The Nature of the \SitAbs, \Cat\ HK, Ca\,{\sc II} IR-triplet, and other Spectral Features in Supernova Type Ia Spectra
Daniel R. van Rossum
Spectra of Supernovae of type Ia (\snia) are commonly interpreted as a continuum with absorption features. The pseudo equivalent width (pEW) and Doppler shift of absorption features like \SitAbs, \Cat\ HK 3750\AA, and the \Cat\ IR-triplet 8150\AA\, measures which are commonly interpreted as the optical thickness and the velocity of a shell of corresponding absorbing material, form the basis of common \snia\ spectral interpretation and classification schemes. In this paper, we examine the nature of spectral features in \snia\ spectra using W7 model spectra, and show that \SitAbs\ and many other features are largely emission dominated instead of absorption dominated. We show that apparent absorption features (like \SitAbs) are frequently just coincidental troughs between two (or more) uncorrelated emission features. The pEW measured between such emission peaks is then only little related to true strength of the presumed absorption feature. It shows how the concepts of ``absorption troughs'' and ``continuum'' can be misleading and should be used with care. Furthermore, using the same model spectra, we demonstrate for different times post explosion how spectral features overlap each other and how together they compose the total spectrum. This overlap distorts individual line profiles and affects measured absorption velocities. With an explosion model atmosphere that is tuned to match specific observations, the method presented in this paper can in principle be used to quantify all these effects and improve the interpretation and informative value of observed \snia\ spectra.
paper.tex
%\documentclass[12pt,preprint]{aastex}
\documentclass[iop]{emulateapj}
\usepackage{graphicx}

\bibliographystyle{apj}

\newcommand{\phx}{\texttt {PHOENIX}}
\newcommand{\snia}{SN Ia}
\newcommand{\sneia}{SNe Ia}
\newcommand{\Mgt}{Mg\,{\sc ii}}
\newcommand{\Sit}{Si\,{\sc ii}}
\newcommand{\SitAbs}{Si\,{\sc ii}\ 6150\AA}
\newcommand{\SitLab}{Si\,{\sc ii}\ 6355\AA}
\newcommand{\St}{S\,{\sc ii}}
\newcommand{\Cat}{Ca\,{\sc ii}}
\newcommand{\Fet}{Fe\,{\sc ii}}

\begin{document}

\title{The Nature of the \SitAbs, \Cat\ HK, Ca\,{\sc II} IR-triplet, and other Spectral Features in Supernova Type Ia Spectra}
\author{Daniel R. van Rossum}
\affil{Department of Astronomy and Astrophysics, University of Chicago, Chicago,IL 60637, USA}
\affil{Center for Computational Science, University of Chicago, Chicago,IL 60637, USA}

\keywords{stars: Supernovae: general - radiative transfer - methods: numerical}

\begin{abstract}
Spectra of Supernovae of type Ia (\snia) are commonly interpreted as a continuum with absorption features.
The pseudo equivalent width (pEW) and Doppler shift of absorption features like \SitAbs, \Cat\ HK 3750\AA, and the \Cat\ IR-triplet 8150\AA\, measures which are commonly interpreted as the optical thickness and the velocity of a shell of corresponding absorbing material, form the basis of common \snia\ spectral interpretation and classification schemes.

In this paper, we examine the nature of spectral features in \snia\ spectra using W7 model spectra, and show that \SitAbs\ and many other features are largely emission dominated instead of absorption dominated.
We show that apparent absorption features (like \SitAbs) are frequently just coincidental troughs between two (or more) uncorrelated emission features.
The pEW measured between such emission peaks is then only little related to true strength of the presumed absorption feature.
It shows how the concepts of ``absorption troughs'' and ``continuum'' can be misleading and should be used with care.
Furthermore, using the same model spectra, we demonstrate for different times post explosion \emph{how} spectral features overlap each other and how together they compose the total spectrum.
This overlap distorts individual line profiles and affects measured absorption velocities.

With an explosion model atmosphere that is tuned to match specific observations, the method presented in this paper can in principle be used to quantify all these effects and improve the interpretation and informative value of observed \snia\ spectra.
\end{abstract}


%-----------------------------------------------------------------------
\section{Introduction}
%{{{
Supernovae of Type Ia (\snia) generally show very similar spectral and photometric behavior (e.g., \citealt{Filippenko97}).
Because of this interesting homogeneity (and the fact that these objects become extremely bright) \sneia\ have proven to be very useful objects to study the universe at large distances \citep{Riess98, Perlmutter99}.
Yet despite the general similarity, there is an even more interesting diversity among \sneia, both photometrical and spectral.
Whereas photometrical diversity has long been studied \citep{Phillips93, Wang03}, good spectroscopic data have only recently become available in large numbers (CfA SN Program \citealt{Blondin12}; Berkeley \snia\ Program \citealt{Silverman12}).

One step towards understanding spectroscopic diversity is finding similarities and defining classes.
A number of spectral classification schemes for \snia\ have been proposed in literature.
These are based on measurements performed on conspicuous \Sit\ and \Cat\ absorption features present in all ``normal'' \citep{Branch93} \snia.
These absorption feature measurements are:
1) The feature strength, expressed as pseudo equivalent width (pEW, see \citealt{Folatelli04}) \citep{Branch06};
2) The line blue-shift, expressed as expansion velocity \citep{Wang09};
3) The rate of change in blue-shift, expressed as velocity gradient \citep{Benetti05}.

It is well known that atomic transition lines formed in a rapidly expanding atmosphere, like \snia, form P-Cyg profiles \citep{Lamers99}.
P-Cyg profiles generally show a superposition of an emission wing that is more or less symmetric around the transition's rest wavelength plus an absorption wing that is blue shifted.
But the precise shape of P-Cyg profiles in relativistically expanding atmospheres, and specifically the ratio of emission to absorption, strongly depends on details of the atmosphere \citep{Hutsemekers93}:
1) The radial extension of the line forming region;
2) The velocity gradient across the line forming region;
3) The wavelength dependence of the ``continuum'' and its change across the line forming region.

Parameterized models of \snia\ atmospheres, like SYNOW \citep{Fisher00, Branch09} or SYNAPPS \citep{Thomas11}, have proven to be very powerful and rewarding tools for the interpretation of \snia\ spectra.
Specifically, these models made it possible to identify lines and study the composition of observed \snia.
However, these models do not account for the geometrical extension of the line forming regions nor the velocity gradients and change of the ``continuum'' flux across the line forming regions involved in shaping the P-Cyg profiles.
Therefore, such models can not be expected to give accurate and physically realistic profile shapes, even though the large number of free parameters usually enables a rather precise reproduction of observed spectra.

Another fact that complicates the interpretation of \snia\ spectra is the large number of transition lines that leave their signature in the spectra in an indistinguishable way (i.e., smeared out by the large expansion velocities of \snia\ atmospheres).
\cite{Kasen06b} have shown that many millions of transitions are needed to calculate the \snia\ opacity accurately.
Even though some of these lines are stronger than others, the background opacity formed by the millions of weaker lines is definitely not constant (see \citealt{Kasen06b}) and will leave its signature on the profiles of those stronger lines.

Clearly, the line profiles in \snia\ spectra contain a large amount of information about the detailed atmospheric conditions and understanding their nature is the key to elevate the use and yields of spectroscopic observations to a higher level.
In this paper we will demonstrate that \SitAbs\ and many other features are in fact emission dominated and that commonly used measures like the pEW and the absorption velocity are little related to the actual properties of the these features.
%}}}

%-----------------------------------------------------------------------
\section{Methods}
%{{{
As described in the previous section, a realistic reproduction of P-Cyg profiles requires model atmospheres that describe the run of physical conditions over the geometrical extent of the line forming region.
In this paper we do not aspire to attain perfect reproductions of observed P-Cyg profiles but rather focus on understanding the general nature of spectral features as they are observed for \sneia.
For this purpose we calculate spectra from model atmospheres in which different atomic transition lines are removed from the opacity and compare them with spectra calculated using the full, unaltered opacity.
The direct comparison between altered and unaltered opacity spectra shows what the signature of the removed atomic transition line(s) on the full spectrum is.

Because the differences between spectra are often small compared to the normal flux level it is important that the model spectra are calculated to very high precision.
Monte Carlo methods for solving the radiation transport problem generate spectra in a statistical way, leading to spectra that are inherently noisy \citep{Lucy99, Kasen06, Kromer09} and consequently are not adequate for this purpose.
Instead, we use \phx, a state-of-the-art, general purpose stellar atmosphere code \citep{Hauschildt92, Hauschildt99, Hauschildt04, Hauschildt06}.
This code solves the special relativistic radiation transport problem with \emph{very high accuracy} using characteristic rays and operator splitting methods \citep{Olson87}.
The time evolution of \snia\ atmospheres can be self-consistently calculated with \phx\ using the radiation energy balance (REB) method \citep{Vanrossum12}.
\phx\ with the REB method can in principle calculate \snia\ atmospheres in NLTE, however the challenging treatment of the huge number of lines that play a role for \sneia\ is work in progress (van Rossum, in preparation).
Therefore, in this paper we use the LTE approximation.

Determining the atmospheric stratification of \snia\ ejecta from explosion models has been a highly active field of research for decades.
Since W7, the very successful phenomenological 1D explosion model by \cite{Nomoto84}, explosion models have continued to improve in physical and numerical respects \citep{Khokhlov91, Plewa04, Kasen09}.
Yet all these improvements have not necessarily made the model end products look more similar to typical \sneia.
Therefore, over time, W7 has become an established 1-dimensional (1D) reference model that is known to reproduce qualitatively the observed spectral properties of \sneia, from a few days to months after the explosion \citep{Kasen06, Kromer09, Vanrossum12}, making this a good starting point for studying the nature of spectral features.

Note that with the limitations of W7 (e.g. being 1D and not specifically tuned to match one certain \snia) it can not be expected to perfectly reproduce observations.
For this reason, we will also not compare results to spectra of individual objects but rather to the \cite{Hsiao07} (hereafter Hsiao07) spectral templates.
%}}}

%-----------------------------------------------------------------------
\section{Results}
%{{{
We examine the signature of individual lines (or groups of lines) by excluding these lines from the opacity and comparing the resulting spectrum to the full, unmodified spectrum.
Figure \ref{fig:0lin_all} shows the W7 model spectrum compared to the Hsiao07 templates at four different times post explosion (pe).
The peak template luminosity is aligned with the peak model luminosity at day 18pe.
A single scaling factor is applied to the unit-less template flux (simultaneously for all times) to globally match the model flux.
Apart from these two tightly constrained parameters, there is no freedom in the comparison.
%The lab wavelengths of the corresponding lines are \Sit\ 6347 and 6371\AA\ (commonly together referred to as 6355\AA), \Cat\ 3934 and 3969\AA, and \Cat\ 8498, 8542, and 8662\AA.
The colored patches indicate the difference (Delta) between spectra calculated with the lines of a certain chemical element removed from the opacity and the full spectrum (with unmodified opacities).
In order to visualize all contributions at once the patches (Deltas) are stacked (starting with the low-Z elements), emission deltas are stacked below the dashed line and absorption deltas above.
Patches below (above) the dashed full spectrum curve represent features in emission (absorption).

Figure \ref{fig:0lin_si2_all} zooms in on the \SitAbs\ feature in Figure \ref{fig:0lin_all}, although shown at different times post explosion to complement the information.
In Figure \ref{fig:0lin_diffeach} the same deltas are shown as in Figure \ref{fig:0lin_all} but now plotted relative to the flux maximum (of the full spectrum) instead of stacked on top of the full spectrum and on top of each other.
As a reference, it also shows the \cite{Bessell90} UBVRI filter functions, arbitrarily normalized.

\begin{figure*}
\centerline{
\begin{minipage}[t]{.49\textwidth}
 \centering
 \includegraphics[width=\textwidth]{w7.0lin.spec.7_all.eps}
 \caption{W7 model spectra (dashed) are compared to the Hsiao07 \snia\ spectral templates (solid) at different times post explosion (pe).
 The comparison uses only one free parameter to simultaneously match the template fluxes.
 The colored patches represent the signature of all atomic transitions of a single chemical element for each color.
 The signatures are individually calculated and subsequently stacked on top of the original spectrum.\\
 The colored areas above (below) the dashed curve show the absorptive (emissive) contributions.
 Apparently, \snia\ are largely \emph{emission dominated} longwards of 5500\AA and absorption features are generally much smaller than a pEW evaluation would suggest.\\ 
 The famous \SitLab\ ``absorption'' feature is poorly described by the model, especially at day 30pe.
 The W7 model (and other models, see text) seem to miss an important emission feature around 5800\AA.
 More care should be taken with the interpretation and designation of the 6150\AA\ trough.
 } \label{fig:0lin_all}
\end{minipage}
\hspace{.2cm}
\begin{minipage}[t]{.49\textwidth}
 \centering
 \includegraphics[width=\textwidth]{w7.0lin_si2.spec.7_all.eps}
 \caption{Zoom in on the \SitAbs\ feature of Figure \ref{fig:0lin_all}, although for different times post explosion: day 6, 8, 12, and 16pe.
 The spectral template flux level is omitted in this plot.\\
 The true absorption signature of this famous \Sit\ feature is much smaller than the total depth of the trough, and decreases over time.
 In these early epochs, the emission wing on the blue side of the 6150\AA\ trough is mainly affected by O, Si, S, Fe, and Co.
 The emission wing on the red side sees contributions from C, Mg, Si, and Co.\\
 \emph{The wavelength of the trough minimum depends not only on the velocity of the \SitLab\ line but also on how other features are blended in (including emission from the \Sit\ 5972\AA\ line).}
 } \label{fig:0lin_si2_all}
\end{minipage}
}
\end{figure*}

\begin{figure}
\centerline{\includegraphics[width=.49\textwidth]{w7.0lin.spec.7_diffeach.eps}}
 \caption{Spectral features of different chemical elements normalized to the unmodified flux peak level in the W7 model (see also Figure \ref{fig:0lin_all}).
 Emission and absorption features are plotted as positive and negative deltas, respectively.
 The dashed curves show the UBVRI \citealt{Bessell90} band filter functions.\\
 The famous \SitAbs\ and \Cat\ HK features, and many others, show emission wings that are stronger than their absorption counterparts.
 pEW definitions that are based on measuring the depth of presumed feature troughs may misrepresent the true feature strength.\\
 The \Cat\ HK feature (like many others) does not present itself isolated but is contaminated with contributions from Si, Co, Ti, and S.
 This complicates both the feature strength and velocity analysis.\\
 In all four epochs shown here, the B-band (more than other bands) contains contributions from many different elements.
 This possibly makes the flux in the B-band more robust against signature strength (abundance) variations between individual \sneia.
 } \label{fig:0lin_diffeach}
\end{figure}

\begin{figure}
 \centerline{\includegraphics[width=.5\textwidth]{si2_v0.eps}}
 \caption{The absorption velocity of the \SitAbs\ feature measured in three ways:
 1) in the model spectrum calculated with full, unmodified opacities (black curve),
 2) in the Si curve of the Delta plot in Figure \ref{fig:0lin_diffeach} (dashed pink curve), and
 3) in a Delta plot created by removing the \SitLab\ feature only (solid pink curve).\\
 The solid pink curve is the most ideal measure of Si absorption velocity as it describes an isolated single feature.
 The dashed pink curve shows that the emission wing of the neighboring \Sit\ 5972\AA\ line (the strength of this line is correlated to the 6355\AA\ line) reduces the apparent absorption velocity, although the effect is small.
 After day 28pe, the \Sit\ trough disappears in the model spectra.
 The black curve does not follow the observed, gradually declining velocity evolutions because of the 5800\AA\ emission feature missing in the model.\\
 The shape of the Si line profile is independent of the presence or absence of that feature, so that the solid pink curve still shows a \emph{realistic run of the true} (as opposed to observed) \emph{Si absorption velocity}.
 } \label{fig:si2_v0}
\end{figure}
\begin{figure}
 \centerline{\includegraphics[width=.5\textwidth]{si2_ew.eps}}
 \caption{The evolution of the \emph{true} equivalent widths (EW) of three spectral features over time.
 Each of the three features has a positive and a negative curve corresponding to absorption and emission, respectively (note the flipped vertical axis).\\
 The EWs are computed using the Deltas from individual lines, i.e. the line profiles are evaluated versus the true run of the continuum flux level over each line profile.\\
 Both the \SitLab\ and the \Cat\ IRT line profiles are emission dominated after the first $\sim$10 days pe.
 The \Cat\ HK line profile is absorption dominated before day 44pe.
 } \label{fig:si2_ew}
\end{figure}

The effects of line blending on the absorption velocity of the \SitAbs\ trough are demonstrated in Figure \ref{fig:si2_v0}.
Figure \ref{fig:si2_ew} presents the \emph{true} equivalent width (EW) values for the isolated \SitLab, \Cat\ IRT, and \Cat\ HK features, calculated versus the true continuum.

\subsection{Absorption features}
\paragraph{\SitAbs\ absorption}
On day 10pe, the absorption trough around 6150\AA\ is dominated by \Sit\ (see also Figure \ref{fig:0lin_si2_all}).
On day 20pe, the \Sit\ feature has mostly changed to emission and is significantly supported by Co emission.
This trend continues on day 30pe, before Fe emission (at nearly the same wavelength) \emph{completely takes over} at day 40pe.                                                                              
\paragraph{\St\ ``W'' absorption}
On day 10pe, the absorption trough around 5700\AA\ is clearly dominated by \St.
On day 20pe, it is heavily blended with Co and Fe emission, and on day 30pe \St\ has vanished.

\paragraph{\Mgt\ absorption}
On day 10pe, the \Mgt\ absorption around 4300\AA\ is blended with Si, S, and Fe absorption.
On day 20pe, Fe absorption and Co emission dominate \Mgt, and on day 30pe \Mgt\ has vanished.

\paragraph{\Cat\ absorption}
The \Cat\ HK absorption around 3750\AA\ at day 10pe sees contributions from Si and S absorption and Ti emission (best seen in Figure \ref{fig:0lin_diffeach}).
On day 20pe and 30pe, the \Cat\ HK trough is significantly deepened by Si absorption, obviously affecting its pEW (note that this problem does not affect the \emph{true} EW values plotted in Figure \ref{fig:si2_ew}).
This also changes the absorption velocity.

The \Cat\ IRT feature appears mainly in emission.

\paragraph{\Fet\ 5000\AA\ blend}
The wide absorption trough around 5000\AA\ is commonly identified as the ``\Fet\ blend'' (e.g., \citep{Folatelli04, Hachinger06}).
Figure \ref{fig:0lin_all} and \ref{fig:0lin_diffeach} show, however, that the Fe is weaker than contributions from othere elements at day 10, 20, and 30pe.
Only as late as day 40pe, this feature is really dominated by Fe absorption.
%}}}

%-----------------------------------------------------------------------
\section{Discussion and Conclusions}
%{{{
\subsection{Absence of an important emission peak}
Generally, W7 provides a reasonably good fit to the Hsiao07 templates.
There is, however, one \emph{very serious mismatch}.
The model does not describe the 5800\AA\ emission peak observed on day 20, 30, and 40pe.
(Note that this finding is in agreement with results from W7 calculated with other radiation transport codes (e.g., \citealt{Kasen06, Kromer09}).)
This is a serious problem since this emission peak is responsible for the blue side of the \SitAbs\ trough and thus affects the \Sit\ pEW measurements.

What makes this problem even more serious is that other \snia\ explosion models do not describe this emission peak either (e.g., \citealt[Figure 2b]{Kasen09}; \citealt[Figure 3]{Roepke12}).
There are two possible reasons for this emission peak to be absent from the models: 1) the chemical element(s) responsible for this observed peak is under-represented in the explosion models, 2) the opacity data for the elements present in the models is missing one or more important transition lines\footnote{
In the scope of this work, we have verified the opacity data used by \phx\ against recent atomic data from R.L. Kurucz (http://kurucz.harvard.edu/).
}.
Note that NLTE effects do affect the shape of \snia\ spectra (\citealt{Baron96}; van Rossum, in preparation) but are unlikely to explain discrepancies as big as this missing emission peak.

Since the origin of this observed emission line is unknown and it does shape the blue side of the \SitAbs\ absorption trough, special care must be taken with the interpretation of the pEW of this famous trough.


\subsection{Blended features}
Line features are known to strongly blend in \snia\ spectra but still it is common to speak of \emph{identified} features, named after the predominant contributor.
Figure \ref{fig:0lin_all} and \ref{fig:0lin_diffeach} show the degree of blending in the model spectra, and the way it changes over time, for each feature in its own way.
Note that the detailed degree of blending depends on the atmospheric conditions, which vary between \snia\ objects.
The W7 results presented here provide a qualitative picture of the blending effects.%!
A \snia\ model atmosphere tuned to fit a certain observation would allow to quantify the degrees of blending using the Delta method presented in this paper.%!
This would be a very effective way to \emph{increase the informative value of observed \snia\ spectra}.%!


\subsection{The \SitAbs\ feature}
Since this is the most widely used feature in the characterization and classification of \snia\ spectra it deserves extra discussion.

The absorption velocity of the \SitLab\ line versus the 6150\AA\ trough is demonstrated in Figure \ref{fig:si2_v0}.
It compares the velocity of isolated Si with the value measured in the full spectrum.
At early times, the \SitAbs\ trough is filled from the blue side by emission from the next bluer \Sit\ line leading to lower velocities, although the effect is small (a few percent).

Figure \ref{fig:si2_ew} shows that from day 26pe on the \Sit\ emission gradually declines.
At the same time the peak is seamlessly taken over by Fe\,{\sc ii} emission at a similar but slightly higher wavelength.
This not only affects the pEW but also makes the \Sit\ velocity at late times appear lower (and its gradient higher) than it really is.

The run of the full spectrum curve in Figure \ref{fig:si2_ew} does not match the observed, gradually declining behavior (e.g., \citealt[Figure 15]{Blondin12}).
Also, the 5800\AA\ emission feature, that is missing in the model (see previous subsection), causes the \Sit\ absorption trough to disappear after day 28pe.
This missing feature apparently plays \emph{a key role} in run of the absorption velocity.

For these reasons, using the \SitAbs\ velocity at late times (e.g., \citealt{Howell06, Blondin12}) is potentially problematic.
The same problems might affect the velocity gradient definition in \cite{Benetti05}.
The ``improved'' velocity gradient definition of \cite{Blondin12} limits using \SitAbs\ velocities to day 10 post maximum brightness (approx. day 28pe for W7) and thus partly avoids these problems.



\subsection{The nature of spectral features in \snia\ spectra}
In this subsection we explain how the concepts of ``absorption troughs'' and ``continuum'' can be misleading and should be used with care.%!

\paragraph{Misinterpretation of ``absorption troughs''}
The \emph{identified} features commonly used for interpretation and classification of \snia\ spectra describe the apparent ``absorption troughs'', with apparent ``continuum'' in between of those troughs.
However, many of the features in \snia\ spectra are predominantly in emission.
Figure \ref{fig:0lin_diffeach} shows that for wavelengths larger than 5400\AA\ the spectral features generally are emission dominated (except possibly \Sit\ at very early times), and on day 20pe (close to peak luminosity, day 18pe) emission contributes significantly to the flux level down to wavelengths as low as 3800\AA.
The widths and depths of these ``absorption troughs'' largely depend on the strengths and separation of the enclosing emission features.

\paragraph{Misinterpretation of ``continuum flux level''}
Estimating a realistic pseudo continuum from a \snia\ spectrum is not trivial.
But since line emission is strong, a ``maximum'' continuum level that connects observed flux peaks is definitely unrealistic for many features at most evolution times.

\paragraph{Redefinition of \snia\ feature strength}
A \snia\ model atmosphere tuned to fit a certain observation would in principle allow to determine the pseudo continuum for any spectral feature using the method presented in this paper.%!
Yet such atmospheres are not generally available.%!

Therefore, instead of the common pEW method, we suggest to measure the strength of spectral features using the height difference of P-Cyg profiles (at least in the region where emission is strong in \snia\ spectra, see above).
The stenght is defined as difference in flux levels at the trough minimum and the peak maximum on the red side of the trough normalized to the mean of these levels.
The emission peak on the blue side of the trough, the strength of which is unrelated to the strength of the feature of interest, can thus be excluded, potentially giving a more accurate strength measure.
Note that this new definition is a dimensionless quantity, not taking into account the width of the feature, which is loosely related to the absorption velocity.
Thus the strength and the velocity measures become potentially more independent.
Application of this redefined strength measure to large \snia\ spectral datasets will show whether this measure improves correlations to other spectroscopic or photometric properties, and how it will affect classification schemes based on line strengths.

In future work we will take the method presented in this paper to a higher level by using model atmospheres that are tuned to match observed \snia\ spectra.
This will be parametrized atmospheres, based on W7 and more modern explosion models.
This way, the details of the atmospheric structure can be directly inferred from observed \snia\ spectra in a physically self-consistent way.
%}}}

\hspace{.5cm}
\acknowledgements
We thank Sean Couch for his helpful comments on the manuscript of this paper.
This research was supported in part by the NSF under grant AST-0909132.

\bibliography{Astro,SNexp,SNobs,RT}

\end{document}

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