 from an exponential
distribution (see Section \ref{scatter} for details on how the
intrinsic scatters and the $e$-folding of the exponential age
distribution were determined). These parameter triples are then
converted to $\hbeta$, \mgb, and Fe5335 index values (index triples),
which are then perturbed using the observed indices error
distributions. These perturbed index triples are then converted back
to parameter triples and the Spearman rank correlation statistic $r_\mathrm{S}$
is calculated for age and [Z/H]. This procedure is carried out 10,000
times. From these Monte Carlo simulations we determine the probability
of obtaining by chance a correlation greater than that found for the
observed data, and we find that there exists a real anti-correlation
between age and [Z/H] (over and above the error-induced correlation)
that is significant at the $>3\sigma$ level.

\subsection{The distributions in the cluster-outskirts}
\label{low den dist}

\begin{figure*}
\includegraphics[width=1.0\textwidth]{figure06.eps}
\caption{Comparison of the SPPs in the cluster-outskirts sample (black
  circles) to those in the cluster sample (red crosses). The marginal
  distributions for the cluster sample are shown as open histograms
  and for the cluster-outskirts sample as black histograms. The dashed
  line in the marginal distributions represents the median parameter
  values in the cluster-outskirts sample and the dotted line that of
  the cluster sample.}
\label{gal params env}
\end{figure*}

The distributions of SPPs in the cluster-outskirts sample (red
crosses) are compared to those from the clusters (black circles) in
Figure \ref{gal params env}. For the sake of clarity, only the errors
for the cluster-outskirts sample are shown. The dashed lines in the
marginal distributions are the median galaxy parameters for the
cluster-outskirts sample and the dotted lines are for the cluster
sample. The galaxies in the cluster-outskirts sample have a median
[Z/H] of $0.21\pm 0.27$ dex, a median age of $6.6^{+8.1}_{-3.6}$ Gyr,
and median $\afe$ of $0.27\pm 0.21$ dex. Compared to the combined
cluster sample we detect no difference in the median values for the
galaxy parameters, although the errors are substantial.

Looking at the distributions we see that there is very little
difference between the galaxies in this sample and those in the
clusters, an assessment that is confirmed by KS testing (a two-sample
2D KS test in the case of the age and [Z/H] distributions and a
two-sample 1D KS test for the $\afe$ distributions).

The fact that the $\afe$ distributions are the same in the clusters
and their outskirts is intriguing, given that the cores of galaxy
clusters are thought to result from regions of high over-density where
the process of star formation occurs rapidly \citep{kauffmann98,
  schindler05, romeo05, delucia06}. Our result is, however, in
agreement with previous studies on the $\afe$ in low- and high-density
environments \citep{kuntschner02, thomas05}. These results suggest
that, in all environments, elliptical galaxies form on similar
timescales.

The often-found differences in low- and high-density environments
between the age \citep{trager00a, poggianti01, kuntschner02,
  terlevich02, caldwell03, proctor04a, thomas05} and [Z/H]
\citep{kuntschner02, thomas05}, with galaxies on average being $\sim
2$ Gyr younger and more metal-rich in lower-density environments, are
not reproduced here. Although, given the size of our errors it is not
surprising that we do not detect such a small difference. These
results are usually found comparing truly isolated galaxies with
cluster galaxies. As the galaxies in the cluster-outskirts sample are
actually drawn from the outer regions of the clusters and the
structures surrounding them (out to a projected radial distance of
$\sim 19$ $h_{70}^{-1}$ Mpc or $\sim10 R_\mathrm{vir}$), it is
possible that the contrast between the average densities of each
environment is not sufficient to reveal any differences in the stellar
populations, and that the change occurs at a lower density threshold
(e.g.\ between isolated galaxies and those in groups/filaments).

\begin{figure}
\includegraphics[width=0.5\textwidth]{figure07.eps}
\caption{The trend of mean age (top), mean [Z/H] (middle), and mean
  $\afe$ (bottom) as a function of cluster virial radius. The mean is
  calculated in bins containing 15 galaxies.}
\label{gal params r}
\end{figure}

Figure \ref{gal params r} shows the variations of the SPPs with
projected cluster-centric distance, normalised to the cluster virial
radius ($R_\mathrm{vir}$). The lines in these plots show the mean
values in bins containing 15 galaxies extending out to almost $10
R_\mathrm{vir}$. The error bars show the standard error of the mean
within each bin. We find no evidence of any significant trends,
although it does appear as if $\afe$ and age increase moving towards
the centre of the cluster, implying that the galaxies in the cluster
cores are older and have shorter star-formation timescales than those
in the cluster outskirts. Despite the lack of clear trends there are
two other interesting aspects of these plots. Firstly, there appears
to be a decrease in [Z/H] outside $\sim2 R_\mathrm{vir}$. Secondly, at
a distance slightly less than the cluster virial radius there appears
to be a dip in the average age. Such a dip would be expected in a
scenario where interaction with the ICM triggers star formation in an
in-falling galaxy, resulting in a stellar population characterised by
a young age and extended star-formation timescale. This decrease in
age begins at a radius $>3 R_\mathrm{vir}$ suggesting that the
influence of the cluster extends to large distances. We also note that
all galaxies with ages $\la 2$ Gyr are located near $R_\mathrm{vir}$
and the age of the youngest stellar population at a given radius
increases moving towards the cluster core.

\section{\boldmath The Parameter--$\sigma$ Relations}\label{scale}

\subsection{\boldmath The cluster relations}
\label{low clus rels}

\begin{figure*}
\begin{center}
\includegraphics[width=1.0\textwidth]{figure08.eps}
\caption{The variations of the SPPs $\log(\mathrm{age})$ (left panel),
  [Z/H] (middle panel) and $\afe$ (right panel) with $\log\sigma$ of
  the galaxies in the cluster sample. All three parameters are
  moderately correlated with $\log\sigma$. The red line in each panel
  is the linear fit to the data.}
\label{param sig}
\end{center}
\end{figure*}

The variations of the SPPs with velocity dispersion are shown in Figure
\ref{param sig}. All parameters are found to be moderately correlated
with a high level of significance; $>5\sigma$ for $\log\mathrm{age}$
and $>4\sigma$ for [Z/H] and $\afe$. Performing linear fits to the
relations we find
\begin{eqnarray}
\log\mathrm{age} & = &
(0.64\pm0.12)\,\log\,\sigma-(0.63\pm0.26) \hspace{1ex},\label{age sig}\\
\mathrm{[Z/H]} & = &
(0.40\pm0.08)\,\log\,\sigma-(0.63\pm0.17) \hspace{1ex},\label{met sig}\\
\mathrm{[\alpha/Fe]} & = &
(0.20\pm0.06)\,\log\,\sigma-(0.17\pm0.13) \hspace{1ex}.\label{alpha sig}
\end{eqnarray}
These fits are shown as red lines in Figure \ref{param sig}. While
these results confirm the existence of the [Z/H]--$\sigma$ and
$\afe$--$\sigma$ relations, the correlation of age with velocity
dispersion found here is not so well accepted (see Section