\section{Introduction}
The last years have seen a new generation of stellar evolution models
emerging, which try to incorporate the influence of rotation on
internal structure and evolution \citep[see e.g.][ and later papers of
these groups]{Meynet.Maeder97, Langer.Heger.ea98, Langer98,
Heger.Langer.ea00}. These studies have been motivated by the
realization that stellar surface abundances in many areas of the
Hertzsprung-Russell diagram require mixing processes in addition to
convection. The effects of rotation in massive stars
are described in detail in \cite{Maeder.Meynet00}, which gives also
many relevant references. In the calculation of rotating stellar
models inevitably the question of angular momentum transport arises,
which cannot yet be described self-consistently from first
principles. Ordinary molecular viscosity is much too small, and other
mechanisms have to be invoked, e.g. turbulence created through shear
instability \citep{Spiegel.Zahn70, Zahn74, Zahn75, Zahn92}, generation
of internal gravity waves \citep{Talon.Kumar.ea02, Talon.Charbonnel03,
Talon.Charbonnel04}, or magnetic torques \citep{Spruit98}.

In this situation the observation of white dwarf rotation can provide
some constraints since it tells us the remaining angular momentum of
the typical 0.6~\hbox{M$\sb{\odot}$}\ interior core of the progenitor at the end of
stellar evolution. If the total angular momentum of the core is
preserved one would expect equatorial velocities in the range of
15~km/s for solar type slow-rotating progenitors to 2000~km/s for a
fast-rotating A0 star, corresponding to rotation periods from 1~h down
to a few minutes (a number useful to remember is that for a 0.6~\hbox{M$\sb{\odot}$}\
white dwarf an equatorial velocity of 15~km/s corresponds to a 1~h
rotation period).

The only detailed calculation with a sophisticated treatment of
angular momentum transfer, which makes a specific prediction for the
final white dwarf rotational velocity is the work of
\citet{Langer.Heger.ea99}. For the 0.8~\hbox{M$\sb{\odot}$}\ white dwarf resulting
from the evolution of a rotating 3~\hbox{M$\sb{\odot}$}\ model they expect an
equatorial rotation of 28\,km/s.

On the other hand -- assuming the core to remain
strongly coupled to the outer expanding envelope in the giant stage --
the rotation of the final white dwarf should be extremely slow, with
periods in the range 30 to 1000 years \citep{Spruit98}. Rotation
periods of hours to days have in a few cases been determined from the
rotational splitting of pulsation frequencies; a list is given in
\cite{Spruit98}, to which could be added a period of 13~hrs for L19-2
and 58~hrs for GD165 \citep{Bradley01}. Periods around one day have
also been measured from variable polarization
\citep{Schmidt.Norsworthy91, Schmidt.Smith95}, but for some of these
magnetic white dwarfs no variation has been observed, implying
rotation periods of at least a century.

The great majority of rotation data for white dwarfs have been
determined spectroscopically, exclusively using the sharp NLTE core of
the \hbox{H$\alpha$}\ line in DA white dwarfs \citep{Pilachowski.Milkey84,
Pilachowski.Milkey87, Milkey.Pilachowski85, Koester.Herrero88,
Heber.Napiwotzki.ea97, Koester.Dreizler.ea98,
Karl.Napiwotzki.ea05}. The sample now contains about 50 stars with
determined $v \sin i$. A few of these objects show a line broadening
suggesting velocities of a few dozen km/s, but in many cases the
unsatisfactory line fits indicate that the reason is not rotation but
magnetic splitting or unknown, as in the case of the ZZ Ceti stars
\citep{Koester.Dreizler.ea98}. In the majority of objects only upper
limits can be determined which typically range from 10 to 30~km/s and
are thus marginally compatible with the result of
\citet{Langer.Heger.ea99}. While this does not rule out that most
white dwarfs could be extremely slow rotators, \cite{Spruit98}, taking
all available empirical evidence and following his idea of strong
core-envelope coupling felt a need for mechanisms, which would {\em
accelerate} the white dwarfs again to the observed values after the
efficient loss of angular momentum in the giant phase.

There is thus clearly a strong motivation to determine more stringent
upper limits using the spectroscopic method. This has become possible
after the detection of a large number of DAZ objects found in the
studies of \cite{Zuckerman.Reid98, Zuckerman.Koester.ea03} and as a
byproduct of the search for Supernova~Ia progenitors
\citep{Napiwotzki.Christlieb.ea03, Napiwotzki.Christlieb.ea01,
Koester.Rollenhagen.ea05}. The Ca\,II~K line in these hydrogen-rich
white dwarfs is deep and quite narrow and should be useful for
measuring much lower rotation velocities than is possible with
\hbox{H$\alpha$}. In this paper we report the results of such a study.

\begin{figure}
\includegraphics[width=6.5cm,angle=90]{3340fig1.ps}
\caption{Ca\,II~K line profile in WD\,1826-045. Dotted: observed
  profile; continuous: non-rotating model after convolution with
  instrumental profile (falls on top of the observation in the line
  center!); dashed and dot dashed with decreasing line depths:
  rotating models with \hbox{$v\,\sin\,i$}\ = 5, 10, 20 km/s
\label{wd1826}}
\end{figure}

\begin{table}[htb!]
\caption{Atmospheric parameters \Teff\ in K, surface gravity \hbox{$\log g$},
  and Ca abundance for the sample. [Ca/H] is the logarithm of the Ca/H
  abundance ratio by numbers. Column Ref gives the source of the
  parameters with Z = \cite{Zuckerman.Koester.ea03}, R =
  \cite{Rollenhagen04} and \cite {Koester.Rollenhagen.ea05}. Columns K
  and S give the number of available spectra from the Keck or the SPY
  samples. {\em Please note that the Ca abundance for HS\,0047+1903 in
\cite {Koester.Rollenhagen.ea05} was based on an erroneous equivalent
  width. The correct abundance is given in Table~\ref{tabres}}
\label{para}}
\centering\tt
\begin{tabular}{lrrrrrr}
\hline
Object & $T_{eff}$ & $\log\,g$ & [Ca/H]& Ref & K & S \\
\hline
WD\,0032-175   &  9235 & 8.0 & -10.2 & Z & 1 & \\
HS\,0047+1903  & 16600 & 7.8 &       & R &   & 1 \\
HE\,0106-3253  & 15700 & 8.0 &  -6.4 & R &   & 2 \\
WD\,0208+296   &  7201 & 7.9 &  -8.8 & Z & 2 & \\
WD\,0235+064   & 11420 & 7.9 &  -9.0 & Z & 1 & \\
WD\,0243-026   &  6798 & 8.2 &  -9.9 & Z & 1 & 2\\
WD\,0245+541   &  5190 & 8.2 & -12.7 & Z & 1 &\\
HS\,0307+0746  & 10200 & 8.1 &  -7.6 & R &   & 1 \\
WD\,0408-041   & 14400 & 7.8 &  -7.1 & R &   & 2\\
WD\,0543+579   &  8142 & 8.0 & -10.3 & Z & 3 & \\
WD\,0846+346   &  7373 & 8.0 &  -9.4 & Z & 2 & \\
WD\,1015+161   & 19300 & 7.9 &  -6.3 & R &   & 2 \\
WD\,1102-183   &  8026 & 8.0 & -10.4 & Z & 1 & \\
WD\,1116+026   & 12200 & 7.9 &  -7.3 & R &   & 2 \\
WD\,1124-293   &  9680 & 8.0 &  -8.5 & Z & 2 & 2 \\
WD\,1150-153   & 12800 & 7.8 &  -6.7 & R &   & 2 \\
WD\,1202-232   &  8619 & 8.0 &  -9.8 & Z & 2 & 2 \\
WD\,1204-136   & 11468 & 8.0 &  -7.7 & Z & 1 & 2 \\
WD\,1208+576   &  5830 & 7.9 & -11.0 & Z & 1 & \\
HE\,1225+0038  &  9400 & 8.1 &  -9.7 & R &   & 3 \\
WD\,1257+278   &  8481 & 7.9 &  -8.1 & Z & 2 & \\
HE\,1315-1105  &  9400 & 8.4 &  -9.2 & R &   & 2 \\
WD\,1337+705   & 20435 & 7.9 &  -6.7 & Z & 3 & \\
WD\,1344+106   &  6945 & 8.0 & -11.1 & Z & 2 & \\
WD\,1407+425   &  9856 & 8.0 &  -9.9 & Z & 1 & \\
WD\,1455+298   &  7366 & 7.6 &  -9.3 & Z & 2 & \\
WD\,1457-086   & 20400 & 8.0 &  -6.3 & R &   & 2 \\
WD\,1614+160   & 17400 & 7.8 &  -7.2 & R &   & 2 \\
WD\,1633+433   &  6569 & 8.1 &  -8.6 & Z & 2 & \\
WD\,1821-131   &  7029 & 8.4 & -10.7 & Z & 1 & \\
WD\,1826-045   &  9480 & 7.9 &  -8.8 & Z & 1 & 1 \\
WD\,1858+393   &  9470 & 8.0 &  -7.8 & Z & 2 & \\
WD\,2115-560   &  9700 & 8.1 &  -7.6 & R &   & 1 \\
HS\,2132+0941  & 13200 & 7.7 &  -7.7 & R &   & 2 \\
WD\,2149+021   & 17300 & 7.9 &  -7.6 & R &   & 2 \\
HE\,2221-1630  & 10100 & 8.2 &  -7.6 & R &   & 2 \\
HS\,2229+2335  & 18600 & 7.9 &  -6.3 & R &   & 2 \\
WD\,2326+049   & 11600 & 8.1 &  -6.9 & Z & 5 & 2 \\
\hline
\end{tabular}
\end{table}

\section{Observations} \label{observations}
The sample consisted of spectra for 38 single cool DA white dwarfs
coming from two different sources. The first is the SPY survey (ESO
Supernova Ia Progenitor Survey) for double-degenerates.  In the search
for variable radial velocities a large number of high-resolution
spectra are obtained with the UVES spectrograph at the ESO VLT Paranal
Observatory, which are useful also for many other
projects. \cite{Rollenhagen04} and \cite{Koester.Rollenhagen.ea05} used this
sample to identify hydrogen-rich white dwarfs with Ca\,II lines and
those objects form the first part of our present sample (excluding
those where the lines are interstellar). Details of the observations
and the reduction procedures are described in
\cite{Napiwotzki.Christlieb.ea03} and \cite{Koester.Napiwotzki.ea01};
we therefore repeat here only the facts relevant for this project. The
observations used a wide slit of 2\farcs 1, because the SPY was
conducted as a filler project for bad weather conditions. This is
important since the nominal resolution with this slit is 18500
corresponding to 0.21~\AA\ at the Ca\,II~K line, however, the actual
resolution can be higher if the seeing disk is smaller than the
slit. The actual seeing during the SPY observations typically varied
between 1\farcs0 to 1\farcs5.

The second source of sample objects are Keck observations of DAs in a
search for DAZs with metals. The observations are described in
\cite{Zuckerman.Reid98} and \cite{Zuckerman.Koester.ea03}. Here the
nominal resolution with a slit width of 1\farcs 15 is 34000,
corresponding to 0.12~\AA\ at Ca\,II~K, but also in this case the actual
resolution might be slightly higher at times of excellent seeing.

For most objects more than one spectrum is available, for some even
spectra from both surveys.  Atmospheric parameters \Teff, \hbox{$\log g$}, and
Ca abundances for the sample are given in
\cite{Koester.Rollenhagen.ea05} and \cite{Zuckerman.Koester.ea03}
(which also show some representative spectra) and are collected in
Table~\ref{para}.  For our analysis the spectra were shifted slightly
to bring the K line in agreement with the laboratory wavelength and
then normalized to a continuum of 1.0 with a straight line fitted to
the continuum approximately 5~\AA\ on both sides of the line center.

\section{Determination of rotation velocities}
Rotation velocities were determined using theoretical model
atmospheres with spectral lines broadened by a rotational broadening
function \citep[see e.g.][]{Unsold68}. Our model grid is calculated
assuming LTE (local thermodynamic equilibrium) with methods and input
physics described in \cite{Finley.Koester.ea97},
\cite{Koester.Wolff00}, and \cite{Homeier.Koester.ea98}. The Ca\,II
lines have the advantage compared to the Balmer lines -- the only
lines used in the past for spectroscopic determinations in white
dwarfs -- that the central Doppler core from thermal broadening is
much narrower because of the higher atomic weight.

On the other hand, the Balmer lines originate from the dominant
element in DAs and therefore only two parameters, \Teff\ and \hbox{$\log g$},
are needed to describe the fitting model. In the case of the K line an
additional parameter is the Ca abundance.  The results given in
\cite{Zuckerman.Koester.ea03} and \cite{Koester.Rollenhagen.ea05} have
been determined using interpolations in pre-calculated tables of
equivalent widths. In the calculations of the theoretical equivalent
widths a fixed interval was assumed, and no attempt was made to match
this exactly in the observed spectra; moreover, the spacing of the
grid of abundances was rather coarse. It is therefore not surprising
that calculated spectra for the ``exact'' atmospheric parameters using
these Ca abundances did not always match the spectra perfectly. We
have then changed the Ca abundance (typically less than a factor of
two) until the EW of the observation agreed exactly with that of the
theoretical model. This is obviously a prerequisite for the
determination of rotational broadening of the profile.

The standard rotational broadening profile uses two free parameters,
the limb darkening coefficient $\beta$ and the projected equatorial
rotation velocity \hbox{$v\,\sin\,i$}. Here $i$ is the inclination of the rotation
axis against the line-of-sight, ranging from 0 to 90 degrees (note
that only this combination can be determined with the spectroscopic
method). Our theoretical models show that $\beta$ varies from
$\approx 0.2$ at the center of the K line to $\approx 1.2$ in the far
wing. Since the parameter has almost no influence on the result we
have continued to use $\beta = 0.15$ which is used by all authors for
\hbox{H$\alpha$}. The best fit value for \hbox{$v\,\sin\,i$}\ and the errors were then
determined with a $\chi^2$ fitting routine very similar to that used
in \cite{Koester.Dreizler.ea98}.  Figure~\ref{wd1826} shows the
observed line in WD\,1826-045 as a typical example together with
theoretical models with different rotational velocities applied. The
figure demonstrates that rotational velocities of a few km/s can
easily be measured from the Ca line.

\begin{table}
\caption{Rotational velocities. When several spectra of the same
  object are of comparable quality, the results do not differ
  significantly and we therefore give only the result from the best
  spectrum. The only exception is WD$2326+049$, a known ZZ Ceti
  variable. [Ca/H] is the Ca
  abundance as determined from fitting the line profile, which is
  sometimes slightly different from that in the original source (see
  Table~\ref{para}; the corrected EW for HS\,0047+1903 used for the
  abundance in this table is 77 m\AA). $v_2 = v\,\sin\,i$ is the rotation
  velocity determined using either the nominal resolution of the
  spectrographs (34000 for Keck, 18500 for SPY), or a slightly higher
  resolution if the line depth cannot be reproduced at nominal
  resolution. In the latter case the values for the resolution are
  given in column Res. Errors are $1\,\sigma$
  errors. $v_1 = (v\,\sin\,i)_{max}$ is an upper limit using the optimum
  resolution as explained in the text
\label{tabres}}
\centering \tt
\begin{tabular}{lrrrr}
\hline
Object &  [Ca/H] & $v_1$ & $v_2$ & Res \\
\hline
WD\,0032-175 & -10.5 & 3.8 & 0.0 $\pm$ 2.8 & 46300 \\
HS\,0047+1903& -6.1  & 11.6& 9.9 $\pm$ 6.1 &       \\
HE\,0106-3253& -5.8 & 6.1 & 0.0 $\pm$ 4.9 & 28100 \\
WD\,0235+064 & -9.2 & 5.1 & 0.0 $\pm$ 4.0 & 39300 \\
WD\,0243-026 & -9.8 & 4.7 & 2.5 $\pm$ 4.2 & 32800 \\
WD\,0245+541 & -11.7 & 0.0 & 0.0 $\pm$ 6.6 & \\
HS\,0307+0746& -7.1 & 9.1 & 5.4 $\pm$ 5.5 & 24600 \\
WD\,0408-041 & -6.6 & 7.1 & 0.0 $\pm$ 5.1 & 24600 \\
WD\,0543+579 & -10.5 & 5.0 & 3.4 $\pm$ 1.7 & 46300 \\
WD\,1015+161 & -5.9 & 10.1 & 2.4 $\pm$ 4.7 & 20700 \\
WD\,1102-183 & -10.3 & 8.1 & 5.9 $\pm$ 1.9 & \\
WD\,1116+026 & -6.5  & 11.4& 3.6 $\pm$ 3.8 & \\
WD\,1124-293 & -8.2 & 5.8 & 0.0 $\pm$ 2.2 & \\
WD\,1150-153 & -6.0 & 14.3 & 9.5 $\pm$ 5.4 & \\
WD\,1202-232 & -9.8 & 5.8 & 0.0 $\pm$ 0.9 & \\
WD\,1204-136 & -7.2 & 11.1 & 9.7 $\pm$ 0.8 & \\
WD\,1208+576 &-10.8 &  8.8 & 6.5 $\pm$ 1.9 & \\
HE\,1225+0038& -9.7 & 9.4 & 0.0 $\pm$ 7.0 & 21300\\
HE\,1315-1105& -9.4 & 6.7 & 1.5 $\pm$ 4.8 & 27100 \\
WD\,1337+705 & -6.7 & 4.4 & 0.0 $\pm$ 1.2 & 41400 \\
WD\,1344+106 & -11.3 & 7.2 & 4.0 $\pm$ 2.5 &  \\
WD\,1407+425 & -9.8 & 8.9 & 6.7 $\pm$ 1.5 & \\
WD\,1457-086 & -6.1 & 9.6 & 0.0 $\pm$ 8.5 & 21900 \\
WD\,1614+160 & -7.2 & 2.2 & 0.4 $\pm$ 4.9 & 35750 \\
WD\,1821-131 & -10.9 & 18.0 & 15.8 $\pm$ 4.6 & \\
WD\,1826-045 & -8.6 & 3.1 & 0.0 $\pm$ 1.3 & 49200 \\
HS\,2132+0941& -7.1 & 8.5 & 0.0 $\pm$ 8.3 & 21260 \\
WD\,2149+021 & -7.7 & 10.7& 0.0 $\pm$ 10.3& 20700 \\
HE\,2221-1630& -7.2 & 4.2 & 2.1 $\pm$ 4.0 & 34200 \\
HS\,2229+2335& -5.9 & 6.5 & 0.0 $\pm$ 5.2 & 27100 \\
WD\,2326+049 & -6.4 & 20.5 & 19.8 $\pm$ 0.5 & \\
             & -6.3 & 15.7 & 14.6 $\pm$ 0.5 & \\
             & -6.3 & 28.2 & 27.6 $\pm$ 0.4 & \\
             & -6.3 & 26.9 & 26.2 $\pm$ 0.3 & \\
             & -6.3 & 28.4 & 28.0 $\pm$ 0.4 & \\
             & -6.4 & 15.3 & 10.7 $\pm$ 2.1 & \\
             & -6.4 & 22.5 & 20.0 $\pm$ 0.9 & \\
\hline
\end{tabular}
\end{table}

\begin{figure}
\includegraphics[width=6.0cm,angle=90]{3340fig2.ps}
\caption{Typical examples for the determination of rotational
  velocities for the Ca\,II~K line, ranging from some of the best
  spectra to marginal. Dotted: observed
  profile; dashed: non-rotating model after convolution with
  instrumental profile; continuous: best fit with rotation.
  In the two top figures the best fit is zero rotation and the models
  fall on top of each other.
\label{goodrot}}
\end{figure}

\subsection{A remark on the standard spectroscopic method}
Assuming that the intensity of the radiation field at the surface of a
star depends only on wavelength $\lambda$ and the angle $\theta$
against the normal, the observational quantity for a non-rotating star
is the disk-averaged intensity
\[ \bar{I} = \frac{1}{\pi}\,\int \int I(\mu,\lambda)\,dx dy
\]
where $\mu = \cos \theta$, and $x,y$ are Cartesian coordinates on the
visible surface of the star in units of the stellar radius $R$.

In the case of a rotating star we have in addition a Doppler shift,
which depends on the position on the disk
\[ \bar{I} = \frac{1}{\pi}\,\int\limits_{-1}^{+1}
      \int\limits_{-\sqrt{1-x^2}}^{+\sqrt{1-x^2}}
      I(\mu,\lambda-\Delta\lambda(x,y))\,dx dy
\]
If the $y$-axis is chosen parallel to the rotation axis,
$\Delta\lambda$ will depend on $x$ only.  This integral can easily be
evaluated numerically if theoretical models are available for the
intensity as a function of $\mu$. It is common practice, however,
to proceed with an analytic evaluation by assuming a limb-darkening
law
\[  I(\mu) = I_c ((1-\epsilon) + \epsilon \mu)
\]
with central intensity $I_c$. This law can also be written relative to
the intensity at the limb $I_l$
\[ I(\mu) = I_l (1 + \beta \mu)
\]
with the limb-darkening coefficient $\beta = \epsilon/(1-\epsilon)$
used above.
Introducing this into the integration over the disk, with $\mu$ a
function of position $x, y$, the integration can be transformed into
a convolution of the disk-averaged intensity for a non-rotating star
\[ \bar{I}_{rot} =
       \int\limits_{-1}^{+1} G(x) \bar{I}(\lambda - x \Delta\lambda_l)
       \,dx
\]
with the maximum shift at the limb $\Delta\lambda_l$, and
\[ G(x) = \frac{4 (1-\epsilon) + \epsilon \sqrt{1-x^2} + \pi \epsilon
  (1-x^2)}{2 \pi (1 - \epsilon/3)}
\]
\cite[see e.g.][]{Unsold68}. This is a large simplification, since
only the theoretical disk-averaged intensity (also called the
astrophysical flux) needs to be calculated, which is always needed for
the analysis of spectra, rotating or not. However, the derivation
makes the assumption that the limb darkening law is independent of
wavelength, in particular that the coefficients $\epsilon$ or $\beta$
are constant over the line profile. This is clearly not the case for
the Ca\,II~K line as mentioned above and it is also not true for
\hbox{H$\alpha$}. We have therefore studied the effect of this assumption by
comparing the exact direct integration with angle-dependent intensities with
the  simplified method using the convolution technique. While this has
been studied before in the literature for other types of stars it has
to our knowledge never been tested for white dwarf spectra.

The result of this test is \citep{Pruin05} that small differences
between the two methods exist in the line cores, but that the error
made when using the simple convolution method is much smaller than
1~km/s, and thus negligible in our study.

\begin{figure}[ht]
\includegraphics[width=21.3cm,angle=90]{3340fig3.ps}
\caption{Seven objects fitted with Lorentz profiles. See text
\label{deep}}
\end{figure}

\begin{table}
\caption{Upper limit to the rotational velocities for the objects with
  deeper observed line profiles than is possible with a non-rotating
  model.  The limits were determined using a Lorentz line profile with
  central intensity zero and equivalent width (EW, in m\AA) equal to
  the observed spectrum
\label{ressix}}
\centering\tt
\begin{tabular}{lcrr}
\hline
Object & $S/N$ & EW & $(v\,\sin\,i)_{max}$ \\
\hline
WD\,0208+296 & 37 & 534  & 8\\
WD\,0846+346 & 24 & 245  & 6\\
WD\,1257+278 & 15 & 375  & 7\\
WD\,1455+298 & 21 & 282  & 8\\
WD\,1633+433 & 19 & 1200 & 13\\
WD\,1858+393 & 27 & 285  & 9\\
WD\,2115-560 & 24 & 292  & 11 \\
\hline
\end{tabular}
\end{table}

\subsection{Observed and theoretical line profiles}
When comparing the observed and theoretical profiles after taking into
account instrumental broadening with the nominal resolution in the
latter, we obtained nearly perfect fits for about half of the
objects {\em without rotational broadening}, implying zero or at least
extremely low rotation. In the other half of objects the observed line
profile had a deeper core than our ``unrotated'' models. Since
rotational broadening can only decrease the central depths, we could
thus not obtain a fit.

What could be the reason for this discrepancy? The line profiles are
of course influenced by the reduction process, e.g. the subtraction of
sky background, to mention just one step which could affect especially
the line cores. Indeed, when we have several spectra of the same
object, there are sometimes differences in equivalent width and also
line depth. Another possibility is on the modeling side: the largest
discrepancies (in seven objects) occur for very deep lines, where the
core is saturated. In these cases the line core originates in the
outermost layers of the atmosphere and it is quite possible that NLTE
effects (not included in our models) will make the model lines deeper,
as is well known for \hbox{H$\alpha$}.

We believe, however, that in most objects the reason is that the
actual resolution of the observation was better than nominal. This is
supported by the finding that in several spectra of the same stars the
equivalent width is constant but the line profile changes, with the
line being deeper and narrower in some of the spectra. This strongly
indicates a changing actual resolution for periods with better
seeing.

Unfortunately we do not know the real resolution for the
observation. In order to be able to nevertheless extract some
information on the rotation velocities we have therefore increased the
resolution (narrowed the instrumental profile) until our models
achieved a good fit without rotation. We have then used our $\chi^2$
routine to determine \hbox{$v\,\sin\,i$}. Obviously the result must be compatible
with zero, but we nevertheless obtain a reasonable upper limit from
the errors provided by the fit. The results obtained for 31 objects
for which this procedure led to a good fit with the nominal or within
a plausible range of resolutions (see below) are shown in
Table~\ref{tabres}. A number in the resolution column indicates the
higher than nominal resolution used in the fit.

There is of course some arbitrariness involved with this method, since
the resolution could actually be even higher, and could also have been
higher in the cases where our zero-rotation models had lines deep
enough. In order to analyze the uncertainty we have assumed a maximum
plausible resolution of 39500 and 65500, equivalent to seeing of
$1\farcs 0$ and $0\farcs 6$, respectively, for SPY and Keck
spectra. The column \hbox{$v\,\sin\,i$} (max) gives the results of the fitting
procedure, which should be considered as very firm upper limits.

Fig.\ref{goodrot} presents four typical examples from spectra with
high to rather low signal-to-noise ratios.

\subsection{Error estimates}
We estimated the errors caused by the uncertainties of $T_{eff}$,
$log\,g$, [Ca/H], $\beta$, and the resolution. The summed
errors from the first four quantities were of the same order of
magnitude as the formal $1\sigma$-error of the $\chi^{2}$-fitting
given in Table~\ref{tabres}. The greatest error was caused by
the uncertainty of the resolution and can be estimated by comparing
the values in the column \hbox{$v\,\sin\,i$}\ and $(\hbox{$v\,\sin\,i$})_{max}$.

\subsection{Objects with very deep line cores}
Even at the assumed maximum increase of the nominal resolution there
remain seven objects which cannot be fitted by unrotated profiles. All
of these objects have strong Ca lines, which are saturated in the
core. In these cases we have used a Lorentz profile with zero central
intensity and a width calculated to reproduce the observed EW. These
profiles were then used to fit the observed profiles, resulting in
firm upper limits to the rotational velocities. These results are
given in Table~\ref{ressix}. The spectra and fits are shown in
Fig.~\ref{deep}, where the dotted lines are the observations, the
dashed lines Lorentz profiles with zero central intensity as an
extreme simulation of the line profile, and the continuous lines are
the ``rotated'' profiles.

\section{Discussion}
21 objects in Table~\ref{tabres} and all 7 in Table~\ref{ressix} are
compatible with zero rotation velocities (within $1\sigma$
errors). Even allowing for a higher than nominal resolution, within
the plausible range, gives upper limits typically smaller than
10~km/s. This is a clear improvement compared to the spectroscopic
determination from the core of \hbox{H$\alpha$}, with many upper limits in the
range 10 - 30~km/s \citep{Koester.Dreizler.ea98,
Karl.Napiwotzki.ea05}. Unfortunately we are aware of only two \hbox{H$\alpha$}\
determinations for objects of our sample, both in
\cite{Koester.Dreizler.ea98}. For WD\,2115$-$560 the upper limit from
\hbox{H$\alpha$}\ is 35~km/s, while we obtain 11~km/s from the Ca line.
The other common object is the ZZ Ceti star WD\,2326$+$049
(G29-38). From \hbox{H$\alpha$}\ a velocity of $45\pm 5$ km/s is obtained, but
the authors note that the fit is not good and the broadening very
likely not due to rotation. From our Ca determination we find a range
of 14-28~km/s from different spectra.

G29-38 has been very extensively studied as a pulsator, both with
photometry and spectroscopy. \cite{van-Kerkwijk.Clemens.ea00} detected
periodic Doppler shifts in the line profiles with amplitudes up to 5
km/s, which they tentatively associated with the horizontal motions of
g-mode oscillations. \cite{Thompson.Clemens.ea03} recently found
Doppler shifts in G29-38 corresponding to radial velocity changes up
to 16.5~km/s. By combining a large number of spectra with close to
zero Doppler shift and comparing with an average spectrum over several
periods they found no significant difference in the line
profiles. They concluded that pulsation cannot be the reason for the
flat-bottom line profiles. This conclusion may be somewhat premature,
however, since the study only demonstrates that even the profile in
one particular pulsation phase does not fit a rotationally broadened
model spectrum. It is plausible that even with an average shift of
zero velocity in that phase different parts of the stellar surface may
show positive as well as negative velocity along the line-of-sight,
leading to zero shift but a broadening of the profile. This can only
be decided with numerical simulations of the pulsating surface
including all shifts and realistic center-to-limb variations.  In any
case the difference between the \hbox{H$\alpha$}\ and Ca results and the
significant variation of the Ca ``rotation velocities'' confirm
convincingly that rotation is not the origin of the broadening.

WD\,2326$+$049 is one of the 10 objects for which we determine a
non-zero rotation velocity for the best fitting model. Two others are
WD\,1150-153 and WD\,1204-136, which according to their atmospheric
parameters fall within or near the ZZ Ceti instability strip.
WD\,1204-136 was found to be non-variable (upper limit 0.3\%) by
\citet{Bergeron.Fontaine.ea04}, while WD\,1150-153 to our knowledge
has not been observed for variability. The latter is on our candidate
list for upcoming time-resolved photometry, but in any case it seems
very likely that the same mechanism -- as yet unidentified -- is
responsible for the line shapes as in the known ZZ Ceti stars
\citep{Koester.Dreizler.ea98}. Four other white dwarfs with non-zero
rotation (WD\,0543$+$579, WD\,1407$+$425, WD\,1344$+$106,
WD\,1821$-$131) are cool and the parameters are based on photometry
only. In the first two cases the parallax is not known and thus the
surface gravity cannot be determined and was fixed at 8.0
\citep{Zuckerman.Koester.ea03}.  However, even in these cases the
rotation velocities are definitely very small, $< 18$ km/s at most.

The results of this paper confirm the findings of all previous
studies, that white dwarfs are slow rotators, and set even more
stringent limits for those objects with Ca lines. Typical rotation
velocities are below 10~km/s and may even be close to zero velocity
for many objects. While this could be just marginally consistent with
conservative evolution for solar-type slow-rotating progenitors, the
result would definitely exclude conservation of angular momentum
for more massive progenitors. How many of such white dwarfs could we
expect in our sample?

Assuming for simplicity a unique initial-final mass relation between
remnant and progenitor, we find from \cite{Weidemann00} that a
0.6~\hbox{M$\sb{\odot}$}\ white dwarf originates from a 2~\hbox{M$\sb{\odot}$}\ main sequence
progenitor. 0.6~\hbox{M$\sb{\odot}$}\ (at \Teff\ = 10000~K) correspond to \hbox{$\log g$}\ =
8.0. Taking the numbers for the atmospheric parameters in
Table~\ref{para} at face value, we have 27 objects with a \hbox{$\log g$}\
determination (versus {\em assuming} 8.00), of which 11 have \hbox{$\log g$}\
$\ge 8.0$ and should have originated from fast-rotating main sequence
stars more massive than 2.0~\hbox{M$\sb{\odot}$}. While the exact numbers may be
uncertain, there is no doubt that these results mean that angular
momentum in the core cannot be preserved. These results also clearly rule out
the prediction by \cite{Langer.Heger.ea99}, which were based on
stellar model calculations without magnetic fields.

While the original study of \cite{Spruit98} predicted very efficient
angular momentum transport through the magnetic field coupling of core
and outer envelope, the new quantitative formulation of magnetic field
generation and magnetic field induced angular momentum transport
developed in \citet{Spruit02} leads to a more limited angular momentum
transport efficiency. First calculations for massive stars
\citep{Heger.Woosley.ea05} including the \citet{Spruit02} magnetic
field physics show these stars evolving far from rigid
rotation. Similar results are expected for low mass stars (Langer
2005, priv. comm.), which would likely bring the predictions
for the white dwarf rotation velocities in agreement with the new
observations.

Our method could in principle be improved further,
if that is necessary to distinguish between theoretical assumptions.
The main reason for the current uncertainty is the unknown
effective resolution due to rather large slits used in the
spectroscopy. With a narrower slit and well controlled resolution an
accuracy of 1-2~km/s could be reached, but the observations would need
a significant amount of observing time on a very large telescope.


\section*{\bf FIGURE CAPTIONS}

{\bf Figure 1}
The distribution of unbinding times $P(\tau,f_{ext})$ for WLC (solid) and Rouse model
(dash-dotted lines) of protein and Poisson approximation $P_u(\tau,f_{ext})$ (dashed lines)
for $f_{ext}=1pN$, $3pN$ and $10pN$ computed for $k_1$$\sim$$z_1/\gamma$ (left) and
$k_1$$\ll$$z_1/\gamma$ (right panels).

\bigskip

{\bf Figure 2}
The joint distribution $P(\tau_1,\tau_2; \Delta t,f_{ext})$ of lifetimes $\tau_1$ and
$\tau_2$ separated by $\Delta t$$=$$1\mu s$ (top), $10\mu s$ (middle) and $500 \mu s$
(bottom panels) for $f_{ext}$$=$$3pN$. The contour plots of $P(\tau_1,\tau_2;\Delta t,f_{ext})$
are shown on the right.

\bigskip

{\bf Figure 3}
Normalized correlation amplitude $D(\tau,\Delta t)/D(\tau,0)$ of equal lifetimes
$\tau$$=$$\tau_1$$=$$\tau_2$ separated by $\Delta t$$=$$1\mu s$ (solid), $10\mu s$ (dash-dotted)
and $500 \mu s$ (dashed lines) for $f_{ext}$$=$$3pN$.

\newpage

\begin{figure}
\includegraphics[width=7.00in]{PRLred1.eps}
\end{figure}

\newpage

\begin{figure}
\includegraphics[width=7.00in]{PRLred2.eps}
\end{figure}

\newpage

\begin{figure}
\includegraphics[width=7.00in]{PRLred3.eps}
\end{figure}

\end{document}


\section{Introduction}

The peak magnitudes of Type Ia supernovae (SNe~Ia) are one of the best
distance indicators at high redshifts, where few reliable distance
indicators are available to study the cosmological parameters.
Beginning with the discovery of SN 1992bi \citep{sn92bi}, the
Supernova Cosmology Project (SCP) has developed search techniques and
rapid analysis methods that allow systematic discovery and follow up
of ``batches'' of high-redshift supernovae. These searches and those
of the High-z SN team have resulted in $\sim100$ published SNe ($0.15
< z < 1.2$), which have been used for measurements of the cosmological
parameters $\rm \Omega_M$ and $\rm \Omega_{\Lambda}$ and to provide
initial constraints on the equation of state of the Universe, $w$
\citep{nature98,schmidt_98,42SNe_98,garn_w_98,riess_acc_98,knop03,barris04,
tonry03,reiss04}.

Spectra have been obtained for as many of the SCP supernovae as
possible, and as close as possible to maximum light. The spectra are
used to determine the redshift of the event and to confirm its
spectral sub-type, both crucial if the supernova is to be used in a
determination of cosmological parameters. In addition the spectra
provide a check that SNe~Ia at high redshift are similar to those at
low redshift, an assumption central to the use of SNe~Ia as
`standardized candles'.  This paper represents a significant
contribution to the amount of published high-redshift SN Ia
spectroscopy \citep{lidman05,barris04,tonry03,coil00,Matheson}.

In this paper we present spectra for a subset of our distant
supernovae, discovered during search campaigns in January and March
1997. Since the goal of this paper is a first quantitative comparison
of high- and low-redshift SN spectra, spectra were chosen where the SN
features were relatively clear. Spectra of 33 objects were taken
during the two campaigns, of which four were found to be clear
broad-lined QSOs and two were ``featureless'' blue objects (possibly BL
Lacs) for which a redshift could not be measured. The other objects
all have measured redshifts and of these about half (13) were rejected
for the purposes of this paper based on one or both of the following
criteria: low signal-to-noise ratio (i.e. less than about 10 per
20\AA\ bin over the observed wavelength range 6000-8000\AA) or large
contamination by the host galaxy, roughly corresponding to a
percentage increase of less than 50\% in $R$-band photometry between
reference and ``new'' images (although the latter is only a rough guide
since the phase of the SN, its location relative to the core of the
galaxy and the seeing all strongly affect how clearly the SN appears
in the spectrum).

The spectra presented here provide the basis for the identification of
these objects as Type Ia, as quoted in \cite{42SNe_98} and
\cite{knop03}.  Future papers by G. Garavini et al. and E. Smith et
al. will present spectroscopic results and analysis on other SCP
datasets.

For each object in our sample we give the redshift and present the
results of matching the spectrum to various supernova templates
(including SNe of Types other than Ia), and hence give spectroscopic
evidence that these distant supernovae are of Type Ia.  The spectral
dates derived from this matching process are compared with the dates
derived from the light curves.  Finally we present the set of
high-redshift SN spectra as a time sequence showing that the spectrum
changes with light curve phase in a similar way at low and high
redshift.

\section{Discovery of the Supernovae}

The observing strategy developed for these high-redshift SN search
runs involved comparison of large numbers of galaxies in each of
$\sim$50 $14.7'\times14.7'$ fields observed twice with a separation of
3--4 weeks \citep{perl96}. This strategy ensures that almost all the
supernovae are discovered before maximum light, and, since it could be
guaranteed that at least a dozen supernovae would be found on a given
search run, the follow-up observing time could be scheduled in
advance. This strategy made it possible to schedule spectroscopy time
on large telescopes while the supernovae were still close to maximum
light, and thus the supernova features can be observed even at
redshifts $>$1. Follow-up photometric measurements were also made of
the supernova light-curves, from which the maximum brightness and date
of maximum is derived.

The supernovae described in this paper were identified on $R$-band
images taken in two searches on the CTIO 4-m Blanco telescope in
January and March 1997. Candidate supernovae were identified on the
difference images (see \cite{perl96} for details on the identification
of candidates). Those supernovae identified as Type Ia were followed
photometrically in the $R$- and $I$-bands at the WIYN telescope at
KPNO, the ESO 3.6m and one (SN 1997ap) was followed with the {\it
Hubble Space Telescope} \citep{nature98}.

Parameterised light-curve fits for all but one of the objects
presented in this paper (SN 1997ag, which does not have sufficient
photometric light curve measurements for cosmological use) are
published in \citet{42SNe_98}.  These SNe form part of the set that
provided the first evidence for the accelerating expansion of the
universe and the presence of some form of dark energy driving this
expansion. Many of these objects (but excluding SN~1997ag, SN~1997G,
SN~1997J and SN~1997S because of poor colour measurements,
$\sigma_{R-I}>0.25$) were also used in the more recent measurements of
cosmological parameters by \citet{knop03}.

\section{Spectroscopic Observations}

The spectra were obtained during two observing runs at the Keck-II
10-m telescope (two nights in January and three in March 1997) and one
run at the ESO 3.6m (1 night in January 1997). The spectroscopic runs
were scheduled to occur within one week of the corresponding search
run at CTIO. At the time of observations, close to maximum light, the
typical magnitudes of the supernova candidates were 22-24 magnitude in
$R$-band and in many cases their host galaxies had similar apparent
magnitudes. In all cases where the galaxy was bright enough, the slit
was aligned at a position angle on the sky such that the
supernova and the center of the host galaxy were both in the
slit. This was done to allow redshift determination from features in
the host galaxy spectrum.

Observations of the spectrophotometric standards HD84937, HD19445 and
BD262606 \citep{oke_abmag_83} were taken at Keck-II, and LTT 1788
\citep{SB83,BS84,Hamuy94} was observed at ESO, in order to obtain
approximate relative flux calibration of the spectra. The standards
were observed at the parallactic angle, although note that the SN
spectra were not necessarily taken at the parallactic angle as
described above. Although the effects of atmospheric dispersion are
small when observing in the red (as is the case for these spectra),
some small wavelength-dependent slit losses may occur.  Therefore the
overall slope of the spectra is not considered to be reliable and the
slope is left as a free parameter in the analysis that follows.

\paragraph{Keck data}
The LRIS spectrograph \citep{B095} with the 300~l/mm grating was used
at the Cassegrain focus of the Keck-II 10m telescope. The spectra
cover the wavelength range 5000 to 10000\AA\ with dispersion of $\sim$
2.5\AA\ per pixel.  Typical exposure times were 0.5 to 1.5 hours.

During the January run, dome flats were taken at the position of
targets when possible. This was necessary because flexure causes the
fringe pattern to shift depending on the zenith distance and rotator
angle. Previous tests showed that dome flats gave marginally better
results compared to internal flats when trying to subtract sky lines
at the red end of the spectrum.  During the March run, bad weather and
technical problems severely limited the amount of usable
time. Therefore, to allow us to observe all the candidates, the
exposure times for each object were reduced and only internal flats
were taken. Hg-Ne-Ar arc spectra for wavelength calibration were
obtained at the same position as the flats.

\paragraph{ESO 3.6m data} The EFOSC1 spectrograph \citep{Buzzoni84}
was used at the cassegrain focus of the ESO 3.6m. The R300 grism was
used, giving a dispersion of 3.5\AA/pixel over the range 5900 to
10000\AA.  Dome flats were used to flat-field the data, and wavelength
calibration was carried out using observations of He-Ar (ESO) arc
lamps.

Table~\ref{snobstab} gives the dates and exposure times for the
observations of the various supernova candidates.

\begin{deluxetable}{lcllcr}
\tablecaption{Summary of spectral observations.\label{snobstab}}
\tablewidth{0pt}
\tablehead{
\colhead{IAU name}&\colhead{R mag\tablenotemark{a}}&\colhead{Telescope}&\colhead{Exp}    &\colhead{Date} \\
         &               &                   &\colhead{(hours)}&\colhead{(UT)} }
\startdata
1997F & 23.9 & Keck-II     & 1.0  & 1997 Jan 12   \\
1997G & 23.7 & Keck-II     & 1.3  & 1997 Jan 13   \\
1997I & 20.9 & ESO 3.6m    & 1.0  & 1997 Jan 13   \\
1997J & 23.4 & Keck-II     & 1.3  & 1997 Jan 13   \\
1997N & 21.0 & ESO 3.6m    & 0.5  & 1997 Jan 13   \\
1997R & 24.4 & Keck-II     & 1.4  & 1997 Jan 13   \\
1997S & 23.6 & Keck-II     & 1.8  & 1997 Jan 13   \\
1997ac & 23.1 & Keck-II    & 0.17 & 1997 Mar 14   \\
1997af & 23.8 & Keck-II    & 0.58 & 1997 Mar 14   \\
1997ag & 23.2 & Keck-II    & 0.25 & 1997 Mar 14   \\
1997ai & 22.3 & Keck-II    & 0.35 & 1997 Mar 13   \\
1997aj & 22.3 & Keck-II    & 0.67 & 1997 Mar 13   \\
1997am & 22.9 & Keck-II    & 0.31 & 1997 Mar 13   \\
1997ap & 24.2 & Keck-II    & 1.5  & 1997 Mar 14   \\
\enddata
\tablenotetext{a}{The magnitudes given are for the supernovae at the
time of the spectroscopic observations, and were estimated from nearby
light-curve photometry points.  These are accurate to about
0.2 mag.
}
\end{deluxetable}

\section{Data Reduction}

The spectra were reduced using the IRAF\footnote{IRAF is distributed
by the National Optical Astronomy Observatories, which are operated by
the Association of Universities for Research in Astronomy, Inc., under
cooperative agreement with the National Science Foundation.} spectral
reduction package. The data were first overscan-subtracted, bias
corrected and flat-fielded using the flats described above.  The
spectra were then extracted to provide 1-D spectra of the targets. In
some cases the supernova was sufficiently offset from the core of its
host galaxy that it was possible to extract the supernova spectrum and
that of the host galaxy separately. In other cases it was only
possible to extract a single spectrum containing a mix of supernova
and host galaxy light.  These spectra were then wavelength calibrated
and flux calibrated.  Except where otherwise stated explicitly, a
correction for atmospheric absorption features was made.

\section{Identification}

SNe~Ia are classified spectroscopically by a lack of hydrogen and the
presence of a strong \ion{Si}{2} feature at $\sim$6150\AA\ in the
rest frame (see \cite{1997ARAA..35..309F} for a review). However, at
redshifts above 0.5 this line moves into the infrared and can no
longer be seen with typical CCDs. Therefore we use the following
criteria to classify a spectrum as a SN~Ia when this line is not seen:

\begin{itemize}

\item Presence of the SII ``W'' feature at $\sim$5500\AA. This is
only seen in Type Ia supernovae (see Figure~\ref{Imax}).

\begin{figure*}
\includegraphics[scale=0.6,angle=270]{f1.eps}
\caption{Comparison of nearby SNe Ia and Ic at maximum light.  SNe Ic can
closely resemble SNe Ia blueward of 4500\AA\ with the exception of the
4000\AA\ \ion{Si}{2} feature.  Also, SNe Ic have a group of Fe
features spanning 4500--5200\AA\ that resemble the same features in a
Type Ia spectrum $\sim$2 weeks after maximum light. The data shown are
SN 1989B \citep{1994AJ....108.2233W}, SN 2003bu (P. Nugent, private
communication), SN 1994I \citep{1996ApJ...462..462C} and SN 1994D
\citep{1996MNRAS.281..263M}.}
\label{Imax}
\end{figure*}

\item Presence of \ion{Si}{2} feature at $\sim$4000\AA. If seen,
this feature indicates that the supernova is definitely of Type Ia,
although for spectra prior to maximum light or for SN 1991T-like
SNe~Ia, this feature can be weak. Note that it is easy to mistake Ca
H\&K from the host galaxy for this feature.

\item The combination of both the light curve age with the temporal
evolution of the \ion{Fe}{2} features at $\sim$4500 and 5100\AA.  In a SN
Ib/c at maximum light, these \ion{Fe}{2} features resemble those in a
SN Ia about two weeks after maximum light (Figure~\ref{Imax}), but the
appearance of a Type Ia spectrum at maximum light is unique.

\end{itemize}

\subsection{Matching Technique}

The spectra in this paper were matched against template SN spectra
using a spectrum matching code developed by \citet{howell02}.  All
fits were performed after rebinning the data to 20\AA\ to allow more
rapid runs of the fitting program.  At a given redshift, the code
computes:
$$\chi^2=\frac{w(\lambda)(O(\lambda)-aT(\lambda)10^{cA_{\lambda}}-bG(\lambda))^2}{\sigma(\lambda)^2},$$
where $w(\lambda)$ is a weighting function, $O$ is the observed
spectrum, $T$ is the SN template spectrum, $G$ is the host galaxy
template spectrum, $A_{\lambda}$ is the redding law, $\sigma$ is the
error on the spectrum, and $a$, $b$, and $c$ are constants that are
varied to find the best fit.  The weighting function was set to be
unity across the spectrum except at the telluric features, which were
given zero weight.  This equation is iterated over a range of
redshifts to find the minimum $\chi^2$ in redshift, host galaxy,
template supernova, and reddening space.  In this analysis, the
reddening law (\citet{card89,ODonn94}) was fixed at $R_V=3.1$,
although the spectrum matching code can handle other values of $R_V$.

When the redshift was well determined from narrow galaxy lines, it was
constrained in the fits.  Likewise, when the host galaxy type was
known from \citet{sullivan03}, it was fixed in the following manner.
For Sullivan type 0, E and S0 galaxies were tested.  For type 1, Sa and
Sb galaxies were allowed, and for type 2, Sb, Sc, and starburst galaxy
types were used.  The galaxy templates subtracted here were those of
\citet{kinney93} which have been smoothed and had their narrow lines
removed.

In three cases, SN 1997ac, SN 1997ai and SN 1997ap, it was not
possible to obtain a host galaxy redshift.  In these cases the
redshift was determined from fits to SN templates.  For each template,
the best-fit redshift was determined by stepping through redshift
space in 0.001 increments. In section 5.3 we describe the estimation
of redshift uncertainties for these cases.

The fitting program is intended to aid human classifiers of SN spectra
--- it is not a replacement for them.  The program returns a list of
fits in decreasing order of goodness-of-fit.  The output is inspected
and a ``best fit'' is chosen.  A good indication of the level of
certainty of a match is the amount of agreement between the best fits.
For data with good signal-to-noise ratio (S/N) and little host
contamination, there is almost always excellent agreement between the
top five fits.

The program is also limited by the available template data.  For
example, less than a dozen SNe have UV spectroscopy near or before
maximum light \citep{IUE}, and core collapse SNe are particularly
underrepresented in terms of UV observations.  Thus at high redshift,
where the observed optical band corresponds to rest-frame UV, it can
be hard to obtain conclusive results.  One other problem area is at
very early times.  From day $-18$ to $-8$ there are few spectra of SNe
Ia, and these are the epochs where SNe Ia show the greatest diversity.
Furthermore, \ion{Si}{2} 4000\AA\ can be weak or absent at very early
times, making a secure classification difficult.  Finally, a week
after maximum light SNe Ib/c and SNe Ia show the greatest similarity
in their spectra (see Figure~\ref{Imax}).  We note that more UV
spectra of SNe Ib/c would be beneficial to the classification
process. With such spectra it would be possible to confirm some cases
which are probable Type Ia but for which Type Ib/c cannot be
eliminated.  If \ion{Si}{2} 6150\AA\ is not in the observed spectral
range, or there is host contamination, classification at this phase
can be harder than at other phases. The list of nearby supernovae used
in the fitting process is given in Table~\ref{library}. A full list of
the dates of these spectra and their references will be presented by
Howell (in preparation).

\begin{deluxetable}{ll}
\tablecaption{A list of the nearby supernovae used as templates in the fitting
procedure. Details of the dates of these spectra and their references
will be presented by Howell (in preparation). The SUSPECT SN archive
\citep{Richardson02} was used in the creation of this
library.\label{library}}
\tablehead{
\colhead{SN Type}& \colhead{Supernova Name}}
\startdata
Ia & SN 1981B, SN 1986G, SN 1989B, SN 1990N, SN 1991T, SN 1991bg, SN 1992A, \\
   & SN 1994D, SN 1996X, SN 1998bu, SN 1999aa, SN 1999ac, SN 1999ao, SN 1999av,\\
   & SN 1999aw, SN 1999be, SN 1999bk, SN 1999bm, SN 1999bn, SN 1999bp, SN 1999ee \\

II & SN 1979C, SN 1980K, SN 1984E, SN 1986I, SN 1987A, SN 1987B, SN 1987K,\\
   & SN 1988A, SN 1988Z, SN 1993J, SN 1993W, SN 1997cy, SN 1999em, SN 1999em\tablenotemark{a} \\

Ibc & SN 1983N, SN 1983V, SN 1984L, SN 1987M, SN 1988L, SN 1990B, SN 1990U,\\
    & SN 1990aa, SN 1991A, SN 1991K, SN 1991L, SN 1991N, SN 1991ar, SN 1994I,\\
    & SN 1995F, SN 1995bb, SN 1996cb, SN 1997C, SN 1997dd, SN 1997dq, SN 1997ei,\\
    & SN 1997ef, SN 1998bw, SN 1998dt, SN 1999P, SN 1999bv, SN 1999di, SN 1999dn,\\
    & SN 1999ex, SN 2000H \\
\enddata
\tablenotetext{a}{Theoretical SN 1999em spectra provided by
Peter Nugent}
\end{deluxetable}

Near maximum light, normal SNe Ia show a characteristic pattern of
absorption features at rest-frame 3800\AA\ (\ion{Ca}{2}), 4000\AA\
(\ion{Si}{2}), 4300\AA\ (\ion{Mg}{2} and Iron-peak lines), 4900\AA\
(various Iron-peak lines), 5400\AA\ (SII 'W'), 5800\AA\ (\ion{Si}{2}
and \ion{Ti}{2}), and 6150\AA\ (\ion{Si}{2}).  If the S/N of the
spectrum is low, typical Ia-indicator lines such as \ion{Si}{2} and
SII may be hard to identify definitively.  However, the overall
pattern of features in a SN Ia at a given epoch is unique, so fits to
the overall spectrum (with sufficient S/N) can allow a secure
classification of the SN type, especially when there is an independent
confirmation of the SN epoch from the light curve.

In summary, this procedure provides a more robust measure of the SN
type than unaided human typing by eye. Furthermore, it uses the shape
of the entire spectrum to classify SNe, not just one or two key
features.  The technique also provides a spectroscopic determination
of the rest-frame date the spectrum was taken (relative to maximum
light), which can be checked against the light curve, yielding an
independent consistency check as shown in the next section.

\subsection{Matching Results}
The results of the fitting procedure are given in Table
\ref{fittable}.  We show the best fit SN template, the type of host
galaxy that was subtracted (if any), the redshift, and the epoch of
the template spectrum.  In addition, we give a ``spectroscopic
epoch,'' which is the weighted average of the epochs from the five
best-fit spectra.  This spectroscopic determination of the epoch of
the SN can be compared with the epoch determined from the light curve,
also presented in Table \ref{fittable}.  Figure \ref{specday} shows
that the two numbers are in good agreement, with an RMS difference of
3.3 days.

\begin{deluxetable}{llrrrlcl}
\tablecaption{Summary of fitting results.\label{fittable}
}
\tablehead{
\colhead{IAU name} &\colhead{$z$} &\colhead{$\tau_{\rm lc}\tablenotemark{a}$} &\colhead{$\tau_{\rm s}\tablenotemark{b}$} & \colhead{$\tau_{\rm t}\tablenotemark{c}$} & \colhead{Match} &\colhead{G\tablenotemark{d}} &\colhead{G\tablenotemark{e}} }
\startdata
1997F  &$0.580 \pm 0.001$ &$ -7.5\pm 0.4 $ & $ -6.5 \pm 4.1$&$ -8$&1999ee &1&Sa \\
1997G  &$0.763 \pm 0.001$ &$ +5.4\pm 1.0 $ & $ +0.7 \pm 5.1$&$ +4$&1999bk &2&-  \\
1997I  &$0.172 \pm 0.001$ &$ -3.3\pm 0.1 $ & $ +1.2 \pm 1.5$&$ +1$&1999bp &1&Sb \\
1997J  &$0.619 \pm 0.001$ &$ +3.6\pm 1.1 $ & $ +0.7 \pm 4.0$&$ +2$&1999bn &0&S0 \\
1997N  &$0.180 \pm 0.001$ &$+15.0\pm 0.1 $ & $+16.7 \pm 2.3$&$+15$&1991T  &2&SB2\\
1997R  &$0.657 \pm 0.001$ &$ -5.6\pm 0.4 $ & $ +0.0 \pm 3.1$&$ -5$&1989B  &1&Sa \\
1997S  &$0.612 \pm 0.001$ &$ +2.1\pm 0.7 $ & $ +3.3 \pm 2.1$&$ +5$&1992A  &-&SB1\\
1997ac &$0.323 \pm 0.005$\tablenotemark{f} &$ +9.4\pm 0.1 $ & $+11.9 \pm 3.4$&$+11$&1998bu &-&-  \\
1997af &$0.579 \pm 0.001$ &$ -5.3\pm 0.3 $ & $ +1.3 \pm 6.2$&$ -2$&1999bp &2&SB6\\
1997ag &$0.592 \pm 0.001$ &$ -1.7\pm 2.0 \tablenotemark{g} $ & $ +1.9 \pm 6.0$&$ -7$&1990N  &-&Sa \\
1997ai &$0.454 \pm 0.006$\tablenotemark{f} &$ +5.2\pm 0.7 $ & $ +4.8 \pm 1.8$&$ +5$&1992A  &-&-  \\
1997aj &$0.581 \pm 0.001$ &$ -4.4\pm 0.3 $ & $ -0.7 \pm 7.5$&$ -3$&1999aa &2&Sb \\
1997am &$0.416 \pm 0.001$ &$+10.0\pm 0.5 $ & $ +7.4 \pm 2.9$&$ +9$&1992A  &2&SB1\\
1997ap &$0.831 \pm 0.007$\tablenotemark{f} &$ -2.3\pm 0.5 $ & $ +0.3 \pm 3.6$&$ -5$&1989B  &2&-  \\
\enddata
\tablenotetext{a}{Spectral epoch relative to $B$ light curve maximum (in the rest frame) as determined from light curve
fitting \citep{knop03}}
\tablenotetext{b}{Spectral epoch from weighted average of 5 best-fit
spectra. The uncertainty quoted is the weighted standard deviation of
the epochs of the 5 best SN Ia fits.}
\tablenotetext{c}{Spectral epoch of best matching comparison template spectrum.}
\tablenotetext{d}{Host galaxy type from \citet{sullivan03}; 0: E-S0, 1: Sa-Sb, 2: Sc and later}
\tablenotetext{e}{Template host galaxy spectrum subtracted from the data for the fits.}

\tablenotetext{f} {Redshift determined from the supernova alone. These
values are derived based on the matching procedure described in
section 5.1 and supercede those in \cite{42SNe_98,knop03}.} The
uncertainties in these redshifts were estimated as described in
section 5.3.

\tablenotetext{g} {This SN has a poorly-constrained light-curve
because only a small number of photometric measurements were made. In
order to estimate the time of maximum it was necessary to constrain
the stretch parameter to s=1 when fitting the light-curve.}

\end{deluxetable}

\begin{figure*}
\epsscale{1.4}
\plotone{f2.eps}
\caption{Comparison of the epoch of the spectrum as determined from
the light curve ($\tau_{lc}$) with that determined from the weighted
average of the 5 best template fits to the spectrum ($\tau_{s}$). In
both cases the epochs are expressed as rest-frame days relative to
maximum light. The dashed line shows the locus $\tau_{lc}$ =
$\tau_{s}$.}
\label{specday}
\end{figure*}

The program also formally returns an estimate of the amount of
reddening (or de-reddening) required to match the observed spectrum,
but this number may not reflect the actual reddening toward the SN.
This parameter accounts for all differences in color between the
observation and the template, so differential slit losses due to
observations not taken at the parallactic angle, wavelength-dependent
seeing, errors in flux calibration, and uncorrected reddening in the
templates, all make an accurate determination of the reddening to the
SN from this spectroscopy alone unfeasible.

\subsection{Measurement of redshifts}

In most cases the slit angle for the spectroscopic observations was
chosen so that light from both the SN and the host galaxy (when
visible on the CTIO images) fell down the slit. Since features in the
galaxy spectra are typically much narrower than those in supernova
spectra it is often possible to obtain a redshift from the galaxy
features, even in cases where the supernova and host galaxy light are
blended. Table~\ref{fittable} summarises the measured redshifts.

In the cases where galaxy features were visible, the centroids of the
lines were measured, and the redshift calculated by taking the mean of
the redshifts derived from the individual lines. The uncertainty in
the mean redshift $z$ is estimated as $\pm 0.001$.

In the cases where there were no identifiable host galaxy features to
determine the redshift (SN 1997ac, SN 1997ai and SN 1997ap), the
redshift was determined from fits to SN templates as described in
section 5.1. In order to estimate the uncertainty in these redshifts
we used the other 11 cases, whose redshifts were known from the host
galaxies, as follows. Each high-redshift spectrum was cross-correlated
with the corresponding best-fit template (using the IRAF
cross-correlation routine ``fxcor'') allowing redshift to vary. In
this sample of 11 cases, we found a mean redshift difference between
SN and host redshift of -0.0012 and an RMS difference of 0.005. Thus
there does not appear to be any significant systematic redshift error
and the uncertainty in redshift is about 0.005. This test measures the
uncertainty in redshift determination when the ``correct'' template is
used. For cases where the redshift is not known from the host galaxy
there is an additional uncertainty from the diversity of velocities
seen in nearby template spectra. To estimate the size of this effect
we took the standard deviation of the velocity of the \ion{Si}{2}
6150\AA\ feature for the SNe Ia presented in Benetti et
al. (2005). Our calculated value of 1300 km s$^{-1}$ corresponds to a
redshift uncertainty of 0.004. Combining these two effects in quadrature
gives an estimated uncertainty in redshift of about 0.006 on
average. However we also note the uncertainties should increase with
redshift (and indeed this general trend is seen in our sample of 11
objects) because the lines become broader, the S/N becomes poorer, and
the spectrum corresponds more to the rest-frame UV where there are
fewer templates available. Thus we use a simple relation of
$0.004\times(1+z)$ to estimate the errors (which gives an error of
0.006 at our average redshift of $z\sim 0.5$).

\section{Results}

Figures~\ref{974spec} to~\ref{9784_spec} show the spectra, all in
$F_{\lambda}$ units with arbitrary normalisation. The top panel shows
the unsmoothed spectrum which typically contains both supernova and
galaxy light.  In the cases where the supernova and galaxy light were
resolved and separate extractions were possible, the top panel shows
the unsmoothed data for the host galaxy.  The lower panels show the
results of the template matching described in the previous section. In
almost all cases clear supernova Ia features are visible in the
high-redshift supernova spectrum.

\begin{figure}
\epsscale{1.0}
\plotone{f3a.eps}
\plotone{f3b.eps}
\caption{(Upper) Unsmoothed spectrum of SN 1997F showing narrow features
used to determine the redshift (note that atmospheric absorption has
not been corrected in this spectrum). (Lower) Rebinned spectrum of SN 1997F
(histogram) after interpolating over narrow galaxy lines and sky
absorption feature at 7600\AA, and subtracting template Sa galaxy,
compared with SN 1999ee at $-8$ days (smooth curve).}
\label{974spec}
\end{figure}

\begin{figure}
\plotone{f4a.eps}
\plotone{f4b.eps}
\plotone{f4c.eps}
\caption{(Upper) Lightly smoothed spectrum of neighbouring (host) galaxy
of SN 1997G (note that atmospheric absorption has not been corrected
in this spectrum). (Lower) Lightly smoothed spectrum of the supernova SN
1997G (lower) rebinned spectrum of SN 1997G (histogram) after interpolating
over poorly-subtracted sky lines and subtracting an Sa galaxy
template, compared with SN 1999bk at +4 days (smooth curve).}
\label{SN1997G_spec}
\end{figure}

\begin{figure}
\plotone{f5a.eps}
\plotone{f5b.eps}
\caption{(Upper) Unsmoothed spectrum of SN 1997I obtained at the ESO 3.6m
showing galaxy lines used to determine the redshift. (Lower) Rebinned
spectrum of SN 1997I (histogram) after interpolating over narrow
galaxy lines and subtraction of an Sb template galaxy spectrum,
compared with SN 1999bp at +1 day (smooth curve).}
\label{970spec}
\end{figure}

\begin{figure}
\plotone{f6a.eps}
\plotone{f6b.eps}
\caption{(Upper) Unsmoothed spectrum of SN 1997J showing the narrow lines
used to determine the redshift. (Lower) rebinned spectrum of SN 1997J
(histogram) after subtraction of template S0 host galaxy, compared
with SN 1999bn at day +2 (smooth curve).}
\label{SN1997J_spec}
\end{figure}

\begin{figure}
\plotone{f7a.eps}
\plotone{f7b.eps}
\caption{(Upper) Unsmoothed spectrum of SN 1997N showing the narrow lines
used to determine the redshift. (Lower) Rebinned spectrum of SN~1997N
(histogram) after interpolating over strong emission lines of [OIII] and
H$\alpha$ and subtracting template SB2 host galaxy light, compared
with SN~1991T at +15 days (smooth curve).}
\label{9710_spec}
\end{figure}

\begin{figure}
\plotone{f8a.eps}
\plotone{f8b.eps}
\plotone{f8c.eps}
\caption{(Upper) Lightly smoothed spectrum of the host (or possible
neighbour) galaxy of SN 1997R (atmospheric absorption has not been
corrected in this spectrum). (Middle) Lightly smoothed spectrum of
SN~1997R. (Lower) Rebinned spectrum of SN~1997R (histogram) after
interpolating over narrow host galaxy lines, rebinning to 20\AA\ and
subtracting template Sa galaxy, compared with with SN~1989B at $-5$ days
(smooth curve).}
\label{SN1997R_spec}
\end{figure}

\begin{figure}
\plotone{f9a.eps}
\plotone{f9b.eps}
\caption{(Upper) Unsmoothed spectrum of SN 1997S showing the narrow
lines used to determine the redshift. (Lower) rebinned spectrum of SN
1997S (histogram) after interpolating over strong emission lines of
[OII] and [OIII], $H\beta$ and $H\gamma$ and subtraction of a SB1 host
galaxy template, compared with SN~1992A at +5 days (smooth curve).}
\label{9739_spec}
\end{figure}

\begin{figure}
\plotone{f10a.eps}
\plotone{f10b.eps}
\caption{(Upper) lightly smoothed spectrum of SN 1997ac. (Lower) rebinned
spectrum of SN 1997ac (histogram) compared with SN 1998bu at + 11 days
(smooth curve).}
\label{SN1997ac_spec}
\end{figure}

\begin{figure}
\plotone{f11a.eps}
\plotone{f11b.eps}
\caption{(Upper) Unsmoothed spectrum of the host of SN~1997af showing the
narrow lines used to determine the redshift. (Lower) rebinned spectrum of
SN~1997af (histogram) after interpolating across narrow lines from the
host galaxy and subtracting an SB6 galaxy template, compared with the
Type Ia SN~1999bp at $-2$ days (smooth curve).}
\label{SN1997af_spec}
\end{figure}

\begin{figure}
\plotone{f12a.eps}
\plotone{f12b.eps}
\caption{(Upper) Lightly smoothed spectrum of SN~1997ag showing narrow
features used to determine the redshift. (Lower) rebinned spectrum
SN~1997ag (histogram) after interpolating over narrow galaxy lines,
compared with the Type Ia SN~1990N at $+7$ days before maximum (smooth
curve).}
\label{9777_spec}
\end{figure}

\begin{figure}
\plotone{f13a.eps}
\plotone{f13b.eps}
\caption{(Upper) Lightly smoothed spectrum of SN~1997ai. (Lower) rebinned
spectrum of SN~1997ai (histogram) compared to the Type Ia SN 1992A at
$+5$ days (smooth curve).}
\label{SN1997ai_spec}
\end{figure}

\begin{figure}
\plotone{f14a.eps}
\plotone{f14b.eps}
\plotone{f14c.eps}
\caption{(Upper) Unsmoothed spectrum of the host of SN 1997aj (a
separate extraction was possible although there are still narrow
galaxy lines in the SN extraction) showing the narrow lines used to
determine the redshift. (Middle) lightly smoothed spectrum of the
supernova SN 1997aj. (Lower) Rebinned spectrum of SN~1997aj
(histogram) after interpolating over narrow host galaxy lines of [OII]
and Ca H\&K and subtracting Sb galaxy light, compared with the Type Ia
SN 1999aa at $-3$ days (smooth curve).}
\label{SN1997aj_spec}
\end{figure}

\begin{figure}
\plotone{f15a.eps}
\plotone{f15b.eps}
\caption{(Upper) Unsmoothed spectrum of the host of SN 1997am showing the
narrow lines used to determine the redshift. (Lower) lightly smoothed
spectrum of SN 1997am.}
\label{9785_spec}
\end{figure}

\setcounter{figure}{14}

\begin{figure}
\plotone{f15c.eps}
\plotone{f15d.eps}
\caption{cont. (Upper) Rebinned spectrum of SN~1997am (histogram) after
interpolating across cosmic rays and narrow lines from the host galaxy
and subtraction of a SB1 galaxy template, compared with the Type Ia
SN~1992A at +9 days (smooth curve). (Lower) previous plot but with the Type
Ia SN~1991T at +10 days for comparison.}
\end{figure}

\begin{figure}
\plotone{f16a.eps}
\plotone{f16b.eps}
\caption{(Upper) Lightly smoothed spectrum of SN 1997ap. (Lower) Rebinned
spectrum of SN 1997ap (histogram) after interpolating across the sky
absorption region at 7600\AA\ compared with the Type Ia SN~1989B at $-5$
days (smooth curve).}
\label{9784_spec}
\end{figure}

\subsection{Notes on Individual Objects}

\paragraph{SN~1997F} (Figure~\ref{974spec}).
This spectrum is a blend of galaxy and SN light (the host galaxy and
SN were not separable in the extraction).  The host galaxy has
$z=0.580\pm 0.001$, based on narrow lines from [OII], H$\gamma$,
G-band and [OIII].  After subtraction of a galaxy template, broad
\ion{Ca}{2} is visible.  In a SN Ia at this epoch ($-7$~days), other
features are weak, so it is difficult to distinguish them given the
S/N of this spectrum.  Though the spectrum is dominated by galaxy
light, two lines of evidence give us confidence that the residual
spectrum is a Type~Ia SN.  Firstly, the epoch determined by the
fitting program ($-6.5$~days) agrees well with the epoch determined
from the light curve ($-7.5$~days).  Secondly, a Type Ia is a much
better fit to the data than any other SN Type.

\paragraph{SN~1997G} (Figure~\ref{SN1997G_spec}).
The supernova is slightly merged with a brighter galaxy which has
$z=0.763\pm 0.001$. SN~Ia features can be seen at the same
redshift. The fact that the two objects were merged together made sky
subtraction difficult, and residuals remain at observed wavelengths
$\lambda_{obs}=5577$\AA\ and 6300\AA. These have been interpolated
across for the Figures.  While we consider a Type Ia identification to
be the most probable for this SN, a Type Ib/c identification cannot be
ruled out due to the lack of definitive \ion{Si}{2} and the poor S/N
redward of $\sim$8200\AA, corresponding to $\sim$4650\AA\ in the rest
frame.  Furthermore, there are few UV spectra of SNe Ib/c, so while
the bumps at 2900\AA\ and 3150\AA\ (rest frame) observed at
$\sim$5110\AA\ and $\sim$5550\AA\ match those in a Ia, the behaviour
of SNe Ib/c in this region is not well studied.

\paragraph{SN~1997I} (Figure~\ref{970spec}).
The redshift of $z=0.172\pm 0.001$ was derived from galaxy lines
(H$\alpha$, H$\beta$, SII, [OIII]). The spectrum is an excellent
match to SN~Ia features, including the 6150\AA\ \ion{Si}{2} feature
seen at $\lambda_{obs} \sim 7200$\AA\ and SII.  The identification
of this SN as a Type Ia is unambiguous, as the \ion{Si}{2} and SII
lines can easily be seen.

\paragraph{SN~1997J} (Figure~\ref{SN1997J_spec}).
Narrow absorption lines in the spectrum (\ion{Ca}{2} H\&K + 4000\AA\
break from the host galaxy) give $z=0.619\pm 0.001$. SN~Ia features at
this redshift are clearly visible, despite being affected by the
galaxy absorption lines. After subtracting a template S0 galaxy
spectrum the features become much more clear.  Here also, the Type Ia
identification is most probable, but not definitive.

\paragraph{SN~1997N} (Figure~\ref{9710_spec}).
The redshift of $z= 0.180 \pm 0.001$ was derived from
narrow galaxy lines (H$\alpha$ [OIII] and possibly SII). SN~Ia
features from $\sim$ 2 weeks after maximum are clearly visible in the
spectrum including the \ion{Si}{2} 6150\AA\ feature seen at
$\lambda_{obs}\sim 7250$\AA.  The spectra of the low-redshift supernovae
SN 1991T and SN 1994D both match this spectrum equally well (see
section~\ref{peculiar}).

\paragraph{SN~1997R} (Figure~\ref{SN1997R_spec}).
The host (or a neighbouring) galaxy 2.6\arcsec\ away has $z=0.657\pm
0.001$ (identified from the Ca~H\&K lines and the 4000\AA\ break). The
supernova shows clear broad features matching a Type Ia at this
redshift including the 4000\AA\ \ion{Si}{2} feature seen at
$\lambda_{obs}\sim 6630$\AA, just redward of the \ion{Ca}{2} feature.
Narrow lines ($\rm H\eta$, H$\rm\delta$ and possibly $\rm H\gamma$)
are also visible in the spectrum, presumably from the host galaxy.

\paragraph{SN~1997S}  (Figure~\ref{9739_spec}).
Narrow lines due to the host galaxy ([OIII], [OII], $\rm H\beta$, $\rm
H\gamma$) give $z=0.612\pm 0.001$. Very clear SN~Ia features matching
this redshift are visible including the \ion{Ca}{2},
\ion{Si}{2}(4000\AA) and \ion{Fe}{2} features.

\paragraph{SN~1997ac} (Figure~\ref{SN1997ac_spec}).
The spectrum shows very clear SN~Ia features
at a redshift of $z=0.323 \pm 0.005$ at $\sim 10$ days after maximum
light. Since there are no clear features due to the host galaxy, the
redshift is based solely on the supernova features. The Si 6150\AA\
feature is clearly visible at an observed wavelength of $\sim$8100\AA,
providing an unambiguous identification of the object as a SN~Ia.

\paragraph{SN~1997af} (Figure~\ref{SN1997af_spec}).
The supernova and host galaxy spectra could not be extracted
separately.  Narrow lines due to the host galaxy of [OII], Ca H\&K,
$\rm H\gamma$, G-band, H$\eta$, H$\theta$, are clear in the spectrum,
giving a redshift of $z=0.579\pm0.001$. The spectrum also shows broad
features consistent with the spectrum of a SN~Ia at the same redshift,
including the small \ion{Si}{2} feature (4000\AA\ rest wavelength) at
an observed wavelength of $\sim6350$\AA.

\paragraph{SN~1997ag} (Figure~\ref{9777_spec}).
The supernova and host light are blended and
it was not possible to obtain separate extractions.  The Ca~H\&K lines
from the host galaxy are visible at $\lambda_{obs}=6266$\AA\ and
6317\AA\ giving a redshift of $z=0.592\pm 0.001$.  SN~Ia features are
clearly visible in the spectrum, especially broad \ion{Ca}{2} and
\ion{Fe}{2}.

\paragraph{SN~1997ai} (Figure~\ref{SN1997ai_spec}).
The spectrum shows very clear SN~Ia  features
at a redshift of $z=0.454\pm 0.006$. Since there are no clear features
due to the host galaxy, the redshift is based solely on the supernova
features.  The lack of host galaxy contamination is confirmed by the
fitting program, which used no galaxy light in its fit of the
spectrum.  The \ion{Si}{2} (6150\AA) feature is clearly visible at an
observed wavelength of $\sim$8900\AA, providing an unambiguous
identification of the object as SN~Ia.

\paragraph{SN~1997aj} (Figure~\ref{SN1997aj_spec}).
The supernova and host were separated by 2.5\arcsec\ on the slit and
their spectra could be extracted separately. The host galaxy spectrum
shows emission lines of [OII], [OIII] and H$\beta$ as well as
absorption lines of $\rm H\gamma$, $\rm H\delta$, $\rm H\theta$, $\rm
H\eta$, G-band and Ca H\& K, at a redshift of $z=0.581\pm 0.001$.  The
supernova spectrum is consistent with that of a Type Ia at $z=0.581$
showing the a strong, broad \ion{Fe}{2} 5000\AA\ feature seen at
$\lambda_{obs} \sim 7900$\AA, but a relatively narrow \ion{Ca}{2}
3800\AA\ feature seen at $\lambda_{obs}\sim 6000$\AA.

\paragraph{SN~1997am} (Figure~\ref{9785_spec}).
Separate extractions of the host galaxy and supernova
were possible since they were separated by 2.3\arcsec\ on the
slit. The host galaxy spectrum shows emission lines of [OII], H$\beta$
and [OIII] at a redshift of $z=0.416\pm 0.001$. The supernova spectrum
is a good match to the normal Type Ia SN~1992A spectrum at day +9 and
clearly shows the presence of the \ion{Si}{2} (6150\AA) feature seen
at $\lambda_{obs}\sim 8710$\AA.  SN 1991T at day +10 is also a good
fit to the spectrum (see Section \ref{peculiar}).

\paragraph{SN~1997ap} (Figure~\ref{9784_spec}).
The redshift of $z=0.831 \pm 0.007$ is based on supernova features
alone. This spectrum was discussed in detail in \citet{nature98}.  The
slightly improved redshift estimate presented here superceeds that of
earlier papers \citep{nature98,42SNe_98,knop03}. We note here that it
was identified as a SN~Ia by the presence of both the blue \ion{Ca}{2}
and \ion{Si}{2} features.

\section{Tests for Evolution}
Although these spectra were taken primarily for the purposes of redshift
measurement and SN classification, they also allow some basic tests of
supernova evolution with redshift.  We show that SNe Ia do not look
dramatically different at high redshift, that they have similar
elemental velocities, and that SNe~Ia fall into the same sub-classes
at high redshift.

\subsection{Spectral morphology}
High-redshift SNe Ia look similar to their low-redshift counterparts, but the
implications for constraints on evolution are complicated.
Figure~\ref{snseq} shows a selection of the high-redshift spectra
(those not significantly contaminated by host galaxy light) plotted in
order of rest frame days past maximum. These are interspersed with the
time sequence for the nearby Branch-normal Type Ia supernova SN
1992A. Note that because the high-redshift spectra are not
spectrophotometric measurements (see sections 3 and 5.2), differences
in overall slope of the spectra should not be considered
significant. Despite the lower S/N of the high-redshift spectra, and
the non-uniform wavelength coverage (since the spectra have been
de-redshifted by different amounts) it is clear that the overall
trends in the spectral evolution with light curve phase are the same
at low and high redshift. Furthermore, Figures \ref{974spec} to
\ref{9784_spec} show that high-redshift SNe resemble well observed
local SNe on an object-by-object basis.

\begin{figure*}
\epsscale{1.9}
\plotone{f17.eps}
\caption{The time sequence of high redshift supernova spectra in order
of rest-frame date relative to maximum light, as determined from the
light curve ($\tau_{lc}$). Spectra of the nearby Type Ia, SN
1992A are interspersed for comparison.}
\label{snseq}
\end{figure*}

\subsection{The incidence of spectroscopically peculiar SNe}
\label{peculiar}

\citet{li01a,li01b} make the case that, while slow-declining,
spectroscopically peculiar SNe like SN 1991T and SN 1999aa represent
20\% of SNe discovered in volume-limited local surveys, they are
strangely absent from high-redshift SN samples.  This is paradoxical,
since these SNe are over-luminous, and so should be seen in {\it
greater} numbers in flux-limited surveys.  The authors present one
possible solution --- that these SNe become spectroscopically normal
with time, and they appear normal by the time spectra are taken.

Perhaps supporting this idea, for two of our cases, both taken at
relatively late times after maximum light, the spectrum of SN 1991T
fits as well as the spectrum of a more normal supernova. SN 1997am has
a light curve stretch (a measure of the rise and decline time of the
supernova light curve, defined in \cite{42SNe_98}) of $s=1.03\pm~0.06$
\citep{knop03}, which is consistent with the stretch of SN 1991T
($s=1.08$). Since the spectrum was taken approximately 10 rest-frame
days after maximum light, by which time SN 1991T itself appears to be
fairly spectroscopically normal, there is no definitive spectroscopic
evidence that SN~1997am is a peculiar Type Ia, but it cannot be ruled
out. In the other case, SN~1997N, the spectrum was taken approximately
16 rest-frame days after maximum. It is possible that this SN looked
spectroscopically similar to SN 1991T at early times although again
there is no definitive spectroscopic evidence for this. The stretch of
SN~1997N is $s=1.03 \pm 0.02$, somewhat lower than that of
SN~1991T. However we note that the nearby SN 1997br also resembled SN
1991T spectroscopically, yet had a fairly normal light curve, with
$\Delta m_{15}(B)=1.00\pm 0.15$, $s=1.04$ \citep{li99}. Garavini et
al. (in preparation) have found definitive evidence for a SN
1991T-like supernova at high redshift. In that case the spectrum was
taken 7 days before maximum when the spectral peculiarities are more
evident.

The explanation for the supposed dearth of SN 1991T-like SNe at high
redshift proposed by Li et al. (that they are in the data set, but
appear spectroscopically normal when spectra are taken) appears to be
part of the answer to the paradox, although the situation may be more
complicated.  These SNe may not be as consistently over-luminous as
first thought \citep{li99,saha01,gibson01}, thus \citet{li01a} may
have over-predicted the expected numbers in a flux-limited survey.

At the other extreme, spectroscopically peculiar under-luminous SNe
like SN 1991bg \citep{filippenko92, leibundgut93, turatto96} and SN
1999by \citep{toth00, howell01a, vink01, hoeflich02} have not been
seen at high redshift.  While it is unlikely that these SNe will be
found in flux-limited searches, it is also possible that these SNe are
from such an old stellar population that they do not exist at $z >
0.5$ \citep{howell01b}.

\subsection{Calcium velocities}
While the depth of absorption lines gives some information about the
quantity of an ion in a SN, the velocity of the line gives information
about the ion's distribution within the photosphere.  Furthermore, the
velocity of the supernova ejecta is related to the overall kinetic
energy of the event. The kinetic energy has a direct influence on the
opacities and hence on the total brightness of the supernova and its
light-curve shape.

In Figure~\ref{ca2vel} we present a plot of the velocities of the
\ion{Ca}{2} minima for the SCP high-redshift supernovae compared
with two well-observed nearby supernovae. We note that while the
absorption feature at 3700\AA\ is dominated by \ion{Ca}{2},
\ion{Si}{2} and various ionization states of the iron-peak elements
\citep[see][]{hatano99} also contribute. For the
three lowest redshift supernovae in the SCP set, the \ion{Ca}{2}
feature is outside the observed wavelength range. Therefore this
analysis is restricted to the SCP supernovae with $z>0.38$.

\begin{figure*}
\epsscale{1.4}
\plotone{f18.eps}
\caption{Comparison of the velocities determined for the minima of the
\ion{Ca}{2} feature for two well observed nearby SNe~Ia (1992A
and 1989B) and the high-redshift SNe presented in this paper.}
\label{ca2vel}
\end{figure*}

The analysis was performed on the spectra after galaxy subtraction had
been done if this was necessary (as described in the previous
section). The \ion{Ca}{2} velocities were estimated by fitting a
Gaussian function to the region 3610\AA\ to 3870\AA\ in the rest
frame, followed by a second iteration using the region from -65\AA\ to
+100\AA\ either side of the previous estimate of the minimum. The
velocity shift was calculated from the wavelength of the minimum by
assuming a rest frame wavelength for this feature of 3945.28\AA.  The
same analysis was done on a spectral series for the normal Type~Ia
SN1992A \citep{kir92a}. Finally the published \ion{Ca}{2} velocities
for the normal Type~Ia supernova 1989B \citep{wells94} are also
shown.

The general trend of decreasing \ion{Ca}{2} velocities as a function
of time and the range in velocities at a given epoch seen in the data
is completely consistent with the range seen in nearby observations.
This result is confirmed by Garavini et al. (in preparation) in a more
recent independent set of high-redshift SN spectra.

Our high-redshift data is also broadly consistent with the recent
study of nearby SNe by \cite{Benetti05}, although the comparison is
complicated by the fact that Benetti et al. measure \ion{Si}{2}
velocities whereas we consider \ion{Ca}{2} velocities. Taking into
account the apparent offset between the two (as can be seen from
SN~1992A and SN~1989B for example), the velocities seen in our
high-redshift sample (including the large value for SN~1997F) are
broadly consistent with the range of velocities seen in nearby
SNe. Because of the larger error bars at high redshift, and only
having a single velocity measurement per SN, it is not possible to
determine into which of the \cite{Benetti05} classes each of our
objects would fall.

\section{Conclusions}

At the present time, the statistical and systematic errors in the
measurement of the cosmological parameters from Type Ia supernovae are
of similar size. As the quality and quantity of high-redshift
supernova observations grows it will become even more important to
constrain the systematic uncertainties. Non-Ia contamination and
evolution are two of the larger potential systematics we currently
face. Here we have presented some of the methods we employ to
reduce/study these uncertainties.

In this paper we have presented spectra for 14 high-redshift
supernovae and demonstrated that the spectra are consistent with Type
Ia.  In three cases at intermediate redshift (SN~1997ac, SN~1997ai and
SN~1997am at $z\sim 0.3-0.5$) in addition to two at lower redshift
(SN~1997I and SN~1997N) we have observed the \ion{Si}{2} 6150\AA\
feature and hence have unambiguously identified the SN as Type Ia. For
higher redshifts this \ion{Si}{2} feature becomes increasingly
difficult to observe as fringing and poor CCD response in the red, and
bright OH lines in the sky background makes the spectra very noisy
redward of $\sim 9500$\AA. Here we have used spectral matching
combined with the identification of specific spectral features to
identify the objects as SN~Ia, the most important being the
identification of \ion{Si}{2} features near 4000\AA.

We have carried out first-order quantitative tests to compare the
high-redshift spectra with their low-redshift counterparts.

We show that the spectral phase determined from spectral matching to
low-redshift Type Ia spectra is in very good agreement with the phase
determined from the high-redshift light curve.  Similarly the spectral
time series shows the same overall trends at low and high
redshift. Finally, quantitative measurements of the Calcium ejection
velocity in the high-redshift spectra are also consistent with those
measured from the spectra of low-redshift Branch-normal Type Ia
SNe. Therefore we have found no evidence for evolution in the
population of Type Ia SNe up to redshifts of $z\sim 0.8$.  While we
cannot prove that all SNe Ia at high redshift are identical to their
low-redshift counterparts, we can say that it is possible to choose a
set of high-redshift SNe Ia that are equivalent to low-redshift
counterparts to within the accuracy of these quantitative tests.

With larger samples, such as those now being collected by the
Supernova Legacy Survey (SNLS, \cite{pritchet05}) and the ESSENCE
project (\cite{Matheson} and references therein) it will be possible
to make more detailed comparisons in smaller redshift bins in the
range $0.2<z<1$ and in subsets based on light curve stretch, galaxy
host type and other factors. Such studies will provide further
confidence in the use of SNe~Ia as distance indicators, and, in cases
where quantitative measurements of spectral features are found to
correlate with luminosity or light curve stretch
(e.g. \citealt{nugseq95,Fol04}), these can be used to reduce scatter
in the Hubble diagram and thus improve the precision to which
cosmological parameters may be measured.

\section*{Acknowledgements}

The authors acknowledge the help of the night assistants and support
staff at the telescopes from which data for this paper were obtained.

We thank the anonymous referee for a very thorough reading of the
paper and helpful suggestions.

This paper makes use of light curve photometry collected at the Cerro
Tololo Inter-American Observatory, which is operated by Association of
Universities for Research in Astronomy, Inc. under a cooperative
agreement with the National Science Foundation. Based in part on
observations obtained at the WIYN Observatory, which is a joint
facility of the University of Wisconsin-Madison, Indiana University,
Yale University, and the National Optical Astronomy Observatory. We
also make use of observations made with the NASA/ESA Hubble Space
Telescope, obtained at the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy,
Inc., under NASA contract NAS 5-26555. These observations are
associated with programs DD-7590 and GO-7336.

This work was supported in part by the Royal Swedish Academy of
Sciences and by the Director, Office of Science, Office of High Energy
and Nuclear Physics, of the US Department of Energy under Contract
No. DE-AC03-76SF00098.  Support for this work was provided by NASA
through grant HST-GO-7336 from the Space Telescope Science Institute,
which is operated by the Association of Universities for Research in
Astronomy, Inc., under NASA contract NAS 5-26555.

\input{ms_preprint.bbl}

\clearpage

\end{document}


\section{Introduction}
SWAS is a complete space-borne radio observatory (\cite{Melnick2000}),
capable of observing the 556.9\,GHz transition of
ortho H$_2 ^{16}$O, among other species, with a velocity resolution of
1\,km\,s$^{-1}$ and a beam size of $3.3\times4.5$\,arcminutes. SWAS began
near-daily monitoring observations of comet 9P/Tempel 1 on 5 June 2005,
and the observations were continued post-impact  until 1 September 2005.
The total water production rate $Q$ of the comet is determined from the
velocity-integrated intensity of the emission detected in the
SWAS spectra (typically being 2 to 3-day co-adds) and employing the
radiative transfer model for water line emission in comets
by \cite{Bensch2004}.
The left panel of Fig.\ \ref{q-vs-time}
shows the SWAS-measured water production rate for the observations
made from 5 June through 9 July. The cometary
activity traced by the water evaporation rate varies by more than a factor
of three, from $3.8\times10^{27}$ to $12.9\times10^{27}$\,
molecules per second (115 to 385 kg\,s$^{-1}$, respectively).
\begin{figure}
\begin{centering}
 \includegraphics[width=40mm,angle=-90]{c_tempel_int_bw.eps}
  \caption{Left panel: Water production rate of comet
9P/Tempel 1, derived from SWAS observations of the  $1_{10}-1_{01}$
transition of o-H$_2 ^{16}$O. The vertical line gives the impact time.
Right panel: SWAS spectrum before and after the
impact. Each spectrum is a co-average of 3 days.}
\label{q-vs-time}
\end{centering}
\end{figure}

No statistically significant increase in the water line emission was detected
by SWAS immediately following the impact. The average velocity-integrated
intensity during the three days after the impact is 0.16$\pm$0.04\,K\,km/s,
virtually identical to the average intensity measured for the three
days before impact (Fig.\ \ref{q-vs-time}, right panel).
This corresponds to a total water production rate
of $(6.6\pm1.5)\times 10^{27}$\,s$^{-1}$.

In order to derive an upper limit on the water released by the impact,
we extended the radiation transfer model to include comets where the
water production rate is time-dependent.
The total water production rate is replaced by $Q=Q_q + Q_b(t)$, where
$Q_q$ is the constant (quiescent) component. $Q_b(t)$ is the
time-variable component, assumed to be a box-car function
for the present simulations. In this case the water production rate is
elevated (but constant) for the duration $\tau$ of the outburst and
returns to the pre-outburst level afterwards. The total number of water
molecules released by the outburst is $N=Q_{\rm b}\times \tau$. An example
for the time evolution of the velocity-integrated intensity derived with this
outburst model is shown in
Fig.\ \ref{outburst-model}.
Preliminary model results for the SWAS observations up to 24\,hrs after
the impact
give a 3$\sigma$ upper limit of $\sim$9$\times 10^{32}$ molecules
($2.7\times 10^4$ tons) for the vaporized water.
This result
is not very sensitive to the assumed duration of the outburst, as
a similar upper limit is obtained for models with
$\tau=$0.5, 2, 8 and 16\,hrs. In addition, a doubling of the
water production rate can be excluded at a $3\sigma$ confidence level
if we assume that the impact created a new, permanently active area on the
comet nucleus. Results from
Odin observations of the ortho-water $1_{10}-1_{01}$ transition made during the
Deep Impact event are presented by D.\ Despois (this volume).

\begin{figure}
\begin{centering}
 \includegraphics[width=40mm,angle=-90]{swas_upper_limits_bw.eps}
  \caption{Left: The velocity-integrated intensity calculated by
an outburst model with $\tau=2$\,hrs and $N=9\times 10^{32}$
is compared to the 3$\sigma$ upper limit derived from the SWAS spectra
observed during each orbital segment of 35\,min duration (error bars).
No emission is detected in individual segments, consistent with the low
$Q_{\rm H_2O}$ measured before impact. Right panel:
the weighted average of line profile for this outburst model
(dashed line) is compared to the SWAS-measured spectrum. The individual
spectra are weighted by the square of integrated intensity predicted by the
model for each orbital segment and the inverse square of the
rms noise in the spectrum.}
\label{outburst-model}
\end{centering}
\end{figure}

\begin{acknowledgments}
Support of the SWAS mission is provided by NASA through SWAS contract
NAS5-30702. Additional support for the SWAS 9P/Tempel 1
observing campaign was obtained by the Deep Impact team. The help and
cooperation of NASA HQ is acknowledged.
\end{acknowledgments}