The Double-peaked Radio Light Curve of Supernova PTF11qcj

The location of PTF11qcj was observed with the Low Resolution Spectrograph-2 (LRS-2) attached to the Hobby–Eberly Telescope (HET) on 2017 February 22 UT. LRS-2 is a twin IFU spectrograph having two arms each: LRS2-B covers the range between 3700–4700 Å (blue arm) and 4600–7000 Å (orange arm) with resolving power of 1900 and 1100, respectively, while LRS2-R covers the range from 6500 to 8400 Å (red arm) and from 8200 to 10500 Å (far-red arm) at a resolution of 1800 in each arm (Chonis et al. 2014, 2016). Each IFU maps a 12'' × 6'' area on the sky covered by 280 fibers. The coverage is complete, so no dithering is required.

We utilized the red arm of LRS2-R to collect spectra in the vicinity of the SN. Figure 2 shows the SDSS r-band frame of the host galaxy of PTF11qcj with the SN position marked in red. The blue rectangle indicates the position of the LRS2-R IFU (it was not completely centered on the SN due to a minor pointing issue). A spectrum at the SN position was extracted by median-combining the signal of the three closest fibers. Wavelength calibration was computed using FeAr spectral lamp observations. Flux calibration was performed by comparing the observed and flux-calibrated cataloged spectra of the standard star HD 84937.

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The final LRS2 spectrum of PTF11qcj is plotted in Figure 3 together with the spectrum obtained with the Keck II-DEIMOS on 2012 March 20 (Corsi et al. 2014). This plot suggests that the broad SN features that were clearly visible in the Keck spectrum 5 yr ago (typical nebular features due to forbidden transitions of neutral oxygen [O i] λλ6300,6364 and ionized [Ca ii] λλ7291,7324), may still be present in the new HET-LRS2 spectrum although with a smaller signal-to-noise. Also, the bright, narrow Hα feature present in both spectra appears basically unchanged (in both strength and width). No broad Hα feature can be identified in the HET-LRS2 spectrum of PTF11qcj (as was the case for the older Keck spectrum during the first radio peak), thus excluding H-rich CSM interaction similar to SN 2014C (Milisavljevic et al. 2015). As discussed in what follows, interaction with an H-poor CSM may explain the late-time radio rebrightening.

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The Very Large Array (VLA) follow-up observations presented here were carried out between 2014 June 1 UT (MJD 56809) and 2016 December 7 UT (MJD 57729). The data were taken at the nominal central frequencies of 2.5, 3.5, 7.4, 13.5, and 16 GHz with a nominal bandwidth of 2 GHz. 3C286 and J1327+4326 were used as flux and phase calibrators, respectively.

The Common Astronomy Software applications (CASA; McMullin et al. 2007)¹¹ was used to calibrate, flag, and image the data. The automated VLA calibration pipeline for CASA was used to calibrate the raw data. Images were formed from the visibility data using the CLEAN algorithm (Högbom 1974). The image size was set to (1024 × 1024) pixels, and the pixel size was determined as one-fourth of the nominal beamwidth. The images were cleaned using natural weighting for 10,000 iterations or until a threshold of ∼0.03 mJy (∼3σ) was reached. The source flux at each epoch was calculated using imstat i.e., as the flux corresponding to the brightest pixel within a circle centered around the PTF position with a radius of 2'' (comparable to the typical R-band seeing of PTF images; Law et al. 2009). Flux errors are calculated as the quadratic sum of the rms map error and a 5% fractional error that accounts for errors in the flux calibration (Weiler et al. 1986; Corsi et al. 2014). We have verified that peak fluxes estimated using imstat are consistent within errors to peak fluxes and total fluxes estimated by using imfit, which fits a two-dimensional Gaussian component to the source. We also note that results from imfit do not show evidence for extended emission, confirming that the radio counterpart of PTF11qcj is a point source up to ≈027, which is the beam size in the VLA A array configuration at 7.4 GHz.

Figure 4 shows the radio light curves of PTF11qcj, with the new data from MJD 56809 onward plotted along with the data from Corsi et al. (2014). The fluxes at each MJD and frequency for the latest observations are reported in Table 1.

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The location of PTF11qcj was imaged by the Advanced CCD Imaging Spectrometer (Garmire et al. 2003) on board Chandra on 2018 February 23 UT and 2018 February 26 UT. A total exposure time of 40 ks was obtained with the location of PTF11qcj positioned at the nominal aimpoint of chip S3 (i.e., ACIS-S). We detect a faint unresolved source at this position, with a 0.5–7.0 keV count rate of (1.6 ± 0.3) × 10⁻³ ct s⁻¹ (corrected for background and the instrument PSF using the task srcflux from the Chandra Interactive Analysis of Observations package v4.9; 90% confidence). Assuming a power-law spectrum with a photon index of Γ ≈ 1.7 and a Galactic N_H column density of ≈10²⁰ cm⁻² (Corsi et al. 2014), this corresponds to an unabsorbed 0.3–8.0 keV flux of (2.2 ± 0.4) × 10⁻¹⁴ erg cm⁻² s⁻¹ (which can be directly compared to the flux measurements obtained during previous epochs and reported in Corsi et al. 2014).

The model in Soderberg et al. (2005) describes the radio emission from SN ejecta interacting with the CSM. The radio flux density at time t and frequency ν is given by:

$$\begin{eqnarray}\begin{array}{rcl}F\left(t,\nu \right) & = & {C}_{f}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{(4{\alpha }_{r}-{\alpha }_{B})/2}\\ & & \times \,{\left[1-{\exp }^{-{\tau }_{\nu }^{\xi }(t)}\right]}^{1/\xi }{\nu }^{5/2}{F}_{3}(x){F}_{2}^{-1}(x),\end{array}\end{eqnarray} \tag{ 1 }$$

with the optical depth given by

$$\begin{eqnarray}&&{\tau }_{\nu }\left(t\right)={C}_{\tau }{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{(3+p/2){\alpha }_{B}+(2p-3){\alpha }_{r}-2(p-2)},\end{eqnarray} \tag{ 2 }$$

where C_f, C_τ are normalization constants, F₂, F₁ are Bessel functions, and x = 2/3(ν/ν_m) where ν_m is the critical synchrotron frequency, t₀ is a reference epoch, t_e is the explosion time, p is the electron energy index, and ξ = [0, 1] describes the sharpness of the spectral break between optically thin and thick regimes (Soderberg et al. 2005). In the above equations, α_r and α_B are the temporal indices of the shock radius r and the magnetic field B, respectively, such that

$$\begin{eqnarray}&&r={r}_{0}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{{\alpha }_{r}},\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&B={B}_{0}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{{\alpha }_{B}},\end{eqnarray} \tag{ 4 }$$

where r₀ and B₀ are the radius and magnetic field at the reference epoch t₀. The expansion of the SN shock is described by:

$$\begin{eqnarray}&&{\alpha }_{r}=\displaystyle \frac{n-3}{n-s},\end{eqnarray} \tag{ 5 }$$

where ρ_ej ∝ r⁻ⁿ is the density profile of the outer SN ejecta, and ${\rho }_{\mathrm{CSM}}\propto {n}_{e}\propto {r}^{-s}$ is that of the shocked CSM (or shocked electron density). The self-similar conditions s < 3 and n > 5 (Chevalier 1982) result in α_r < 1.

In the standard scenario, the magnetic energy density and the relativistic electron energy density are constant fractions, _B and _e, respectively, of the post-shock energy density. Under these assumptions:

$$\begin{eqnarray}&&{\alpha }_{B}=\displaystyle \frac{(2-s)}{2}{\alpha }_{r}-1,\end{eqnarray} \tag{ 6 }$$

and the minimum Lorentz gamma factor and the critical synchrotron frequency of the radiating electrons read:

$$\begin{eqnarray}&&{\gamma }_{m}={\gamma }_{0}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{2({\alpha }_{r}-1)},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\nu }_{m}={\nu }_{m,0}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{(10{\alpha }_{r}-s{\alpha }_{r}-10)/2}.\end{eqnarray} \tag{ 8 }$$

The electron number density within the shocked CSM is given by:

$$\begin{eqnarray}&&{n}_{e}=\displaystyle \frac{p-2}{p-1}\displaystyle \frac{({\epsilon }_{e}/{\epsilon }_{B}){B}_{0}^{2}}{8\pi {m}_{e}{c}^{2}{\gamma }_{m,0}}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{-s{\alpha }_{r}}\,{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 9 }$$

where m_e is the electron mass, and c is the speed of light. The SN progenitor mass-loss rate reads:

$$\begin{eqnarray}&&\dot{M}=\displaystyle \frac{8\pi {n}_{e}{m}_{p}{r}_{0}^{2}{v}_{w}}{\eta }{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{{\alpha }_{r}(2-s)},\end{eqnarray} \tag{ 10 }$$

where m_p is the proton mass and v_w is the wind velocity, while η describes the thickness, r/η, of the radiating shell at radius r. Finally, the ejecta energy reads:

$$\begin{eqnarray}&&E=\displaystyle \frac{4\pi {r}_{0}^{2}{B}_{0}^{2}}{\eta 8\pi {\epsilon }_{B}}{\left(\displaystyle \frac{t-{t}_{e}}{{t}_{0}}\right)}^{5{\alpha }_{r}-s{\alpha }_{r}-2}.\end{eqnarray} \tag{ 11 }$$

Hereafter, we work under the equipartition hypothesis and set _e = _B = 0.33. We note that departures from equipartition would imply _e/_B ≳ 1 and likely point to _B ≲ 0.33, thus increasing both the shocked electron number density (and, in turn, the estimated mass-loss rate; see Equations (9)–(10)), and the energy budget (Equation (11)).

As is evident from Equations (1)–(2) and (4)–(8), the observed flux at a given frequency ν and time t depends on C_f, C_τ, p, s, α_r, ${\nu }_{m,0}$, ξ, and t_e. Since C_f, C_τ can be expressed in terms of r₀ and B₀ (see Equations (6)–(8) in Soderberg et al. 2005), the observed flux ultimately depends on r₀, B₀, p, s, α_r, ${\nu }_{m,0}$, ξ, and t_e, which we determine by comparison with the observed data using a χ² minimization procedure (see Corsi et al. 2014). In modeling the radio emission from PTF11qcj, following Corsi et al. (2014), we set t₀ = 10 days and ${\nu }_{m,0}\approx 1\,\mathrm{GHz}$.

In a scenario in which PTF11qcj is powered by a central engine, we may interpret its double-peaked radio light curves as a combination of radio emission from an uncollimated (nonrelativistic) SN shock interacting with a dense CSM (as described in Corsi et al. 2014), and that of an off-axis relativistic jet entering our line of sight at late times, as the jet decelerates and spreads sideways (Waxman 2004). The jet dynamics in the relativistic regime is described by the Blandford–Mckee solution (Blandford & McKee 1976), and in the late nonrelativistic regime by the Sedov-von Neumann–Taylor solution (Taylor 1946).

Because analytical solutions for the dynamics of a spreading and decelerating relativistic jet cannot fully capture the details (sideways expansion and transition to the nonrelativistic regime) of the blast wave evolution, hereafter we use the high-resolution relativistic hydrodynamic simulations by Zhang & MacFadyen (2009) and van Eerten et al. (2012) for a jet expanding in a constant density ISM. These two-dimensional simulations include the transition from the relativistic to nonrelativistic regime, which is essential to accurately model the GRB outflow at late times (Zhang & MacFadyen 2009). The observed flux can be modeled as a function of eight parameters: the isotropic equivalent kinetic energy of the explosion, E_iso; the circumburst medium number density, n_ISM; the jet half-opening angle, θ₀; the observer's angle, θ_obs; the fraction of internal energy in the shock going into magnetic fields, _B; the fraction of internal energy going into accelerating electrons, _e; the fraction of electrons shock-accelerated in a power-law energy distribution, ξ_N ∼ 1; and the power-law index of the accelerated electron energy distribution, p. These parameters are determined by comparison with the observed fluxes (at each observed frequency and time) using a χ² minimization procedure (van Eerten et al. 2012).

In Table 2 and Figure 4 we report fit results for the second radio peak of PTF11qcj within the synchrotron SSA scenario described in Section 3.1 (Model 1). We impose a smooth radial evolution of the SN shock, i.e., we require that r₀ and α_r remain unchanged with respect to what was found during the first radio peak (Model 0), and attempt to model the second radio peak by varying the wind density profile (s and, in turn, α_B; see Equation (5)) and the magnetic field (B₀). This is justified by the consideration that a change in these parameters at fixed r₀ and α_r effectively corresponds to a change in CSM density as F_ν ∝ B⁴ ∝ n²_e (Wellons et al. 2012). We also allow for variations in ξ. The model was fit to the data obtained after MJD 56429, which is around the time rebrightening occurs, yielding a ${\chi }^{2}/\mathrm{dof}=575/35$ (see Table 2). We note that including the rising data points in the fit results in B₀ ≈ 1.9, s ≈ 0.87 and ξ ≈ 0.17, and a ${\chi }^{2}/\mathrm{dof}=1703/49$.

Table 2.  Best-fit Parameters for the Standard SSA Model Described in Section 3.1

Parameter	Model 0	Model 1
r0 (cm)	1.0 × 1016	1.0 × 1016
ξ	0.24	0.19
αr	0.81	0.81
te	55842	55842
s	2.0	1.13
B0 (G)	6.5	3.2
p	3.0	3.0
αB	−1.0	−0.64
${\gamma }_{m,0}$	7.4	11
αγ	−0.38	−0.38
${n}_{e,0}\,({\mathrm{cm}}^{-3})$	1.4 × 105	2.3 × 104
${\alpha }_{{n}_{e}}$	−1.6	−0.91
${\dot{M}}_{0}\,({M}_{\odot }\,{\mathrm{yr}}^{-1})$	9.8 × 10−5	1.7 × 10−5
${\alpha }_{\dot{M}}$	0.0	0.70
${E}_{k,0}$ (erg)	7.1 × 1048	1.7 × 1048
${\alpha }_{{E}_{k}}$	0.42	1.1
χ2/dof	1825/90	575/35

Note. Model 0 is the best fit for the first peak in the radio light curves, with fixed p = 3, s = 2, and t_e = 55842. Model 1 is the best fit for the late-time rebrightening phase. For Model 1, B₀, s, and ξ are varied.

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The physical parameters derived from Model 1 best-fit results are shown in Figure 5. When deriving the mass-loss rate and energy implied by the best-fit results (from Equations (10)–(11)), we assume a shell thickness of η = 10. Previous studies have assumed shell thicknesses in between η = 4 (e.g., Soderberg et al. 2006; Wellons et al. 2012) and η = 10 (Li & Chevalier 1999; Frail et al. 2000; Soderberg et al. 2005). This range is consistent with shell widths of η ≈ 3–10 measured via VLBI observations (Chevalier 1982; Bietenholz et al. 2003; Bartel et al. 2007; Roy et al. 2009). As is evident from Figure 5, the best fit (Model 1 in Table 2) requires an increase in energy and mass-loss rate at the start of the second radio peak due to the flattening of the density profile (i.e., smaller value of s in Model 1 than in Model 0; see the top left and bottom right panels of Figure 5). This is similar to what was observed in, e.g., the CSM-interacting SN 2003bg (Soderberg et al. 2006).

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In Figure 6 we show the uncertainties in the best-fit results for s and B₀, which shows that during the second radio peak a flattening in the CSM profile is also required (compared to the first peak). We note that a flattening in the density profile may be attributed to passage through a termination shock (e.g., Chevalier et al. 2004), although the simplified analytical model used here does not allow us to properly account for other sources of possible density profile variations such as, e.g., clumpiness in the stellar wind. More generally, the fact that the best-fit model to the second peak favors a value of s that is different from the standard s = 2 supports the hypothesis of a time-varying mass loss.

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We use the off-axis afterglow model described in Section 3.2 to model the second radio peak in the PTF11qcj light curve. In our fits, _e = _B = 0.33, and p are held fixed. For a fireball expanding in a constant density medium, we find a best fit with an explosion energy of E_iso ≈ 7 × 10⁵² erg, θ₀ ≈ 0.3 rad, ISM density ${n}_{\mathrm{ISM}}\approx 3\times {10}^{-5}\,{\mathrm{cm}}^{-3}$, ${\theta }_{\mathrm{obs}}\approx 0.6\,\mathrm{rad}$, and a ${\chi }^{2}/\mathrm{dof}\,=1167/36$ (Figure 4, dashed curve). The ISM density predicted by this model is rather low compared to typical long GRBs, which have ${n}_{\mathrm{ISM}}\gtrsim 1\,{\mathrm{cm}}^{-3}$. However, low ISM densities although peculiar are not unseen in long GRB broadband afterglow modeling (Kumar & Panaitescu 2000; Laskar et al. 2014, 2015; Alexander et al. 2017). We note that setting ${n}_{\mathrm{ISM}}\,=1\,{\mathrm{cm}}^{-3}$ (as typical for long GRBs) and allowing θ₀ and θ_obs to vary, returns a best fit with E_iso ≈ 2 × 10⁵³ erg, θ₀ ≈ 0.05 rad, θ_obs ≈ 0.17 rad, and a ${\chi }^{2}/\mathrm{dof}=4154/37$.

Our past (Corsi et al. 2014) and recent observations of PTF11qcj with Chandra all yielded detections. Here, we compare our X-ray data set with the predictions from the CSM-interacting SN and off-axis GRB scenarios described in Sections 4.1 and 4.2.

For each of our X-ray observations, we conservatively calculate the implied 1 keV flux density taking into account errors on both the X-ray spectral index (as estimated in Corsi et al. 2014), and on the measured count rate. More specifically, at each epoch we calculate the implied 1 keV flux range by extrapolating the full ±1σ range on the measured count rate using a power law of index spanning the range ${\rm{\Gamma }}={1.6}_{-0.71}^{+0.81}$ (Corsi et al. 2014). This yields the 1 keV fluxes (dots) and the large error bars shown in Figure 7. We then compare these extrapolated data points with specific model predictions.

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Within the SSA model for CSM-interacting SNe, we expect the radio-to-X-ray spectral index to be $\beta \lesssim -(p-1)/2$, and thus β ≲ −1 for p ≈ 3 (see Table 2). In Figure 7 we plot the SSA models shown in the bottom right panel of Figure 4, extrapolated to 1 keV by using β = −1. As is evident from this figure, and as already noted by Corsi et al. (2014), the SSA model for the first radio peak (black solid line in Figure 4) can explain the observed X-ray emission if no spectral break is present between the radio and X-rays (black solid line in Figure 7).

On the other hand, our latest Chandra data implies an X-ray flux that falls above the extrapolation to 1 keV of the best-fit SSA model for the second radio peak (dotted lines in Figures 4 and 7). The 1 keV flux predictions from the off-axis GRB model discussed in Section 4.2 (dashed lines in Figures 4 and 7) are instead consistent with this data point. Given the large measurement errors, as we discuss in the next section, VLBI observations are likely needed to securely confirm (or rule out) the off-axis GRB hypothesis.