VARIABLE SODIUM ABSORPTION IN A LOW-EXTINCTION TYPE Ia SUPERNOVA^*,^†

Our high-resolution observing campaign for SN 2007le began on 2007 October 20 with the High Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) on the Keck I 10 m telescope. Over the following three months, we obtained a total of five HIRES spectra of the SN. The seeing was often poor (≳1'') during the observations, and the overall conditions varied significantly over the course of the many observing runs. The data were obtained with a range of different spectrograph setups, including spectra with both the blue and red cross-dispersers, and achieved signal-to-noise ratios (S/Ns) ranging between 11 and 113 per pixel. Most of the spectra used a 70 × 086 slit, yielding a spectral resolution of R ≈ 52, 000, but one spectrum was obtained with a wider (70 × 115) slit and R ≈ 42, 000. All of the HIRES spectra cover the Ca H & K, Na D, and Hα lines. We reduced the Keck data using the IDL data reduction package for HIRES (version 2.0) developed by J. X. Prochaska and collaborators (R. Bernstein et al. 2009, in preparation).²⁰

We also obtained one spectrum of SN 2007le with the High-Resolution Spectrograph (HRS; Tull 1998) on the Hobby–Eberly Telescope (HET) on 2007 November 4. The spectrograph was in its R = 60, 000 mode, with a 2''-diameter fiber and the 316 line/mm grating centered at 6948 Å, providing nearly complete wavelength coverage from 5100 to 8850 Å. We obtained 3 spectra totaling 3000 s of exposure time and reached a combined S/N of 16 per pixel. The HRS data were reduced in IRAF²¹ with the echelle package using standard procedures. A comprehensive summary of all the high-resolution data is presented in Table 1.

SN 2007le was the target of extensive photometric follow-up observations with the 0.76 m Katzman Automatic Imaging Telescope (KAIT; Li et al. 2000; Filippenko et al. 2001), continuing for approximately three months until the SN went into conjunction with the Sun. We obtained post-explosion images in 2008 August after the SN had faded in order to subtract the host-galaxy light. We used the daophot package (Stetson 1987) in IRAF to perform point-spread function photometry of SN 2007le relative to various field stars in the KAIT images, which were calibrated on five photometric nights with KAIT and the Nickel 1 m telescope at Lick Observatory.

We also obtained low-resolution spectra of SN 2007le with the Low Resolution Imaging Spectrometer (LRIS; Oke et al. 1995) on the Keck I telescope on 2007 October 15, October 16, November 11, November 12, and December 12, and with the Kast spectrograph (Miller & Stone 1993) on the 3 m Shane telescope at Lick Observatory on 2007 November 2, November 18, and December 1. These data were reduced in IRAF and IDL following normal procedures (for details, see Foley et al. 2003; Matheson et al. 2000; Horne 1986).

We display BVRI light curves of SN 2007le in Figure 1. We fitted the photometric data with the latest version of the multicolor light curve shape method (MLCS2k2; Jha et al. 2007) to determine the parameters of the SN. We find that the time of B-band maximum was 2007 October 25.65 (JD = 2,454,399.15), with an uncertainty of 0.09 day. The derived line-of-sight extinction to the SN is A_V = 0.71 ± 0.06 mag, with an extinction law of R_V = 2.56 ± 0.22 (the Milky Way foreground reddening is 0.033 mag, corresponding to A_V = 0.11 mag; Schlegel et al. 1998). The distance modulus to SN 2007le is m − M = (32.35 − 5log [H₀/(70 km s⁻¹ Mpc⁻¹)]) ± 0.06 mag, giving the SN an absolute magnitude of M_V = (−19.34 + 5log H₀/70) ± 0.09. The luminosity/light curve-shape parameter is Δ = −0.14 ± 0.02, and the MLCS2k2 χ² value of 163.0 for 106 degrees of freedom indicates an acceptable fit (much of the χ² comes from the I-band data, which significantly differ from the fit).

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In Figure 2, we show the spectrum of SN 2007le at an epoch of 10.3 days before maximum light. Analysis with the Superfit spectral fitting code of Howell et al. (2005) indicates that SN 2007le is most similar to spectra of the highly reddened type Ia SN 2002bo at 11 days before maximum from Benetti et al. (2004), while the SNID package (Blondin & Tonry 2007) finds a best fit to early spectra of the "golden standard" type Ia SN 2005cf (Wang et al. 2009). Spectra of SN 2002bo and SN 2005cf were included in the databases used by both fitting packages; the differences between the two lie in their fitting methods and treatment of color information (see Blondin & Tonry 2007), but since SN 2002bo and SN 2005cf were similar events in many respects (e.g., line velocities) we do not regard these fit results as a significant disagreement. Like SN 2002bo (and SN 2006X; Wang et al. 2008a), SN 2007le exhibits a high-velocity component to the Ca ii near-infrared triplet absorption lines, although such features are very common in SNe Ia that are observed at sufficiently early times (Mazzali et al. 2005b). Both SN 2006X and SN 2002bo were highly polarized (Wang et al. 2006, 2007; Wang & Wheeler 2008), suggesting that polarization measurements of SN 2007le might be very interesting as well.

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In order to classify SN 2007le among the various known subgroups of SNe Ia, we analyzed the spectra according to the prescriptions of Benetti et al. (2005) and Branch et al. (2006, 2009). The widths of the absorption features near 5750 Å and 6100 Å correspond to the broad line (BL) group of Branch et al. but the equivalent widths are not too far from those of the core normal group. We measured a velocity gradient in the Si ii λ6355 line of 83 ± 3 km s⁻¹day⁻¹, placing SN 2007le marginally within the high velocity gradient (HVG) class of Benetti et al. (2005), although this gradient is not very much larger than is seen in some low velocity gradient SNe. SN 2006X and SN 2002bo both exhibited significantly larger gradients.

The host-galaxy Na D absorption lines of SN 2007le are located at observed wavelengths between 5930 and 5940 Å. This region of the spectrum unfortunately contains a number of telluric absorption lines from water molecules (e.g., Moore et al. 1966). Before searching for variations in the intrinsic SN line profiles, we therefore must remove the telluric features, which will change with atmospheric conditions (e.g., Wade & Horne 1988; Matheson et al. 2001).

On every night except that of the second-epoch observations (day 0, 2007 October 25), we obtained at least one spectrum of a hot, rapidly rotating star to serve as a telluric standard. These spectra were reduced and normalized to a flat continuum level in the same manner as the SN data. Because the telluric standards do not contain any atmospheric features in the region of interest, any absorption seen must necessarily be of telluric origin. From the telluric standard observations we constructed telluric absorption line model spectra using a list of known telluric lines.²² Pixels in the model spectra that were at the wavelength of telluric lines were set equal to the value of those pixels in the telluric standard star spectra, and all other pixels were set to unity. We then selected the telluric model obtained at the most similar airmass and time for each SN spectrum.²³ The telluric models were further adjusted to match the air mass of the SN observations with the following scaling: model_adjusted = (model_original)^b, where b = air mass_SN/air mass_standard. Finally, we divided the reduced SN spectra by their respective telluric absorption models to produce clean SN spectra. The strength of the absorption lines that overlap with the wavelength of the host-galaxy Na D absorption in the telluric models ranges from a few percent to 10%, not nearly large enough to account for the absorption profile changes described in Section 3.3.

We display the high-resolution spectra of SN 2007le around the host-galaxy Na D lines in Figure 3. At least one component of the absorption (near 5931.5 Å and 5937.5 Å in the D₂ and D₁ lines, respectively) strengthens over the course of the observations, increasing from a total depth of ∼0.6 in the first spectrum to nearly saturated three months later.²⁴ The full absorption profile is complex, with ∼7 distinct velocity components visible. Fitting the absorption with a set of Gaussians reveals that at least two additional components blended with the strongest absorption features are needed in order to remove significant residuals, and even then the fits to the high-S/N spectra are not statistically satisfactory. Fits of multiple blended Gaussians do not produce unique results, however, so we cannot study the physical conditions in the variable absorption component(s) accurately with this technique.

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Instead, wefirst integrate directly over the entire absorption profile to determine the equivalent width (EW) of the D₁ and D₂ lines in each spectrum. The measured equivalent widths are listed in columns (2) and (3) of Table 2. Over the course of the observations, both lines show an increase in the EW of slightly more than 100 mÅ. The total EWs in the day −5 spectrum are 894 ± 2 mÅ and 649 ± 2 mÅ for the D₂ and D₁ lines, respectively. By the final spectrum at day +84, the EWs are 1006 ± 5 mÅ and 766 ± 5 mÅ. For the four blue most components of the absorption, we rule out any variation at high significance; the EWs of those components agree in all six epochs to a level of a few mÅ. To reduce the uncertainty in the estimate of the change in EW, we can therefore leave these components out of the integration and sum the absorption starting from the peak at 5931 Å (5937 Å for the D₁ line) instead of from the blue edge of the absorption at 5929.5 Å (5935.5 Å). With this smaller wavelength range we find a total increase in EW between day −5 and day +84 of 106 ± 5 mÅ and 105 ± 5 mÅ for the D₂ and D₁ lines (last two columns of Table 2).

It is worth noting that all of the Na D absorption features, including the variable component, are blueshifted relative to the strongest absorption component at 5931.8 Å (2130 km s⁻¹), which we presume corresponds to the local interstellar medium (ISM) velocity (see Sections 3.6 and 3.7). In SN 2006X and SN 1999cl, the variable absorption was also blueshifted compared to the ISM absorption (Patat et al. 2007a; Blondin et al. 2009), perhaps providing a clue as to the origin of the material responsible for the varying absorption.

Having determined the EW of the variable absorption, we would now like to constrain the column density and line width of the absorbing gas. To isolate the changing component of the absorption profile, we examine the difference between each spectrum and the first one. These difference spectra are displayed in Figure 4. At day 0 (maximum light), we do not detect any obvious difference signal from 5 days earlier, despite the possible small increase in EW reported for the D₁ line in Table 2. Between day −5 and day +10, however, a clear Gaussian line emerges in the difference spectrum (Figure 4, second row from the top). As expected from the EW measurements above, the strength of this line increases through day +84, and we also find that the Doppler parameter of the line grows from 4.2 ± 0.7 km s⁻¹ in the (day +10 minus day −5) spectrum to 10.7 ± 0.2 km s⁻¹ in the (day +84 minus day −5) spectrum (see Table 3). The line in the (day +10 minus day −5) difference spectrum appears close to symmetric, but stronger hints of asymmetry in the later difference spectra suggest that some of this increase in line width is resulting from the addition of a second blended absorption component.

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We see in Table 3 that the change in the EW of the D₂ line is very similar to the change in the EW of the D₁ line at all epochs. At 1σ significance, $0.78 \le \Delta \mbox{EW}_{\rm {D}_{1}}/\Delta \mbox{EW}_{\rm {D}_{2}} \le 1.03$ over the full set of difference spectra. Following Spitzer (1978, Equations (3)–(49) to (3)–(51)), we can use a curve-of-growth analysis to determine the observed EW for any combination of column density and line width. Using the derived Doppler parameters for the difference spectra given in Section 3.3, we calculate the change in EW that would be observed for a small increase in column density for all column densities between 10⁹ cm⁻² and 10^14.5 cm⁻². For a given line width, these calculations demonstrate that increasing the equivalent width of the D₂ and D₁ lines by the same amount is only possible for a narrow range of column densities, as illustrated in Figure 5. For b = 4.2 km s⁻¹, we find that 12.0 ⩽ log N_Na i ⩽ 12.3, for b = 5.3 km s⁻¹, we find that 12.1 ⩽ log N_Na i ⩽ 12.4, and for b = 10.7 km s⁻¹, we find that 12.4 ⩽ log N_Na i ⩽ 12.7. We therefore conclude that the column density of the varying component of the absorption is N_Na i ≈ 2.5 × 10¹² cm⁻². Depending on the Doppler parameter, such a column density corresponds to a total Na D₂ EW of 200–300 mÅ for this component, consistent with the results shown in Table 2.

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It is evident from Figure 5 that the column densities indicated by the first two difference spectra (blue dashed and red dotted curves) are not consistent with the column density preferred by the third difference spectrum (gray dot-dashed curve). We suspect that the reason for this inconsistency is that the third difference spectrum contains multiple blended components, making the measured line width larger than the true line width as noted in Section 3.3. However, without a higher-resolution spectrum that separates the two components, we do not have conclusive evidence in favor of this interpretation.

The Keck/HIRES spectra extend far enough to the blue that we also detect Ca ii H & K absorption lines from both the Milky Way and the host galaxy, although the S/N of the day +90 spectrum in the blue is too low to be useful. We compare the host-galaxy absorption profile in Ca K and Na D₂ in Figure 6, where a close correspondence between the absorption components in each species is visible.

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Unlike the Na lines, though, we detect no statistically significant changes in the Ca absorption profile with time. Even nearly three months after maximum light, the profile shape (Figure 7) and total EW (Table 4) match those from before maximum light within the uncertainties. However, while changes in the Ca EW as large as those seen in the Na lines can be ruled out, the low S/N of the late-time spectra prevents us from being able to place strong constraints on smaller variations.

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In addition to the Ca H & K and Na D lines, we also searched the spectra for the additional absorption features identified by Patat et al. (2007a) and Cox & Patat (2008) in SN 2006X. In the highest S/N (day −5) spectrum, we detect the CH⁺ 3957.70 Å and 4232.55 Å lines at a velocity of 2130 km s⁻¹, exactly matching the velocity of the strongest Na D absorption. Diffuse interstellar bands are also visible at 5780 Å and 6283 Å (rest wavelengths). We do not detect Ca i at 4226.73 Å, CH at 4300.30 Å, the CN vibrational band, or K i at 7699 Å.

All six of our high-resolution spectra cover the expected wavelength of the redshifted Hα line. However, the fiber spectrum from the HET is not suitable for investigating possible Hα emission from the SN because of the lack of local sky subtraction, so we only consider the five high-resolution Keck spectra in this section.

We detect a narrow Hα emission line at the position of the SN at all epochs of our observations. The HIRES spectra were obtained with a slit 7'' long and either 086 or 115 wide, and faint Hα emission from the host galaxy is visible all the way along the slit. However, careful sky subtraction reveals that there is also additional emission that is spatially unresolved and coincident with the position of the SN (see Figures 8 and 9). This line is also at a slightly different velocity than the extended host-galaxy emission.

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From the HIRES data, we measure the EW, velocity, and line width of the Hα emission at the position of the SN; the results are listed in Table 5. At v = 2139 km s⁻¹, the Hα emission is offset by 16 km s⁻¹ from the velocity of the variable sodium line (and has a higher velocity than any of the main sodium components), but is similar to the host-galaxy rotation velocity at the radius of the SN measured from a long-slit spectrum by Afanasyev et al. (1992). We then use the observed light curve and a comprehensive set of low-resolution spectra of SN 2007le from the CfA Supernova Archive to derive absolute flux calibrations for the spectra. In order to remove contaminating host-galaxy light (since the CfA spectra were obtained with a wide slit), the spectra are first scaled to the observed V-band magnitudes of the SN and then warped to match the B magnitudes as well. From these flux-calibrated spectra, we determine the SN continuum level at the position of Hα (6610 Å), scaling the spectra up and down as necessary to account for the change in the SN's R magnitude between the time of observation for each high-resolution spectrum and the nearest low-resolution spectrum (the time differences are less than two days for days −5 and +12, six days for day 0, and five and eleven days for days +84 and +90, respectively). We then convert the high-resolution EWs to fluxes in absolute units with the measured continuum levels.

In principle, the Hα emission could originate either from an H ii region projected within 140 pc (corresponding to the ≈1'' slit width) of the explosion or from the SN itself. In the case of an H ii region, one would expect other lines such as [O iii] λλ4959, 5007, [N ii] λλ6548, 6583, and [S ii] λλ6717, 6731 to be visible depending on the S/N. In the day +84 spectrum, which is the best for this purpose because of its relatively high S/N at red wavelengths and the faint magnitude of the SN at late times, we indeed detect Hβ and both lines of the [N ii] and [S ii] doublets. [O iii], though, which is generally the brightest of the forbidden metal lines in H ii regions (and comparable to or brighter than Hβ), remains undetected at all epochs. [O iii] fluxes more than an order of magnitude weaker than Hα are uncommon in H ii regions (e.g., Kennicutt et al. 2003; Magrini et al. 2007).

On the other hand, if the Hα emission is related to the SN, changes in the Hα flux with time would seem likely (although since such emission has never been detected, its exact characteristics are unknown). From the day −5 spectrum through the day +12 spectrum (spanning about 2.5 weeks), we detect no variability in the flux. In the day +84 spectrum more than two months later, however, the Hα flux is ∼40% higher than at day +12. If the comparison is made only between days +12 and +84, the significance of this change is just 2σ, but if we combine the measurements from days −5, 0, and +12 together then the day +84 increase is significant at the 4σ level. The low-S/N day +90 spectrum is consistent with both the early time and day +84 measurements. Taken as a whole, we consider these results to be a tentative indication of Hα emission from SN 2007le. More conclusive evidence of the nature of the Hα emission awaits late-time imaging and spectroscopy to determine whether or not there is an H ii region coincident with the position of the SN.

If the Hα emission is associated with the SN, its luminosity is related to the amount of gas present. The observed Hα emission has a luminosity of 1.6 × 10³⁵ erg s⁻¹ (for the average Hα flux of 1.5 × 10⁻¹⁶ erg cm⁻² s⁻¹). If we follow the calculations of Leonard (2007), such a luminosity corresponds to ∼0.025 M_☉ of solar-abundance material in the circumstellar environment, in reasonable agreement with theoretical predictions (Marietta et al. 2000; Mattila et al. 2005; Pakmor et al. 2008). However, those computations were specifically aimed at modeling Hα emission at very late times (day +380) and may not apply to the much earlier epochs at which we observed SN 2007le. Using the simple model presented by Patat et al. (2007a) as an alternative suggests a hydrogen mass of ∼0.001 M_☉ for the observed Hα luminosity. The presence of hydrogen in the CSM has also been indirectly inferred from the broadening of absorption features in the earliest spectra of SNe Ia (Mazzali et al. 2005a). In this scenario, ∼0.005 M_☉ of material with a solar composition is necessary to provide the electron density that is required to trap photons and cause absorption in lines such as Ca H & K or Si ii λ6355 (Mazzali et al. 2005b; Tanaka et al. 2008).

If the changes in the Na D absorption lines are a result of recombination, then the spectra tell us directly that the recombination timescale is ∼10 days (e.g., Figure 3). Since

$$\begin{equation} \tau _{\rm recomb} = \frac{1}{\alpha n_{e}}, \end{equation} \tag{ 1 }$$

where α, the Na i recombination coefficient, is 1.43 × 10⁻¹³ cm³ s⁻¹ at T = 10⁴ K (Badnell 2006; see Section 4.2 for a justification of this temperature range), we find that the electron density must be n_e ≈ 8 × 10⁶ cm⁻³. (For T = 2 × 10³ K, α = 6.18 × 10⁻¹³ cm³ s⁻¹, and the required electron density will be lower by a corresponding factor.) We note that such a high electron density necessitates that hydrogen must be partially or mostly ionized; no other plausible source could provide so many electrons (as mentioned in Section 3.7, Mazzali et al. 2005a suggest that electrons provided by hydrogen can also explain the origin of high-velocity features, as were present in SN 2007le, at early times). Even with full ionization, though, the physical density required in the absorbing material is quite high.

In order for a significant fraction of neutral sodium to be present, the photoionization rate should be comparable to or slower than the recombination rate, or alternatively, the recombination time should be shorter than the photoionization time. Following Murray et al. (2007), we write

$$\begin{equation} \tau _{\rm ion} = \frac{4 \pi r^{2} \langle h\nu \rangle }{a_{\rm Na\,{\scriptscriptstyle I}} L_{\rm UV}}, \end{equation} \tag{ 2 }$$

where r is the distance between the SN and the absorbing material, a_Na i ≈ 10⁻¹⁹ cm² is the photoionization cross section (Verner et al. 1996), L_UV is the UV luminosity of the SN, and 〈hν〉 corresponds to the photon energy required to ionize a Na i atom. We do not have UV photometry of SN 2007le, but to the degree that UV light curves of SNe Ia are fairly homogeneous (Brown et al. 2009), it is reasonable to substitute another similar SN. We therefore use the Swift light curve of the well-observed normal SN Ia 2007af from Brown et al. to estimate the UV luminosity. SN 2007af reached a peak Vega magnitude of ∼16.2 in the Swift UVW2 filter a few days before the B-band maximum. The UVW2 filter has a central wavelength of 1928 Å, a full width at half-maximum intensity (FWHM) of 657 Å, and a photometric zero point of 17.35 ± 0.03 mag (Poole et al. 2008). We use the distance modulus of SN 2007af, 32.06 mag (Simon et al. 2007), to deduce an absolute magnitude of M_UVW2 = −15.86, which suggests an apparent peak magnitude for SN 2007le of 16.49. With the zero point of the UVW2 filter, this magnitude translates to a Swift count rate of 2.2 counts s⁻¹. Poole et al. (2008) provide a conversion between the count rate and the flux density that depends only slightly on the source spectrum (∼6.1 × 10⁻¹⁶ erg cm⁻² s⁻¹ Å⁻¹), so integrating over the filter bandwidth yields a flux of 8.8 × 10⁻¹³ erg cm⁻² s. We therefore estimate a UVW2 luminosity for SN 2007le of 9 × 10⁴⁰ erg s⁻¹. Note that this bandpass corresponds reasonably closely to the wavelengths that can ionize Na i atoms most efficiently (λ < 2412 Å). Using Equation (2), we can now calculate that the ionization timescale is

$$\begin{equation} \tau _{\rm ion} = 1.1 \left(\frac{r}{10^{16} \; \rm{cm}} \right)^{2} \mbox{s}. \end{equation} \tag{ 3 }$$

For r ≲ 3 pc, then, τ_ion ≪ τ_recomb, and this estimate suggests that the Na should be essentially fully ionized around maximum light.

Although the UV luminosity of the SN declines with time, the Swift data show that the decline is rather slow: ∼1 mag in 10 days (Brown et al. 2009). Even several weeks after peak, when the high-resolution spectroscopy demonstrates that significant changes in the Na D line profile have already occurred, Equations (1) and (3) indicate that recombination (at the assumed densities) should only be occurring at relatively large radii, calling into question the hypothesis that circumstellar material is responsible for the varying Na absorption. These simple calculations are in qualitative agreement with the results of Chugai (2008) and indicate that more sophisticated modeling is required.

Because the simple ionization and recombination timescale arguments do not provide obvious answers as to why the Na i column density is changing, we now consider more detailed photoionization models of the SN environment. Starting with a synthetic UV spectrum for a SN Ia from Nugent et al. (1995) as the ionizing source, we carry out calculations with the photoionization code Cloudy (Ferland et al. 1998, ver. 07.02). We use the UV spectra of SN 2001eh and SN 2001ep (Sauer et al. 2008) to normalize the total number of UV–IR photons (1500–12000 Å) in the model spectrum.²⁶ The model spectrum is only available for the time of maximum brightness, so we assume that the light curve at all UV wavelengths follows the SN 2007af UVW2 light curve from Brown et al. (2009). (We have to combine observations of multiple SNe here because there are few nearby SNe Ia with both good UV light curves and UV spectroscopy.) This assumption means that the calculations are not explicitly time dependent, but we do not expect non-equilibrium effects to produce qualitative changes in the results discussed below. We also note that, while at longer UV wavelengths the model can be compared to observed spectra for validation, at the critical far-UV wavelengths that ionize hydrogen no such comparison with real data is possible. Borkowski et al. (2009) have recently carried out very similar calculations to investigate the possibility of detecting CSM interactions in spectra obtained before maximum light.

Guided by the hypothesis that recombination is responsible for the varying Na i column density, a few iterations of the model demonstrate that very large hydrogen densities are needed to produce significant Na recombination (for a solar abundance ratio). We therefore assume a hydrogen density of n = 2.4 × 10⁷ cm⁻³ (somewhat higher than the electron density inferred in Section 4.1). We then calculate the Na i, Ca ii, and H ii fractions as a function of time and distance away from the SN. The results are illustrated in Figure 10. For distances greater than ∼0.1 pc the fraction of Ca ii ions is essentially constant with time, while the Na i fraction increases by approximately an order of magnitude over a time span of 30 days. Closer to the SN, the Ca becomes more highly ionized (and the Ca ii fraction varies with time) and the amount of neutral Na is negligible, supporting the idea that the source of the absorption must be located at distances larger than 10¹⁶ cm.

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Over the range of parameter space where these calculations indicate that the Na i column density will vary and the Ca ii column density will not, the EW of the Na D₂ line is predicted to be less than or equal to the EW of the Ca K line. At distances of less than 1 pc, the Ca absorption should be ≳100 times as strong as the Na absorption. Yet, we observe that EW_Na D is larger than EW_{Ca H&K} at all epochs (see Tables 2 and 4 and Figure 7). If the model described above is correct, we therefore require that nearly all of the Ca atoms in the CSM are locked up in dust grains, as is also typical in dense interstellar clouds (e.g., Spitzer 1954; Howard et al. 1963; Herbig 1968; Morton 1975). In the immediate vicinity of a SN, the destruction of dust grains is obviously a possibility. However, given sublimation temperatures of ∼1500 K for silicate and carbonaceous grains (Draine & Salpeter 1979), calculations of grain temperatures with Cloudy using the dust grain model of van Hoof et al. (2004) indicate that dust can survive at distances greater than 0.05 pc (unshaded region in Figure 10) over the relevant time period.

The results presented in Section 4.2 point to distances of ∼0.1 pc for the absorbing material. Figure 10 demonstrates that the EW of the Ca ii absorption would remain constant for distances as small as ∼0.02 pc, but the fraction of neutral sodium so close to the SN is negligible. Distances greater than 1 pc are also allowed by the photoionization modeling, but then the ionization fraction is very low (as a result of the weak UV field), requiring extremely high physical densities to produce large numbers of electrons. Of course, at such distances the galactic UV field may contribute significantly to the photoionization rate, rendering our calculations no longer applicable. At d = 0.1 pc, the gas temperature is ∼5000 K, and densities of n_H ≈ 2.4 × 10⁷ cm⁻³ are required to produce significant Na recombination. With such a high density, substantial amounts of dust could be present, and strong depletion of Ca atoms onto dust grains would explain the large observed ratio of Na i to Ca ii. Given the derived density and a solar Na abundance of 12 + log(Na/H) = 6.17 (Asplund et al. 2005), the total density of Na atoms is n_Na = 35 cm⁻³. The Na i fraction under these conditions is ∼5 × 10⁻⁴, yielding n_Na i = 1.8 × 10⁻² cm⁻³. In order to obtain a column density of ∼2.5 × 10¹² cm⁻², as derived in Section 3.4, the path length through the absorbing cloud must be ∼1.4 × 10¹⁴ cm. If such a clump of material were roughly spherical its mass could be quite small (∼2 × 10⁻⁷ M_☉), but a clump of radius 10¹⁴ cm at a distance of 10¹⁷ cm would only occult a small fraction of the SN photosphere (radius ∼10¹⁵ cm) after maximum light and thus could not be responsible for an absorption line as deep as that observed in Figure 3. In order to produce a covering fraction closer to unity, there must either be large numbers of small clumps, or perhaps a larger sheet of material with perpendicular dimensions of ∼10¹⁷ cm (the rather high densities involved probably rule out the full spherical shell model proposed by Patat et al. 2007a). The total CSM mass along the line of sight (presumably more CSM is present in other directions) would then be in the range of 10⁻⁵–10⁻² M_☉ depending on the geometry.

Is it plausible for such a cloud (or clouds) to exist in the SN progenitor system? The measured velocity of the variable absorbing component is 2123 km s⁻¹, while the local host galaxy ISM velocity near the position of the SN is ≈2130 km s⁻¹. Assuming that the velocity measured for the local ISM is representative of the progenitor system, then the variable absorption component has a line-of-sight velocity of ∼10 km s⁻¹ relative to the progenitor itself. At a constant velocity of 10 km s⁻¹, material starting near the center of the progenitor system would take ∼10⁴ yr to reach a radius of 0.1 pc, but factoring in the gravitational deceleration the actual travel time would be ∼3000 yr if the material originated in the red giant wind or ∼10 yr if it was blown off from the white dwarf (which requires a much higher initial velocity). Regardless, this timescale is certainly far shorter than the age of the progenitor system or its evolutionary timescale, so producing CSM at such distances does not appear to be problematic. The survival of dense clumps similar to what we are proposing here, though, may be an issue: the thermal pressure in such a cloud is quite large, suggesting that the cloud should expand and dissipate on timescales of decades unless the average CSM pressure is also very high. We do not currently have enough information to determine whether such clumps of material could form from the wind of the companion alone, the interaction of the wind with surrounding material, or a nova eruption on the surface of the white dwarf.