The late-time light curve of the type Ia supernova SN 2011fe

TL;DR

This study analyzes late-time optical light curves of supernova SN 2011fe, revealing deviations from simple radioactive decay models and evidence for positron escape and infrared contributions, providing insights into supernova energy mechanisms.

Contribution

It presents densely sampled late-time optical data and models the bolometric light curve considering positron escape and infrared effects, highlighting the presence of $^{57}$Ni and constraining its ratio.

Findings

Light curve decline faster than radioactive decay models predict between 450-600 days.

Evidence supports positron escape and infrared contributions affecting the light curve.

Detection of $^{57}$Ni decay chain with a high mass ratio, but no significant $^{55}$Co detection.

Abstract

We present late-time optical $R$-band imaging data from the Palomar Transient Factory (PTF) for the nearby type Ia supernova SN 2011fe. The stacked PTF light curve provides densely sampled coverage down to $R\simeq22$ mag over 200 to 620 days past explosion. Combining with literature data, we estimate the pseudo-bolometric light curve for this event from 200 to 1600 days after explosion, and constrain the likely near-infrared contribution. This light curve shows a smooth decline consistent with radioactive decay, except over ~450 to ~600 days where the light curve appears to decrease faster than expected based on the radioactive isotopes presumed to be present, before flattening at around 600 days. We model the 200-1600d pseudo-bolometric light curve with the luminosity generated by the radioactive decay chains of $^{56}$Ni, $^{57}$Ni and $^{55}$Co, and find it is not consistent with models that have full positron trapping and no infrared catastrophe (IRC); some additional energy escape other than optical/near-IR photons is required. However, the light curve is consistent with models that allow for positron escape (reaching 75% by day 500) and/or an IRC (with 85% of the flux emerging in non-optical wavelengths by day 600). The presence of the $^{57}$Ni decay chain is robustly detected, but the $^{55}$Co decay chain is not formally required, with an upper mass limit estimated at 0.014 M$_{\odot}$. The measurement of the $^{57}$Ni/$^{56}$Ni mass ratio is subject to significant systematic uncertainties, but all of our fits require a high ratio >0.031 (>1.3 in solar abundances).

Tables

Table 1. Table 1: Literature R 𝑅 R -band and near-IR photometric data for SN 2011fe

MJD	Phase	Telescope	Filter	Magnitudeb	Magnitude	Reference
	(days)a				system
$56049.50$	$234.93$	LBT	$J$	$17.18\left(0.10\right)$	Vega	Shappee et al. (2016)
$56257.50$	$442.76$	LBT	$J$	$17.42\left(0.04\right)$	Vega	Shappee et al. (2016)
$56332.50$	$517.70$	LBT	$J$	$18.23\left(0.18\right)$	Vega	Shappee et al. (2016)
$56337.48$	$522.67$	LBT	$R$	$22.14\left(0.02\right)$	Vega	Shappee et al. (2016)
$56419.46$	$604.59$	LBT	$R$	$21.74\left(0.03\right)$	Vega	Shappee et al. (2016)
$56440.81$	$625.92$	1m Crimean Astrophysical Observatory	$R$	$22.00\left(0.12\right)$	Vega	Tsvetkov et al. (2013)
$56441.84$	$626.95$	1m Crimean Astrophysical Observatory	$R$	$22.07\left(0.11\right)$	Vega	Tsvetkov et al. (2013)
$56449.21$	$634.32$	LBT	$R$	$21.91\left(0.02\right)$	Vega	Shappee et al. (2016)
$56453.26$	$638.36$	LBT	$R$	$21.95\left(0.02\right)$	Vega	Shappee et al. (2016)
$56637.53$	$822.48$	LBT	$R$	$23.19\left(0.04\right)$	Vega	Shappee et al. (2016)
$56743.00$	$927.87$	GMOS North	$r$	$24.01\left(0.14\right)$	AB	Kerzendorf et al. (2014)
$56812.35$	$997.17$	LBT	$R$	$24.31\left(0.08\right)$	Vega	Shappee et al. (2016)
$56837.28$	$1022.08$	LBT	$R$	$24.54\left(0.08\right)$	Vega	Shappee et al. (2016)
$56839.29$	$1024.08$	LBT	$R$	$24.27\left(0.06\right)$	Vega	Shappee et al. (2016)
$56939.00$	$1123.71$	HST	F600LP	$24.84\left(0.02\right)$	Vega	Shappee et al. (2016)
$56939.00$	$1123.71$	HST	F110W	$24.30\left(0.04\right)$	Vega	Shappee et al. (2016)
$56939.00$	$1123.71$	HST	F160W	$22.37\left(0.02\right)$	Vega	Shappee et al. (2016)
$57041.51$	$1226.14$	LBT	$R$	$25.34\left(0.20\right)$	Vega	Shappee et al. (2016)
$57071.51$	$1256.11$	LBT	$R$	$25.46\left(0.14\right)$	Vega	Shappee et al. (2016)
$57118.00$	$1302.57$	HST	F600LP	$25.55\left(0.05\right)$	Vega	Shappee et al. (2016)
$57118.00$	$1302.57$	HST	F110W	$25.28\left(0.09\right)$	Vega	Shappee et al. (2016)
$57118.00$	$1302.57$	HST	F160W	$23.12\left(0.02\right)$	Vega	Shappee et al. (2016)
$57135.19$	$1319.75$	LBT	$R$	$25.51\left(0.18\right)$	Vega	Shappee et al. (2016)
$57163.23$	$1347.76$	LBT	$R$	$26.07\left(0.26\right)$	Vega	Shappee et al. (2016)
$57217.00$	$1401.52$	HST	F600LP	$25.82\left(0.04\right)$	Vega	Shappee et al. (2016)
$57217.00$	$1401.52$	HST	F110W	$25.43\left(0.10\right)$	Vega	Shappee et al. (2016)
$57217.00$	$1401.52$	HST	F160W	$23.41\left(0.04\right)$	Vega	Shappee et al. (2016)
$57389.53$	$1573.88$	LBT	$R$	$26.39\left(0.41\right)$	Vega	Shappee et al. (2016)
$57425.49$	$1609.81$	LBT	$R$	$26.24\left(0.42\right)$	Vega	Shappee et al. (2016)
$57437.50$	$1621.81$	HST	F600LP	$26.51\left(0.09\right)$	Vega	Shappee et al. (2016)
$57437.50$	$1621.81$	HST	F110W	$26.15\left(0.19\right)$	Vega	Shappee et al. (2016)
$57437.50$	$1621.81$	HST	F160W	$24.09\left(0.09\right)$	Vega	Shappee et al. (2016)
aAssuming maximum light at MJD 55814.30 (Mazzali et al., 2014).
b1-$\sigma$ uncertainties in parentheses.

Table 2. Table 2: SN 2011fe spectroscopy log

Date (UT)	Phase (days)a	Telescope	Instrument	Wavelength Coverage (Å)	Reference
2012 04 02	$205$	Lick 3m	KAST	$3440-10268$	Mazzali et al. (2015)
2012 04 23	$226$	Lick 3m	KAST	$3438-10176$	Mazzali et al. (2015)
2012 04 27	$230$	Keck-I	LRIS	$3200-9991$	Mazzali et al. (2015)
2012 05 01	$250$	LBT	Lucifer	$11700-13100$	Mazzali et al. (2015)
2012 05 01	$250$	LBT	Lucifer	$15500-17400$	Mazzali et al. (2015)
2012 05 01	$250$	LBT	Lucifer	$20500-23700$	Mazzali et al. (2015)
2012 05 26	$259$	WHT	ISIS	$3498-9490$	Mazzali et al. (2015)
2012 06 25	$289$	WHT	ISIS	$3426-10268$	Mazzali et al. (2015)
2012 07 17	$311$	Lick 3m	KAST	$3458-10254$	Mazzali et al. (2015)
2012 08 23	$348$	Lick 3m	KAST	$3488-10238$	Mazzali et al. (2015)
2013 04 08	$576$	Keck-II	DEIMOS	$4450-9637$	Graham et al. (2015)
2014 05 01	$964$	Keck-I	LRIS	$3077-10328$	Graham et al. (2015)
2014 06 23	$1034$	LBT	MODS1	$3100-10495$	Taubenberger et al. (2015)
aAssuming maximum light at MJD 55814.30 (Mazzali et al., 2014).

Table 3. Table 3: The SN 2011fe pseudo-bolometric luminosity light curve.

Phase	${\log }_{10}\left(L\right)$	Source
(days)	(erg s-1)	(see Sec. 3.1 for details)
$205.00$	$40.676\left(0.035\right)$	Spec - Mazzali et al. (2015)
$212.92$	$40.616\left(0.024\right)$	Phot - $R_{\mathrm{P48}}$
$226.00$	$40.532\left(0.033\right)$	Spec - Mazzali et al. (2015)
$237.99$	$40.514\left(0.034\right)$	Phot - $R_{\mathrm{P48}}$
$259.00$	$40.312\left(0.037\right)$	Spec - Mazzali et al. (2015)
$262.75$	$40.308\left(0.025\right)$	Phot - $R_{\mathrm{P48}}$
$287.20$	$40.151\left(0.026\right)$	Phot - $R_{\mathrm{P48}}$
$289.00$	$40.118\left(0.042\right)$	Spec - Mazzali et al. (2015)
$311.00$	$40.002\left(0.039\right)$	Spec - Mazzali et al. (2015)
$311.91$	$39.991\left(0.027\right)$	Phot - $R_{\mathrm{P48}}$
$331.12$	$39.872\left(0.027\right)$	Phot - $R_{\mathrm{P48}}$
$348.00$	$39.779\left(0.045\right)$	Spec - Mazzali et al. (2015)
$354.90$	$39.732\left(0.028\right)$	Phot - $R_{\mathrm{P48}}$
$420.32$	$39.333\left(0.030\right)$	Phot - $R_{\mathrm{P48}}$
$440.69$	$39.218\left(0.031\right)$	Phot - $R_{\mathrm{P48}}$
$456.88$	$39.102\left(0.031\right)$	Phot - $R_{\mathrm{P48}}$
$486.84$	$38.892\left(0.032\right)$	Phot - $R_{\mathrm{P48}}$
$516.39$	$38.663\left(0.033\right)$	Phot - $R_{\mathrm{P48}}$
$522.67$	$38.546\left(0.034\right)$	Phot - Shappee et al. (2016)
$536.90$	$38.548\left(0.034\right)$	Phot - $R_{\mathrm{P48}}$
$562.54$	$38.430\left(0.035\right)$	Phot - $R_{\mathrm{P48}}$
$576.00$	$38.399\left(0.035\right)$	Spec - Graham et al. (2015)
$585.51$	$38.450\left(0.036\right)$	Phot - $R_{\mathrm{P48}}$
$604.59$	$38.320\left(0.037\right)$	Phot - Shappee et al. (2016)
$610.98$	$38.390\left(0.037\right)$	Phot - $R_{\mathrm{P48}}$
$625.92$	$38.278\left(0.037\right)$	Phot - Tsvetkov et al. (2013)
$626.95$	$38.257\left(0.037\right)$	Phot - Tsvetkov et al. (2013)
$634.32$	$38.247\left(0.038\right)$	Phot - Shappee et al. (2016)
$638.36$	$38.230\left(0.038\right)$	Phot - Shappee et al. (2016)
$822.48$	$37.604\left(0.045\right)$	Phot - Shappee et al. (2016)
$927.87$	$37.382\left(0.050\right)$	Phot - Kerzendorf et al. (2014)
$964.00$	$37.273\left(0.051\right)$	Spec - Graham et al. (2015)
$1034.00$	$37.056\left(0.058\right)$	Spec - Taubenberger et al. (2015)
$1123.71$	$36.897\left(0.066\right)$	Phot - Shappee et al. (2016)-HST
$1302.57$	$36.602\left(0.075\right)$	Phot - Shappee et al. (2016)-HST
$1401.52$	$36.453\left(0.087\right)$	Phot - Shappee et al. (2016)-HST
$1621.81$	$36.163\left(0.100\right)$	Phot - Shappee et al. (2016)-HST

Table 4. Table 4: Radioactive decay constants and energies per decay.

Nucleus	${\lambda }_A$	$q^{\gamma }$	$q^x$	$q^l$
	${\mathrm{days}}^{-1}$	[keV]	[keV]	[keV]a
${}_{}{}^{56}\mathrm{Co}$	8.975e-3	3606.0	1.587	124.61
${}_{}{}^{57}\mathrm{Co}$	2.551e-3	121.6	3.6	17.814
${}_{}{}^{55}\mathrm{Fe}$	6.916e-4	–	1.635	3.973
aIncluding $e^{+}$, Auger $e^{-}$ and IC $e^{-}$

Table 5. Table 5: The best-fitting parameters for the modelling of the pseudo-bolometric light curve.

	$M\left(56\right)$	$M\left(57\right)$	$M\left(55\right)$	$M\left(57\right)/M\left(56\right)$	$M\left(57\right)/M\left(56\right)$	$t_{56}^l$	$t_{57}^l$	${\chi }_{\mathrm{DOF}}^2$	DOF
Model	$\left(M_{☉}\right)$	$\left(M_{☉}\right)$	$\left(M_{☉}\right)$		${{(}^{57}\mathrm{Fe}{/}^{56}\mathrm{Fe})}_{☉}$a	$\left(\mathrm{days}\right)$	$\left(\mathrm{days}\right)$
Case 0	$0.179\left(0.003\right)$	$0.004\left(0.001\right)$	$0.0\left(0.0\right)$	$0.022\left(0.003\right)$	$0.963\left(0.114\right)$	—	—	$24.09$	36
Case 1	$0.461\left(0.041\right)$	$0.014\left(0.005\right)$	$0.0\left(0.0\right)$	$0.031\left(0.011\right)$	$1.322\left(0.473\right)$	$233\left(30\right)$	$886\left(427\right)$	$2.47$	34
Case 1, excluding 550-650d	$0.442\left(0.035\right)$	$0.015\left(0.007\right)$	$0.0\left(0.0\right)$	$0.023\left(0.016\right)$	$1.47\left(0.731\right)$	$249\left(32\right)$	$812\left(551\right)$	$2.25$	25
Case 2	$0.355\left(0.017\right)$	$0.021\left(0.003\right)$	$0.0\left(0.0\right)$	$0.059\left(0.008\right)$	$2.571\left(0.346\right)$	—	—	$2.08$	33
Case 2, excluding 550-650d	$0.357\left(0.018\right)$	$0.022\left(0.003\right)$	$0.0\left(0.0\right)$	$0.060\left(0.009\right)$	$2.627\left(0.391\right)$	—	—	$1.96$	24
aAssuming ${\left({}_{}{}^{57}\mathrm{Fe}{/}^{56}\mathrm{Fe}\right)}_{☉}=0.023$ (Asplund et al., 2009)

Table 6. Table 6: PTF Photometry for SN 2011fe.

MJD	Phase (days)a	Telescope	Filter	Mag(AB)	$\Delta$Mag(AB)
$56016.138$	$201.59$	P48	PTF48R	$16.034$	$0.009$
$56016.201$	$201.65$	P48	PTF48R	$16.045$	$0.016$
$56016.263$	$201.72$	P48	PTF48R	$16.058$	$0.008$
$56017.170$	$202.62$	P48	PTF48R	$16.085$	$0.009$
$56017.214$	$202.67$	P48	PTF48R	$16.077$	$0.009$
$56017.245$	$202.70$	P48	PTF48R	$16.073$	$0.008$
$56017.463$	$202.91$	P48	PTF48R	$16.050$	$0.008$
$56017.507$	$202.96$	P48	PTF48R	$16.064$	$0.008$
…	…	…	…	…	…
…	…	…	…	…	…
…	…	…	…	…	…
aMJD${}_{\max }{}^{}=55814.38$, calculated by SiFTO

Equations

$${}^{56}\mathrm{Ni}\stackrel{{\mathrm{t}}_{1/2}=6.08\mathrm{d}}{\to }{}^{56}\mathrm{Co}\stackrel{{\mathrm{t}}_{1/2}=77.2\mathrm{d}}{\to }{}^{56}\mathrm{Fe}$$

$${}^{57}\mathrm{Ni}\stackrel{{\mathrm{t}}_{1/2}=35.6\mathrm{h}}{\to }{}^{57}\mathrm{Co}\stackrel{{\mathrm{t}}_{1/2}=271.2\mathrm{d}}{\to }{}^{57}\mathrm{Fe}$$

$${}^{55}\mathrm{Co}\stackrel{{\mathrm{t}}_{1/2}=17.53\mathrm{h}}{\to }{}^{55}\mathrm{Fe}\stackrel{{\mathrm{t}}_{1/2}=999.67\mathrm{d}}{\to }{}^{55}\mathrm{Mn}$$

$$\begin{gathered} L_A(t)=\frac{2.221}{A}\frac{{\lambda }_A}{\mathrm{day}{\mathrm{s}}^{-1}}\frac{M(A)}{M_{\text{sun}}}\frac{q_A^l{f}_A^l(t)+q_A^{\gamma }{f}_A^{\gamma }(t)+q_A^x}{\mathrm{keV}} \\ \times \exp (-{\lambda }_{A}t)\times 10^{43}\mathrm{erg}{\mathrm{s}}^{-1}, \end{gathered}$$

$$f_A^{l,\gamma }(t)=1-\exp \left[-{\left(\frac{t_A^{l,\gamma }}{t}\right)}^2\right],$$

$$F_{\mathrm{nir}}(t)=\frac{L_{\mathrm{nir}}(t)}{L_{\mathrm{opt}}(t)+L_{\mathrm{nir}}(t)}$$

$$L_{\mathrm{model}}(t)=\left[L_{55}(t)+L_{56}(t)+L_{57}(t)\right]\times \left[1-F_{\mathrm{nir}}(t)\right]$$

$$L_{\mathrm{model}}(t)=\left[L_{55}(t)+L_{56}(t)+L_{57}(t)\right]\times \left[1-F_{\mathrm{non}-\mathrm{opt}}(t)\right]$$

Full text

††pubyear: 2016††pagerange: The late-time light curve of the type Ia supernova SN 2011fe–6

1 Introduction

Type Ia supernovae (SNe Ia) have assumed a major role in cosmology as standardisable candles over the last 20 years (Riess et al., 1998; Perlmutter et al., 1998) and continue to attract considerable interest as astrophysical phenomena in their own right. Although decades of intensive study have led to significant breakthroughs in our understanding of these systems, many questions remain unanswered (see the review of Howell, 2011). Perhaps the most important is the underlying physical mechanism that leads to the explosion (Hillebrandt & Niemeyer, 2000; Hillebrandt et al., 2013), including the configuration of the progenitor system. There is a general theoretical and observational consensus that SNe Ia are caused by the explosive thermonuclear burning of the degenerate material (Hoyle & Fowler, 1960) of a carbon-oxygen white dwarf star (Nugent et al., 2011; Bloom et al., 2012) in a binary system, with the light curve primarily sustained by the radioactive decay of 56Ni synthesised in the explosion (Colgate & McKee, 1969; Kuchner et al., 1994; Churazov et al., 2014). The nature of the companion star to the white dwarf, however, remains uncertain (see review of Maoz et al., 2014).

The two most widely-discussed classical scenarios for the progenitor system are the single degenerate scenario (SD), where the binary consists of the accreting white dwarf with a non-degenerate companion star (Whelan & Iben, 1973), and the double degenerate Scenario (DD), where the explosion originates from the ‘merger’ of two white dwarfs (Iben & Tutukov, 1984; Webbink, 1984). There continues to be significant debate in the literature as to the relative frequency of the two channels. A promising approach to distinguish the competing scenarios is the study of SNe Ia at late times via extremely late-time photometry (Röpke et al., 2012).

The shape of a SN Ia bolometric light curve is driven by the energetic output of the radioactive decay, and the opacity of the (homologously) expanding ejecta into which the decay energy is deposited. The principle contributor for the first few hundred days after explosion is 56Co, generated from the decay of the most abundant radioactive synthesised nuclide, 56Ni. Most of the energy in the decay of 56Co is emitted in the form of $\gamma$-rays and the rest in charged leptons, such as positrons and electrons, both of which are thermalised in the expanding ejecta, creating optical and near-infrared (near-IR) photons. As the ejecta expand and the column density decreases as $t^{-2}$, the ejecta become transparent to the high-energy $\gamma$-rays which increasingly escape thermalisation. This leads to an observed light curve decline that is faster than that expected from the radioactive decay, until, after about 75–250 days (Childress et al., 2015), the contribution of positrons/electrons becomes dominant (e.g., Arnett, 1979) and the bolometric light curve is expected to settle on to a decline broadly matching that of 56Co. This assumes that the majority of charged leptons get trapped in the ejecta; the extent to which these get trapped is uncertain (e.g., Cappellaro et al., 1997; Milne et al., 2001), but models with full positron/electron trapping are consistent with the late-time optical/near-IR observations of several SNe Ia (e.g., Sollerman et al., 2004; Stritzinger & Sollerman, 2007; Leloudas et al., 2009; Graur et al., 2016).

At even later epochs, other longer-lived decay chains may also contribute, in particular the chains of 57Ni and 55Co (Seitenzahl et al., 2009; Röpke et al., 2012). Both chains have decay steps with half-lives of several hundred days, longer than 56Co, leading to a slowly declining energy input and thus a predicted flattening in very late-time ($>$500–600 days) SN Ia light curves. The first evidence for such nuclides has been claimed for the nearby SN Ia SN 2012cg (Graur et al., 2016), and a slow-down in the decline of the optical/near-IR light curves has been observed in other SNe Ia after $\sim$600 days (Cappellaro et al., 1997; Sollerman et al., 2004; Leloudas et al., 2009). Spectral modelling of the late-time SN 2011fe spectra also seems to require additional energy input from 57Co (Fransson & Jerkstrand, 2015).

In principle, an accurate determination of the mass of these other longer-lived nuclides provides a promising diagnostic tool to distinguish between different explosion mechanisms and, in turn, different progenitor scenarios. Theoretical modelling shows that models with burning at higher central white dwarf densities predict higher $\rm{{}^{57}Ni}/\rm{{}^{56}Ni}$ mass ratios, since $\rm{{}^{57}Ni}$ is a neutron-rich isotope that is produced in greater abundance in neutron rich environments, such as near-Chandrasekhar mass explosion models. Lower central density models, such as white dwarf mergers, generally predict lower ratios (for example, see Röpke et al., 2012; Seitenzahl et al., 2013).

A major complication is the accurate determination of the bolometric light curve. A robust theoretical prediction is the ‘infrared catastrophe’ (IRC), a predicted thermal instability that occurs at $\sim$500 d after maximum light (Axelrod, 1980). During the nebular phase up until the onset of any IRC, cooling of the SN ejecta results from optical and near-IR transitions, dominated by Fe ii and Fe iii. However, as the ejecta expand and the temperature decreases, at $\sim$2000 K the cooling is instead predicted to become dominated by fine structure lines in the mid-IR with excitation temperatures of $\sim$500 K. This is predicted to result in rapid cooling, and the thermal emission moving from the optical/near-IR to the mid/far-IR; only $\sim$5% of the deposited energy would emerge in the optical/near-IR at 1000 days after maximum light (Fransson & Jerkstrand, 2015).

This rapid decrease in luminosity in the optical/near-IR has never been observed in SN Ia light curves (e.g., Sollerman et al., 2004; Leloudas et al., 2009), which is not consistent with predictions from model light curve codes (Kozma & Fransson, 1998a, b). The reason is unclear. One possibility is that at least part of the SN ejecta is kept at temperatures above that at which the IRC occurs due to clumping in the ejecta (Leloudas et al., 2009). Lower density regions cool more slowly and may therefore remain above the limit for the IRC, thus maintaining some flux output in the optical/near-IR. Another possibility is that optical/near-IR luminosity can be maintained by a redistribution of UV flux into the optical wavebands, due to non-local scattering and fluorescence (Fransson & Jerkstrand, 2015). Other possible complications include ‘freeze-out’ (Fransson & Kozma, 1993), where the assumption that radiated luminosity is equal to the deposition of energy from radioactive decay, breaks down.

The number of SNe Ia with the high-quality late-time data needed to make progress with these issues remains small: events must be both very nearby and (ideally) suffer from only minimal absorption from dust along the line of sight, keeping them visible in the optical until very late times. Moreover, it is advantageous if the SN is located in the outskirts of its host galaxy, avoiding possible crowding from nearby stars and simplifying the photometry at late times. In this paper, we study the late-time light curve of SN 2011fe (Nugent et al., 2011), a low-extinction and spectroscopically normal SN Ia in the face-on spiral galaxy M101 located at a distance of 6.4 Mpc (Shappee & Stanek, 2011). The event has a remarkably rich data set, making it the most well-studied SN Ia to date (see Kasen & Nugent, 2013, for a review of the early scientific results from studies of SN 2011fe), with excellent spectroscopic (Nugent et al., 2011; Parrent et al., 2012; Pereira et al., 2013; Mazzali et al., 2014, 2015; Taubenberger et al., 2015; Graham et al., 2015; Shappee et al., 2013; Patat et al., 2013a; Foley & Kirshner, 2013) and photometric (McClelland et al., 2013; Matheson et al., 2012; Richmond & Smith, 2012; Munari et al., 2013; Tsvetkov et al., 2013; Zhang et al., 2016; Shappee et al., 2016) coverage. Some late-time photometric studies of SN 2011fe have already been performed (Kerzendorf et al., 2014; Zhang et al., 2016; Shappee et al., 2016), and here we present new data on SN 2011fe from late-time monitoring by the Palomar Transient Factory (PTF), combined with other ground-based and Hubble Space Telescope (HST) data up until around 1600 days after peak brightness.

In Section 2, we briefly introduce SN 2011fe and describe the new measurements of its light curve from PTF and the data reduction techniques employed. In Section 3, we present the late time PTF light curve and describe how we construct a pseudo-bolometric light curve. We introduce our analysis framework in Section 4, including correcting for photons emitted at near-IR wavelengths, and present models for the bolometric light curve, including evidence for the 57Ni decay chain. In Section 5, we discuss the implications of our study. Throughout this paper, we adopt the AB magnitude system and a Hubble constant of $H_{0}=70\,\text{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$.

2 Data

Our first task is to compile a complete photometric and spectroscopic data set on SN 2011fe. In this section, we discuss SN 2011fe itself, and then the new observations presented in this paper, i.e., a new late-time light curve from the PTF survey. We also describe the significant amount of photometric and spectroscopic data that we use from the literature.

2.1 SN 2011fe and the PTF light curve

SN 2011fe was discovered by PTF at MJD 55797.2, at an apparent magnitude of $g=17.3$ mag. Fits to the early time data give an inferred epoch for the emergence of the first photons of MJD $55796.687\pm 0.014$ (Nugent et al., 2011). The SiFTO light curve fitter (Conley et al., 2008) measures a stretch of $0.98\pm 0.01$ and a $B-V$ colour at maximum light of $-0.07\pm 0.02$ mag (Mazzali et al., 2014), together with an epoch of maximum light in the rest-frame $B$-band of $55814.30\pm 0.06$ (2011 September 10.4 UT); all phases in this paper are given relative to this epoch. These numbers are all indicative of a normal SN Ia. The SALT2 light curve fitter (Guy et al., 2007) gives similar results (Pereira et al., 2013). The Milky Way extinction was $E(B-V)_{\mathrm{MW}}=0.009$ mag, and a measurement of the host galaxy extinction along the line-of-sight to the SN gives $E(B-V)_{\mathrm{host}}=0.014\pm 0.002$ mag (Patat et al., 2013b).

The PTF (Law et al., 2009; Rau et al., 2009) was a rolling, wide-field transient survey, operating from 2009 to 2012, using the CFH12k camera (Rahmer et al., 2008) mounted on the Samuel Oschin 48-in telescope at the Palomar Observatory (P48). The typical cadence of the survey was from hours up to 5 days, with data mostly taken in a Mould $R$-band filter $R_{\mathrm{P48}}$ interspersed with some $g$-band ($g_{\mathrm{P48}}$) observations. The limiting magnitude in good conditions was around $R_{\mathrm{P48}}\simeq 20.5$ mag. Although SN 2011fe exploded during the PTF survey, it continued to be monitored after 2012 by the iPTF experiment (the intermediate Palomar Transient Factory). The iPTF coverage began on 2013 January 1, at a phase of $\simeq 480$ d. All the PTF and a portion of the iPTF data we analyse are publicly available through the Infrared Science Archive at IPAC111http://irsa.ipac.caltech.edu/.

For the PTF light curve of SN 2011fe, we employ a pipeline used extensively in earlier PTF papers (see, e.g., Ofek et al., 2014; Firth et al., 2015, for particular details). The pipeline is based around a classical image-subtraction concept. A deep reference image is constructed using data prior to the first detection of SN 2011fe, which is then photometrically aligned, astrometrically registered and re-sampled to each image containing the SN light (the science image). The point spread function (PSF) of the resampled and science images are determined from isolated point sources in the images, and the image with the best seeing is degraded to match the PSF in the worst seeing image, using a similar technique to that of Bramich (2008). The reference image is then subtracted from the science image, producing a ‘difference’ image containing only objects that have changed their flux level between the reference and science image epochs.

The SN position is then measured from the difference images that have a very high signal-to-noise (S/N) of the SN, and then PSF-fitting photometry measured at this fixed position in all images after the SN was first detected, calibrating to nearby stars from the Sloan Digital Sky Survey (York et al., 2000) Data Release 12 (Alam et al., 2015). Note that SN 2011fe was bright enough to saturate the PTF camera when around peak brightness, and these saturated images are discarded from our analysis. The bulk of the data were taken in the $R_{\mathrm{P48}}$ filter (although $g_{\mathrm{P48}}$ data were also taken around maximum light), and thus we restrict the PTF analysis to $R_{\mathrm{P48}}$ in this paper. Our PTF light curve of SN 2011fe is given in Table 6, covering 819 $R_{\mathrm{P48}}$ images over PTF and iPTF, with typical science, reference and difference images, shown in Fig. 1.

2.2 Co-adding the PTF data

SN 2011fe is clearly detected in individual PTF images up until around 400 d after maximum light. After these epochs, the data measured from the individual images, with typical 60 s exposure times, are mostly formal non-detections (i.e., $<3$ $\sigma$ significance in any individual frame). However, much deeper limits can be reached by co-adding the PTF data to increase the sensitivity. This technique has previously been used on PTF data to successfully detect pre-cursor events to core collapse SNe that lie below the formal detection limit of PTF (Ofek et al., 2013, 2014; Corsi et al., 2014; Strotjohann et al., 2015).

There are several choices as to how to perform this co-adding. The individual science images could be co-added prior to image subtraction; the individual difference images could be co-added prior to the photometric measurement; or the co-adding can be performed on the individual photometric data points produced by the pipeline. The first two of these options have a significant disadvantage, in that the PSF size and shape in the individual images may change significantly over the averaging period, which may last up to several tens of nights. Thus we instead choose to average the individual flux measurements made from each PTF observation.

In order to test the fidelity of the subtraction pipeline at these faint flux levels, we insert artificial SNe (‘fakes’) into the unsubtracted science images, and then test how accurately these fluxes can be recovered after image subtraction and averaging of the PSF photometric measurements. We inserted 2300 fake point sources at random positions, but with each fake having the same magnitude in each frame chosen to lie over the range $19<R_{\mathrm{P48}}<24$. We then run the image subtraction pipeline on all images, and measure the PSF flux of the fake SNe in each individual frame. These individual measurements are then averaged together to give a final recovered magnitude, exactly as for the real SN 2011fe light curve. The result of this test is shown in Fig. 2.

The pipeline generally performs well. For the epoch range we are particularly interested in ($+$200 to $+$700 days), when SN 2011fe is expected to be at $R_{\mathrm{P48}}\simeq 19-22$, we calculate a mean offset of $-0.006$ with a $0.019$ 1 $\sigma$ standard deviation. The outliers on the distribution are found mainly on the fainter side of the magnitude space, especially at $\rm{mag}>21$, where naturally the pipeline underperforms.

2.3 Other data

As well as the new PTF light curve data for SN 2011fe, we also use complementary photometric data from other sources. In particular, we use data from Tsvetkov et al. (2013) from the Crimean Astrophysical Observatory out to $\sim$600 d, and imaging from the Gemini-N telescope from Kerzendorf et al. (2014) at around $\sim$900 d, for which we make no attempt to correct for any background light and use it as published. We also use photometric data from the Large Binocular Telescope (LBT) at $\sim$500-1600 d and HST imaging from the Shappee et al. (2016) HST program at epochs $>$1000 d (HST proposals 13737 and 14166).

These data are compiled together in Table 1. Note that, unlike the PTF and the LBT light curve, none of these data are based on difference imaging, i.e. there is the possibility of contamination from background light from M101. However, deep pre-explosion HST imaging reveals no objects within 0.2” at the position of the SN to a 3-$\sigma$ limit of 26.6 mag, in F606W (Li et al., 2011), and thus the late-time HST data will be only negligibly contaminated by background light.

Our analysis also requires spectroscopic data, for the calculation of $s$-corrections, in order to correct the magnitudes estimated from different telescopes and bandpasses to a common system ($R_{\mathrm{P48}}$), and pseudo-bolometric luminosities. A significant amount of spectral data on SN 2011fe at late-time already exist, and we use optical data from the Lick Observatory 3-m telescope, the 4.2-m William Herschel Telescope (WHT), and the 10-m Keck-I and Keck-II telescopes, and near-IR data from LBT. We take all spectra from the WISeREP archive222http://wiserep.weizmann.ac.il/(Yaron & Gal-Yam, 2012), and details are presented in Table 2.

3 The late-time light curve of SN 2011fe

In Fig. 3, we present the late-time photometry of SN 2011fe obtained from PTF. The PTF coverage is shown from around $+200$ d when the supernova was $R_{\mathrm{P48}}\sim 16$, through to $+1200$ d, when the SN was not detected even in measurements averaged over several weeks. There was no PTF coverage from around $+620$ d to $+780$ d, as the field was not included in the iPTF survey during this period. We also show the other ground- and space-based optical data, where we have $s$-corrected to the $R_{\mathrm{P48}}$ filter using the closest available spectrum. Such a procedure assumes that our spectra match the SEDs of the SN on the photometric epochs. We check this in Fig. 4, where we compare the late-time photometric $V$-$R$ colour of SN 2011fe with the equivalent colour calculated from the available spectra. The photometric and ‘spectral’ colours are consistent, with no indication of a mis-match between the colours calculated from the spectra and those from the photometry.

We identify three distinct regions of interest in the $R_{\mathrm{P48}}$ light curve. The first corresponds to 200 to 400 d, the second to 400 to 600 d and the third to 600 to 1600 d. During the first and third regions, the light curve is declining linearly in magnitude space (but with different slopes), qualitatively consistent with a light curve powered by radioactive decay. During the first period, we calculate a slope of $1.53\pm 0.09$ mag per 100 days (consistent with similar studies of SNe Ia in the $R$ band, see Milne et al., 2001; Lair et al., 2006) and in the third region a slope of $0.37\pm 0.12$ mag per 100 days. In the second region, the light curve appears to deviate from a simple linear radioactive decay, showing a faster decline for $\sim$100 d followed by an apparent flattening over the next $\sim$50 d. The final PTF measurement is consistent with the Tsvetkov et al. (2013) $R$-band measurement at 625 d.

3.1 A pseudo-bolometric light curve

A more detailed and quantitative investigation requires a bolometric light curve covering wavelengths from the ultraviolet (UV) to the infrared (IR). However, there are few published observations in the near-IR of SN 2011fe at these late phases: the latest spectroscopic observation is at $\simeq$250 d (Mazzali et al., 2015), while for photometric observations, available data include LBT ($J$ band) and HST imaging, both presented in Shappee et al. (2016). Thus a true bolometric light curve is not possible to directly construct, and we restrict ourselves to constructing a bolometric light curve based only on the optical data where good spectral coverage does exist, which we hereafter refer to as a ‘pseudo’-bolometric light curve. We then correct for the likely near-IR contribution in our models, based on data from SN 2011fe and other events, when analysing this pseudo-bolometric light curve. We describe the pseudo-bolometric light curve in this section.

Table 2 describes the published spectral data available, and three of these late-time spectra, at $348$ d, $576$ d and $1034$ d, are shown in Fig. 5. We introduce two pseudo-bolometric box filters, a wide ‘BFW’ filter from 3550-9500 Å, and a slightly narrower ‘BFN’ filter covering 4500-9500 Å. The narrower BFN filter is designed for the narrower wavelength coverage of the $576$ d spectrum. These two filter responses are also shown in Fig. 5, together with the $R_{\mathrm{P48}}$ filter response. The spectral features clearly evolve over these phases, principally with the disappearance of the [Fe iii] line at 4700 Å. Most relevant for the $R_{\mathrm{P48}}$ filter is the weakening of the 7200 Å feature relative to the other stronger lines at bluer wavelengths (see Taubenberger et al., 2015).

We flux-calibrate the spectra in Table 2 by estimating the $R_{\mathrm{P48}}$ magnitude on the epoch of the spectral observation using linear interpolation. For the $+576$ d spectrum that coincides with the apparent break in the light curve, we average the photometry from $+550$ to $+650$ d instead of using the linear interpolation. We rescale each spectrum so that it reproduces the estimated $R_{\mathrm{P48}}$ apparent magnitude when integrated through that filter, and then measure the pseudo-bolometric fluxes by integrating through the BFW or BFN filters as appropriate. This is, essentially, a flux-space ‘bolometric correction’ for the $R_{\mathrm{P48}}$ data determined on each epoch on which a spectrum exists.

We then extend the bolometric corrections made on epochs on which spectra exist, to any epoch on which we have $R_{\mathrm{P48}}$ data. To do this we check the stability of the bolometric correction by plotting the ratio of the flux in $R_{\mathrm{P48}}$ to the flux in the BFW and BFN filters in Fig. 6. The two ratios evolve slowly as a function of time, with the relative flux in the $R_{\mathrm{P48}}$ filter generally increasing. The exception to the trend is the data point from the spectrum at $+576$ d in the BFN filter, where there is an apparent drop in the ratio. This is temporally coincident with the apparent break in the $R_{\mathrm{P48}}$-band light curve (Fig. 3).

We fit a simple smooth spline function to the BFN ratio data to interpolate over this time period, superimposed on a linear fit to the remaining epochs for interpolation at other epochs (Fig. 6). We scale this spline fit to the BFW data, and combined with a linear fit to the other BFW data. This process allows us to translate the $R$-band data on any epoch into a pseudo-bolometric luminosity, and thus construct a pseudo-bolometric light curve. This light curve is shown in Fig. 8 and presented in Table 3.

4 Analysis

We now turn to the analysis of the radioactive decay chains that may explain the shape and evolution of the pseudo-bolometric light curve of SN 2011fe. Our goals in this section are two-fold. The first goal is to confirm that the shape of the pseudo-bolometric light curve can be explained in terms of the radioactive decay chains that are expected to be present, including 57Co and 55Fe (e.g., Seitenzahl et al., 2014; Graur et al., 2016; Shappee et al., 2016). The second goal is to attempt to robustly determine the relative amounts of 56Co, 57Co and 55Fe that are required to reproduce the pseudo-bolometric light curve evolution. We first introduce the framework of our modelling and the assumptions, then discuss our procedures for adjusting the model luminosity to account for near-IR contributions and finally fit the pseudo-bolometric light curve.

4.1 Analysis framework

Several different decay chains may contribute at different levels during the time period that we are sampling. These produce not only $\gamma$-rays and positrons but also electrons and X-rays that can be thermalised and deposit their energy in the expanding SN ejecta. At the late-time epochs that we study here, when the SN ejecta become increasingly optically thin, the delay between the energy deposition from the radioactive decay and the emission of optical radiation becomes short, and the light curve time-scale becomes driven by the various decay chains.

We consider three decay chains that are likely to contribute to the bolometric output:

[TABLE]

[TABLE]

[TABLE]

where the half-life is given in each case. Note that in each of these chains, one of the decay steps is significantly longer than the other; thus we make the approximation that only 56Co, 57Co and 55Fe are important in our late-time ($t>200$ d) light curve analysis (we neglect other decay chains such as 44Ti as they will be sub-dominant for many years; e.g., Seitenzahl et al., 2009).

Under these approximations, the bolometric luminosity $L$ for a given decay chain is

[TABLE]

where $t$ is the time since explosion, $A$ is the atomic number of the decay chain, $M(A)$ is the initial mass synthesised, $\lambda_{A}$ is the decay constant, $q_{A}^{l}$, $q_{A}^{\gamma}$ and $q_{A}^{x}$ are the average energies per decay carried by the (electrically charged) leptons, $\gamma$-rays and X-rays, and $f_{A}^{l}$ and $f_{A}^{\gamma}$ are the fractions of the leptons and $\gamma$-rays absorbed in the ejecta, i.e. that contribute to the luminosity. Under the assumption of homologous expansion, these fractions are time-dependent and are given by

[TABLE]

where $t_{A}^{l,\gamma}$ corresponds to the time when the optical depth, $\tau_{A}^{l,\gamma}$, for positrons/leptons or $\gamma$-rays becomes unity, i.e. $\tau_{A}^{l,\gamma}=\left(t_{A}^{l,\gamma}/t\right)^{2}$. In this framework, the effective opacity for each species, $k_{A}^{l,\gamma}$, is proportional to $t_{A}^{2}$. Under the leptonic contribution channel, we include positrons, Auger and internal conversion electrons.

The values for the decay energies and constants are presented in Table 4, and can be found at the National Nuclear Data Center333http://www.nndc.bnl.gov.

In our analysis, we make the following simplifications:

We fix $t_{56}^{\gamma}=35$ d. This parameter is not well-measured by our late-time pseudo-bolometric light curve, but is well-constrained in the literature: Sollerman et al. (2004), Stritzinger & Sollerman (2007) and Leloudas et al. (2009) have calculated $t_{56}^{\gamma}$ for SN 2000cx, SN 2001el and SN 2003hv to be $t_{56}^{\gamma}=31.5$, 35, and 32.5 d respectively, while Zhang et al. (2016) infer $t_{56}^{\gamma}=34.5$ d for SN 2011fe. We include the (small) contribution to the heating from 56Co $\gamma$-rays even at very late times. 2. 2.

We fix $t_{57}^{\gamma}=160$ d. This parameter is, again, not well-measured. The main $\gamma$-ray contribution from 57Co are the low-energy 14.4, 122.1 and 136.5 keV lines. For the 14.4 keV, we assume complete trapping. For the 122.1 and 136.5 keV ones, we use a value for $t_{57}^{\gamma}$ that suggests a 10% trapping at 500 days. 3. 3.

We treat $t_{56}^{l}$, $t_{57}^{l}$ and $t_{55}^{l}$ as free parameters in some of the fits, i.e., we do not assume full trapping of the leptonic channel. 4. 4.

We assume complete trapping of the low energy X-rays.

4.2 Non-optical contribution

Our pseudo-bolometric light curve (Section 3.1) is based on optical data only. We must therefore account for the radiation emerging at non-optical wavelengths by either correcting the data, or by adjusting the model. Here we choose to adjust the model, keeping our data as close to the observations as possible.

In Fig. 7, we present the total near-IR contribution for SN 2000cx, SN 2001el and SN 2003hv based on broad-band imaging (Sollerman et al., 2004; Stritzinger & Sollerman, 2007; Leloudas et al., 2009). We also use near-IR estimates based on nebular spectra of 7 SNe Ia taken from Maguire et al. (in preparation) and observed in the optical and near-IR with the Very Large Telescope XShooter spectrograph. Finally, we include a point for SN 2011fe obtained by integrating the available near-IR spectrum at 250 d (Table 2), the near-IR LBT data (plotted with upward arrows, since they are only $J$-band observations) and the HST data from Shappee et al. (2016) at later phases. Note for the literature SNe Ia the rapidly increasing near-IR flux contribution over 400–600 d, as flux is presumably redistributed from the optical to the near-IR. This generally leads to optical light curves declining more rapidly than near-IR light curves over these epochs due to the decreasing temperature in the ejecta and the consequently increased importance of the near-IR [Fe ii] lines (e.g. Sollerman et al., 2004; Leloudas et al., 2009).

The calculation of the near-IR contributions of SN 2000cx, SN 2001el and SN 2003hv were performed as follows. The published light-curve magnitudes were extinction corrected and converted to fluxes, with linear interpolation for the missing epochs. The optical fluxes were used to adjust the available spectra of SN 2011fe so that the spectra reproduce the optical colours, using the ‘mangling’ technique of Hsiao et al. (2007) and Conley et al. (2008), and then integrated through the ‘BFW’ filter (or the ‘BFN’ filter, adding the appropriate correction from Section 3.1), thus calculating the optical part of the bolometric light curve. The near-IR $JHK$ fluxes were integrated over their effective filter area. The near-IR contribution is then defined as:

[TABLE]

While the absolute contributions differ from SN to SN, the general shape is similar, indicating a colour evolution over $\sim$300–600 d that shifts flux from the optical to near-IR wavelengths. We fit a simple functional form, based on the logistic or sigmoid function, to these data to model this near-IR contribution as a function of time.While a more complex near-IR colour evolution is possible, leading to a local maximum at $\simeq$650 d, seen in SN 2003hv (Fig. 8 of Leloudas et al., 2009), we will retain this simple functional form, mainly because the last data point of SN 2003hv is based on extrapolating the light curve, under the assumption that the J-H and H-K colours do not change, making the final calculation highly speculative.

The final model luminosity, $L_{\mathrm{model}}(t)$, that we fit to our pseudo-bolometric light curve is then

[TABLE]

where $L_{55}$, $L_{56}$ and $L_{57}$ are given by equation (4), and $F_{\mathrm{nir}}$ is determined as detailed above.

We also estimate the contribution of mid-IR emission over 500–600 d from the Spitzer light curve obtained with the Infrared Array Camera (IRAC) channels 1 and 2 (at 3.6 and 4.5 $\micron$ respectively) presented in Johansson et al. (2014). We can only estimate an upper limit, but find a total mid-IR contribution of $\sim$1 per cent, indicating that little flux is emitted in the mid-IR region at these epochs. We defer a discussion of the far-IR region (and a possible IRC) to the next section. Finally, we note that the contribution at UV wavelengths smaller than 3500 Å is expected to be small, since it is dominated by optically thick lines, and the only SN 2011fe nebular UV spectrum shows little UV flux (Friesen et al., in prep).

4.3 Analysis of the bolometric light curve

The late-time pseudo-bolometric light curve of SN 2011fe derived in Section 3.1 is shown in Fig. 8. In this section, we consider various model fits to this light curve and attempt to constrain the various fit parameters.

We begin by fitting equation 7, including all three decay chains (equation 1 to equation 3), directly to our data, assuming full trapping of positrons/electrons and no flux emerging at $\lambda>2.5$ µm (Fig. 8), which we refer to as Case 0. The best-fitting parameters are reported in Table 5. This provides a very poor fit to the data with a reduced $\chi^{2}$ (the $\chi^{2}$ per degree of freedom) of $\chi^{2}_{\mathrm{DOF}}\simeq 24$ for 36 degrees of freedom. The model cannot simultaneously fit the data over 200–400 d and over 900–1600 d, and prefers a low 56Ni mass ($M(56)=0.18$ $M_{\sun}$).

This $M(56)$ is significantly lower than other independent estimates. Mazzali et al. (2015) estimated $M(56)=0.47\pm 0.05$ M*☉* from nebular spectroscopy. Pereira et al. (2013) used the bolometric light curve around peak brightness to estimate $M(56)=(0.44\pm 0.08)\times(1.2/\alpha)$ M*☉, where $\alpha=1.2\pm 0.2$ is the ratio of bolometric to radioactivity luminosities. (Instead assuming $\alpha=1.0\pm 0.2$ would give $M(56)=0.53\pm 0.11$ M☉*.) This suggests that the model needs a way to lose energy outside of the optical/near-IR wavebands. We next consider two ways in which this could occur.

For Case 1, which we refer to as the ‘positron/electron escape case’, we allow the positron/electron escape fractions ($t_{56}^{l}$, $t_{57}^{l}$) to be free parameters in the fit. We then find (Table 5) $M(56)=0.461\pm 0.041$ $M_{\sun}$, $M(57)=0.014\pm 0.004$ $M_{\sun}$ and $M(55)=0.0$ $M_{\sun}$, with a much improved fit quality ($\chi^{2}_{\mathrm{DOF}}\simeq 2.5$ for 34 degrees of freedom, Fig. 9). This fit provides strong evidence for the presence of the 57Ni decay chain. The fit also requires a considerable positron/electron escape over the phase range of 200–600 d; for example, at 500 d $\simeq$75 per cent of the leptonic energetic output escapes the ejecta. The inferred trapping functions for this fit are shown in Fig. 10.

Our inferred 56Ni mass, $M(56)=0.461$ $M_{\sun}$, is consistent with other independent estimates (see above). The small difference compared to that from nebular spectroscopy (around 3 per cent) could be attributed to some part of the spectral energy distribution that is not covered by our pseudo-bolometric filter or near-IR correction. Generally, however, this fit provides evidence that our pseudo-bolometric light curve and the near-IR modelling are accounting for most of the emitted photons, assuming that the charged leptons are able to escape the ejecta.

We note that from around +500 d onwards, the optical bolometric light curve begins to decline faster, before beginning to flatten just before +600 d. This behaviour, particularly the flattening, is not well reproduced by our simple model, although the faster decline occurs at the same time as flux is being redistributed to the near-IR (Fig. 7). We therefore also fit the bolometric light curve excluding the phase range of 550–650 d, where the underlying behaviour is not well-understood and shows a different apparent decay rate. The parameters are again reported in Table 5 – these are close to the original fit values, but with an improved $\chi^{2}_{\mathrm{DOF}}$.

Although we have accounted for the non-optical luminosity in our fitting up to 2.5 $\micron$, there are strong theoretical reasons to expect significant mid/far-IR luminosity at late-times due to the IRC. Therefore, in Case 2 we use a different approach for the non-optical contribution. Motivated by the fact that there has not been any strong observational evidence for positron/electron escape in SNe Ia, we investigate the ‘IRC case’, where a substantial amount of flux is shifted to far-IR wavelengths. For this, we fit a model where we force complete positron/electron trapping for the full bolometric light curve; in other words, we fix $f_{A}^{l}$ in equation (4) to 1. This final model luminosity then will be

[TABLE]

where $L_{55}$, $L_{56}$ and $L_{57}$ are given as before, by equation (4), and $F_{\mathrm{non-opt}}$ corresponds to the total non-optical contribution, for which we assume the same logistic functional form as in Fig. 7, but with a limit at 200 d to be $\simeq$1 per cent. We then fit for $M(56)$, $M(57)$, $M(55)$ and the total non-optical contribution. The result is plotted in Fig. 11 and again reported in Table 5.

For this fit, we find $M(56)=0.355\pm 0.017$ $M_{\sun}$, $M(57)=0.021\pm 0.003$ $M_{\sun}$ and $M(55)=0.0$ $M_{\sun}$ with $\chi^{2}_{\mathrm{DOF}}\simeq 2$ for 33 degrees of freedom. We note this fit prefers a smaller $M(56)$, lower than previous estimates, indicating that at the earlier part of our bolometric light curve there still is 25 per cent of the flux that we still do not probe. Fixing instead the $M(56)$ mass to $M(56)=0.47$ $M_{\sun}$, and allowing the non-optical contribution at 200 d to be unconstrained, we find $M(57)=0.028\pm 0.003$ $M_{\sun}$ and a non-optical contribution at 200 d equal to 25 per cent, with $\chi^{2}_{\mathrm{DOF}}\simeq 2.04$ for 34 degrees of freedom.

Our inferred non-optical contribution increases to $\simeq 85$ per cent at 600 d. This is consistent with Fransson & Jerkstrand (2015), who estimated an 80 per cent non-optical/NIR contribution at 1000 d, although their models indicate that this flux is mainly emitted in the mid-IR. In our analysis, we have assumed a constant evolution of the non-optical contributions after +600 d, the approximate expected epoch of any IRC. This assumption may not be correct, and a colour evolution that decreases the near-IR flux and increases the mid/far-IR one is certainly possible, but not constrained by our data. We note that the Spitzer mid-IR flux from Johansson et al. (2014) is very small, although in the event of an IRC, the theoretical prediction is that the fine-structure lines will be redward of the mid-IR Spitzer filter bandpass.

5 Discussion

We now investigate other physical mechanisms that may contribute to the behaviour of the bolometric late-time light curve, and then discuss the implications of our study in the context of SN Ia explosion models and physics.

5.1 Other contributions to the late-time luminosity

Although our simple model provides a good fit to most of our pseudo-bolometric light curve data, we cannot rule out other sources of luminosity that may effect the late-time light curve. We discuss these in turn (see also a similar discussion in the context of SN 2012cg in Graur et al., 2016).

5.1.1 A light echo

The flattening of the light curve of SN 2011fe from around 600 d could be explained by the presence of a dust cloud near the SN that scatters a fraction of the SN light towards the observer, a mechanism known as ‘light echo’. Due to the different light travel times, the spectrum we observe would then be an integrated combination of the scattered early-time spectra combined with the direct line-of-sight late-time spectrum. This behaviour has been demonstrated for several SNe Ia (e.g. Schmidt et al., 1994; Cappellaro et al., 2001; Drozdov et al., 2015), where the bolometric light curves evolve in a manner consistent with 56Co up to $>$500–600 d, at which point the light curves flatten. The spectra also evolve from classical SN Ia nebular spectra over $\sim$300–500 d, to spectra showing a blue continuum with broad absorption and nebular emission features super-imposed at $\sim$500–600 d. Schmidt et al. (1994) and Cappellaro et al. (2001) can reproduce the late-time SN Ia spectra with a model light-echo spectrum, constructed by co-adding early-time spectra multiplied by a power-law scattering function, $S_{\lambda}\propto\lambda^{-\alpha}$, with $\alpha=1,2$ (for a more detailed approach see Marino et al., 2015).

While the late-time photometric behaviour of SN 2011fe partly resembles a light echo (e.g., the flattening at $\sim$600 d), a light echo is unlikely to be responsible. The late-time spectra are considerably different to a blue-continuum light echo spectra. We demonstrate this following the procedure of Schmidt et al. (1994) and Cappellaro et al. (2001). We used the available early-time spectra of SN 2011fe from WISEREP, spanning $-5$ to $+75$ d, and after weighting them by the integrated luminosity at their observed epochs, correcting for reddening and using scattering laws as above, we co-add them constructing an expected light echo spectrum. In Figure 12, we compare the late-time SN 2011fe spectrum with the two synthesised light echo spectra for the two different scattering laws with $\alpha=1,2$. We observe no similarities between the observed and synthetic spectra, and thus conclude that a light echo cannot significantly influence the SN 2011fe light curve. A similar study has been performed by Shappee et al. (2016), reaching similar conclusions.

5.1.2 Circumstellar material

A possible explanation for the brightness of SN 2011fe at these late epochs is that the light curve is contaminated by interaction of the ejecta with circumstellar material (CSM) around the explosion. While a complete investigation of these possibilities is beyond the scope of this paper, we note that we do not see hydrogen in any of the late-time spectra, indicating little interaction with a CSM, with no hot black-body component and no narrow/intermediate emission lines of any element.

5.1.3 A surviving companion

Finally, we consider whether a surviving binary companion could also contribute significant luminosity at late times. If we fit the bolometric light curve with a simple 56Co decline law and a constant luminosity component, simulating a surviving companion, we find $L_{\mathrm{companion}}=7.7\pm 0.6\times 10^{2}$ $L_{\sun}$, with $\chi^{2}_{\mathrm{DOF}}\simeq 8.6$ for 36 degrees of freedom. This upper limit would be consistent with a red giant or main-sequence star of $\sim 6$ M*☉*, but such a possibility is excluded by the analysis of the HST pre-explosion imaging by Li et al. (2011) and stellar evolution timescales (however, less massive companions could have an increased luminosity due to interaction with the SN ejecta). Thus, again, this possibility seems unlikely, again as was also found in Shappee et al. (2016).

5.2 Implications

5.2.1 The infrared catastrophe or positron/electron escape?

We next investigate whether an infrared catastrophe (IRC) may have occurred in SN 2011fe, and discuss the implications of this phenomenon regarding the late light curve evolution. As discussed in the introduction, the IRC is the result of a rapid cooling in the ejecta, with the thermal emission moving from the optical/near-IR to the mid/far-IR. There is no published direct evidence for a dramatic IRC in SN 2011fe, either in the spectroscopy, where prominent optical emission lines are still seen (e.g., Taubenberger et al., 2015), or in other photometric studies (Kerzendorf et al., 2014).

However, in this paper, we have shown that the pseudo-bolometric light curve is difficult to reconcile with the evolution expected from the 56Ni and 57Ni decay chains with models involving complete or nearly-complete trapping of the charged leptons. Such a model can be made to be consistent with the data in the presence of significant positron/electron escape.

We note that such a positron escape is disfavoured, since it would be difficult to explain the distribution of the observed positron signal from the Milky Way. Crocker et al. (2016) determine the required age of a putative stellar source of the Galactic positrons, constrained by the recently revised positron annihilation signal of the bulge, disc, and nucleus of our Milky Way (Siegert et al., 2016). Crocker et al. (2016) require the putative stellar source of the positrons to have an age of  3-6 Gyr and argue strongly against normal SNe Ia (with typical ages of 300 - 1000 Myr) as the main source of positrons in our Galaxy.

Therefore, taking into account the lack of any evidence for very high positron/electron escape from SNe Ia, we favour the interpretation of the late time light curve evolution as a redistribution of flux from the optical/NIR to the mid/far-IR, similar to an occurrence of an IRC.

5.2.2 Evidence for $\rm{{}^{57}Ni}$ and $\rm{{}^{55}Co}$

The luminosity produced by the 57Ni decay chain is unambiguously detected in our analysis; fits based just on 56Ni cannot reproduce the observed evolution. In principle, a measurement of the ratio of the synthesised 57Ni to 56Ni masses ($M(57)$/$M(56)$) can possibly disentangle potential explosion mechanisms, and thus provide some insight into the progenitor system. A recent demonstration on the SN Ia SN 2012cg (Graur et al., 2016) claimed $M(57)/M(56)=1.87^{+0.52}_{-0.48}$ (here we give all ratios in terms of the solar value of $\left({}^{57}\mathrm{Fe}/^{56}\mathrm{Fe}\right)_{\sun}=0.023$; see Asplund et al., 2009), increasing to a higher value of $2.96^{+0.83}_{-0.78}$ when enforcing $M(55)=0$.

However, such measurements are clearly susceptible to assumptions made in the modelling, and thus systematic uncertainties. Our fit ratios range from $M(57)/M(56)=1.3$–2.5, i.e. super-solar. These values are broadly consistent with the ratios predicted for the one-dimensional W7 (1.7; Iwamoto et al., 1999), the two-dimensional (1.4-1.9; Maeda et al., 2010) and three-dimensional (1.3; Seitenzahl et al., 2013) delayed detonation models, but not with the violent merger (1.1; Pakmor et al., 2012) model.

While the systematic uncertainties of our modelling are large, the (simple) models that we have investigated indicate a high $M(57)/M(56)$ ratio, in contradiction with Shappee et al. (2016), pointing to an explosion mechanism with relatively high central density of synthesised $\rm{{}^{56}Ni}$. This kind of explosion mechanisms are associated with SD progenitor channels (however, see Fenn et al. 2016).

In all of our fits, we use a physical prior in our models that all of the synthesised masses must be greater than or equal to zero. The fits including 55Co hit this limit for $M(55)$, precluding the calculation of uncertainties in this parameter. Relaxing this assumption gives a very small negative value for $M(55)$ with an uncertainty of $6.9\times 10^{-3}$ M*☉* and $1.6\times 10^{-2}$ M*☉* for ‘Case 1’ and ‘Case 2’, respectively. The corresponding upper limit ratios are then $M(55)/M(57)=0.47$ and $0.56$; the corresponding predicted ratios for the two-dimensional delayed detonation models are $0.4-0.6$ (Maeda et al., 2010), for the three-dimensional delayed detonation model (Seitenzahl et al., 2013) the prediction is $0.7$, and for the violent merger model (Pakmor et al., 2012; Röpke et al., 2012) $0.25$.

6 Conclusions

In this paper we have presented a late-time optical $R$-band light curve for the type Ia supernova SN 2011fe, measured using data from the Palomar Transient Factory. Combining this light curve with other published photometric data from ground-based telescopes and the Hubble Space Telescope, and with ground-based spectroscopy, we have estimated the pseudo (‘optical’) bolometric light curve out to 1600 days after peak brightness. We also construct a model for the likely bolometric output in the near-IR. We analyse this light curve using a simple model of energy deposition from three possible decay chains (56Ni, 57Ni and 55Co). Our main findings are:

• •

The bolometric light curve for SN 2011fe can be broadly explained by the radioactive inputs from the 56Ni and 57Ni decay chains. The presence of the 57Ni decay chain is required and is robustly detected in our analysis.

• •

The PTF $R$-band light curve has a noticeable rapid decline followed by a plateau over 500-600 days. While the rapid decline can be explained by a cooling ejecta and the emergence of [Fe ii] lines producing increased near-IR flux, our simple models are unable to adequately explain the plateau feature in the light curve.

• •

Our pseudo-bolometric light curve is not consistent with models that have full trapping of the produced charged leptons and no infrared catastrophe (IRC). An additional route for energy escape outside of the optical/near-IR is required.

• •

The bolometric light curve is consistent with models that allow for positron/electron escape. In this case, our best-fitting initial 56Ni mass is $M(55)=0.461\pm 0.041$ $M_{\sun}$, consistent with independent estimates from nebular spectroscopy analysis and Arnett’s Law. However this model requires 75 per cent of the charged leptons to escape by 500 d.

• •

Alternatively, our data are also fully consistent with a model that has complete positron/electron trapping but does allow for a redistribution of flux to the mid-far IR, similar to an IRC (Axelrod, 1980). In this case, around 85 per cent of the total bolometric luminosity must be escaping at non-optical wavelengths by day 600 and onwards.

• •

For both of these scenarios, the amount of 57Ni we estimate is relatively large, compared to popular explosion models. This large inferred mass indicates high central density explosion environments, mainly predicted from SD progenitor channels and WD collisions.

• •

Including contributions from the 55Co decay chain does not improve the quality of the bolometric light curve fits, although this is not well constrained by the dataset. We estimate an upper limit of 55Co mass of $1.6\times 10^{-2}$ M*☉*.

This paper has highlighted the significant systematic uncertainties that are involved in modelling the late-time light curves of SNe Ia, even within a very simple framework. Even though high-quality data were obtained for SN 2011fe during the late phases, a denser time sampling of photometric and spectroscopic late-time observations spanning the optical and infrared are needed for future similar events, particularly over the phase range of 400–800 days, where the ionisation state appears to change and a sharp colour evolution can occur (including mid and far-IR data where possible). We also neglected dependent effects in our modelling, such as freeze-out. While these effects are predicted to be small out to at least 900 days (Fransson & Jerkstrand, 2015), more detailed modelling may provide an improved interpretation of the very late-time light curve. Repeating this study for a larger sample of events will allow us to precisely measure and compare the nickel masses obtained from peak and late times, extracting information on the energetic output of the radioactive decay and confirm whether positron escape and/or an IRC is occurring.

Acknowledgments

The authors thank Claes Fransson and Anders Jerkstrand for helpful discussions regarding the infrared catastrophe and its theoretical implications.

We acknowledge support from EU/FP7-ERC grant No. [615929] and the Weizmann-UK ‘Making Connections’ program. This research was supported by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe”. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 091.D-0764, 092.D-0632 and 096.D-0627. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with programs #13737 and #14166. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

Appendix A Photometric data