Simulating galaxy clusters : the ICM and the galaxy populations
\speaker{Alessio D. Romeo}
%\thanks{A footnote may follow.}
Univ. di Catania
Italy and TAC Copenhagen
Denmark
Cosmological LambdaCDM TreeSPH simulations of the formation and evolution of galaxy groups and clusters have been performed. The simulations include: star formation, chemical evolution with non-instantaneous recycling, metal dependent radiative cooling, strong star burst and (optionally) AGN driven galactic super winds, effects of a meta-galactic UV field and thermal conduction. We report results on the temperature and entropy profiles of the ICM, the X-ray luminosity, cold fraction, M/L and IMLR ratios in gas and stars, metal abundances and gradients. Besides, the properties of the galaxy populations in the two richest clusters are discussed: global star formation rates of the cluster galaxies, the total K-band luminosity, the galaxy luminosity functions at z=0 and their redshift evolution, the colour-magnitude relation (``red sequence'') as resulting from metallicity effects, and the role of the IMF in reproducing colours and abundances of the stellar populations. Finally the contribute (20-40\%) to cluster light from the intra-cluster stars and the cD galaxy has been investigated: surface brightness profiles, mean colours and chemical abundances, kinematics (velocity dispersion and distributions) of the IC stellar populations.
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\title{Simulating galaxy clusters : the ICM and the galaxy populations}

\ShortTitle{Simulating galaxy clusters: ICM and galaxies}

\author{\speaker{Alessio D. Romeo}\\ %\thanks{A footnote may follow.}\\
        Univ. di Catania, Italy and TAC Copenhagen, Denmark\\
        E-mail: \email{aro@ct.astro.it}}

\author{{Jesper Sommer-Larsen}\\
        TAC, Nordita and NBI, Copenhagen, Denmark\\
        E-mail: \email{jslarsen@tac.dk}}

\author{{Laura Portinari}\\
        TAC Copenhagen, Denmark and Tuorla Obs, Finland\\
        E-mail: \email{lporti@utu.fi}}

\abstract{Cosmological LambdaCDM TreeSPH simulations of the formation and evolution
of galaxy groups and clusters have been performed. The simulations include:
star formation, chemical evolution with non-instantaneous recycling, metal
dependent radiative cooling, strong star burst and (optionally) AGN driven
galactic super winds, effects of a meta-galactic UV field and thermal
conduction.
We report results on the temperature and entropy profiles of the ICM, the
X-ray luminosity, cold fraction, M/L and IMLR ratios in gas and stars, metal
abundances and gradients. Besides, the properties of the galaxy
populations in the two richest clusters are discussed: global star
formation rates of the cluster galaxies, the total K-band luminosity, the
galaxy luminosity functions at z=0 and their redshift evolution, the
colour-magnitude relation
(``red sequence'') as resulting from metallicity effects, and the role of
the IMF in reproducing colours and abundances of the stellar populations.
Finally the contribute (20-40\%) to cluster light from the intra-cluster
stars and the cD galaxy has been investigated: surface brightness
profiles, mean colours and chemical abundances, kinematics (velocity
dispersion and distributions) of the IC stellar populations.}

\FullConference{Baryons in Dark Matter Halos\\
                 5-9 October 2004\\
                 Novigrad, Croatia}

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\begin{document}


\section{Introduction}

%The basic scenario describing the thermal behaviour of the ICM calls
%for a gravity-only driven heating of the gas by adiabatic compression,
%followed by hydrodynamical shocks arisen during the accretion of infalling 
%material.
%It is by now well known how this simple model basically fails in reproducing
%the actual scaling properties observed ($L_X-T$, $S-T$, $M-T$ relations)
%Physical mechanisms able to explain the break of self--similarity in the
%properties of ICM have been extensively proposed during last ten years, and
%mainly fall in the two broad classes of non--gravitational heating and
%radiative cooling. 
%However, all cluster simulations with cooling and star formation
%reveal a significant drawback: cooling alone makes a large fraction (30 up
%to 55\%) of gas get converted into a cold phase (either cold gas or stars)
%within the virial radius at $z=0$, in contradiction with a value less than %10-20\% of the baryon fraction locked into stars indicated by observations. 
Non--gravitational heating and radiative cooling have been extensively 
recoursed to in the literature in order to complete the basic gravity-only driven scenario describing the thermal behaviour of the ICM, which 
is unable to reproduce
the scaling relations observed between $L_X$, $T$ and $S$.

So far few hydrodynamical simulations have been performed which were able
to pursue at the same time and in a self--consistent way the target of
realistically modelling the ICM dynamics by including both non-gravitational
heating/cooling (complete with feedback/star formation) {\it and} metal
enrichment directly springing from chemically following up the stellar feedback itself 
%Borgani \etal 2001 and 2002, Tornatore \etal 2003, McCarthy 
%\etal 2004 , and 
(e.g. Tornatore \etal 2004).
Moreover, only recently
%, due to advances in computing capabilities
%as well as in the detail in the physical modelling, cluster
simulations have reached a level of sophistication adequate to even trace
star formation and related effects in individual galaxies, coupled with 
the chemical enrichment of the ICM by galactic winds.

In this series of papers, we present 
%for the first time (to our knowledge) 
an analysis
of the properties of the galaxy population of clusters as predicted directly
from cosmological simulations that follow detailed baryonic 
physics, gas dynamics and galaxy formation and evolution. The resolution
for such N--body + hydrodynamical simulations cannot reach the extent of  
resolving galaxies at the faint end of the luminosity function 
($M_B$$\ga$--16). On the other hand, our simulations aim at
describing in a self--consistent way the
hydrodynamical response of the ICM to star formation, stellar
feedback and chemical enrichment. 

%\section{The Code and Simulations}

Our cosmological N-body simulation has been performed using the FLY code 
(Antonuccio \etal, 2003) on a standard
$\Lambda$CDM model 
%($\Omega_0$=0.3, $\Omega_{\Lambda}$=0.7, $\Omega_b$=0.04, $h=0.7$, %$\sigma_8$=0.9) with $128^3$ collisionless DM-only particles filling 
%an initial box of 150$h^{-1}$ Mpc at an initial redshift $z_i=39$. 
The selected clusters and groups 
%(listed in Tab. 1) 
isolated from the cosmological simulation,
as identified at $z=0$, have 
been resampled and resimulated (adding the baryon particles according to
a $f_b=0.12$) by means of an improved version of the Tree-SPH 
code described in Sommer-Larsen, G\"otz \& Portinari (2003; SLGP). 
%All DM particles within
%a virial radius at $z=0$ have been traced back to the initial condition at %$z_{i}$,
%when one SPH particle has been added per each DM one, with a mass of $m_{SPH}=
%f_b\cdot m_{DM}^0$ (with $f_b=0.12$).
The mass resolution reached inside the resampled
lagrangian subvolumes from the cosmological set was such that 
$m_{DM}=1.8\cdot 10^9 h^{-1} M_{\odot}$
and $m_{g}=m_*=2.5\cdot 10^8 h^{-1} M_{\odot}$.

Most of the results hereby presented refer to the two larger systems:
``sub-Coma'' ($M_{vir}=12.4\cdot 10^{14}M_{\odot}$, $<kT_{ew}>\simeq$ 6~keV) and
``Virgo'' ($2.8\cdot 10^{14}M_{\odot}$, 3~keV), though four more smaller
clusters and groups have been simulated which span the range 1-2~keV.

The adopted TreeSPH code schematically includes the following features:
conservative solution of entropy equation;
metal-depending atomic radiative cooling;
star formation, according to either an Arimoto-Yoshii (AY) or a Salpeter IMF;
feedback as starburst (SN II) driven galactic winds;
chemical evolution with non-instantaneous recycling of gas and heavy
elements (H, He, C, N, O, Mg, Si, S, Ca, Fe);
thermal conduction;
meta-galactic, redshift-dependent UV field.
All the runs implement the ``Super-Wind" (SW) scheme, characterized by
high efficiency parameters $\epsilon_{SF}$ and $f_{wind}$: the latter
is amplified by a factor 2 or 4 in those runs simulating the effect of
stronger non-stellar feedback (AGN). On the top of this model, it has
been added either thermal conduction (at $\frac{1}{3}$ Spitzer value (COND
runs), or pre-heating in form of 0.75/1.50/50cm$^2$~keV injected per
particle at $z$=0 (PH runs). Finally, we ran as a test the WFB simulation,
which adopts a normal feedback scheme with only early winds at low values of
$f_{wind}$ (as described in SLGP), and the ADIABATIC run without cooling nor
star formation at all. 
%\begin{table}[t]
%\caption{Structural properties of selected clusters at $z$=0
%\label{t:1}}
%\small
%\begin{tabular}{lcccc}
%\hline\hline
%Cluster \# & $M_{tot}$ & $M_{vir}$ & $R_{vir}$ & $<kT_{ew}>$ $^{\rm a}$ \\
%%$^{\rm c}$ & $M_{b,tot}$ $^{\rm d}$ & $N_*$ & $N_{cg}$ $^{\rm b}$ & $N_{hg}$ %$^{\rm c}$ 
%%& $N_{b,tot}$ $^{\rm d}$\\
%  & [10$^{14}$M$_{\odot}$] & [10$^{14}$M$_{\odot}$] & [Mpc] &[keV]~\\
%\hline
%``Coma" & ~~14.0 &12.38 & 2.90 & 6.00\\
%``Virgo" & ~~3.05 & 2.77 & 1.62 & 3.06\\
%202 & ~~1.26 & 1.26 & 1.29 & 2.24\\
%215 & ~~1.17 & 1.03 & 1.18 & 1.48\\
%550 & ~~0.51 & 0.48 & 0.96 & 1.00\\
%563 & ~~0.50 & 0.49 & 0.94 & 1.10\\
%\hline \hline
%\multicolumn{5}{l}{$^{\rm a}$ Temperatures referred to the "standard" run %SW-AY}\\ 
%\end{tabular}
%\end{table}

\section{Results on the ICM}
\subsection{Thermal properties}

%\begin{figure}[h!]
%\centering \includegraphics[width=70mm]{TTr.eps}
%%  \vspace*{174pt}
%  \caption{Temperature radial profiles of ``Virgo'' and ``Coma'', normalized to %their mean emission-weighted temperature. {\it Top left}: ``Virgo'' SW-AY, %SWx2FB, 
%SWx4FB; {\it top right}: ``Virgo'' SW-AY, SW-Sal, WFB-Sal; {\it bottom left}: 
%``Virgo'' SW-AY, PH 0.75, PH 1.5, PH 50 and ADIAB.; {\it bottom right}: %``Virgo'' SW-AY
%and COND., ``Coma'' SW-AY and COND. Data points from De Grandi \& Molendi 
%(2002), 
%%give the average deprojected profile over a sample of nearby clusters, 
%both CF ({\it dash-shadowed})
%and non-CF ({\it point-shadowed}); also shown some individual CF clusters.}
%\label{TTr}
%\end{figure}

%In Fig.~\ref{TTr} we compare our simulated temperature profiles of ``Virgo"
%with the results by De Grandi \& Molendi (2002), which analysed spatially
%resolved data for a sample of rich nearby clusters with BeppoSAX, both %cold--core and non, finding that their
%profiles are characterized by an isothermal core extending out to $\sim 0.2 %R_{180}$, beyond which they steeply drop following a power-law, which is %flatter
%for the cold--core systems.
%
%In the four panels the separate role of feedback,
%IMF, pre--heating and conduction may be assessed by varying the relevant
%parameters and keeping the SW scheme with AY IMF as standard reference for all. %All the runs fairly follow the observed decline
%from $\sim 0.25 R_{vir}$ outwards. However the formation of a core around 
%$0.1 R_{vir}$ is not satisfactorily modelled: 
%temperature keeps rising inwards even beyond $0.1 R_{vir}$ and only veers off %to decline at around $0.05 R_{vir}$, which is too central
%especially if compared with more rapid descent of cold--core clusters.

%\subsection{Entropy}

Observations of X-ray surface brightness profiles in poor clusters and 
groups of galaxies had hinted at the existence of isentropic cores or 
``entropy floor'' 
at around 100 keV$\cdot$cm$^2$ (e.g. Ponman \etal, 2003), associated with a
reduced gas central density. Systems at all scales, and in particular low-mass ones, show evidence for
excess entropy compared to that expected from pure gravitational heating by compression.
%However, the existence of isentropic cores is still controversial.
In Fig.~\ref{TSr} the inner pattern of temperature and entropy for particles in ``Virgo'' cluster is shown: 
%from which it is evident that there is not a proper isentropic core or entropy %floor: 
most models exhibit central entropy values of
$\sim$ 10-20 keV$\cdot$cm$^2$, which may get agreed with inferred values
in cold--core clusters; 
%(Voit \etal 2003); 
instead a plateau has formed only in two distinct cases, namely the WFB metal-independent scheme with Salpeter IMF and the 4-times SW scheme with AY IMF. The former has less efficient cooling and therefore lets more high-entropy gas settle down in a core.



 The SWx4 model ends up with forming a floor around 100 keV$\cdot$cm$^2$ at $z=0$. 

\begin{figure}
\centering \includegraphics[width=70mm]{TSr.eps}
%  \vspace*{174pt}
  \caption{Temperature ({\it left}) and Entropy ({\it right}) profiles of 
``Virgo'', all models: same colours as in  Fig.~3.}
\label{TSr}               
\end{figure}

%\subsection{X-ray Luminosity}

%Measurements of ICM temperature by X-ray
%satellites demonstrated that the observed systems scatter around the curve %$L_X\propto T^\alpha$ with $\alpha \simeq 3$, the largest deviations from which
%are presented by those systems whose emission is strongly associated with a
%cold core, or former ``cooling--flow" clusters. Moreover, there is evidence 
%that low-mass systems ($T\lsim 1$ keV) may
%show an even steeper relation (Helsdon \& Ponman 2000).

\begin{figure}
\centering \includegraphics[width=70mm]{LT.eps}
%  \vspace*{174pt}
  \caption{X-ray luminosity - emission weighted temperature scaling 
relation from the inner 1 Mpc.}
\label{LT}
\end{figure}

In Fig. \ref{LT} the relation $L_X$-$T$ at $z\simeq 0$ is shown: 
%where the temperature is 
%the emission-weighted one over the whole cluster central emitting 
%region (1 Mpc). 
the observed (open) points exhibit the characteristic double-slope
behaviour; as expected, the adiabatic runs lie above this line.
Overall, the predicted luminosities are higher than the observed one; both an enhanced
feedback mechanism and a top--lighter IMF, however, can indipendently concur in 
lowering the X-ray emission from the ICM: the latter effect seems to be dominant,
since the runs with Salpeter IMF and Weak Feedback, which result in a low
level of metal enrichment as well, are those placed lowest. All in all, when strengthening the feedback up to 4 times and using a top--heavy
IMF, a satisfactory normalization seems to be reached. 
%at least for moderate to rich clusters. 
One could conclude then that
previous simulations successful at reproducing both
the L-T relation and the surface brightness profiles and the rather high
entropy floor as well, might have underestimated the contribution of metals to
radiative cooling by adopting simple primordial cooling functions.

%\subsection{The cluster cold component}
Baryonic mass of clusters is largely dominated by the hot ICM mass,
yet it is the (stellar) mass in galaxies playing the major role
to the purpose of chemical enrichment of ICM itself.
A measure of galaxy formation efficiency is
the ``cold fraction'', defined as the fraction of baryons in the form
of cold gas and stars. 
%The cold fraction is a crucial constraint for cluster simulations, as 
It plays
a major role in assessing the mechanisms responsible for the ``excess
entropy'': the larger the cold fraction (hence star formation),
%the larger the role played by galaxy and star formation which removes 
the more low entropy gas is removed from
the core; with a small cold fraction, instead, strong (pre)heating or
feedback are needed to enhance the entropy level.
The radial profiles of the cold fraction are shown in Fig.~\ref{FcR}  
for our two largest simulated clusters:
%, compared to the observed profiles of Roussel {\it et al.} (2000). 
the AY-SW simulations (including the conduction
and preheating cases) are those that best reproduce the overall level, or equivalently the $\frac{M_{ICM}}{L}$ ratio, within the virial radius.
The Salpeter runs instead, 
especially the weak feedback case, result in too large $f_c$, while
the enhanced feedback cases predict significantly too low values.
It is a remarkable result that our simulations with SN feedback 
(with a top-heavy IMF, but without invoking additional sources of energy input 
via enhanced feedback) 
fairly succeed in reproducing the low fraction of baryons in the cold phase. Indeed it is a common feature of all hydrodynamical simulations with 
cooling and star formation ending up with a too large fraction of gas 
to get converted into stars (30 up
to 55\%, against observed 10-20\%). 
When including also non--gravitational (pre)-heating, 
star formation gets attenuated by preventing gas from cooling at high
redshift, hence causing a decrease in the cold fraction.
\begin{figure}
\centering \includegraphics[width=70mm]{FcR.eps}
%  \vspace*{174pt}
  \caption{Cold gas plus stellar radial profiles of the two most massive clusters 
(``Coma'': {\it dot-dashed} line). {\it Crosses}: observed cold fraction 
profiles, derived from the $\frac{M_*}{M_{ICM}}$ data of Roussel {\it et al.} 
(2000).}
\label{FcR}
\end{figure}

\subsection{Metal enrichment}

The presence of heavy elements in the hot intra-cluster gas calls 
for the processing of a significant fraction of the ICM inside galaxies. 
%with a tight link between the stellar and gaseous components of clusters.
In our simulations, the super--winds connected to the SNII feedback 
prescription lead to the enrichment of the ICM by 
dispersal of the chemical elements produced by the star particles.
\begin{figure}
\centering \includegraphics[width=70mm]{FehVC.eps}
%  \vspace*{174pt}
  \caption{Iron gradients in the ICM of the two most massive simulated
clusters. Data from De Grandi {\it et al} (2004) refer to cold-core 
({\it open circles}) 
and non cold-core ({\it filled squares}) clusters.}
\label{FehVC}
\end{figure}
Fig.~\ref{FehVC} shows the simulated iron abundance profiles, 
compared
to data for cool core and non-cold core clusters. Our simulated clusters
are best compared to cold-core clusters. Evidently, the simulated
gradients are steeper than the observed ones. This seems to be a general
problem in simulations, 
%(e.g.\ Tornatore {\it et~al.}\ 2004), 
probably related
to the fact that star formation is very concentrated toward
the inner region of the cD galaxy. 
%Another important constraint on the chemical evolution of clusters is the
The ICM is also enriched by $\alpha$--elements, mainly produced by
SNII and specifically traced by silicon abundances: 
We found a trend of supersolar
[Si/Fe] increasing with radius in all our simulations. 

%\begin{figure}
%\centering \includegraphics[width=70mm]{evolICM.eps}
%%  \vspace*{174pt}
%\caption{Radial profiles of Iron ({\it top}) and Silicon ({\it bottom}) 
%abundances in the ICM at z=0,1,2. {\it Left}: ``Virgo'' ({\it solid}: AY-SW, 
%{\it dashed}: Sal-SW); {\it right}: ``Coma''. $R_{vir}$ evaluated at z=0.}
%\label{SiFez}
%\end{figure}

%Fig.~\ref{SiFez} (top panels) shows that the overall iron abundance, and 
Moreover, it has also been found that the overall iron abundance in the ICM 
of simulated clusters is unvaried from $z=0$
to $z=1$, in very good agreement with observations. 
%(Tozzi {\it et~al.}\ 2003).
Since the bulk of the stars in the cluster are formed at $z \geq$2, 
at $z$=2 we predict much lower iron metallicities,
but such redshift range is not probed by observations yet.
%The lower panels in the figure show the evolution in 
As expected, [Si/Fe]
decreases in time due to the delayed iron contribution of SNIa 
with respect to the bulk of the silicon production (from SNII).


\begin{figure}
\centering \includegraphics[width=70mm]{lf_rel.eps}
%  \vspace*{174pt}
  \caption{{\it Top panels}: B--band luminosity functions at $z$=0
for the standard (AY-SW) ``Virgo'' ({\it left}) and ``Coma'' ({\it right})
simulations, compared with the observed composite LF of Trentham 1998, {\it triangles}. 
{\it Bottom panels}:
K--band LFs for the same clusters, compared to the observed average LF by
Lin {\it et~al.}\ (2004); the dotted line marks the limit $M_K \leq -21$ where 
the observational estimate is considered reliable.}
\label{fig:LF_rel}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results on cluster galaxies}

\subsection{Luminosity Functions}
\label{sect:lf}
Each star particle represents a Single Stellar Population (SSP), 
with individual stellar masses distributed
according to a particular IMF. We 
keep record of the age and the metallicity 
of each of these SSPs and hence 
the global luminosities and colours of the member galaxies are computed.
%, as the sum
%of the contribution of their constituent star particles. 
We identify for the
standard (SW-AY) runs 42 galaxies in ``Virgo'' and 212 galaxies in ``Coma''.

In Fig.~\ref{fig:LF_rel} 
we compare the B--band and K--band luminosity function (LF) 
of our simulated cluster galaxies, exluding the cD, 
to two observational composite LFs. 
%by Trentham (1998), and by Lin {\it et al.} (2004). 
The number of resolved individual galaxies 
quickly drops for objects fainter than $M_B \sim$ -17, both in ``Virgo'' and in
``Coma'', due to the resolution limits.
Although we are missing the dwarf galaxies that largely dominate
in number, we are able to describe the bulk of the stellar mass and of the
luminosity in clusters, which is dominated by galaxies around $L_*$. 
%--- though in the
%simulations, too much of the luminosity and stellar mass expected in
%objects around $L_*$, is instead accumulated in the central cD.
In the significative  magnitude range, the shape of the predicted and 
observed LF is directly comparable. 
Yet, the simulated LF is steeper than the observed one, namely we 
underestimate
the relative number of bright galaxies, at least within the SW model here
shown.
We have found that increasing the feedback strength
results in decreasing the masses of all galaxies, and hence, in particular,
the number of bright galaxies. Alongside this, the LF in the Salpeter 
cases is broader in luminosity, due the lower stellar feedback
with respect to the AY IMF, allowing a larger accumulation of stellar 
mass in galaxies. 

%The LF in clusters can evolve due to two effects:
%passive luminosity evolution from the aging of the stellar populations
%in the galaxies, and mass evolution due to dynamical effects such as mergers, 
%tidal stripping and dynamical friction.
%Cluster galaxies are known to consist of old stellar populations, with the
%bulk of their star formation occurring at $z$$\gg$1. However,
%they may not have been completely assembled by $z\sim 1$, and they 
%may still undergo one or two significant mergers since then, or grow by gas 
%accretion (van Dokkum {\it et al.}, 1999). 

\begin{figure}
\centering \includegraphics[width=70mm]{lf_mf_z1.eps}
%  \vspace*{174pt}
\caption{{\it Top panels:} Luminosity functions in absolute number of
galaxies per {\it B}-band magnitude bin for the standard (AY-SW) ``Virgo'' 
and ``Coma'' simulations, at $z=0$ ({\it solid 
lines}) and $z=1$ ({\it dashed lines}); the dotted lines represents
the expected $z=1$ LF if pure passive evolution of the stellar populations is
applied to the $z=0$ LF. {\it Middle panels:} Same, for the K-band luminosity
functions.
{\it Bottom panels:} Mass function of cluster
galaxies at $z=0$ and $z=1$.}
\label{fig:LFz}
\end{figure}

In Fig.~\ref{fig:LFz} we show the evolution of the B and K band LFs 
of our ``Virgo'' and ``Coma'' clusters. The $z=1$ LF is shifted to brighter 
magnitudes with respect to the distribution at $z=0$.
Both passive luminosity evolution and dynamical mass evolution 
are seen to play a role. In particular, 
since the bulk of the stars in our galaxies are formed at $z \gsim 2$ 
(see Fig.~\ref{fig:redsequence}, bottom panel),
luminosity dimming is an important effect.\\
The LF inferred at $z=1$ as expected from pure passive evolution of
the $z=0$ LF (dotted line
in Fig.~\ref{fig:LFz}) matches very closely the actual LF in the $z=1$ 
frame. 
%(apart from the absence of a few bright galaxies). 
This indicates that luminosity dimming of the stellar populations
drives most of the LF evolution and the fading to fainter 
magnitudes from $z=1$ to $z=0$ for the simulated galaxy population. 
%in line with some recent results (Kodama \& Bower, 2003).
In the brightest luminosity bins of our LF, some additional,
dynamical effect seems to be required. 
To assess mass evolution effects, we plot in the lower panels of
Fig.~\ref{fig:LFz} the mass function of cluster 
galaxies at $z=1$ and at $z=0$, which appear indeed very
similar.
``Over--merging'' onto the central cD in cosmological simulations can be 
responsible for depleting the number of galaxies at the bright end of the LF
(Springel {\it et~al.}\ 2001). Yet the results of a test simulation at 8 times 
higher mass and two times better force resolution we performed do allow us to
disregard this interpretation.

\begin{figure}
\centering \includegraphics[width=70mm]{redsequence.eps}
%  \vspace*{174pt}
\caption{{\bf (a--b)} Colour--magnitude relation at $z$=0 for galaxies in the
simulated ``Coma'' cluster, compared to the observed relation with its scatter 
({\it solid} and {\it dashed} lines, from Bower {\it et~al.}\ 1992; 
{\it dotted} lines, from Terlevich {\it et~al.}\ 2001).
The extension of the dashed/dotted lines
indicates the magnitude range actually probed observationally, while the
solid line is an extrapolation of their fit to lower magnitudes.
{\bf (c)} Metallicity--luminosity relation for the ``Coma'' galaxies 
(full symbols for mass--averaged stellar metallicity, open symbols for 
luminosity--weighted metallicity); the dotted line is a linear fit.
{\bf (d)}: Age--luminosity relation for the ``Coma'' galaxies (mass--average
and luminosity--weighted stellar ages); the dotted line is a linear fit.}
\label{fig:redsequence}
\end{figure}


\subsection{The Red Sequence}
%
The light and stellar mass in clusters of galaxies is dominated by bright,
massive ellipticals, which are known
to form a tight colour-magnitude relation, or Red Sequence (Gladders {\it et~al.}\ 1998). In Fig.~\ref{fig:redsequence}ab, 
we compare the colour--magnitude relation for our ``Coma'' cluster galaxies,
excluding the cD, to two observed Red Sequences of Coma. 
There is overall agreement, although our Red Sequence is not as extended
as the observed one, and it also 
%(which reaches magnitudes brighter that $M_V=-23$), 
%due to the lack of simulated galaxies at the bright end of the luminosity 
%function already discussed. The simulated Red Sequence also
displays a larger scatter. 
%than the observed Coma Red Sequence, which remains 
%very narrow down to $M_V \sim -18$ (dashed and dotted lines).
The colour--magnitude relation is classically interpreted as a 
mass--metallicity relation. 
%(Kodama \& Arimoto 1997). In our simulations, 
The bulk of the stars in the simulated galaxies are formed at $z\ga$2, 
hence our galaxies are essentially coeval (Fig.~\ref{fig:redsequence}d) 
and their colour--magnitude relation is mostly a metallicity effect, as
it can be seen from combining these results with the 
metallicity--luminosity relation 
displayed in Fig.~\ref{fig:redsequence}c.
%both in terms of mass-weighted metallicity and of (V--band) 
%luminosity--weighted metallicity. 
%The latter is systematically slightly lower than the actual 
%mass-averaged stellar metallicity, as expected since more metal--poor 
%populations tend to be brighter, skewing the luminosity--weighted metallicity 
%to lower values; the trend with galaxy luminosity is however
%maintained. Although metal-rich objects seem to exist at all 
%luminosities, the fraction of metal--poor galaxies (as well as the scatter in
%metallicity) increases with decreasing galaxy luminosity; the average 
%stellar metallicity decreases for fainter galaxies (dotted line), which 
%ultimately drives the simulated colour--magnitude relation.

%Fig.~\ref{fig:redsequence}d shows the average stellar ages
%for the galaxies. All the galaxies consist of old stellar populations,
%with average ages between 7 and 12 Gyrs. The brighter galaxies are essentially
%coeval, 9--10~Gyr old. 
%The luminosity--weighted ages are in general very close to the mass
%averaged ages, which implies a small scatter of stellar ages within the
%individual galaxies: extended tails or recent episodes of star formation, 
%would strongly skew the luminosity--weighted estimate to younger ages.
%The scatter in age among the galaxies is around 30\%, with a mild trend 
%of decreasing age with increasing luminosity (dotted line), but the effect 
%is much smaller that the systematic variation in metallicity (a factor of 10,
%Fig.~\ref{fig:redsequence}c). This confirms that the color--magnitude 
%relation of our simulated galaxies is driven by a mass-metallicity relation.
%The mild age trend, which tends to act in the
%opposite direction (making brighter galaxies bluer) is probably responsible of
%the fact that our colour-magnitude relation are somewhat less steep than
%the observed ones.

Moreover, the simulated Red Sequence appears to be a robust prediction
of our simulations, quite unaffected by the adopted physical prescriptions 
other than the chosen IMF. 
%The IMF sets the zero--point of the simulated 
%Red Sequence, via the typical stellar metallicity attainable in cluster
%galaxies; but the observed slope of the colour--magnitude relation
%is well reproduced by all simulations.

\bibliographystyle{plain}
\begin{thebibliography}{99}
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MNRAS submitted, astro-ph/0404445 (Paper~II) 

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Tornatore L., Borgani S., Matteucci F., Recchi S., Tozzi P., 
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\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Intra-cluster stars}

Current hierarchical or ``bottom-up'' structure formation theories
predict that %the stellar haloes (the ``field'' stars) 
the field star populations of haloes of galaxies like the 
Milky Way should consist partly of stars originally born in small 
proto-galaxies and later tidally stripped from these by the main galaxy
or through interaction with other proto-galaxies. The halo stars resulting from tidal stripping
or disruption of a proto-galaxy will stay localized in phase-space for
a long period and several such ``streams'' of halo stars have been
detected in the haloes of the Milky Way and M31 (e.g., Helmi et al. 1999;
Ferguson et al. 2002).

We define the system of IC stars as the stars located at cluster-centric
distances $r_{\rm{cD}}\le r \le r_{\rm{vir}}$ and {\it not} inside of
the tidal radius of any galaxy in the cluster. The tidal radius 
for each galaxy is taken to be the Jacobi limit
%
\begin{equation}
r_J = \left(\frac{m}{3M}\right)^{1/3}\hspace{-1mm}D ~~~,
\end{equation}
%
where $m$ is the mass of stars, gas and dark matter in the galaxy (inside of
$r_J$), $D$ is the distance from the cluster center to the galaxy and 
$M$ is the total mass of the cluster inside of $D$ 
\citep[e.g.,][]{BT87}.  The cD
itself is effectively just the inward continuation of the system of IC
stars and the division between IC stars and cD is somewhat arbitrary
(we hence below quote intracluster star fractions for $r_{\rm{cD}}$=80 as well
as 40 kpc). 
The above definition of IC stars is conservative, since
we calculate the tidal radii on the basis of the $z$=0 frame 
cluster-centric distances of the galaxies. For any 
galaxy which has been through at least one peri-center passage,
this tidal radius should be taken as a firm upper limit.
Moreover, some IC stars will be inside of the tidal
radius of one of the cluster galaxies just by chance. As the total ``tidal
volume'' of all the cluster galaxies is found to be a few tenth of a 
percent of the virial volume of the cluster this effect can be neglected,
however.

Using the above definition we find that the intracluster stars
constitute 21 and 28\% of the stellar mass
inside of the virial radius for clusters C1 and C2,
respectively (adopting $r_{\rm{cD}}$=40 kpc these numbers increase 
slightly to 28 and 34\%). For comparison with observations it is more 
relevant to look at stellar luminosities. Since ages and metallicities are
available for all star particles, the photometric properties are 
straightforward to calculate treating each star particle as a Single
Stellar Population. We find that 
the IC stars contribute only 9 and 11\% of the cluster B-band luminosities
for C1 and C2 respectively. This is significantly less than the
observational estimate of $L_{\rm{B,IC}}/L_{\rm{B,tot}}\ga$0.2 
\citep[e.g.,][]{A04}. The likely reason for this discrepancy is an excess
of stars formed fairly recently at the center of the cD and dominating
the B-band luminosity. After correcting for these,
$L_{\rm{B,IC}}/L_{\rm{B,tot}}$=0.29 and 0.42 for
the two clusters in better agreement with observations (for 
$r_{\rm{cD}}$=40 kpc we obtain $L_{\rm{B,IC}}/L_{\rm{B,tot}}$=0.42 and 0.50, 
respectively), though somewhat high compared to the values found by
\cite{F.04a} for non-cD clusters.

\begin{figure}[ht!]
%\epsfxsize=\columnwidth
\centering \includegraphics[width=70mm]{zborn.ps}
\caption{Mean redshift of formation of the cD + IC stars (solid lines) and
stars in galaxies (dashed lines) at $z$=0. Results for cluster C1 
(``Virgo-like'') are shown by thin lines and for C2 (``mini-Coma'') by
heavy lines. Statistical uncertainties are $\Delta \overline{z}_f \lsim $ 0.05. }
\label{fig1}
\end{figure}


In Figure \ref{fig1} we show the mean (spherically averaged) redshift
of formation, $\overline{z}_f$, of the cD + IC stars (solid lines) and stars
in galaxies except the cD (dashed lines) as a function of radial distance 
from the
center of the cD for clusters C1 and C2, respectively. For both clusters
the average formation redshift of the IC stars is $\overline{z}_{f,IC}\sim$3 at
$r\ga$100-200 kpc. The stars in galaxies (except the cD) are on average
somewhat younger with $\overline{z}_{f,gal}\sim$2.5. This is to be expected, since
the bulk of the IC stars originate in (proto) galaxies, which have been
partly or fully disrupted through tidal stripping in the main cluster
potential or by galaxy-galaxy interactions. In contrast, the galaxies
still remaining at $z$=0 have potentially been able to continue forming stars 
out of remaining cold gas or gas recycled
by evolved stars and subsequently cooled to star-forming temperatures
and densities. Still, due to ram-pressure stripping 
of the hot and dilute gas reservoir in galactic haloes 
and other effects, the
star-formation rate in the galaxies decreases significantly from $z$=2 to 0,
 
considerably more so than in field galaxies.

\begin{figure}[h]
%\epsfxsize=\columnwidth
\centering \includegraphics[width=70mm]{ICabunr.ps}
\caption{Spherically averaged iron abundance of cD+IC stars (solid lines)
and stars in cluster galaxies (dashed lines) for clusters 
C1 (thin lines) and C2 (thick lines), respectively.}
\label{fig5}
\end{figure}

\begin{figure}
%\epsfxsize=\columnwidth
\centering \includegraphics[width=70mm]{ICabunratr.ps}
\caption{Spherically averaged oxygen-to-iron abundance ratio of cD+IC stars 
(solid lines) and stars in cluster galaxies (dashed lines) for 
clusters C1 (thin lines) and C2 (thick lines), respectively. The limiting
case for pure SNII enrichment [O/Fe]=0.55 is shown by the thin, dotted line
(results for $r<$20 kpc are not shown for clarity).}
\label{fig6}
\end{figure}

In Figure \ref{fig5} we show the spherically averaged iron abundance of the 
cD+IC stars as well as of the stars in cluster galaxies as a function
of cluster-centric distance. 
Iron is super-solar in the cD+IC stars at
$r\la$100 kpc and decreases to about half solar at large $r$; 
the stars in cluster galaxies follow a similar trend.
The fairly large overall iron abundance of the IC stars, as compared to,
e.g., stars in the halo of the Milky Way, reflects that it is the 
galaxies (past and present) which have to enrich the 5-10 times more massive
hot intracluster medium (ICM) to an iron abundance of about 1/3 solar
(Paper I). %\citep{R.04}. 

\begin{figure}[t!]
%\epsfxsize=\columnwidth
\centering \includegraphics[width=70mm]{rho_virgo.ps}
\caption{Spherically averaged density profiles for cluster C1 of cD+IC
stars (solid line), dark matter (dot-dashed line), stars in cluster galaxies
(dashed line) and, with arbitrary normalization, {\it number} density of
cluster galaxies (dotted line). The result of applying the correction for the
excess, young central stars is shown by the thin, dotted line.}  
\label{fig11}
\end{figure}
In Figure 11 we show for cluster C1 the spherically averaged density
distributions of the cD+IC stars (with and without the correction for the 
excess, young central stars), the stars in cluster galaxies, the dark
matter and, with an arbitrary normalization, the {\it number} density of
cluster galaxies. As suggested previously, the number density profile of 
cluster galaxies in the inner parts of the cluster as well as the dark matter 
density profile in general are significantly
flatter than the density profile of cD+IC stars. For the cluster galaxies
this is a manifestation of the transformation of galaxies to cD+IC stars
by tidal disruption in the inner parts of the cluster. 

Moreover, it follows from Fig.\ref{fig11} that the average (mass) density of stars
in cluster galaxies only exceeds that of the IC stars beyond about half
the virial radius - this is in good agreement with the findings of
\cite{M.04}. 


\end{document}


astro-ph0504348.extracted_bib
\begin{thebibliography}{99}
%  \bibitem{...} ....

\bibitem{}
Antonuccio-Delogu V., Becciani U., Ferro D., 2003, Comput. Phys. Commun. 
155, 159

\bibitem{}
De Grandi S., Ettori S., Longhetti M., Molendi S., 2004, A\&A 419, 7

\bibitem{}
Gladders M.D., Lopez-Cruz O., Yee H.K.C., Kodama T., 1998, ApJ 501, 571

\bibitem{} 
Lin Y., Mohr J., Stanford S.A., 2003, ApJ 591, 749 

\bibitem{}
Ponman T.J., Sanderson A.J.R., 
Finoguenov A., 2003, MNRAS 343, 331	

\bibitem{}
Portinari L., Moretti A., Chiosi C., Sommer-Larsen J., 2004, ApJ 604, 579

%\bibitem{}
%Romeo A.D., Sommer-Larsen J., Portinari L., Antonuccio V., 2004, 
%submitted to MNRAS (Paper~I) - and references therein

\bibitem{}
Romeo A.D., Portinari L., Sommer-Larsen J., 2004, 
MNRAS submitted, astro-ph/0404445 (Paper~II) 

\bibitem{}
Roussel H., Sadat R., Blanchard A., 2000, \aap 361, 429

\bibitem{}
Sommer-Larsen J., G\"otz M., Portinari L., 2003, ApJ 596, 47 (SLGP)

\bibitem{}
Sommer-Larsen J., Romeo A.D., Portinari L., 2004, MNRAS in press,
astro-ph/0403282 (Paper~III) 

\bibitem{}
Tornatore L., Borgani S., Matteucci F., Recchi S., Tozzi P., 
2004, MNRAS 349, L19

\bibitem{}
Trentham N., 1998, MNRAS 295, 360

\end{thebibliography}