Shedding light on the Type Ia supernova extinction puzzle: dust location found
M. Bulla A. Goobar S. Dhawan


: multi-purpose gravitational lens modelling software package
    Adam Amaracor2
    .
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The colour evolution of reddened Type Ia supernovae can place strong constraints on the location of dust and help address the question of whether the observed extinction stems from the interstellar medium or from circumstellar material surrounding the progenitor. Here we analyse BV photometry of 48 reddened Type Ia supernovae from the literature and estimate the dust location from their B-V colour evolution. We find a time-variable colour excess  for 15 supernovae in our sample and constrain dust to distances between 0.013 and 45 pc (4×10^16-10^20 cm). For the remaining supernovae, we obtain a constant  evolution and place lower limits on the dust distance from the explosion. In all the 48 supernovae, the inferred dust location is compatible with an interstellar origin for the extinction. This is corroborated by the observation that supernovae with relatively nearby dust (≲ 1 pc) are located close to the center of their host galaxy, in high-density dusty regions where interactions between the supernova radiation and interstellar clouds close by are likely to occur. For supernovae showing time-variable , we identify a potential preference for low  values, unusually strong sodium absorption and blue-shifted and time-variable absorption features. Within the interstellar framework, this brings evidence to a proposed scenario where cloud-cloud collisions induced by the supernova radiation pressure can shift the grain size distribution to smaller values and enhance the abundance of sodium in the gaseous phase. 


dust, extinction – circumstellar matter – supernovae: general




§ INTRODUCTION



In the past couple of decades, Type Ia supernovae (SNe Ia) have been used as cosmological distance indicators and led to the discovery of the accelerated expansion of the Universe <cit.>. A key to such breakthrough discovery was the realization that the absolute brightness dispersion of SNe Ia can be reduced by exploiting two main empirical relations. The first is the well-known “width-luminosity” relation, which corrects for the fact that intrinsically brighter SNe Ia decline more slowly after peak <cit.>. The second one, usually referred to as the “colour-brightness” relation, accounts for the fact that fainter SNe Ia are typically redder than brighter SNe Ia <cit.>. Despite their importance for cosmological studies, the physical origin of these two relations has not yet been settled (see  for a review).

Intervening dust along the line of sight to the observer can dim the SN radiation and make it redder, thus possibly explaining the observed “colour-brightness” relation. However, both sample studies <cit.> and the analysis of individual SNe <cit.> clearly indicate that the extinction seen in many SNe Ia is peculiar compared to the one in our Galaxy. Specifically, its wavelength dependence (the so-called extinction law) is much steeper than the one observed in the Milky Way (MW). This is usually parametrised by the total-to-selective extinction ratio , where A_X is the extinction in the band X, with increasingly steeper extinction laws corresponding to increasingly smaller . While R_V∼3 is typical for dust in our Galaxy <cit.>, much lower values are often found in SNe Ia (R_V≲2, see references above). Independent confirmations of low  come from spectropolarimetry of some reddened SNe Ia <cit.>.

The relatively low  values have been taken as evidence for circumstellar (CS) material surrounding the progenitor at the time of explosion <cit.>. In particular, multiple-scattering of photons within a CS shell can increase the relative extinction at shorter wavelengths and thus steepen the resulting extinction law. The CS scenario would be consistent with the claimed detection of CS gas from blue-shifted and time-variable sodium and potassium features <cit.> and with the observed similarities between the continuum polarization of some reddened SNe Ia and that of proto-planetary nebulae <cit.>. 




Another possible interpretation is that the low  values are caused by dust in the interstellar (IS) medium. This would require IS clouds in the SN host galaxies to be characterized by smaller dust grains compared to those seen in our Galaxy <cit.>, possibly as a consequence of cloud-cloud collisions induced by the SN radiation pressure <cit.>. Evidences supporting the IS scenario were given by <cit.> who found a strong correlation between the extinction of SNe Ia and the strength of a diffuse interstellar band at 5870 Å. In addition, a recent study by <cit.> demonstrated how the colour evolution of reddened SNe Ia can put strong constraints on the location of dust. In particular, the time-evolution of the colour excess  in SN 2006X and SN 2014J was found to be consistent with dust in the IS medium.

Understanding the origin of dust extinction is also important to constraint the progenitor system(s) of SNe Ia since the amount of CS material is predicted to vary among different scenarios. While SNe Ia are widely-accepted to arise from thermonuclear explosion of carbon-oxygen white dwarves in binary systems, whether the companion star is a non-degenerate star (“single-degenerate”, SD, scenario) or another white dwarf (“double-degenerate”, DD, scenario) is still hotly debated (see e.g.  and references therein). In the SD scenario, the high mass transfer preceding the explosion creates high-density CS material with which the SN radiation is likely to interact. In contrast, a cleaner local environment is expected from DD models. 

In this paper, we use the technique introduced by <cit.> and estimate dust locations in 48 SNe Ia from the literature. We describe our dataset in Section <ref> and outline the approach used to estimate  for both models and observations in Section <ref>. We then present our results in Section <ref> and discuss their implication for the CS/IS debate in Section <ref>. Finally, we draw conclusions in Section <ref>.



§ DATA



Photometric data of the 48 SNe Ia analysed in this work are taken from the literature. In particular, photometry for 38 SNe is obtained from either the CfA supernova program <cit.>, the Lick Observatory Supernova Search (LOSS, ) or the Carnegie Spectroscopic Project (CSP, ). Photometry references for all the SNe are reported in Table <ref>. The time of maximum in the B-band, t_max(B), the decline in the B-band magnitude from peak to 15 d after, , and the MW colour excess, E(B-V)_MW, are calculated using snoopy <cit.> and also reported for each SN in Table <ref>.





§ ESTIMATING COLOUR EXCESS



In this Section, we outline the approach use throughout the paper to estimate colour excess values .

As discussed below, extracting colour excess values from both dust models (Section <ref>) and observations (Section <ref>) require some assumptions on the intrinsic SN light curves. For most of the SNe in our sample, the pristine light curves are constructed from the spectral template of <cit.>. For SN 2003hx, SN 2007cf and SN 2014J, photometric data that are relevant to capture the full  evolution and to provide strong constraints on the dust localization extend to epochs not covered by the Hsiao template (≳85 d). For these particular cases, we take the low-extinction <cit.>, apparently normal `plain vanilla' <cit.> SN 2011fe as representative of a pristine SN Ia. In particular, we adopt photometric data of SN 2011fe from <cit.> as these extend to 325 d after peak. 

Here we stress that SN 2011fe is used only when necessary to properly characterize the  evolution of the reddened SNe, while preferring a template constructed from a sample of SNe Ia (as the Hsiao template is) for the analysis of most of the objects. We note, however, that the choice of the template is important, with templates extending to later epochs typically leading to stronger constraints on the dust localization (see discussion in Section <ref>).




 §.§ Models



In this work, we use the Monte Carlo code described by <cit.> and publicly available at <https://github.com/mbulla/dust3d>. The code is similar to those in <cit.> and <cit.>, and simulates the propagation of Monte Carlo photons through a dust region. In particular, it keeps track of the accumulated time-delay of each photon compared to non-interacting photons, Δ t. The dust region is modelled as a spherical shell at a given distance ζ from the SN, with an optical depth τ_λ set to give a desired colour excess along the direct line-of-sight (DLOS), . For the range of  values used in this work, 0.3 mag≤  ≤1.75 mag, optical depths in the B- and V-band are in the range 1.2≤τ_B≤6.8 and 0.9≤τ_V≤5.2, respectively. Dust properties are taken from <cit.> and corresponds to type dust[In terms of  time evolution, our models are not very sensitive to the specific dust composition (see figs. 2 and 3 of  and discussion in Section <ref>).]. 

Photons escaping the dust shell are collected and used to construct a light curve after interaction with dust. As described by <cit.>, this is done by using a SN template (see above) and taking into account the different time-delays Δ t of photons escaping the dust shell. When comparing models to data in Section <ref>, templates are stretched in time by a convenient amount s so that  for the constructed light-curve matches the one observed for the SN under investigation,  (see Table <ref>). At each time, the extinction A_X in any given X-band is then computed as the difference between the template and the constructed light-curve, and the desired colour excess calculated as =A_B-A_V.



 §.§ Observations







For the SNe in our sample,  are calculated by comparing the observed B-V colours to those from the adopted template. The procedure is shown in Fig. <ref> for the specific case of SN 1999ee and summarized in the following steps:


  * Photometric data points are averaged over 10-day windows following <cit.>; 

  * The SN template is stretched in time by a factor s (see above);

  * B-V colours are calculated for both the observations and the template;

  * The total (host + MW) colour excess is calculated by subtracting the template B-V from the observed B-V; 
  * The MW colour excess of the SN (Table <ref>) is subtracted to find the desired extragalactic colour excess .

This procedure is analogous to the one adopted in <cit.> and <cit.> but does not require to fit photometric data for both  and R_V. This is justified by the relatively small impact of different dust compositions (and thus corresponding R_V) on the predicted colour excess values (see fig. 3 of  and discussion in Section <ref>). The different approaches were tested on SN 2006X and found to give similar results.

  



Uncertainties on the extracted  includes propagation of errors on photometric data, σ_data, and intrinsic SN colour variations, σ_intr. The latter is estimated by comparing B-V colours of individual low-reddening SNe Ia from the CSP sample <cit.> to those predicted by the Hsiao template. In particular, we follow <cit.> and use SNe that are either in E/S0-type galaxies, located away from dusty regions or with no detection of Na I D lines in early-time spectra. This is shown in Fig. <ref> for 8 SNe Ia with <0.05 mag <cit.> and with a range of observed  spanning the one in our sample of 48 reddened SNe Ia (0.8-1.6 mag). After stretching light curves for the 8 SNe Ia to match the same  of the Hsiao template, the root-mean-square deviations (RMSD) about the template are typically smaller than ∼ 0.1 mag. For epochs earlier than -15 d, instead, the RMSD is large (0.3 mag) and due to the difference in colour between SN 2011fe and the Hsiao template. Nonetheless, we note that this high RMSD has no impact on our analysis since no SN in our sample has available photometry at these early epochs. The inferred RMSD values are taken as representative of the intrinsic SN colour variations and are reported in Table <ref>. These are in good agreement with those estimated by <cit.>. We notice that the intrinsic B-V colour variations reach ∼ 0.1 mag at ∼30-40 d after B-band maximum (when the B-V curve is at its peak) but they are typically smaller (≲ 0.05 mag) at earlier and later epochs. This point will be important in Section <ref> when constraining the dust distance from the  temporal evolution of some SNe Ia.





§ RESULTS




As described by <cit.>, no temporal variations in the colour excess are expected for dust relatively distant from the SN since the percentage of photons removed from the DLOS by dust – and thus the extinction – is the same at all epochs. In contrast, a time-variable colour excess is expected for relatively nearby dust due to the simultaneous arrival to the observer of non-interacting photons and of photons scattered by dust. Compared to non-interacting photons, scattered photons have travelled longer and were thus emitted earlier with a different brightness and colour. As a result of the intrinsic SN colour evolution (see Fig. <ref>), scattered photons are typically bluer than non-interacting photons at early epochs, while redder than non-interacting photons at late epochs. Therefore – after an initial phase of constant colour excess when the contribution of scattered photons is negligible – the colour excess evolution is characterized by a first drop (i.e. decreasing reddening) and then by a later rise (i.e. increasing reddening).

In this section, we analyse the 48 SNe Ia in our sample and infer dust distance values through the comparison between observed and predicted . We find 15 SNe to be characterized by a time-variable colour excess (Section <ref>), while the remaining 33 SNe to be consistent with a constant colour-excess evolution (Section <ref>). Following <cit.>, we extract distance values for the first sub-sample while distance lower limits for the second sub-sample.

Best-fit dust models, and corresponding most-probable distances ζ, are identified via a simple χ^2 test. The extracted dust distance values, together with the inferred  and , are summarized in Table <ref> and Table <ref>. For SNe showing a time-variable , we report the most-probable distance values and their 1σ-uncertainties; for SNe compatible with a constant  evolution, instead, we report 3σ lower-limits on the dust distance.




 §.§ Dust distance estimates



Fig. <ref> and Fig. <ref> show the extracted  values for the 15 SNe Ia with time-variable extinction, together with the inferred best-fit dust models.

The 15 SNe span a wide range of , going from the highly-reddened SN 2014J (1.40 mag) to SN 2007le (0.34 mag).

For 13 of the 15 SNe, we are able to capture only the initial constant phase and the drop in . This is because the rise in  predicted in our models is not captured either by the available photometry or by the adopted template (see discussion in Section <ref>). The inferred locations of dust for these SNe span a range between 0.7 (SN 1996bo) and ∼ 45 pc (SN 2014J). The full evolution is instead captured by the remaining two SNe. In the case of SN 2007cs, this is due to the combination of relatively nearby dust (∼ 1 pc) and good time-sampling of the light curve (extending to ∼150 d after peak). In the case of SN 2003hx, instead, the dust is quite nearby the explosion site and thus the  evolution is faster. Specifically, our models suggest that dust is located at 0.013 pc. In Section <ref>, we discuss the possibility that the relatively nearby dust toward this SN is suggestive of CS material connected to the progenitor system.



 






We note that the distance locations estimated here for SN 2006X and SN 2014J differ from those in <cit.>. This is due to the use of different photometric data for these SNe. For SN 2006X, we adopt data from <cit.> and obtain a distance value of ζ=17.3_-1.0^+1.3 pc, slightly smaller than the value obtained by <cit.> using data from <cit.>, ζ=22.1_-3.8^+6.0 pc. For SN 2014J, we use photometry from <cit.> extending to ∼150 d after maximum, that is to much later epochs than those covered by photometry from <cit.> used in <cit.>. While a roughly constant  was found in <cit.> and only a 1σ lower limit[Note that 1σ lower limits were reported in , while here we adopt a more appropriate and conservative choice of 3σ for the lower limits (see Table <ref>).] of ζ>38 pc was placed on the dust distance, here we identify a drop in  starting from ∼ 40 d and constraint the presence of dust at ζ=44.6_-2.0^+3.3. This demonstrates the importance of late-time photometry to constraint dust distances with our technique.






 §.§ Dust distance lower limits



For the 33 SNe Ia showing a constant colour excess, we estimate lower limits on the dust distance. Unfortunately, photometry for many of these SNe is not good enough (e.g. sparse sampling, lack of early-/late-time epochs) to unambiguously exclude dust located at distances inferred in Section <ref>. For instance, about half of these SNe have rather poor photometry that results into relatively weak constraints (ζ≳0.3-1 pc), while SNe with better photometry (e.g. SN 1999e in Fig. <ref>) are more constraining. 
In particular, SNe with photometry extending to relatively late epochs (≳40 d post peak) are typically characterized by strong constraints on the dust location (ζ≳5 pc). Again, this highlights the importance of late-time photometry. Nevertheless, comparisons with models suggest that dust for these SNe is likely to be located at pc-scale distances.











§ DISCUSSION



In Section <ref>, we extracted dust distance values for the 48 SNe Ia in our sample. The  evolution for most of the SNe is consistent with dust located at ζ≳1 pc. In one case, instead, dust is estimated to be much closer to the SN, at distances where CS material connected to the progenitor might be present (ζ∼0.013 pc, SN 2003hx). Here we discuss the implications of our results for explaining the origin of extinction in SNe Ia. In Section <ref>, we first compare the dust distance values obtained in this study to those available in the literature for some of the SNe in our sample. In Section <ref>, we then provide evidences that dust seen in all 48 SNe is located within IS clouds. In Section <ref>, we finally discuss how the “cloud-cloud collisions” scenario proposed by <cit.> is able to explain our findings within the IS framework and to answer many open questions connected to SNe Ia suffering extinction by dust.



 §.§ Comparison to previous studies





  §.§.§ Light echoes



An approach to estimate the location of dust in SNe is through the detection of light echoes, namely light scattered by dust into the DLOS <cit.>. Instead of using early-time photometry as done in this work, light echo measurements rely on imaging observations and focus on later times (≳ 1 year). At these epochs, the SN flux along the DLOS has faded considerably and the delayed signal from light scattered by dust produces a ring/arc on the plane of the sky. Simple geometrical arguments are then used to work out the dust location from the time-delay and position of the appeared light echo. This approach was applied to a handful of SNe Ia in the past, including four objects in our sample: SN 1998bu, SN 1995E, SN 2006X and SN 2014J. 

Two distinct light echoes were detected towards SN 1998bu <cit.>, one estimated to be at 10±3 pc from the SN and the other one between ∼ 100 and 200 pc. Our lower limit on the dust distance to SN 1998bu is ζ>11.7 pc, fully consistent with the outer echo while only marginally with the inner one. Dust located at 207±35 pc was instead inferred from light echo measurements of SN 1995E <cit.>, in agreement with our lower limit of ζ>7.7 pc. 

A light echo towards SN 2006X was detected by <cit.> at ∼ 300 d after peak, and the dust responsible for the echo was located at ∼27-120 pc away from the SN. Using images at later epochs (680 d), <cit.> constrained the same echo to be at 26.3±3.2 pc. Analysis by <cit.> on even later epochs (∼1400 d) resulted in a larger estimate for the SN-dust distance, 80±5 pc. Our calculations suggest that dust responsible for the bulk of extinction seen in SN 2006X is located at ζ=17.3_-1.0^+1.3 pc from the SN, 2.6σ away from the findings of <cit.>. Using  instead of type dust in our models gives ζ=20.1_-1.4^+1.3 pc, in better agreement with the light echo estimate (1.8σ). 

Light echo detections towards SN 2014J were reported by <cit.> and <cit.>. Specifically, both studies identify an arc-like echo created by dust at ∼200-300 pc and a more diffuse echo originated by an extended dust layer with an inner radius of ∼ 100 pc. Localization of possible dust at distances closer than ∼ 70 pc were not possible due to difficulties in properly removing the DLOS SN flux within 0.3 arcsec from the SN position. This range includes the location of dust estimated for SN 2014J in this work, ζ=44.6_-2.0^+3.3, highlighting the advantage of our approach to identify dust at relatively nearby distances.

Dust distance estimates presented in this work for SN 1998bu, SN 1995E, SN 2006X and SN 2014J are compatible to those inferred for the same SNe from light echo detections. Both approaches exploit the information from light scattered by dust, but light echo studies focus on late-time imaging observations while our technique on photometry at earlier epochs. As shown above, this difference results in our technique being more sensitive to relatively-nearby dust (within a few tens of pc) while light echo measurements to relatively-far dust (distances larger than a few tens of pc).










  §.§.§ Time-evolution of polarization



Dust sufficiently close to the SN is predicted to induce time-variability in the polarization signal <cit.>. The lack of time-variability in the continuum polarization of SN 1986G, SN 2006X and SN 2014J was taken by <cit.> as evidence that the bulk of extinction suffered by these SNe stems from IS clouds. In the case of SN 2014J, a similar conclusion was reached by <cit.> analysing optical and near-infrared imaging polarimetry. However, observations by <cit.> and <cit.> were taken at relatively early epochs, between ∼ 10 d before and ∼ 35 d after peak brightness. A recent study by <cit.> analyses imaging polarimetry of SN 2014J at later epochs and claims a variation in the polarization signal occurring 277 d after maximum light. This is interpreted as due to ∼10^-6M_⊙ of CS dust located at ∼ 0.2 pc from the SN. Owing to the small mass, this dust shell is expected to contribute only marginally to the extinction <cit.> and therefore not to be captured by the approach presented here.



  §.§.§ Time-evolution of sodium, potassium and calcium








The detection of time-variable Na I, K I and Ca II features in SNe Ia can be interpreted as arising in either (i) CS gas subject to ionisation and recombination effects <cit.> or in (ii) IS gas with covering factor changing with time due to the expansion of the underlying SN photosphere (usually referred to as geometric effects, ). 

Time-variable features were found for three SNe Ia in our sample. Variations in the Na I D profiles were reported for SN 2006X <cit.> and SN 2007le <cit.>, while changes in a K I spectral feature were seen in SN 2014J <cit.>. If attributed to outflowing CS gas (see below), the corresponding distances for the absorbing material would be ∼ 0.003-0.03, 0.1 and 2-5 pc for SN 2006X, SN 2007le and SN 2014J, respectively. Our technique places dust at larger distances from the explosion in the case of SN 2006X (17.3 pc), SN 2007le (5.9 pc) and SN 2014J (44.6 pc). 


We note that – with the exception of SN 2006X – all the detection of time-variable absorption features in the literature are also compatible with geometric effects. In particular, the observed changes in the equivalent widths of these lines are of the order of those induced by the SN photosphere expanding in a fractal patchy IS medium <cit.>. If attributed to geometric effects, changes observed in SN 2007le and SN 2014J would be qualitatively consistent with our findings of material at 5.6 and 44.6 pc, respectively. For SN 2014J, our estimate would also agree with the one inferred by <cit.> from the non-detection of time-variable Na I D (a few ∼ 10 pc away from the explosion, see their figure 14). For geometric effects to work in the case of SN 2006X, where equivalent widths of Na I D were as high as ∼ 300 mÅ, one would instead need to invoke a highly non-homogeneous IS medium (see also  and discussion in Section <ref>).

Time-variability was searched but not detected in low-resolution spectra of 9 SNe Ia in our sample <cit.>: SN 1997dt, SN 1998bu, SN 1998dm, SN 1999cl[Changes in the Na I D features were originally reported for SN 1999cl by <cit.>, but later found to be due to measurements errors <cit.>.], SN 2002bo, SN 2003cg, SN 2007S, SN 2007bm and SN 2007ca. Our study suggests that dust is located at interstellar scales (≳ 1 pc) for these SNe, and therefore the lack of time-variability can be interpreted as due to the SN photosphere expanding into a rather homogenous interstellar medium (i.e. no detectable geometric effects).









 §.§ Evidence for interstellar extinction



In Section <ref>, we analysed the  evolution for all the SNe Ia in our sample and using dust models inferred the location of dust in each of them. Here we discuss what these numbers tell us in terms of the origin of the extinction.

In particular, we frame our discussion in terms of whether the dust responsible for the extinction is connected (CS) or not connected (IS) with the SN progenitor system.




 curves consistent with no-time variability are found for 33 SNe in our sample and the dust inferred to be at distances larger than ∼ 0.3 pc (Table <ref>). As discussed in Section <ref>, however, photometry for about half of these SNe is rather poor and does not extend to late-enough epochs to place strong constraints on the dust location. SNe with good photometry, e.g. those with last epoch taken later than 40 d after maximum, are more constraining and associated with dust at ζ≳5 pc .

These distances are consistent with the extinction arising within the IS medium from dust not connected to the SN progenitor system.

 curves showing time-variability are found for 15 SNe in our sample (Table <ref>). For most of these SNe, our simulations indicate that dust is located between 1 and 50 pc and therefore that the origin of the observed extinction is likely within IS clouds. SN 2003hx is instead predicted to have dust much closer to the explosion (0.013 pc). Given the proximity of dust, it is tempting to think that the extinction for this SN stems from CS material associated to the progenitor system. However, the proximity of dust might also be reflecting extinction by IS clouds in a relatively high-density region of the host galaxy. In the following, we try to investigate this in more details by looking at the location of this SN.



The location of SN 2003hx is interesting in the sense that this SN occurred very close to the center of the host galaxy. This is shown in Fig. <ref>, where we plot the dust distance estimates vs the SN location for SN 2003hx and the remaining 14 SNe with time-variable .

The SN position is calculated as R/R_25, where R is the projected galactocentric distance and R_25 parametrises the size of the galaxy and is defined as the radius at which the surface brightness of a spiral galaxy falls off to a level of 25 mag/arcsec^2 in the B-band. Values for R_25 are taken from the Asiago Supernova catalogue <cit.>. Although R/R_25 probes a projected rather than a real distance to the galactic centre, Fig. <ref> shows that SN 2003hx not only is the object in our sample with dust closer to the explosion but also the one that exploded closer to the center of the galaxy (R/R_25≲0.1). 

A potential trend of increasing dust distance with increasing galactocentric radius is found when looking at the other 14 SNe. This is especially true when removing SN 2002bo, SN 2007cs, SN 1986G and SN 2014J that are known to occur in peculiar regions of their host galaxy, specifically within dust lanes. In particular, SNe at intermediate distance from the galactic centre (0.1≲ R/R_25≲0.2) are associate with dust between ∼ 1 and ∼ 5 pc, while those at larger distances (R/R_25≳0.2) with dust around 10 pc. The observed trend is what one would expect if the extinction in all the 15 SNe is located within IS clouds, with the chances of interacting with a dust layer increasingly closer to the SN site becoming higher as one moves from the outskirts to the center of the galaxy.

We conclude that the extinction in all the 48 SNe Ia is likely to originate from dust in the IS medium not connected to the SN progenitor system.





 §.§ A scenario for interstellar dust



The extinction observed for the 48 SNe Ia in our sample is likely to occur within IS clouds. Here we discuss a possible scenario in the IS framework that can account for most of the open questions connected to reddened SNe Ia, namely the origin for the low  values, the unusually large sodium abundance, the preference for blue-shifted absorption features and the time-variability of sodium and potassium features (see Section <ref>). These open questions, together with their possible interpretation within the IS scenario, are summarised in Table <ref> and discussed in the following.

In a recent study, <cit.> suggested that strong radiation pressure from a SN explosion can accelerate dust grains to high terminal velocities and induce collisions between the cloud closest to the SN and another cloud in its vicinity. Specifically, grains of size a at distance ζ from the SN will be accelerated to velocities

    v ∼ 171 (L_bol/10^8 L_⊙)^1/2(ζ/100 pc)^-1/2(a/10^-5 cm)^-1/2km s^-1

where L_bol is the bolometric luminosity of the SN (see also ). Hence, even relatively distant dust clouds (ζ∼100 pc) can be accelerated to rather large velocities (∼200 km s^-1) for bolometric luminosities of the order of L_bol∼10^8 L⊙. For a typical SN Ia, this luminosity is reached within a few days after explosion (∼ 15 d before peak, ).

The peculiarly low  values seen in some SNe Ia are explained by this scenario in terms of a shift in grain sizes from larger to smaller values caused by the cloud-cloud collisions <cit.>. In other terms, it is the SN itself that affects the IS environment and causes the dust properties to change from known (i.e. MW-like) to peculiar. Supporting this picture are observations by <cit.> which indicate that dust at the location of SN 2014J in M82 was normal  (R_V∼3, see their figure 6) a few years before explosion.

Moreover, the cloud-cloud collision scenario holds promises for explaining the Na column-density excess observed in some reddened SNe Ia. Following <cit.>, this can be defined as

    Δ N_Na = N_Na - 10^13.180 + 1.125 log A_V  ,

where N_Na is the observed sodium column density while the second term the one expected from the extinction A_V if the dust-to-gas relation seen in the MW is valid. In the scenario proposed by <cit.>, the collisions between clouds not only leads to a change in the grain size distribution but also enhances the abundance in the gaseous phase of sodium (and other atoms), thus explaining why Na column densities are unusually strong in some reddened SNe Ia.




Given the above arguments, a correlation might be expected between  and Δ N_Na in reddened SNe Ia. In Fig. <ref>, we show these two parameters for a sample of 18 SNe Ia[Note that here we include only SNe Ia from the literature displaying a Na excess, i.e. those with Δ N_Na>0.] from the literature <cit.>. 
While we use values taken from the literature for most of the SNe, A_V for the 7 SNe that are also in our sample are calculated using the  from Tables <ref> and <ref>. We do not see a clear correlation between  and Δ N_Na in Fig. <ref>. However, we notice a preference for SNe Ia with lower  to have larger Δ N_Na. It is intriguing that most of the SNe characterized by low  and high Na abundances are those in our sample for which we estimate the dust to be at pc-scale distances to the explosion (SN 2002bo, SN 2006X, SN 2007le, SN 2008fp and SN 2014J). This should be a natural prediction in the scenario proposed by <cit.>, where the efficiency of the collision mechanism is expected to be higher for closer compared to more distant dust clouds. 

The acceleration of dust clouds towards the observer can also explain the observed preponderance of blue-shifted absorption features in the spectra of some reddened SNe Ia. In particular, velocities estimated from equation (<ref>) show a reasonable match to those observed in SNe Ia (∼ 100 km s^-1). It is also encouraging that blue-shifted absorption features were identified in almost all the events in Fig. <ref> showing low  and high sodium excess, namely SN 2002bo, SN 2006X, SN 2006cm, SN 2007le, SN 2008fp and SN 2014J <cit.>. <cit.> already pointed out that the SNe in their sample with unusually strong Na are all associated with blue-shifted absorptions. In contrast to the interpretation provided here, however, this was taken as evidence for CS material since the presence of outflowing material connected to the progenitor system was the only natural explanation for the observed blue-shifted features[Note that the scenario of <cit.> had not been proposed yet at the time the study by <cit.> was conducted.].

As discussed in Section <ref>, the observed time-variabilities in Na and K features can only be explained in the IS framework by invoking some geometric effects. While most of the variations seen in SNe Ia are compatible with the projected SN photosphere expanding in a fractal patchy IS medium <cit.>, those observed in SN 2006X would require a highly non-homogeneous IS medium <cit.>. It is again encouraging that all the normal SNe Ia showing time-variability in either Na or K (SN 2006X, SN 2007le, SN 2014J) have dust in relatively nearby interstellar clouds according to our analysis, at distances where the covering factor of IS clouds can be significant and rapidly varying with time.

In summary, our analysis suggests that low  values, unusually strong Na abundances and blue-shifted and time-variable absorption features are likely to originate from IS clouds relatively nearby to the explosion (pc-scale distances) and experiencing collisions induced by the SN radiation pressure <cit.>.




§ CONCLUSIONS



We have studied the temporal evolution of the colour excess  in 48 SNe Ia. Comparing the observed evolution to that predicted by dust models from <cit.>, we constrained the location of dust ζ in each SN. We summarise the main findings of our analysis in the following:




  * 15 SNe in our sample show time-variable . We predict dust to be located at distances in the range  pc for 4 of these SNe, while between ∼1 and 50 pc for the remaining SNe;


  * 33 SNe in our sample are consistent with no time-variability in . For these SNe, we place lower limits on the dust distance that range from ζ≳ 0.3 to ζ≳ 40 pc. We highlight how the rather poor light-curve sampling, and specifically the lack of late-time epochs, in about half of these SNe result in poor constraints on the dust location;


  * The inferred dust locations for all the 48 SNe Ia are consistent with the observed extinction arising within IS clouds. This is corroborated by the fact that  SNe interacting with dust relatively nearby (ζ≲1 pc, at possible CS scales) are those in our sample that are the closest (in projection) to the center of their host galaxy, where high-density dust environments are expected;


  * Comparing our results to those in the literature, we find that SNe interacting with interstellar dust at pc-scale distances show a preference for low , unusually strong Na absorption and blue-shifted and time-variable absorption features;


  * In the IS framework, we interpret the last point as a possible evidence supporting the cloud-cloud collision scenario by <cit.>. Relatively nearby IS clouds can be accelerated towards the observer (hence blue-shifted absorption) by strong SN radiation pressure and then collide with clouds in their vicinity, thus causing the destruction of big into smaller grains (hence low Rv) and the enhancement of sodium in the gaseous phase (hence unusually Na absorption). In the IS scenario, the observed time-variabilities of sodium and potassium would instead be explained by the change in the covering factor of the IS gas following the expansion of the underlying SN photosphere (namely by invoking geometric effects);


  * We do not find any evidence for CS dust, thus tentatively supporting progenitor scenario with rather clean local environments (e.g. double-degenerate systems).



In this study, we highlighted the power of photometry to place constraints on the location of dust towards SNe Ia, showing in particular the importance of both early- and late-time data. Surveys like the ongoing Zwicky Transient Facility (ZTF, ) will be valuable in finding reddened and nearby SNe Ia at early epochs, allowing for dedicated photometric follow-ups to sample the light-curves of these SNe in the months following the explosion. This will allow us to test the results presented here on an even larger sample, possibly solving the questions about the origin of the extinction in SNe Ia.



§ ACKNOWLEDGEMENTS


The authors are thankful to the anonymous reviewer for his/her valuable comments. This research has made use of the CfA Supernova Archive, which is funded in part by the National Science Foundation through grant AST 0907903. The authors acknowledge support from the Swedish Research Council (Vetenskapsrådet) and the Swedish National Space Board.





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