Studying the small scale ISM structure with supernovae
F. Patat^{1}  N. L. J. Cox^{2,3}  J. Parrent^{4,5}  D. Branch^{4}
1  European Organization for Astronomical Research in the Southern
Hemisphere (ESO), K. Schwarzschildstr. 2, 85748 Garching b.
München, Germany
2  Herschel Science Centre, European Space Astronomy Centre, ESA, PO
Box 78, 28691 Madrid, Spain
3  Institute of Astronomy, K.U. Leuven, Celestijnenlaan 200D, 3001
Leuven, Belgium
4  Department of Physics and Astronomy. University of Oklahoma,
Norman, OK 73019, USA
5  Department of Physics and Astronomy, Dartmouth College, Hanover, NH
037553528, Germany
Received 23 December 2009 / Accepted 25 February 2010
Abstract
Aims. In this work we explore the possibility of
using the fast expansion of a type Ia supernova photosphere to
detect extragalactic ISM column density variations on spatial scales
of 100 AU
on time scales of a few months.
Methods. We constructed a simple model which
describes the expansion of the photodisk and the effects of a patchy
interstellar cloud on the observed equivalent width of Na I
D lines. Using this model we derived the behavior of the equivalent
width as a function of time, spatial scale and amplitude of the column
density fluctuations.
Results. The calculations show that isolated, small (100 AU)
clouds with Na I column densities exceeding
a few 10^{11} cm^{2}
would be easily detected. In contrast, the effects of a more realistic,
patchy ISM become measurable in a fraction of cases, and for
peaktopeak variations larger than 10^{12} cm^{2}
on a scale of 1000 AU.
Conclusions. The proposed technique provides a
unique way to probe the extragalactic small scale structure, which is
out of reach for any of the methods used so far. The same tool can also
be applied to study the subAU Galactic ISM structure.
Key words: supernovae: general  ISM: clouds  ISM: structure  ISM: general
1 Introduction
For many years it was accepted that the minimum size for the column density fluctuations in the Galactic ISM is around 1 pc ( AU). The common understanding was that although subparsec structures do exist, only a tiny fraction of the column density could be ascribed to these small scales (Dickey & Lockman 1990). However, the pioneering VLBI work by Dieter et al. Romney (1976), and the later confirmation by Diamond et al. (1989) demonstrated the existence of significant fluctuations over scales of 20 AU. These findings were confirmed by 21 cm absorption measurements against highvelocity pulsars (Frail et al. 1991,1994), which showed that the H I column density varies significantly over scales between 5 and 110 AU, with 10%15% of the cold neutral gas distributed in AUsized structures (Frail et al. 1994). However, some more recent radio observations on the same pulsars (Weisberg & Stanimirovic 2007) have shown that the variations are far smaller than those originally found by Frail et al. (1991,1994).
These studies were followed by a series of works looking at the variations of Ca II and/or Na I column densities along the lines of sights to close binaries or high proper motion stars (see Crawford 2003; Lauroesch 2007, for a review). Similar investigations were carried out for molecular gas (CH, CH^{+}, and CN; Pan et al. 2004; Rollinde et al. 2003), and diffuse interstellar bands (Cordiner et al. 2005). An alternative method is the study of interstellar absorptions along the lines of sight to stellar clusters, like M92 (Andrews et al. 2001) and Cen (Van Loon et al. 2009), or the Magellanic Clouds (e.g. André et al. 2004). For a general review on the small ISM structures in our Galaxy the reader is referred to Haverkorn & Goss (2007).
In this article we present an independent technique to analyze extragalactic ISM structure on spatial scales of about 100 AU. The proposed method is based on the extremely high expansion velocity displayed by a supernova (SN) photosphere (10^{4} km s^{1} or 5.7 AU day^{1}). A type Ia SN reaches a photospheric radius of 10^{15} cm (100 AU) in two weeks from the explosion, and expands at a rate of 6 to 3 AU day^{1} during the first two months of its evolution. If the typical size of the fluctuations in an intervening cloud is much larger than 10^{15} cm, then the associated absorption features will not evolve with time. On the contrary, if the ISM is patchy on comparable scales, the column density fluctuations will translate into measurable variations of the corresponding absorption features.
After introducing a simple model for the calculation of timedependent line equivalent widths for a given cloud geometry (Sect. 2), we present the results of MonteCarlo simulations (Sect. 3), and discuss the applicability of the method and the effects on the observations of type Ia's (Sect. 4). Appendix A gives the details on the derivation of the composite equivalent width.
Figure 1: Relevant quantities used in the text. The underlying column density map was generated using a powerlaw spectrum with (see Sect. 2.1). 

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2 A simple model
Let us fix a reference polar coordinates system (r, )
whose
origin is located at the explosion center (see
Fig. 1).
As seen from a far observer (the distance
between the SN and the observer is assumed to be much larger than that
between the SN and the intervening material), the SN will appear as an
expanding photodisk which extends to the ejecta boundary radius
.
We then consider a cloud placed in front of the SN,
at a distance large enough that the explosion has no effect on its
physical conditions (i.e. >10 pc, see Simon et
al. 2009. See
also Sect. 4
here), and we
indicate with
the cloud column density in the species
under consideration. Finally, we introduce
as the
timedependent surface brightness profile of the photodisk at the
wavelength of interest^{}.
This function is normalized as follows
so that the total continuum flux emitted by the photodisk along the line of sight is equal to unity. If g(N,b) is the curve of growth for the given transition and Doppler parameter b, the equivalent width (EW) produced by an infinitesimal cloud column with cross section is .
If we neglect the small contribution by photons scattered by the cloud
into the line of sight, the total equivalent width is then computed
integrating the contribution of each single infinitesimal element over
the photodisk (see Appendix A for the details):
So far we have considered the possibility that the Doppler parameter b can change across the cloud. However, in the lack of evidence for significant variations over the scales of interest (see for instance Welty & Fitzpatrick 2001), in the following we will assume that b is constant across the relevant portion of the cloud. We will briefly discuss the effects of a spacedependent Doppler parameter in Sect. 3.
With the aid of the outlined procedure, one can follow the time evolution of EW for any input cloud column density map, provided the photodisk's expansion law is known.
In the assumption of homologous expansion (see for instance
Jeffery &
Branch 1990),
the radius of the photosphere
is
obtained
multiplying the photospheric velocity
by
the time t elapsed from the explosion. For
we
adopted a best fit to values computed via SYNOW modeling of the
spectroscopically normal SN 1994D, a standard type Ia
event (Branch et
al. 2005).
During the photospheric phase (t<100 days),
is
well approximated by an exponential law, and the
best fit relation takes the following form:
where is expressed in AU and t in days from the explosion. This law shows that the change in the photodisk dimensions is significant during the first two months: from day 10 to day +50 (counted from maximum light) increases by more than a factor of 5 (60 AU to 320 AU).
As for the photodisk surface brightness profile, this was computed using a modified version of the spectrum synthesis code SYNOW^{} (Branch et al. 2005). The profile, obtained from best fits of SN 1994D spectra for the Na I D restframe wavelength, rapidly drops at during the early phases (see Fig. 2). As time goes by, a significant fraction of the flux (up to 16% on day +15) is emitted above the photosphere, and it is due to scattering by the broad Na I D doublet intrinsic to the SN (see for instance Jeffery & Branch 1990). At all epochs, for , which we used as the effective external boundary of the ejecta. To include the time dependence we tabulated for a number of epochs (10, 4, +7, +15, +28 and +50 days from B maximum) and subsequently used a linear interpolation to derive the profile at any given epoch. The time from B maximum light was converted into tusing the rise time of SN 1994D (18 days; Vacca & Leibundgut 1996). In view of the lack of very early spectra, we conservatively assumed that for at t=0.
Figure 2: Surface brightness profiles derived from SYNOW best fits of SN 1994D spectra on four different epochs (10, 4, +7 and +15 days from B maximum). 

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2.1 Cloud generation
We explored two possible cases: i) an isolated, homogeneous, spherical cloud with radius and offset with respect to the line of sight; ii) a patchy sheet with an input spectrum for the column density fluctuations.
In the first case, the column density profile is , where is the projected distance from the cloud center ( ), and N_{0} is the column density corresponding to a ray going through the center of the cloud. The average column density is N_{0}. As the case of an isolated, small cloud is probably quite unrealistic, it may be rather regarded as a simplified model of an overdensity on an otherwise homogeneous sheet.
For the patchy sheet we adopted a procedure similar to the one described by Deshpande (2000). Molecular clouds (Elmegreen et al. 1996), the diffuse ionized component (Cordes et al. 1991), and H I (Stanimirovic et al. 1999; Deshpande et al. 2000) have a fractal structure, characterized by a powerlaw behavior. Therefore, if k=l_{0}/l is the wavenumber corresponding to a given spatial scale l (where l_{0} is the maximum spatial scale under consideration), the power spectrum of the fluctuations can be written as . For the large scale structure of H I in the SMC 3 (Stanimirovic et al. 1999) and for the cold atomic gas in the Galaxy (Deshpande et al. 2000). Since the variations observed on scales of 100 AU and below can be explained in terms of a single power spectrum description (Deshpande 2000), we computed the column density maps using a powerlaw with . After generating a power spectrum map P(k_{x},k_{y}) in the Fourier plane (with k=(k^{2}_{x} +k^{2}_{y})^{1/2}), we derive the real and imaginary parts of the Fourier transform of N(x,y) using as the modulus of the complex numbers, while phases are generated as random numbers uniformly distributed between 0 and 2. The column density map is then obtained antitransforming into ordinary space. Given the radius of the photosphere during the time interval typically covered by observations (<400 AU), we adopt l_{0}=1024 AU.
Once the twodimensional column density map is generated, it is renormalized to have an average column density and peaktopeak fluctuations (with ). An example cloud is presented in Fig. 1. If is the column density difference between two points separated by a distance l, the numerical simulations show that for the RMS value of is 10% of the maximum variation on the same spatial scale. Furthermore, the differences at higher wavenumbers decrease proportionally to , as predicted by theory (Deshpande 2000). This implies that the fluctuations expected on scales of 100 AU are 40% of those observed on scales of 1000 AU, which for Na I can reach a few 10^{12} cm^{2} (Andrews et al. 2001). Therefore, a significant amount of structure is expected on spatial scales comparable to the typical photospheric radius.
2.2 Measurement error estimates
The ability of detecting small EW variations hinges on the precision to which EWs can be measured. In turn, this relates to the instrumental setup and the signaltonoise ratio SNR reached on the adjacent continuum per resolution element. To estimate the expected RMS uncertainty we have used the following formula (Chalabaev & Maillard 1983; see their Eq. A.10), derived for photon noisedominated conditions and with no assumption on the line profile:where is the line integration range, is the dispersion (Å pix^{1}), is the number of pixels corresponding to the integration range, F is the average flux within the absorption, is the average continuum level, and is the associated uncertainty. In addition to the parameters directly related to the instrumental setup, the application of this analytical expression requires the knowledge of other quantities which depend on the line profile. To compute them for a given column density, we generate synthetic Voigt profiles with input velocity parameter and a sampling equal to . This is then convolved with a Gaussian profile with a given full width halfmaximum (FWHM) to mimic the instrumental broadening. Using this profile we calculate the line integration range , which we define as the wavelength interval in which the equivalent width is 99% of total. Within this interval we finally derive the average line flux F. As for the continuum , we assume this is computed on two symmetric, equally extended regions on either side of the absorption. If is the total extension of this region, then the number of pixels used to determine is , so that . In the computations we have used Å which, given the featureless nature of the SN pseudocontinuum, is a realistic value^{}.
Table 1: Estimated Equivalent Width RMS errors computed using Eq. (1) ( SNR = 100 and N=10^{12} cm^{2}).
The results obtained for N=10^{12} cm^{2}, and for different values of FWHM, b (1, 3 and 5 km s^{1}corresponding to EW of 58, 113 and 138 mÅ respectively), and are presented in Table 1. As the RMS errors are inversely proportional to SNR, these values can be readily scaled to different signaltonoise ratios. These results have been checked against MonteCarlo simulations and were found to be consistent to within a few 0.1 mÅ. Incidentally, this questions the need for a revision of the Chalabaev & Maillard formula discussed by Vollmann & Eversberg (2006).
In the following we will consider an equivalent width variation detectable if . For a typical case where FWHM=7 km s^{1}, b=1 km s^{1}, Å pix^{1}, and , this turns into a 5 detection limit mÅ ( mÅ for b=5 km s^{1}).
3 Results of simulations
Although the model can be used for any interstellar absorption line, in the following we present the results obtained for Na I D_{2}, because it is a strong transition, it falls in a region almost free of telluric absorption features, and in a spectral interval where most optical, highresolution spectrographs have their maximum sensitivity.
Figure 3: Examples of simulated Na I D_{2}EW variation as a function of time for a spherical, homogeneous cloud with offset (upper panel) and AU ((lower panel). The various curves refer to different cloud radii (8, 16, 32, 64, 128, 256, 512 and 1024 AU from bottom to top). In all cases cm^{2} and b=1 km s^{1}. 

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3.1 Isolated spherical cloud
Example EW evolutions for two different cloud offsets (0 and 64 AU) and a number of cloud radii are presented in Fig. 3 up to 2 months after maximum light. In general, the maximum variability is expected when the cloud is close to the center of the photodisk. The maximum variation is achieved for cloud radii between 64 and 128 AU. Also, small clouds are better detected during the early phases (when their size is comparable to that of the photosphere), while the detection of large clouds requires a larger time span. If the cloud is too large ( AU), then the EW variation is not sufficiently ample to be detected. With a minimum set of two observations taken 10 days apart in the premaximum phases, for typical highresolution setup, a 5 detection limit of 4.4 mÅ, b=1 km s^{1} and cm^{2}, the simulations show that one is able to detect clouds with between 16 and 128 AU up to a maximum offset of 64 AU. For larger offsets the cloud starts to intersect the photodisk when its size is too large and the corresponding covering factor is too small. Besides implying probably unrealistic density contrasts, increasing the column density does not enhance the detectability of a small size cloud, since this rapidly becomes totally opaque. We note that, while this causes the saturation of the covering factor, it does not produce a saturated profile in the emerging absorption line.
A distinctive feature of small ( AU), offset clouds is an EW growth followed by a decrease during the premaximum light phase (Fig. 3, lower panel). This is due to the growth of the covering factor as the photodisk starts intersecting the offcentered knot. Once a maximum value is reached, the subsequent increase in the photodisk size causes the covering factor to drop. Although this mechanism can produce an absorption feature which grows in strength and then disappears on timescales of a month, this is expected to happen only during the premaximum epochs, when the relative increase in the photodisk surface is very fast. Larger clouds placed at larger offsets ( AU) produce features which start appearing around maximum light, but keep steadily increasing in strength up to several months after maximum.
Finally, for central column densities (or density contrasts) smaller than cm^{2}, the EW variations are always below the detection limit.
3.2 Patchy clouds
Because of the stochastic geometry of the powerlaw clouds, their effects were evaluated using a statistical approach. For a given cloud realization we computed and derived the absolute peaktopeak variation over the whole time interval 10 to +50 days. To mimic a more realistic situation, we did this also on two subsets of datapoints including only postmaximum observations (0, +10, +20, +30, +40, +50 and 0, +10, +20, +30 days). As statistical estimators we computed the median absolute peaktopeak variation over the full range ( ), the semiinterquartile range and the 99th percentile ( ). For the three time ranges we finally estimated the 5 detection probabilities P_{t} ( ), P_{0} () and P_{30} (0 ). The results are presented in Table 2 for different values of , and b. For each parameter set 5000 cloud realizations were computed (blank values indicate detection probabilities ). indicates the EW corresponding to and the input Doppler parameter.
Visual inspection of a number of realizations showed that the EW(t)curves are smooth, with typical timescales of the order of 10 days. The variation rate is systematically larger at early epochs and decreases as time goes by. The simulations show that peaktopeak column density fluctuations smaller than 10^{11} cm^{2} on a scale of 1000 AU do not produce any measurable effects. Even with high (100) signaltonoise ratio observations, starting 10 days before maximum and covering the first two months of the SN evolution, the detection probability is below 2%. This grows significantly when the fluctuations exceed cm^{2}. Incidentally, this implies that SNe suffering higher extinction are expected to display more pronounced variations (see also Chugai 2008). Because of the saturation effect, the EW variations are more marked for larger values of the Doppler parameter. Finally, for column densities exceeding cm^{2} the detection probability decreases due to line saturation.
Table 2: Results of MonteCarlo simulations for powerlaw clouds with .
To study the effect of a spatially variable Doppler parameter, we have run a set of simulations in which b is allowed to fluctuate around the average value across the cloud. We have tentatively modeled the Doppler parameter map using the same algorithm and spatial scales spectrum adopted for the column density generation (see Sect. 2.1). We remark that this is not meant to reproduce real physical conditions, but only to estimate the consequences of velocity dispersion fluctuations (for instance, b and were left completely independent). The MC runs (with 1 km s^{1}) show that for cm^{2} there is practically no difference with respect to the case with constant Doppler parameter. The differences start to be significant at cm^{2}, for which the typical variations and detection probabilities become much larger.
This was to be expected, since in the quasilinear regime attained at low column densities the effect tends to average out. On the contrary, as one enters the nonlinear part of the curve of growth, the largest variations are produced by the regions of the cloud where the Doppler parameter is higher (i.e. less subject to saturation), thus skewing the distribution towards more marked EW fluctuations. The exact behavior depends on the way the velocity dispersion varies across the cloud and how this (if any) relates to the column density fluctuations. However, the conclusion that a variable Doppler parameter enhances the detection probability is of general validity.
4 Discussion and conclusions
The simulations presented in this paper indicate that marked time effects on the measured EWs are expected for small ( AU), isolated clouds with cm^{2} and for small offsets ( AU). However, the existence of such structures is seriously questioned in terms of pressure equilibrium arguments and the yet unknown processes that would produce them (see Heiles 1997, and references therein). Frail et al. (1994) detected maximum variations that range from 10^{19} to cm^{2} on scales between 5 and 100 AU. For a Galactic Na I/H I ratio (Ferlet et al. 1985) this turns into between and cm^{2}. These large changes have been interpreted as arising within ubiquitously distributed small structures. However, this picture has been questioned by Deshpande (2000), who has convincingly shown that the observations are consistent with a single powerlaw description of the ISM, down to AU scales. In these circumstances, peaktopeak variations cm^{2}are expected on scales of AU. Our calculations (see Table 2) show that for a type Ia SN observed under the most favorable conditions ( on all epochs, spanning from 10 to +50 days) these would appear in less than 10% of the cases for km s^{1}. This fraction increases to 80% for km s^{1}, but in all cases it is mÅ. These small variations imply negligible changes in E(BV), but can have some effect on observing programmes studying the evolution of Na I features possibly arising in the circumstellar environment of type Ia progenitors (Patat et al. 2007; Simon et al. 2009). However, we note that these values are more than a factor of 10 smaller than the Na I D variations detected in the type Ia SN 2006X (Patat et al. 2007), which were attributed to the ionization effects induced by the SN on its circumstellar environment. In contrast, no statistically significant variations were detected for the CN, CH, CH^{+}, Ca I lines and DIBs associated to an interstellar cloud in the host galaxy (Patat et al. 2007; Cox & Patat 2008). In the only other well studied case published so far (SN 2007le), the EW of four Na I D components remained constant to within a few mÅ during six epochs spanning about 3 months (Simon et al. 2009). In this time range changed approximately from 100 to 400 AU, and the lack of evolution is in line with the predictions of our model for a power spectrum ISM and definitely excludes the presence of small, isolated clouds with sizes comparable to . Although the available data are still scanty, the multiepoch, highresolution campaigns which are being conducted for the study of type Ia progenitors will provide a more statistically significant sample.
All the discussion so far is based on the assumption that the physical conditions of the ISM are not modified by the SN explosion, so that all variations in the absorption lines are due to pure geometric effects. Indeed, a type Ia SN can produce changes in the ionization balance of lowionization species (like Na I or K I) up to quite large distances, of the order of 10 pc (Patat et al. 2007; Chugai 2008; Simon et al. 2009). Given the UV flux predicted for these distances (Simon et al. 2009), the ionization timescale of Na I is expected to be about 120 days. Since the electron density in the ISM is low, the recombination time is extremely long. Therefore, under the assumption of a constant ionizing field, one would expect the amount of neutral Na to decrease with timescales of several months, hence mimiking small scale structure effects. However, the UV flux of a type Ia SN decreases significantly after maximum light (a factor 20 in the first 40 days; Brown et al. 2009) implying that the maximum distance is probably less than 10 pc. Given the range of possible distances to an interstellar cloud within the host galaxy, this suggests that significant Na I column density variations in the ISM induced by the SN radiation field are improbable.
Another important fact is that, because of the much higher ionization potential of Ca II, and the strong UV line blocking present in type Ia spectra, the EW of the ubiquitous H&K lines becomes insensitive to the SN radiation field already at distances of a few 0.1 pc (Simon et al. 2009). In contrast, in the case of a geometrical origin, all species are expected to show synchronous variations, although possibly with different amplitudes and time scales (Lauroesch & Meyer 2003). Therefore, a comparison between the behaviors of Na I and Ca II should allow one to disentangle between geometrical and ionization effects, similar to what has been proposed by Patat et al. (2007).
The method we presented enables the study of small scale structure in the extragalactic ISM, which is out of reach for any of the techniques deployed so far. In this respect we note that the same method can be in principle applied to the Galactic ISM to study its subAU structure, for which no direct measurements are available yet. Although probably requiring very high signaltonoise ratios, this technique might put important constraints on the very small scale structure. In this article we have discussed the case of the strong, easily detectable Na I D lines. However, other weaker lines (e.g., K I, Ca I, Ca II) can be used, especially when the Na I D lines are saturated. In general, the simultaneous study of different atomic/diatomic lines along the lines of sight to SNe will contribute to get a more detailed picture of the physical conditions of the ISM in the small scales regime.
Appendix A: Calculation of composite equivalent width
Let us consider an extended photodisk, a cloud placed in front of it
and an absorption line with profile function ,
normalized so that
If N is the spacedependent column density of the cloud (atoms cm^{2}) for the atomic species under consideration, the monochromatic line optical depth is , where K is the frequencyintegrated line opacity (cm^{2}atom^{1}). Let us now consider an infinitesimal element of the photodisk in the polar reference system introduced in Sect. 2 (see Fig. 1), and define as the radial profile of continuum surface brightness for the photodisk. For the sake of simplicity we assume that the continuum I_{0} is constant across the line profile and is normalized to unity:
With these settings, the monochromatic intensity contributed by the infinitesimal photodisk element to the total observed intensity can be written as
For a distant observer, to whom the photodisk will appear as an unresolved source, the total line intensity profile is obtained integrating all the infinitesimal contributions over the photodisk:
Given the definition of equivalent width
the composite equivalent width can be expressed as follows:
Because of the normalization of , the previous relation can be rewritten as
Now, the inner integral is the equivalent width one would observe if the physical system were composed only by the infinitesimal cloud element and the corresponding portion of the photodisk:
which implies that the composite equivalent width is the weighted sum of the equivalent widths produced within each infinitesimal cloud element. If g(N,b) is the curve of growth for the given transition (where b is the Doppler parameter that characterizes the line profile), then Eq. (A.1) can be reformulated as:
Acknowledgements
The authors wish to thank L. TacconiGarman, S. Stanimirovic and A. Deshpande for their kind help. The authors are also grateful to an anonymous referee for the useful comments and suggestions.
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Footnotes
 ... interest^{}
 We assume has no azimuthal dependence, i.e. that the SN is spherically symmetric. For a type Ia SN this is a reasonable assumption (Wang & Wheeler 2008).
 ... SYNOW^{}
 Note that SYNOW, like some other SN spectrum synthesis codes (see e.g. Mazzali & Lucy 1993), has zero limb darkening.
 ... value^{}
 For the purposes of measuring the EW of an interstellar line, the continuum definition in a SN spectrum is much easier than in a stellar spectrum, where the presence of other intrinsic features may contaminate the adjacent regions.
All Tables
Table 1: Estimated Equivalent Width RMS errors computed using Eq. (1) ( SNR = 100 and N=10^{12} cm^{2}).
Table 2: Results of MonteCarlo simulations for powerlaw clouds with .
All Figures
Figure 1: Relevant quantities used in the text. The underlying column density map was generated using a powerlaw spectrum with (see Sect. 2.1). 

Open with DEXTER  
In the text 
Figure 2: Surface brightness profiles derived from SYNOW best fits of SN 1994D spectra on four different epochs (10, 4, +7 and +15 days from B maximum). 

Open with DEXTER  
In the text 
Figure 3: Examples of simulated Na I D_{2}EW variation as a function of time for a spherical, homogeneous cloud with offset (upper panel) and AU ((lower panel). The various curves refer to different cloud radii (8, 16, 32, 64, 128, 256, 512 and 1024 AU from bottom to top). In all cases cm^{2} and b=1 km s^{1}. 

Open with DEXTER  
In the text 
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