A MULTI-WAVELENGTH INVESTIGATION OF THE RADIO-LOUD SUPERNOVA PTF11qcj AND ITS CIRCUMSTELLAR ENVIRONMENT

On 2011 November 1 (hereafter all times are given in UTC unless otherwise stated), we discovered PTF11qcj in an R-band image from the 48 inch Samuel Oschin telescope at Palomar Observatory (P48), which is routinely used by the Palomar Transient Factory (PTF¹⁵; Law et al. 2009; Rau et al. 2009). The SN was also detected by our autonomous discovery and classification framework called Oarical (Bloom et al. 2012). PTF11qcj is located at $\alpha = 13^{{\rm h}}13^{{\rm m}}41\buildrel{\mathrm{s}}\over{.}51$, δ = +47°17'570 (J2000), at an angular distance of ≈068 from the nucleus of the galaxy SDSS¹⁶ J131341.57+471757.2, at z = 0.0287 (Figure 1). Assuming H₀ = 71 km s⁻¹ Mpc⁻¹, Ω_M = 0.27, Ω_Λ = 0.73, this redshift correspond to a luminosity distance of d_L≅124 Mpc. The SN was visible at a magnitude of g ≈ 17.6 in a previous exposure taken in g-band on 2011 October 23. It was not detected in an R-band image taken with the P48 on 2011 August 17.

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Subsequent observations with the P48 were conducted with the Mould-R and Gunn-g filters (Figure 1). Photometry (Table 2) was performed relative to the SDSS r-band and g-band magnitudes of stars in the field (York et al. 2000), using our custom pipeline which performs image subtraction followed by point spread function (PSF) photometry on stacks of PTF images extracted from the PTF Infrared Processing and Analysis Center (IPAC) archive (Maguire et al. 2012; Ofek et al. 2012, 2013c). The flux residuals from individual subtracted images were binned, and then converted to magnitudes. The errors were estimated from the standard deviation of the photometric measurements in each bin (Figure 2).

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Multi-color optical (gri) optical light curves were also obtained using the Palomar 60 inch telescope (P60; Cenko et al. 2006) and the RATCAM optical imager on the robotic 2m Liverpool Telescope (LT; Steele et al. 2004) located at the Roque de Los Muchachos Observatory on La Palma. P60 and LT photometry was extracted using the same pipeline described above (Table 2).

After the discovery of PTF11qcj with P48, we triggered a spectroscopic follow-up campaign¹⁷ (Figure 3). A first spectrum was taken on 2011 November 5 using OSIRIS on the Gran Telescopio Canarias¹⁸ (GTC), with a 1'' long-slit and the R300R grism (≈5000–10000 Å). The exposure time for this observation was ≈240 s. Two other spectra were obtained on 2011 November 7 and on 2011 December 21 using ISIS on the William Herschel Telescope¹⁹ (WHT), with a 1'' long-slit, the R300B grating plus ≈4500 Å central wavelength on the blue side; the R158R grating plus ≈7500 Å central wavelength on the red side. The exposure times were of 600 s and 1800 s, respectively for the two observations. On 2011 November 26 and 2011 December 31, we obtained two spectra using LRIS (Oke et al. 1995) mounted on the Keck-I 10 m telescope.²⁰ For both spectra we used a 1'' slit, with the 400/8500 grating plus ≈7850 Å central wavelength on the red side, and the 400/3400 grism on the blue side. The exposure times were 1200 s for the blue side and 720 s for the red side, on 2011 November 26; 900 s for the blue side and 840 s for the red side, on 2011 December 31. Finally, on 2012 March 20, we took a last spectrum with DEIMOS (Faber et al. 2003) mounted on the Keck-II 10 m telescope, using a 08 slit, and the 600ZD (600/7500) grism centered at 7000 Å. The wavelength coverage was 4350–9650 Å, the exposure time was 2 × 1200 s, and the spectral resolution was ≈3.4 Å.

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Our first spectrum obtained on 2011 November 5 with the GTC (see Figure 3) is found to be most similar to a type Ic SN by the Superfit software (Howell et al. 2005). A comparison with template spectra shows relatively high velocities (around 22,000 km s⁻¹ derived from the Ca ii IR triplet minimum), that are driving Superfit to find the BL SN 1998bw (at ≈19 day post-maximum) as the best fit (Figure 3, red line). However, more normal type Ic SN spectra (e.g., SN 1994I, black line in Figure 3) could also provide an acceptable fit, provided these are artificially blue shifted by ∼5000 km s⁻¹ (dotted line in Figure 3). In fact, the absorption features profiles are more similar to SN 1994I than to SN 1998bw. Aside from nebular Hα emission from the host galaxy, no strong narrow lines are seen in this first spectrum.

The WHT spectrum taken 2 days later (Figure 4, black line) extends to bluer wavelengths. No narrow lines are seen except for common host galaxy lines (Balmer series, O[iii] and O[ii]), and the continuum shape is red (with a prominent decreasing trend beyond ≈5500 Å) as is common for SNe of type Ic. As the SN flux declines, most notably in the spectra taken on and after 2011 November 26, the SN signal is strongly mixed with underlying emission from the host galaxy.

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Given the classification as Ic-BL SN, we commenced, in addition to our optical spectroscopic follow-up, an extensive radio, X-ray, and IR follow-up campaign (Sections 2.4–2.6).

We searched for evidence of pre-explosion activity in the PTF images taken before the discovery date of PTF11qcj. To this end, we constructed a reference frame from 30 images taken over the period 2013 March–May (≈16 months after the SN explosion, when PTF11qcj was no longer detected by P48). We geometrically resampled this reference image to the coordinate frame of every image of the PTF11qcj field taken before and during the SN explosion (hereafter, the "science images"). We matched the PSFs of the resampled reference image and of the science images using a non-parametric kernel (a process that usually results in the degradation of the reference image quality; e.g., Bramich 2008), and subtracted the reference image from each science image. Note that in this process the science frames are never geometrically resampled. PSF photometry was then performed on the difference images at the position of PTF11qcj, which was measured accurately from the difference images in which the SN is present, with the shape of the PSF measured from either the pre-subtraction science or reference images. This results in a flux and flux uncertainty computed at the position of PTF11qcj for every science image, regardless of whether the SN was present or not (see also Ofek et al. 2013c).

In Figure 5 we show the light curve data obtained for the pre-SN season of 2009 May–2010 September (top panel), and for the post-SN season of 2012 January–2013 January (bottom panel). In each of these figures, the black circles show measurements made using PSF fitting at the SN position on the individual difference images; the green area shows the weighted-mean flux in time bins, and the extent of the error on the mean; finally, the blue area shows the typical range of PSF counts obtained by performing PSF photometry on other galaxies in the PTF11qcj field, having similar magnitude as the PTF11qcj host (±2 mag), and at the same isophotal radius as the one at which PTF11qcj occurred in its own host. Thus, while the blue areas in these figures track any global image subtraction problems, the green areas track departures in the mean flux from zero and allow us to assess the presence of any pre-SN activity.

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As evident from Figure 5, during the pre-SN season (top panel) we find tentative evidence for periods of activities in between 2009 May and July (see also panel (B) in Figure 6), and between 2010 May and June (see also panel (D) in Figure 6). These active periods are separated by a period of non-detection, between 2009 November and 2010 May (see also panel (C) in Figure 6). Specifically, we note that during the active period of 2009 May–July, the possible precursor from PTF11qcj is as bright as the SN was during 2012 April–May (this is also evident comparing panels (B) and (F) in Figure 6). We finally checked whether any evidence for excess flux was still present during the post-SN season (after the SN itself faded below the P48 detection limit). As evident from the bottom panel of Figure 5, at t ≳ 1 yr after explosion no excess flux is detected at the position of PTF11qcj. This further supports the evidence for precursor activity during the pre-SN season.

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On 2011 November 15, we started a long-term monitoring campaign of PTF11qcj (along with calibrators J1327+4326 and 3C 286) with the Karl G. Jansky Very Large Array²¹ (VLA; Perley et al. 2009) in its D, DnC, C, CnB, and A configurations, under our Target of Opportunity programs.²² VLA data were reduced and imaged using the Common Astronomy Software Applications package.

The light curves of PTF11qcj at frequencies of 2.5 GHz, 3.5 GHz, 5 GHz, 7.4 GHz, 13.5 GHz, 16 GHz are reported in Table 3. The VLA measurement errors are a combination of the rms map error which measures the contribution of small unresolved fluctuations in the background emission and random map fluctuations due to receiver noise, and a basic fractional error (here estimated to be of ≈5%) which accounts for inaccuracies of the flux density calibration (see, e.g., Weiler et al. 1986; Ofek et al. 2011).

We also observed the field of PTF11qcj (together with the test calibrator J1203+480) using the Combined Array for Research in Millimeter-wave Astronomy²³ (CARMA), at a frequency of 93 GHz. Bad weather caused de-coherence which resulted in a limited follow-up of this source with CARMA. Nevertheless, the data collected on 2011 November 19 and 2011 November 26²⁴ both resulted in a detection of PTF11qcj (Table 3).

We have carried out an X-ray monitoring campaign of PTF11qcj with Chandra²⁵ (Weisskopf et al. 2002) and Swift²⁶ (Gehrels et al. 2004). All our Swift-XRT observations yielded non detections (see Table 4), while Chandra detected PTF11qcj in three epochs.²⁷ The results of our X-ray follow-up are reported in Table 4.

We reduced Swift data adopting an on-source circular aperture of 72 and assuming that 37% of the photons are within this aperture. In addition to the results obtained for single-epoch observations (see Table 4), we also provide the upper-limit obtained by co-adding all of the Swift non-detections.

For Chandra observations, we adopted a circular aperture of 2'' fixed on the optical position of PTF11qcj. The background has been estimated from nearby portions of the image. Net source counts have been derived by subtracting from the counts measured in the on-source region the number of counts measured in the background region, scaled by the ratio of the on-source to background region areas. We caution that the background estimated from nearby portions of the image does not include any diffuse or point sources associated with the host galaxy itself. In this sense, the count rates indicated in Table 4 are upper-limits on the flux from PTF11qcj. Since the first epoch is brighter than the third, we can however be confident that we have seen some emission from PTF11qcj during the first epoch.

We observed the position of PTF11qcj with Spitzer²⁸ (Fazio et al. 2004) on two epochs²⁹ (on 2012 March 28.747 and 2012 June 25.643; Table 5). Data were reduced using the standard Spitzer/IRAC pipeline. Subsequently, aperture photometry was performed using a radius of 4 pixels (i.e., 24) for PTF11qcj, and an annulus of radii 8–16 pixels (i.e., 48 to 96)for the sky background. Calibration and aperture corrections were as per the zero points listed in the Spitzer/IRAC handbook. However, there is clearly a host galaxy contribution at the position of PTF11qcj (see Figure 7): for this reason, we consider the flux measurement obtained with Spitzer as an upper-limit to the IR flux of PTF11qcj.

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On 2012 March 28, we also observed the field of PTF11qcj in K_s-band with the Palomar 200 inch telescope (P200). The contribution of the host galaxy is clearly evident at the position of PTF11qcj in the P200 image (Figure 1, lower-right panel). Performing aperture photometry with respect to two reference stars in the field (2MASS 13134736+4719100 and 13135239+4717152), we derive a K_s-band photometric data point that, as for the Spitzer observations, we consider as an upper-limit to the SN flux in this band (Table 5).

As discussed in Section 2.2, our first spectrum of PTF11qcj shows relatively high velocities (around 22,000 km s⁻¹). The spectral similarity between PTF11qcj at ≈13 days after g-band discovery (2011 November 5, or 55870 MJD), and the Ic-BL SN 1998bw at ≈19 days after peak (Figure 3), suggests that we discovered PTF11qcj around peak time or shortly thereafter. We thus infer an explosion date of 55830 MJD ≲ t_e ≲ 55850 MJD, assuming an explosion-to-peak timescale of ≈7–20 days, as derived from well studied Ib/c SN samples (e.g., Figure 5 in Drout et al. 2011).

The continuum shape observed in the spectrum of PTF11qcj taken on 2011 November 26 (Figure 4, orange line), is flat or even rising toward the blue side down to ≈3500 Å. Indeed, this spectrum and the ones taken after 2011 November 26 show the SN signal strongly mixed with underlying emission from the host galaxy. We have attempted to subtract this contribution using a spectrum of the host galaxy obtained by the SDSS survey.³⁰ We scaled the host spectrum until the strong emission lines approximately matched in intensity the lines observed in our SN spectra, rebinning the spectra as needed to match the line width, and then subtracted the host spectrum (resampled onto the SN spectrum wavelength scale) from our observed spectra. These host-subtracted spectra are plotted in Figure 8.

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We run the Superfit program on the host-subtracted spectra and find a reasonable match to spectra of SNe Ib/c of comparable ages. The earlier-epoch spectra seem to fit SNe of the Ic-BL class, while later-epoch ones are better matched by more normal SNe Ib/c. We stress that the host-subtraction procedure we attempted is quite rough due to the irregular nature of the host and an apparent bright star-forming knot seen in pre-explosion SDSS imaging, co-located with the SN. Still, in view of the matches found between the host-subtracted spectra and observations of other events, it is certainly reasonable to assume that PTF11qcj evolved in a similar way to other SNe Ib/c.

Finally, the late-time spectrum (2012 March 20; Figure 4, purple line) is nebular, with O i 6300 Å and Ca ii emission lines. After subtracting the host contribution, this spectrum appears quite similar to late spectra of SN 1998bw, again within the large uncertainties related to host contamination (Figure 9).

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In order to estimate the host-galaxy extinction toward the SN, we measured the Na D absorption doublet in our high signal-to-noise and high-resolution Keck/DEIMOS spectrum obtained on 2012 March 20. We detect weak features consistent with the Na D doublet at the host redshift, for which we measure the following equivalent width values using the IRAF/splot routine: EW_5890 Å = 0.087 Å, and EW_5896 Å = 0.265 Å. Since the 5890 Å component is always stronger (opposite what we see), and the FWHM of the 5896 Å component is almost three times larger than that of the 5890 Å line, we assume that the 5896 Å doublet component is probably contaminated by a noise fluctuation. Using the relation of Poznanski et al. (2012, see their Figure 7) we estimate an extinction of E_{B − V} = 0.03 (A_R ∼ 0.08 mag) using the 5890 Å component. We thus apply this small correction, together with the correction for Galactic extinction, to our photometry plotted in Figure 2.

In recent years pre-SN eruptions have been detected from type IIn SNe (e.g., Mauerhan et al. 2013; Prieto et al. 2013; Ofek et al. 2013c) and type Ibn SNe (e.g., Nakano et al. 2006; Pastorello et al. 2007; Foley et al. 2007). The total ejected mass in these outbursts is estimated to be a fraction of a solar mass (e.g., Immler et al. 2008; Ofek et al. 2013b, 2013c). However, pre-SN eruptions were never detected before from other types of SNe.

Our pre-explosion images of PTF11qcj show tentative evidence for the existence of pre-explosion activity, particularly in terms of a precursor event reaching m_r ≈ 22.3 mag during 2009 May–July, ≈2.5 yr before the SN discovery. Marginal evidence for further pre-SN activity (m_r ≈ 23 mag) is found during 2010 May–June, ≈1.5 yr before discovery.

At the distance of PTF11qcj (d_L≅124 Mpc), correcting for local (see Section 3.1) and Galactic (A_r = 0.024 mag; Schlafly & Finkbeiner 2011) extinction at the SN position, m_r ≈ 22.3 mag (m_r ≈ 23 mag) corresponds to M_r ≈ −13.3 mag (M_r ≈ −12.6 mag). For comparison, the precursor observed 2 yr before explosion of the prototype Ibn SN 2006jc reached M_r ≈ −14 mag (Pastorello et al. 2007). Based on our pre-SN images, we can exclude a precursor as bright as the one observed for SN 2006jc during the epochs covered by our observations (see Figure 5).

As discussed in Section 3.1, we likely discovered PTF11qcj around peak time. Based on our first detection with P48 (m_r ≈ 17.6 mag), we can constrain the peak r-band magnitude to be m_{r, peak} ≲ 17.6 mag. Correcting for extinction (Galactic plus local), our upper-limit on the peak r-band magnitude of PTF11qcj (m_{r, peak} ≲ 17.6 mag) corresponds to an absolute r-band peak magnitude of M_{r, peak} ≲ −18 mag. This limit is compatible with the average R-band peak magnitudes found in recent systematic studies of Ib/c SNe (Drout et al. 2011).

In the 15 days following the P48 discovery, the P48 r-band light curve of PTF11qcj decreased by Δm_r ≈ 0.9 mag. The optical light curve of the GRB-associated SN 1998bw dropped by 1.1 mag in the first 15 days after peak (McKenzie & Schaefer 1999). Note that for PTF11qcj the 15 day magnitude drop is calculated with respect to the discovery time rather than the peak time; however, we likely discovered the PTF11qcj around peak. In Figure 2, we show a comparison between the PTF11qcj r-band light curve, and the light curves of the Ic SN 1994I (Richmond et al. 1996), and the Ic-BL SN 1998bw (Clocchiatti et al. 2011), scaled to the distance of PTF11qcj. The explosion time of PTF11qcj is taken as ≈15 days before peak, where the last is assumed to be coincident with our g-band discovery (as suggested by our spectral analysis). A rise time of ≈15 days is a factor of ≈1.3 shorter than the rise time of SN 1998bw (≈20 days). Indeed, the SN 1998bw R-band light curve compressed by a factor of ≈1.33 (Figure 2, orange dashed-line) seems to match the one of PTF11qcj.

The early-time (t ≲ 60 days) evolution of Ib/c SNe is usually referred to as the photospheric phase. This phase is characterized by a high optical depth, and a post-maximum light curve decay rate slower than the ⁵⁶Ni-to-⁵⁶Co decay. During this phase, we can use the light curve properties to estimate the physical parameters of the SN explosion (Arnett 1982): the ⁵⁶Ni mass, the ejecta mass, and the kinetic energy. From Figure 22 in Drout et al. (2011), for Δm_{15, r} ≈ 0.9 mag and M_{r, peak} ≲ −18 mag, we roughly estimate M_Ni ≳ 0.1 M_☉ and (E_K/10⁵¹ erg)^−1/4(M_eje/M_☉)^3/4 ≈ 1.2. For a photospheric velocity of 22,000 km s⁻¹ (Section 3.1), these values imply (e.g., Equations (1) and (2) in Drout et al. 2011) M_eje ≈ 2.5 M_☉ and E_K ≈ 8 × 10⁵¹ erg. While these are very tentative estimates due to the uncertainties in the PTF11qcj peak-time and magnitude, these values are compatible with the ones observed for other Ib/c SNe. Particularly, the energy is consistent with what observed for BL-Ic and engine-driven events (Drout et al. 2011).

Our radio monitoring campaign reveals that PTF11qcj is among the most radio-luminous Ib/c SNe. With a peak luminosity of L_5GHz ≈ 10²⁹ erg s⁻¹ Hz⁻¹, PTF11qcj is comparable to the GRB-associated SN 1998bw (Figure 10; Kulkarni et al. 1998). From the presence of incoherent radio emission, assuming a non-relativistic source, we can derive a lower-limit on the size of the emitter (Θ), by imposing that the brightness temperature T_B does not exceed the equipartition value of T_{B, eq} ≈ 10¹¹ K (Readhead 1994; Kulkarni et al. 1998):

$$\begin{eqnarray} T_{B}&=&\frac{c^2f_\nu }{2k_B\nu ^2\pi \Theta ^2}\lesssim 10^{11}\,{\rm K}\Rightarrow \Theta \gtrsim 2.1\nonumber\\ &&\times\left(\frac{f_{\nu }}{\mu {\rm Jy}}\right)^{1/2}\left(\frac{\nu }{\rm GHz}\right)^{-1}\,\mu {\rm as,} \end{eqnarray} \tag{ 1 }$$

where k_B is Boltzmann's constant, ν is the observing frequency, f_ν is the observed flux density, and Θ is the angular diameter of the emitting region. As evident from the above equation, the largest lower limits are obtained for the lowest frequencies (and highest fluxes). Adopting the flux measured at 2.5 GHz around day 100 (when the low-frequency radio flux is around peak), we get $\Theta _{\rm 100\,{\rm days}}\gtrsim 33\,\mu$as. At the distance of PTF11qcj, this corresponds to a size of 2r ≳ 6 × 10¹⁶ cm, or an average expansion speed $r/{\rm 100\,{\rm days}}\gtrsim 0.1$ c. We note that typical average speeds measured for the fastest moving ejecta of non GRB-associated Ib/c SNe are in the range (0.03–0.3)c (Berger et al. 2003; Wellons et al. 2012).

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A remarkable feature of PTF11qcj is that its radio light curves do not show a smooth evolution (see Figure 10). This behavior, while difficult to model, is not completely unusual. Indeed, Soderberg et al. (2006b) found that ≈50% of radio SNe show evidence for abrupt light curve variability, including abrupt steepening, abrupt rise, or episodic variations (≲ than a factor of 2 in flux) of the radio light curve. These are typically explained as inhomogeneities in the CSM. We caution, however, that data corresponding to the most abrupt flux variation observed for PTF11qcj—a factor of ≈2 in about 1 day between ≈55949.6 and ≈55950.4 MJD—were collected in 2012 while the VLA antennas were being moved from the D to the C configuration.

Radio SNe are known to emit non-thermal synchrotron radiation. According to the model proposed by Chevalier (1982), the relativistic electrons and amplified magnetic fields necessary for synchrotron emission are produced in the SN blast wave during the interaction with an ionized CSM. The last is usually assumed to be emitted via a constant mass-loss rate, constant velocity wind (i.e., $\rho _{\rm CSM}= \dot{M_w}/(4\pi v_wr^2)$) from a massive progenitor. The observed radio emission is characterized by a smooth turn-on first at higher frequencies, and later at lower frequencies, usually explained as temporally decreasing self-absorption as the shock propagates toward lower density regions. The initial turn on may also be related to free–free absorption in the ionized CSM (Chevalier 1998). While external free–free absorption from a homogeneous medium is predicted to cause an exponential rise of the low-frequency radio flux, a power-law is characteristic of self-absorption, internal absorption from thermal absorbing gas mixed into the synchrotron emitting gas, or free–free absorption from a clumpy external medium (Weiler et al. 1990).

In Figure 12 we show the observed spectral indices β (with f_ν∝ν^β) for PTF11qcj, as derived from VLA observations in adjacent sub-bands. The observed spectral indices at early times do not exceed the limiting value of 2.5 expected in the case of synchrotron self-absorption. In what follows, we thus assume the simplest "standard" synchrotron-self absorption model. However, we stress that alternative scenarios invoking internal free–free absorption, or free–free absorption from a clumpy CSM, may also explain such spectral indices and/or contribute to the attenuation mechanisms during the earlier part of our observations (Chevalier 1998; Weiler et al. 1990).

To estimate the physical parameters of PTF11qcj, we follow the formulation by Soderberg et al. (2005) for synchrotron self-absorbed emission arising from sub-relativistic SN ejecta expanding in a smooth CSM with a power-law density distribution. Given the observed variability of PTF11qcj, we do not expect smooth hydrodynamic models to provide a perfect fit to its radio observations. Nevertheless, we attempt such modeling to provide a tentative estimate of the bulk properties of PTF11qcj and its surrounding environment. Hereafter, we consider all the data points collected with the VLA before 56101 MJD. On and after this epoch, a radio re-brightening is evident (possibly associated with an enhanced density CSM; Figures 11 and 13). Since late-time radio observations of PTF11qcj aimed at confirming and tracking this phase of enhanced emission are on-going at the time of writing, here we concentrate on the interpretation of the earlier epochs.

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Within the synchrotron self-absorbed model, at any time t the observed synchrotron emission originates from a shell of shock-accelerated electrons, with radius r and thickness r/η. The shell expands spherically while following a self-similar evolution in the interaction with a smooth CSM. The electrons are accelerated into a power-law energy distribution N(γ)∝γ^−p, with γ ≳ γ_m, and carry a fraction _e of the energy density of the ejecta. A fraction _B of such energy goes into magnetic fields. The temporal evolution of the radius (r), magnetic field (B), minimum Lorentz factor (γ_m), and of the electrons-to-magnetic field energy ratio (_e/_B), is parameterized as follows (Soderberg et al. 2005, 2006a):

$$\begin{eqnarray} r= r_0 \left(\frac{t-t_e}{t_0}\right)^{\alpha _r} & B=B_0 \left(\frac{t-t_{e}}{t_0}\right)^{\alpha _B} \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} \gamma _m=\gamma _{m,0}\left(\frac{t-t_{e}}{t_0}\right)^{\alpha _{\gamma }} & \frac{\epsilon _e}{\epsilon _B},=\mathfrak {F}_0 \left(\frac{t-t_{e}}{t_0}\right)^{\alpha _{\mathfrak {F}}}, \end{eqnarray} \tag{ 3 }$$

where t₀ is an arbitrary reference time that we set to day 10 since explosion, and t_e is the explosion time of the SN. In the above equations, α_r = (n − 3)/(n − s) (Chevalier 1982, 1996), where n characterizes the density profile of the outer SN ejecta (ρ_SN∝(r/t)⁻ⁿ), and s characterizes the density profile of the radiating electrons within the shocked CSM (n_e∝r^−s).

In the "standard" scenario (Chevalier 1996), the magnetic energy density (U_B∝B²) and the relativistic electron energy density (U_e∝n_eγ_m) are assumed to be a fixed fraction (i.e., $\alpha _{\mathfrak {F}}=0$) of the total post-shock energy density (U∝n_e < v > ²). With these assumptions (Soderberg et al. 2005, 2006a),

$$\begin{equation} U_e\propto U \Rightarrow \alpha _{\gamma }=2(\alpha _r-1), \end{equation} \tag{ 4 }$$

and

$$\begin{equation} U_B \propto U \Rightarrow \alpha _B=\frac{(2-s)}{2}\alpha _r-1. \end{equation} \tag{ 5 }$$

With the above parameterization, the flux density from the uniform shell of radiating electrons reads (Soderberg et al. 2005, 2006a):

$$\begin{eqnarray} f_{\nu }&=&C_f\left(\frac{t-t_e}{t_0}\right)^{(4\alpha _r-\alpha _B)/2}(1-\exp (-\tau _{\nu }^\xi (t)))^{1/\xi }\left(\frac{\nu }{\rm 1\,GHz}\right)^{5/2} \nonumber\\ &&\times F_3(x)F^{-1}_2(x)\,{\rm mJy}, \end{eqnarray} \tag{ 6 }$$

where x = 2/3(ν/ν_m), and

$$\begin{equation} \nu _m=\gamma^{2}_m \frac{e B}{2\pi m_e c}=\nu _{m,0}\left(\frac{t-t_e}{t_0}\right)^{2\alpha _{\gamma }+\alpha _B} \end{equation} \tag{ 7 }$$

is the characteristic synchrotron frequency of electrons with Lorentz factor γ_m; F₂ and F₃ are integrals of the modified Bessel function of order 2/3 (see Equation (A11) in Soderberg et al. 2005); ξ parameterizes the sharpness of the spectral break between optically thick and thin regimes (Soderberg et al. 2005, 2006a), and

$$\begin{eqnarray} \tau _{\nu }(t)&=&C_{\tau }\left(\frac{t-t_{e}}{t_0}\right)^{(p-2)\alpha _{\gamma }+(3+p/2)\alpha _B+\alpha _r+\alpha _{\mathfrak {F}}}\nonumber\\ &&\times\left(\frac{\nu }{\rm 1\,GHz}\right)^{-(p+4)/2}F_2(x) \end{eqnarray} \tag{ 8 }$$

is the optical depth (Soderberg et al. 2005, 2006a).

As evident from Equations (6)–(8), the observed spectral and temporal evolution of the radio emission depends on the parameters (C_f, C_τ, ν_{m, 0}, t_e, p, ξ, α_r, s). Note that the three parameters C_f, C_τ, and ν_{m, 0} are functions of the four normalization constants r₀, B₀, γ₀, $\mathfrak {F}_0$³¹ (and of the source distance d_L). Thus, while C_f, C_τ, and ν_{m, 0} can be determined by comparison with the data, another constraint is needed to derive r₀, B₀, γ_{m, 0}, $\mathfrak {F}_0$ from the fitted values of C_f, C_τ, and ν_{m, 0}. Following common practice in radio SNe studies, we set s = 2 (as expected for a wind density profile), p ≈ 3, and ν₀ ≈ 1 GHz. This reduces the number of free parameters to five, (C_f, C_τ, t_e, ξ, α_r), or four when the explosion date t_e is known.

In the case of PTF11qcj, we first set t_e ≈ 55842 MJD (as suggested by our optical observations; see Sections 3.1 and 3.3). Using a χ² minimization procedure and the simplified four-parameter model described above, we obtain the results reported in Table 1 (Model 1). The model predictions are shown in Figures 11–13(dotted lines). From our radio best fit results and by Equations (2) and (3) here and Equations (6)– (8) in Soderberg et al. (2005), we derive the evolution of r, B, and γ_m (Table 1). The radial evolution is within the expected range of 0.67 ≲ α_r ≲ 1.0 (Chevalier 1996, 1998), and it implies an ejecta velocity of dr/dt ≈ 0.34c at day 10. A velocity of ≈0.3 c at day 10, is somewhat higher than the typical range of (0.1–0.25) c for normal Ib/c SNe (e.g., Berger et al. 2003), but significantly slower than SN 1998bw, for which the bulk Lorentz factor was estimated to be Γ ∼ 2 on a similar timescale (Kulkarni et al. 1998; Li & Chevalier 1999). We also note that the implied velocity of ≈0.28 c at the time of our first spectrum of PTF11qcj, is consistent with the lower-limit estimate derived from the Ca triplet (≈22, 000 km s⁻¹), while the implied size at day 100 (r ≈ 7 × 10¹⁶ cm) is consistent with the lower limit (r ≳ 3 × 10¹⁶ cm at day 100) derived in Equation (1).

Table 1. Radio Model Parameters for a Standard Synchrotron Self-absorbed Model with s = 2 and ν_{m, 0} = 1 GHz

	Model 1 (Fixed te)	Model 2 (Free te)
Cf	≈7.1 × 10−4	≈1.9 × 10−3
Cτ	≈1.8 × 107	≈9.3 × 106
ξ	≈0.23	≈0.17
αr	≈0.80	≈0.80
te (MJD)	55842	≈55857
r0 (cm)	${\approx }1.1\times 10^{16}\mathfrak {F}_0^{-1/17}(\eta /10)^{1/17}$	${\approx }1.7\times 10^{16}\mathfrak {F}_0^{-1/17}(\eta /10)^{1/17}$
B0 (G)	${\approx }6.7\,\mathfrak {F}_0^{-1/17}(\eta /10)^{4/17}$	${\approx }5.1\,\mathfrak {F}_0^{-1/17}(\eta /10)^{4/17}$
αB	−1	−1
γm, 0	${\approx }7.3\,\mathfrak {F}_0^{2/17}(\eta /10)^{-2/17}$	${\approx }8.3\,\mathfrak {F}_0^{2/17}(\eta /10)^{-2/17}$
αγ	−0.40	−0.40
ne, 0 (cm−3)	${\approx }1.4\times 10^{5}\mathfrak {F}_0^{13/17}(\eta /10)^{10/17}$	${\approx }7.6\times 10^{4}\mathfrak {F}_0^{13/17}(\eta /10)^{10/17}$
$\alpha _{n_{e}}$	≈ − 1.6	≈ − 1.6
$\dot{M}_0$ (M☉ yr−1)	${\approx }1.2\times 10^{-4}\mathfrak {F}_0^{11/17}(\eta /10)^{-5/17}(v_w/10^3\,{\rm km\,s^{-1}})$	${\approx }1.4\times 10^{-4}\mathfrak {F}_0^{11/17}(\eta /10)^{-5/17}(v_w/10^3\,{\rm km\,s^{-1}})$
$\alpha _{\dot{M}}$	0	0
EK, 0 (erg)	${\approx }9.3\times 10^{48}(\epsilon _{e}/0.33)^{-1}\mathfrak {F}_0^{12/17}(\eta /10)^{-6/17}$	${\approx }1.9\times 10^{49}(\epsilon _{e}/0.33)^{-1}\mathfrak {F}_0^{12/17}(\eta /10)^{-6/17}$
$\alpha _{E_K}$	0.40	0.40

Notes. All quantities marked with a ₀ are normalized to t − t_e = 10 days. The first five rows give the values of the fitted parameters. The other rows give the values of physical quantities derived by Equations (2)–(3), (4)–(5), (9)–(11), and Equations (6)– (8) in Soderberg et al. (2005).

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The number density of the emitting electrons, progenitor mass loss rate, and total energy of the ejecta, then follow from the relations (Soderberg et al. 2005, 2006a):

$$\begin{eqnarray} n_e&=&\frac{p-2}{p-1}\frac{B^2_0}{8\pi }\frac{\mathfrak {F}_0}{m_ec^2\gamma _{m,0}}\left(\frac{t-t_{e}}{10\,\rm {\rm days}}\right)^{2\alpha _B-\alpha _{\gamma }}\nonumber\\ &=& n_{e,0}\left(\frac{t-t_{e}}{10\,\rm {\rm days}}\right)^{\alpha _{n_e}} \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} \dot{M}&=&\frac{8\pi }{\eta }n_{e,0}m_pr_0^2v_w\left(\frac{t-t_{e}}{10\,\rm {\rm days}}\right)^{2\alpha _B-\alpha _{\gamma }+2\alpha _r}\nonumber\\ &=&\dot{M}_0\left(\frac{t-t_{e}}{10\,\rm {\rm days}}\right)^{\alpha _{\dot{M}}} \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} E_{K}&=&\frac{4\pi \mathfrak {F}_0}{\eta }r^3_0\frac{B^2_0}{8\pi \epsilon _e}\left(\frac{t-t_{e}}{10\,\rm {\rm days}}\right)^{2\alpha _B+3\alpha _{r}}\nonumber\\ &=&E_{K,0}\left(\frac{t-t_{e}}{10\,\rm {\rm days}}\right)^{\alpha _{E_k}} \end{eqnarray} \tag{ 11 }$$

for a nucleon-to-electron density ratio of 2 (as appropriate for Wolf–Rayet (W-R) winds). The values derived for these quantities within Model 1 are also reported in Table 1.

In terms of total energy in the radio emitting material, PTF11qcj ranks as one of the most energetic Ib/c SNe, comparable to the GRB-associated SN 1998bw (Kulkarni et al. 1998), and similar to other radio SNe that show evidence for CSM density variations (e.g., SN 2003bg; Soderberg et al. 2006a). The mass-loss rate estimates for GRBs and engine-driven SNe such as 1998bw are typically $(\dot{M}/10^{-5} \,M_{\odot }\,{\rm yr}^{-1})\times ({\rm {v_w}/10^3\,km\,s^{-1}})^{-1}\approx 0.1\hbox{--}1$ (Panaitescu & Kumar 2002; Yost et al. 2003; Chevalier et al. 2004). PTF11qcj lies at the higher end of the range observed for local W-R stars, $\dot{M}\approx (0.6\hbox{--}9.5)\times 10^{-5}\,M_{\odot }\,{\rm yr}^{-1}$ (Cappa et al. 2004), and is similar to normal type Ib/c SNe with prominent variations in the radio light curves (such as SNe 2004cc and 2004gq; Soderberg et al. 2006a; Wellons et al. 2012).

As underlined before, we do not expect the simplified model used in our analysis to provide a perfect fit to (nor to represent a complete physical interpretation of the) PTF11qcj radio emission. Indeed, our high χ²/dof = 1790/90 is similar to what obtained in other analyses of complex radio SN light curves (e.g., Soderberg et al. 2006a; Krauss et al. 2012). So the results of our fit should be regarded as a tentative insight into the properties of the PTF11qcj fastest ejecta and CSM.

The match between Model 1 and data is most problematic during the earlier observations at the lowest frequencies (2.5–3.5 GHz): we caution, however, that at such low frequencies the data are most affected by RFI, so the uncertainties on the measured fluxes may be underestimated.

We note that it is unlikely that the high value of the χ² derived for the radio model is entirely dominated by interstellar scattering and scintillation (ISS). Indeed, considering the data collected at t ⩽ 56057 MJD (i.e., in between ≈38 and 215 days since 55842 MJD), we derive a modulation index with respect to our best fit model flux predictions (f_{ν, pred}),

$$\begin{equation} m_{{\rm obs}}(\nu)=\frac{\sqrt{\left<(f_{\nu,{\rm obs}}-f_{\nu,{\rm pred}})^{2}\right>-\left<\sigma ^{2}_{\nu,{\rm obs}}\right>}}{\left<f_{\nu,{\rm obs}}\right>} \end{equation} \tag{ 12 }$$

(where f_ν are the measured flux densities and σ_ν their measured uncertainties), of m(ν) ≈ 0.1–0.3 for ν = (2.5–16) GHz (with the highest modulation observed at the lowest frequencies). Using the maps provided by Walker (2001) and the PTF11qcj Galactic coordinates (l = 112°, b = 69°), we roughly estimate ν₀ ≈ 5 GHz for the ISS transition frequency between the strong and weak scattering regimes (thus, our lowest frequency observations are in the strong scattering regime), and Θ₀ ≈ 5 μas for the Fresnel angle at the transition frequency. The expected value for the modulation index in the refractive strong scattering reads (e.g., Walker 1998, 2001; Kulkarni et al. 1998; Cenko et al. 2013):

$$\begin{eqnarray} &&m_{p}(\nu)=\left(\frac{\nu }{\nu _0}\right)^{17/30}\left(\frac{\Theta _r}{\Theta }\right)^{7/6}, \end{eqnarray} \tag{ 13 }$$

with

$$\begin{equation} \Theta _r=\Theta _0\left(\frac{\nu _0}{\nu }\right)^{11/5}. \end{equation} \tag{ 14 }$$

Using the above equations, m_p(2.5 GHz) ≈ 0.3 requires Θ ≈ 45 μas around day 130, or r ≈ 4 × 10¹⁶ cm, which is a factor of ≈2 smaller than the radius at day 130 implied by our radio fits. Moreover, flux variations are observed up to the highest frequencies of our observations (16 GHz), that fall in the weak scattering regime (and so the observed modulations at such frequencies cannot be explained by ISS).

Due to the uncertainties in the explosion date of PTF11qcj, we also performed a fit removing the constraint on the explosion time (Model 2; see Table 1 and the solid lines in Figures 11–13). In this case the fit drives t_e around the discovery time of PTF11qcj, 55857 MJD, and $\chi ^2_r$/dof = 1717/89. We note, however, that in this last model the velocity of the fastest ejecta is rather high, dr/dt ≈ 0.52c at day 10. Assuming PTF11qcj exploded around 55842 MJD (see Section 3.1), the delayed onset of the radio emission may be interpreted as interaction with a shell whose inner radius is located at a distance r ≳ 22, 000 km s⁻¹ × 15 days ≈ 3 × 10¹⁵ cm. This is comparable to the distances that may have been reached by material ejected during the pre-SN activity event probed by our P48 pre-discovery images (see Section 3.2), if such material was ejected at speeds v_w ≳ 400 km s⁻¹.

An element common to both the above fit results is the low value of ξ ≈ 0.2, a parameter that measures the sharpness of the spectral break between optically thick and thin regimes. Indeed, we see that at the higher frequencies the late-time spectral index of PTF11qcj appears to remain ≳ − 1, or show an abrupt increase (this is evident during our latest two VLA observations; Figures 12 and 13) rather than transition toward a value of ≈ − 1. Recently, Björnsson (2013) has shown that the broadening observed in the radio spectra and/or light curves of some type Ib/c SNe may be a direct indication of inhomogeneities in the CSM.

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The increase in spectral index observed for PTF11qcj at t ≳ 300 days (during the re-brightening phase) in the 2.5–13 GHz range is reminiscent of cases like the normal type Ic SN 2004cc, for which light curve variations associated with an increase of the spectral peak have been observed (Wellons et al. 2012), or the Ic-BL SN 2007bg, for which an increase in the flux density accompanied by an increase in spectral index was observed around day 300 and interpreted as an absorption turn-on associated with a sharp CSM density enhancement presumably due to evolution of the progenitor wind (Salas et al. 2013). Indeed, this event seems most similar to PTF11qcj (see also Figure 10). According to our radio modeling of the earlier data, the re-brightening occurs at r ≳ 1.7 × 10¹⁷ cm.

Finally, we note that Moriya et al. (2013) have recently investigated the link between LBV as SN progenitors and the appearance of episodic light curve modulations in the radio light curves of the SN events, showing how these early modulations have the potential to probe the progenitor's mass loss history immediately before the SN explosion. A long-term monitoring campaign of PTF11qcj aimed at tracking the properties of its late-time re-brightening is on-going at the time of writing.

We performed a spectral fit of our three observations of PTF11qcj with Chandra (Figure 14). We used a power-law model with fixed N_H, and power-law index Γ (where N_phot(E)∝E^−Γ) free to vary but constrained to be the same across the three epochs. Flux normalizations were set free to vary for each epoch. This way, we derive $\Gamma =1.70^{+0.81}_{-0.72}$ (90% confidence; Cstat/dof = 5.18/3) for N_H = 10²⁰ cm⁻² (Galactic column density in the direction of PTF11qcj). We achieve satisfactory X-ray spectral fits for fixed N_H and power-law photon index unchanging across the three epochs, as long as N_H is less than a few × 10²¹ cm⁻². Even if the CSM is not ionized, this upper-limit on N_H is consistent with the mass-loss (and radius at day 100) derived from our radio analysis. Indeed, for a given $\dot{M}$ and r, the implied N_H (from r to infinity) reads (e.g., Ofek et al. 2013a):

$$\begin{eqnarray} N_{{\rm H}}&\approx& 10^{20} \left(\frac{\dot{M}}{1.5\times 10^{-4} \,M_{\odot }\,\rm yr^{-1}}\right)\left(\frac{v_w}{10^3{\rm \,km\,s^{-1}}}\right)^{-1}\nonumber \\ &&\times \left(\frac{r}{7\times 10^{16}\rm \,cm}\right)^{-1}\,{\rm cm^{-2}}. \end{eqnarray} \tag{ 15 }$$

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From the above spectral analysis, we derive the corresponding unabsorbed X-ray fluxes for the three epochs (see the last column in Table 4). If we conservatively assume that all the flux measured during the last of our Chandra observations is entirely due to the contribution of the host galaxy, and subtract such flux from the flux measured during the first epoch, then we get that the X-ray emission from PTF11qcj is ≳ (4.5 ± 2.9) × 10⁻¹⁵ erg cm⁻² s⁻¹, which corresponds to an X-ray luminosity ≳ (8 ± 5) × 10³⁹ erg s⁻¹ (0.3–8 keV; cps-to-flux conversion computed assuming the best fit Γ ≈ 1.7, and an N_H ≈ 10²⁰ cm⁻²).

Hereafter, we consider three main mechanism to explain PTF11qcj X-ray observations (Chevalier & Fransson 2006): synchrotron emission; thermal (free–free) bremsstrahlung emission from material in the circumstellar shock or in the ejecta reverse shock; and inverse Compton scattering of photospheric emission photons by relativistic electrons.

The radio-to-X-ray spectral index estimated comparing the 16 GHz flux of PTF11qcj around the time of our first Chandra observation (F_16 GHz ≈ 6.5 mJy) with the X-ray one ($F_{\rm X}\gtrsim 4.5\times 10^{-15}\,\rm {erg\,cm^{-2}\,s^{-1}}/(2.4\times 10^{17}\rm {\,Hz})\approx 2\times 10^{-6}$ mJy), is Γ − 1 ≈ 0.9. The last is in agreement with both the X-ray spectral index derived from the analysis of our Chandra data, and the radio synchrotron self-absorbed model discussed in the previous section. Indeed, within the last model, the optically thin portion of the synchrotron spectrum is predicted to have a photon index of Γ − 1 ≈ (p − 1)/2 ≈ 1.

Based on the above considerations, synchrotron emission would seem to be the most straightforward interpretation of the PTF11qcj X-ray observations. A complication (not unique to the case of PTF11qcj) arises from the fact that the synchrotron cooling break is expected to steepen the spectral index between the radio and X-ray bands. For PTF11qcj, this would cause the X-ray flux extrapolated from our radio data to fall below the one measured by Chandra. However, Ellison et al. (2000) have suggested that a cosmic-ray dominated shock can flatten the synchrotron spectrum at high frequencies, and Chevalier & Fransson (2006) have invoked such a flattening to explain the X-ray emission of SN 1994I in the context of synchrotron emission.

Within the thermal emission hypothesis, X-rays are produced while the forward shock plows into the CSM and/or by the reverse shock heating the ejecta. Thus, the X-ray luminosity depends on the density of the emitting material, which then cools by free–free emission. For an r⁻² wind density profile, the free–free luminosity can be estimated as (e.g., Chevalier & Fransson 2001; Sutaria et al. 2003; Soderberg et al. 2005, 2006a; Ofek et al. 2013b):

$$\begin{eqnarray} L_X &\approx& \frac{C}{690} \times 3\times 10^{39}\left(\frac{\dot{M}}{1.5\times 10^{-4} \,M_{\odot }\,{\rm yr^{-1}}}\right)^{2}\nonumber \\ &&\times\left(\frac{v_w}{10^3\,{\rm km\,s^{-1}}}\right)^{-2} \left(\frac{r}{5\times 10^{15}\,\rm cm}\right)^{-1}\,{\rm erg\,s^{-1}},\quad \end{eqnarray} \tag{ 16 }$$

where C = 4(1 + (n − 3)(n − 4)²/4) accounts for both the reverse and forward shock contribution.³² As evident from the above equation, for typical values of n = 5–12.5 (i.e., C ≈ 6–690 or α_r ≈ 0.67–0.9; Chevalier 1982) and $\dot{M}\lesssim 1.5\times 10^{-4}\,M_{\odot }\,{\rm yr^{-1}}$  (v_w/10³ km s⁻¹; as typically found in Ib/c SNe; Wellons et al. 2012), L_X ≳ (8 ± 5) × 10³⁹ erg s⁻¹ at 100 days requires r (100 days) ≲ 5 × 10¹⁵ cm. This is smaller than the r (100 days) ≳ 3 × 10¹⁶ cm derived from Equation (1), and r (100 days) ≈ 7 × 10¹⁶ cm derived from the radio modeling (see Table 1).

In the inverse Compton scenario, the X-ray luminosity can be estimated as follows (e.g., Björnsson & Fransson 2004; Soderberg et al. 2005, 2006a):

$$\begin{equation} \frac{L_{X}}{L_{\rm radio}}\approx \frac{U_{\rm ph}}{U_B}, \end{equation} \tag{ 17 }$$

where

$$\begin{equation} U_B=\frac{B^{2}}{8\pi } \end{equation} \tag{ 18 }$$

and (Soderberg et al. 2006a)

$$\begin{equation} U_{\rm ph}\approx 0.4 \left(\frac{L_{\rm bol}}{10^{42}\,\rm erg\,s^{-1}}\right)\left(\frac{t}{\rm 1\,{\rm day}}\frac{v}{c}\right)^{-2}\,{\rm erg\,cm^{-3}}. \end{equation} \tag{ 19 }$$

For PTF11qcj, using our late-time P48 photometry and applying a tentative bolometric correction of ≈ − 0.5 (for an SN temperature of ≈10⁴ K; Corsi et al. 2012), we estimate L_bol ≈ 10⁸ L_☉ during the first Chandra epoch. Also, at this time, we estimate v = dr/dt ≈ 0.2 c and B ≈ 0.7 G based on the radio model discussed in Section 3.4. Thus, we get U_ph ≈ 4 × 10⁻⁴ erg cm⁻³ and U_B ≈ 2 × 10⁻² erg cm⁻³, or U_ph/U_B ≈ 0.02. Since L_radio ≈ 2 × 10³⁹ erg s⁻¹ (based on the 16 GHz flux of ≈6.5 mJy during the first Chandra observation), we derive L_X ≈ 4 × 10³⁷ erg s⁻¹, which underestimates the observed X-ray luminosity by a factor of ≳ 100.

In conclusion, synchrotron emission seems to be the easiest explanation for the X-ray counterpart to PTF11qcj but, as discussed before, a mechanism flattening the spectrum at high frequencies (such as a cosmic-ray dominated shock) would be required. Free–free emission could explain the observed counterpart only for mass-loss rates in excess of the highest values found for SNe of type Ib/c ($\dot{M}\gtrsim 1.5\times 10^{-4}\,M_{\odot }$ yr⁻¹); the inverse Compton scenario seems hard to reconcile with the observed X-ray counterpart to PTF11qcj, unless the magnetic field energy is much lower than what estimated from the radio fit (which assumes equipartition; see also Horesh et al. 2013).

Given the radio and X-ray detections of PTF11qcj, we have searched the Fermi GRB catalog for GRBs with explosion date in between the time of our first detection of PTF11qcj, and a month earlier. No GRBs were found within such time frame, with positions compatible with PTF11qcj. The non-relativistic speeds suggested by the simplest radio model (Section 3.4) are in agreement with the lack of a GRB detection.

From a tentative comparison with the light curves of off-axis low-luminosity GRBs expanding in a constant density environment available in the literature (van Eerten & MacFadyen 2011), we deduce that fitting the PTF11qcj radio and X-ray emission within the simplest off-axis GRB scenarios, may be difficult to achieve. Moreover, despite there is some amount of controversy in the literature concerning whether the contribution from a GRB counter jet could cause a late-time light curve bump (e.g., Li & Song 2004; Wang et al. 2009; van Eerten & MacFadyen 2011), the late-time radio re-brightening observed in PTF11qcj is a feature that challenges the simplest off-axis models.

We detected an IR counterpart to PTF11qcj in both the Spitzer (Figure 7) and P200 (Figure 1) images. The extrapolation from the visible light spectral energy distribution suggests that a late-time IR excess may be associated with PTF11qcj (Figure 15). A continuum IR emission can be expected from hot dust—either newly formed hot dust, or pre-existing dust in the CSM that is heated by the SN. Several heating mechanisms are possible. Pre-existing dust, for example, may be collisionally heated by hot shocked gas, or radiatively heated by either the peak SN luminosity or the late-time optical emission from circumstellar interaction (e.g., Draine & Salpeter 1979; Draine 1981; Dwek 1983, 1985).

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Detailed modeling of the dust components and heating mechanisms is hard to accomplish with only upper limits. Nonetheless, we can get some tentative insight into the properties of the dust using our Spitzer and P200 observations. The following analysis should be considered as an order of magnitude estimate of the dust properties around PTF11qcj, rather than an accurate measurement.

The simplest scenario usually invoked to explain IR echoes observed in type II SNe is that of radiative heating of pre-existing dust by the SN optical emission. The SN peak luminosity is expected to vaporize the dust out to a vaporization radius r_vap, and heat the inside of this shell to a temperature roughly equal to the vaporization temperature, T_vap ≈ 2000 K for graphite (hereafter, we limit our discussion to graphite grains only since our data do not allow us to distinguish between, e.g., graphite and silicate). Alternatively, the dust may be formed at a radius r_echo larger than the vaporization radius, and heated to a temperature T_echo ≲ T_vap.

For simple dust populations composed entirely of graphite with a single grains size, we can estimate the echo (vaporization) radius (r_echo/vap) of a dust shell at T_echo (T_vap ≈ 2000 K), using the estimated peak bolometric luminosity of PTF11qcj (L_{bol, p}), and the fact that the equilibrium dust temperature is set by balancing the energy absorbed by a spherical dust grain of radius a (Fox et al. 2010):

$$\begin{eqnarray} &&\nonumber L_{\rm abs}(a)=4\pi r^{2}_{\rm SN}\frac{\pi a^{2}}{4\pi r^{2}_{\rm echo/vap}}\int \pi B_{\nu }(T_{\rm SN}) Q_{\nu }(a) d\nu =\\ &&=\frac{L_{\rm bol,p}}{\sigma T^{4}_{\rm SN}}\frac{ a^{2}}{4 r^{2}_{\rm echo/vap}}\int \pi B_{\nu }(T_{\rm SN}) Q_{\nu }(a) d\nu, \end{eqnarray} \tag{ 20 }$$

and the energy emitted by the dust:

$$\begin{equation} L_{\rm rad}(a)=4\pi a^{2} \int \pi B_{\nu }(T_{\rm echo/vap}) Q_{\nu }(a) d\nu. \end{equation} \tag{ 21 }$$

In the above equations we have assumed wavelength dependent emissivities/absorption Q_ν(a) (Draine & Lee 1984; Laor & Draine 1993); B_ν(T) is the Planck black-body function; σ is the Stefan–Boltzmann constant; T_SN ≈ 10⁴ K is the effective SN temperature; and r_SN the effective SN emitting radius. The equality condition yields (Fox et al. 2010):

$$\begin{equation} r_{\rm echo/vap}(a)=\sqrt{\frac{3}{64}\frac{L_{\rm bol,p}}{(\pi \rho _d a \sigma T_{\rm SN}^4)} \frac{\int B_{\nu }(T_{\rm SN})Q_{\nu }(a)d\nu }{\int B_{\nu }(T_{\rm echo/vap})k_{\nu }(a)d\nu }}, \end{equation} \tag{ 22 }$$

where k(ν) = Q_ν(a)/(4/3ρ_da) is the mass absorption coefficient, and ρ_d = 2.25 g cm⁻³ (Smith et al. 2008) is the assumed dust density.

To estimate PTF11qcj peak bolometric luminosity we use the observed P48 peak magnitude corrected for Galactic and local host galaxy extinction, and apply a bolometric correction of −0.5 mag (for an SN temperature of ≈10⁴ K; Corsi et al. 2012). This way we get L_{bol, p} ≈ 10^9.2 L_☉ (but note that our observations do not allow us to precisely constrain the optical peak). Thus, from Equation (22):

$$\begin{eqnarray} &&r_{\rm vap}\approx (1\hbox{--}5)\times 10^{16}\,{\rm cm}. \end{eqnarray} \tag{ 23 }$$

The above range is calculated for dust grains of sizes (0.03–0.3) μm, and the smaller r_vap are for the larger dust grains (note that in Equation (22) r_vap ∼ a⁻¹).

In the radiative heating scenario, the shock radius (r_s) sets a lower limit for the dust radius (since any dust within this radius would be independently heated or destroyed by the forward shock). At the time of our first Spitzer observation:

$$\begin{equation} r_s \approx (0.3\hbox{--}1)\times 10^{17}\,{\rm \,cm}. \end{equation} \tag{ 24 }$$

The smallest value above is calculated assuming an average speed of 22, 000 km s⁻¹ as indicated by the optical spectra of PTF11qcj (Section 3.1), while the largest value is derived from the radio model (radio emission probes the fastest ejecta; see Section 3.4). Moreover, if the dust shell is located at or beyond the vaporization radius, light travel time effects cause the thermal radiation from the dust grains to reach the observer over an extended period of time (thus the term "IR echo"). So the duration of the IR echo sets a scale for the dust radius (Fox et al. 2010, 2011):

$$\begin{equation} c \times \Delta t_{\rm IR} \approx 2 \times r_{\rm IR}. \end{equation} \tag{ 25 }$$

Using the fact that in the case of PTF11qcj Δt_IR ≳ 100–200 days, we deduce r_IR ≳ (1–3) × 10¹⁷ cm (which is indeed larger than, or at the least comparable to, r_s—this assures self-consistency of the IR echo scenario).

Since r_vap < r_IR, we conclude that the dust should be at T_echo < T_vap in order for the IR echo scenario to be self-consistent. Using again Equation (22), and setting the condition r_echo ≈ r_IR ≳ (1–3) × 10¹⁷ cm, we derive the maximum temperatures that the dust can reach when heated by PTF11qcj optical peak:

$$\begin{equation} T_{\rm echo} \lesssim 580\hbox{--}1500\,{\rm K}, \end{equation} \tag{ 26 }$$

where the smallest temperature is for dust of size 0.3 μm located at a dust echo radius of r_echo ≈ 3 × 10¹⁷ cm, while the largest temperature is for dust grains of 0.03 μm located at r_echo ≈ 10¹⁷ cm. We note that temperatures in the range of 500–770 K are typically found in modeling type IIn SNe emission with graphite dust grains of sizes a ≈ 0.01–0.3 μm (Fox et al. 2011).

For optically thin dust with mass M_d, located at a distance d_L from the observer, and thermally emitting at a single equilibrium temperature T_echo (note that the IR emission of SNe such as SN 2006jd and SN 2005ip was best fit by a multi-component dust model as opposed to a single component one; here we limit our discussion to a single component—a similar approach has been used for other SNe with limited data-sets; see e.g., Fox et al. (2011)), the total flux reads (see, e.g., Fox et al. 2010):

$$\begin{equation} F_{\nu }=\frac{M_d}{4/3\pi \rho _d a^{3}}\frac{L_{\nu }(a)}{4\pi d^2_L}=\frac{M_d B_{\nu }(T_{\rm echo})k_{\nu }(a)}{d^2_L}, \end{equation} \tag{ 27 }$$

where L_ν(a) is defined in Equation (21). As shown in Figure 15, for dust grains of sizes 0.03–0.3 μm we tentatively estimate dust masses in the range M_d ∼ 10⁻⁵–10⁻³ M_☉. For comparison, dust masses in the range M_d ≈ 2 × 10⁻⁵–5 × 10⁻² M_☉ are typically invoked for type IIn SNe with IR emission (Fox et al. 2011).

The estimates derived in this section are subject to the large uncertainties related to the limitation of our IR data-set. However, a light echo scenario from pre-existing dust, presumably emitted by the SN progenitor before the SN itself, seems to offer a consistent explanation.