SN 2010jl: OPTICAL TO HARD X-RAY OBSERVATIONS REVEAL AN EXPLOSION EMBEDDED IN A TEN SOLAR MASS COCOON

The Palomar Transient Factory (PTF¹⁵; Law et al. 2009; Rau et al. 2009) detected SN 2010jl (PTF 10aaxf) on 2010 November 13.4, 10 days after its discovery by Newton & Puckett (2010). The PTF data-reduction pipeline is presented by R. Laher et al. (in preparation), and the photometric calibration is described by Ofek et al. (2012a, 2012b). The PTF light curve of this SN and the All Sky Automated Survey (ASAS) prediscovery data points from Stoll et al. (2011) are presented in Figure 1 and listed in Table 1. ASAS first detected the SN on 2010 September 10, about 15 days prior to I-band maximum light—soon after its solar conjunction.

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The first-year PTF flux measurements taken before MJD 55,760 show a clear power-law decline (Section 5); the second-year flux measurements obtained between MJD 55760 and MJD 56,070 (Figure 1) are consistent with an exponential decay (i.e., ∝exp(− t/τ_exp), where t is the time and τ_exp is the exponential timescale). We find that the best-fit exponential timescale is τ_exp = 129.8 ± 1.5 day (χ²/dof = 0.7/15), where the uncertainty is estimated using the bootstrap technique (Efron 1982; Efron & Tibshirani 1993). We note that this is longer than the timescale expected from ⁵⁶Co decay (∼111 day). Were this decay produced by ⁵⁶Ni decay to ⁵⁶Co and finally ⁵⁶Fe, then at least 27 M_☉ of ⁵⁶Ni would be required, which is unlikely. Moreover, at later times the decay rate becomes significantly slower than the exponential decay expected from radioactive material (see Figure 1). Therefore, a more reasonable interpretation is that the SN light curve is powered by interaction of the SN shock with CSM. Interestingly, the second-year and third-year data (MJD >56,070) are also roughly consistent with a power-law decay. The power-law fits to the light-curve data are shown in Figure 1 and discussed in Section 5.

Zhang et al. (2012) reported on a flattening in the optical light curve of the SN, starting about 90 days after maximum light. However, this flattening is somewhat larger than we measure in the PTF R-band data (A possible theoretical explanation for the flattening is discussed in Section 5.1). We note that Zhang et al. (2012) did not use image subtraction, due to the lack of a reference image, and instead they subtracted an estimate of the host light obtained from the Sloan Digital Sky Survey (SDSS) image. According to the SDSS source catalog, the magnitude of the underlying host galaxy is r≅15.5, which is about 30% of the SN flux at about 200 days after maximum light. This may be large enough to contaminate their photometry. Given that the inconsistency is small (∼0.05 mag), we speculate that the late-time flattening reported by Zhang et al. (2012) is due to contamination from the host galaxy.

SN 2010jl was observed spectroscopically by the PTF collaboration on several occasions. A log file of the observations is presented in Table 2. The data will be electronically released via the WISeREP Web site¹⁶ (Yaron & Gal-Yam 2012). Selected spectra of SN 2010jl are shown in Figure 2.

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Inspection of the spectra of SN 2010jl shows that the Hα line consists of several components. The narrowest features we detected are Hα, Hβ, and He i P Cygni lines, with a velocity difference between the peak and minimum of ∼70 km s⁻¹ (see also Smith et al. 2012). The Hα profile in the spectra can be decomposed into a Lorentzian and a Gaussian, where the Gaussian has a velocity width of σ ≈ 300 km s⁻¹. Alternatively, the early-time spectra can be decomposed into three Gaussians, in which the widest Gaussian has velocity width σ ≈ 4000 km s⁻¹. At late times, about six months after maximum light, the Hα line develops some asymmetry; it is discussed by Smith et al. (2012) and attributed to dust formation. We fitted a blackbody spectrum to the spectroscopic measurements as a function of time, and the derived blackbody temperatures and radii are shown in Figure 3.

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The Ultra-Violet/Optical Telescope (UVOT; Roming et al. 2005) on board the Swift satellite (Gehrels et al. 2004) observed SN 2010jl on several occasions. The data were reduced using standard procedures (e.g., Brown et al. 2009). Flux from the transient was extracted from a 3'' radius aperture, with a correction applied to put the photometry on the standard UVOT system (Poole et al. 2008). The resulting measurements, all of which have been converted to the AB system, are listed in Table 1 and are shown in Figure 1. We caution that these results have not incorporated any contribution from the underlying host galaxy and may therefore overestimate the SN flux at late times. Specifically, the UVOT measurements in Figure 1 near 900 days are heavily contaminated by an underlying star-forming region in the host galaxy.

We fitted a blackbody spectrum to the UVOT measurements as a function of time, and the results are shown in Figure 4. In the fits we corrected the flux measurements for Galactic extinction, assuming E_{B − V} = 0.027 mag (Schlegel et al. 1998) and R_V = 3.08 (Cardelli et al. 1989). We note that we also tried to fit the blackbody spectrum with E_{B − V} as a free parameter and verified that the best fit is obtained near the Schlegel et al. (1998) value for E_{B − V}. The Swift-derived blackbody temperature shows some indications that it is rising in the first ∼200 days after maximum light. However, we caution that deviations from a blackbody caused by spectral lines that are not dealt with in the broadband observations, as well as deviations from a blackbody spectrum (see Section 5.2) and metal-line blanketing, can affect the derived temperature and radius. Therefore, we argue that the quoted temperatures are likely only a lower limit on the effective temperatures.

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These temperature measurements differ from those obtained using the spectroscopic observations (Section 2.3). However, due to metal-line blanketing and given that the spectral peak is too blue to be probed by visible-light spectra, we consider both the spectroscopic and UVOT observations to be lower limits on the temperature. The temperature evolution based on the visible-light spectra is opposite to that based on the UVOT observations. However, both evolutions seen in Figures 3 and 4 are very moderate. In Section 5.1 we investigate the effect of this uncertainty on our results, and in Section 5.2 we discuss the nature of the decrease in the blackbody radius at late times.

NuSTAR is the first hard X-ray focusing satellite (Harrison et al. 2013). Its broad energy range (3–79 keV) allows us to determine the previously unconstrained temperature of the hardest component of the X-ray spectrum. NuSTAR observed SN 2010jl on 2012 October 6, roughly 2 yr after the discovery of the SN. We obtained a usable exposure time of 46 ks. This was the first SN observed by the NuSTAR "supernovae and target-of-opportunity program." Spectra and images were extracted using the standard NuSTAR Data Analysis Software (NuSTARDAS version 0.11.1) and HEASOFT (version 6.13). XSPEC (Arnaud 1996, version 12.8) was used to perform the spectral analysis in combination with the XMM data. A summary of the high-energy observations of SN 2010jl is given in Table 3.

Shortly after we obtained the NuSTAR observations, we triggered XMM-Newton for a target-of-opportunity observation (see Table 3) with the goal of determining the bound-free absorption utilizing XMM's good low-energy X-ray response. The observation was carried out during 2012 November 1 for 13 ks, resulting in a usable exposure time of ∼10 ks for the MOS1 and MOS2 detectors and ∼4 ks for the PN detector, after filtering out periods of high background flaring activity. The Science Analysis System software (SAS, version 12) was used for data reduction. Spectral analysis combined with the NuSTAR data was performed using XSPEC version 12.8.

Chandra observed the location of SN 2010jl on five epochs (PIs Pooley, Chandra; Chandra et al. 2012a). All the observations except one are public.

Inspection of the Chandra images shows emission from the SN position, as well as from another source only about 2'' east of the SN (Figure 5). In order to make sure that the Chandra flux measurements are compatible with the other X-ray observations, we used a relatively large aperture of radius 3''. This extraction aperture contains light from the nearby source. The background was extracted from an annulus with an inner (outer) radius of 20'' (40''). The observations are plotted in Figure 6 and presented in Table 3.

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In addition, there are multiple weak sources located within the source extraction regions of XMM and NuSTAR (Figure 5). We use the Chandra observation to determine their mean flux and spectrum, and as an additional (known) component while fitting the spectra from NuSTAR and XMM. The Chandra data were analyzed using XSPEC¹⁷ V12.7.1 (Schafer 1991). The Galactic neutral hydrogen column density in the direction of SN 2010jl is N_H = 3 × 10²⁰ cm⁻² (Dickey & Lockman 1990). All of the nearby sources were fitted jointly with an absorbed power law assuming Galactic absorption. The fit resulted in a photon power-law index of Γ = 1.375 and a flux of 6.3 × 10⁻⁶ photon cm⁻² s⁻¹ in the energy range 0.3–10 keV (χ²/dof =12.5/12). The spectra and the contamination by the nearby sources are discussed and modeled in Section 3.

The Swift X-Ray Telescope (XRT; Burrows et al. 2005) observed SN 2010jl on multiple epochs since the SN discovery. For each Swift/XRT image of the SN, we extracted the number of X-ray counts in the 0.2–10 keV band within an aperture of 9'' radius centered on the SN position. This aperture contains ∼50% of the source flux (Moretti et al. 2004). The background count rates were estimated in an annulus around the SN location, with an inner (outer) radius of 50'' (100''). The log of Swift-XRT observations, along with the source and background X-ray counts in the individual observations, is listed in Table 4. The binned Swift-XRT observations are presented in Figure 6 and listed in Table 5.

Table 5. Binned Swift-XRT Data

〈MJD〉	Range		CR	Exp.	〈E〉
(day)	(day)	(day)	(counts ks−1)	(ks)	(keV)
55,505.5	−0.4	3.6	$2.50_{-0.37}^{+0.43}$	36.02	4.76 ± 0.28
55,512.6	−0.7	4.9	$2.22_{-0.41}^{+0.49}$	26.08	5.62 ± 0.33
55,523.4	−3.3	2.8	$1.47_{-0.64}^{+1.0}$	6.79	5.00 ± 0.48
55,532.7	−3.0	3.0	$3.02_{-0.94}^{+1.3}$	6.63	5.15 ± 0.70
55,675.9	−0.3	3.2	$7.00_{-1.2}^{+1.4}$	9.43	4.53 ± 0.30
56,219.9	−12.9	1.4	$7.28_{-1.2}^{+1.4}$	10.72	3.06 ± 0.31
56,326.9	−13.8	16.7	$7.21_{-0.87}^{+1.0}$	18.85	2.72 ± 0.23
56,355.5	−0.0	0.0	$4.61_{-3.0}^{+6.1}$	0.87	2.88 ± 1.94
56,380.3	−0.0	0.0	$7.07_{-0.94}^{+1.1}$	15.83	2.80 ± 0.23
56,429.1	−2.4	4.4	$6.76_{-0.79}^{+0.89}$	21.60	2.92 ± 0.21

Notes. SN 2010jl binned Swift-XRT light curve. 〈MJD〉 is the weighted mean modified Julian day of all the observations in a given bin, where the observations are weighted by their exposure times. Range is the time range around 〈MJD〉 in which the light curve (Table 4) was binned. CR is the counts rate along with the lower and upper 1σ uncertainties. The source count rates are corrected for extraction aperture losses (50%). 〈E〉 is the mean energy of photons within the 0.2–10 keV range and the standard error of the mean.

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A brief outline of the model is as follows. After the SN shock moves beyond the stellar surface, it propagates in an optically thick CSM and some of its kinetic energy is converted into optical photons (UV to IR). The relevant source of opacity is mainly Thomson scattering, which is independent of wavelength. If the Thomson optical depth τ is large enough, the photons are trapped and the shock energy is mediated by photons—photons diffuse out, scattering upstream electrons and accelerating them. A radiation-mediated shock "breaks down" or "breaks out" (i.e., radiation escapes ahead of the shock) when photons diffuse ahead of the shock faster than the shock propagates. This happens when τ ≈ c/v_s (Weaver 1976; and see discussion for the case of wind breakout in Ofek et al. 2010). Here v_s is the shock velocity, and c is the speed of light.

Katz et al. (2011) and Murase et al. (2011) showed that if there is a sufficiently large amount of CSM above the shock-breakout radius, the shock will transform from being radiation mediated to collisionless (i.e., the photons are no longer trapped). At this time the shock (and ejecta) is moving through the CSM and its kinetic energy is converted to radiation at a rate of $\epsilon (\rho v_{{\rm s}}^{2}/2)(4\pi r_{{\rm s}}^{2}v_{{\rm s}})$, where is the efficiency, ρ is the CSM density, and r_s is the shock radius (e.g., Svirski et al. 2012). The time dependences of r_s and v_s, while the ejecta and CSM are interacting, are known from self-similar solutions of the hydrodynamical equations (Chevalier 1982).

Later on, when the shock runs over a mass of CSM equivalent to the ejecta mass, the shock will go into a new phase of either conservation of energy if the density is low enough and the gas cannot cool quickly (i.e., the Sedov–Taylor phase), or conservation of momentum if the gas can radiate its energy by fast cooling (i.e., the snow-plow phase). In either case, the light curve in this final stage will be characterized by a steeper decay rate (Svirski et al. 2012).

The observables in this approach are the light-curve rise time, the luminosity and its decay rate, the time of power-law break in the light curve, and the shock velocity at late times as measured from the hard X-ray observations. These observables allow us to solve for the shock radius and velocity as a function of time, the CSM density profile, and the total mass; they also provide a consistency test.

An SN explosion embedded in CSM with optical depth in excess of ∼c/v_s, where c is the speed of light and v_s is the SN shock velocity, will have a shock breakout within the optically thick CSM. The analytical theory behind this was presented by Ofek et al. (2010), Chevalier & Irwin (2011), Balberg & Loeb (2011), Ginzburg & Balberg (2012), Moriya & Tominaga (2012), and Svirski et al. (2012), while simulations of such scenarios were presented by Falk & Arnett (1973, 1977), among others. Here we review the theory and extend it to a general CSM power-law density profile and general ejecta velocity power-law distributions.

Following Chevalier (1982), we assume that the expanding ejecta have a spherically symmetric power-law velocity distribution of the form

$$\begin{equation} \rho _{{\rm ej}}=t^{-3}\left (\frac{r}{tg}\right)^{-m}. \end{equation} \tag{ 1 }$$

Here ρ_ej is the ejecta density, t is the time, r is the radius, m is the power-law index of the velocity distribution, and g is a normalization constant. This model is justified because the outer density profile of massive stars can likely be approximated as a power law (e.g., Nomoto & Sugimoto 1972). We expect m ≈ 10 for progenitor stars with a radiative envelope, and m ≈ 12 for progenitor stars with a convective envelope (e.g., Matzner & McKee 1999). We assume that the ejecta are expanding into a CSM with a spherically symmetric power-law density profile of the form

$$\begin{equation} \rho _{{\rm csm}} = Kr^{-w}, \end{equation} \tag{ 2 }$$

where w is the power-law index and K is the normalization.¹⁹ In a wind profile, w = 2, $K=\dot{M}/(4\pi v_{{\rm CSM}})$ is called the mass-loading parameter with units of g cm⁻¹ (where v_CSM is the CSM or wind velocity), and $\dot{M}$ is the mass-loss rate. We note that even if the CSM is ejected in a single outburst, we expect the CSM to spread over a wide range of radii since the ejecta probably have a wide range of velocities. Given these assumptions, Chevalier (1982) showed that the forward-shock radius is given by

$$\begin{eqnarray} r_{{\rm s}} & = & \left (\frac{Ag^{m}}{K}\right)^{1/(m-w)} t^{(m-3)/(m-w)} \cr \ms & \equiv & r_{0} \left (\frac{t}{t_{0}}\right)^{(m-3)/(m-w)}, \end{eqnarray} \tag{ 3 }$$

where A is a constant derived from the self-similar solution. The second part of the equation simply absorbs the coefficients into arbitrary r₀ and t₀. By differentiating Equation (3), we get the forward-shock velocity as a function of time,

$$\begin{eqnarray} v_{{\rm s}} & = & \frac{m-3}{m-w} \left (\frac{Ag^{n}}{K}\right)^{1/(m-w)} t^{(w-3)/(m-w)} \cr\ms & \equiv & v_{0} \left (\frac{t}{t_{0}}\right)^{(w-3)/(m-w)}, \end{eqnarray} \tag{ 4 }$$

where

$$\begin{equation} v_{0}\equiv \frac{m-3}{m-w} \frac{r_{0}}{t_{0}}. \end{equation} \tag{ 5 }$$

The shock breakout in a CSM environment occurs when the Thomson optical depth is

$$\begin{equation} \tau \approx c/v_{{\rm bo}}, \end{equation} \tag{ 6 }$$

where v_bo is the shock velocity at breakout (e.g., Weaver 1976). The expression for the Thomson optical depth, assuming w > 1, is

$$\begin{equation} \tau =\int _{r_{{\rm s}}}^{\infty }{\rho \kappa dr}=\frac{\kappa K}{w-1} r_{{\rm s}}^{1-w}, \end{equation} \tag{ 7 }$$

where r_s is the forward-shock radius and κ is the opacity. We note that for w = 2, Balberg & Loeb (2011) showed that the total optical depth (taking into account the reverse-shock contribution) is a factor of 1.55 times larger. Chevalier (2013) argues that at relatively late times, if the CSM density is low, the reverse shock may dominate the X-ray emission. In this case the effective optical depth may be even higher. Effectively, this uncertainty can be absorbed into the uncertainty in the opacity κ, which is discussed in Section 5. We note that our main conclusions do not depend on the late-time observations. From Equations (6) and (7) we can derive an expression for K,

$$\begin{eqnarray} K & \approx & \frac{c}{v_{{\rm bo}}\kappa } (w-1)r_{{\rm bo}}^{w-1} \cr & = & \frac{c}{\kappa }(w-1)\left (\frac{m-w}{m-3}\right)^{w-1}v_{{\rm bo}}^{w-2}t_{{\rm bo}}^{w-1}, \end{eqnarray} \tag{ 8 }$$

where the last step is obtained using Equation (5). Here r_bo, v_bo, and t_bo are the radius, velocity, and timescale of the shock breakout, respectively (replacing r₀, v₀, and t₀).

The integrated CSM mass within radius r or time t, assuming w < 3 and star radius r_* ≪ r, is given by

$$\begin{eqnarray} M & = \int _{0}^{r}{4\pi r^{2} Kr^{-w}dr} = \frac{4\pi K}{3-w} r^{3-w} \cr\ms & = \frac{4\pi K}{3-w} \left (\frac{m-w}{m-3}\right)^{3-w} v_{{\rm bo}}^{3-w} t_{{\rm bo}}^{(3-w)^{2}/(m-w)} t^{(3-w)(m-3)/(m-w)}. \nonumber \\ \end{eqnarray} \tag{ 9 }$$

Assuming fast cooling, following the shock breakout the kinetic energy is converted into radiation (bolometric luminosity) at a rate of

$$\begin{equation} L = 2\pi \epsilon r_{{\rm s}}^{2}\rho v_{{\rm s}}^{3}. \end{equation} \tag{ 10 }$$

The value of the efficiency factor, , is discussed in Section 5. We note that Equation (10) assumes that v_s ≫ v_CSM. Substituting the expressions for r_s (Equation (3)), ρ (Equation (2)), and v_s (Equation (4)) into Equation (10), we get

$$\begin{equation} L = L_{0}t^{\alpha }, \end{equation} \tag{ 11 }$$

where

$$\begin{equation} \alpha \equiv \frac{(2-w)(m-3)+3(w-3)}{m-w} \end{equation} \tag{ 12 }$$

and

$$\begin{equation} L_{0}\equiv 2\pi \epsilon K r_{{\rm bo}}^{2-w}v_{{\rm bo}}^{3}t_{{\rm bo}}^{-\alpha }. \end{equation} \tag{ 13 }$$

Using Equation (5), we can remove r_bo from Equation (10) and get

$$\begin{equation} L_{0} = 2\pi \epsilon K \left (\frac{m-w}{m-3}\right)^{2-w} v_{{\rm bo}}^{5-w} t_{{\rm bo}}^{2-w-\alpha }. \end{equation} \tag{ 14 }$$

Equation (11) was derived by Svirski et al. (2012) for the special case of w = 2 and m = 12, 7, 4.

Equation (11) provides a description of the light curve following the shock breakout, assuming w < 3 and m > 4 (for radiative shock). However, another condition is that w ⩾ 2. The reason is that if w < 2, then the diffusion timescale diverges, and therefore the shock will break out near the edge of the CSM. In this case we will not see a light curve with a power-law decay (i.e., Equation (11)) lasting for a long period of time as seen in Figure 1. Therefore, w < 2 is not a relevant solution for SN 2010jl. Figure 9 presents the value of α as a function of m and w. We are not aware of a relevant self-similar solution²⁰ for w > 3.

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Equation (11) is correct only if the shock is in the fast-cooling regime. The free–free cooling timescale is

$$\begin{equation} t_{{\rm ff,cool}} \approx 1.8\times 10^{15} \left (\frac{T}{10^{8}\,{\rm K}}\right)^{1/2} \left (\frac{n}{1\,{\rm cm}^{-3}}\right)^{-1} Z^{-2}\,{\rm s}, \end{equation} \tag{ 15 }$$

where Z is the atomic number of the atom and n is the particle density given by

$$\begin{equation} n=\frac{K}{\langle \mu _{{\rm p}}\rangle m_{{\rm p}}} r^{-w}, \end{equation} \tag{ 16 }$$

where 〈μ_p〉 is the mean number of nucleons per particle (mean molecular weight) and m_p the proton mass. The criterion for fast cooling is that t_{ff, cool} ≲ t. Therefore, for timescales of a year (3 × 10⁷ s), fast cooling requires n ≳ 6 × 10⁷ cm⁻³.

Several other important relations can be derived. By rearranging Equation (8), we get

$$\begin{equation} t_{{\rm bo}} = \left [ \frac{c}{\kappa K} (w-1) \right ]^{1/(1-w)} \frac{m-3}{m-w} v_{{\rm bo}}^{(w-2)/(1-w)}. \end{equation} \tag{ 17 }$$

From Equation (14) we find

$$\begin{equation} K = \frac{L_{0}}{2\pi \epsilon } \left (\frac{m-w}{m-3}\right)^{w-2} v_{{\rm bo}}^{w-5} t_{{\rm bo}}^{\alpha +w-2}, \end{equation} \tag{ 18 }$$

and by substituting Equation (18) into Equation (17) we get

$$\begin{equation} t_{{\rm bo}} = \left [ 2\pi \epsilon \frac{m-w}{m-3} (w-1)\frac{c}{\kappa L_{0}} v_{{\rm bo}}^{3} \right ]^{1/(\alpha -1)}, \end{equation} \tag{ 19 }$$

or alternatively

$$\begin{equation} v_{{\rm bo}} = t_{{\rm bo}}^{(\alpha -1)/3} \left [ 2\pi \epsilon \frac{m-w}{m-3} (w-1)\frac{c}{\kappa L_{0}} \right ]^{-1/3}. \end{equation} \tag{ 20 }$$

These relations suggest that in SNe that are powered by interaction we expect to detect correlations between the SN rise time, its peak luminosity, and shock velocity. We note that this can be used to test the hypothesis that the super-luminous SNe (see review in Gal-Yam 2012) are powered by interaction of their ejecta and CSM (e.g., Quimby et al. 2011). As far as we can tell, such correlations are not expected in the context of other models (e.g., Kasen & Bildsten 2010). Furthermore, by inserting Equations (8) and (20) into Equation (9), we get the total CSM mass swept by the shock up to time t as a function of the observables (e.g., L₀, t_bo),

$$\begin{eqnarray} M & = & 4\pi c \left (\frac{m-w}{m-3}\right)^{5/3} \frac{(w-1)^{2/3}}{3-w} \left (\frac{2\pi c \epsilon }{L_{0}}\right)^{-1/3} \kappa ^{-2/3}\cr \ms && \times t_{{\rm bo}}^{[(m-w)(3w-4) + (w-3)(3w-6)+(2-w)(m-3)]/[3(m-w)]} \cr\ms && \times t^{(3-m)(w-3)/(m-w)}. \end{eqnarray} \tag{ 21 }$$

For the specific case of w = 2 and m = 10 we can write this as

$$\begin{eqnarray} M & \approx & 15.1 \left (\frac{\epsilon }{0.25}\right)^{-1/3} \left (\frac{L_{0}}{10^{46}\,{\rm erg\,s}^{-1}}\right)^{1/3} \left (\frac{\kappa }{0.34\,{\rm cm}^{2}\,{\rm gr}^{-1}}\right)^{-2/3}\cr & &\times \left (\frac{t_{{\rm bo}}}{20\,{\rm day}}\right)^{2/3} \left (\frac{t}{365\,{\rm day}}\right)^{7/8}\,{M}_{\odot }. \end{eqnarray} \tag{ 22 }$$

We note that L₀ is the luminosity evaluated at time of 1 s rather than t_bo (see definition in Equation (11)). Additional relations can be derived, including relations that depend on v_bo and/or the integrated luminosity (i.e., ∫Ldt = L₀t^{α + 1}/[α + 1]), rather than on L₀. However, some of these relations are algebraically long, and we do not provide them here.

At later times when the mass of the CSM accumulated by the ejecta is equivalent to the ejecta mass, the light curve evolves in a different way than described so far. In the case of fast cooling (i.e., cooling timescale [e.g., Equation (15)] is shorter than the dynamical timescale), the system enters the snow-plow phase. While at these times the reverse shock is absent and the formalism of Chevalier (1982) does not apply, we can obtain the correct time dependence by using an artificial value of m (no longer related to the ejecta profile). In this snow-plow phase the light curve evolves effectively with m = 4 regardless of the value of w (see Svirski et al. 2012). The reason is that, while in this case the energy is radiated away, the momentum is conserved, and from momentum conservation ρr³v ≈ constant, we get ρ∝v⁻⁴, hence m = 4. If the shock is slowly cooling, we enter the Sedov-Taylor phase and the light curve will drop rapidly.

Figure 9 suggests that for m = 4, α ≈ −3/2, with relatively weak dependence on the value of w. However, the exact value of α is sensitive to the value of m, and for m slightly lower than 4, α can change dramatically. In any case, once the swept-up CSM mass is comparable to the ejected mass, we expect substantially more rapid decline of the bolometric emission.

Given the CSM density profile, we can calculate some additional properties. The column density, assuming w > 1, between radius r and infinity (i.e., the observer) is

$$\begin{equation} N = \int _{r}^{\infty }{\frac{K}{\langle \mu _{{\rm p}}\rangle m_{{\rm p}}} r^{-w}dr} = \frac{K}{\langle \mu _{{\rm p}}\rangle m_{{\rm p}} (w-1)}r^{1-w}. \end{equation} \tag{ 23 }$$

The free–free optical depth between the shock region and the observer is given by (e.g., Lang 1999, Equation (1.223); Ofek et al. 2013a)

$$\begin{eqnarray} \tau _{{\rm ff}} & \approx & 8.5\times 10^{-28} T_{{\rm e},4}^{-1.35} \nu _{10}^{-2.1} \int _{r}^{\infty }{n_{{\rm e}}^{2}dr} \cr & \cong & 8.5\times 10^{-28} T_{{\rm e},4}^{-1.35} \nu _{10}^{-2.1} \frac{K^{2}}{\langle \mu _{{\rm p}}\rangle ^{2} m_{{\rm p}}^{2} (2w-1)} r^{1-2w},\nonumber \\ \end{eqnarray} \tag{ 24 }$$

where T_{e, 4} is the electron temperature in units of 10⁴ K and ν₁₀ is the frequency in units of 10 GHz. Note that r is measured in cm, and that the last expression is valid for w > 1/2. If τ_ff ≫ 1, a radio signal is not expected (see Murase et al. 2013 for discussions).

NuSTAR opens the hard X-ray band for discovery. Specifically, the shock temperatures associated with typical SN shock velocities (∼10⁴ km s⁻¹) are above 10 keV. Therefore, if the shock is in an optically thin region, the X-ray temperature constitutes a reliable measurement of the shock velocity. The shock velocity depends on the shock temperature (kT) and, assuming an equation of state with γ = 5/3 and an equilibrium between the electrons and protons, is given by (e.g., Gnat & Sternberg 2009)

$$\begin{eqnarray} v_{{\rm sh}} & \approx & \sqrt{\frac{16 kT}{3 \langle \mu _{{\rm p}}\rangle m_{{\rm p}}}} \cr & \approx & 2920 \left (\frac{\langle \mu _{{\rm p}}\rangle }{0.6}\right)^{-1/2} \left (\frac{kT}{10\,{\rm keV}}\right)^{1/2}\,{\rm km\,s}^{-1}. \end{eqnarray} \tag{ 25 }$$

If equilibrium between the electrons and protons is not present, as expected in SN remnants (e.g., Itoh 1978; Draine & Mckee 1993; Ghavamian et al. 2013), then Equation (25) gives a lower limit on the shock velocity. We note that the expected equilibrium timescale between the protons and electrons is of order $6\times 10^{8}(v_{{\rm s}}/3000\,{\rm km\,s}^{-1})^{3}n_{{\rm e}}^{-1}$ s, where n_e is the electron density in cm⁻³ (i.e., roughly given by Equation (16); Ghavamian et al. 2013).

However, if the Thomson optical depth is larger than a few, the X-ray emission becomes more complicated. Katz et al. (2011) and Murase et al. (2011) showed that after the shock breakout in a wind CSM environment, the shock transforms from being radiation dominated to collisionless, and hard X-ray emission should be generated. However, Chevalier & Irwin (2012) and Svirski et al. (2012) argued that the hard X-ray photons will be Comptonized to lower energies, and that when the optical depth is large the X-ray spectrum will have a cutoff above an energy of ∼m_ec²/τ². According to Svirski et al. (2012), the observed energy cutoff of the X-ray photons will be

$$\begin{equation} k_{{\rm B}} T_{{\rm x,obs}} \approx \min {\left [\frac{m_{{\rm e}}c^{2}}{\tau ^{2}}, \frac{3}{16}\mu _{{\rm p}} m_{{\rm p}} v_{{\rm s}}^{2} \right ]}, \end{equation} \tag{ 26 }$$

where the second term in the minimum function is the shock temperature from Equation (25).

Ignoring bound-free absorption, Svirski et al. (2012) estimated that the X-ray luminosity is roughly given by

$$\begin{equation} L_{{\rm X}}(t) \approx L(t) \frac{T_{{\rm x,obs}}(t)}{T_{{\rm e}}(t)} \min \left [1, \frac{\epsilon ^{{\rm ff}}}{\epsilon ^{{\rm IC}}}(t)\right ]. \end{equation} \tag{ 27 }$$

Here ^ff and ^IC are the free–free and inverse-Compton cooling efficiencies, respectively (see Chevalier & Irwin 2012; Svirski et al. 2012), and T_e is the electron temperature (Equation (25)). Equation (27) neglects the effect of bound-free absorption and therefore should be regarded as an upper limit. Furthermore, we note that there is no agreement between different theoretical models on the exact X-ray spectral and flux evolution.

Chevalier & Irwin (2012) define²¹ an ionization parameter as ξ = L/(nr²). This definition is only valid when material above the shock is optically thin. When the optical depth (Equation (7)) is larger than unity, one needs to take into account the fact that the photons diffuse out slower than the speed of light. Since the effective outward-diffusion speed of the photons is ∼c/τ, we define the ionization parameter as

$$\begin{equation} \xi \sim \frac{L}{nr^{2}} \max \left [ \tau, 1 \right ]. \end{equation} \tag{ 28 }$$

However, we stress that this is only an order-of-magnitude estimate of the ionization parameter. Chevalier & Irwin (2012) argue that if the ionization parameter is larger than ∼10⁴, then all the metals (which dominate the bound-free absorption) will be completely ionized, and for ξ ≳ 10² the CNO elements will be completely ionized. Here, an important caveat is that it is not clear if the estimate of Chevalier & Irwin (2012) is valid for high optical depth.

In our model, the rise time is governed by the shock-breakout timescale, and the light curve following shock breakout is given by Equation (11) with m ≈ 10–12 at early times and m ≈ 4 at late times. Alternatively, m ≈ 10–12 and w changes with radius. As a reminder, we note that the value of m at early times is related to the polytropic structure of the stellar envelope (e.g., Matzner & McKee 1999), while m = 4 at late times is obtained from conservation of momentum (Section 4.2).

Figure 1 suggests that the light curve of SN 2010jl can be described as a broken power law, with the break between 180 and 340 days after maximum light. Since both Figure 3 and Figure 4 suggest that the temperature in the first year was roughly constant and close to 9000 K, the bolometric correction is rather small²² and constant. Here we adopt a constant bolometric correction of −0.27 mag, which corresponds to a blackbody spectrum with T = 9000 K. We apply this bolometric correction to the PTF R-band data to obtain the bolometric light curve. Later we test the stability of our solution to this assumption.

A power-law fit depends on the temporal zero point, which in our model is roughly the time of maximum luminosity minus the shock-breakout timescale. However, since the shock-breakout timescale is related to the light-curve rise time, and since we do not have good constraints on the light-curve rise time, we have to estimate the shock-breakout timescale in a different way. Therefore, we fitted the first-year PTF luminosity measurements with a power law of the form L_{0, obs}([t + t_bo]/t_bo)^α, where t is measured relative to the ASAS I-band maximum light (MJD 55,494). Figure 10 shows the fit χ², as well as α₁, as a function of t_bo. Here α₁ is the power-law index of the bolometric light curve in the first year after maximum light. The black arrows indicate t_bo at which the first ASAS detection was obtained, and t_bo derived by fitting the first three ASAS I-band measurements with a t² law (e.g., Nugent et al. 2011). The fit prefers t_bo ≈ 10 day, but t_bo ≲ 25 day is acceptable, while the ASAS early detection indicates t_bo > 15 day.

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Given specific values of κ, m, α, w, and L₀, Equation (19) shows that there is a relation between t_bo and v_bo. Moreover, based on Figure 10 we know that 15 ≲ t_bo ≲ 25 day. Figure 11 shows the solutions of Equation (19) as a function of t_bo and v_bo for various values of m, given the measured values of α₁ (and hence w) and L₀ as a function of t_bo (i.e., Figure 10). Also shown, in blue contours, are lines of equal CSM mass within the break radius (M_br). Here the break radius is defined as the radius of the shock at 300 days—roughly when the observed break in the power-law light curve is detected. Regardless of the exact values of m, t_bo, and v_bo, Figure 11 shows that the CSM mass M_br ≳ 10 M_☉ (see also Equations (21) and (22)). It also suggests that M_br ≲ 16 M_☉, but the upper limit is somewhat weaker due to several uncertainties that are discussed next.

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Assuming = 1/4 and κ = 0.34 cm² g⁻¹, Table 7 presents the measured values of L₀ and α and the calculated values of w, K, r_bo, v_bo, and M_br, as a function of the assumed t_bo and m. Regardless of the exact value of t_bo, we find that 2 ≲ w ≲ 2.2—close to the value expected from a wind profile. We note that our approach allows us to measure w rather than assume its value. For the rest of the discussion we will adopt t_bo = 20 days and m = 10. In this case, the value of K is translated to a mass-loss rate of

$$\begin{equation} \dot{M}\approx 0.8 \frac{v_{{\rm CSM}}}{300\,{\rm km\,s}^{-1}}\,{M}_{\odot }\,{\rm yr}^{-1}, \end{equation} \tag{ 29 }$$

where we normalized the CSM velocity by the highest-velocity Gaussian component in the spectra. This tremendous mass-loss rate is discussed in Section 6.

Figure 11 assumes = 1/4. The reason for this choice is that it is expected that a shock propagating through the CSM will convert only the thermal energy stored in the ejecta to radiation. The thermal energy of the ejecta is roughly half of its kinetic energy (e.g., Nakar & Sari 2010). In addition, since the CSM is optically thick, at early times half the photons probably diffuse inward (and will be released at later times), therefore taking the efficiency roughly another factor of two down. However, at late times we expect the efficiency to increase to about 1/2—therefore, may change slowly with time. Indication for this may be detected as a small deviation from the power-law decay in the first year (Figure 1). We note that the exact value of has a relatively small effect on the results. For example, assuming = 0.1 ( = 0.5) gives M_br > 15 M_☉ (M_br > 8 M_☉).

Another assumption that goes into Figure 11 is that the bolometric correction in the first year is constant. However, as seen in Figures 1 and 4, there are some indications for variations in the bolometric correction. Given this uncertainty, we investigated the effect of variable bolometric correction on our results. Specifically, we assumed that the effective temperature of the photosphere evolves as T = T_bo(t/t_bo)^β, where T_bo is the observed temperature at shock breakout (see Section 2). Assuming T_bo = 9000 K and t_bo = 20 day, we corrected our light curve according to the bolometric correction we get from the temperature, and we investigated the effect of β on our results. We find that for −0.2 < β < 0.1 the estimate on M_br does not change by more than 20%. Figures 3 and 4 suggest that |β| ≲ 0.1. Another unknown factor is the opacity κ. Increasing κ to 0.5 cm² g⁻¹ will set M_br ≳ 8 M_☉.

The entire analysis presented here assumes that the CSM and ejecta have spherical symmetry. This is likely not the case (e.g., Patat et al. 2011). However, an order-of-unity deviation from spherical geometry will not change the results dramatically since the integrated luminosity depends on the total mass of the CSM. In order for the results (and specifically the M_br estimate) to change significantly, an extreme geometry is probably required. We cannot rule out such a scenario. However, given that our model explains the observed broken-power-law behavior, finds values of m and w that are consistent with expectations, and successfully predicts the observed shock velocity (see also Sections 5.2 and 5.3), we conclude that our description is correct. Another important point may be the clumpiness of the CSM. However, if the Chevalier (1982) solutions are still valid on average, our results are correct, as they depend on the global (average) properties of the CSM and ejecta. Therefore, we conclude that our main result that the mass in the CSM of SN 2010jl is in excess of about 10 M_☉ is robust. Finally, we note that Svirski et al. (2012) predict that at early times the color temperature will evolve slowly with time. This is roughly consistent with the observations of SN 2010jl.

Around 300 days after maximum light, the optical light curve of SN 2010jl shows a break in its power-law evolution, and the R-band power-law index becomes α₂ ≈ −3. The change in power-law slope at late times may have three possible explanations: (1) we reached the snow-plow phase, and therefore m changes to about 4 (Svirski et al. 2012); (2) the shock became slow cooling and therefore the light curve drops rapidly; and (3) the shock reached the end of the CSM, or in other words, the CSM density profile became steeper than r⁻². Next, we will test these possibilities and find that the snow-plow phase option is the most likely. We note that the measurement of M_br is not affected by the nature of the break.

Our solution suggests that the CSM density at r_br is ∼10⁹ cm⁻³. Given this very high density, the shock must be fast cooling and option (2) can be ruled out (Equation (15); see also Figure 12, panel (d)). Assuming m = 10, Equation (12) suggests that in order to get the observed value α₂ ≈ −3, we require w ≈ 5. However, the Chevalier (1982) self-similar solutions are invalid for w > 3. Nevertheless, the steep value of α₂ probably means that if m ≈ 10, w > 3. We note that in this case, the shock will accelerate, and at late times we expect v_s ≳ 4000 km s⁻¹ (Figure 12). This is somewhat higher than the velocity suggested by our NuSTAR observations (see Section 5.3).

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Given the solution presented in Figure 11 (using α₁), integration of the mass to the break radius gives M_br ≳ 10 M_☉. Normal SN explosions have an ejecta mass that is similar, to an order of magnitude, to our derived CSM mass. Therefore, it is likely that the ejecta collected a CSM mass that is equivalent to its own mass and the system reached the snow-plow phase, and hence there is a natural explanation to the change in α without changing w (at least not in a major way). Of course, it is possible that during t_br the values of both m and w are changing. This idea requires a coincidence between two independent phases, and therefore we will not discuss it further.

Assuming w = 2 and m = 4, we expect α₂ ≈ −3/2 (see also Svirski et al. 2012), while we observed α₂ ≈ −3. There are several possibilities to explain this. First, at late times (a year after peak brightness) there may be significant evolution in the bolometric correction. Interestingly, the late-time spectra (see Figure 2) suggest that the SN becomes bluer at late times. We note, however, that these late-time measurements are affected by the underlying star-forming region and are therefore uncertain. In addition, the missing radiation may be emitted in the X-ray band. We find that if the intrinsic unabsorbed X-ray luminosity of the SN is ∼20 times higher than observed, the contribution of the X-ray luminosity to the bolometric light curve will modify α₂ to about −3/2.

A second possibility is that the system is approaching the slow cooling stage and some of the energy is not released efficiently as optical photons. Our estimate suggests that at late times the cooling timescale is increasing to about 10% of the dynamical timescale (Figure 12). Therefore, it is possible that the shock starts to be nonradiative, hence explaining the steeper than expected power-law slope. To summarize the issue, we suggest that the most likely explanation to the discrepancy between the observed and predicted value of α₂ is that at late times there is a substantial bolometric correction, and possibly the shock is becoming nonradiative. Unfortunately, we do not have reliable multi-band or spectroscopic observations during the second year.

Based on our simple model, Figure 12 shows the evolution of the various parameters as a function of time. Panel (b) indicates that even at late times, about 3 yr after maximum light, the density of the CSM at the shock radius is of order a few times 10⁸ cm⁻³. Interestingly, the Thomson optical depth above the shock, 3 yr after maximum light, is decreased to roughly unity. This may explain why the visible-light spectrum of the SN is becoming bluer, as the region heated by the shock is becoming more exposed and the photons emitted in the shock region are affected by less and less processing. The free–free optical depth [panel (f)] above the shock at 10 GHz, 3 yr after maximum light, is τ_ff ≈ 10⁵, assuming that the electron temperature above the shock is 10⁴ K. Therefore, naively, radio emission is not expected in the near future. However, if the electron temperature just above the shock is significantly higher and the CSM cocoon is terminated at a few times the shock radius, then τ_ff can be small enough and radio emission would be detected. Finally, we note that the cooling timescale divided by the hydrodynamical timescale [panel (d)] suggests that at late times, the system may approach slow cooling, so some energy losses (not in optical radiation) are expected.

An interesting point to note is that Figures 3 and 4 show that at late times the effective blackbody radius is decreasing. Svirski et al. (2012) argue that at late times the fraction of the energy released in X-rays is increasing (as seen in SN 2010jl). In this case, the optical photons will deviate from a blackbody spectrum as fewer photons are available in the optical, and this can generate an apparent decrease in the effective blackbody radius. In general, this effect should caution against the use of blackbody fits to estimate the photospheric radius of such explosions.

We still do not have a good theoretical understanding of the expected X-ray spectral evolution from optically thick sources (e.g., Katz et al. 2011; Chevalier & Irwin 2012; Svirski et al. 2012). Another problem is that the X-ray spectral observations are hard to model. The reasons are the low number of photons, contamination from nearby sources, and the degeneracy between the free parameters in the various models. Nevertheless, it is interesting to compare the rough expectations with the observations. Given these issues, our approach is to use the model we constructed based on the optical data to make some predictions for the X-ray band, and to compare the X-ray observations with these predictions. Especially interesting are the NuSTAR+XMM observations, which cover a large energy range and were taken when the Thomson optical depth is expected to be relatively low, τ ≈ 3. Here we discuss the bound-free absorption, the X-ray flux, and the X-ray spectrum.

Figure 12 shows that the predicted column density above the shock is very large, ∼10²⁶ cm⁻² during the shock breakout, dropping to ∼10²⁵ cm⁻² during our XMM+NuSTAR observations. These predicted column densities are larger, by about two orders of magnitude, than the bound-free column densities suggested by Chandra et al. (2012a) at early times. A plausible explanation is that the CSM above the shock is ionized by the SN radiation field. Indeed, panel (g) in Figure 12 suggests that at early times the ionization parameter (Equation (28)) is >10² erg cm s⁻¹, and possibly as high as ∼10⁴ erg cm s⁻¹. Such a large value is enough to ionize all the metals in the CSM (Chevalier & Irwin 2012). However, at late times, the ionization parameter is only ∼10² erg cm s⁻¹, which may leave some bound electrons in heavy elements.

The next simple test is to use the order-of-magnitude estimate in Equation (27) to predict the X-ray flux as a function of time. The prediction is shown in Figure 6 as a gray line. At early times, about 100 days after the SN maximum visible light, the prediction is consistent with the observations. About a year later, the X-ray prediction is a factor of four higher than the observations, while around 2.5 yr after maximum visible light, the predicted X-ray luminosity is a factor of two higher than observed. We note that Equation (27) is an order-of-magnitude estimate of the luminosity in the entire X-ray band (including soft and hard (>10 keV) X-rays), and that it does not take into account the bound-free absorption, which, even if not very high, still can affect the emission of soft X-rays considerably. For example, for N_H = 10²² cm⁻², the bound-free optical depth (e.g., Morrison & McCammon 1983) at 0.5 keV (1 keV) is 7.3 (2.4), which will decrease the observed X-rays at this energy by a factor of 1600 (11).

According to Svirski et al. (2012), at early times we expect that the cutoff energy will be around m_ec²/τ², while when the optical depth decreases to roughly a few, we expect that the cutoff energy will represent the shock temperature (Equation (26)). Figure 13 shows the predicted cutoff energy as a function of time. Also plotted are the NuSTAR+XMM measured X-ray temperatures based on the various fits (Table 6) and assuming temperature equilibrium between the ions and the electrons. If equilibrium is not present, then our measurement is only a lower limit on the shock velocity.

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Figure 13 suggests that the NuSTAR+XMM observation measures the shock temperature, and hence the shock velocity. The three models in Table 6 in which the faint nearby sources were removed suggest a shock velocity with 1σ confidence interval in the range 1900–4500 km s⁻¹. We suggest that the most physically motivated model is the power-law model with exponential cutoff, in which the power-law index is set to Γ = 1. The reason is that below the cutoff energy we speculate that free–free processes, with a spectrum n(E)∝E⁻¹, will dominate the emission (Svirski et al. 2012). This model suggests an exponential cutoff energy $kT=10.1_{-1.6}^{+2.1}$ keV, which translates to v_s ≈ 2900 ± 300 km s⁻¹. However, if the ions and the electrons are not in equilibrium, all we can say is that the shock velocity is larger than ∼2000 km s⁻¹. This measured shock velocity is in agreement with the predicted shock velocity of ∼2600 km s⁻¹ (Figure 12, panel (h)). Under the assumption that the SN is powered by interaction, by comparing the kinetic energy to the integrated luminosity, the X-ray-derived velocity, along with the integrated bolometric luminosity, can be used to roughly determine the CSM mass. While lacking the exact prefactors we derived in Section 4.2, we obtain an order-of-magnitude estimate of the CSM mass—∼10 M_☉.

We estimate that during the NuSTAR+XMM observation the ionization parameter was ∼10² erg cm s⁻¹. According to Chevalier & Irwin (2012), this value is not enough to ionize all of the metals. Therefore, our estimate of the ionization parameter is in conflict with the value of the bound-free column density we deduced from the NuSTAR+XMM observations. Possible solutions include the existence of even harder photons in the shock, or that the estimate of the effective ionization parameter at high optical depth is wrong.

Given the difficulties in modeling the early-time data obtained by Chandra, we attempt to fit these observations with a power law having an exponential cutoff as predicted by Equation (26) (Figure 13)—1.5 keV and 15 keV, for the Chandra first and second epochs, respectively. While the fit to the second epoch has an acceptable C-statistic (see Table 6), fitting the first epoch while freezing the cutoff energy at 1.5 keV failed. Given the unknowns associated with the X-ray emission at such high optical depth (τ ≈ 20), we do not consider this to be a problem for our model.

We note that the marginal detection of the Kα line in the first Chandra epoch (Section 3) is not naturally explained in our model. In the context of our model, the Kα line must be generated at relatively large radii where the optical depth is low.

The spectra of SN 2010jl show a variety of emission lines. Based on spectropolarimetric observations, Patat et al. (2011) suggested that the Balmer lines form above the photosphere. Therefore, the emission from the Balmer lines will not constitute a good estimate of the mass in the CSM (see discussion in Ofek et al. 2013c). Smith et al. (2012) show that the line shape evolves with time, presumably due to the formation of dust.

Nevertheless, the width of the Balmer lines gives us an estimate of the CSM velocity. This is important in order to estimate when the CSM was ejected from the SN progenitor. Given the velocities of the Balmer lines of SN 2010jl (between ∼70 km s⁻¹ and ∼300 km s⁻¹; Section 2.2), and the typical radii of the CSM of ∼2 × 10¹⁶ cm, we estimate that the CSM was ejected from the progenitor ∼10–100 yr prior to the explosion. Given this prediction, we searched for archival images at this sky location. PTF images of the SN location taken about 200 days prior to explosion did not reveal any pre-explosion outburst; see E. O. Ofek et al. (in preparation) for details.