\chapter{The \emph{K} Band Luminosity Function of High Redshift Clusters of Galaxies.} \label{ch:klf} This chapter contains the main body of the paper `The \emph{K} band galaxy luminosity functions of three massive high redshift clusters of galaxies'. This paper will be submitted to the Monthly Notices of the Royal Astronomical Society to be considered for publication. For the sake of clarity the title, author list, keywords and acknowledgements all appear in \ref{ap:klf}. The references appear in the main list at the end of the thesis. %\documentclass[usenatbib,useAMS]{mn2e} %\usepackage{psfig} %\title{The \emph{K} band galaxy luminosity functions of three massive %high redshift clusters of galaxies.} %\author[S.C. Ellis and L.R. Jones] % {S.C.~Ellis\thanks{E-mail: sce@star.sr.bham.ac.uk} and L.R.~Jones \\ % School of Physics and %Astronomy, University of Birmingham, Birmingham, B15 2TT, UK. \\} %\date{Accepted .....................; Received .....................; %in original form .......................} %\input{/data/sce/latex/newcom.tex} %How to use natbib and labels %See section \ref{data} %\protect\citet{hen92} %(\protect\citealt{hen00}) %\begin{document} %\maketitle %\begin{abstract} \section{Abstract.} \emph{K} band observations of the galaxy populations of three high redshift ($z=0.8$--$1.0$), X-ray selected, massive clusters are presented. The observations reach a depth of $K \simeq 21.5$, corresponding to $K^{*}+3.5$ mag. The evolution of the galaxy properties are discussed in terms of their \emph{K} band luminosity functions and the \emph{K} band Hubble diagram of brightest cluster galaxies. The bulk of the galaxy luminosities, as characterised by the parameter $K^{*}$ from the \citet{sch76} function, are found to be consistent with passive evolution with a redshift of formation of $z_{f}\approx 1.5$--2. This is consistent with observations of other high redshift clusters, but may be in disagreement with galaxies in the field at similar redshifts. A good match to the shape of the Coma cluster luminosity function is found by simply dimming the high redshift luminosity function by an amount consistent with passive evolution. %The shape of the luminosity function at high redshift is not %significantly different from that of the Coma cluster, again in %agreement with passive evolution. The evolution of the cumulative fraction of $K$ band light as a function of luminosity shows no evidence of merger activity in the brighter galaxies. %is also consistent with passive evolution.% if brightest cluster galaxies are excluded. The evolution of the brightest cluster galaxies (BCGs) is tested by their \emph{K} band Hubble diagram. The scatter in the Hubble diagram allows for a range of evolutionary histories. The fraction of total cluster light contained in the BCGs is large compared to Coma, suggesting that they are already very massive with no need to hypothesise significant mergers in their futures. %\end{abstract} %\begin{keywords} %galaxies:clusters:general -- galaxies:evolution -- galaxies:formation %\end{keywords} \section{Introduction.} The evolutionary history of galaxies in clusters remains a subject of debate. The two most common explanations of massive galaxy formation and evolution are those of monolithic collapse (e.g.\ \citealt{egg62}) and hierarchical merging (e.g.\ \citealt{col94}). In the monolithic collapse scenario all the galaxies (and the stars therein) are formed in a single burst and subsequently evolve passively (along the main sequence) with no further star formation. Such a model will result in a very homogeneous population of galaxies since their ages and metallicities will all be close to identical, with a small degree of scatter reflecting the variations in initial mass function at formation. There is a significant body of observational evidence that luminous early type cluster galaxies are indeed remarkably homogeneous. For example, the observed tightness of the colour-magnitude relation (e.g.\ \citealt{vis77}, \citealt{bow92b}) is naturally explained by such a homogeneous population. In the merging model galaxies form by a series a mergers within a hierarchical model of structure formation (e.g.\ \protect\citealt{kau93}). The hierarchical scenario presents a radically different evolution than in the monolithic collapse picture. Bursts of star formation, related to mergers, may occur and a more gradual increase in the number of massive galaxies in a cluster would then occur as small galaxies merge to form larger ones. %However, the hierarchical scenario can also explain the tightness of the colour-magnitude relation provided that enough time has elapsed since the last major merger. If galaxy merging is almost dissipationless then significant star formation will not occur (since if most gas is already in the form of stars there will be little radiative loss of energy, \protect\citealt{ben92}). This is more likely to be the case in massive galaxies. Note also that in herarchical merging as opposed to random merging higher mass galaxies have a higher chance of merging due to dynamical friction (\protect\citealt{kau98a}) and their location in the denser, central parts of the cluster. Although the ages of the majority of the stars in galaxies in both scenarios are similar (in order to explain the tigntness of the colour-magnitude relation and the evolution of the fundamental plane) the mass as a function of look-back time will be quite different, i.e.\ the epoch of the assembly of massive galaxies is not the same as the epoch of star formation in the merging case. The hierarchical model will exhibit negative evolution of mass as a function of redshift (i.e.\ at higher redshifts the galaxies will be less massive on average), whereas in the monolithic collapse model galaxies will have a constant mass with redshift. The negative evolution expected from hierarchical models will be most apparent in the most massive galaxies, which are predicted to have assembled more recently. \citet{kau98b} claim that this is the case in the deep field samples of \citet{son94} and \citet{cow96}, with strong evidence of the absence of massive galaxies at $z \sim 1$, which would require large amounts of dust obscuration in the monolithic collapse picture. %\subsection{\emph{K} band luminosity functions.} If one is interested in the stellar mass of the galaxies, as opposed to their star formation rate, then the \emph{K} band is a good choice in which to make observations since the light at such wavelengths originates mainly from the longer lived stars of the main sequence (see e.g.\ figure~1 of \protect\citealt{kau98b}), and has the added advantages that the k-correction differences between galaxies of different spectral types are small in \emph{K} and the Galactic extinction by dust is small. Thus computing \emph{K} band luminosity functions for clusters of galaxies should give a useful measure of the mass distribution of galaxies within clusters (see section~\ref{sec:klf}). %\citet{dia01} show that in a merging model massive galaxies in clusters are predicted to have formed at high redshift with very little evolution in the number of galaxies since $z=0.8$. This is a consequence of the fact that clusters are associated with regions of high overdensity, and it is expected from hierarchical models that structures will first collapse in such regions. %Parametrizing the luminosity functions with a \citet{sch76} function allows an estimate of $K^{*}$ as a function of redshift. This a useful probe with which to distinguish between the two evolutionary scenarios. In the hierarchical scenario evolution of $K^{*}$ would be expected in the sense that if galaxies are merging to form more massive galaxies with time, then there should be fewer massive galaxies at high redshifts than at present and thus $K^{*}$ should be dimmer at high $z$. In the monolithic collapse model all evolution is due to passive evolution, with the strongest effects on $K^{*}$ being simply due to the distance to the galaxies. See e.g.\ figure 3 of \citet{kau98b}. Although this paper concerns the galaxies in the field more than it does galaxies in clusters the principle ideas are still relevant here. %It is with the aim of distinguishing between these possibilities that we present the \emph{K} band luminosity functions of the three high redshift, massive clusters of galaxies described in section~\ref{sec:gals}. %\subsection{Brightest cluster galaxies.} Brightest cluster galaxies (BCGs) are known to have a very limited variation in absolute magnitude which historically led to their nomination as a candidate for a cosmological standard candle with which to directly measure the curvature of the Universe (\protect\citealt{san72a}, \protect\citealt{san72b}.) Controversy over their suitability as standard candles is believed to be due to environment, with BCGs in high $L_{\mathrm{X}}$ clusters exhibiting much less scatter than those in low $L_{\mathrm{X}}$ systems (\protect\citealt{bro02}, \protect\citealt{bur00}). However the evolution of BCGs with redshift is now the primary interest of research on BCGs, since there is much evidence that they are a special case in the evolution of galaxies within clusters. For instance, it is known that BCGs often do not follow the same luminosity function as other galaxies in clusters (\protect\citealt{sch76}, \protect\citealt{dre78}, \protect\citealt{bha85}), most probably due to the fact that they have a peculiar formation history. Therefore the evolution of BCGs can provide a different and complementary study of galaxy formation theories compared with the general cluster population. The \emph{K} band Hubble diagram of BCGs provides an efficient measure of the evolution of BCGs, and we present our results of this in section~\ref{sec:bcg}. Throughout this paper we have used a cosmology of $H_{0}=70$ km s$^{-1}$ Mpc$^{-1}$ in a flat universe with $\Omega_{\mathrm{M}}=0.3$ and $\Omega_{\Lambda}=0.7$, except where stated. \section{Data.} \label{data} \subsection{The Clusters.} \label{sec:gals} We present a study of three of the most massive ($\sim 10^{15}$M$\odot$ \protect\citealt{mau03a}, \protect\citealt{mau03b}), high redshift clusters known. They are thus ideal probes of galaxy evolution within clusters. % In a hierarchical model galaxies are predicted to first form in regions with the highest overdensities which merge over time with other systems to become massive clusters. Such massive clusters at high redshift are very rare and we have an unusual opportunity to study the galaxy populations of rich, distant clusters and compare results with local massive clusters such as Coma. The high redshift of the clusters should make any evolution in the galaxy populations easier to observe. Two of the clusters (ClJ1226 and ClJ1415) appear relaxed based on their X-ray morphologies and one of them (ClJ0152) is probably in a state of merging. Thus we also have a small selection of different environments. % The cluster X-ray properties are consistent with little or no evolution when compared to local clusters. All three clusters were discovered in the Wide Angle {\sc Rosat} Pointed Survey (WARPS, \protect\citealt{sch97}). Details are given below on the X-ray properties of each cluster along with a summary of the infrared data obtained. \subsubsection{ClJ0152} \label{sec:0152} The cluster ClJ0152.7-1357 (\protect\citealt{ebe00}, \protect\citealt{del00}, \protect\citealt{mau03a}) is at a redshift of $z=0.833$ and has a bolometric X-ray luminosity of $1.6 \pm 0.2 \times 10^{45}$ ergs s$^{-1}$. It is composed of two major subclumps which are probably gravitationally bound (\protect\citealt{mau03a}). The separation of the subclumps is $\approx$1.5 arcmin (or 720 kpc), suggesting that they may be in the early stage of a very massive merger event. Individually the subclumps have X-ray temperatures and luminosities of $L_{\mathrm{X}}=1.0 \pm 0.2 \times 10^{45}$ ergs s$^{-1}$, $T_{\mathrm{X}}=5.5^{+0.9}_{-0.8}$ keV and $L_{\mathrm{X}}=5.8^{+1.1}_{-0.9} \times 10^{44}$ ergs s$^{-1}$, $T_{\mathrm{X}}=5.2^{+1.1}_{-0.9}$ keV respectively. Thus, each clump considered individually is still a massive cluster with $M_{\mathrm{total}}\approx 6 \times 10^{14}$ M$_{\odot}$ and $M_{\mathrm{total}}\approx 5 \times 10^{14}$ M$_{\odot}$ for the northern and southern clumps respectively (see \protect\citealt{mau03a}). \subsubsection{ClJ1226} ClJ1226.9+3332 (\protect\citealt{ebe01}, \protect\citealt{mau03b}, \protect\citealt{cag01}) is at a redshift $z=0.888$. It has a bolometric X-ray luminosity of $L_{\mathrm{X}}=5.3^{+0.2}_{-0.2} \times 10^{45}$ ergs s$^{-1}$, an X-ray temperature of $T_{\mathrm{X}}=11.5^{+1.1}_{-0.9}$ keV and $M_{\mathrm{total}}=1.4^{+0.2}_{-0.2} \times 10^{15}$ M$_{\odot}$ (\protect\citealt{mau03b}), making it one of the most massive clusters known at high redshifts. The cluster appears remarkably relaxed in X-rays indicating little dynamic activity. \subsubsection{ClJ1415} ClJ1415.1+3612 is at the highest redshift of the three clusters with $z=1.03$ (\protect\citealt{per02}). Although at present we have only {\sc ROSAT} X-ray data for the cluster so sub-structure may not be resolved, it exhibits smooth X-ray contours suggesting it may be dynamically relaxed. It has a bolometric X-ray luminosity of $L_{\mathrm{X}} = 2 \times 10^{45}$ ergs s$^{-1}$, and is thus likely to be a very massive cluster. \subsection{Observations.} \label{sec:klfobs} All three clusters were observed at the 3.8m United Kingdom Infrared Telescope ({\sc UKIRT}) on Mauna Kea, Hawaii using the {\sc UFTI} camera, a $1024 \times 1024$ pixel HgCdTe array, with a pixel size of 0.091 arcseconds. Observations were made with the K98 and J98 filters (50\% cut-offs 1.17--1.33$\mu$m and 2.03--2.37$\mu$m respectively) with an exposure time of 66s per frame (or 60s per frame for a minority of observations made in service time) and the seeing was typically $\approx 0.5$ arcseconds in $K$. The total exposure times in $K$ are listed in table~\ref{tab:calibration} along with the average seeing for each cluster. A spatial dithering of 15 arcsec was employed. Since the field of view of 90x90 arcsec covered only a fraction of each cluster, mosaics of different sizes for each cluster were used. On each night standard stars (\protect\citealt{haw01}) were observed in order to calibrate the zero-point, air mass coefficient, and colour coefficient. Some nights had patchy cloud cover, and thus yielded inaccurate photometric calibration. Such nights were deemed to be non-photometric, and a calibration was obtained via observations of the same fields on photometric nights. % The data from non-photometric nights could still be used by comparing the count rates of bright objects viewed through clouds with same objects observed on the photometric nights and applying a scaling to obtain the correct exposure. Non-photometric data were then weighted according to their shorter effective exposure times. Observations were also made of one or more offset fields in the vicinity of each cluster on the sky, chosen as a reasonably `blank' part of a digitised sky survey image (since at these redshifts, the clusters are also blank on digitised sky survey images). The purpose of these fields was to provide a means of estimating the contamination from foreground and background galaxies when constructing luminosity functions. The offset fields were typically 4 arcmin (1.9 Mpc) from the clusters. Observations of ClJ0152 were made in service time, on the nights of 15th and 20th October 2000, 24th December 2000, 28th December 2000 and 16th January 2001. % The cluster consists of two sub-clumps that appear to be in a state of merging, hereafter The northern and southern clumps will be referred to as field A and field B. The nights of 16th January, 24th December and 28th December were found to be non-photometric and were treated accordingly. The areas observed were $1'50'' \times 1'39''$ and $1'51'' \times 1'50''$ for fields A and B respectively, corresponding to a physical size of $836 \times 752$kpc and $843 \times 836$ kpc. ClJ1226.9+332 was observed on the 18th, 19th and 20th April 2001. The night of the 18th had patchy cloud cover and was deemed non-photometric. %calibrate the photometry independently. The observations covered an area of $2'47'' \times 3'1''$, corresponding to a physical size of $1.3 \times 1.4$ Mpc. Cl1415 was observed on the 10th, 11th, 12th April 2002. All nights were photometric with an average seeing in \emph{K} of 0.52''. The observations covered an area of $2'2'' \times 2'44''$, corresponding to an area of $1.0 \times 1.3$ Mpc. For this cluster, with the highest redshift, two offset fields were observed in order to reduce the uncertainties in the field subtraction. For the other clusters, one offset field was observed. \subsection{Data Reduction.} \label{data:reduction} Each observation of a particular field consisted of a number of dithered frames of the object plus a dark frame containing only the instrumental background. The data were dark-subtracted, flat-fielded and combined %reduced using standard {\sc iraf} procedures. A bad-pixel mask was applied to exclude any defective pixels. Flat-field frames were produced from the object frames, using the median vlue of each pixel %was achieved by dividing each object frame by a flat-frame, which in turn was produced by combining the median value of each object frame after applying a 4$\sigma$ clipping and scaling each frame by its modal value to account for sky brightness variations. Each frame had its background sky value subtracted, using a $3\sigma$-clipped mean as an estimate of the sky value. The relative offsets of each frame were then measured using bright stars in the field of view. Finally the frames were shifted and combined using the mean value of each pixel, again after applying a 4$\sigma$ clipping around the median value. %No %scaling was used in combining the frames as they had already been %background subtracted and were all at the same exposure. % MAYBE STICK THIS BACK IN FOR THESIS % %The data for each object were reduced using %the following procedure. %\begin{itemize} %\item The dark frame was subtracted from each object frame in order to eliminate the %contribution due to instrumental noise. %\item A bad pixel mask was applied to each object frame in order to exclude any %defective part of the UFTI CCD array from the reduction. %\item The object frames were combined using the median value of each pixel in %order to create a flat frame. As part of this process a sigma-clipping %routine was applied using a value of 4$\sigma$ to exclude any unusually %high or low pixel values and each frame was scaled by the modal value of %its pixels. %\item The flat frame was normalised by taking its mean value and then using it as a divisor on %the flat frame. %\item Each object frame was then divided by the normalised flat, thus removing %the effects of varying sensitivities of pixels across the CCD. %\item The background due to the brightness of the sky in the IR was subtracted %from each frame. This was estimated by measuring the mean and standard %deviation of each frame and then excluding anything greater than 3$\sigma$ %either side of the frame and retaking the mean --- this value was then %used as an estimate of the sky. %\item Finally the frames were shifted and combined using the mean value of each pixel, again %after applying a 4$\sigma$ clipping around the median of each frame. No %scaling was used in combining the frames as they had already been %background subtracted and were all at the same exposure. The relative offsets of each frame were measured %using bright stars in the field. %\end{itemize} %All stages of this procedure were made using the {\sc iraf} software package. \section{Photometry.} \label{photometry} \subsection{Standard stars.} \label{standards} The photometry of the galaxies was calibrated using standard stars with well determined magnitudes, observed on the same nights as the cluster fields. The true magnitude of any celestial body is taken to be \begin{equation} m_{{\rm true}} = ZP - 2.5 {\rm log} (CR) + A{\rm sec}z + B(J-K)_{{\rm true}} \label{eqn:mag_klf} \end{equation} where $ZP$ is the zero-point magnitude, $CR$ is the count rate, $A$ is the coefficient of extinction per unit airmass, $z$ is the zenith angle (sec$z$ therefore being the airmass) and $B$ is the colour coefficient necessary due to differences in the response of the combination of camera and filters used and the standard star system. %true colour, given by $(J-K)_{{\rm true}}$. %Standard stars of varying colour and at varying airmasses were observed and %for each one a value of $2.5 {\rm log} \left( {\rm \frac{counts}{exposure}} \r%ight)$ was measured using the {\sc qphot} task % within {\sc iraf}. %The values of $ZP$, $A$ and $B$ were calibrated using the standard stars %from a particular night and in a particular band as %follows. %A plot of %$M_{{\rm true}} + 2.5 {\rm log} \left( {\rm \frac{counts}{exposure}} \right) %- A{\rm sec}z$ vs.\ $ (J-K)_{{\rm true}}$ %will give a straight line with an %intercept equal to $ZP$ and a slope equal to $B$. %Similarly a plot of %$M_{{\rm true}} + 2.5 {\rm log} \left( {\rm \frac{counts}{exposure}} \right) - B(J-K)_{{\rm true}}$ vs.\ ${\rm sec}z$ %will give a straight line with intecept $ZP$ and slope $A$. %Therefore by applying a sensible staring values for $A$ (such as the avearge values %see e.g. {\scriptsize {\sf % http://www.jach.hawaii.edu/JACpublic/UKIRT/astronomy/exts.html}} or %\citealt{kri87}) $B$ may be determined, and using this value of $B$ a more %accurate value for $A$ may be heralded. This procedure can be iterated %until stable values of $A$ and $B$ are found. The zeropoints were found by %averaging the value of %\begin{equation} %M_{{\rm true}} + 2.5 {\rm log} \left( {\rm \frac{counts}{exposure}} \right) - B(J-K)_{{\rm true}} - A{\rm sec}z %\end{equation} Values of $ZP$, $A$ and $B$ were determined from observations of standard stars taken on each night. Standard stars were selected from the \emph{UKIRT} faint standards list (\protect\citealt{haw01}). Typically each night 6 stars of varying colours were observed between 1 and 3 times each at differing airmasses. Whilst the value of $A$ and $ZP$ may vary from night to night, the value of the colour coefficient $B$ should be almost constant. Thus a single value of $B$ was measured from all the photometric nights combined. The value we obtained is consistent with zero, as expected from the filter design (\protect\citealt{tok02}). An independent value of $ZP$ was measured for each night, and for most nights, an independent value of $A$ was measured. Where this was not possible a value of $A$ %It was not always possible to fit slopes for each night to determine $A$ from other nights was used and checked for consistency. All were found to be consistent. In any case the low extinction in the K band and the low airmasses at which most observations were performed combine to make this a relatively small correction ($<$0.2 mag). %Values of $A$ and $ZP$ were determined separately for each cluster's set of observations where as $B$ was determined from an average of all the observations (after checking that it was consistent between them all). The parameters are listed in table~\ref{tab:calibration}. Note that the final photometric error, as estimated from the standard stars, is less than 0.1 mag.% (see table~\ref{tab:calibration}). \begin{table*} \centering \caption{Calibration parameters for $K$ band.} \begin{tabular}{lllll} & ClJ0152A & ClJ0152B & ClJ1226 & ClJ1415 \\ \hline $A$ & $-0.150^{+}_{-}0.110$ &$-0.150^{+}_{-}0.110$ & $-0.178^{+}_{-}0.052$ & $-0.088$ \\ $B$ & \multicolumn{4}{c|} {$-0.0043^{+}_{-}0.094$} \\ $ZP$ &22.248$\pm 0.018$& 22.248$\pm 0.018$ & $22.367\pm 0.008$ & 22.422$\pm 0.044$ \\ Mean sec$z$ & 1.445 & 1.403 &1.210 & 1.160 \\ Exposure/ s & 3882 & 3942 & 23946 & 17820 \\ Seeing/ arcsec & 0.53 & 0.53 & 0.57 & 0.52 \\ \label{tab:calibration} \end{tabular} \end{table*} \subsection{Galaxies.} \label{galaxies} The {\sc SExtractor} software of \citet{ber96} was used to search for objects in each field. To detect objects a threshold value per pixel must be chosen along with a minimum number of connected pixels. Because the final images are constructed from a jittered pattern of images, the depth of observation varied across the final image, being at its deepest in the centres and shallowest at the edges. Therefore detection of objects was done in two regions for each image, using different detection parameters, in order to get as deep as possible in the centre of the image whilst avoiding spurious detections at the edges of the image. Typically the significance of the detections were 3.5$\sigma$ or 4$\sigma$ in the centres of the images and $7\sigma$ at the edges, where $\sigma$ is the background RMS determined from counts over the whole image. (This is not the case for the central region of ClJ1415, when $\sigma$ was determined from the central region alone). Reliability of the object detection, and in particular the handling of overlapping objects, was checked by eye. Faint pixels surrounding deblended objects are assigned to one of the sub-objects with a probability based on the expected contribution at that pixel from each of the deblended objects (see \protect\citealt{ber96}). Counts were measured using an adaptive aperture based on Kron's algorithm (\citealt{kro80}), and also in a circular aperture with radius chosen to maximise the signal to noise. For each object a value of stellarity was also measured (see \citealt{ber96}). To determine the magnitudes of the objects equation~\ref{eqn:mag_klf} was used with the following complication. It is unknown to start with what the true colours of the objects are, therefore the magnitudes cannot be determined. To circumvent this problem an approximate magnitude was measured in the \emph{J} and \emph{K} bands neglecting the colour term in equation~\ref{eqn:mag_klf}. This allowed an approximate colour to be determined. The approximate colour is then multiplied by a correction factor previously determined using the same technique to measure the approximate colours of standard stars and their true colours. The average value of $\frac{(J-K)_{{\rm true}}}{(J-K)_{{\rm approx}}}$ was 0.982. Note that for the determination of colours, magnitudes derived from fixed, circular apertures were used, whereas the adaptive aperture magnitudes were used to derive pseudo-total magnitudes. The use of adaptive aperture magnitudes using {\sc SExtractor}'s Kron radius should avoid the problem described by \citet{and02} of underestimating fluxes for galaxies with low central surface brightness. The overall reliability of the photometry was checked by comparing the field galaxy counts derived from the offset fields with deeper published results (see fig.~\ref{fig:fieldcounts}). There is generally good agreement down to our limiting magnitudes (see below). %\subsubsection{Star-galaxy distinguishment} Star--galaxy discrimination was determined using {\sc SExtractor's} stellarity parameter. A cut-off of 0.8 was selected to delimit the two classes of objects with those objects with values greater than 0.8 being excluded as stars. This value was confirmed by examination of the radial profiles of detected objects. It was found that objects with a stellarity greater than 0.8 had an almost constant Gaussian FWHM close to the value of the seeing, whereas objects with stellarity less than 0.8 had more extended profiles. The results are insensitive to the precise value of this cut-off as there were very few stars in each field. %To decide which value of stellarity was appropriate to distinguish stars %from galaxies a plot of stellarity vs. magnitude was made, The stars in this plot occupy a line across %the top and a cut-off of 0.8 was elected to delimit the two classes of objects. %\begin{figure} %\begin{minipage}{12cm} %\psfig{file=stellarity.eps,width=8.5cm,height=6.5cm,angle=270} %\caption{Stellarity as a function of magnitude.} %\label{fig:stellarity} %\end{minipage} %\end{figure} \section{Luminosity functions.} \label{sec:klf} \emph{K} band luminosity functions (KLFs) were calculated separately for each cluster of galaxies. This was done in the following way. Firstly, some of the brightest galaxies have been spectroscopically confirmed as cluster members, so where there are no other bright galaxies in the field, no subtraction is necessary. At fainter magnitudes, the number of galaxies, $N_{\mathrm{cl}}$, in a bin of width $\Delta M$ was counted, with $\Delta M=1$ for ClJ0152 and ClJ1415 and $\Delta M=0.75$ for ClJ1226. The contamination from foreground and background galaxies was estimated by counting the number of galaxies, $N_{\mathrm{back}}$ in the offset field(s) associated with the cluster. Thus the number of galaxies in each bin is given by, \begin{equation} \label{eqn:nbin} N_{\mathrm{bin}} = N_{\mathrm{cl}}-N_{\mathrm{back}} \times \left( \frac{A_{\mathrm{cl}}}{A_{\mathrm{back}}} \right) \end{equation} where $A_{\mathrm{cl}}$ is the area of the cluster field and $A_{\mathrm{back}}$ is the area of the offset field. Note that because two different detection thresholds have been used for each field the faintest bins are from a smaller area than the brighter bins (since only a small area was deep enough to detect objects of this magnitude). Therefore the detections from the smaller area have been scaled by an appropriate factor to bring them into line with the detections from the total area. The corresponding error on the number of galaxies in each bin is found by summing in quadrature the error on $N_{\mathrm{cl}}$ and the error on $N_{\mathrm{back}}$, \begin{equation} \label{eqn:errorn_bin} \sigma_{\mathrm{Nbin}} = \sqrt{\sigma_{\mathrm{Ncl}}^{2} + \sigma_{\mathrm{Nback}}^{2}} \end{equation} where $\sigma_{\mathrm{Ncl}}$ is the Poissonian error (\protect\citealt{geh86}). The background error, $\sigma_{\mathrm{Nback}}$, has a Poissonian term and a term taking into account the clustering properties of galaxies, viz., \begin{equation} \label{eqn:error_back} \sigma_{\mathrm{Nback}} = \sqrt{N_{\mathrm{back}}} \times \sqrt{1+\frac{2\pi\it{N}A_{\omega}\theta_{\mathrm{c}}^{2-\delta}}{2-\delta}} \end{equation} where $\it{N}$ is the number density of galaxies in the bin and $\theta_{\mathrm{c}}$ is the angular radius such that $\Omega=\pi\theta_{\mathrm{c}}^{2}$ where $\Omega$ is the solid angle of the background field. The parameters $A_{\omega}$ and $\delta$ describe the angular correlation function of galaxies such that, \begin{equation} \label{eqn:ang_corr} \omega(\theta)=A_{\omega}\theta^{\delta}. \end{equation} (\protect\citealt{pee80}, pg.175). The value of $\delta=-0.8$ was used and $A_{\omega}$ was found using the formula \begin{equation} \label{eqn:aomega} \mathrm{log}_{10}A_{\omega}=7.677 - 0.3297r_{\mathrm{lim}} \end{equation} derived by fitting to the data in table~1 of \citet{bra95}. Following \citet{djo95} we used $r_{\mathrm{lim}}-K_{\mathrm{lim}}\sim 3$. Our limiting \emph{K} magnitude was determined by plotting the number density of galaxies from the offset fields as a function of magnitude and determining the drop-off point when compared to large area field surveys. Figure~\ref{fig:fieldcounts} shows the number density of galaxies in our four offset fields.ClJ1226 and ClJ1415 were found to be complete to $K_{\mathrm{lim}} = 21.5$ and ClJ0152 was found to be complete to $K_{\mathrm{lim}} = 20.0$. \begin{figure} \centerline{\psfig{file=klf/field2.eps,width=0.8\columnwidth,angle=270}} \caption{Number density of field counts as a function of magnitude. Open circles are from the combined offset fields of ClJ1226 and ClJ1415, open squares are from the offset field of ClJ0152, closed circles from \protect\citet{hua97}, squares from \protect\citet{sar97} and stars from \protect\citet{djo95}.} \label{fig:fieldcounts} \end{figure} Luminosity functions for the three clusters are shown in figure~\ref{fig:klf0152}, \ref{fig:klf1226} and \ref{fig:klf1415} along with the best fitting Schechter functions as described below. \begin{figure} \centerline{\psfig{file=klf/0152klf.eps,width=0.8\columnwidth,angle=270}} \caption{Binned \emph{K} band luminosity function of ClJ0152. Also shown is the best fitting Schechter function. The solid line excludes the BCGs (one from each sub-clump)and the dashed line includes the BCGs. The open symbol (offset slightly for clarity) shows the effect of excluding the BCGs from each sub-clump.} \label{fig:klf0152} \end{figure} \begin{figure} \centerline{\psfig{file=klf/1226klf.eps,width=0.8\columnwidth,angle=270}} \caption{\emph{K} band luminosity function of ClJ1226. The best fitting Schechter functions including (dashed line) and excluding (solid line) the BCG are shown. } \label{fig:klf1226} \end{figure} \begin{figure} \centerline{\psfig{file=klf/1415klf.eps,width=0.8\columnwidth,angle=270}} \caption{\emph{K} band luminosity function for ClJ1415. Schechter function fits are as for Fig \protect\ref{fig:klf1226}.} \label{fig:klf1415} \end{figure} \subsection{Fitting Schechter functions.} The parametric \citet{sch76} luminosity function is of the form, \begin{equation} \label{eqn:schechter_lum} \frac{\mathrm{d}\phi}{\mathrm{d}L}\mathrm{d}L=\phi^{*^{'}}(\frac{L}{L^{*}})^{\alpha}e^{-(\frac{L}{L^{*}})}\mathrm{d}(L/L^{*}) \end{equation} where $L^{*}$ is the characteristic luminosity at which the function turns over to the faint end which has slope $\alpha$. The normalisation of the function is given by $\phi^{*}$. The corresponding function in terms of \emph{K} band magnitudes is \begin{equation} \label{eqn:schechter_mag} \mathrm{d}\phi=\phi^{*}10^{0.4(K^{*}-K)^{\alpha+1}}e^{-10^{0.4(K^{*}-K)}}\mathrm{d}K \end{equation} where $K^{*}$ is the characteristic \emph{K} band magnitude. Note that $\phi^{*}$ in equation~\ref{eqn:schechter_mag} is not identical to the $\phi^{*^{'}}$ in equation~\ref{eqn:schechter_lum}, but in fact differs by a factor $-0.4\mathrm{ln}10$. Equation~\ref{eqn:schechter_mag} was fit to the KLFs of all 3 clusters using the maximum likelihood technique of \citet{cas79}. It was found that $\alpha$ could not be constrained and therefore, following \citet{dep99} we fix at $\alpha = -0.9$ which is the value for the KLF of the Coma cluster (\protect\citealt{dep98}) and of the field (\protect\citealt{gar97}). The characteristic magnitude, $K^{*}$, was fit as a free parameter and $\phi^{*}$ was determined by ensuring that the integral of the function down to the limiting magnitude was equal to the total number of galaxies to the same magnitude. It has been noticed since early studies of cluster luminosity functions that the brightest cluster galaxies are a special class of object (\protect\citealt{pea69}) and that a better fit to the luminosity function can often be made by excluding them from the fit (\protect\citealt{sch76}, \protect\citealt{dre78}). Therefore Schechter functions were fit both including the BCGs and excluding them. Note that in the case of ClJ0152 we have excluded two BCGs, one from each sub-clump of the system. The best fitting parameters are given in table~\ref{tab:schechter_fits}, along with the $\chi^{2}$ probability. The best fitting Schechter functions, including and excluding the BCGs, are also shown in figures~\ref{fig:klf0152},\ref{fig:klf1226} and \ref{fig:klf1415}. \begin{table*} \centering \caption{Best fitting parameters of a Schechter function for each cluster with $\alpha=-0.9$.} \label{tab:schechter_fits} \begin{tabular}{lllllll} & \multicolumn{3}{c}{Including BCGs} & \multicolumn{3}{c}{Excluding BCGs} \\ & $K^{*}$ (lower limit, upper limit) & $\phi^{*}$ & Prob($\chi^{2}$) & $K^{*}$ (lower limit, upper limit) & $\phi^{*}$ & Prob($\chi^{2}$) \\ \hline Cl0152 & 17.59 (17.26,17.90) & 47.89 & 0.57 & 17.76 (17.43,18.07) & 50.31 & 0.81\\ Cl1226 & 17.79 (17.52,18.03) & 57.37 & 0.01 & 18.04 (17.79,18.28) & 61.01 & 0.03 \\ Cl1415 & 17.96 (17.63,18.26) & 42.53 & 0.96 & 18.08 (17.75,18.38) & 43.80 & 0.78\\ Combined & 17.81 (17.51,18.07) & 44.69& 0.07 & 17.98 (17.68,18.26) & 45.59 & 0.16 \\ \end{tabular} \end{table*} This is not apparent from the results listed in table~\ref{tab:schechter_fits} in which good fits are found in all cases for every cluster except ClJ1226. For the case of ClJ1226 fits are poor regardless of whether the BCG is included or not, this is due to the high number count in the second faintest bin. A visual inspection of figure~\ref{fig:klf1226} would suggest that the BCG is not well fit by a Schechter function. \subsection{Evolution of $K^{*}$} To evaluate the evolution in $K^{*}$, models of galaxy evolution were made using the synthetic stellar population (SSP) libraries of \citet{bru03}. The parameters of the model are the star formation history, the initial mass function and the metallicity. For all models considered here a single burst of star formation was assumed to occur (but at different epochs), the initial mass function was that of \citet{sal55} and the metallicity was assumed to be solar (i.e.\ $Z=2\%$). Models were constructed for passive evolution in the following way. Spectral energy distributions (SEDs) were drawn from the \citet{bru03} libraries for different ages. A redshift of formation was chosen, corresponding to an age $t=0$. Then for the adopted cosmology redshifts were calculated for the rest of the ages. The SEDs were redshifted to the appropriate redshift for each age and multiplied by the filter transmission curve. Thus the flux of the SSP could be calculated for each redshift. The fluxes were converted to apparent magnitudes and scaled appropriately, e.g.\ for the evolution of $K^{*}$ the models were normalised to $K^{*}$ of the Coma cluster (see below). No-evolution models are made by simply choosing an SED of a particular age and integrating the redshifted spectrum under the filter transmission curves. Passively evolving models with redshifts of formation $z_{\rm{f}}=1.5$, $z_{\rm{f}}=2$ and $z_{\rm{f}}=5$ were made along with a no evolution model of a 10Gyr old population. The results are shown in figure~\ref{fig:klf_evol}. The models were scaled to the Coma cluster with $K^{*}=10.9$ at $z=0.0231$ (\protect\citealt{dep99}). \begin{figure} \centerline{\psfig{file=klf/kstar.eps,width=0.8\columnwidth,angle=270}} \caption{The evolution of $K^{*}$. The circles are data from this paper. The squares are from \protect\citet{dep99}, open symbols being low $L_{\mathrm{X}}$ systems and closed symbols being high $L_{\mathrm{X}}$ systems.} \label{fig:klf_evol} \end{figure} It is clear that the trend seen in \citet{dep99} is followed here and extended to higher redshifts. All values of $K^{*}$ are brighter than predicted by no evolution, and are consistent with passive evolution models. The high redshift points seem most consistent with $z_{f}\simeq 1.5$ although the errorbars are large enough to allow a range of $z_{f}$. %Note that a redshift of formation of five is consistent with recent observations of the Gunn-Peterson effect (e.g.\ \protect\citealt{bec01}) which may be interpreted as due to galaxy formation at $z_{f}\sim 5$. Although it was not possible to constrain $\alpha$ for any individual cluster, in order to verify the choice of $\alpha=-0.9$ we have combined the counts from all three clusters into a general luminosity function and fitted a Schechter function leaving $\alpha$ as a free parameter. The combined luminosity function was generated by finding a weighted average of the counts in each bin of the individual clusters where the weighting factor was the reciprocal of $\phi^{*}$. In fact the $\phi^{*}$ values for each cluster are similar, so the weighting has little effect. The combined luminosity functions, excluding and including BCGs, with the best fitting Schechter functions are shown in figures~\ref{fig:klfcomb} and \ref{fig:klfcomb_wbcg} respectively. The best fitting values were $K^{*}=18.53$ and $\alpha=-0.54$, when the BCGs were not included and $\alpha=-0.94$ and $K^{*}=17.83$ when they were. The confidence limits, excluding BCGs, are shown in figure~\ref{fig:klfcombconf} where it can be seen that the constraint on $\alpha$ is not strong, ranging from $\alpha \approx -0.15$ -- $-0.9$. There is a large degree of degeneracy between $K^{*}$ and $\alpha$ as is evident from the shape of the contour in figure~\ref{fig:klfcombconf}. Consequently when fitting a Schechter function, $K^{*}$ may be misleading if the faint end slope is poorly constrained. To quantify this effect the combined luminosity function was fit with $\alpha=-0.9$. The results are given in table~\ref{tab:schechter_fits}. The result (excluding BCGs) is a brighter value of $K^{*}$, as expected from the shape of figure~\ref{fig:klfcombconf}. Consequently, if $\alpha\approx -0.54$ for high redshift clusters then the evolution in $K^{*}$ may not be as strong as seen in figure~\ref{fig:klf_evol}, and values of $z_f\approx 2-5$ would be more consistent with the $K^*$ measurements. However, if the last bin of the combined KLF of figure \ref{fig:klfcomb} is excluded then $\alpha=-0.99 \pm 0.4$. Thus, it is unclear whether the measured value of $\alpha=-0.54$ is in artefact due to incompleteness in the faintest bin, and therefore we elect to use the value of $\alpha=-0.9$ for the individual fits to clusters. \subsection{Comparison with the Coma cluster} The $K$ band luminosity function of the Coma cluster, shown in figure~\ref{fig:klfcomb} was derived from the $H$ band luminosity function of \citet{dep98} using their given value of $H-K=0.22$ and to a magnitude limit for which all galaxies had spectroscopic membership confirmation. The Coma luminosity function of \citet{and00} covers a smaller area but is in good agreement with that of \citet{dep98} (except for a dip in one bin). For the high redshift clusters, the conversion from apparent to absolute magnitude was made using %$M_{K}=K-5{\rm log}D_{L}+5$, where $D_{L}$ is the luminosity distance, and then a \emph{k}-correction calculated for a 10Gyr old synthetic stellar population from the libraries of \citet{bru03}. The typical size of the \emph{k}-correction was $\sim0.7$ mag. Corrections due to Galactic extinction are negligible and were not applied. The offset of $K\approx$1.2 mag between the Coma KLF and the combined high redshift KLF indicates the degree of evolution. The same degree of evolution is also seen in Fig 5 as the difference between the observed values of $K^*$ and the no-evolution predictions. % in magnitude between the two luminosity functions is in agreement with the offset between the data points and the no evolution model of figure \ref{fig:klf_evol}. The predicted evolution of the Schechter function, assuming a passively evolving model formed at $z_{\rm{f}}=2$, is shown by the dotted line in figures~\ref{fig:klfcomb} and \ref{fig:klfcomb_wbcg}. This appears to be a reasonable fit to the Coma luminosity function suggesting most differences between the systems, at least at the bright end of the luminosity function, can be accounted for by passive evolution, as there is no obvious change in the shape of the functions. \begin{figure} \centerline{\psfig{file=klf/klfcomb2.eps,width=0.8\columnwidth,angle=270}} \caption{The combined luminosity function of all three clusters shown by solid circles, excluding BCGs. Its best fitting Schechter function is the solid line, whilst the dotted line shows the expected evolution at the redshift of Coma assuming passive evolution and $z_{\rm{f}}=2$ . The number of galaxies in each bin is a weighted average. The Coma $K$ band luminosity function derived from \protect\citet{dep98} is shown by open squares. See text for details} \label{fig:klfcomb} \end{figure} \begin{figure} \centerline{\psfig{file=klf/klfcomb_conf2.eps,width=0.8\columnwidth,angle=270}} \caption{Confidence limits at 68 \% on the best fitting parameters for the combined luminosity function when BCGs are excluded.} \label{fig:klfcombconf} \end{figure} \begin{figure} \centerline{\psfig{file=klf/klfcomb_wbcg.eps,width=0.8\columnwidth,angle=270}} \caption{The combined luminosity function of three clusters including BCGs. Symbols are as in figure~\ref{fig:klfcomb}.} \label{fig:klfcomb_wbcg} \end{figure} \subsection{Integrated light functions.} \label{sec:fracl} An interesting and complementary measure of galaxy evolution within clusters is provided by their integrated light function. This is calculated from the luminosity function using %\newfont{\afont}{cmdunh10} \begin{eqnarray} \label{eqn:fracl} %{\rm d} {\sf IL}( 1.9 \times 10^{44}$ ergs s$^{-1}$ 0.3 -- 3.5 keV) BCG photometry of \citet{bro02}. The high $L_{\mathrm{X}}$ clusters from figure~1 of \citet{bro02} are shown as the open circles. The BCGs in the current study, shown as solid points, extend to higher redshifts. Note that there are two BCGs shown for ClJ0152, one for each sub-clump. The magnitudes of the BCGs were measured in a metric aperture of radius, $r_{\mathrm{m}}=12.5h^{-1}$kpc for comparison with \citet{bro02}. \begin{figure} \centerline{\psfig{file=klf/khubble.eps,width=0.8\columnwidth,angle=270}} \caption{\emph{K} band Hubble diagram for BCGs. Solid points are from this paper, hollow points are from \protect\citet{bro02}.} \label{fig:khubble} \end{figure} It can be seen from figure~\ref{fig:khubble} that the BCGs in ClJ1226 and ClJ1415 are somewhat brighter than no-evolution predictions but in good agreement with passive evolution models. The BCGs of ClJ0152 are fainter relative to the models than any other BCGs in figure~\ref{fig:khubble} at $z>0.4$, perhaps indicative that the BCGs are not fully formed. Indeed as ClJ0152 is probably a merging system (\protect\citealt{mau03a}) it may be supposed that if the BCGs merge they will produce a brighter BCG more consistent with the passive evolution predictions. In contrast ClJ1226 has a relaxed X-ray morphology (\protect\citealt{mau03b}, \protect\citealt{ebe01}), and is one of the brightest BCGs at $z>0.4$ relative to the models. % The other bright high redshift BCG (at z=0.55, $K$=14.9 from Brough et al),is also in a relaxed cluster ???? These results are suggestive that the evolution of BCGs at high redshifts may be related to the dynamical state of the cluster they inhabit, although confirmation awaits a larger sample. Table~\ref{tab:bcg_fracl} lists the contributions from the BCG to the total $K$ band light for each cluster. The total luminosity in each cluster was calculated from the luminosity functions, as for the integrated light functions, to a depth of $L/L^* \approx 0.06$. In the case of Coma a correction was made to account for the smaller fraction of the virial radius observed. This was calculated using the number density profile of \citet{ken82} to determine the ratio of the number of galaxies within the observed area to that within $0.37r_{{\rm vir}}$, the average area observed for the high redshift sample. The correction was a factor of 1.96. The luminosity of the Coma BCG, NGC4874, was calculated using the 2MASS total magnitude $K_{{\rm S}}=8.86$, where we have assumed that the difference in wavebands is negligible. Note that there are two galaxies of similar brightness in Coma, NGC4874 and NGC4884, we have taken NGC4874 as the BCG here as it resides at the centre of the X-ray emmision. \begin{table} \centering \caption{Fraction of the light in the BCGs to total cluster light. The values for the BCGs in ClJ0152 are the fractions in each sub-clump independently.$^{*}$ extrapolated from 0.17, see text. } \label{tab:bcg_fracl} \begin{tabular}{lll} Cluster & Fraction & Area/ $r_{\rm{vir}}$\\ \hline CLJ0152A & 0.12 & 0.29\\ ClJ0152B & 0.10 & 0.34\\ ClJ1226 & 0.16 & 0.41\\ ClJ1415 & 0.11& 0.43\\ Coma & 0.06 & 0.37$^{*}$ \end{tabular} \end{table} The results show that the fraction of light in the BCGs at high redshift is greater than, or equal to, the fraction in Coma. %The fractions of light in the ClJ0152 BCGs are the smallest, but %considered together would be similar to that of Coma. Note also that %the total luminosity is that of the whole cluster, i.e.\ the two %sub-clumps are considered together. If the sub-clumps were considered %separately the fraction of light in the BCGs would be roughly twice %the value listed. The subclumps in ClJ0152 have been treated independently, since each has a BCG. %****These results show that the reason the BCGs in ClJ0152 are fainter % than passive evolution predictions is indeed because they contain % a smaller fraction of the cluster light, rather than because, for example, the % cluster itself is of low mass. All three clusters have a large fraction of light in their BCGs compared to that of Coma. This may be due in part to uncertainties in the corrections applied in order to compare the clusters which were originally measured to slightly different depths and covered different areas, although the variations in radius would produce only small variations in total $K$ band light ($\sim$20\% using the Coma light profile). A physical interpretation is that the high redshift BCGs are already of similar or greater mass to that of Coma with no need to hypothesise any significant mergers in their future evolution. Note that if the BCGs in ClJ0152 were to merge, as was hypothesised to explain their positions in the Hubble diagram, the fraction of light in the BCG would remain roughly similar, for although the expected luminosity of the BCG would be approximately twice the value of each galaxy as measured, the total luminosity of the cluster would also approximately double. Thus their positions in the Hubble diagram may be regarded as being due to their location in rather less luminous clusters. Note that ClJ1226 and ClJ0152 are very massive clusters and ClJ1415 is also very X-ray luminous and therefore likely to be massive. Galaxies in such regions of high overdensity may have a significantly earlier epoch of assembly than in Coma, according to hierarchical models of structure formation. %Thus the galaxies in these clusters would be expected to be very old. \section{Discussion and conclusions} \label{sec:dicuss} The evolution of the galaxy populations of three high redshift clusters of galaxies has been studied. The bulk evolution of the galaxies, as characterised by $K^{*}$, is found to be consistent with passive evolution with a redshift of formation $z_{\mathrm{f}}\sim\ 1.5$--2. Further evidence for passive evolution is seen in the similarity of the shape of the high-redshift luminosity function with that of Coma, and in the consistent shapes of the integrated light functions. Purely passive evolution of early-type galaxies is consistent with several other studies including the evolution of the $K$ band luminosity function (\protect\citealt{dep99}), evolution of the fundamental plane in terms of mass-light ratios (\protect\citealt{van98}), and studies of the scatter of the colour-magnitude relation (see e.g.\ \protect\citealt{ell97}, \protect\citealt{sta98}). %Note also that a high redshift of %formation such as %$z_{\rm{f}}=5$ is consistent with redshifts of formation inferred from recent observations of the Gunn-Peterson effect %(\protect\citealt{bec01}). %A high redshift of formation of \emph{the stars in} early-type galaxies is consistent with evoutionary studies based on the fundamental plane. The work of \citet{van98} on the evolution of the mass to light ratio to $z=0.83$ indicates a redshift of formation of $z_{f}>2.8$ for $\Omega_{\rm{M}}=0.3$, $\Omega_{\rm{\Lambda}}=0$. Similarly studies of the scatter of the colour-magnitude diagram indicate $z_{\rm{f}}>3$ (see e.g.\ \protect\citealt{van01b}). When discussing formation it is important to distinguish between the epoch at which the stars in the galaxies were formed and the epoch at which the galaxies were assembled. The studies of the fundamental plane and the colour-magnitude relation refer to the epoch of star formation. If merging were a dissipationless process then it would be possible to have no extra star formation as a result of a merger and thus the age of the stars within a galaxy could be older than the age of galaxy assembly. A study of the cluster of galaxies MS 1054-03 at $z=0.83$ is presented by \citet{van99} in which there is observed a high fraction of merging red galaxies. Very little star formation is seen in the merging galaxies constituting evidence that the galaxies are in fact somewhat younger than the stars that reside within them. %The increase of spiral-fraction with redshift (\protect\citealt{dre97}, \protect\citealt{cou98}, \protect\citealt{van00}, \protect\citealt{van01a}) and the Butcher-Oemler effect of increasing blue fraction (\protect\citealt{but78}, \protect\citealt{but84}, \protect\citealt{fai02}) also provide evidence for evolution of morphological type. Is such merging reflected in the evolution of the luminosity function? The $K$ band luminosity of a galaxy is very nearly independent of star-formation, but reflects the mass of the old stars within the galaxy. Thus $K$ magnitudes are a good measure of the stellar mass of a galaxy. \citet{dia01} give predictions of the evolution of the KLF from semi-analytic models of dissipationless, hierarchical structure formation. The models show that there is very little evolution of the number of massive galaxies in clusters since $z=0.8$. The galaxies are assembled at high redshift and evolve passively with little subsequent merging after $z=0.8$. A detailed comparison with these models is however not possible because the degree of luminosity evolution in the models is not sufficiently accurate for massive galaxies (Diaferio et al 2001). It is clear from figures \ref{fig:klf_evol} and 6 that the bulk of galaxies in our sample were brighter in the past than predicted from no-evolution models, by $K \approx$ 1.2 mag at $z=0.9$. A direct comparison of the high-redshift $K$ band luminosity function with that of Coma suggests that the two are very similar in shape and the differences in $M_{K}^{*}$ may be reconciled by pure luminosity evolution. The fading with time by $K \approx$ 1.2 mag is consistent with passive evolution from a formation epoch $z_{f}\approx 2$. In the monolithic collapse picture this would be expected as the galaxies have always been present but were intrinsically brighter in the past due to their stars being younger. The models of \citet{dia01} show that this is also the case for massive cluster galaxies in a hierarchical scenario since most merging takes place early on the history of the cluster. %Since in a merging model the bright end of the luminosity function would show a deficit of bright galaxies resulting in a steeper slope (see fig. 3 of \citealt{kau98b}) and hence a dimmer $K^{*}$ as compared to a no-evolution model, in order to reconcile the result of \ref{fig:klf_evol} there would also need to be a significant amount of passive evolution to brighten the galaxies and hence $K^{*}$. In a merging model passive evolutionary processes will still be present, and thus $K^{*}$ would still appear brighter than no-evolution predictions. A probe of `extrapassive' processes is the shape of the luminosity function. It is found that $\alpha$ is consistent with that of Coma although it is poorly constrained here. Perhaps a stronger test for extrapassive processes is the shape of the integrated light function. We remove the effects of pure luminosity evolution to investigate any changes in the distribution of light with galaxy luminosity, such as would be caused by merging. %The cumulative fraction of light in the cluster is normalised to the %total light, and the luminosity of the galaxies are normalised to %$L^{*}$. Thus any evolution that solely affects the luminosity of %the galaxies is removed from the plot, and all that remains is any %evolution that affects the number counts of galaxies within each bin, %such as merging. The lack of any major changes seen in figure~\ref{fig:fracl} is suggestive that passive evolution alone is responsible for the evolution measured in $K^{*}$, when the BCGs are excluded. We conclude that the luminosity evolution of bright galaxies in massive clusters is consistent with pure passive evolution, but note that this may be consistent with hierarchical models if most merging takes place at high redshifts. %\emph{Because we have constructed our luminosity function in the $K$ band it is relatively insensitive to star formation, and reflects more strongly the stellar mass of the galaxies. The merging of two galaxies will thus result in a brighter galaxy. If merging happened preferentially between more massive galaxies, as a result of e.g.\ dynamical friction, it is possible that a break in the luminosity function could result, such as the break often seen between first and second ranked cluster galaxies. If merging happens between galaxies of all masses then the shape of the luminosity function would not necessarily be altered although its normalisation and $K^{*}$ would be, with $K^{*}$ being shifted to brighter magnitudes as the number of massive galaxies increased. For the case of MS 1054-03 it is seen that the mergers are preferentially found in the outskirts of the cluster and are most probably taking place in cold infalling sub-clumps (\protect\citealt{van99}). \emph{If} this is a typical result and we can assume that mergers do not happen preferentially between systems of a given mass then it would be expected that evolution of $K^{*}$ in a merging scenario would be greater than that predicted in a monolithic collapse scenario, assuming the same age of formation. Thus in a merging scenario the observed degree of evolution of $K^{*}$, would result from a younger age of assembly than in a monolithic collapse scenario.} This is in contrast to the conclusions of \citet{kau98b} and \citet{kau96} who find evidence for a deficit of massive galaxies in the field at $z\approx 1$ (although see Im et al 2000 for different results). Therefore environment may have an effect on the mode of evolution. Indeed it is expected from simulations (\protect\citealt{kau93}, \protect\citealt{bau96}, \protect\citealt{dia01}) that the typical redshift of assembly of an elliptical will be higher in a rich cluster than in the field, due to the fact that structures collapse earlier in denser environments. Note that although there is a strong case that the observed evolution in $K^{*}$ may be attributed almost entirely to passive evolutionary processes, the interpretation of this result as being due to a redshift of formation $z_{f}=1.5-2$ is less secure. The models used are for a single burst of star-formation for a stellar population with a Salpeter initial mass function having a solar metallicity. Age-redshift relations are calculated for an assumed flat cosmology with $H_{0}=70$km s$^{-1}$ Mpc$^{-1}$ and $\Omega_{\rm{M}}=0.3$ and $\Omega_{\Lambda}=0.7$. All of these assumptions affect the resultant redshift of formation and so there is clearly some slack in the interpretation of the evolution. Future work based on the colours and morphologies of the member galaxies will provide stronger constraints on the epoch of formation. %may be gained from studies of the evolution of the colours of the galaxies. Such work will be presented in a future paper with $J$and $K$ for all three clusters and $VIR$ for ClJ1226. %It is worth noting that the field sizes are typically $\sim 1$Mpc on a side. Therefore we are restricted to examining the cores of the clusters. The work of \citet{whi91} and \citet{whi93} on the reexamination and reinterpretation of the morphology-density relation of \citet{dre80} shows that the morphological mix of galaxies depends on the cluster-centric radius. The evolution of this relation (\protect\citealt{dre97}) holds up to $z=0.5$. \citet{dri98} show that the dwarf to giant galaxy ratio increases in the outer parts of clusters and \citet{and01} shows that the shape of the luminosity function is sensitive to the area from which it was measured. Thus comparison between KLFs and integrated light functions from clusters with different mean projected densities, and covering different fractions of the virial radius may be misleading if one is interested in measuring the contribution from dwarf galaxies. In this paper we are concerned only with the evolution of massive early-type galaxies and since our cluster field sizes are comparable comparisons between clusters should be robust. %FIELD LOWER VELS EASIER MERGING. MORPH-DENSITY RELN - DIFF TYPE OF GALS DIFF EVOL HISTORY. The evolution of BCGs is consistent with that found by \citet{bro02}. Figure \ref{fig:khubble} exhibits a degree of scatter in the evolution of BCGs at high redshift with some BCGs being consistent with no-evolution predictions, of which ClJ0152 is an example, and others being consistent with passive evolution. This can be interpreted as showing that some BCGs are fully formed at high redshift e.g.\ ClJ1226, whereas others would need to undergo merging between $z=1$ and the present to reconcile them with local BCGs. This is supported by the ratios of $K$ band light in the BCGs to total cluster light. The ratio in Coma is not larger than the ratio in the high redshift clusters. Thus brightening of the BCGs with time, as would be expected from such processes as cannibalism, is not observed, suggesting that the BCGs, at least of ClJ1226 and ClJ1415, are already fully formed. %are broadly similar for all the high redshift clusters and the Coma cluster. %The BCGs in ClJ0152 contain less light than the BCG of Coma supporting the idea that they will undergo further mergers in the course of their evolution. The fraction of light in the BCGs of ClJ1226 and ClJ1415 is greater than that of Coma. This is contrary to any expected brightening of the BCGs toward lower redshift as would be expected from such processes as cannibalism, suggesting that the BCGs of ClJ1226 and ClJ1415 are already fully formed. %*****The BCGs of ClJ0152 are dimmer than passive evolution predictions and contain a relatively small amount of the total cluster light, suggesting they will undergo more merging as they evolve, which is perhaps expected from the apparent dynamical state of the cluster. On the other hand, ClJ1226 appears relaxed in X-rays (\protect\citealt{mau03a}, \protect\citealt{ebe01}) and has the brightest BCG of our small sample containing the largest fraction of cluster light. Together these results are suggestive that the evolution of BCGs is related to the overall dynamical state of the cluster. %\section*{Acknowledgments.} %The authors would like to thank warmly both Ben Maughan, for his role %in observing at UKIRT %and discussions about the X-ray properties of the clusters, and Harald %Ebeling, for providing all the galaxy redshifts. UKIRT staff have been %very efficient and helpful; service time %observations were performed by some of them. %The authors would also like to thank Sarah Brough for providing the %data for figure~\ref{fig:khubble}, and %Stefano Andreon and Antonaldo Diaferio for their help. SCE %acknowledges a PPARC studentship. This research has made use of the NASA/ IPAC Infrared % Science Archive, which is operated by the Jet Propulsion % Laboratory, California Institute of Technology, under % contract with the National Aeronautics and Space % Administration. % Thank UKIRT. %\bibliographystyle{mn2e} %\bibliography{clusters} %\end{document}