Late-time Flattening of Type Ia Supernova Light Curves: Constraints from SN 2014J in M82

The astronomical community widely agrees that luminous hydrogen-poor Type Ia supernovae (SNe) explosions are powered by the thermonuclear runaway of (≥1 M_⊙) carbon/oxygen white dwarfs (WDs, Hoyle & Fowler 1960). The accretion-induced explosion fuses ∼0.1–1.0 M_⊙ of radioactive ⁵⁶Ni. Type Ia SNe cosmology uses these SNe as the most accurate distance indicators at redshifts out to z ∼ 2 (Riess et al. 1998; Perlmutter et al. 1999; Riess et al. 2016). Amazingly, this accuracy is achieved without knowing the exact nature of the progenitors.

Prior to maximum luminosity, the light curve of Type Ia SNe is powered by the energy generated by the decay of explosion-synthesized radioactive nuclei. The reprocessing in the ejecta converts the energy to longer wavelengths. The decay chain of ${}^{56}\mathrm{Ni}\to {}^{56}\mathrm{Co}\to {}^{56}\mathrm{Fe}$ provides the main source of energy deposition into the ejecta of Type I SNe (Arnett 1982). During the early phases, the optically thick ejecta trap the energy. The dominant process is Compton scattering of γ-rays produced by the decay ${}^{56}\mathrm{Ni}+{e}^{-}\to {}^{56}\mathrm{Co}+\gamma +{\nu }_{e}$, (t_1/2 ∼ 6.08 days), which allows energy to escape as an X-ray continuum or be absorbed by the material in the ejecta via the photoelectric effect (see Milne et al. 1999; Penney & Hoeflich 2014 for comprehensive reviews). The produced ⁵⁶Co decays to stable ⁵⁶Fe, and the ⁵⁶Co decay process, with a half-life t_1/2 ∼ 77 days, dominates after ∼200 days, when the expanding ejecta become more and more optically thin, and the column density decreases as t⁻² (e.g., Arnett 1979; Chan & Lingenfelter 1993; Cappellaro et al. 1997; Milne et al. 1999). Eighty-one percent of the ⁵⁶Co decays via electron capture (${}^{56}\mathrm{Co}+{e}^{-}\to {}^{56}\mathrm{Fe}+\gamma +{\nu }_{e}$), and the remainder decays through annihilation of high-energy positrons in the ejecta (${}^{56}\mathrm{Co}\to {}^{56}\mathrm{Fe}+{e}^{+}+\gamma +{\nu }_{e}$).

Observations at extremely late phases provide unique opportunities to examine various models exploring the effects of a magnetic field. As long as energy deposition is dominated by positrons being completely trapped by the magnetic field, the slope of the bolometric light curve should match the ⁵⁶Co decay rate. On the other hand, Milne et al. (1999) suggested a "radially combed" magnetic field, or even a magnetic-field-free situation (as no magnetic field in radial directions will lead to an increasing fraction of positron escape), would cause the light curve to decline faster than the rate of ⁵⁶Co decay. The discrepancy between the "trapping scenario" with a confining magnetic field and the case without a magnetic field can be as significant as 2 mag in the photometric light curves from 400–800 days (see Figure 9 of Milne et al. 1999). Similar variations of the late-time light curves have been found by Penney & Hoeflich (2014) based on measuring positron transport effects and their dependency on the magnetic field with late-time line profiles. As the SN envelope undergoes homologous expansion, the morphology of the magnetic field remains but the Larmor radius increases linearly with time, such that the fraction of escaped photons would exhibit a time-dependence due to the variations of the magnetic field and the light curve should decline faster than the rate of ⁵⁶Co decay.

Additionally, different effects of nucleosynthesis can be testable through the very late photometric evolution of Type Ia SNe and may be used to discriminate between different explosion models. Two of the most favorable explosion channels: a delayed detonation in a Chandrasekhar-mass WD (Khokhlov 1991) and a violent merger of two carbon-oxygen WDs (Pakmor et al. 2011, 2012), will result in late-time light curves behaving differently due to different amounts of ejecta heating from ⁵⁷Co and ⁵⁵Fe (Röpke et al. 2012). The decline rate of the light curve at extremely late times provides a unique opportunity, therefore, to test the enigmatic explosion mechanisms of Type Ia SNe.

Increasing evidence shows the flattening of Type Ia SN light curves around 800–1000 days, i.e., SN 1992A (∼950 days; Cappellaro et al. 1997), SN 2003hv (∼700 days; Leloudas et al. 2009), and SN 2011fe (∼930 days; Kerzendorf et al. 2014). This flattening cannot be explained even by complete trapping of the ⁵⁶Co positrons. Seitenzahl et al. (2009) suggested that additional heating from the Auger and internal conversion electrons, together with the associated X-ray cascade produced by the decay of ${}^{57}\mathrm{Co}\to {}^{57}\mathrm{Fe}$ (${t}_{1/2}\,\approx 272\,\mathrm{days}$) and ${}^{55}\mathrm{Fe}\to {}^{55}\mathrm{Mn}$ (${t}_{1/2}\approx 1000\,\mathrm{days}$), will significantly slowdown the decline of the light curve.

Recently, Graur et al. (2016) carried out an analysis of the light curve of SN 2012cg as late as ∼1055 days after the explosion and excluded the scenario in which the light curve of SN 2012cg is solely powered by the radioactive decay chain ${}^{56}\mathrm{Ni}\to {}^{56}\mathrm{Co}\to {}^{56}\mathrm{Fe}$, unless there is an unresolved light echo ∼14 mag fainter than the SN peak luminosity. Another very careful study on the late-time evolution of SN 2011fe has already extended the observing effort to an unprecedented 1622 days past the B-band maximum light (Shappee et al. 2016). This analysis has clearly detected the radioactive decay channel powered by ⁵⁷Co, with a mass ratio of $\mathrm{log}{(}^{57}\mathrm{Co}{/}^{56}\mathrm{Co})=-{1.62}_{-0.09}^{+0.08}$. This abundance ratio is strongly favored by double degenerate models that require a lower central density. The detection of ⁵⁵Fe is still unclear at these late epochs (Shappee et al. 2016). Another study based on the pseudo-bolometric light curve for the SN 2011fe has measured the mass ratio of ⁵⁷Co to ⁵⁶Co to be 1.3–2.5 times the solar value, which is broadly consistent with the ratios predicted for the delayed-detonation models (Dimitriadis et al. 2017). Additionally, spectroscopic information on the nearby SN 2011fe has been obtained at 981 days (Graham et al. 2015) and 1034 days (Taubenberger et al. 2015). Strong energy input from the radioactive decay of ⁵⁷Co is required, without which the optical spectrum would be underproduced by a factor of ∼4 (Fransson & Jerkstrand 2015). The produced mass ratio of ⁵⁷Ni to ⁵⁶Ni , which gives a strong constraint on the Type Ia SN explosions, is found to be roughly 2.8 and 2 times the solar ratios for SN 2011fe and SN 2012cg, respectively (Fransson & Jerkstrand 2015; Graur et al. 2016).

Recently, Graur et al. (2017) proposed a new model-independent correlation between the stretch of SNe and the shapes of their late-time light curves based on the shapes of the light curve of four type Ia SNe measured at > day 900, i.e., SN 2012cg (Graur et al. 2016), SN 2011fe (Shappee et al. 2016), SN 2014J (this work) and SN 2015F (Graur et al. 2017). They indicated that ⁵⁷Co may be underproduced in subluminous type Ia SNe. This correlation provides a novel way to test various physical processes driving the slowdown of the type Ia SN light curves ∼900 days after explosion.

SN 2014J was first discovered on January 21.805 UT by Fossey et al. (2014) in the very nearby starburst galaxy M82 (3.53 ± 0.04 Mpc, Dalcanton et al. 2009). Later observations constrained the first light of the SN to January 14.75 UT (Goobar et al. 2014; Zheng et al. 2014). This date is consistent with the early rising recorded by the 0.5 m Antarctic Survey Telescope (AST) during its test observations (Ma et al. 2014), as well as with other pre-discovery limits reported by various groups (Denisenko et al. 2014; Gerke et al. 2014; Itagaki et al. 2014). SN 2014J reached its B-band maximum on February 2.0 UT (JD 2,456,690.5) at a magnitude of 11.85 ± 0.02 (Foley et al. 2014). Follow-up photometric and spectroscopic observations have been made by various groups (Lundqvist et al. 2015; Bonanos & Boumis 2016; Srivastav et al. 2016; Johansson et al. 2017). The strength of γ-ray lines (Churazov et al. 2014; Diehl et al. 2015) and an analytic model fit to the pseudo-bolometric light curve (Srivastav et al. 2016) of SN 2014J suggest that ∼0.5–0.6 M_⊙ of ⁵⁶Ni was synthesized in the explosion. In this paper, we present our late-time Hubble Space Telescope (HST) photometric observations of SN 2014J and fit both the F606W (broad V) band and an estimate of the pseudo-bolometric luminosity evolution with the Bateman equation considering the luminosity contributed by the decay of ⁵⁶Co, ⁵⁷Co, and ⁵⁵Fe. In addition to following a similar approach to that presented in Graur et al. (2016), we provide a careful astrometric analysis of the time evolution of the position and profile of the SN 2014J point source at very late epochs.

We imaged SN 2014J with the HST Advanced Camera for Surveys/Wide Field Channel (HST ACS/WFC) during six visits (V1–V6) under multiple HST programs: GO-13717 (PI: Wang), GO-14139 (PI: Wang), and GO-14663 (PI: Wang), i.e., V1 ∼ day 277, V2 ∼ day 416, V3 ∼ day 649, V4 ∼ day 796, V5 ∼ day 983, and V6 ∼ day 1181 relative to its B-band maximum at a magnitude of 11.85 ± 0.02 on February 2.0 UT (JD 2,456,690.5, Foley et al. 2014). Figure 1 shows the field around SN 2014J. A log of observations is presented in Table 1. Exposures obtained with different ACS visual polarizers and in different filter combinations and visits have been aligned through Tweakreg in the Astrodrizzle package (Gonzaga et al. 2012).

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The throughput of each ACS/WFC polarizer being used by the Synphot¹¹ synthetic photometry does not match the values determined from on-orbit calibrations. We corrected the polarizers' throughput with the values deduced by on-orbit calibrations (i.e., Table 12 of Cracraft & Sparks 2007; also see Biretta et al. 2004). Following the three polarizer cases described in earlier works by Sparks & Axon (1999), we deduced the Stokes vectors from the observations. In this work, we only discuss the observed flux from SN 2014J, and the intensity maps (Stokes I) are the only required input parameter for this analysis.

$$\begin{eqnarray}&&I=\displaystyle \frac{2}{3}[r({POL}0)+r({POL}60)+r({POL}120)],\end{eqnarray} \tag{ 1 }$$

where $r({POL}0)$, etc. are the count rates in the images obtained through the three polarizers. The polarimetric properties of SN 2014J at different late phases will be discussed in a future work.

After ∼600 days past maximum light, the SN became sufficiently dim and the count rates at the central pixels of the SN point-spread function (PSF) became comparable to the bright part of the nebulosity close to the SN. The field shows that the SN lies at one end of a dark lane, and just west of a bright patch of nebulosity. A background subtraction procedure significantly diminishes the time-invariant signals and improves the photometry of evolving faint sources. Unfortunately, we found no pre-SN Hubble images, either with or without the polarizers, showing the same region using filters compatible with our observations. Images obtained on 2006 March 29 (program #10776; PI:Mountain) with HST ACS/WFC in the F435W, F555W, and F814W were used as background templates for our F475W, F606W, and F775W exposures, respectively. For each band, the background templates have been scaled and subtracted from the intensity map. The templates have been scaled according to the average flux of four local bright sources [(R.A. = 9:55:40.98, decl. = +69:40:27.16); (R.A. = 9:55:41.99, decl. = +69:40:21.60); (R.A. = 9:55:42.84, decl. = +69:40:31.42); (R.A. = 9:55:43.95, decl. = +69:40:35.47)].

Photometry of SN 2014J was conducted with a circular aperture of 015 (3 pixels in the ACS/WFC FOV) with aperture corrections according to Hartig (2009) and Sirianni et al. (2005). The photometry was performed using the IRAF¹² apphot package. The residual of the background was estimated by the median pixel value of an annulus around the SN. Compromising between determining the local background residual with nearby pixels and excluding the contamination from resolved interstellar light echoes (Yang et al. 2017), we choose the inner and outer radii as 12 (24 pixels) and 15 (30 pixels) for V1 and V2, and 045 (9 pixels) and 075 (15 pixels) for V3, V4, V5, and V6. Table 2 presents the AB magnitudes of SN 2014J at the six late epochs.

This photometry strategy has been carried out considering that extremely nonuniform background structures dominate the error budget in the late phases of the SN 2014J photometry, especially after V4. For the scientific consideration of this study, which is testing the models for the light curve evolution at very late phases, the major concern in the data reduction procedure is to obtain the correct decline rate of the SN light curves. We conducted a sanity check to test the reliability of our measurement by performing photometry on differenced images from our observations obtained at different epochs. Observations on V3 ∼ day 649 were subtracted from the observations on V4, V5, and V6. This directly measures the differential fluxes and therefore the light curve decline rate. The divergence of magnitude between this estimation and the photometry on scaled and background subtracted images are most significant in V6 when the SN is faintest, which gives ∼0.01, 0.04, and 0.05 mag differences in F475W, F606W, and F775W, respectively. This difference is ≲0.01 in V4 and V5. We conclude that our photometry is reasonable based on the agreement between these two approaches, and the differences represent the systematic uncertainties introduced in the use of subtraction templates acquired with different filters. The photometric uncertainties we quote include this difference, the Poisson noise of the signal, the photon noise of the background, the readout noise contribution (3.75 electrons/pixel for ACS/WFC), and the uncertainties in the aperture corrections. These quantities were added in quadrature. The decline rates between all the epochs, calculated from photometry shown in Table 3 and measured using this sanity check, agree within ∼2% and are smaller than the photometric uncertainties.

Table 3.  HST Late-time Light Curve Decline Rate of SN 2014J

Perioda $\setminus$Filter	F475W	F606W	F775W	Pseudo-bolometric
(Days)	(Δmag/100 days)	(Δmag/100 days)	(Δmag/100 days)	(Δmag/100 days)
277–416	1.511 ± 0.003	1.532 ± 0.004	1.079 ± 0.004	1.432 ± 0.044
416–649	1.245 ± 0.002	1.024 ± 0.003	1.370 ± 0.003	1.219 ± 0.038
649–796	0.611 ± 0.006	0.646 ± 0.009	0.721 ± 0.009	0.640 ± 0.064
796–983	0.483 ± 0.009	0.540 ± 0.018	0.429 ± 0.011	0.503 ± 0.040
983–1181	0.301 ± 0.015	0.387 ± 0.035	0.474 ± 0.030	0.358 ± 0.055

Note.

^aApproximate days after B maximum, 2014 February 2.0 (JD 245 6690.5).

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We correct our measurements for both the interstellar dust extinction in the SN host galaxy and the Galactic extinction toward SN 2014J. In fact, any imperfection in the extinction correction will only affect the individual magnitudes but not the decline rates of the light curves. A peculiar extinction law R_V ∼ 1.4 toward the SN 2014J line of sight has been suggested by many studies (Amanullah et al. 2014; Foley et al. 2014; Goobar et al. 2014; Brown et al. 2015; Gao et al. 2015). In this study, we adopt R_V = 1.44 ± 0.03 and A_V = 2.07 ± 0.18 mag from Foley et al. (2014) for the extinction from the host galaxy and R_V = 3.1 and $E(B-V)=0.054\,\mathrm{mag}$ for the Galactic extinction following Foley et al. (2014) based on Dalcanton et al. (2009) and Schlafly & Finkbeiner (2011). Extinction in the F475W, F606W, and F775W bands has been calculated for each component using a reddening law from Cardelli et al. (1989) with the corresponding R_V value. Both components are added to account for the total extinction toward SN 2014J for each HST ACS bandpass.

In this section, we will test different mechanisms powering the late-time light curve, and whether the light curve behavior is consistent with the prediction for the delayed-detonation and the violent merger scenarios following a similar procedure to Graur et al. (2016) for SN 2012cg. We assume that the ejecta do not interact with any circumstellar material.

3.1. Pseudo-bolometric Light Curve

The pseudo-bolometric light curve for SN 2014J was calculated over a wavelength range from 3500 to 9000 Å based on our multi-band optical photometry. We briefly summarize the steps as follows:

• (1)  
  Based on the lack of significant spectral evolution of SN 2011fe compared to a spectrum at 593 days (Graham et al. 2015), we assume the MODS/LBT spectrum of SN 2011fe at 1016 days (Taubenberger et al. 2015) represents the major spectral features of SN 2014J on V3 ∼ day 649, V4 ∼ day 796, V5 ∼ day 983, and V6 ∼ day 1181. The spectrum was retrieved from the WISeREP archive.¹³
• (2)  
  We then perform synthetic photometry on this spectrum for the F475W, F606W, and F775W bands.
• (3)  
  We calculate the differences between the synthetic photometry of the SN 2011fe spectrum and our extinction-corrected, observed photometry of SN 2014J.
• (4)  
  We calculate the scale factors between the observed and synthetic magnitudes in each filter.
• (5a)  
  We warp the spectrum using a second-order polynomial fit to the scale factors determined at the effective wavelength for each filter.¹⁴
• (5b)  
  Alternatively, for each epoch, we fit a single wavelength-independent grayscale across all wavelengths.
• (6)  
  We iterate steps (2)–(5) until the synthetic and observed photometry match to better than 0.02 mag in each filter for (5a), or the mean difference between the synthetic and the observed photometry converges to its minimum value for (5b), for which the standard deviation among the three filters is 0.11 mag.

The pseudo-bolometric luminosity for each epoch was obtained by integrating the scaled spectrum returned from (5a) or (5b) over the wavelength range 3500–9000 Å. The errors on the pseudo-bolometric light curve were computed through a Monte Carlo re-sampling approach using the photometric errors. The warping in (5a) aims at iteratively producing spectra consistent with the photometry, which follows a procedure very similar to that described in Shappee et al. (2016), while the scaling in (5b) is less sensitive to the extrapolation of the polynomial correction to the spectrum. The pseudo-bolometric luminosities calculated from (5a) are on average 13% higher than those from (5b). This discrepancy results from the construction of pseudo-bolometric light curves. For the scientific consideration of our study, this systematic difference does not affect the measurement of the abundance ratio affecting the decline rate of the SN luminosity. After correcting this discrepancy, the pseudo-bolometric luminosities calculated from these two approaches agree within 8% at all epochs, compatible with the uncertainties of the Monte Carlo approach. The error used in fitting the ratio of the isotopes has been estimated by adding this difference to the uncertainties obtained from the Monte Carlo approach in quadrature. The pseudo-bolometric luminosity of SN 2014J is listed in Table 2. The optical pseudo-bolometric luminosity at t ∼ 277 days after the B-band maximum (log L ≈ 40.28) is roughly consistent with the UVOIR bolometric luminosity at t ∼ 269 days (log L ≈ 40.35) estimated from Figure 8 of Srivastav et al. (2016). Our analysis of the bolometric evolution of SN 2014J is based on the bolometric luminosity obtained with (5b). Qualitatively similar results have been obtained by duplicating the entire analysis based on (5a) as follows.

In Figure 2 we present the spectra constructed using the warping procedure (left panel) and with grayscaling (right panel). For comparison, in each upper panel, we overplot the bandpass monochromatic flux calculated as the product Total Counts × PHOTFLAM,¹⁵ where PHOTFLAM is the inverse sensitivity (in erg cm⁻² s⁻¹ Å⁻¹) representing a signal of 1 electron per second. The lower panels present the total bandpass throughput curve (HST + ACS)¹⁶ for our F475W, F606W, and F775W observations. The spectra on the left panel are iterated to agree quantitatively with the photometry. Visual differences between the monochromatic bandpass flux and the spectra arise because the PHOTFLAM used for the SED assumes a smooth AB spectrum, which differs from the SN spectrum (see Brown et al. 2016 for a comprehensive discussion).

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3.2. Radioactive Decay

In the left panels of Figure 3, we present the F475W, F606W, and F775W-band luminosities of SN 2014J after correction for extinction. In addition to fitting the pseudo-bolometric light curve after ∼650 days with the contribution from three decay chains: ${}^{56}\mathrm{Co}\to {}^{56}\mathrm{Fe}$, ${}^{57}\mathrm{Co}\to {}^{57}$Fe, and ${}^{55}\mathrm{Fe}\to {}^{55}\mathrm{Mn}$ (an "all isotopes" model), we also fit the same model to our F606W-band observations. Here, we have assumed that after ∼500 days the F606W-band, which is centered at wavelength 5888.8 Å and with a width¹⁷ of 2570 Å, captures the dominant Fe features ([Fe ii] around 4700 and 5300 Å, blended [Fe ii]λ7155 and [Ni ii]λ7378 around 7200 Å; Taubenberger et al. 2015), and is proportional to the bolometric light curves as V-band observations (Milne et al. 2001).

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Limited by a small number of visits, we approximate the "all isotopes" model with two free parameters: the mass ratio $M{(}^{57}\mathrm{Co})/M{(}^{56}\mathrm{Co})$, and a scale factor to match the F606W photometry (or the pseudo-bolometric luminosity) with the model-calculated values. Using the solution to the Bateman equation that describes the abundances and activities in a decay chain as a function of time (following Seitenzahl et al. 2014), and by counting the decay energy carried by charged leptons and X-rays, the luminosity contribution from a single decay chain gives:

$$\begin{eqnarray}{L}_{A}(t) & = & 2.221\displaystyle \frac{C}{A}\displaystyle \frac{{\lambda }_{A}}{\mathrm{day}\ {{\rm{s}}}^{-1}}\displaystyle \frac{M(A)}{{M}_{\odot }}\displaystyle \frac{{q}_{A}^{l}+{q}_{A}^{X}}{\mathrm{keV}}\exp (-{\lambda }_{A}{t}_{e})\\ & & \times {10}^{43}\,\mathrm{erg}\ {{\rm{s}}}^{-1},\end{eqnarray} \tag{ 2 }$$

where C is a scaling factor, A gives the corresponding atomic number, λ_A is the inverse mean lifetime (${\lambda }_{A}={\tau }_{A}^{-1}\,=\mathrm{ln}(2)/{t}_{1/2,A}$), M(A) is the total mass of a certain decaying element, q^l_A and q^X_A are the average energies per decay carried by charged leptons and X-rays, respectively, and t_e is the time since explosion. Due to the limited data points in our late-time photometry, we used a ratio of $M{(}^{57}\mathrm{Co})/M{(}^{55}\mathrm{Fe})\approx 0.8$ (model rpc32; Ohlmann et al. 2014). The values of λ_A, ${q}_{A}^{l}$ and ${q}_{A}^{X}$ used here are sourced from Table 1 of Seitenzahl et al. (2009) and Table 2 of Seitenzahl et al. (2014). We justify our assumptions as follows. (1) The total deposition function is determined by both the net deposition functions for γ-rays and positrons. The γ-rays produced by the annihilation of the positrons are subject to both deposition functions. By simply assuming the radioactive source is confined to the center of a spherical distribution of ejecta yields a fraction $1-{e}^{-{\tau }_{\gamma }}$ of the energy produced by γ-rays would be left behind in the ejecta (Swartz & Wheeler 1991). The γ–ray optical depth τ_γ drops significantly as t⁻² and we neglect contributions from γ-rays because the SN ejecta became transparent to γ-rays at t ≳ 500 days (Milne et al. 2001). (2) Limited by a small number of photometric points, we begin by fitting Equation (2) assuming full trapping of positrons/electrons. In other words, we assume positrons, electrons, and X-rays are fully trapped, instantaneously deposited, and radiate their energy. One should also note that very recently, Dimitriadis et al. (2017) found that the late-time bolometric light curve of SN 2011fe is consistent with both models: either a model that allows for positron/electron escape, or a model that has complete positron/electron trapping but allows for redistribution of flux to the mid-far-IR.

The luminosity contribution from each decay channel is shown in Figure 3. The total luminosity given by these decay chains is represented by the pink dashed line. In the left panel, we show that a mass ratio of $M{(}^{57}\mathrm{Co})/M{(}^{56}\mathrm{Co})={0.065}_{-0.004}^{+0.005}$ gives the best fit to the "all isotopes" model based on the F606W-band observations after t ∼ 500 days (V3–V6). The dot-dashed gray lines show the model including the luminosity from ⁵⁶Co decay and possible reflections from an unresolved t⁻¹ light echo (see Graur et al. 2016). In the right panel, we show the same trend in a similar fitting based on the pseudo-bolometric light curve, with the mass ratio given as $M{(}^{57}\mathrm{Co})/M{(}^{56}\mathrm{Co})={0.066}_{-0.008}^{+0.009}$. We also tested the same abundance ratio, using a fit based on the pseudo-bolometric light curve constructed with the warped spectrum (procedure 5a in Section 3). A similar mass ratio of $M{(}^{57}\mathrm{Co})/M{(}^{56}\mathrm{Co})\,={0.078}_{-0.010}^{+0.011}$ was obtained.

3.3. Light Echoes?

If light echoes dominate the late time signal from the SN, we may expect a significant profile change or centroid drift if the circumstellar matter is distributed at sufficiently large distances from the SN. Light scattered by dust at such distances can produce measurable distortions to the image profiles if the scattered light dominates the total observed flux. At the distance of SN 2014J, 1 lt-yr corresponds to 0.17 HST ACS/WFC pixels. Depending on the dust distribution, we may expect the stellar profiles to become non-point-like, or the centroid of the stellar profile to drift at late times. We have checked the stellar profiles and found no significant deviations from a point source at all epochs of our observations. In the following, we provide a comprehensive check on the centroid position of the SN.

The barycenters of the stars and H ii regions around SN 2014J were measured to estimate a possible change in the relative position of the light emission of the SN. The precision is limited by the scarcity of stars in the immediate vicinity of the SN, as well as the uncharacterized field distortions caused by ACS/WFC polarizers (see, i.e., Section 5.3 of Gonzaga et al. 2012). Figure 4 presents the apparent shift in position measured from our observations in F475W and F606W. The R.A. and decl. were calculated using the image from V3, with the SN at the origin of the coordinates. The gray arrows show the vector difference of the originally measured positions of the source on two different epochs. The black arrow shows the same vector after a 2D linear regression to remove the dependence on R.A. and decl., which may be caused by residual errors of astrometric calibrations. The linear regression was found to be able to reduce the shift significantly in all cases. The reference objects for astrometric comparisons were selected within a radius of 500 pixels of the position of the SN. The FWHM of the objects was restricted to be less than 8 pixels. Only a small number objects in the earliest epoch V1 satisfy these criteria, due to the relatively short exposure time.

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In Figure 4, the upper panels present the measurements based on the highest S/N F475W-band exposures, and the lower panels present the same figures for F606W. For V5 and V6, when the the SN became sufficiently dim, to minimize the effect of the local background, the centroid of the SN was determined based on scaled and background subtracted images. For instance, in the upper row, the first panel presents the comparison between V3 and V1. The SN (red dot) exhibits an apparent motion of 0079 (gray arrow); after linear regression with the R.A. and decl., this reduces to 0029 (black arrow). This is in agreement with all the other objects in the field, which show an average distance shift of 0022 and an rms of 0014. The second panel presents the comparison between V3 and V2. The SN exhibits an apparent drift in position of 0020; after linear regression this reduces to 0.016. The field objects exhibit an average drift of 0036 and an rms of 0023. This implies that the position drift of the SN is significantly lower than the average of the field objects. The third to the fifth panels present the comparison between V3 and V4, V3 and V5, V3 and V6, respectively. After linear regression with R.A. and decl., using the stars around the SN, the drift of the SN compared to the average drift ±rms gives: 0015 versus 0.024 ± 0016, 0008 versus 0.021 ± 0016, and 0031 versus 0.027 ± 0017, respectively. An upper bound on the centroid position drift of the SN between V3 and another epoch is thus observed to be the sum of the SN drift and the rms of the drift measured from field objects. In each of these cases, this upper bound has been found to be larger than the average drift of the field objects, which implies that there is no apparent position drift of the SN. Similar results were obtained for F606W-band exposures. In all cases, we have not observed a significant position drift of the SN. The only exception is the 0077 versus 0031 ± 0018 in V3 compared to V6, F606W. Considering no drift was found in the same epoch of F475W and the low signal-to-noise ratio of the F606W observation, we do not consider significant drift of the SN in V6. The absence of such drift sets a strong constraint on the nature of the late-time emission from SN 2014J. If the significant flattening in F606W-band and pseudo-bolometric light curves is due to light echoes, the dust must lie within 0017 of the SN.

Here, we address the possibility of an unresolved light echo within the PSF of the late-time source at the SN position. Our photometry allows us to measure the $F475W\mbox{--}F606W$ and $F606W\mbox{--}F775W$ colors at very late phases. We also compared the late-time color evolution of SN 2014J with SN 2011fe, which does not exhibit evident flux contribution from the light echoes. Light-echo flux is dominated by the light of the SN around its peak, and scattering by dust favors blue light. At extremely late phases, when light from the SN may no longer dominate over the scattered light echoes, the color of the integrated flux can appear to be bluer by a few tenths of a magnitude (Rest et al. 2012; Graur et al. 2016). A redder color measured at a very late time, therefore, would suggest the absence of a light echo. In Figure 5, we present a comparison of the late-time color evolution of SN 2014J and SN 2011fe. The B- and V-band AB magnitudes of SN 2014J were calculated with pysynphot using the grayscaled spectrum introduced in (5b) in Section 3. Systematic differences between the synthetic photometry in F475W and F606W on the grayscaled and the HST photometry were included when calculating the error in the B − V color of SN 2014J.

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The B − V color of SN 2014J from t ∼ −8 to 269 days has been calculated based on the photometry of Srivastav et al. (2016). The B − V color curves of SN 2011fe at early (Zhang et al. 2016) and at late (Shappee et al. 2016) phases are shown for comparison. We note that from days ∼140 to 500, the B − V color of SN 2014J appears to be bluer than it was around the maximum light (see, i.e., Figure 5). A similar effect can be expected if SN 2014J was contaminated by light echoes. The color of SN 2014J at days ∼650 to 1200 is redder, however, in B − V by ∼0.3 mag, and the color evolution of SN 2014J also shows a similar trend to that of SN 2011fe at the same phase. Spectra of SN 2011fe at day ∼1000 detected no trace of a light echo (Graham et al. 2015; Taubenberger et al. 2015). Thus, the similarity in the late-time color evolutions of SN 2014J and SN 2011fe, together with our astrometric analysis, lead us to argue that the luminosity measurement of SN 2014J was not contaminated by a light echo at days ∼650 to 1200.

Table 3 shows the decline rate of the light curves at different epochs. Before t ∼ 600 days, the SN dims more rapidly than the light curve powered solely by the ⁵⁶Co decay. The γ-ray energy deposition becomes no longer significant after ∼200 days, therefore a substantial fraction of the flux may be shifting out of the optical bands into the infrared. Similar behavior has been discussed in the case of SN 2011fe (Kerzendorf et al. 2014) and SN 2003hv (Leloudas et al. 2009). After t ∼ 600 days, a slower decay can be identified in all the F475W, F606W, and F775W-bandpasses.

Some observations of nearby type Ia SNe show that their bolometric light curves at late phases follow the ⁵⁶Co decay channel (Cappellaro et al. 1997; Sollerman et al. 2004; Lair et al. 2006; Stritzinger & Sollerman 2007; Leloudas et al. 2009). These observations suggest that a turbulent, confining magnetic field traps the positrons, resulting in local energy deposition (see Chan & Lingenfelter 1993; Milne et al. 1999, 2001; Penney & Hoeflich 2014). In contrast, ⁵⁶Co positron escape has been suggested in some cases (Milne et al. 1999, 2001). As the ejecta expand over time, the pre-configured magnetic field weakens to the point that the Larmor radius exceeds the size of the turbulence (see Penney & Hoeflich 2014).

The late-time pseudo-bolometric decline rate of SN 2014J during day 277 to day 416 (1.432 ± 0.044 mag per 100 days) and day 416 to day 649 (1.219 ± 0.038) is larger than the predicted decay rate of radioactive ⁵⁶Co (0.98 mag per 100 days). This may be caused by the positron escape, which would produce a faster decay rate. A similar decline rate can also be seen in the quasi-bolometric light curve of SN 2014J at ∼days 238–269 (i.e., ∼1.3 mag per 100 days; Srivastav et al. 2016). Qualitatively speaking, at these intermediate epochs, the contributions from γ-rays may still be non-negligible since the SN ejecta may not have become transparent to γ-ray photons.

We fit both the F606W-band and a "pseudo-bolometric" light curve using Bateman's equation for the luminosity contribution of the ⁵⁶Co, ⁵⁷Co, and ⁵⁵Fe decay channels. The best fit to the pseudo-bolometric light curve and the F606W-band light curve give mass ratios of $M{(}^{57}\mathrm{Co})/M{(}^{56}\mathrm{Co})={0.065}_{-0.004}^{+0.005}$ and ${0.066}_{-0.008}^{+0.009}$, respectively. Assuming the same mass ratio yields for isotopes of the same iron-group elements (see Graur et al. 2016, based on Truran et al. 1967 and Woosley et al. 1973), our measurements correspond to ∼3 times the $M{(}^{57}\mathrm{Fe})/M{(}^{56}\mathrm{Fe})$ ratio of the Sun (i.e., ∼0.0217, see Table 3 of Asplund et al. 2009). This is higher than the solar ratio of ∼1.8 predicted for the W7 model (calculated from Table 3 of Iwamoto et al. 1999), and the solar ratio of ∼1.7 predicted for the near-Chandrasekhar-mass 3D delayed-detonation model N1600 (calculated from Table 2 of Seitenzahl et al. 2013). The $M{(}^{57}\mathrm{Fe})/M{(}^{56}\mathrm{Fe})$ ratio in our measurements is also higher compared to the ratios of ∼2 and ∼1.1 suggested by the late-time quasi-bolometric light curve analysis on SN 2012cg (Graur et al. 2016) and SN 2011fe (Shappee et al. 2016). A higher-metallicity progenitor could decrease the production of ⁵⁶Ni and result in a higher $M{(}^{57}\mathrm{Ni})/M{(}^{56}\mathrm{Ni})$ ratio (Seitenzahl et al. 2013). An enhancement of neutron excess due to electron captures in the deflagration wave could lead to the same effect.

It has been suggested that beyond ∼500 days in the ejecta, energy is shifted from the optical and near-infrared to the mid- and far-infrared (referred to as the infrared catastrophe; Axelrod 1980, and see Fransson et al. 1996; Fransson & Jerkstrand 2015). The V or optical luminosity may not represent the actual behavior of the bolometric light curves. This has never been observed so far in any type Ia SNe (e.g., Sollerman et al. 2004; Leloudas et al. 2009; Kerzendorf et al. 2014; McCully et al. 2014; Graur et al. 2016; Shappee et al. 2016). However, Dimitriadis et al. (2017) suggested that the evolution of SN 2011fe, around 550 to 650 days, is consistent with both a model that allows for positron/electron escape and a model allowing for a redistribution of flux from the optical to the mid-far-infrared. In our study, we fitted the F606W-band and optical bolometric luminosity after ∼650 days and did not consider the infrared catastrophe. Future studies based on a larger sample will be able to help distinguish these two possible scenarios.

As suggested by Kerzendorf et al. (2017), although the flattening of the late-time light curves of SN 2014J can be explained by additional energy input from the decay of ⁵⁷Co, we concede that one cannot draw strong conclusions from the current observation due to the uncertain physical processes. The determination of a precise isotopic abundance requires detailed modeling of the processes. Another mechanism that may plausibly explain the late-time luminosity-flattening is the survival of the donor WD after the explosion. A small amount of ⁵⁶Ni-rich material synthesized by the primary WD's explosion at low velocities might remain gravitationally bound and captured by the surviving WD companion (Shen & Schwab 2017). The lack of electrons on the surface of the donor WD significantly reduces the decay rates of ⁵⁶Ni and ⁵⁶Co more than electron capture (Sur et al. 1990; da Cruz et al. 1992). The radioactive decay is delayed, thus the surviving WD can be another source of late-time type Ia SN luminosity. Future observations of type Ia SNe at extremely late phases will be important for understanding the physical processes at this late stage and further testing the explosion mechanisms of type Ia SNe.

In summary, our multi-band photometry of SN 2014J out to 1181 days past the B-band maximum light clearly detected the flattening due to extra luminosity contributions other than the decay of ⁵⁶Co. We conclude that the high $M{(}^{57}\mathrm{Ni})/M{(}^{56}\mathrm{Ni})$ ratio estimated from the late-time luminosity evolution of SN 2014J favors a near-Chandrasekhar-mass explosion model such as W7 of Iwamoto et al. (1999). Any significant circumstellar light echoes beyond 0.3 pc on the plane of the sky can be excluded by our astrometric analysis. The observations strongly suggest additional heating from internal conversion and Auger electrons of ${}^{57}\mathrm{Co}\to {}^{57}$Fe; however, one should be cautious on the high mass ratio of ⁵⁷Ni to ⁵⁶Ni. Systematic uncertainties from the SED construction procedure, especially the missing information from NIR observations and the interpolation of the SED based on limited bandpass coverage should not be ignored (i.e., see Brown et al. 2016). Additionally, the reliability of approximating the bolometric luminosity evolution after t ∼ 650 days with the F606W-band emission requires more careful justification.

The authors are grateful to Dave Borncamp and the HST ACS team for fixing the distortion correction issues in the ACS/WFC polarized images. We would like to thank the anonymous referee and Or Graur for the very helpful discussions and constructive suggestions that improved the paper. Some of the data used in this study were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. This work also made use of the Weizmann interactive supernova data repository (WISeREP). The supernova research by Y. Yang, P. J. Brown, and L. Wang is supported by NSF grant AST-0708873. P. J. Brown was partially supported by a Mitchell Postdoctoral Fellowship. Y. Yang and M. Cracraft also acknowledge support from NASA/STScI through grant HST-GO-13717.001-A, grant HST-GO-13717.001-A, HST-GO-14139.001-A, and HST-GO-14663.001-A. The research of Y. Yang is supported through a Benoziyo Prize Postdoctoral Fellowship. The research of J. Maund is supported through a Royal Society University Research Fellowship. L. Wang is supported by the Strategic Priority Research Program "The Emergence of Cosmological Structures" of the Chinese Academy of Sciences, Grant No. XDB09000000. L. Wang and X. Wang are supported by the Major State Basic Research Development Program (2013CB834903), and X. Wang is also supported by the National Natural Science Foundation of China (NSFC grants 11178003 and 11325313).