors are uninformative). The reason we took this precaution—which turned out to be unnecessary—is that in the case of non-Gaussian or nonsymmetric marginalized posterior distribution, the m.a.p. may not fall necessarily close to the median of the posterior distribution. In such cases, computing the position of the m.a.p. can be problematic and challenging, all the more when the χ² is noisy and full of local minima. The main challenge is then to choose an initial combination of parameter values to give a minimization algorithm (most minimization algorithms require an initial combination of parameters from which to start their search for the minimum). Here, we made the following choice of initial conditions (a reasonable yet somewhat arbitrary choice): we took the 10 sets of parameters from the chain corresponding to the lowest χ². In the current case, because the posterior distribution is reasonably close to a Gaussian and is well-behaved, the m.a.p. coincides with the median of the posterior distribution.

When errors are noted, they correspond to the 1σ limits of the marginalized posterior distributions.

The interpretation of the late-time spectra of SNe IIn should be made with the complexity caused by CSM interaction in mind. In "normal" SNe II, the ejecta that become optically thin at late times are heated from the inside by two sources of energy: (1) remaining thermal deposition from the original heat of the explosion and (2) radioactivity. The late, or "nebular," spectrum shows no clear continuum, and its emission lines reflect the expansion of the ejecta in which they formed. In SNe IIn, the CSM interaction may continue to dominate the spectrum at late times, e.g., because the ejecta is heated by the shock wave propagating backward from the CSM into its outermost layers (Chevalier & Fransson 2003).

The two late-time spectra of PTF 12glz look similar (the latest spectrum is shown in Figure 3): they show a weak continuum in red, a pseudo-continuum presumably formed by the superposition of narrow [Fe ii] emission lines in the blue (e.g., Kiewe et al. 2012), and several broad emission lines. The temperatures and radii derived by fitting a blackbody curve to the observed spectrum are listed in Table 4 and compared to the parameters derived from photometry in Figure 7. Both spectra show broad Balmer emission lines. The derived speeds and redshifts are presented in Table 4. Figure 4 shows the line and the best fit for the earliest spectrum and Figure 5 shows them for the latest spectrum. The Hα lines are best fit by a linear combination of an intermediate-width Gaussian component with FWHM ≈ 2000 km s⁻¹ that is offset to the blue by ≈250 km s⁻¹ relative to the galaxy rest frame and a broad Gaussian component with FWHM ≈ 8000 km s⁻¹ that is offset to the blue by ≈1000 km s⁻¹ relative to the galaxy rest frame. The intermediate-width component could come from the shocked gas in a cold dense shell forming at the contact discontinuity between the decelerated ejecta and the shocked CSM, which is reheated by X-rays and UV radiation from the shock. The broad component velocities are too high to originate from the CSM: they are rather characteristic ejecta velocity values. We deduce from these velocities that the ejecta has—at least partially—emerged through the optically thick layers of the CSM.

Both late spectra also show emission from the the Ca ii IR triplet. This is commonly seen in late-time-interacting SNe, e.g., in the type IIn SN 2005ip (Boles et al. 2005; Stritzinger et al. 2012), the type Ic SN 2007dio (Kuncarayakti et al. 2018) and the type Ia-csm PTF 11kx (Dilday et al. 2012; Silverman et al. 2013; Graham et al. 2017), as well as in other types of SNe.

We retrieved science-ready imaging data from the several surveys, summarized in Table 5: the Galaxy Evolution Explorer (GALEX) general release 7 (GR7) (Martin et al. 2005), the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS; PS1) data release 1 (DR1) (Flewelling et al. 2016) and the Sloan Digital sky survey data release 9 (DR9) (SDSS; Ahn et al. 2012). We used the software package LAMBDAR (Lambda Adaptive Multi-Band Deblending Algorithm in R; Wright et al. 2016) that is based on Bourne et al. (2012) to perform multiband matched aperture photometry (i.e., taking into account different pixel scales and point-spread functions). The absolute flux calibration was done against instrument-specific zero points (for details on the photometry, see S. Schulze et al. 2019, in preparation).

We modeled the SED of the host galaxy with the software packages Le Phare on the web portal GAZPAR¹⁹ (GAlaxy photometric redshifts (Z) and physical PARameters). We modeled the data using a grid of templates based on Bruzual & Charlot (2003) stellar population-synthesis models with the Chabrier initial-mass-function (Chabrier 2003), a star formation history that is approximated by a declining exponential function, and a Calzetti et al. (2000) dust attenuation curve.

Figure 6 shows the measured galaxy photometry and the best fit. The SED can be adequately described by a template with a stellar mass of $\mathrm{log}M/{M}_{\odot }={9.0}_{-0.2}^{+0.7}$, a star formation rate of ${1.9}_{-0.7}^{+4.1}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, and negligible attenuation $\left[E\left(B-V\right)=0\right]$ (χ²/number of filters = 1.3/8). Using the parameterization of the mass–metallicity relation by Andrews & Martini (2013), we estimate the galaxy metallicity to be ∼0.4–0.5 solar.

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Taking advantage of the multiple-band photometry coverage, we derived the temperature and radius of the blackbody that best fits the photometric data at each epoch (after correcting for redshift and extinction, interpolating the various data sets to obtain data coverage of coinciding epochs and deriving the errors at the interpolated points with Markov chain Monte Carlo (MCMC) simulations). The derived best-fit temperatures T_BB and radii r_BB are shown in Figure 7. For the temperature MCMC fit, we adopted a broad, uninformative (flat) prior T ∈ [5000, 25000]. The edge values of the prior were chosen to contain the range of temperatures observed, e.g., in the SNe IIn sample by Taddia et al. (2013). The best-fit temperatures T_BB should be seen as a lower limit on the temperature, because the spectra of SNe typically show a deficit of flux due to line-blanketing by metal lines.

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In most epochs, this method implies fitting a blackbody spectrum to a two-point SED. Comparison with results derived from more constraining data detailed in Section 3.1—either spectroscopy (shown in red) or a combination of observed and synthetic photometry (shown in green)—suggests that this method leads to a slight overestimation of the radius and underestimation of the temperature.

The temperature T_BB drops from ≈15,000 K to ≈8000 K during the rise and is stable at ≈6000 K during the decay phase. The decrease of T_BB at early times is well fitted by a power law tⁿ with index n = −0.6 and is consistent with the temperature evolution observed in the sample by Taddia et al. (2013). However, PTF 12glz is relatively hot compared to the SNe IIn of this sample, where temperatures span between 11,500 K and 5500 K, and compared to other well studied SNe IIn (e.g., 2006gy, Ofek et al. 2007; Smith & McCray 2007; SN 2005ip, Smith et al. 2009; SN 2010jl, Ofek et al. 2014a).

The derived radius grows by an order of magnitude, from ${r}_{\mathrm{BB}}\approx 6\times {10}^{14}$ cm to ${r}_{\mathrm{BB}}\approx 3\times {10}^{15}$ cm during the rising phase and is ≈4 × 10¹⁵ cm during the light-curve decay phase. Such a rise is puzzling within a picture where the optically thick CSM is supposed to mask the shock and the expanding material. To our knowledge, the measured blackbody radius of all SNe IIn observed to date either stalls after a slight increase (e.g., 2005kj, 2006bo, 2008fq, 2006qq, Taddia et al. 2013; 2006tf, Smith et al. 2008), or stays relatively constant at early times (e.g., SN 2010jl Ofek et al. 2014a), or even supposedly shrinks (e.g., SN 2005ip; SN2006jd, Taddia et al. 2013). Whereas a constant radius is consistent with the continuum photosphere being located in the unshocked optically thick CSM, the possible presence of clumps in the CSM that may expose underlying layers has been invoked by Smith et al. (2008) to interpret observations of a stalling or shrinking radius.

In our case, the velocity at which r_BB grows, reaches ≈7000 km s⁻¹, a value too large to trace the unshocked layers of the CSM or even the reverse shock. One may naively think that if the CSM was optically thin, then the observed radius would grow. However, in this case, it would be hard to convert efficiently the hard X-ray produced by the collisionless shock into optical radiation. In Section 5 we propose one possible geometrical solution to this puzzle.

To investigate the effect of variations in the extinction and mimic the impact of errors in E_B−V on this results, we reran the analysis for the two extreme values of the parameter R ≡ A(V)/E_B−V mentioned in Fitzpatrick (1999): 2.2 and 5.8. We observed an offset in the radius and temperature, but our qualitative result—a growth in the radius at a velocity ⪆7000 km s⁻¹ and a decrease in the temperature—is maintained.

Based on the measurement of r_BB and T_BB, we were able to derive the luminosity ${L}_{\mathrm{BB}}=4\pi {R}^{2}\sigma {T}^{4}$ of the blackbody fits, shown in Figure 8. Since PTF 12glz was not observed at peak luminosity, we can only derive a lower limit on the peak luminosity ${L}_{\mathrm{BB},\mathrm{peak}}\gt 4\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. This makes PTF 12glz more luminous than SN 2008fq—the brightest SN of the Taddia et al. (2013) sample—and brighter than all but one SNe of the sample by Ofek et al. (2014c). This suggests that PTF 12glz is at the bright end of the SNe IIn luminosity range.

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We fitted the light curve during the rise time with a function of the form

$$\begin{eqnarray}&&L={L}_{\max }\{1-\exp [({t}_{0}-t)/{t}_{c}]\},\end{eqnarray} \tag{ 1 }$$

(where t₀ is the time of zero flux, L_max is the maximum bolometric luminosity, and t_c is the characteristic rise time of the bolometric light curve). This allowed us to estimate the epoch at which the extrapolated light curve is crossing zero, which is used throughout this paper as the reference time t₀ (MJD) = 56097.58.

As shown in Figure 8, L_BB decreases about twice as slowly as the 0.98 mag/100 d decline rate characteristic of the radioactive decay of ⁵⁶Ni to ⁵⁶Co and then to ⁵⁶Fe. If this decline was produced by the radioactive decay of ⁵⁶Ni, a nickel mass of at least $14\,{M}_{\odot }$ would be required to reach the bolometric luminosity of PTF 12glz (see Figure 8). The evolution of L should be taken cautiously, because at late time, a blackbody model for the SED may not be valid anymore, and so the temperatures and radii used to calculate L are less reliable. Although the late spectra analyzed in Section 3.1 suggest that the ejecta may have emerged through the CSM at late times, the slow decline of L_BB hints at the fact that interaction may still play an important role in the radiation budget (this may happen through radiation from the reverse shock, or through processed radiation through the edge of the wind, or if the ejecta has only partially emerged through the CSM and is still interacting with it in some places).

A very crude estimation of the swept-up CSM mass can be obtained by assuming that the ejecta mass is comparable to the mass of the CSM and that the bolometric luminosity L is accounted for by the conversion of the kinetic energy of the ejecta into radiation:

$$\begin{eqnarray}&&{\int }_{t}L(t){dt}\lesssim \displaystyle \frac{1}{2}\,{M}_{\mathrm{CSM}}{v}_{e}^{2},\end{eqnarray} \tag{ 2 }$$

where M_CSM is the mass of the CSM and v_e is the velocity of the ejecta. To an order of magnitude, this estimate will not be very different than estimates based on more realistic treatment of the hydrodynamics (e.g., Ofek et al. 2014a). We use the width FWHM ≈ 8000 km s⁻¹ of the broad Hα component at late time (see Section 3.1) as an approximation of the ejecta velocity v_e. From the measured bolometric luminosity shown in Figure 8, ${\int }_{t}L(t){dt}\approx {10}^{51}$ erg. Substituting this value in Equation (2) gives M_CSM ≳ 2 M_⊙.

In interacting SNe, the optically thick CSM masks the explosion, which may leave an ambiguity about the underlying explosion type. In particular, SNe Ia and core-collapse SNe exploding inside a thick CSM would result in similar observational signatures. The spectra would look similar at early times (as long as the explosion is masked by the CSM) and at very late times, when Ni has decayed and there is no more energy to illuminate the ejecta and create absorption lines in the spectrum. Here, the high value of the bolometric luminosity excludes the possibility that PTF 12glz is a masked SN Ia.

Another order of magnitude estimate of the mass can be obtained by assuming a wind density profile, ${\rho }_{\mathrm{CSM}}(r)=K\,{r}^{-2}$ and using the photon diffusion timescale (e.g., Ofek et al. 2010),

$$\begin{eqnarray}&&{t}_{d}\sim \displaystyle \frac{\kappa K}{c},\end{eqnarray} \tag{ 3 }$$

where κ is the CSM opacity. We assume t_d ∼ t_c, where t_c is the characteristic rise time of the bolometric light curve. We estimate t_c by fitting the rising part of the bolometric light curve with the exponential function defined in Equation (1) and assuming κ ≈ 0.34 cm² g⁻¹. We obtain t_c ∼ 20 day, and K ≈ 1.5 × 10¹⁷ g cm⁻¹, which corresponds to high values of the density ${\rho }_{\mathrm{CSM}}\gtrsim {10}^{-14}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ (or a particle density of n ∼ 10¹⁰ cm⁻³, assuming a mean number of nucleons per particle $\langle {\mu }_{p}\rangle =0.6$) at the radii shown in Figure 7.

Measuring K allows us to estimate the mass of the CSM swept up by the ejecta, M_CSM(r) ∼ 4πKr (assuming that the CSM extends on much higher scales than the stellar radius). Using the highest early-time radius r_BB, shown in Figure 7, as a lower limit of the maximum size of the shell of CSM surrounding the explosion, gives M_CSM ≳ 3 M_⊙, in good agreement with the estimates obtained with Equation (2).

For a wind density profile, $K=\dot{M}/(4\pi {v}_{w})$, where v_w is the velocity of the CSM. Therefore, measuring K also gives us an estimation of the mass-loss rate. Using the width of the narrow Hα component during rise time, FWHM ≈ 200 km s⁻¹, as a proxi of v_w (see Table 4), we obtain a large mass-loss rate $\dot{M}$ ∼ 0.6 M_⊙ yr⁻¹, higher than the mass-loss rates observed in Taddia et al. (2013) and Kiewe et al. (2012) and comparable to the mass-loss rate of, e.g., iPTF13z (Nyholm et al. 2017). Combining these estimates suggests that the CSM mass was ejected on a timescale of 1–10 years prior to the SN explosion.

We wish to emphasize that we made several simplifying assumptions in this section. In particular, (1) we assumed a spherical symmetry of the CSM, (2) we assumed a wind profile of the CSM, and (3) we assumed that the kinetic energy of the ejecta converts efficiently into radiation. Therefore, the numbers above have to be considered as order of magnitude estimates.

At late times, the broad component of the Hα Balmer line is blueshifted by ≈1000 km s⁻¹ relative to the galaxy rest frame. The intermediate component is also blueshifted, but by a velocity of ≈250 km s⁻¹, which is consistent with a typical stellar velocity within the galaxy (see Table 4 and Figure 5). Several explanations have been proposed to explain the blueshift of emission line profiles in interacting SN. One possible explanation is the formation of dust, which increasingly blocks the receding parts of the ejecta (e.g., as proposed in the case of the Type IIn SN 2010 jl, Smith et al. 2012; Gall et al. 2014). In our case, the blueshift of the broad component does not grow with time (see Table 4), meaning that if there is dust, it may have formed before the epoch of the first nebular spectrum. In our case, testing the wavelength dependency of the blueshift—a blueshift caused by dust would be stronger at bluer wavelengths—is tricky because the blended iron emission lines mask the structure of the Hβ line profile. We tried to apply several filters on this area, to separate the possible broad Balmer components from Fe ii blend structures, but the results remained inconclusive.

Other explanations have been proposed for the blueshift of emission lines. Fransson et al. (2014), for example, attributed the blueshift of emission lines to radiative acceleration of the preshock gas by the SN radiation, whereas Smith et al. (2012) proposed a geometric explanation.

We have written a three-dimensional computer program in python, SLAB-Diffusion, available online (Soumagnac 2018, Codebase: https://github.com/maayane/SLAB-Diffusion), in order to calculate the propagation of photons through a slab and simulate the main observables. The simple geometry we consider is illustrated in a cartoon in Figure 9. Following a similar approach as in Ginzburg & Balberg (2014), we replaced the hydrodynamical description of the SN explosion by a stationary model of the shock breakout. This is equivalent to neglecting the expansion of the gas due to the explosion and modeling the interaction between the shock and the CSM as an instantaneous deposition of energy in the slab.

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The assumption of an instantaneous release of energy is justified when t_d/t_e ≫ 1, where t_e is the timescale over which the energy of the ejecta is converted into radiation by deceleration, and t_d is the photon diffusion timescale Equation (3). The timescale t_e is given by

$$\begin{eqnarray}&&{t}_{e}=\displaystyle \frac{{R}_{c}}{v},\end{eqnarray} \tag{ 4 }$$

where R_c is the radius at which the accumulated CSM mass is comparable to the ejecta mass and v is the ejecta velocity. Since t_d/t_e = τv/c, the condition t_d/t_e ≫ 1 is satisfied, and the instantaneous energy deposition approximation is valid, as long as τ ≫ c/v ∼ 30. This is indeed the case for the model parameters we infer for PTF 12glz: a wind velocity v = 200 km s⁻¹ (Section 3.1) with a mass-loss rate $\dot{M}$ ∼ 0.6 M_⊙ yr⁻¹ (Section 3.5), yielding ${R}_{c}\sim {10}^{15}$ cm and τ ∼ 200.

We assumed that the problem can be treated accurately within the diffusion approximation (i.e., assuming that τ ≈ 1 occurs close to the surface). Hence, the energy density u within the slab is described by the diffusion equation:

$$\begin{eqnarray}&&\displaystyle \frac{\partial u}{\partial t}={\rm{\nabla }}\cdot (D{\boldsymbol{\nabla }}u),\end{eqnarray} \tag{ 5 }$$

where D = c/(3κρ) is the diffusion coefficient. Here we explored three density profiles: a constant density profile, ρ, and two functions of the ${{\boldsymbol{e}}}_{z}$ coordinate: $\rho \propto | z{| }^{-1}$ and ρ ∝ z⁻², where the origin of the z-axis is at the center of the slab. We leave the treatment of the angular dependency of D to further extensions of this work.

At the boundaries, the energy escapes from the slab and the flux ${\boldsymbol{f}}$ is linked to the density of energy through

$$\begin{eqnarray}&&| | {\boldsymbol{f}}| | =\alpha {cu},\end{eqnarray} \tag{ 6 }$$

where α ≈ 1/4 (the case α = 1/4 corresponds to isotropic radiation). We discretized Equation (5) in a Cartesian, three-dimensional grid (illustrated in a cartoon in Figure 9), using an explicit forward Euler scheme with ΔtD/(Δd)² < 0.1 where Δd is defined for each coordinate and is the size of the mesh in each direction ${{\boldsymbol{e}}}_{x}$, ${{\boldsymbol{e}}}_{y}$, and ${{\boldsymbol{e}}}_{z}$. We assumed that D does not depend on the wavelength of the photons (i.e., we made the so-called gray approximation). We solved a dimensionless version of Equation (5):

$$\begin{eqnarray}&&\displaystyle \frac{\partial u}{\partial t^{\prime} }={\rm{\nabla }}\cdot (D^{\prime} (z){\boldsymbol{\nabla }}u),\end{eqnarray} \tag{ 7 }$$

where t' = tD(h)/h² and D'(z) = D(z)/D(h). In this case the boundary conditions are

$$\begin{eqnarray}&&| | {\boldsymbol{f}}| | =\alpha {vu},v=c/{v}_{d},\end{eqnarray} \tag{ 8 }$$

where v_d = D(h)/h. A slab with a width h = 10¹⁶ cm, an opacity κ = 0.34 cm² g⁻¹, and a constant mass density ρ = 1 × 10⁻¹⁶ g cm⁻³ (corresponding, e.g., to 3 M_⊙ of CSM in a slab with L_x = L_y = 8 h), corresponds to the unitless velocity v ≈ 1 and a diffusion time t_d = h²3κρ/c = 94 days.

In order to minimize the effect of the finite size of the grid along ${{\boldsymbol{e}}}_{x}$ and ${{\boldsymbol{e}}}_{y}$, we took several precautions. We chose the dimensions of the grid along the ${{\boldsymbol{e}}}_{x}$ and ${{\boldsymbol{e}}}_{y}$ directions so that L_x = L_y and L_x ≫ h. We checked that L_x is large enough compared to h so that a change in L_x does not affect the results. Equation (8) only describes the boundary conditions along the ${e}_{{\boldsymbol{z}}}$ direction. We checked that we can apply reflective boundary conditions (i.e., ${\boldsymbol{f}}={\boldsymbol{0}}$ at the boundaries) or absorbing boundary conditions (i.e., u = 0 at the boundary) along the ${{\boldsymbol{e}}}_{x}$ and ${{\boldsymbol{e}}}_{y}$ directions, without affecting the results. We also checked for convergence of the code with respect to the time steps.

As photons diffuse in the slab, they reach the z = h/2 surface visible to the observer (see Figure 9). In Figure 10, we show the evolution of the total flux of energy escaping from the surface of a slab with v = 1, in response to an instantaneous deposition of energy at t = 0. We use the full width at half maximum (FWHM) of the energy density u at the surface of the slab as a proxy for the radius seen by the observer (below we present another proxy for r_BB). In Figure 11, we show that the FWHM grows in time for all three checked density profiles ρ = Const., $\rho \propto | z{| }^{-1}$ and ρ ∝ z⁻². We checked that varying the parameter v does not change this qualitative result.

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We would like to check whether our model can reproduce the decrease of the blackbody temperature T_BB observed in Figure 7, in addition to producing a growing radius. Here again, given the simplicity of our model geometry, we are interested in the evolution of T_BB rather than trying to fit its actual values. By modeling each cell of the z = h/2 surface as a blackbody with temperature T ∝ u(x, y)^1/4 and summing up all the cell spectra, we can compute the overall spectrum of the surface. The resultant spectrum is well represented by a blackbody spectrum, which allows us to deduce the blackbody temperature of the surface. This strategy also provides an additional way to recover the growing radius, by using:

$$\begin{eqnarray}&&L={\int }_{S}f\,{ds}=\sigma {T}_{\mathrm{BB}}^{4}4\pi {r}_{\mathrm{BB}}^{2}.\end{eqnarray} \tag{ 9 }$$

In Figure 12, we show the evolution of the temperature T_BB and radius r_BB, assuming h = 1 × 10¹⁵ cm, a constant mass density ρ = 3 × 10⁻¹⁴ g cm⁻³ and an input energy of 10⁵¹ erg. Using this energy (calculated in Equation (2)) as the energy initially deposited in the slab, is equivalent to assuming that all the energy radiated by the SN explosion is released by the time it starts diffusing out through the CSM. This is only correct within the assumption that the characteristic timescale for diffusion is much larger than the characteristic timescale for interaction between the ejecta and the CSM, which we showed is a correct assumption in the case of PTF 12glz. Figure 12 shows that the aspherical geometry of the slab allows us to recover the increase of the radius r_BB and the decrease of the temperature T_BB observed in Figure 7.

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