PTF11kx: A Type Ia Supernova with Hydrogen Emission Persisting after 3.5 Years

We obtained images of PTF11kx with the Low Resolution Imaging Spectrometer (LRIS; Oke et al. 1995) at Keck Observatory on 2014 September 24: three 220 s exposures with the g filter and three 180 s exposures with the R_s filter. In Figure 1, we show these images, coregistered to an HST Advanced Camera for Surveys (ACS) image taken 2012 December 27 with the F814W filter (program GO-13024; PI J. Mulchaey). We do not see any source at the location of PTF11kx in either LRIS filter, and by comparing to SDSS catalogs we estimate the limiting magnitude at the location of PTF11kx in its host galaxy to be $g\gtrsim 22$ mag.

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Despite this nondetection, we used an offset star to obtain a spectrum of PTF11kx with the DEep Imaging Multi-Object Spectrograph (DEIMOS; Faber et al. 2003) at Keck Observatory on 2014 October 02 (1342 days after peak brightness). We acquired three 900 s exposures using the 1200 lines mm⁻¹ grating, the GG455 order-blocking filter, the 10 slit with a position angle of 968 east of north, and a central wavelength of 6100 Å. This configuration produces a scale of 0.33 Å per pixel, an FWHM resolution of 1.1–1.6 Å, and wavelength coverage of 4500–7000 Å. These spectra were reduced using routines written specifically for Keck LRIS/DEIMOS in the Carnegie Python (CarPy) package. The two-dimensional (2D) images were flat fielded, corrected for distortion along the y-axis (rectified), wavelength calibrated, cleaned of cosmic rays, and sky subtracted. An example of one of our 2D spectra in the region of Hα is shown in Figure 2, in which we see narrow emission lines from a spatially extended source and a broad component from a point source. We defined the apertures, extracted the one-dimensional (1D) spectra of the point source (i.e., PTF11kx), and combined them. Spectra of standard stars were used to determine the position, trace, and aperture size, as well as the sensitivity function. We deredden the spectrum for the effects of Milky Way extinction assuming $E(B-V)=0.052$ mag (Schlegel et al. 1998).

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The resulting 1D spectrum is shown in the left plot of Figure 3, in the observed wavelength for a limited wavelength range, outside of which the spectrum is mostly featureless. The trace of the continuum emission from PTF11kx is not seen (e.g., Figure 2), and so we do not see the blue continuum flux increase blueward of 5700 Å as reported by S13. We do detect a broad, blueshifted Hα emission line, as well as narrow features of Hα, Hβ, [N ii] $\lambda \lambda$ 6548, 6583, [S ii] $\lambda \lambda$ 6717, 6731, and [O iii] λ5007 (note that Hβ and [O iii] are very faint, noisy features that we have opted not to plot, as they fall outside the wavelength range of Figure 3). We do not detect He i λ5876 or λ7065, but S13 saw the former in their 680 day spectrum from 2013 January 09.

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Based on the 2D version of our spectrum shown in Figure 2 and discussed below, the broad component of Hα is from a point-like source because it is not spatially extended along the slit like the narrow features; the broad Hα can be attributed entirely to the SN, but the narrow features are from underlying host-galaxy emission. Although the redshift of the host is z = 0.0466, from the superposed narrow emission lines of Hα, [N ii], and [S ii] we determine that the correct redshift to apply is $z=0.0461\pm 0.0001$. This is a velocity of $-150\ \mathrm{km}\,{{\rm{s}}}^{-1}$ with respect to the host galaxy's recession velocity and accounts for galaxy rotation, as PTF11kx is located on the approaching side of this inclined spiral (D12). We do not see any of the common nebular features from the underlying SN Ia (e.g., [Fe ii], [Fe iii], [Ni ii], or [Co iii]), but we would not expect to. In the ∼1000 day spectrum of the normal SN Ia 2011fe, the flux at $\lambda \approx 6500$ Å was $\sim 5\times {10}^{-19}$ erg s⁻¹ cm⁻¹ Å⁻¹ (Graham et al. 2015). At the distance of PTF11kx, this would be $\sim 5\times {10}^{-22}$ erg s⁻¹ cm⁻¹ Å⁻¹, far below the flux in the broad Hα emission and essentially undetectable. In Sections 2.1.1 and 2.1.2, we interpret the narrow and broad components of Hα, respectively.

2.1.1. The Narrow Hα Feature

It is difficult to quantify the change in the narrow Hα feature because it is contaminated by flux from the host galaxy. For example, the spatial distribution of the narrow emission along the slit (i.e., along the y-axis) in Figure 2, as emphasized with the pink contour lines, shows no peak at the position of PTF11kx in 2014 like it did in 2013. This suggests that the narrow line is dominated by host emission in 2014, unlike earlier epochs when it is, in part, produced by the SN+CSM interaction (S13; D12). To further investigate the nature of the narrow-line evolution, for both the 2014 and 2013 epochs we measure three properties: (1) the ratio of the integrated, pseudocontinuum-subtracted fluxes in Hα and [N ii], F(Hα)/F([N ii]); (2) the EW of Hα, EW$({\rm{H}}\alpha );$ and (3) the peak wavelength of Hα, $\lambda ({\rm{H}}\alpha )$. For properties 1 and 2, the pseudocontinuum is determined from the regions immediately adjacent to, and on both sides of, each emission line. The results are presented in Table 1.

Table 1.  Parameters of the Narrow Hα Feature

Date	Parameter	Value
2013 Sep 01	F(Hα)/F([N ii])	4.5 ± 0.9
	EW(Hα)	15.9 ± 3.2 Å
	λ(Hα)	6563.8 ± 1.2 Å
2014 Oct 02	F(Hα)/F([N ii])	6.0 ± 1.2
	EW(Hα)	16.2 ± 3.2 Å
	λ(Hα)	6563.2 ± 0.6 Å

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In our narrow-line analysis, we find that the flux ratio F(Hα)/F([N ii]) increases from 2013 to 2014 (although still within 1σ uncertainties), perhaps due to the change in relative contributions from the host and SN+CSM interaction to these lines. We find that the EW remains the same at ∼16 Å, which is higher than the previously recorded low of ∼7 Å at 393 days by S13. Since the flux from the underlying host galaxy is not expected to vary, this may suggest that there is still some narrow Hα contribution from the SN in 2014—but it could also be attributed to a much higher SN continuum at 393 days.

Between 2013 and 2014, we find that the peak of the narrow Hα line appears to have shifted ∼0.6 Å blueward; although this is within 1σ uncertainties, we consider the physical implications of such a blueshift. By reflecting the blue side of the narrow Hα onto the red side (see the thin gray lines in the left panel of Figure 4), we show that the narrow Hα is less symmetric in 2014 than in 2013; specifically, the red-side flux is depressed relative to the blue side, and this effect is more significant in 2014 than 2013. Such behavior is commonly attributed to the formation of dust in the system, because material that produces the redshifted emission is moving away from the observer and is on the far side of the system, and therefore has a larger line-of-sight column density of absorbing material. We do not expect an asymmetric narrow Hα line if the emission is from the host galaxy alone, as there is not typically a direct correlation between the line-of-sight distance and velocity of the emitting material. This asymmetry might be further evidence of the narrow Hα line being formed at least in part by ongoing SN+CSM interaction, and not entirely dominated by host emission. However, a close examination of the pink contour lines in the 2014 2D spectrum (left-hand image in Figure 2) reveals a more likely explanation: the spatial structure over the extraction aperture has contrived to produce this asymmetry in 2014 compared to 2013, as the velocity profile appears dominated by spatially extended characteristics.

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2.1.2. The Broad Hα Feature

We acquired two epochs of photometry with Spitzer's Infrared Array Camera (IRAC; Fazio et al. 2004) at 3.6 and 4.5 μm, summarized in Table 3. The first epoch was obtained 105 days before our Keck+DEIMOS spectrum, and the second epoch 1.3 years after. In Figure 5, we show the Spitzer 3.6 μm image obtained at 1237 days, coregistered with an HST image of PTF11kx, and find IR emission at the location of PTF11kx that is securely detected above the host-galaxy background. These Spitzer observations suggest the presence of warm dust in the PTF11kx system; together with our optical observations, we attempt to constrain the origin and heating mechanism of the dust and distinguish whether it is newly formed or pre-existing.

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In Figure 6, we present the spectral energy distribution (SED) of PTF11kx and compare it to other SNe Ia-CSM, 2002ic and 2005gj, from Fox & Filippenko (2013). We find that the IR SED of PTF11kx has a shape and luminosity that is similar to those of these other two examples, and that its late-time decline in IR luminosity is similar to that of SN 2002ic. We fit a blackbody to the IR SED and calculate a dust mass and temperature, which are also summarized in Table 3. As in Fox & Filippenko (2013), we constrain the origin and heating mechanism of the dust by assuming a spherically symmetric, optically thin dust shell. This yields the blackbody radius for the dust of ${R}_{\mathrm{dust}}$, given by

$$\begin{eqnarray}&&{R}_{\mathrm{dust}}={\left(\displaystyle \frac{{L}_{\mathrm{dust}}}{4\pi \sigma {T}_{\mathrm{dust}}^{4}}\right)}^{1/2},\end{eqnarray} \tag{ 1 }$$

where ${T}_{\mathrm{dust}}=587$ ${\rm{K}}$ is determined from a blackbody fit to the IR SED and ${L}_{\mathrm{dust}}$ is derived by integrating that blackbody, both of which are listed in Table 3, and σ is the Stefan–Boltzmann constant. For the first epoch at 1237 days we find that ${R}_{\mathrm{dust}}\approx 5\times {10}^{16}$ cm, or ∼0.015 pc (note that this is a minimum estimate). For the second epoch at 1818 days, we find that ${R}_{\mathrm{dust}}$ is the same within the error bars (i.e., the dust temperature is about the same and the flux uncertainties are $\sim 100 \%$). Since ${R}_{\mathrm{dust}}\approx 5\times {10}^{16}$ $\mathrm{cm}$ is consistent with the vaporization radius for a typical SN Ia peak luminosity of 10¹⁰ L_⊙ assuming 0.1 μm grains (as demonstrated by Figure 8 of Fox et al. 2010), it is more likely to be newly formed than pre-existing. However, we cannot rule out that (for example) significant clump-enabled self-shielding helped to preserve existing dust. Furthermore, we note that the blueshift in the broad Hα line—commonly interpreted as a signature of dust formation in the line-forming region—was exhibited with the same peak wavelength since at least 124 days after peak brightness (S13), which means that any dust created post-explosion started forming quite early.

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To investigate the potential heating mechanism of the dust, we consider that this radius is also consistent with the extent of the CSM, ∼5 × 10¹⁶ $\mathrm{cm}$ (S13, and the second paragraph of Section 2), and that even the bulk of the slower SN ejecta traveling at ∼5000 km s⁻¹ has passed this distance by 1200 days. This means that the emission from the CSM interaction is no longer dominated by the forward shock, as also discussed in Section 2.1.2, which is the typical source of radiation to heat the dust in, for example, SNe IIn (Fox et al. 2013). Furthermore, the inferred optical luminosity required to radiatively heat dust at the distance and temperature we observe for the PTF11kx system is ${L}_{\mathrm{opt}}\approx {10}^{8}$ ${L}_{\odot }$. Assuming a bolometric absolute magnitude of 4.75 for the Sun, we estimate that the optical apparent magnitude of the PTF11kx system would have to have been ∼21.2 mag. This is brighter than our estimated limiting magnitude of $g\gtrsim 22$ at 1334 days derived in Section 2.1, suggesting that the heating mechanism is more likely to be collisional than radiative, which agrees with the fact that we have identified this dust as residing in the vicinity of the post-shock CSM. As an additional note, we point out that while some models of the post-shock companion star predict an increase in luminosity, it is typically $\lt {10}^{4}$ ${L}_{\odot }$, and in some cases the secondary actually becomes significantly fainter (Podsiadlowski 2003; Pan et al. 2012, 2013; Noda et al. 2016). Even the maximum possible increase in luminosity seen in these models is too low for the dust to be heated by emission from the shocked companion.

Therefore, unlike most SNe IIn and the SNe Ia-CSM 2002ic and 2005gj, in which a massive pre-existing dust shell is radiatively heated by the forward shock (Fox et al. 2013; Fox & Filippenko 2013), we conclude that the dust in PTF11kx is likely to be newly formed and collisionally heated, although we cannot rule out contributions from pre-existing dust and radiative heating (e.g., from the reverse shock). In terms of dust formation, PTF11kx may be similar to the Type Iax SN 2014dt, for which a late-time IR excess indicated 10⁻⁵ M_⊙ of newly formed dust, but pre-existing dust from the nondegenerate companion could not be ruled out (Fox et al. 2016).

We acquired a single epoch of UV imaging with HST's Wide Field Camera 3 (WFC3) under Cycle 24 snapshot program GO-14779 (PI: M. L. Graham). This program is designed to look for the UV signature of late-onset CSM interaction in a large sample of SNe Ia. Most of the objects targeted by this survey were chosen to be one to three years past peak brightness in Cycle 24. Despite its advanced age, we included PTF11kx because of its history of energetic CSM interaction. The images were obtained on 2016 December 1 UT (2133 days past peak brightness) with the F275W filter and have a total integration time of 858 s, which we split into two exposures to mitigate contamination by cosmic rays. We use the pipeline processed "drc" images, which have been drizzled, corrected for charge-transfer efficiency, and combined.

In Figure 7, we compare our day 2133 UV image to the day 698 optical HST image shown also in Figures 1 and 5. No point source is seen at the location of PTF11kx in its host galaxy. To estimate an upper limit for our observation, we use a forced flux measurement with an aperture of $r=0\buildrel{\prime\prime}\over{.} 4$, which encloses $86.2 \%$ of the flux, and the AB zeropoint for this aperture of 24.0 mag.⁷ Based on this analysis, we find a limiting magnitude of ${m}_{{\rm{F}}275{\rm{W}}}\gt 23.5$ for PTF11kx at 2133 days past peak brightness. We convert this to a luminosity and find an upper limit of ${L}_{\mathrm{UV}}\lt {10}^{7}$ ${L}_{\odot }$, indicating that no new, late-onset, energetic CSM-interaction episode is occurring.

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The mass and distribution of the CSM inferred from observations of PTF11kx suggests that the progenitor was a white dwarf with a red giant companion in a recurrent-nova system. The physical picture is that there are at least two kinematically distinct regions of CSM (which we call "shells," though they are of unknown angular extent), with a slower shell lying much farther from the SN than the nearer shell. The evidence for this includes the multiple velocity components in its absorption features and the erratic behavior of the narrow-line emission (S13) from photoionized material beyond the shock front, which suggests the shock interaction is variable. The impact of the inner shell is seen, but not the second.

There are several models for recurrent-nova systems. Moore & Bildsten (2012, hereafter MB12) model the creation of CSM shells in a symbiotic recurrent-nova system, where a white dwarf accretes material from the wind of a donor star. In contrast, recurrent novae from cataclysmic variable systems involve mass accretion from a companion star overflowing its Roche lobe (i.e., a close binary system). The timescales, accretion efficiency, and total mass lost can be significantly different between these two types of systems; in particular, accretion from a wind is much less efficient and symbiotic systems can generate a higher CSM mass. In the 1D models of MB12, after the ejecta from a nova eruption have swept up the wind material and the shock has cooled and decelerated, the resulting CSM shell has a thickness of ${\rm{\Delta }}{R}_{\mathrm{CSM}}\approx 0.1\ {R}_{\mathrm{CSM}}$, where ${R}_{\mathrm{CSM}}$ is the distance from the progenitor star to the CSM. However, in that model, shells of material from separate outbursts stack to create a more extended CSM. Harris et al. (2016) find that a collision with concentric, kinematically distinct nova shells creates a light curve that is significantly longer than the sum of the light curves of two individual shells. We will keep in mind these models of CSM interaction as we proceed through the next sections with estimates of the CSM physical parameters.

D12 use high-resolution spectra of PTF11kx obtained at peak brightness, before the onset of interaction, to measure the EW (${\mathrm{EW}}_{\mathrm{Ca}{\rm{II}}}$) of the blueshifted, saturated Ca ii H&K lines (${\lambda }_{\mathrm{Ca}{\rm{II}},{\rm{H}}}=3968.47\,\mathring{\rm A}$, ${\lambda }_{\mathrm{Ca}{\rm{II}},{\rm{K}}}=3933.66$ Å), from which they estimate the column density of calcium to be ${N}_{\mathrm{Ca}{\rm{II}}}\approx 5\times {10}^{18}$ cm⁻². They show that, assuming the CSM is of solar abundance and in a spherical shell of $R\approx {10}^{16}$ $\mathrm{cm}$, this implies a total CSM mass of ${M}_{\mathrm{CSM}}\approx 5.36$ ${M}_{\odot }$. As this is much larger than expected for a recurrent-nova progenitor system, D12 infer that the CSM must be distributed in a ring geometry instead of a spherically symmetric shell. Since this is a significant conclusion regarding the progenitor system, we reassess this measure of column density from the Ca ii absorption feature in the pre-maximum spectra after reconsidering the appropriate regime on the curve of growth and incorporating turbulence effects, with additional analysis of the Na i D lines and the optical depth of the CSM (all of which are described in detail in the Appendix). We find that the column density of Ca ii could be as low as ${N}_{\mathrm{Ca}}\approx {10}^{16}$ cm⁻², more than two orders of magnitude lower than estimated by D12. Under the assumption of solar abundance (i.e., that the number ratio of Ca to H is approximately (1–5) × 10⁵), this is a total particle column density of ${N}_{\mathrm{CSM}}\approx 5\times {10}^{21}$ cm⁻².

In order to translate a column density into a particle volume density, we must first estimate the radial extent of the CSM. The integrated luminosity of broad Hα decreases steeply around 500 days after explosion, indicating the end of interaction (the end of energy deposition), i.e., that the shock has swept over the CSM. Using this observation and the inferred impact time of ∼40 days after explosion, one can estimate the CSM extent from Equation (6) of Harris et al. (2016) to be ${\rm{\Delta }}{R}_{\mathrm{CSM}}\approx 4{R}_{\mathrm{imp}}$, where ${R}_{\mathrm{imp}}$ is the inner radius of the CSM. This is in agreement with the estimate of S13, as presented in Section 2. We note, however, that it neglects possible re-energization from a collision with the second shell (Harris et al. 2016).

We combine our revised CSM column density and this CSM depth to estimate that the CSM particle volume density is ${n}_{\mathrm{CSM}}={N}_{\mathrm{CSM}}/{\rm{\Delta }}{R}_{\mathrm{CSM}}\approx 1.3\times {10}^{5}$ cm⁻³. We consider ${N}_{\mathrm{CSM}}\approx {10}^{23}$ cm⁻², as estimated by D12, to be an upper limit that gives ${n}_{\mathrm{CSM}}\lesssim 1.0\times {10}^{7}$ cm⁻³.

If we assume a solar abundance (average mass per atomic particle of ∼1.3 times the proton mass) and a spherical shell geometry for the CSM in the PTF11kx system, our revised estimates for the density and radial extent of the material lead to a total mass estimate of ${M}_{\mathrm{CSM}}\approx 0.06$ ${M}_{\odot }$. This value is significantly lower than the estimate of D12 by about two orders of magnitude, but similar to derived estimates of CSM mass for other SNe Ia such as Kepler (Katsuda et al. 2015) and SN 2003du (Gerardy et al. 2004). It is much larger than the amount of CSM estimated by mass build-up models (2 × 10⁻⁶ ${M}_{\odot }$, MB12) and may indicate a phase of rapid mass loss by the companion in the years before the white dwarf explodes. Together with the dust-mass estimate in Table 3, this implies a gas-to-dust ratio of ∼15 in the post-shock CSM of PTF11kx. This is lower than the typical ratio of 100–200 for the CSM of stars undergoing mass loss (which is similar to the ratio for the interstellar medium), but not implausible as other SNe have been observed to form unexpectedly large dust masses (e.g., Matsuura et al. 2015; Wesson et al. 2015).

As previously discussed, we interpret the late-time observations of PTF11kx at $\gt 500$ days as indication that the forward shock has passed over the bulk of the CSM and is no longer powering the broad ${\rm{H}}\alpha$ emission. However, the reverse shock lives on and continues to produce ionizing photons that propagate into the cooled, fast CSM. Thus, some of the observed broad ${\rm{H}}\alpha$ luminosity may be the result of continued photoionization and photorecombination powered by the reverse shock. To investigate this, we evolve a model like that described by Harris et al. (2016) with an impact time of 50 days after explosion, CSM density ${10}^{-18}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ (i.e., intermediate between our estimate and that of D12), and CSM depth ${\rm{\Delta }}R=4{R}_{\mathrm{imp}}$. This CSM has a mass of ∼0.3${M}_{\odot }$, nearly the same as the outer ejecta, so the reverse shock enters the inner ejecta at about the time the forward shock overtakes the edge of the CSM. The SN ejecta profile has the same properties as in Harris et al. (2016), and for these CSM properties its outer edge velocity is at ∼20,000 km s⁻¹.

To estimate the plausibility that the reverse shock powers ${\rm{H}}\alpha$ through photoionization equilibrium, here we naively estimate that one ${\rm{H}}\alpha$ photon is produced per ionizing photon absorbed. Then, the broad ${\rm{H}}\alpha$ luminosity is related to the ionizing photon production rate, ${\dot{Q}}_{\mathrm{ion}}$, simply as

$$\begin{eqnarray}&&{L}_{{\rm{H}}\alpha }={E}_{{\rm{H}}\alpha }{\dot{Q}}_{\mathrm{ion}},\end{eqnarray} \tag{ 2 }$$

where E_Hα is the energy of a ${\rm{H}}\alpha$ photon. The ionizing photon production rate is defined by

$$\begin{eqnarray}&&{\dot{Q}}_{\mathrm{ion}}={\int }_{\mathrm{Ry}/h}^{{\nu }_{\mathrm{abs}}}\displaystyle \frac{{L}_{\nu ,{ff}}(\nu )}{h\nu }d\nu ,\end{eqnarray} \tag{ 3 }$$

where ${\nu }_{\mathrm{abs}}$ is the frequency at which the bound−free optical depth of the CSM is 2/3 (i.e., single scatter) and depends on the structure of the CSM. In fact, ${\dot{Q}}_{\mathrm{ion}}$ is rather insensitive to ${\nu }_{\mathrm{abs}}$ because the frequency dependence of the free–free emission is ${L}_{\nu ,{ff}}\propto {e}^{-h\nu /{kT}}$ (where h is Planck's constant, k the Boltzmann constant, and T the gas temperature), and so is nearly constant up to $h\nu \approx {kT}$. In our simulations ${kT}\gtrsim 10$ $\mathrm{keV}$ and ${\nu }_{\mathrm{abs}}\lesssim 10$ $\mathrm{keV}$, even at early times, so the gamma-rays being produced in the SN core easily free-stream through the CSM and do not contribute to the ${\rm{H}}\alpha$ luminosity (see also our discussion about gamma-ray free-streaming in Section 2.1.2). To determine ${L}_{\nu ,{ff}}$, we use ${L}_{\nu ,{ff}}=4\pi {j}_{\nu ,{ff}}V$ and calculate the emission ${\epsilon }_{\nu ,{ff}}=4\pi {j}_{\nu ,{ff}}$ in each resolution element using Rybicki & Lightman's (1979) Equation 5.14(b), where V is the volume of the resolution element. We assume that Z = 7, that at all times the shocked ejecta are fully ionized and free of hydrogen, and that the electrons are thermally distributed.

Based on this simulation, we find that the late-time X-ray (i.e., Swift 0.2–10 $\mathrm{keV}$ band) luminosity is similar to the observed 1342 day ${\rm{H}}\alpha$ luminosity for PTF11kx (peaking at ${L}_{{\rm{X}}}\approx {10}^{40}$ $\mathrm{erg}\ {{\rm{s}}}^{-1}$ in Figure 8), but with a slower evolution that is more similar to that of SN 1988Z in Figure 4. We also find that the UV luminosity absorbed by the CSM is similar to the observed 1342 day ${\rm{H}}\alpha$ luminosity, which is below the upper limit from our HST UV imaging in Section 2.3. However, we emphasize that it is inappropriate to directly compare this free–free X-ray luminosity to the observed ${\rm{H}}\alpha$ luminosity because high-energy photons must be reprocessed to produce Hα luminosity.

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We can quantitatively show that reprocessing of the X-rays by photoionization and recombination is insufficient to power the observed Hα luminosity. As an upper limit to the possible ${\rm{H}}\alpha$ luminosity that could be generated in our model with appropriate reprocessing, we calculate ${\dot{Q}}_{\mathrm{ion}}$ from each mass resolution element in the shocked ejecta and convert it to L_Hα. In Figure 8, this is represented by the green and orange curves. Because we do not include cooling in our hydrodynamic evolution, for reference we show the budget from just the inner 30% of the shocked ejecta as well as from the entirety of the shocked ejecta. To actually estimate the reprocessing from the shocked CSM in our simulations requires calculating the absorbed fraction of these photons. To do this, we first estimate the frequency at which the bound−free optical depth is 2/3, using the hydrogenic approximation for the bound−free cross-section,

$$\begin{eqnarray}&&{\sigma }_{\mathrm{bf}}=\displaystyle \frac{\pi {q}_{e}^{2}}{{m}_{e}c{\nu }_{\mathrm{Ry}}}{(\nu /{\nu }_{\mathrm{Ry}})}^{-3},\end{eqnarray} \tag{ 4 }$$

where ν is the photon frequency, ${\nu }_{\mathrm{Ry}}=13.6\,\mathrm{eV}/h$, and other constants are defined in Appendix A.2. We then calculate the ionizing photon production rate up to this frequency and compare it to the hydrogen photorecombination rate of the whole CSM, which is analogous to a Strömgren sphere calculation for ionization state, and take the minimum of the two values to be the number of recombinations occurring per second. This is shown as the bold black curve in Figure 8; it represents ionizing photons from all of the shocked ejecta, and we find that it falls short of the available budget once the CSM begins to freely expand, owing to the decline in optical depth. The luminosity decline is more rapid than is observed for PTF11kx, like the reverse-shock-dominated SN 1993S (see discussion in Section 2.1.2).

Based on our analysis of this model, we conclude that although the CSM can produce X-rays with a luminosity similar to our ${\rm{H}}\alpha$ observations, photoionization and recombination in the CSM cannot explain the observed Hα, and reprocessing of the reverse shock's high-energy photons cannot reproduce the observed ${\rm{H}}\alpha$ light curve unless that reprocessing is nearly 100% efficient. This suggests that another reprocessing mechanism must be responsible. At this point, it is important to note that we have not incorporated real radiation transport in our model, or considered options such as Case C recombination, where the Balmer continuum is optically thick (Xu et al. 1992), or collisional excitation from the SN ejecta interacting with a multitude of low-mass CSM shells, as suggested for SNe Ia-CSM by Silverman et al. (2013b). In the last two circumstances, the signature observation is a large Balmer decrement (i.e., the ratio of Hα to Hβ luminosity $\gt 10$), and S13 observed PTF11kx's Balmer decrement to increase at late times, peaking at ∼15 at ∼680 days. Unfortunately, the signal-to-noise ratio of our day 1342 DEIMOS spectrum is too low at the blue end to derive a useful upper limit on Hβ, so we cannot confirm whether the Balmer decrement remains high. However, since we do find that most of the X-rays are absorbed for the conditions of interest, we think the most likely scenario is that the X-rays are thermalized and heat the CSM, leading to collisionally excited Hα emission.

As a final note, we consider the impact of using a CSM density in between that of D12 and our own estimate. A lower density would simply lead to a lower predicted ${\rm{H}}\alpha$ luminosity, but the higher density of D12 implied a CSM mass that was too large and caused them to infer a disk-like geometry, which is also problematic. If we raise the model's CSM density, we could get an ${\rm{H}}\alpha$ emission boost, but a decrease in emitting volume (and possibly viewing-angle effects) could cause an overall reduction in the luminosity compared to spherically symmetric, lower density models. Furthermore, in a model with a 3 M_⊙ CSM and ∼1.4 M_⊙ ejecta, the reverse shock will sweep over all the ejecta before the forward shock has swept over the CSM, likely rendering moot the question of reverse-shock ionization at late times. The impact of CSM characteristics like density and geometry require a more complicated modeling process than could be included in this work, but we do consider one such special case in the next section.

D12 suggested that the CSM geometry of PTF11kx was a disk instead of a shell. We consider the situation in which the CSM is a relatively dense ring, and the SN ejecta sweep over and past it, compressing and heating the material. In this situation, the interaction would be continuous, and instead of being swept up the disk might survive, embedded in the ejecta, and encountering denser, lower velocity ejecta material as time passes. In the SN Ia ejecta, the density as a function of initial white dwarf radius (${r}_{\mathrm{WD}}$) is given by ${\rho }_{\mathrm{ej}}\propto {r}_{\mathrm{WD}}^{-10}$ if ${r}_{\mathrm{WD}}^{}\gt {r}_{t}$, and ${\rho }_{\mathrm{ej}}\propto {r}_{\mathrm{WD}}^{-1}$ if ${r}_{\mathrm{WD}}^{}\lt {r}_{t}$, where r_t is a transition radius in the white dwarf, and the velocity of ejecta at the transition radius is typically ∼5000 $\mathrm{km}\ {{\rm{s}}}^{-1}$ (Kasen 2010; Harris et al. 2016). A dense ring of CSM at $R\approx {10}^{16}$ cm would experience this transition from fast, low-density ejecta to slow, higher-density ejecta at around 230 days after explosion. This ∼230 day timescale matches the evolution of the broad Hα luminosity presented by S13 (their Figure 3), where the integrated flux increases until >250 days and then remains at about that level before declining. While this coincidence is interesting and may reflect the general physical reality of timescales, distances, and velocities for the CSM−ejecta interaction, it is not enough evidence to confirm a ring geometry—for example, a CSM distributed in dense clumps may also survive to encounter the post-transition material. Ultimately, we consider a full model of interaction between SN ejecta and alternative geometries of CSM to be beyond the scope of this work.

We briefly speculate on an alternative physical interpretation for the PTF11kx observations. Like D12 and S13 before us, we have interpreted the Ca ii and Hα narrow emission that arises ∼50 days after explosion to represent the photoionization of CSM by high-energy emission from the fastest SN Ia material interacting with the inner edge of this CSM. In this interpretation, the speed of the fastest SN Ia ejecta sets the distance between the progenitor star and the CSM to be 10¹⁶ $\mathrm{cm}$. Our alternate scenario is that the CSM is photoionized by high-energy emission from the SN Ia ejecta impacting the companion star. Since most binary evolution models result in a mass-donating secondary star that is very close to the white dwarf, this impact occurs almost immediately. In this scenario, the speed of light sets the distance between the progenitor star and the CSM to be $\sim {10}^{17}$ $\mathrm{cm}$ (close enough to be considered CSM and not interstellar medium, $\lt 1$ pc). The interaction of SN Ia ejecta with a companion star at close range is predicted to release 0.1–2 $\mathrm{keV}$ X-ray emission with a total luminosity of ${L}_{{\rm{X}}}\approx {10}^{44}$ $\mathrm{erg}\ {{\rm{s}}}^{-1}$ (Kasen 2010). CSM with a column density of ${N}_{\mathrm{CSM}}\approx {10}^{21}$–10²³ cm⁻², as we have inferred for the PTF11kx system, would be optically thick to this radiation (Veigele 1973) and thus able to absorb and re-emit as the narrow Hα emission observed by D12 and S13. The interaction of SN Ia ejecta with a companion star also predicts an optical/UV emission "bump" on the light curve within days of explosion (Kasen 2010), and two such detections for SNe Ia have been reported (Cao et al. 2015; Marion et al. 2016), but PTF11kx was discovered too late to confirm or deny a similar early-time excess.

This proposed CSM at 10¹⁷ $\mathrm{cm}$ from the white dwarf would be impacted by fast ejecta with a velocity of 2 × 10⁴ km s⁻¹ at ∼580 days after peak brightness, and indeed S13 shows an increase in the narrow Hα line emission between observations on days ∼430 and 680 (see their Figure 4). In this scenario, we postulate that the appearance of broad Hα at 40 days after peak brightness—previously attributed to the shocked CSM—is instead from the stripped hydrogen-rich envelope of the companion star. The ablation and stripping of material from a star's outer layers by SN ejecta was initially investigated by Wheeler et al. (1975). Marietta et al. (2000) advanced this work with high-resolution simulations and found that almost the entire envelope of giant-star companions ends up embedded in the iron layers of the SN Ia ejecta, moving with a velocity of $v\lt 1000$ km s⁻¹. The observational signature of this is Hα emission with FWHM ≲ 22 Å, visible once the optical depth of the SN Ia material has dropped and photons from deep in the ejecta can escape, typically several months after explosion. Initial searches for Hα in the nebular-phase spectra of SNe Ia yielded only upper limits (Leonard 2007; Shappee et al. 2013; Lundqvist et al. 2015), with one recent survey reporting a tentative detection (Maguire et al. 2016). Although the velocity and FWHM of PTF11kx's broad Hα are consistent with predictions for an impacted companion (which would be on the near side because of the blueshift in peak wavelength), the relatively early appearance of this feature is not. The latter could perhaps be explained by a companion star that is significantly farther away from the primary. However, binary evolution models that result in a distant secondary also show that during the time needed for the stars to move apart, the leftover core of a post-mass-transfer red giant star has time to evolve into a white dwarf itself—and would have no envelope to lose (Hachisu et al. 2012). Earlier visibility of Hα from a stripped envelope might also be attributable to differences in the radial density distribution of the ejecta, but we consider the detailed models necessary to confirm this to be beyond the scope of this work.

Ultimately, we find that detailed models are not necessary because we can identify three major detracting arguments for this speculative scenario. (1) CSM photoionized by a single, short burst of X-ray emission would then steadily and quickly recombine, whereas Figure 1 of D12 reveals further variability (most evident in the Ca ii), and S13 show continued variable narrow-line emission. These observations are best explained by the SN Ia shock proceeding through a nonuniformly distributed mass of CSM. (2) Emission from the stripped envelope would be powered by the radioactive decay and decline accordingly, whereas we observe the broad Hα to continue to increase in luminosity up to ∼500 days after peak brightness. (3) The near-peak optical light curve of PTF11kx remained brighter for longer than most SNe Ia, and its early-time spectra exhibited contributions from blackbody emission that nearly overwhelmed the typical SN Ia features (D12, S13). If the CSM was not impacted until $\gt 500$ days, PTF11kx would have exhibited a more typical early-time evolution. This is the strongest argument against our hypothetical scenario.

We conclude that our alternative model is not supported by the observations, but it was worthwhile to consider given the ongoing attempts to find signatures of a nondegenerate companion's stripped material in SNe Ia. We concede that it remains difficult to distinguish a H-rich CSM from a H-rich companion (see, e.g., Marion et al. 2016 versus Shappee et al. 2016); as mentioned above, early light-curve observations help to constrain direct interaction between the ejecta and a nearby companion star. In the future, being able to differentiate these effects will lead to a better understanding of the late stages of evolution of SN Ia progenitor systems.

In high-resolution Keck spectra obtained around the time of peak brightness, before the onset of interaction, presented by D12 (their Figure 2), the Na i D 1 and 2 absorption features (${\lambda }_{\mathrm{Na}{\rm{I}}{\rm{D}}1}=5895.92$ Å and ${\lambda }_{\mathrm{Na}{\rm{I}}{\rm{D}}1}=5889.95$ Å) show three components. All three appear to be unsaturated and are at velocities ${v}_{\mathrm{Na}}\approx -65$, 0, and $+50\,\mathrm{km}\ {{\rm{s}}}^{-1}$ with respect to the host-galaxy reference frame. The EW of the blueshifted line increases after peak brightness, an effect that is commonly interpreted as the recombination of Na i after an initial photoionization by the early-time SN's UV photons (e.g., Patat et al. 2007). The blueshifted, saturated lines of Ca ii H&K are also centered at ${v}_{\mathrm{Ca}}\approx -65\,\mathrm{km}\ {{\rm{s}}}^{-1}$, and we know that they are from CSM because they transition to emission features upon interaction with the SN ejecta (D12). This implies a physical coincidence of the Na and Ca in the CSM. (As an aside, the two components of Na i D at ${v}_{\mathrm{Na}}\approx 0$ and $+50\,\mathrm{km}\ {{\rm{s}}}^{-1}$ that do not change with time are likely generated by the host galaxy's interstellar material; from them, we estimate the total column density of the interstellar material to be ${N}_{\mathrm{Na}}\approx {10}^{12}$ cm⁻² along the line of sight to PTF11kx).

The column density of recombined Na i in the ${v}_{\mathrm{Na}}\approx -65\,\mathrm{km}\ {{\rm{s}}}^{-1}$ component at ∼20 days after peak is ${N}_{\mathrm{Na}}\approx {10}^{12}$ cm⁻². In Section 3.2 we show that the column density of calcium is ${N}_{\mathrm{Ca}}\gtrsim {10}^{16}$ cm⁻¹, and since calcium and sodium have similar number fractions in solar-abundance material, this suggests that only a very small fraction of the sodium has recombined. This is to be expected because the ionization energies for Na i and Ca i are quite similar, ${E}_{\mathrm{Na}{\rm{I}}}=5.14$ ${E}_{\mathrm{Ca}{\rm{I}}}=6.11$ $\mathrm{eV}$, but the recombination timescale for Ca ii is longer. Similar observations of Na i D recombination and a nonevolving Ca ii feature were presented for SN Ia 2006X by Patat et al. (2007). In fact, if the CSM environment is $T\approx {10}^{4}$ ${\rm{K}}$ as predicted by MB12, most of the sodium and calcium would be ionized even without the influence of the SN Ia radiation. With only a small amount of sodium in the ground state and available for photoionization by the young SN's UV photons, only a small amount of recombination is expected.

This small amount of recombined Na i represents the CSM that is hot but perhaps not very turbulent (otherwise, it may be collisionally ionized). This feature is unsaturated and has a line width of $\mathrm{FWHM}\approx 40\,\mathrm{km}\ {{\rm{s}}}^{-1}$, or ${\rm{\Delta }}{\lambda }_{\mathrm{FWHM}}\approx 0.8\,\mathring{\rm A}$, which is related to the temperature of the emitting material under the assumption of thermal broadening by

$$\begin{eqnarray}&&{\rm{\Delta }}{\lambda }_{\mathrm{FWHM}}=\sqrt{\left(\displaystyle \frac{8{k}_{B}T\mathrm{ln}2}{{{mc}}^{2}}\right)}{\lambda }_{0},\end{eqnarray} \tag{ 6 }$$

where k_B is the Boltzmann constant (${k}_{{\rm{B}}}=1.38\,\times {10}^{-16}$ $\mathrm{erg}\,{{\rm{K}}}^{-1}$), T is the temperature of the line-forming material, and m is the atomic mass of the element creating the line (${m}_{\mathrm{Ca}}=40$ and ${m}_{\mathrm{Na}}=23$ $\times 1.66\times {10}^{-24}$ ${\rm{g}}$). Based on the observed line width of Na i, we estimate that $T\approx 8\times {10}^{5}$ ${\rm{K}}$ in the recombined Na i, which we note is hotter than predicted by MB12 for wind material swept up into shells of CSM, $T\approx {10}^{4}$ ${\rm{K}}$. We then use this same equation to estimate the thermal broadening of Ca ii at this temperature, and find FWHM $\approx 30\,\mathrm{km}\ {{\rm{s}}}^{-1}$. Assuming the absorption feature is Gaussian, the component of the standard deviation from thermal broadening in the Ca ii line is ${b}_{\mathrm{therm}}\approx 13$ $\mathrm{km}\ {{\rm{s}}}^{-1}$.

The specific intensity (I_ν) in an absorption line as a function of frequency (ν) is given by

$$\begin{eqnarray}&&{I}_{\nu }={I}_{0}(\nu ){e}^{-\tau (\nu )},\end{eqnarray} \tag{ 7 }$$

where I₀ is the continuum (or pseudocontinuum, the flux level on either side of the absorption line) and $\tau (\nu )$ is the optical depth. The optical depth is related to the column density, N, of the absorbing material and its absorption cross-section, $\sigma (\nu )$, via

$$\begin{eqnarray}&&\tau (\nu )=N\sigma (\nu );\end{eqnarray} \tag{ 8 }$$

thus, we can derive the column density directly from the Ca ii line without using curve-of-growth relations if we know $\sigma (\nu )$. The absorption cross-section of an atomic line is given by

$$\begin{eqnarray}&&\sigma (\nu )=\displaystyle \frac{\pi {q}_{e}^{2}}{{m}_{e}c}{f}_{\mathrm{osc}}\phi (\nu )=(1.7\times {10}^{-2}\,{\mathrm{cm}}^{2}\,\mathrm{Hz})\,\phi (\nu ),\end{eqnarray} \tag{ 9 }$$

where q_e is the electron charge, m_e is the electron mass, f_osc is the line oscillator strength (0.682 for the Ca ii K line), and $\phi (\nu )$ is the line profile. We assume that the profile is Gaussian owing to the thermal (and turbulent) broadening of a Lorentzian,

$$\begin{eqnarray}&&\phi (\nu )=\displaystyle \frac{1}{\sqrt{\pi }{\rm{\Delta }}\nu }\exp \left[-{\left(\displaystyle \frac{\nu -{\nu }_{0}}{{\rm{\Delta }}\nu }\right)}^{2}\right],\end{eqnarray} \tag{ 10 }$$

where ${\rm{\Delta }}\nu$ is the line width and ${\nu }_{0}$ is the central line frequency (${\nu }_{\mathrm{Ca}{\rm{II}},{\rm{K}}}=7.63\times {10}^{14}\,\mathrm{Hz}$).

In the spectrum, we look for the frequency ${\nu }_{1}$ at which $\tau =1$ (i.e., ${I}_{\nu }=0.37\,{I}_{0}$) and thus $\sigma ({\nu }_{1})={N}^{-1}$. For the saturated Ca ii line in D12, $\tau =1$ at ∼400 km s⁻¹, so ${\nu }_{1}\approx {\nu }_{\mathrm{Ca}}+{10}^{12}\,\mathrm{Hz}$. Using FWHM $\lesssim \,350\,\mathrm{km}\,{{\rm{s}}}^{-1}$, as in Section 3.2, the line width is $b=\mathrm{FWHM}/2.35$, which when translated into frequency gives ${\rm{\Delta }}\nu \lesssim 3.8\times {10}^{11}\,\mathrm{Hz}$. With these numbers, we find that $\phi ({\nu }_{1})=1.4\,\times \,{10}^{-15}\,\mathrm{Hz}$, $\sigma ({\nu }_{1})=2.3\,\times \,{10}^{-17}\,{\mathrm{cm}}^{2}$, and that the column density of Ca ii is ${N}_{\mathrm{Ca}{\rm{II}}}\approx 4\times {10}^{16}$ cm⁻². This is lower than the column density estimate of D12 (assuming the square-root regime of the curve of growth) and slightly higher than our own estimated lower limit assuming the flat regime of the curve of growth presented above.