SUPER-LUMINOUS TYPE Ic SUPERNOVAE: CATCHING A MAGNETAR BY THE TAIL

The strong tendency for these SL-SNe to be hosted in faint galaxies appears not to be a bias, which suggests a straightforward way of finding them in large volume, wide-field searches. With the Pan-STARRS1 survey, we have been running the "Faint Galaxy Supernova Survey" (FGSS) which is aimed at finding transients in faint galaxies originally identified in the Sloan Digital Sky Survey (SDSS) catalog (Valenti et al. 2010).

The Pan-STARRS1 optical system uses a 1.8 m diameter aspheric primary mirror, a strongly aspheric 0.9 m secondary and three-lens corrector and has 8 m focal length (Hodapp et al. 2004). The telescope illuminates a diameter of 3.3 deg and the "GigaPixel Camera" (Tonry & Onaka 2009) comprises a total of 60 4800 × 4800 pixel detectors, with 10 μm pixels that subtend 0.258 arcsec (for more details, see Magnier et al. 2013). The PS1 filter system is described in Tonry et al. (2012b), and is similar to but not identical to the SDSS griz (York et al. 2000) filter system. However, it is close enough that cataloged cross-matching between the surveys can identify high amplitude transients. In this paper, we will convert all the PS1 filter magnitudes g_P1r_P1i_P1z_P1y_P1 to the SDSS AB magnitude system as the bulk of the follow-up data were taken in SDSS-like filters (see Section 3 for more details).

The PS1 telescope is operated by the PS1 Science Consortium (PS1SC) to undertake several surveys, with the two major ones being the "Medium Deep Field" survey (e.g., Botticella et al. 2010; Tonry et al. 2012a; Gezari et al. 2012; Berger et al. 2012) which was optimized for transients (allocated around 25% of the total telescope time) and the wide-area 3π survey, allocated around 56% of the available observing time. As described in Magnier et al. (2013), the goal of the 3π survey is to observe the portion of the sky north of −30 deg declination, with a total of 20 exposures per year across all five filters for each field center. The 3π survey plan is to observe each field center four times in each of g_P1r_P1i_P1z_P1y_P1 during a 12 month period, although this can be interrupted by bad weather. As described by Magnier et al. (2013), the four epochs in a calendar year are typically split into two pairs called Transient Time Interval (TTI) pairs which are single observations separated by 20–30 minutes to allow for the discovery of moving objects. The temporal distribution of the two sets of TTI pairs is not a well-defined and straightforward schedule. The blue bands (g_P1, r_P1, i_P1) are scheduled close to opposition to enhance asteroid discovery with g_P1 and r_P1 being constrained in dark time as far as possible. The z_P1 and y_P1 filters are scheduled far opposition to optimize for parallax measurements of faint red objects. Although a large area of sky is observed each night (typically 6000 deg²), the moving object and parallax constraints mean the 3π survey is not optimized for finding young, extragalactic transients in a way that the PTF and La Silla-QUEST projects are. The exposure times at each epoch (i.e., in each of the TTI exposures) are 43 s, 40 s, 45 s, 30 s, and 30 s in g_P1r_P1i_P1z_P1y_P1. These reach typical 5σ depths of roughly 22.0, 21.6, 21.7, 21.4, and 19.3 as estimated from point sources with uncertainties of 02 (in the PS1 AB magnitude system described by Tonry et al. 2012b).

The PS1 images are processed by the Pan-STARRS1 Image Processing Pipeline (IPP), which performs automatic bias subtraction, flat fielding, astrometry (Magnier et al. 2008), and photometry (Magnier 2007). These photometrically and astrometrically calibrated catalogs produced in MHPCC are made available to the PS1SC on a nightly basis and are immediately ingested into a MySQL database at Queen's University Belfast. We apply a tested rejection algorithm and cross match the PS1 objects with SDSS objects in the DR7 catalog¹⁴ (Abazajian et al. 2009). We apply the following selection filters to the PS1 data (all criteria must be simultaneously fulfilled).

• 1.  
  PS1 source must have 15 < g_P1 < 20 or 15 < r_P1 < 20 or 15 < i_P1 < 20 or 15 < z_P1 < 20.
• 2.  
  SDSS counterpart must have 18 < r_SDSS < 23.
• 3.  
  Distance between PS1 source and SDSS source must be <3 arcsec.
• 4.  
  PS1 mag must be 1.5 mag brighter than SDSS (in the corresponding filter).
• 5.  
  The PS1 object must be present in both TTI pairs and the astrometric recurrences be <03. Objects with multiple detections must have rms scatter <01.
• 6.  
  The PS1 object must not be in the galactic plane (|b| > 5°).

All objects are then displayed through a Django-based interface to a set of interactive Web sites, and human eyeballing and checking takes place. We use the star–galaxy separation in SDSS to guide us in what may be variable stars (i.e., stellar sources in SDSS which increase their luminosity) or extragalactic transients (i.e., quasi-stellar objects (QSOs), active galactic nuclei (AGNs), and SNe). While the cadence of the PS1 observations is not ideal for detections of young SNe we have found many SNe in intrinsically faint galaxies. As of 2013 January, spectroscopically confirmed objects include 34 QSOs or AGNs and 41 SNe. Several of these have been confirmed as SL-SNe. Pastorello et al. (2010) presented the data of SN 2010gx recovered in 3π images, and in many cases the same object is detected by the Catalina Real-Time Transient Survey (CSS/MSS) and PTF. As we are not doing difference imaging and only comparing to objects in the SDSS footprint, there are some cases where objects are reported by PTF or CRTS and interesting pre-discovery epochs are available in PS1. As 3π difference imaging is not being carried out routinely, we often use the PTF and CRTS announcements to inform a retrospective search. In this paper, we present five SL-SNe Ic which were either detected through the PS1 FGSS, or were announced in the public domain. A sixth object (PTF12dam≡PS1-12arh) is discussed in a companion paper (M. Nicholl et al., in preparation). In all five cases follow-up imaging and spectroscopy was carried out as discussed below.

For all the SNe listed here (and throughout this paper) we adopt a standard cosmology with H₀ = 72 km s⁻¹, Ω_M = 0.27, and Ω_λ = 0.73. There is no detection of Na i interstellar medium (ISM) features from the host galaxies, nor do we have any evidence of significant extinction in the hosts from the SNe themselves. This suggests that the absorption in host is low and we assume that extinction from the host galaxies is negligible. Although we do detect Mg ii ISM lines from the hosts in some cases, there is no clear correlation with these line strengths and line of sight extinction. In all cases only the Milky Way foreground extinction was adopted.

PTF10hgi was first discovered by PTF on 2010 May 15.5 and announced on 2010 July 15 (Quimby et al. 2010). The spectra taken by PTF on May 21.0 UT and June 11.0 UT were reported as a blue continuum with faint features typical of SL-SNe Ic. Another spectrum obtained by PTF on July 7.0 UT was similar to PTF09cnd at three weeks past peak brightness. Initiated by Quimby et al. (2010), ultraviolet (UV) observations were obtained with Swift+UVOT in 2010 July, and we analyze those independently later in Section 3. PTF10hgi lies outside the SDSS DR9¹⁵ area (Ahn et al. 2012), hence was not discovered by our PS1 FGSS software. However, after the announcement (Quimby et al. 2010), we recovered it in PS1 images taken (in band r_P1) on 2010 May 18 and on four other epochs around peak magnitude (in bands g_P1r_P1i_P1), listed in Table A1.

We detect a faint host galaxy in deep griz-band images taken with the William Herschel Telescope on 2012 May 26 and the Telescopio Nazionale Galileo on 2012 May 28. At a magnitude r = 22.01 ± 0.07, this is too faint to affect the measurements of the SN flux up to +90 days. There are no host galaxy emission lines detected in our spectra, hence the redshift is determined through cross-correlation of the spectra of PTF10hgi with other SNe at confirmed redshift indicating a redshift of z = 0.100 ± 0.014, corresponding to a luminosity distance of d_L ∼ 448 Mpc. The Galactic reddening toward the SN line of sight is E(B − V) = 0.09 mag (Schlegel et al. 1998).

SN 2011ke was discovered in the CRTS (CSS110406:135058+261642) and PTF surveys (PTF11dij) with the earliest detections on 2011 April 6 and March 30, respectively (Drake et al. 2011; Quimby et al. 2011c). We also independently detected this transient in the PS1 FGSS (PS1-11xk) on images taken on 2011 April 15 (Smartt et al. 2011). However, earlier PS1 data show that we can determine the epoch of explosion to around 1 day, at least as far as the sensitivity of the images allow. On MJD 55649.55 (2011 March 29.55 UT) a PS1 image (r_P1 = 40 s) shows no detection of the transient to r ≃ 21.17 mag. Quimby et al. (2011c) report the PTF detection on 2011 March 30 (MJD = 55650) the night after the PS1 non-detection at g ≃ 21. PS1 detections then occurred one and three days after this on 55651.6 and 55653.6 (in i_P1 and r_P1), respectively. The photometry is given in Table A2. This is the best constraint on the explosion epoch of SL-SNe to date, allowing the rise time and light curve shape to be confidently measured. The object brightened rapidly, by ∼3 mag in the g band in the first 15 days and 1.7 mag in the subsequent 20 days. It was classified as SL-SNe Ic by both Drake et al. (2011) and Quimby et al. (2011c); their spectra obtained on May 8.0 and 11.0 UT, respectively, showed a blue continuum with faint features, similar to PTF09cnd about one week past maximum light (Quimby et al. 2011c).

We found a nearby source in the SDSS DR9 catalog (g = 21.10 ± 0.08, r = 20.71 ± 0.08), which is the host galaxy as confirmed by deep griz images taken with the William Herschel Telescope on 2012 May 26 at a magnitude g = 21.18 ± 0.05, r = 20.72 ± 0.04. The host emission lines set the SN at z = 0.143, equivalent to a luminosity distance of d_L = 660 Mpc. The Galactic reddening toward the position of the SN is E(B − V) = 0.01 mag (Schlegel et al. 1998).

PTF11rks was first detected by PTF on 2011 December 21.0 UT (Quimby et al. 2011a). Spectra acquired on December 27.0 UT and 31.0 UT showed a blue continuum with broad features similar to PTF09cnd at maximum light, confirming it as an SL-SN Ic. A non-detection in the r band on December 11 UT prior to discovery is also reported, setting a limit of >20.6 mag at this epoch. Quimby et al. (2011a) detailed a brightening of 0.8 mag in the r band in the first six days after the discovery. Prompt observations with Swift revealed a UV source at the optical position of the SN, but no source was detected in X-rays at the same epochs. There are no useful early data from PS1 for this object.

The host galaxy is listed in the SDSS DR9 catalog with g = 21.59 ± 0.11 mag and r = 20.88 ± 0.10 mag. Confirmation of these host magnitudes as achieved with our deep gr-band images taken with the William Herschel Telescope on 2012 September 22 at a magnitude g = 21.67 ± 0.07 and r = 20.83 ± 0.05. The emission lines of the host and narrow absorptions consistent with Mg ii λλ2796, 2803 doublet locate the transient at z = 0.19, corresponding to a luminosity distance of d_L = 904 Mpc. The Galactic reddening toward the position of the SNe is E(B − V) = 0.04 mag (Schlegel et al. 1998).

SN 2011kf was first detected by CRTS (CSS111230:143658+163057) on 2011 December 30.5 UT (Drake et al. 2012). The spectra taken by Prieto et al. (2012) on 2012 January 2.5 UT and 17.5 UT reveal a blue continuum with absorption feature typical of a luminous Type Ic SN.

The closest galaxy is ∼23'' S/W of the object position and is hence too far to be the host. There is no obvious host coincident with the position of this SN in SDSS DR9. We detect a faint host galaxy in deep gri-band images taken with the William Herschel Telescope on 2012 July 20. At a magnitude r = 23.94 ± 0.20, this is too faint to affect the measurements of the SN flux out to +120 days. The redshift of the object has been determined to be z = 0.245 from narrow Hα and [O iii], equivalent to a luminosity distance of d_L = 1204 Mpc. The foreground reddening is E(B − V) = 0.02 mag from Schlegel et al. (1998).

SN 2012il was first detected in the PS1 FGSS on 2012 January 19.9 UT (Smartt et al. 2012) and also independently discovered by CRTS on 2012 January 21 (CSS120121:094613+195028; Drake et al. 2012). On January 29 UT we obtained a spectrum of the SN, which resembled SN 2010gx four days after maximum light. The merged PS1 and CRTS data suggest a rise time of more than two weeks, different from that of SN 2010gx (Pastorello et al. 2010) but similar to PS1-11ky (Chomiuk et al. 2011). An initial analysis of observations from Swift revealed a marginal detection in the u, b, v, and uvm2 filters, with no detection in the uvw1 and uvw2 filters, or in X-rays (Margutti et al. 2012). However, our re-analysis of the Swift data reveals a detection above the 3σ level in the uvw1 and uvw2 filters (see Table A7). No radio continuum emission from the SN was detected by the EVLA (Chomiuk et al. 2012).

In our astrometrically calibrated images the SN coordinates are $\alpha =09^{\rm h}46^{\rm m}12\buildrel{\mathrm{s}}\over{.}91$ $\pm \; 0\buildrel{\mathrm{s}}\over{.}05$, δ = +19°50'2870 ±005 (J2000). This is within 012 of the faint galaxy SDSS J094612.91+195028.6 (g = 22.13 ± 0.08, r = 21.46 ± 0.07). The emission lines of Hα, Hβ and [O iii] from the host give a redshift of z = 0.175, corresponding to a luminosity distance of d_L = 825 Mpc. The Galactic reddening toward SN 2012il given by Schlegel et al. (1998) is E(B − V) = 0.02 mag, three times lower than the value reported by Margutti et al. (2012).

3.1.1. PTF10hgi

Pre-peak observations are available only in the r band, suggesting a rise time comparable to SN 2011ke. PTF10hgi shows a bell-like-shaped light curve around peak. The post-maximum light curve shows a constant decline in all the bands until 40 days. After 40 days, the decline rate of PTF10hgi changes to have a slope similar to the decays shown by the other SNe. The change in the i-band slope is not as evident, while the z-band light curve is also dissimilar to the other bands. The magnitudes beyond 90 days are evaluated using the template subtraction with 646–648 day epochs as template images.

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3.1.2. SN 2011ke

3.1.3. PTF11rks

3.1.4. SN 2011kf

3.1.5. SN 2012il

In calculating absolute magnitudes (and subsequently bolometric magnitudes), we have assumed negligible internal host galaxy reddening for all the objects, and applied only foreground reddening, with the values reported in Section 2. No Na i D absorption features due to gas in the hosts were observed. However, we cannot exclude possible dust extinction from the hosts, therefore the absolute magnitudes reported here are technically lower limits. Given that the hosts are all dwarf galaxies, and the transients have quite blue spectra around peak, it appears that any correction would be small. We computed k-corrections for each SN using the spectral sequence we have gathered. For photometric epochs for which no spectra were available, we determined a spectral energy distribution (SED) using the multi-color photometric measurements available. This SED was then used as a spectrum template to compute the K-corrections. Comparisons between the two methods (K-correction directly from spectra, or with the photometric colors) showed no significant differences. We also determined K-corrections for SN2010gx using the spectral method and using photometric colors. Again we found consistency between the two methods. After applying foreground reddening corrections and K-corrections we estimated absolute rest-frame peak magnitudes (cf. Table 1).

In Figure 3, we compare the rest-frame g-band absolute light curves (in the AB system) of the SNe studied here with those of other low-z super-luminous events and the well-studied Type Ic SN 1998bw (Galama et al. 1998; McKenzie & Schaefer 1999; Sollerman et al. 2000; Patat et al. 2001). The epochs of the maxima were computed with low-order polynomial fits and by comparison of the light curves and their color evolution with those of other SL-SNe, and are listed in Table 1. The absolute peak magnitudes of PTF11rks and PTF10hgi are fainter than the bulk of SL-SNe, although they are still ∼2 mag brighter than SN 1998bw. Interestingly, the two faintest SL-SNe Ic display different decline rates to each other, PTF11rks is similar to SN 2010gx whereas PTF10hgi decreases at a slower rate. The other three objects have peak magnitudes comparable with that of SN 2010gx (M_g ≈ −21.67) and show a similar decline. The decline slope changes in 4 of the 5 objects after 50 days in the rest frame (while for the other, SN 2011ke, our data do not constrain it). The light curves then settle on a tail resembling the decay of ⁵⁶Co. This is apparent in Figure 3 as the tails of the SL-SNe are similar to that of SN 1998bw which is known to be powered by ⁵⁶Ni. The light curves follow the ⁵⁶Co decay within errors of 10%, the biggest discrepancy is for SN 2011kf which falls more rapidly between 100–200 days. SN 2007bi (Gal-Yam et al. 2009; Young et al. 2010) also followed the ⁵⁶Co decay at late times, but with a tail that is ∼2 mag brighter than the SL-SNe Ic, and with a much slower overall evolution. We also note that the light curves of our objects flatten at slightly different epochs, with the tail for SN 2011ke commencing ∼10 days after the last point in the light curve for SN 2010gx.

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We computed rest-frame color curves, after accounting for the reddening and redshift effects of time dilation and K-correction. The color curves are useful probes of the temperature evolution of the SNe. We also calculated rest-frame color evolution of SN 2010gx, the only other SL-SN Ic with a good coverage in SDSS filters at a similar redshift. In Figure 4, the SL-SNe show a constant color close to g − r = 0 from the pre-peak phase to ∼15 days. This evolution is similar to the color evolution of the higher redshift PS1-10ky in the observed bands i_P1–z_P1 (Chomiuk et al. 2011).

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The constant color until 15 days implies that the SED does not evolve over these epochs. Up to maximum light, the spectra of these SL-SNe appear to be blue, with the only strong features being the O ii lines in this range covered by the gr filters (Pastorello et al. 2010; Quimby et al. 2011b; Chomiuk et al. 2011; Leloudas et al. 2012). Hence this could be due to an approximately constant temperature. This is also illustrated in Figure 8 of Chomiuk et al. (2011) for the higher redshift PS1-10ky. Close to peak the O ii lines disappear, leaving the spectra featureless for ∼10 days, while after peak the temperature begins to decrease (see Section 5.1). A monotonic temperature decline between 14,000 K and 12,000 K (blackbody peak 2100 Å ≲ λ ≲ 2500 Å) in objects with featureless spectra does not strongly affect the color evolution for λ ≳ λ_peak, as the slopes of blackbodies at these two temperatures are quite similar. To detect differences in temperature between a 12,000 K and a 14,000 K blackbody requires color curves which sample rest wavelengths below 3800 Å, such as the g − z color of PS1-10ky. Indeed this object did show an increase in g − z.

After this early period of constant color, the g − r color increases, reaching another phase of almost constant value at ∼40 days, perhaps indicating a decrease in the cooling rate. There are some exceptions to this behavior, as seen for PTF11rks and PTF10hgi. The g − r color of PTF11rks increases earlier and with a steeper slope than the other SNe, but unfortunately our data stop before the possible second period of constant color. In contrast, the g − r color for PTF10hgi increases much more slowly, and only reaches the possible late constant phase at ∼80 days.

The r − z colors of the sample show a roughly constant increase from peak to ∼50–60 days, when the color evolution appears to flatten. The two exceptions are again PTF11rks and PTF10hgi; the former increases in r − z more rapidly than the other objects, whereas the latter does not become as red, and experiences a clear decrease in r − z after 80 days. The r − z color evolution of SN 2010gx is similar to that of SN 2011ke and PTF10hgi.

In Figure 5 the evolution of the temperature is plotted. This is derived from a blackbody fit to the continuum of our spectra (see Section 5), and compared to those of SN 2010gx and SN 2007gr. We also fit color temperatures at rest frame with a blackbody and the measurements are in good agreement with those from spectra.

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Only PTF11rks has a good temperature coverage around peak, whereas our spectroscopic data are not well sampled at that phase. While we cannot clearly confirm the apparent constant temperature seen in SN 2010gx until ∼10 days, the epochs at either side of ±10 days are suggestive of a roughly constant temperature phase. After ∼10 days a clear decline in temperature is seen, with a rate of decline of ∼2500 K over 10 days. This decline continues until the SN reaches a constant temperature of ∼6000 K prior to, and during, the pseudo-nebular phase.

The expansion velocities measured for PTF11rks, SN 2011ke, SN 2012il, PTF10hgi, and SN 2011kf are reported in Table 3, and are compared with those of SN 2010gx and the standard Type Ic SN 2007gr (Hunter et al. 2009) in Figure 11. In the photospheric phase these were derived from the minima of the P Cygni profiles and their errors were established from the scatter between several independent measurements. During the phase in which the objects appear to be transitioning to the nebular phase (i.e., beyond about 50 days) the velocities were computed as the FWHM of the emission lines (we will call this the pseudo-nebular phase). These are not tracers of the photospheric velocity and are reported for completeness; they will be discussed further in Section 6.2. We used Fe ii λ5169, Mg iiλ 4481, and O i λ7775 to measure velocities during the photospheric phase, and Ca ii H&K and Mg i] λ4571 during the pseudo-nebular phase. The Fe ii velocity evolution monotonically declines for all transients, ranging from ∼18,000 km s⁻¹ around peak to ∼8000 km s⁻¹ at 50 days. After this epoch we have only PTF10hgi spectra, and they still show clear absorption components of Fe ii. The decline in velocity is faster and it reaches ∼4600 km s⁻¹ at the last epoch of 80 days, which is quite similar to the Fe ii velocity of SN 2007gr. Mg ii in the photospheric phase is always seen at higher velocity than Fe ii, and decreases linearly by ∼1000 km s⁻¹ every 10 days. The O i velocity is comparable with that of Fe ii, with the exception of SN 2011ke where it is slower than all other ions (albeit with large uncertainties due to the low S/N). Ca ii and Mg i] appear after ∼40 days from maximum light and show the same intensity and velocity of ∼11,000 km s⁻¹ in the entire sample.

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The PTF10hgi spectra after 60 days show Ca ii absorption at late epochs, similar to normal Type Ic SNe. Figure 11 shows evidence for decreasing line velocity with time, especially from 10 days post-peak as seen for SN 2010gx (previously reported as a comparison in Chomiuk et al. 2011). Our analysis shows a clear sign of change in the rate of photospheric expansion (30–40 days), with a decline which resembles that typically observed during the photospheric phase of a CCSN explosion.

Alternative models have been proposed: the spin down of a rapidly rotating young magnetar (Kasen & Bildsten 2010; Woosley 2010; Dessart et al. 2012); interaction of the SN ejecta with a massive (3–5 M_☉) C/O-rich CSM (Blinnikov & Sorokina 2010); shock breakout from dense mass loss (Chevalier & Irwin 2011); or pulsational pair instability in which collisions between high-velocity shells are the source of multiple, bright optical transients (Woosley et al. 2007).

The first scenario considered is the pulsational pair-instability model, where the luminosity is powered by the collision of shells of material ejected at different times by the pulsations (see Chatzopoulos & Wheeler 2012 for the implementation of this on SL-SNe Ic). The outbursts are expected to be energetic, reaching very high peak luminosities and creating hot (T_eff ≈ 25,000 K), optically thick photospheres (Woosley et al. 2007). PS1 has occasionally observed the explosion sites of the five SL-SNe presented here, and SN 2010gx on 3–5 occasions (per event) in one of g_P1r_P1i_P1z_P1y_P1 filters, reaching typical AB magnitudes of 22, 21.6, 21.7, 21.4, and 19.3 (see Section 2). This would correspond to absolute magnitudes of roughly −16 to −19. We found no detection of any previous outburst to these magnitude limits in the 1–2 yr periods before explosion. Typically there were 5–10 epochs of images from PS1. This is not a constant monitoring period, and it does not rule out that pre-explosion outbursts occur, but we have found no evidence for them.

The second scenario is that of circumstellar interaction proposed by Chevalier & Irwin (2011) in which the interaction between the ejecta and the CSM converts the ejecta's kinetic energy into radiation. This requires the diffusion radius of the SN to be about the radius of the stellar wind expelled by the SN progenitor. Based on this model of Chevalier & Irwin (2011), Chomiuk et al. (2011) determined the dense wind and ejecta parameters for the observed z ≃ 1 SL-SNe. This requires a mass-loss rate of 6 M_☉ in the last year before the SN explosion, with an outer radius of around 40,000 R_☉, and ejecta mass of 10 M_☉ (consistent with the estimates reported in Moriya & Tominaga 2012). The rise times of our low-z SNe are similar to that of PS1-10awh (from Chomiuk et al. 2011)—around 10–30 days which would lead to similar estimates of masses. The detection of He i in SN 2012il at ∼53 days is perhaps some hint that the dense CSM wind could be plausible, since it is difficult to find any known massive star example in the local universe which has a dense and extended wind comprising 6 M_☉ of C+O material only. Although the S/N in the NIR is low, the He i line is relatively broad, indicating that it arises from ejecta. Although most SL-SNe have similar rise times which are feasible in the diffusion model of Chevalier & Irwin (2011), the light curves of SN 2010gx (Pastorello et al. 2010), PTF11rks and SN 2012il are not symmetric, which one would expect in this dense wind scenario. However, Ginzburg & Balberg (2012) showed as the rise and fall times can be different. They also showed that the tail phase can be explained as diffusion from the inner layers which can slow the decline. We have not investigated in this scenario in depth, but a combination of this model plus a ⁵⁶Ni tail would be unlikely because of the necessity of full γ-ray trapping (see Section 4).

A variant of the previous scenario was proposed by Blinnikov & Sorokina (2010) claiming high luminosities from a radiative shock in a massive C–O shell. The shells have radii and density profiles that are similar to the dense wind of Chevalier & Irwin (2011), with a density gradient of ρ(r)∝r^−1.8 and radii of the order of 10⁵ R_☉. Considering a total ejecta mass M_ej ≲ 1 M_☉, which collides with a shell of mass 5 M_☉, these models have initial energies (2–4 × 10⁵¹ erg) higher than those retrieved by the expansion velocities during the pseudo-nebular phase in our set of spectra (∼1 × 10⁵¹ erg). Moreover, the light curves are too shallow and they are not able to reproduce the SN 2010gx decline after 30 days post-maximum light. We find no unequivocal signs of interaction in the spectra of the objects.

In Sections 3.1 and 4 we have presented detections of the SNe at later times than published to date, due to our focus on low-redshift candidates. We detect a flattening of the light curve and a tail phase in four out of five transients and this slope appears to be consistent with ⁵⁶Co decay. In summary, these interaction scenarios can reproduce the peak energy and diffusion time. However, the ejecta/shell velocity should be lower (by about a factor of two) than those observed. In addition, one still needs another power source for the late time luminosity that we now detect in these SL-SNe Ic.

As shown previously by Chatzopoulos et al. (2009), Pastorello et al. (2010), Quimby et al. (2011b), Chomiuk et al. (2011), Chatzopoulos et al. (2012), and Leloudas et al. (2012), the light curve peaks cannot be fit with a physically plausible ⁵⁶Ni diffusion model like normal SNe Ic, and our similarly shaped light curves result in the same conclusion. If the tail phase was actually due to ⁵⁶Co powering, then approximately 1–4 M_☉ of ⁵⁶Ni would be required, but this would not be enough to power the peak luminosity solely through radioactive heating. In Figure 12 we also show the best-fitting ⁵⁶Ni-powered models, under the assumptions that the ⁵⁶Ni mass must be <50% ejecta mass, and that the ejecta velocities are less than 15,000 km s⁻¹. The first assumption is based on the implausibility of a pure ⁵⁶Ni ejecta; Umeda & Nomoto (2008) found typical ⁵⁶Ni masses which are at most 20% of the ejecta, while if the ejecta was comprised of more ⁵⁶Ni than this, we would expect to see spectra dominated by Fe-group rather than intermediate-mass elements. The velocity constraint is motivated by the observed velocities in our sample. Thus we derived kinetic energies of 6.2 ≲ E(10⁵¹ erg) ≲ 15.0, ejecta masses of 5.9 ≲ M_ej(M_☉) ≲ 13.0, and ⁵⁶Ni masses of $2.9\lesssim M_{\rm ^{56}{\rm Ni}}$(M_☉) ≲ 6.0 (see Appendix D.5 for further details).

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From the fits, it appears that no physical and consistent solutions for ⁵⁶Ni heating can be determined, as found by previous authors (Pastorello et al. 2010; Quimby et al. 2011b; Chomiuk et al. 2011; Leloudas et al. 2012). One could invoke a combination of CSM interaction to explain the peak luminosity and then ⁵⁶Ni masses of 1–4 M_☉ to account for the tail phases, but, as discussed above, this requires full γ-ray trapping and somewhat fine tuning of the two scenarios to work in unison.

Kasen & Bildsten (2010), Woosley (2010), and Dessart et al. (2012) have already proposed that a rapidly spinning magnetar can deposit its rotational energy into an SN explosion and significantly enhance the luminosity. This appears to be an appealing scenario as the model is fairly simple, and this additional power source can potentially transform a canonical Type Ic SN into an SL-SN Ic. To investigate this further and quantitatively compare our extensive light curves with this model, we have derived semi-analytical diffusion models. We use standard diffusion equations derived by Arnett (1982) and add magnetar powering (as in Kasen & Bildsten 2010) to fit the light curves of our five objects. A full description can be found in Appendix D. Assuming full trapping of the magnetar radiation,¹⁸ the ejecta mass M_ej, explosion energy E_k, and the opacity κ only influence the bolometric light curve through their combined effect on the diffusion timescale parameter (see Appendix D):

$$\begin{equation} \tau _{\rm m} = 10\, {\rm d} \left(\frac{M_{\rm ej}}{1\, M_\odot }\right)^{3/4}{ \left(\frac{E_{\rm k}}{10^{51}\, {\rm erg}}\right)^{-1/4}\left(\frac{\kappa }{0.1\, {\rm cm^{2}\,g^{-1}} }\right)}^{1/2}. \end{equation} \tag{ 2 }$$

The magnetar luminosity depends on two parameters, the magnetic field strength B₁₄ (expressed in terms of 10¹⁴ G) and the initial spin period P_ms (in milliseconds). Combined with the explosion date t₀, we therefore have four free parameters to fit. Table 4 lists the best-fit parameters for each object, and Figure 12 shows the fits. As the χ² fitting gives good matches to models without⁵⁶Ni, we have no need to introduce ⁵⁶Ni as an additional free parameter. All the models have M(⁵⁶Ni) = 0 M_☉ and we investigated the sensitivity to the assumed ⁵⁶Ni mass by recomputing the fits including 0.1 M_☉ of ⁵⁶Ni in the ejecta (the typical ⁵⁶Ni yield in core-collapse SNe). We find virtually the same fit parameters as we do without the nickel. As can be seen from Figure 13 (top), the two magnetar models (B₁₄ = 5, P_ms = 5, M_ej = 5 M_☉) with (red dashed) and without ⁵⁶Ni (black solid) are similar. The late decline rate in the magnetar model (>100 days) is actually quite similar to the ⁵⁶Co decay rate as shown in Figure 13 (bottom), but fully trapped γ-rays are required. As Figure 13 shows, this full trapping is different from the typical light curve of a radioactivity-powered Type Ib/c SN in which full trapping is not observed. Recently, Dexter & Kasen (2012) showed that fall-back accretion can give a similar asymptotic behavior of the light curve (L_t∝t^−5/3) which is a scenario that would need further investigation.

Table 1. Main Properties of the SN Sample

	PTF10hgi	SN 2011ke	PTF11rks	SN 2011kf	SN 2012il
Alternative	PSO J249.4461+06.20815	PS1-11xk, PTF11idj,		CSS111230:	PS1-12fo
names		CSS110406:		143658+163057	CSS120121:
		135058+261642			094613+195028
α (J2000.0)	$16^{\rm h}37^{\rm m}47\buildrel{\mathrm{s}}\over{.}08$	$13^{\rm h}50^{\rm m}57\buildrel{\mathrm{s}}\over{.}78$	$01^{\rm h}39^{\rm m}45\buildrel{\mathrm{s}}\over{.}49$	$14^{\rm h}36^{\rm m}57\buildrel{\mathrm{s}}\over{.}64$	$09^{\rm h}46^{\rm m}12\buildrel{\mathrm{s}}\over{.}91$
δ (J2000.0)	06°12'2935	26°16'4240	29°55'2687	16°30'5717	19°50'2870
z	0.100	0.143	0.190	0.245	0.175
Peak g (mag)	−20.42	−21.42	−20.76	−21.73	−21.56
E(B − V) (mag)	0.09	0.01	0.04	0.02	0.02
Lgriz peak (× 1043 erg s−1)	2.09	4.47	3.24	6.45	≳4.47
Light curve peak (MJD)	55326.4 ± 4.0	55686.5 ± 2.0	55932.7 ± 2.0	55925.5 ± 3.0	55941.4 ± 3.0
Host r (mag)	−16.50	−18.42	−19.02	−16.52	−18.18

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Table 4. Best-fit Parameters for Magnetar Modeling of the Bolometric Light Curves and $\chi ^2_{\rm red}$ Value on the Left; Derived Parameters on the Right

Object	τm	B14	Pms	t0	$\chi ^2_{\rm red}$		Emag	Mej	$V_{\rm core}^{\rm final}$
	(days)			(MJD)			(1051 erg)	(M☉)	(km s−1)
SN 2011ke	35.0	6.4	1.7	55650.65	1.8		6.9	8.6	12400
SN 2011kf	15.4	4.7	2.0	55920.65	5.8		5.0	2.6	19600
SN 2012il	18.4	4.1	6.1	55918.56	3.9		0.5	2.3	10500
PTF10hgi	26.2	3.6	7.2	55322.78	0.5		0.4	3.9	7700
PTF11rks	21.0	6.8	7.5	55912.11	5.0		3.6	2.8	16500
SN 2010gx	32.4	7.4	2.0	55269.22	3.9		5.0	7.1	11900

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The light curves are quite well reproduced with appropriate choices of parameters, and the tail phase luminosities that we measure can also be explained with this model. The diffusion timescale parameters are between 15 and 35 days, which corresponds to ejecta masses of 2–9 M_☉ for κ = 0.1 cm² g⁻¹ (see Appendix D for further details about κ) and

$$\begin{equation} E_{\rm k}=10^{51} + \frac{1}{2} (E^{\rm mag}-E^{\rm rad}) {\rm (erg)}, \end{equation} \tag{ 3 }$$

where E ^mag is the total energy of the magnetar and E ^rad is the total radiated energy of the SN. We use a factor of 1/2 for an approximation of the average kinetic energy over the magnetar energy input phase, which we show in Appendix D.4 produces good agreement with more detailed time-dependent calculations of E_k.¹⁹ These ejecta masses are consistent with the ones derived for radioactivity-powered Type Ib/c ejecta (Ensman & Woosley 1988; Shigeyama et al. 1990; Valenti et al. 2011; Drout et al. 2011; Eldridge et al. 2013). Furthermore, Figure 14 shows that the evolution of photospheric velocities and temperatures match the observed ones reasonably well. While the velocity and temperature evolution as estimated from our models are crude, this good agreement is an important test for the physical self-consistency of the model. From the light curve fits to ejecta mass and kinetic energy, we estimate the ejecta velocities (V_core; see Equation (D11)) which we compare to the observed velocities of the emission lines (Ca H&K and Mg i] λ4571; see Table 3), finding them reasonably similar.

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From the plot of bolometric light curves shown in Figure 6, it appears that there is a gap between the faintest SL-SNe Ic and the brightest normal Type Ic. One possibility is that this is an observational bias. As SL-SNe Ic are intrinsically rare (Quimby et al. 2011b), we find them at moderate redshifts simply due to the large survey volume required to find them. If intermediate objects were also as rare, then we would require wider field searches to encompass more local volume as they may evade detection at the higher redshifts of known population of SL-SNe. An alternative explanation is that the mechanism powering the SL-SNe has a minimum energy. From the semi-analytic magnetar model of Kasen & Bildsten (2010), which we have used to match our light curves, we see that the peak magnetar luminosity is inversely proportional to the square of the magnetic field (Equation (4) of Kasen & Bildsten 2010). Hence, the question arises, could the apparent lower luminosity limit to the SL-SNe Ic be caused by some physical upper limit to the magnetic field in a magnetar? For our faintest SL-SNe, and using a minimum plausible ejecta mass of 1 M_☉, we determine an upper limit to the magnetar B-field of B < 1.4 × 10¹⁵ G from Equation (4) of Kasen & Bildsten (2010). If we consider that Kasen & Bildsten (2010) assume an angle between the B field and the spin axis of the magnetar of 45°, then a factor of two higher than this value could be a plausible maximum. We hence adopt B < 3 × 10¹⁵ G.

The most conservative limit we can set is that the magnetic energy in the magnetar must be less than the gravitational binding energy of the neutron star (Chandrasekhar & Fermi 1953). This implies that

$$\begin{equation} B<10^{18} { \left(\frac{M_{{\rm NS}}}{1.4\, M_{\odot }}\right)} {\rm \left(\frac{R_{{\rm NS}}}{10\, {\rm km}}\right)}^{-2} \, {\rm G} \end{equation} \tag{ 4 }$$

which does not set a particularly stringent upper limit on the B-field. This limit is consistent with the B values retrieved from all the galactic magnetars studied so far which have B ∼ 10¹⁴–10¹⁵ G (Woods & Thompson 2006). However, magnetic fields are known to be a possible source of braking in stars (Meynet et al. 2011), while the magnetar models require an extreme magnetic field and fast rotational period. An explanation for both a small rotational period and a large magnetic field could be a large-scale helical dynamo that is possible when the rotation period is comparable to the timescale of the convective motions (Duncan & Thompson 1992).

The opacity due to electron scattering is

$$\begin{equation} \kappa _{\rm es} = 0.2 \left(\frac{\bar{Z}/\bar{A}}{0.5}\right)\left(\frac{x_{\rm e}}{\bar{Z}}\right)\, \mbox{cm}^2\, \mbox{g}^{-1}, \end{equation} \tag{ D9 }$$

where $\bar{Z}$ is the mean atomic charge, $\bar{A}$ is the mean atomic mass, and x_e = n_e/n_nuclei is the ionization fraction. For any chemical mixture excluding hydrogen, $\bar{Z}/\bar{A} \approx 0.5$, which we can assume since there is no trace of hydrogen lines in the spectra of these objects. The radiation temperature in a radiation-pressure-dominated uniform sphere of internal energy E(t) is

$$\begin{eqnarray} T(t) &=& \left(\frac{E(t)}{a \frac{4\pi }{3} V_{\rm ej}^3 t^3}\right)^{1/4} \nonumber\\ &\sim & 10^5\ {\rm K} \left(\frac{E(t)}{10^{51}\ \mbox{erg}}\right)^{-1/8}\left(\frac{M}{1\ M_\odot }\right)^{3/8} \left(\frac{t}{10 \,{\rm days}}\right)^{-3/4}.\nonumber\\ \end{eqnarray} \tag{ D10 }$$

In LTE, and at relevant densities here, the ionization fraction of a pure oxygen plasma at T = 10⁵ K is x_e ∼ 6, for carbon it is x_e ∼ 4, and for Fe is x_e ∼ 10. It should thus be a reasonable approximation to use $x_{\rm e}/\bar{Z} = 0.5$. With this choice κ_es = 0.1 cm² g⁻¹, comparable to the value κ = 0.08 that Arnett (1982) uses.

In rapidly expanding media, line opacity can become comparable to or exceed electron scattering opacity (Karp et al. 1977), which complicates analytical light curve modeling. A common simplistic approach to take line opacity into account is to put a minimum value on the opacity. In previous works, values ranging between 0.01 and 0.24 cm² g⁻¹ have been used (e.g., Herzig et al. 1990; Swartz et al. 1991; Young 2004; Bersten et al. 2011, 2012). The implication of line opacity is that κ is likely to stay in the 0.01–0.2 cm² g⁻¹ range throughout the evolution, and κ = 0.1 cm² g⁻¹ is probably not off by more than a factor of a few at any given time.

The spectral and directional distribution of the magnetar radiation is highly uncertain. At early times, most radiation will undoubtedly be absorbed, but at later times this is not certain (Kotera et al. 2013). X-rays are efficiently absorbed for a long time by photoionization from inner shell electrons, but gamma-rays see a smaller opacity and start escaping after a few weeks for typical ejecta parameters.

Here, we assume full trapping of the magnetar radiation. Most of the energy input occurs over a few weeks after explosion, and it is a reasonable assumption that most of the energy emerges as X-rays, as in the Crab pulsar, for instance (Weisskopf et al. 2000).

To estimate the photospheric velocities and temperatures for a given solution for ejecta mass, we approximate the ejecta density structure with a core + envelope morphology. The velocity of the homogenous core (which contains almost all the mass to force consistency with the diffusion calculation) is determined from

$$\begin{equation} \frac{3}{10}M_{\rm ej} V_{\rm core}^2 = E_{\rm k} \end{equation} \tag{ D11 }$$

and its density is given by

$$\begin{equation} \rho _{\rm core}(t) = \frac{M_{\rm ej}}{\frac{4\pi }{3} V_{\rm core}^3 t^3}. \end{equation} \tag{ D12 }$$

Outside we attach an envelope with density profile

$$\begin{equation} \rho (t,V) = \rho _{\rm core}(t)\left(\frac{V}{V_{\rm core}}\right)^{-\alpha }. \end{equation} \tag{ D13 }$$

The optical depth of the envelope is

$$\begin{eqnarray} \tau _{\rm env} &=& \int _{R_{\rm core}}^\infty \rho _{\rm core}\left(\frac{r}{R_{\rm core}}\right)^{-\alpha } \kappa dr \end{eqnarray} \tag{ D14 }$$

$$\begin{eqnarray} &=& \frac{\rho _{\rm core} R_{\rm core} \kappa }{\alpha - 1} \end{eqnarray} \tag{ D15 }$$

$$\begin{eqnarray} &=& \frac{\tau _{\rm core}}{\alpha -1}, \end{eqnarray} \tag{ D16 }$$

where

$$\begin{equation} \tau _{\rm core}(t) = \kappa \rho _{\rm core}(t)V_{\rm core} t. \end{equation} \tag{ D17 }$$

One should in general distinguish between the photosphere and the thermalization layer, where the temperature of the continuum is determined. The position of these depends on the total opacity κ and the absorption opacity κ_a, respectively. However, due to our simple treatment such detail is unwarranted, and we approximate them to be the same, R_*. As long as the atmosphere is optically thick (τ_env > 1), R_*(t) is found from solving

$$\begin{equation} \int _{R_*(t)}^\infty \kappa \rho (r,t) dr = 1, \end{equation} \tag{ D18 }$$

which gives

$$\begin{equation} R_*(t) = R_{\rm core}(t)\left(\frac{\alpha -1}{\tau _{\rm core}(t)}\right)^{\frac{1}{1-\alpha }}. \end{equation} \tag{ D19 }$$

If the photosphere has receded into the core, the expression is instead

$$\begin{equation} R_*(t) = R_{\rm core}(t) - \frac{1 - \frac{\tau _{\rm core}}{\alpha -1}}{\kappa \rho _{\rm core}}. \end{equation} \tag{ D20 }$$

Typical density profiles in Ib/c models show α ∼ 10 (e.g., Figure 1 in Kasen & Bildsten 2010), which is the value we use here.

From the photospheric radius R_*(t) we retrieved the velocity at the photosphere (V_phot)

$$\begin{equation} V_{\rm phot}(t)=V_{\rm core}\frac{R_*(t)}{R_{\rm core}(t)}. \end{equation} \tag{ D21 }$$

The effective temperature is found from application of the blackbody formula, using the model luminosities at corresponding times.

To test the applicability of our approach, we perform three tests. The first one is to compare light curves, photospheric velocities, and temperatures to the radiation hydrodynamical simulations of Kasen & Bildsten (2010). Figure 17 shows three representative examples. For these comparisons, we use Equation (3) to relate magnetar energy to ejecta kinetic energy, which shows itself a good approximation. Table A9 shows the derived parameters of M_ej, B₁₄, P_ms, and Δt for a series of simulations in Kasen & Bildsten (2010). The magnetar properties (B₁₄ and P_ms) are reliably recovered to within 10%–20% of their input values, whereas the ejecta mass can vary, tending to be too low in the fits. It is, however, always within a factor two of the correct value.

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The second test is to derive parameters for the Type IIb SN 2011dh, using the approach described above (fixing E_k = 10⁵¹ erg, κ = 0.1, α = 10). The best fit is shown in Figure 18. The recovered values M_ej = 2.2 M_☉ and M_56Ni = 0.08 M_☉ agree well with the results from Bersten et al. (2011), who used radiation hydrodynamical simulations.

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While for the third one, we tried to test our code on Type Ic SNe 1998bw (Galama et al. 1998; McKenzie & Schaefer 1999; Sollerman et al. 2000; Patat et al. 2001) and 2007gr (Valenti et al. 2008; Hunter et al. 2009) to explore its limits. We fitted the first 200 days with the magnetar model (fixing E_k = 10⁵¹ erg, κ = 0.1, α = 10), recovering M_ej = 1.8 M_☉ and B₁₄ = 9.8 G and P_ms = 18.9 ms for SN 1998bw and M_ej = 1.3 M_☉ and B₁₄ = 22.9 G and P_ms = 47.6 ms for SN 2007gr. Instead, trying to fit the entire data set, we recover M_ej = 19.9 M_☉ and B₁₄ = 16.6 G and P_ms = 0.5 ms for SN 1998bw and M_ej = 2.5 M_☉ and B₁₄ = 26.0 G and P_ms = 42.6 ms for SN 2007gr. The best fit to those data are shown in Figure 19 along with the fit of the radioactive decay. The applicability of the Arnett model to derive parameters for radioactivity-driven SNe has also been demonstrated by Valenti et al. (2011). The results imply that at those energies the magnetar contribution is more important after 150–200 days, highlighting the importance of obtaining late-time data.

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For the ⁵⁶Ni models we computed γ-ray trapping as in Arnett (1982). The parameters are reported in Table A10.