mirror of
https://asciireactor.com/otho/phy-4660.git
synced 2024-11-24 21:25:08 +00:00
1920 lines
107 KiB
TeX
1920 lines
107 KiB
TeX
|
|
\documentstyle[epsfig,natbib2,natbibmnfix]{mn}
|
|
|
|
\newcommand{\be}{\begin{equation}}
|
|
\newcommand{\ee}{\end{equation}}
|
|
\newcommand{\bea}{\begin{eqnarray}}
|
|
\newcommand{\eea}{\end{eqnarray}}
|
|
\newcommand{\bc}{\begin{center}}
|
|
\newcommand{\ec}{\end{center}}
|
|
\newcommand{\ol}[1]{ {\overline{#1}}}
|
|
\newcommand{\lu}{\,h^{-1}{\rm kpc}}
|
|
\renewcommand{\vec}[1]{ {\bmath #1} }
|
|
\newcommand{\e}{{\rm e}}
|
|
\newcommand{\dd}{{\rm d}}
|
|
|
|
|
|
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
|
|
|
|
|
|
\newcommand\aj{AJ}% % Astronomical Journal
|
|
\newcommand\araa{ARA\&A}% % Annual Review of Astron and Astrophys
|
|
\newcommand\apj{ApJ}% % Astrophysical Journal
|
|
\newcommand\apjl{ApJ}% % Astrophysical Journal, Letters
|
|
\newcommand\apjs{ApJS}% % Astrophysical Journal, Supplement
|
|
\newcommand\ao{Appl.~Opt.}% % Applied Optics
|
|
\newcommand\apss{Ap\&SS}% % Astrophysics and Space Science
|
|
\newcommand\aap{A\&A}% % Astronomy and Astrophysics
|
|
\newcommand\aapr{A\&A~Rev.}% % Astronomy and Astrophysics Reviews
|
|
\newcommand\aaps{A\&AS}% % Astronomy and Astrophysics, Supplement
|
|
\newcommand\azh{AZh}% % Astronomicheskii Zhurnal
|
|
\newcommand\baas{BAAS}% % Bulletin of the AAS
|
|
\newcommand\jrasc{JRASC}% % Journal of the RAS of Canada
|
|
\newcommand\memras{MmRAS}% % Memoirs of the RAS
|
|
\newcommand\mnras{MNRAS}% % Monthly Notices of the RAS
|
|
\newcommand\pra{Phys.~Rev.~A}% % Physical Review A: General Physics
|
|
\newcommand\prb{Phys.~Rev.~B}% % Physical Review B: Solid State
|
|
\newcommand\prc{Phys.~Rev.~C}% % Physical Review C
|
|
\newcommand\prd{Phys.~Rev.~D}% % Physical Review D
|
|
\newcommand\pre{Phys.~Rev.~E}% % Physical Review E
|
|
\newcommand\prl{Phys.~Rev.~Lett.}% % Physical Review Letters
|
|
\newcommand\pasp{PASP}% % Publications of the ASP
|
|
\newcommand\pasj{PASJ}% % Publications of the ASJ
|
|
\newcommand\qjras{QJRAS}% % Quarterly Journal of the RAS
|
|
\newcommand\skytel{S\&T}% % Sky and Telescope
|
|
\newcommand\solphys{Sol.~Phys.}% % Solar Physics
|
|
\newcommand\sovast{Soviet~Ast.}% % Soviet Astronomy
|
|
\newcommand\ssr{Space~Sci.~Rev.}% % Space Science Reviews
|
|
\newcommand\zap{ZAp}% % Zeitschrift fuer Astrophysik
|
|
\newcommand\nat{Nature}% % Nature
|
|
\newcommand\iaucirc{IAU~Circ.}% % IAU Cirulars
|
|
\newcommand\aplett{Astrophys.~Lett.}% % Astrophysics Letters
|
|
\newcommand\apspr{Astrophys.~Space~Phys.~Res.}% % Astrophysics Space Physics Research
|
|
\newcommand\bain{Bull.~Astron.~Inst.~Netherlands}% % Bulletin Astronomical Institute of the Netherlands
|
|
\newcommand\fcp{Fund.~Cosmic~Phys.}% % Fundamental Cosmic Physics
|
|
\newcommand\gca{Geochim.~Cosmochim.~Acta}% % Geochimica Cosmochimica Acta
|
|
\newcommand\grl{Geophys.~Res.~Lett.}% % Geophysics Research Letters
|
|
\newcommand\jcp{J.~Chem.~Phys.}% % Journal of Chemical Physics
|
|
\newcommand\jgr{J.~Geophys.~Res.}% % Journal of Geophysics Research
|
|
\newcommand\jqsrt{J.~Quant.~Spec.~Radiat.~Transf.}% % Journal of Quantitiative Spectroscopy and Radiative Trasfer
|
|
\newcommand\memsai{Mem.~Soc.~Astron.~Italiana}% % Mem. Societa Astronomica Italiana
|
|
\newcommand\nphysa{Nucl.~Phys.~A}% % Nuclear Physics A
|
|
\newcommand\physrep{Phys.~Rep.}% % Physics Reports
|
|
\newcommand\physscr{Phys.~Scr}% % Physica Scripta
|
|
\newcommand\planss{Planet.~Space~Sci.}% % Planetary Space Science
|
|
\newcommand\procspie{Proc.~SPIE}% % Proceedings of the SPIE
|
|
\newcommand\rmaa{Rev. Mex. Astron. Astrof.}% % Rev Mexicana Astron Astrof
|
|
|
|
|
|
|
|
|
|
|
|
\title{Hydrodynamical simulations of cluster formation with central
|
|
AGN heating}
|
|
|
|
|
|
\author[D.~Sijacki et al.]
|
|
{\parbox{18cm}{D.~Sijacki\footnotemark[1] and
|
|
V.~Springel}\vspace{0.3cm}\\
|
|
Max-Planck-Institut f\"{u}r Astrophysik,
|
|
Karl-Schwarzschild-Stra\ss{}e 1, 85740 Garching bei M\"{u}nchen, Germany}
|
|
|
|
\begin{document}
|
|
|
|
\maketitle
|
|
\begin{abstract}
|
|
We analyse a hydrodynamical simulation model for the recurrent
|
|
heating of the central intracluster medium (ICM) by active galactic
|
|
nuclei (AGN). Besides the self-gravity of the dark matter and gas
|
|
components, our approach includes the radiative cooling and
|
|
photoheating of the gas, as well as a subresolution multiphase model
|
|
for star formation and supernova feedback. Additionally, we
|
|
incorporate a periodic heating mechanism in the form of hot, buoyant
|
|
bubbles, injected into the intragalactic medium (IGM) during the
|
|
active phases of the accreting central AGN. We use simulations of
|
|
isolated cluster halos of different masses to study the bubble
|
|
dynamics and the heat transport into the IGM. We also apply our
|
|
model to self-consistent cosmological simulations of the formation
|
|
of galaxy clusters with a range of masses. Our numerical schemes
|
|
explore a variety of different assumptions for the spatial
|
|
configuration of AGN-driven bubbles, for their duty cycles and for
|
|
the energy injection mechanism, in order to obtain better
|
|
constraints on the underlying physical picture. We argue that AGN
|
|
heating can substantially affect the properties of both the stellar
|
|
and gaseous components of clusters of galaxies. Most importantly, it
|
|
alters the properties of the central dominant (cD) galaxy by
|
|
reducing the mass deposition rate of freshly cooled gas out of the
|
|
ICM, thereby offering an energetically plausible solution to the
|
|
cooling flow problem. At the same time, this leads to reduced or
|
|
eliminated star formation in the central cD galaxy, giving it red
|
|
stellar colours as observed.
|
|
\end{abstract}
|
|
\begin{keywords}
|
|
methods: numerical -- galaxies: clusters: general
|
|
-- cooling flows -- cosmology: theory.
|
|
\end{keywords}
|
|
|
|
\section{Introduction}
|
|
|
|
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
|
|
\footnotetext[1]{E-mail: deboras@mpa-garching.mpg.de }
|
|
|
|
Clusters of galaxies are the largest virialised objects in the Universe and
|
|
are thought to contain a representative fraction of baryons \citep[]{White93}.
|
|
Most of these baryons can be found in the diffuse gas of the intracluster
|
|
medium (ICM), which is directly observable in X-rays, making clusters of
|
|
galaxies an almost ideal laboratory for studying the physical processes that
|
|
shape galaxies and halos in the Universe. Clusters of galaxies are also a
|
|
useful cosmological probe \citep[for a recent review see][]{Voit2004}, and
|
|
therefore have been a prime target for theoretical modelling early on, both
|
|
numerically and analytically.
|
|
|
|
A first order approximation for the ICM is to represent it as an ideal,
|
|
non-radiative gas. This leads to the predications of scale invariant relations
|
|
between X-ray luminosity, mass and temperature \citep[]{Kaiser86}. However,
|
|
it has long been established that the observed relations do not agree in
|
|
detail with these assumptions, e.g.~the observed $L_{\rm X}$--$T$ relation is
|
|
much steeper than expected based on this simple model. In addition, recent
|
|
observations with radio and X-ray telescopes have revealed a stunning
|
|
complexity of the ICM physics, including phenomena such as cold fronts, radio
|
|
ghosts, cluster turbulence, and apparently nearly uniform enrichment to high
|
|
metallicity.
|
|
|
|
However, possibly the most puzzling observational fact is the ``cooling-flow''
|
|
problem. Since the cooling time in the central regions of galaxy clusters is
|
|
smaller than the age of the clusters themselves, a central inflow of cool gas
|
|
is expected to occur \citep[e.g.][]{FabianNulsen77, Cowie77, Fabian94}. The
|
|
rate of gas cooling can be estimated by energetic arguments if one assumes
|
|
that X-ray cooling radiation is fed by the thermal reservoir of the hot
|
|
cluster plasma. Based on this, the estimated rate of accretion onto the
|
|
central galaxy is rather high in many cases
|
|
\citep[e.g.][]{Fabian84,White94,WJF97, Allen00, Allen2001}, implying that a
|
|
significant amount of gas cooler than $1-2\,{\rm keV}$ should be present in the
|
|
centre. However, up to the present time, optical and X--ray observations have
|
|
failed to detect the required amount of this cool gas, suggesting
|
|
that it is simply not there \citep[e.g.][]{McNamara00, Peterson01,Tamura01,
|
|
Balogh01, Kaastra01, Edge01, Edge02, Edge03, Fabian01, Boehringer2002,Peterson03,Salome2004}. The
|
|
low current star formation rates of central galaxies
|
|
\citep[e.g.][]{O'Connell89, Johnstone87,Allen95} provide additional support
|
|
for the absence of strong cooling flows. Apparently, there must be a physical
|
|
process that offsets the radiative cooling in the centre, preventing the gas
|
|
from falling out of the ICM in a cooling flow.
|
|
|
|
|
|
Theoretical studies have therefore often invoked some sort of
|
|
non--gravitational heating to explain the cluster scaling relations
|
|
\cite[e.g.][]{Kaiser91, Navarro95, Bower97, TozziN01, Borgani01, VoitBryan01,
|
|
Babul02, Voit02, Voit03, OhBenson03, Tornatore03, Borgani04}. The main
|
|
unsolved issue in these models remains the origin and nature of the physical
|
|
sources that cause the extra--heating of the ICM. Perhaps the most obvious
|
|
heat source is supernovae associated with star formation, but it seems
|
|
questionable that they are able to supply the required amount of feedback
|
|
energy. Curiously, radiative cooling alone may also account for the steepness
|
|
of the $L_{\rm X}$--$T$ relation by eliminating gas more efficiently in
|
|
low-mass systems \citep[e.g.][]{Lewis00, VoitBryan01, Muanwong01, Yoshida02,
|
|
WuXue02, Voit02, McCarthy04}, but this produces a drastic overprediction of
|
|
the amount of cold gas (apart from a problem with the $L_{\rm X}$--$T$
|
|
zero-point) and is therefore disfavoured. Models that self-consistently
|
|
incorporate SNe heating and radiative cooling processes are also only found to
|
|
have limited success \citep[e.g.][]{Borgani04}. The over--cooling problem
|
|
therefore remains unsolved. Another problem is posed by simulated temperature
|
|
profiles, which typically exhibit a trend to increase towards the cluster
|
|
centre, in disagreement with observational inferences \cite[e.g.][]{ Allen01,
|
|
DeGrandiM02}.
|
|
|
|
|
|
An array of different physical hypothesis have been proposed to solve the
|
|
cooling-flow paradox, including thermal conduction, magnetic fields, cosmic
|
|
rays, and hot buoyant bubbles from AGN jets. Thermal conduction may in
|
|
principle offset central cooling losses by inducing a heat current from outer,
|
|
hotter regions of clusters \citep[e.g.][]{Narayan01}, provided the
|
|
conductivity is not strongly suppressed by tangled magnetic fields. Analysis
|
|
of static cluster models with conduction have been able to provide good
|
|
matches to observed temperature profiles in some cases
|
|
\citep[e.g][]{Voigt02, Zakamska03, Voigt03} but detailed
|
|
self-consistent numerical simulations which followed conduction still
|
|
encountered the cooling flow problem \citep[e.g.][]{Jubelgas04, Dolag04},
|
|
making it questionable whether this can be the real solution.
|
|
|
|
The more widely favoured hypothesis is instead that the central AGN may supply
|
|
the required amount of energy. Accretion onto supermassive black holes is
|
|
thought to liberate of order $\sim 10\%$ of the accreted rest mass energy,
|
|
implying that even for low accretion rates onto a supermassive black hole,
|
|
offsetting the cooling flows is energetically quite possible. In fact, such
|
|
accretion powers high-redshift quasars, the most luminous sources in the
|
|
universe. Quasar activity is likely to be triggered by mergers of galaxies,
|
|
where cold gas is forced to the nuclei by gravitational tidal forces. This
|
|
accretion and the associated quasar feedback has recently been incorporated
|
|
into simulations, and shown to play a potentially important role in shaping
|
|
the properties of elliptical galaxies \citep[]{Springel2005}.
|
|
|
|
In clusters of galaxies, however, it seems clear that the central AGN activity
|
|
that causes bubbles is of a different nature, and needs not be triggered by
|
|
galaxy mergers. Observationally, many clusters of galaxies show evidence for
|
|
X-ray cavities filled with radio plasma \citep[e.g.][]{Owen2000,Blanton01}, which are
|
|
thought to be inflated by relativistic jets from the AGN. Theoretically, it has
|
|
been shown that these bubbles may rise buoyantly and raise some of the
|
|
central cool gas \citep[e.g.][]{Churazov01}, allowing it to mix with
|
|
the hotter gas further out. Together with the accompanying mechanical and
|
|
possibly viscous heating, this can then constitute an efficient feedback
|
|
mechanism.
|
|
|
|
|
|
In this paper, we focus on the phenomenology of this bubble feedback, without
|
|
addressing the small scale physics of the accretion onto the black hole. This
|
|
extends earlier simulation studies which all employed hydrodynamical mesh
|
|
codes, but which focused exclusively on highly idealised cluster models
|
|
\citep[e.g.][]{Churazov01, Quilis01, Ruszkowski02, Churazov02,
|
|
Brueggen02,Brueggen02b,Brueggen03,Nulsen03, DVecchia04, Hoeft04}. A first
|
|
goal of our work is to demonstrate that such simulations are also possible with the
|
|
smoothed particle hydrodynamics (SPH) technique, and give results
|
|
consistent with earlier studies. This is important because
|
|
the Lagrangian nature of SPH is ideal for cosmological simulations of
|
|
structure formation, and if applicable for bubble feedback, will allow us to
|
|
carry out the first self-consistent cosmological simulations with
|
|
AGN-driven bubble heating. An equally important goal of our simulations is to gain new
|
|
insights into the efficiency of bubble feedback associated with AGN for
|
|
modifying the thermodynamic state of the ICM and the properties of cluster
|
|
galaxies over the course of cosmic history. Our modelling can hence inform
|
|
semi-analytic models of galaxy formation that have just begun to include AGN
|
|
feedback \citep[e.g.][]{Croton05}, and provide crucial input for
|
|
future hydrodynamic simulations that try to incorporate the growth of
|
|
supermassive black holes both from the quasar- and the radio-mode.
|
|
|
|
|
|
The outline of this paper is as follows. In Section~\ref{MET}, we describe the
|
|
characteristics of our simulation code and the numerical method adopted to
|
|
introduce bubble heating. In Section~\ref{ISO} we analyse the AGN heating in
|
|
isolated galaxy halos, spanning a wide range in mass, and we present some
|
|
Chandra-like photon images of simulated bubbles. The effects of AGN
|
|
heating in cosmological simulations of galaxy cluster
|
|
formation is discussed in Section \ref{AGN_cosmo}. Finally, in Section
|
|
\ref{DIS} we discuss successes and limitations of our model, and we present
|
|
our conclusions.
|
|
|
|
|
|
\section{Methodology} \label{MET}
|
|
|
|
\subsection{Basic code properties}
|
|
|
|
Our simulations have been performed with the parallel TreeSPH-code {\small
|
|
GADGET-2} \citep{Gadget2,SYW01b}. We use the `entropy formulation' for SPH
|
|
suggested by \citet{SH02}, which manifestly conserves both energy and entropy
|
|
when adaptive smoothing lengths are used. Besides gravitational and
|
|
hydrodynamical processes, we include radiative cooling of the gas component,
|
|
together with heating by a spatially uniform, time dependent UV background
|
|
modelled as in \citet{KWH96}. The gas consists of an optically thin primordial
|
|
plasma of hydrogen and helium. In addition, a multiphase subresolution model
|
|
for the treatment of star formation and associated feedback mechanisms has
|
|
been adopted \citep{SH03}. In this model, stars form from dense cold gas
|
|
clouds assumed to reside at pressure equilibrium in a surrounding hot phase of
|
|
the interstellar medium. Supernova explosions heat the hot medium and
|
|
evaporate cold clouds, thereby providing a self-regulation cycle for star
|
|
formation, and a net pressurisation for the highly overdense ISM.
|
|
Additionally, we use a simple prescription for metal enrichment, which assumes
|
|
that each star-forming gas element can be locally approximated by a closed box
|
|
model in which perfect and instantaneous mixing of metals between cold clouds
|
|
and ambient gas occurs, as explained in detail in \cite{SH03}.
|
|
|
|
|
|
|
|
\subsection{Phenomenological description of AGN heating in clusters}
|
|
|
|
Besides considering the physical processes already implemented in {\small
|
|
GADGET-2}, we have implemented for this study a new model that accounts for
|
|
heating by the central AGN in clusters of galaxies. This model does not
|
|
attempt to provide a fully self-consistent ab initio treatment of the complex
|
|
physical processes related to accretion onto supermassive black holes in
|
|
clusters and the associated AGN activity. Rather, we try to mimic the observed
|
|
phenomenology of hot bubbles in clusters directly in our simulations, without
|
|
addressing the jet physics that presumably inflates the bubbles in the first
|
|
place. We therefore assume as a starting point that such bubbles are generated
|
|
during phases in which an AGN is ``switched on'', and introduce them into the
|
|
IGM in a phenomenological fashion. This allows us to study how the bubbles
|
|
affect the properties of the central ICM as a function of their
|
|
characteristics, in particular with respect to distributing their energy
|
|
content to the surrounding cooler gas.
|
|
|
|
|
|
For definiteness, we assume in our model that a certain amount of thermal
|
|
energy is injected in the form of centrally concentrated bubbles spaced in
|
|
uniform time intervals. We parameterise this scheme in terms of the AGN duty
|
|
cycle, the amount of energy $E_{\rm bub}$ injected, and by the radius $R_{\rm
|
|
bub}$ and distance $d_{\rm bub}$ of the buoyant bubbles from the cluster
|
|
centre, respectively.
|
|
|
|
|
|
We first test our scheme for AGN-heating on isolated, axisymmetric halo
|
|
models. These systems are clean laboratories which permit us to compare
|
|
directly with analogous modelling in the literature \citep[e.g.][]{Churazov01,
|
|
Quilis01, DVecchia04}, and hence to evaluate whether SPH is suitable for
|
|
such simulations. Moreover, these simplified models give us the possibility
|
|
to explore straightforwardly and with comparatively low computational cost a
|
|
large number of cases. In this way we can investigate the importance of
|
|
different physical parameters of the bubbles, thus constraining their
|
|
dynamical evolution and the heat transport into the ICM.
|
|
|
|
As a second step, we apply the model for bubble heating to fully
|
|
self-consistent cosmological simulations of galaxy cluster formation. Here, we
|
|
also investigate different redshift-dependent energy injection schemes,
|
|
allowing us to gain some insight in how the AGN activity influences the
|
|
hierarchical galaxy cluster growth and the characteristics of the central
|
|
cluster galaxy, and to elucidate the relative importance of AGN heating with
|
|
respect to the other physics included. We consider a set a galaxy clusters
|
|
spanning a range in mass because we expect the efficiency of bubble heating
|
|
to have a significant mass dependence.
|
|
|
|
|
|
Both for isolated halos and in cosmological simulations, we explored
|
|
two different schemes for spatially placing the bubbles around the
|
|
cluster centres. In the first scheme, the bubbles are introduced randomly
|
|
within a sphere with a radius given by $d_{\rm bub}$ around the centre, while in the second
|
|
approach, two symmetric bubbles are placed randomly along a fixed axis
|
|
of length $2\times d_{\rm bub}$, which has an orientation preserved
|
|
in time during subsequent bubble events. The latter hence mimics a situation
|
|
where the AGN jet that inflates the bubbles has directional stability over
|
|
time, which could arise due to some coupling with the host galaxy's angular
|
|
momentum, for example. At the present time there is no clear evidence either
|
|
way concerning what is the preferred scenario, therefore our main aim is to
|
|
investigate the possible differences in the ICM properties between these two
|
|
bracketing scenarios.
|
|
|
|
\subsection{Constraining the model parameters}
|
|
|
|
Our choice for the values of $R_{\rm bub}$ and $d_{\rm bub}$ has been guided
|
|
by observational constraints on X--ray cavities in clusters, and also by the
|
|
values typically adopted in previous numerical works, for easier comparison.
|
|
For simplicity, we restricted most of our simulations to the case where the
|
|
values of $R_{\rm bub}$ and $d_{\rm bub}$ depend only on the mass of the halo
|
|
under consideration, and on the redshift in the case of cosmological
|
|
simulations. Specifically, we adopted \be
|
|
\label{Rbub(M_200)}
|
|
R_{\rm bub} \, \propto \, M_{200}(z)^{1/3}\times
|
|
\frac{1+z}{(\Omega_{0m}(1+z)^3 + \Omega_{0\Lambda})^{1/3}}\,, \ee where
|
|
$M_{200}(z)$ is the virial mass of the host galaxy cluster at given redshift
|
|
of AGN activity, and the same scaling has been adopted for $d_{\rm bub}$. For
|
|
the simulations of isolated halos, we used the same dependence of $R_{\rm
|
|
bub}$ and $d_{\rm bub}$ on cluster mass, setting $z=0$.
|
|
|
|
We study multiple bubble injection events in order to analyze how AGN heating
|
|
couples with radiative cooling losses over a sufficiently long time interval.
|
|
Thus, our modeling requires prescriptions both for the AGN duty cycle and for
|
|
the time evolution of the energy content stored in the bubbles. However, most
|
|
of the observed AGN-driven bubbles are found at low redshifts
|
|
\citep[e.g.][]{Birzan04}, and only recently some observational evidence for
|
|
X--ray cavities in more distant galaxy clusters has been found
|
|
\citep{McNamara05}. Therefore, the properties and presence of radio-bubbles at
|
|
higher redshifts, and their evolution with time, are observationally rather
|
|
unconstrained. We hence limit ourselves in this work to simple parametric
|
|
prescriptions for the evolution of $E_{\rm bub}$, derived from basic
|
|
theoretical considerations and empirical laws, which hopefully bracket
|
|
reality. Typically, we started injecting bubbles at redshift $z=3$, which is
|
|
the epoch that roughly corresponds to the peak of the comoving quasar space
|
|
density, but we also tested an earlier epoch given by $z=6$ for the start of
|
|
the bubble activity. For our modeling of the evolution of $E_{\rm bub}$ with
|
|
time, we adopted two scenarios with rather different behaviour. In the first
|
|
one, most of the energy is released at late epochs, while in the second one,
|
|
the bubble energy is coupled more closely to an assumed BH accretion rate
|
|
(BHAR) model for the growth of the black hole population as a whole, such that
|
|
the energy release is more pronounced at high redshifts.
|
|
|
|
More specifically, our first model is loosely motivated by the
|
|
Magorrian relationship, which implies $M_{\rm BH} \propto
|
|
\sigma^4$. A relation between the bubble mechanical luminosity
|
|
and the black hole accretion rate, $\dot M_{\rm BH}$, can be derived
|
|
by assuming that only a small fraction of the total bolometric
|
|
luminosity thermally couples with the ICM. Hence, $L_{\rm bub} = f
|
|
\times L_{\rm bol} = f \times \epsilon \dot M_{\rm BH}c^2$. The factor
|
|
$f$ sets the efficiency of thermal coupling with the ICM, and is
|
|
typically assumed to lie in the range of 1-5\%, while $\epsilon$ is
|
|
the radiative efficiency factor. Assuming that the mechanical
|
|
luminosity for Eddington-limited accretion is directly proportional to
|
|
the black hole mass, the energy content $E_{\rm bub}$ of the bubbles
|
|
is then proportional to $M_{200}(z)^{4/3}$, provided the mass of the
|
|
central cluster galaxy scales self-similarly with the cluster mass.
|
|
Hence, it follows that in this model the bubble energy content is
|
|
determined by the mass assembly of the host galaxy cluster with time.
|
|
|
|
In our second scenario, we instead relate the amount of bubble energy to the
|
|
average growth rate of supermassive central black holes. To describe the
|
|
latter, we employ an estimate of the BHAR by \cite{DiMatteo03}, who give an
|
|
analytic fit \be \dot \rho(z) \,= \, \epsilon_{\rm BH} \,\frac{b \,
|
|
\rm{exp}\it[a(z-z_m)]}{b - a + a\,\rm{exp} \it[b(z-z_m)]} \, , \ee for their
|
|
numerical results, with the parameters $a=5/4$, $b=3/2$, $z_m=4.8$, and $
|
|
\epsilon_{\rm BH} = 3 \times 10^{-4} {\rm M}_\odot \, {\rm yr}^{-1} \, {\rm
|
|
Mpc}^{-3}$. Thus, for every duty cycle of AGN activity we can directly
|
|
relate $\dot M_{\rm BH}$ with $E_{\rm bub}$ in the following way, \be
|
|
\label{Ebub_BHAR_eq} \frac{E_{\rm bub}}{E_{\rm norm}} = f \times \epsilon \times c^2
|
|
\int_{z_1}^{z_2} \dot \rho(z)\, {\rm d}z \, , \ee where a normalisation
|
|
factor, $E_{\rm norm}$, has been introduced which we set such that the total
|
|
energy injected over all duty cycles is the same in our two schemes. We note that the different temporal evolution of the BH mass in this
|
|
approach implies a significantly reduced energy content of the bubbles at low
|
|
redshifts. Finally, there is still one free constant of integration which we
|
|
choose by requiring that the assumed mass of the black hole is the same at
|
|
$z=0$ in both scenarious.
|
|
|
|
A number of observational and theoretical works
|
|
\citep[e.g.][]{Birzan04,Sanderson2005,McNamara05,Nulsen05,Nulsen2005,
|
|
Donahue2005,Voit2005} have constrained the plausible time interval between
|
|
two successive bubble injection episodes to be of order of $\Delta t_{\rm
|
|
bub}\sim 10^8$yrs. Clearly, $\Delta t_{\rm bub}$ could vary both for
|
|
clusters of different mass and also in time, especially if the bubble activity
|
|
is triggered by a self-regulated mechanism that operates between AGN feedback
|
|
and the cooling flow. Nevertheless, given our simple phenomenological
|
|
approach and lack of any better observational constraints, we adopt the value
|
|
of $\Delta t_{\rm bub}\,=\, 10^8$yrs for all of our cluster simulations,
|
|
independent of the cosmological epoch.
|
|
|
|
|
|
While some of our prescriptions for bubble parameters are motivated by
|
|
``quasar-like'' phenomena, our models are really meant to reflect a mode of
|
|
feedback by supermassive black holes different from that of ordinary quasars.
|
|
Instead of being triggered by mergers and being fueled with dense and cold ISM
|
|
gas, the bubbles are a model for the radio activity observed in clusters.
|
|
Note that there are also newly emerging theoretical models
|
|
\citep[e.g.][]{Croton05,Churazov2005} on how both quasar activity at higher
|
|
redshifts and AGN-driven radio bubbles at lower redshifts can be described
|
|
within a common unified framework. We will discuss this possibility in more
|
|
detail in our conclusions.
|
|
|
|
|
|
\section{AGN heating of isolated galaxy clusters} \label{ISO}
|
|
|
|
|
|
We here analyse simulations of isolated halos, consisting of a static NFW dark
|
|
matter halo \citep{NFW96, NFW97} with a gaseous component that is initially in
|
|
hydrostatic equilibrium and chosen to follow a density distribution similar to
|
|
the NFW dark matter profile, but slightly softened at the centre according to
|
|
\be \rho_g(r) \, = \,\frac{f_b \, \delta_0 \, \rho_{\rm
|
|
crit}}{(r+r_0)/r_s(1+r/r_s)^2}, \ee where $r_0$ is a parameter introduced
|
|
to mimic a gas core radius. The baryonic fraction is given by $f_b$, while
|
|
$\rho_{\rm crit}$ is the critical density, $\delta_0$ is the characteristic
|
|
overdensity and $r_s$ is the scale radius. The gas follows the equation of
|
|
state of an ideal monoatomic gas with adiabatic index $\gamma = 5/3$.
|
|
Besides, a certain amount of angular momentum has been imposed that can be
|
|
quantified by the dimensionless spin parameter of a halo, \be \lambda \, = \,
|
|
\frac{J|E|^{1/2}}{GM^{5/2}}, \ee where $J$ represents the angular momentum,
|
|
$M$ is the halo mass, and $E$ its total energy.
|
|
|
|
|
|
The boundary conditions were chosen to be vacuum, i.e. both density and
|
|
pressure are initially zero outside the virial radius (defined here as the
|
|
radius enclosing a mean density equal to $200\,\rho_{\rm crit}$). We have
|
|
simulated halos with a wide range of masses, with virial radii and
|
|
concentration parameters as listed in Table \ref{tab_simpar_iso}. The baryonic
|
|
fraction, $f_b=0.12$, the spin parameter, $\lambda=0.05$, and $r_0=0.02
|
|
R_{200}$ were kept fixed for all the halos. When evolved without radiative
|
|
cooling, these initial models are perfectly stable for more than $1/4$ of the
|
|
Hubble time, as we explicitly checked. This is the timespan we subsequently
|
|
consider in all our non-trivial simulations, both for the case with cooling
|
|
and star formation only, and also for the case with additional AGN heating.
|
|
|
|
%%
|
|
\begin{table*}
|
|
\bc
|
|
\begin{tabular}{cccccccc}
|
|
\hline
|
|
\hline
|
|
$M_{200}$ [$\,h^{-1}{\rm M}_\odot\,$] & $R_{200}$ [$\,h^{-1}{\rm kpc}$] & $c$ & $N_{\rm gas}$ & $m_{\rm gas}$ [$\,h^{-1}{\rm M}_\odot\,$] & $\epsilon$ [$\,h^{-1}{\rm kpc}\,$] \\
|
|
\hline
|
|
$ 10^{12}$ & 206 & 12.0 & $3 \times 10^{5}$ & $4.0 \times 10^5$ & 1.0 \\
|
|
$ 10^{13}$ & 444 & 6.5 & $3 \times 10^{5}$ & $4.0 \times 10^6$ & 2.0 \\
|
|
$ 10^{14}$ & 957 & 8.0 & $3 \times 10^{5}$ & $4.0 \times 10^7$ & 5.0 \\
|
|
$ 10^{15}$ & 2063 & 5.0 & $3 \times 10^{5}$ & $4.0 \times 10^8$ & 10.0 \\
|
|
$ 10^{15}$ & 2063 & 5.0 & $1 \times 10^{6}$ & $1.2 \times 10^8$ & 6.5 \\
|
|
\hline
|
|
\hline
|
|
\end{tabular}
|
|
\caption{Numerical parameters of the isolated galaxy clusters. The
|
|
first two columns give the virial mass and radius of the halos,
|
|
evaluated at $200\, \rho_{\rm crit}$. The assumed values for the
|
|
concentration parameter are in the third column, while the initial
|
|
number and the mass of the gas particles is shown in the fourth and
|
|
the fifth columns, respectively. The mass of the star particles is
|
|
half that of the gas particles, because we set the number of
|
|
generations of star particles that a gas particle may produce to
|
|
two. Note that there are no parameters for the dark matter particles
|
|
in these run, because we modelled the dark halo with a static NFW potential.
|
|
Finally, in the last column, the gravitational softening length
|
|
$\epsilon$ for the gas and star particles is given.
|
|
\label{tab_simpar_iso}}
|
|
\ec
|
|
\end{table*}
|
|
%%
|
|
|
|
For the $10^{15} h^{-1}{\rm M}_{\odot}$ isolated cluster, our fiducial set of
|
|
AGN heating parameters is (if not explicitly stated otherwise) $E_{\rm bub} =
|
|
5 \times 10^{60}\,{\rm erg}$, $R_{\rm bub} = 30\,h^{-1}{\rm kpc}$, $d_{\rm
|
|
bub} = 50\,h^{-1}{\rm kpc}$, and a duty cycle of $\Delta t_{\rm bub} =
|
|
10^8{\rm yrs}$, all kept fix in time. The thermal energy injected in the form
|
|
of bubbles has been estimated using the simple relations given in Section
|
|
\ref{MET}, assuming a $\sim (5 \times 10^8 - 3 \times 10^9) {\rm M}_{\odot}$
|
|
black hole in the cluster centre (depending on the thermal-coupling efficiency
|
|
factor $f$). For the halos of lower mass, we assumed that the thermal content
|
|
of the bubble is proportional to $M_{200}^{4/3}$, in analogy to our first
|
|
scenario for scaling the bubble energy in a cosmological setting. This energy
|
|
scaling is motivated by the well established observational relation between
|
|
the black hole mass and the velocity dispersion of the stellar component of
|
|
the bulge \citep[e.g.][]{Tremaine02}, given by \be M_{\rm BH} = (1.5 \pm 0.2)
|
|
\times 10^8\, {\rm M}_{\odot} \ \bigg(\frac{\sigma}{200 \ {\rm km \
|
|
s^{-1}}}\bigg)^{4.02 \pm 0.32}, \, \ee and on the hypothesis that the
|
|
central cluster galaxy scales self-similarly with the mass of the cluster
|
|
itself. Even though these assumptions are certainly very restrictive, they
|
|
provide us with a definite model that allows a straightforward interpretation
|
|
of the trends with mass, and supply some guidance for what to expect in full
|
|
cosmological simulations. We also note that the recent numerical work of
|
|
\cite{DiMatteo03} suggests that, once the $M_{\rm BH}-\sigma$ relation is
|
|
established with time, the black hole mass is proportional to $M_{\rm
|
|
DM}^{4/3}$ of the host galaxy.
|
|
|
|
|
|
\subsection{AGN heating of a massive galaxy cluster}
|
|
|
|
In this section we concentrate on the effects of bubble heating on an isolated
|
|
galaxy cluster of mass $10^{15} h^{-1}{\rm M}_{\odot}$, while we will discuss
|
|
the relative importance of AGN heating as a function of mass in the next
|
|
section. In Figures~\ref{Tmap_iso} and \ref{Tmap_isobig}, we show maps of the
|
|
projected mass-weighted temperature of the $10^{15} h^{-1}{\rm M}_{\odot}$
|
|
galaxy cluster, focusing on the central regions in order to highlight the
|
|
morphology of the bubbles with different injection schemes and at various
|
|
evolutionary stages. Both figures were obtained for simulations with cooling
|
|
and star formation, and with AGN feedback of the same strength. However, in
|
|
the left panel of Figure~\ref{Tmap_iso}, the bubbles were placed randomly
|
|
within a sphere of radius $d_{\rm bub}$, while the remaining three panels
|
|
illustrate the case of two symmetrical bubbles injected simultaneously,
|
|
containing half of the energy each, and with the injection axis preserved with
|
|
time for different bubble cycles.
|
|
|
|
|
|
|
|
In Figure \ref{Tmap_isobig}, we show results for simulations with the same
|
|
feedback energy as in Figure~\ref{Tmap_iso}, but this time the initial radius
|
|
of the bubbles was two times larger and equal to $60\,h^{-1}{\rm kpc}$. After
|
|
being injected, the bubbles rise due to buoyancy and start to assume more
|
|
elongated, ``pancake-like'' shapes, as clearly visible in the left panel of
|
|
Figure~\ref{Tmap_iso}. They continue to rise until the surrounding gas entropy
|
|
becomes comparable to their entropy content, at which point they settle into
|
|
an equilibrium and dissolve slowly with time. While rising, they push the
|
|
intracluster gas above them, and also entrain some of the cooler central gas,
|
|
transporting it to larger radii.
|
|
|
|
A closer comparison of Figures~\ref{Tmap_iso} and \ref{Tmap_isobig} makes it
|
|
clear that the smaller bubbles with their significantly higher energy per
|
|
particle result in more pronounced mushroom-like structures. Nevertheless,
|
|
they do not shock the surrounding gas which, on the very top of the bubbles,
|
|
forms cold rims. At late evolutionary stages, corresponding roughly to a
|
|
quarter of the Hubble time, (see the right panel of Figure \ref{Tmap_isobig}),
|
|
a characteristic bipolar outflow is visible as a result.
|
|
|
|
\begin{figure*}
|
|
\centerline{
|
|
\hbox{
|
|
\psfig{file=figs/bcsf_Tm_iso_nn.0010.eps,width=9truecm,height=8truecm}
|
|
\hspace{0.3truecm}
|
|
\psfig{file=figs/jbcsf_Tm_iso10_6_nn.0010.eps,width=9truecm,height=8truecm}
|
|
}}
|
|
\caption{Projected mass-weighted temperature maps of the central
|
|
regions of an isolated galaxy cluster of mass $10^{15} h^{-1}{\rm
|
|
M}_{\odot}$. In the left panel, bubbles have been introduced with a random
|
|
placement inside a spherical region, while in the right panel, a
|
|
``jet-like'' injection of bubbles is shown where two bubbles are placed
|
|
opposite of each other, and subsequent generations of bubbles are injected
|
|
along the same spatial axis. $E_{\rm bub}$, $R_{\rm bub}$ and $d_{\rm bub}$
|
|
in both cases are the same, and given by $5 \times 10^{60} {\rm erg}$,
|
|
$30\,h^{-1}{\rm kpc}$ and $50\,h^{-1}{\rm kpc}$, respectively. The maps have
|
|
been constructed for times of $\sim 1.4\, {\rm Gyr}$ and $\sim 0.8\, {\rm
|
|
Gyr}$ after the beginning of the runs.}
|
|
\label{Tmap_iso}
|
|
\end{figure*}
|
|
|
|
|
|
\begin{figure*}
|
|
\centerline{
|
|
\hbox{
|
|
\psfig{file=figs/jbcsf_Tm_iso10_6.0022.eps,width=9truecm,height=8truecm}
|
|
\hspace{0.3truecm}
|
|
\psfig{file=figs/jbcsf_Tm_iso10_6.0041.eps,width=9truecm,height=8truecm}
|
|
}}
|
|
\caption{Time evolution of the isolated $10^{15} h^{-1}{\rm M}_{\odot}$
|
|
galaxy cluster with a jet-like AGN heating. The models are the same as in
|
|
Figure~\ref{Tmap_iso}, only the bubbles have two times bigger radii,
|
|
namely $60\,h^{-1}{\rm kpc}$. It can be noticed how the morphology in the
|
|
central cluster region changes due to bubble-induced motions, from $\sim
|
|
1.8\,{\rm Gyr}$ (left panel) to $\sim 3.3\,{\rm Gyr}$, when a well-defined
|
|
bipolar outflow is clearly visible (right panel).}
|
|
\label{Tmap_isobig}
|
|
\end{figure*}
|
|
%%
|
|
|
|
|
|
|
|
|
|
In Figure \ref{Prof_iso}, we analyse the global gas properties of the
|
|
cluster in terms of radial profiles of density, temperature and
|
|
entropy of the hot gas component, i.e.~gas of the cold interstellar
|
|
medium is not included in the plots\footnote{The `cold' gas
|
|
component has been defined here as all gas cooler than $1\, \rm{keV}$
|
|
and with a density higher than the star formation density
|
|
threshold.}. Bubble injection modifies the inner $100\,h^{-1}{\rm
|
|
kpc}$ substantially, reducing the density and increasing the
|
|
temperature profile. Accordingly, the entropy of the central gas
|
|
particles is changed as well, and an entropy floor is formed. The
|
|
lower right panel of Figure~\ref{Prof_iso} shows the mean gas inflow
|
|
rate in the central $30\,h^{-1}{\rm kpc}$. After a relatively brief
|
|
period of time, AGN heating regulates, in a stable fashion, the flow
|
|
of gas towards the centre, preventing the unrealistically high mass
|
|
deposition rates of a fully developed cooling flow, which can reach up
|
|
to $1200\,{\rm M}_{\odot}{\rm yr}^{-1}$ in the case without bubble
|
|
heating. Even though a repeated injection of bubbles along the same
|
|
spatial axis (``jet-like'') is somewhat more efficient than a random
|
|
placement within a sphere, the gas profiles have very similar trends
|
|
in both cases, indicating the robustness of the results with respect
|
|
to these details of the bubble injection scheme, at least in
|
|
situations free of secondary effects due to infalling structures and
|
|
mergers.
|
|
|
|
|
|
\begin{figure*}
|
|
\centerline{\vbox{
|
|
\hbox{
|
|
\psfig{file=figs/Deniso_csfbn26.eps,width=8.truecm,height=7.5truecm}
|
|
\hspace{1.0truecm}
|
|
\psfig{file=figs/Tiso_csfbn26.eps,width=8.truecm,height=7.5truecm}
|
|
}
|
|
\hbox{
|
|
\psfig{file=figs/Siso_csfbn26.eps,width=8.truecm,height=7.5truecm}
|
|
\hspace{1.0truecm}
|
|
\psfig{file=figs/meanMinflowiso_csfbn26.eps,width=8.truecm,height=7.5truecm}
|
|
}
|
|
}}
|
|
\caption{Radial profiles of gas density (upper left panel),
|
|
temperature (upper right panel) and entropy (lower left panel) of
|
|
the isolated $10^{15} h^{-1}{\rm M}_{\odot}$ halo. Blue lines: run
|
|
with cooling and star formation. Red lines: AGN feedback mechanism
|
|
also included (random placement of bubbles). The vertical dotted
|
|
lines denote $R_{200}$. Lower right panel: Mean mass inflow rate in
|
|
the central $30\,h^{-1}{\rm kpc}$ as a function of time, normalised
|
|
to the Hubble time. It can be seen that the mass deposition rate onto
|
|
the central galaxy is substantially reduced with AGN heating and
|
|
stabilised at $\sim 150\,{\rm M}_{\odot}{\rm yr}^{-1}$ after a
|
|
relatively brief period of time.}
|
|
\label{Prof_iso}
|
|
\end{figure*}
|
|
%%
|
|
|
|
|
|
In Figure \ref{Soundwave}, we show unsharped masked maps of the X--ray
|
|
emissivity of one of our cluster models. The X-ray emission has been estimated
|
|
using the bremsstrahlung approximation, \be \label{Lx_eq} L_{\rm X} = 1.2 \times
|
|
10^{-24} \frac{1} {\mu^2 m_p^2} \sum_{i=0}^{N_{\rm gas}} m_{{\rm gas},i} \
|
|
\rho_{i} \ T_i^{1/2} \quad [{\rm erg\,s^{-1}}]. \ee The unsharp-masking has
|
|
been performed by subtracting from the original projected $L_{\rm X}$-map the same
|
|
map smoothed on a $100\,h^{-1}{\rm kpc}$ scale. A large number of
|
|
centrally concentrated ripples are clearly visible in the result. These
|
|
ripples in the X--ray emissivity are sound waves generated by the expansion of
|
|
the bubble after the thermal energy is injected. The sound waves
|
|
travel through
|
|
the cluster, and if the IGM has a residual viscosity they can be dissipated,
|
|
providing a nonlocal heating of the central cluster volume. We note that we
|
|
have explored different scales over which the smoothing in the unsharped
|
|
masked technique is performed, obtaining the sharpest and most prominent
|
|
features for smoothing scales corresponding to approximately $100\,h^{-1}{\rm
|
|
kpc}$, which by order of magnitude agrees with the dimension of the ripples
|
|
themselves.
|
|
|
|
|
|
We find that the ripples reach distances of $\sim 800\,h^{-1}{\rm kpc}$ after
|
|
$1\,{\rm Gyr}$, translating to a velocity of order $\sim 10^3\,{\rm
|
|
km\,s^{-1}}$, which matches the expected sound speed in the ICM of this
|
|
cluster. At larger radii, the sound waves are not detectable any more. Note
|
|
that we also expect that their velocity drops strongly in the outskirts of the
|
|
cluster, where the temperature and hence the sound speed decline.
|
|
|
|
|
|
|
|
Upon closer inspection, it can be seen that the ripples are actually slightly
|
|
offset from the cluster centre, with their midpoint directly matching the
|
|
initial coordinates of the injected bubble. Moreover, the ripples
|
|
progressively lose their intensity at larger radii, both due to a $1/r^2$
|
|
dilution of their intensity, and to a lesser extent, due to a damping caused
|
|
by the residual viscosity of our SPH scheme. Note that some level of
|
|
numerical viscosity is instrinsic to all SPH schemes, even though we are
|
|
modelling an ideal gas. Quantifying the exact magnitude of the resulting
|
|
effective viscosity is not trivial, also because it depends on the spatial
|
|
resolution achieved in the simulations. However, recent observations of
|
|
optical H$\alpha$-filaments \citep{Fabian03} suggest that the gas in the
|
|
central region of the cool-core cluster Perseus might be quite viscous, rather
|
|
then turbulent \cite[but see][for estimates of gas turbulence on smaller
|
|
scales]{Ensslin2005}. If ICM viscosity is relevant, it would imply a high rate
|
|
of dissipation of the energy contained in the sound waves at small radii. This
|
|
physical viscosity could be easily higher than the numerical viscosity we have
|
|
in our simulations. Naturally, it is then desirable to treat the dissipation
|
|
process accurately, which requires an SPH discretization of the Navier-Stokes
|
|
equation combined with an assumed level of physical Spitzer-viscosity.
|
|
Recently, first mesh-based studies of isolated clusters with viscosity and
|
|
bubble heating have appeared
|
|
\citep[e.g.][]{Ruszkowski04,Reynolds05,Brueggen05}. We plan to investigate
|
|
this theoretical issue in a forthcoming study.
|
|
|
|
|
|
\begin{figure}
|
|
\bc
|
|
\centerline{\includegraphics[width=9.5truecm,height=8.5truecm]{figs/Lxun_iso_bcsf_016.eps}}
|
|
\caption{Unsharped masked map of the X--ray luminosity in the central
|
|
region of the $10^{15}\, h^{-1} {\rm M}_{\odot}$ isolated halo, at
|
|
time $\sim 2.2\,{\rm Gyr}$. The unsharped masking has been performed
|
|
by subtracting a smoothed map from the original projected X-ray
|
|
emissivity map, with the smoothing scale set to $ 100\,h^{-1}{\rm
|
|
kpc}$. It can be seen that the AGN bubble heating generates a number
|
|
of sound waves, which could gradually release their energy to the ICM
|
|
if they are viscously damped on their way to the cluster outskirts.}
|
|
\label{Soundwave}
|
|
\ec
|
|
\end{figure}
|
|
|
|
%%
|
|
|
|
|
|
%%
|
|
In Figure~\ref{BubbleEntropy}, we show the locus of selected particles in the
|
|
$\log S - \log R$ plane, at a time equal to one quarter of the Hubble time
|
|
(which marks the end of most of our isolated simulations). We selected only
|
|
particles that at least once belonged to one of the injected bubbles. For
|
|
easier comparison, the mean radial entropy profile of the AGN heated cluster
|
|
(the same as in Figure~\ref{Prof_iso}) is plotted as a red dot-dashed line.
|
|
The dots shown for each individual bubble particle have been colour-coded
|
|
according to their relative temperature. The particles of a recently injected
|
|
bubble have the highest temperature values. As expected, their entropy values
|
|
lie substantially above the average entropy of the cluster for the same range
|
|
of radii, implying that the bubble will rise due to buoyancy. We thus expect
|
|
these bubble particles to reach larger radii and to lose some of their thermal
|
|
energy content during the rise, and this expectation is borne out by the
|
|
cooler particles from older bubbles.
|
|
|
|
The radius at which the bubble particles attain the mean cluster entropy level
|
|
is set by their initial density and thermal content after injection, by their
|
|
capacity to shed some of the energy to the surrounding ICM, by their
|
|
radiative cooling efficiency, and by the amount of mixing. Given
|
|
that the different bubbles at various
|
|
epochs have very similar initial temperature and mass,
|
|
Figure~\ref{BubbleEntropy} implies that the bubbles reduce their temperature
|
|
almost by one order of magnitude, from the injection instant to the final
|
|
equilibrium position. If we sum up the total energy injected over the entire
|
|
simulated time, and assume that it gets thermalized over the whole cluster, we
|
|
obtain that the gas particle temperature is increased by $\sim 0.2\,{\rm
|
|
keV}$, which roughly corresponds to the bubble temperature decrement
|
|
mentioned above when the mass fraction of the bubbles is taken into account.
|
|
Thus, it appears that the radiative cooling is not severe inside the bubbles,
|
|
even though in the cluster as a whole it approximately balances the AGN
|
|
feedback mechanism.
|
|
|
|
We extended our investigation by considering `jet-like' injection of bubbles
|
|
and also a scenario in which the bubbles are inflated in a continuous fashion
|
|
over some time interval $t_{\rm inj}$. We tried values from $t_{\rm
|
|
inj}=5\times10^7{\rm yrs}$ to $t_{\rm inj}=5\times10^8{\rm yrs}$, which is
|
|
significantly longer than the sound crossing time over the scale $R_{\rm bub}$
|
|
of the bubble. The maximum radius reached by the bubbles is essentially
|
|
invariant in all of these cases, yielding $\sim 250-300\,h^{-1}{\rm kpc}$ at
|
|
the final simulated epoch. Also, the heating efficiency of buoyant bubbles
|
|
remains very similar, although the bubbles are somewhat more energetic in the
|
|
continuous injection scheme, presumably because cooling losses are reduced
|
|
here due to the expansion of the bubble before the bulk of the energy us
|
|
released. This is directly reflected in an even lower mass deposition rate
|
|
onto the central object, which always occurs in a stable fashion with time,
|
|
where the gas cooling inflow is balanced by the AGN heating rate.
|
|
|
|
It is important to point out that the entropy content of the bubbles and the
|
|
maximum distance they can reach from the cluster centre depend upon the
|
|
equation of state assumed for the gas belonging to the bubbles. In all our
|
|
models bubbles have been simulated assuming the equation of state of an ideal
|
|
gas. However, radio observations indicate that AGN-driven bubbles contain
|
|
relativistic particles which possibly dominate over the thermal pressure
|
|
component, implying a softer equation of state. Moreover, the energy contrast
|
|
of the individual bubbles is not very high in our approach, resulting in a
|
|
gentle ICM heating, without presence of significant shocks. In fact, most of
|
|
the observations of AGN-heated clusters point out that strong bubble-induced
|
|
shocks appear to be absent, although recently a few clusters with moderate
|
|
shocks in connection with AGN activity have been discovered
|
|
\citep{Fabian2003,Nulsen05,McNamara05}. Therefore, due to the assumptions of
|
|
our model, the maximum possible distance reached by the buoyant bubbles in the
|
|
cluster atmosphere may be underestimated with respect to the case where the
|
|
relativistic particle component is modelled as well.
|
|
|
|
\begin{figure}
|
|
\bc
|
|
\centerline{\includegraphics[width=8.6truecm,height=8.3truecm]{figs/Siso_csfbn25_bub.eps}}
|
|
\caption{Mean radial entropy profile of the $10^{15}\, h^{-1}{\rm
|
|
M}_{\odot}$ isolated halo. The result for the simulation with AGN
|
|
feedback is given by the red dot-dashed line. The dots show the
|
|
positions of the bubble particles, and they are colour-coded
|
|
according to their temperature.}
|
|
\label{BubbleEntropy}
|
|
\ec
|
|
\end{figure}
|
|
|
|
%%
|
|
|
|
|
|
%%
|
|
An important question is whether the bubbles are capable of raising cold gas
|
|
from the cluster centre and mixing it with higher entropy gas at larger radii.
|
|
Note that the X--ray observations of central metal abundance gradients put a
|
|
constraint on the amount of gas mixing \citep{Boehringer04} in the centre. In
|
|
order to address this issue, we analysed the gas metallicity distribution in
|
|
the clusters. Without AGN feedback mechanism, all metals produced in the
|
|
$10^{15}\, h^{-1} {\rm M}_{\odot}$ isolated halo are enclosed in the star
|
|
forming region and are confined to the very centre, where the density is
|
|
sufficiently high to allow star formation. The bubble heating instead
|
|
produces both a reduction of the star formation rate by heating some of the
|
|
gas that would otherwise end-up in the cold interstellar medium, and a
|
|
spreading of metals away from the cluster centre. Moreover, in our simple
|
|
model, the metals produced by the central stars are partly entrained and
|
|
transported along with the bubbles to larger radii. The spatial distribution
|
|
of the stars themselves is unaffected by the bubbles, however. It is important
|
|
to note that the metal mixing in our model due to the bubbles represents a
|
|
lower limit on the induced additional mixing, because fluid dynamical
|
|
processes that produce small-scale mixing tend to be underresolved in
|
|
cosmological simulations.
|
|
|
|
|
|
|
|
\subsection{Efficiency of bubble heating in halos of different mass}
|
|
|
|
The radiative cooling times of halos depend on their mass, both because of
|
|
their different virial temperatures, and because of the temperature dependence
|
|
of the cooling rate. Given that the masses of supermassive black holes, and
|
|
hence our assumed bubble feedback, have a mass dependence as well, we expect a
|
|
complex interplay between the different heating and cooling processes, and as
|
|
a result a mass-dependent efficiency of the feedback. This non-linear dynamics
|
|
can be best studied with detailed numerical analysis. To this end, we
|
|
simulated isolated halos for a range of masses, starting from $10^{12}\,
|
|
h^{-1}{\rm M}_{\odot}$ and reaching up to $10^{15}\, h^{-1}{\rm M}_{\odot}$,
|
|
with the characteristics listed in Table~\ref{tab_simpar_iso}. The case of the
|
|
most massive halo has been discussed in some detail in the previous section,
|
|
so that we can restrict ourselves here to highlight the differences that occur
|
|
when the mass of the systems is lowered. The study of smaller halos gives also
|
|
direct insight into the question of how bubble feedback may affect the
|
|
hierarchical assembly of present-day massive clusters.
|
|
|
|
In our numerical simulations, the coupled dynamics resulting from
|
|
cooling, subsequent star formation and a given heating mechanism is
|
|
quite complex. The introduction of a certain amount of heating can in
|
|
special situations even trigger an increase of the net cooling
|
|
rate. For example, let us consider star-forming gas with a density
|
|
higher than the density threshold set for star formation. If this cold
|
|
gas component receives an amount of thermal energy from bubble heating
|
|
that is insufficient to bring it back into the hot phase (where most
|
|
of the intracluster gas resides), then a counterintuitive process may
|
|
occur. In this case, the thermal energy injection prevents the cold
|
|
gas component from forming stars, but the local gas density will
|
|
remain comparatively high, which in turn stimulates even larger
|
|
radiative cooling losses. Thus, such a gentle heating, especially in
|
|
low mass systems where radiative cooling is more pronounced, can in
|
|
extreme cases even stimulate an increase of the cold gas component.
|
|
|
|
Analysing diagnostic phase-space diagrams like the $\log \rho - \log T$ plane,
|
|
one can notice that for the $10^{15}\,h^{-1} {\rm M}_{\odot}$ halo the
|
|
relative quantity of central cool gas is low, and it is promptly heated once
|
|
bubble injection is switched on. Moreover, the energy per bubble particle is
|
|
sufficiently high to push the gas into the ``hot-phase'' (upwards and to the
|
|
left in the diagnostic diagram, as illustrated for the $10^{14}\,h^{-1} {\rm
|
|
M}_{\odot}$ halo in Figure~\ref{Diagnostic_10^14}) and the star formation at
|
|
late simulated epochs is completely quenched.
|
|
|
|
%%
|
|
Examining the $10^{14}\,h^{-1}{\rm M}_{\odot}$ isolated halo, we find that
|
|
bubbles are still very efficient in reducing the star formation rate,
|
|
e.g.~after the time $t_{\rm H}/4$, only 14\% of stars are formed with respect to the
|
|
run without AGN heating, but the total amount of cold gas, both in the
|
|
central regions and out to $R_{200}$, remains very similar. The diagnostic
|
|
diagram for this cluster is shown in Figure~\ref{Diagnostic_10^14}. The small
|
|
orange dots are the gas particles for the run without AGN feedback, and they
|
|
maintain practically the same position even when bubble heating is included.
|
|
The big blue dots are those gas particles that belong to the `cold phase',
|
|
here defined as having temperatures less than $1\,{\rm keV}$ and densities
|
|
higher than the density threshold set for star formation (as indicated by the
|
|
vertical dotted line). With AGN heating included, the red dots denote the
|
|
bubble particles, while the green star symbols give the locations of cold
|
|
phase gas particles. Two different features due to the presence of bubble
|
|
feedback are readily apparent. First, the cold gas fraction for the
|
|
intermediate temperature range, from $5 \times 10^{5}\,{\rm K}$ to $10^7\,{\rm
|
|
K}$, is substantially reduced in the case with feedback, because for most of
|
|
these particles it is possible to transport them back to the hot phase. In
|
|
contrast, along the line in the lower-right part of the diagram, there are
|
|
more particles when bubble heating is included. This can be explained by the
|
|
fact that this line is determined by the multiphase structure of the ISM,
|
|
given here in terms of an effective mass-weighted mean temperature, which is a
|
|
combination of the temperature of cold gas clouds and the one of the hot ISM
|
|
component \citep[see][]{SH03}. Hence, while the feedback mechanism reduces the
|
|
number of stars formed, it produces a higher amount of interstellar gas with
|
|
very low temperatures. Note that a large fraction of these cold gas particles
|
|
have been a part of a bubble at some earlier epoch.
|
|
|
|
|
|
\begin{figure}
|
|
\bc
|
|
\centerline{\includegraphics[width=8.6truecm,height=8.3truecm]{figs/DD_14.eps}}
|
|
\caption{Phase-space diagram of gas temperature versus gas density for
|
|
the $10^{14}\, h^{-1}{\rm M}_{\odot}$ galaxy cluster. Small orange
|
|
dots are the particles outside the cold star-forming region, while big
|
|
blue dots denote the gas particles in the run with no feedback, at
|
|
high overdensities and with temperatures below $1\,{\rm keV}$. Green
|
|
star symbols are the particles satisfying the same criteria, but when
|
|
the AGN heating is included. Finally, the position of the bubble
|
|
particles is given by red dots.}
|
|
\label{Diagnostic_10^14}
|
|
\ec
|
|
\end{figure}
|
|
%%
|
|
|
|
In the lower mass system of mass $10^{13}h^{-1} {\rm M}_{\odot}$, the final
|
|
number of stars is also reduced, this time by only 13\%, and, as well as in
|
|
the previous case, the cold gas fraction is essentially unchanged. In the most
|
|
extreme case of the $10^{12}\,h^{-1} {\rm M}_{\odot}$ halo, the bubble heating
|
|
of the ICM is radiated away on a very short timescale without producing any
|
|
substantial modification.
|
|
|
|
|
|
We considered different injection assumptions in order to test whether this
|
|
effect can be alleviated and the bubble heating efficiency can be increased.
|
|
Focusing on the most difficult case of the $10^{12}\,h^{-1}{\rm M}_{\odot}$
|
|
halo, we explored a range of injection energies, from $5 \times 10^{56}\,{\rm
|
|
erg}$ per bubble, to $3 \times 10^{57}\,{\rm erg}$. Also, we injected
|
|
bubbles according to different schemes: in a spatially correlated `jet-like'
|
|
fashion, or by inflating the bubbles gradually for $t_{\rm
|
|
inj}=5\times10^7 {\rm yrs}$, i.e.~by
|
|
releasing the energy slowly instead of instantaneously. Finally, we also
|
|
tried a model where we artificially prevented bubble particles from cooling
|
|
for $10^7-10^8 {\rm yrs}$, motivated by some observational evidences
|
|
that the bubbles contain a non-thermal component, which would then be able to
|
|
maintain its pressure longer. From these experiments we conclude that relating
|
|
the bubble energetics to the underlying dark matter potential by invoking an
|
|
assumed relation with its black hole mass, does not in general provide stable
|
|
solutions for an efficient elimination of cooling flows in low mass systems.
|
|
Unrealistically high bubble energies are required in order to offset the
|
|
cooling flow, and also their thermal content has to be fine-tuned to a
|
|
restricted range of values, otherwise AGN heating may easily become so strong
|
|
that bubbles blow out substantial amounts of mass from the halo or cluster
|
|
potential well.
|
|
|
|
This clearly indicates an important deficiency of bubble models like the one
|
|
studied here. Since there is no internal trigger for bubble activity, a
|
|
self-regulation loop is missing, but this will be required for stability in
|
|
more general situations. Therefore, a more detailed physical scenario for the
|
|
triggering of bubble activity is needed which couples the local physics of
|
|
the cooling flow with the activity and energetics of AGN feedback.
|
|
We will discuss a number of possibilities for this in Section \ref{DIS}.
|
|
|
|
%%
|
|
\subsection{Observational X--ray features of simulated bubbles}
|
|
|
|
In the last few years, a growing number of observations performed with
|
|
the Chandra X-ray telescope have found the presence of
|
|
so called X--ray cavities in central galaxy cluster regions
|
|
\citep[e.g.][]{McNamara00, Fabian00, Sanders02, Mazzotta02,
|
|
Birzan04}. These observations support a scenario where relaxed
|
|
cooling-flow/cooling-core clusters are heated due to the presence of a
|
|
supermassive black hole at their centres. Hence, it is interesting to
|
|
see whether the simulated bubbles produced by our model have
|
|
morphologies comparable to the observed ones.
|
|
|
|
In order to perform this comparison accurately, it is necessary to produce
|
|
artificial emissivity maps which are as similar as possible to realistic
|
|
observations based on a finite exposure time on an X--ray
|
|
telescope. Recently, a similar analysis has also been performed by
|
|
\cite{Brueggen05}. To this
|
|
end, we processed selected simulation outputs with the {\small X-MAS} software
|
|
package. A final result of this code is a photon event file quite similar to
|
|
the one an observer would acquire with the Chandra telescope in ACIS-S3 mode.
|
|
The instrument background has been included in our images by taking into
|
|
account an appropriate blank-sky background file \citep[]{Markevitch01}. A
|
|
detailed description of the {\small X-MAS} simulator can be found elsewhere
|
|
\citep[e.g.][]{Gardini04}; here we limit the description to the subsequent
|
|
analysis steps performed and the significance of the maps obtained.
|
|
|
|
|
|
We have generated event files for different exposure times, ranging from
|
|
$10\,{\rm ks}$ to $1\,{\rm Ms}$, both for the runs with and without AGN
|
|
feedback. We then selected a number of different energy bands, performed a
|
|
Gaussian smoothing on a range of scales, and finally produced unsharp masked
|
|
images to search for evidence of systematic departures of the flux from the
|
|
mean. In Figure~\ref{XMAS_maps}, we show photon images of the central region
|
|
of the $10^{15}\,h^{-1} {\rm M}_{\odot}$ isolated halo\footnote{We decided to
|
|
perform this analysis on an isolated cluster in order to minimise other
|
|
features in the X--ray emissivity that would have been imprinted by
|
|
possible substructures or merger events. Further discussion of this issue in
|
|
the cosmological framework is given in Section~\ref{S2_24}.}, both in a case
|
|
with and without additional AGN heating. The physical scale of the maps
|
|
corresponds to $\sim 670\,{\rm kpc}$ (2048pix), the energy band has been
|
|
chosen to be $\Delta E = [0.3,1.5]\,{\rm keV}$, and the maps have been
|
|
smoothed by summing the pixel fluxes in bins of 4 pixels. For this $\Delta E$,
|
|
the instrument background is minimised and the features due to the presence of
|
|
the bubbles are more evident.
|
|
|
|
The first panel of Figure~\ref{XMAS_maps} shows a photon image of the
|
|
AGN-heated cluster after $100\,{\rm ks}$ of exposure time, before
|
|
applying any smoothing. The rest of the plots have been created by
|
|
Gaussian smoothing them first on a small scale (3pix), then
|
|
re-smoothing the obtained image on a bigger scale (15pix), and finally
|
|
unsharp masking the two smoothed images (i.e. subtracting off the
|
|
15pix smoothed version). The smoothing scales have been selected to
|
|
maximise flux departures from the mean. The second and the third panel
|
|
illustrate how the bubbles introduce emissivity irregularities for two
|
|
different exposure times, $100\,{\rm ks}$ and $1\,{\rm Ms}$,
|
|
respectively. Finally, the fourth panel presents the cluster photon
|
|
image after an exposure time of $100\,{\rm ks}$ and with no AGN
|
|
feedback.
|
|
|
|
It is clear that the bubbles generate characteristic fluctuations in
|
|
the photon counts, both creating bright features and X--ray
|
|
depressions. The typical dimension of these irregularities is $\sim
|
|
50\,{\rm kpc}$, very similar to the size of the bubbles
|
|
themselves. The hot spots can be associated with the most recent
|
|
bubble events, containing particles still significantly hotter than
|
|
the surrounding ICM (as can be also seen from
|
|
Figure~\ref{BubbleEntropy}), whereas the depressions in photon counts
|
|
can be explained with previous bubble episodes. These peculiarities in
|
|
emissivity are completely absent in the galaxy cluster without AGN
|
|
feedback, indicating that they are real features and not artifacts
|
|
produced by counting statistics or our analysis. The only feature that
|
|
is present in the fourth panel of Figure~\ref{XMAS_maps} is the
|
|
central excess due to the prominent cooling flow of $ \sim 100\,{\rm
|
|
kpc}$ size in diameter. Based on these results we conclude that in a
|
|
relaxed galaxy cluster, departures from the mean flux stemming from
|
|
bubbles with characteristics as given by our model can be detected,
|
|
provided the exposure times are long enough, and provided that other
|
|
sources of photon count deviations are absent or negligible.
|
|
|
|
\begin{figure*}
|
|
\centerline{
|
|
\hbox{
|
|
\psfig{file=figs/100ks.z.mm_4_ckg.eps,width=4truecm,height=4truecm}
|
|
\hspace{0.3truecm}
|
|
\psfig{file=figs/diff_100ks.z.mm_4_ckg_3_15.eps,width=4truecm,height=4truecm}
|
|
\hspace{0.3truecm}
|
|
\psfig{file=figs/diff1Ms_mm_ckg_3_15.eps,width=4truecm,height=4truecm}
|
|
\hspace{0.3truecm}
|
|
\psfig{file=figs/diff_mm_ckg_3_15.eps,width=4truecm,height=4truecm}
|
|
}}
|
|
\caption{Artificial photon images for the $10^{15}\, h^{-1}{\rm
|
|
M}_{\odot}$ isolated galaxy cluster obtained with the {\small X-MAS}
|
|
software package. In all the panels, photons have been selected in
|
|
the energy band $\Delta E = [0.3, 1.5]\,{\rm keV}$, maps were binned
|
|
in 4pix/bin, and the physical scale of the maps is $\sim
|
|
670\,{\rm kpc}$. The cluster emissivity map before applying any
|
|
smoothing is illustrated in the first panel, while all the other
|
|
panels have been obtained by unsharp masking previously smoothed
|
|
images, as explained in the text. The second and the third panels
|
|
show qualitatively very similar features. They are for the run with
|
|
AGN feedback and differ only in the exposure time. Finally, the
|
|
fourth panel shows how the same cluster appears when the bubble
|
|
heating is absent.}
|
|
\label{XMAS_maps}
|
|
\end{figure*}
|
|
%%
|
|
|
|
|
|
\section{Effect of AGN bubble heating in cosmological simulations} \label{AGN_cosmo}
|
|
|
|
%%
|
|
\subsection{Simulation characteristics}
|
|
As a next step in our analysis, we consider the importance of AGN feedback in
|
|
full cosmological simulations of cluster formation. To this end, we selected a
|
|
number of galaxy clusters with a wide range of masses from a parent dark
|
|
matter simulation, and resimulated them with higher resolution including gas
|
|
dynamics. Our hydrodynamical simulations account for cooling and star
|
|
formation, and additional AGN heating as well. In a subset of our runs, we also
|
|
included galactic winds powered by star formation, as implemented by
|
|
\cite{SH03}.
|
|
Models with winds provide a better description of some galaxy cluster
|
|
properties, e.g.~the distribution of metals, but in order to be able to
|
|
cleanly identify the effects of bubble heating, we focus most of our
|
|
analysis on simulations without winds. Where appropriate, we will
|
|
however briefly discuss any changes of our results when winds are also
|
|
included.
|
|
|
|
|
|
Our primary series of simulations consist of resimulations of a cluster
|
|
extracted from the GIF $\Lambda$CDM simulation \citep{K99}. We selected the
|
|
second most massive galaxy cluster from this simulation and constructed higher
|
|
resolution initial conditions for it using the ``Zoomed Initial
|
|
Conditions'' technique \citep{T97}. We carried out two runs with different
|
|
resolution in order to test numerical convergence of our results. These
|
|
clusters are equivalent to the ones used by \citet{Springel01}, but with gas
|
|
(from now on we will refer to these runs as S1 and S2, respectively). The
|
|
simulations have been evolved from an initial redshift of $z_{\rm ini}=30$ for
|
|
S1, and $z_{\rm ini}=50$ for S2, producing 25 outputs uniformly spaced in the
|
|
logarithm of the expansion factor. Additionally, we selected two other
|
|
clusters with final virial masses substantially smaller (g676) and bigger (g1)
|
|
than the S1/S2-cluster. These clusters have been extracted form a cosmological
|
|
$\Lambda$CDM simulation of box-size $479\, h^{-1}{\rm Mpc}$
|
|
\citep{Yoshida01,Jenkins01}, and again have been resimulated with the ZIC
|
|
technique at higher resolution \citep{Dolag2004}.
|
|
|
|
The simulation of the g1 galaxy cluster includes several other smaller
|
|
systems in the high-resolution region which we also included in our analysis.
|
|
Tables~\ref{tab_simpar} and~\ref{tab_GCpar} provide a summary of the
|
|
main properties of our set of simulated galaxy clusters. In all runs,
|
|
the cosmological parameters were that
|
|
of a flat concordance $\Lambda$CDM model, with $\Omega_{m}=0.3$,
|
|
$\Omega_{\Lambda}=0.7$, $\Omega_b=0.04$, a normalisation of the power spectrum
|
|
given by $\sigma_8=0.9$, and a Hubble constant at the present epoch of $H=70 \
|
|
{\rm km} \ {\rm s}^{-1} \ {\rm Mpc}^{-1}$.
|
|
|
|
\begin{table*}
|
|
\bc
|
|
\begin{tabular}{crrccccc}
|
|
\hline
|
|
\hline
|
|
Simulation & $N_{\rm HR}$ & $N_{\rm gas}$ & $m_{\rm DM}$
|
|
[$\,h^{-1}{\rm M}_\odot\,$] & $m_{\rm gas}$ [$\,h^{-1}{\rm
|
|
M}_\odot\,$] & $z_{\rm start}$ & $z_{\rm end}$ & $\epsilon$
|
|
[$\,h^{-1}{\rm kpc}\,$] \\
|
|
\hline
|
|
S1 & $450088$ & $450088$ & $5.96\times 10^{9}$ & $0.92\times 10^{9}$
|
|
& $30$ & $0$ & $14.5$ \\
|
|
S2 & $1999978$ & $1999978$ & $1.18\times 10^{9}$ & $0.18\times 10^{9}$
|
|
& $50$ & $0$ & $8.5$ \\
|
|
g676 & $314518$ & $314518$ & $1.13\times 10^9$ & $0.17\times 10^9$ &
|
|
$60$ & $0$ & $5.0$ \\
|
|
g1 & $4937886$ & $4937886$ & $1.13\times 10^9$ & $0.17\times 10^9$ &
|
|
$60$ & $0$ & $5.0$ \\
|
|
|
|
\hline
|
|
\hline
|
|
\end{tabular}
|
|
\caption{Numerical parameters of the cosmological galaxy cluster simulations used in this
|
|
study. The values listed from the second to the fifth column refer
|
|
to the number and to the mass of high resolution dark matter
|
|
particles and of gas particles. Note that the actual values of
|
|
$N_{\rm gas}$ and $m_{\rm gas}$ vary in time due to star formation.
|
|
The last three columns give the initial and final redshifts of the
|
|
runs, and the gravitational softening length $\epsilon$.
|
|
\label{tab_simpar}}
|
|
\ec
|
|
\end{table*}
|
|
|
|
|
|
\begin{table*}
|
|
\bc
|
|
\begin{tabular}{cccccc}
|
|
\hline
|
|
\hline
|
|
Cluster & $R_{\rm 200}$ [$\,h^{-1}{\rm kpc}\,$] & $M_{\rm 200}$
|
|
[$\,h^{-1}{\rm M}_\odot\,$] & $T_{\rm mw}$
|
|
[$K$] & $T_{\rm ew}$ [$K$] & $L_{\rm X}$ [$\,\rm ergs^{-1}\,$] \\
|
|
\hline
|
|
S1 & $2427$ & $9.98\times10^{14}$ & $5.2\times10^7$ &
|
|
$8.7\times10^7$ & $9.8\times10^{44}$ \\
|
|
S2 & $2466$ & $1.05\times10^{15}$ & $5.1\times10^7$ &
|
|
$8.8\times10^7$ & $9.9\times10^{44}$ \\
|
|
g676 & $1176$ & $1.13\times10^{14}$ & $1.4\times10^7$ &
|
|
$2.6\times10^7$ & $1.6\times10^{43}$ \\
|
|
g1\_a & $2857$ & $1.63\times10^{15}$ & $7.3\times10^7$ &
|
|
$1.3\times10^8$ & $1.0\times10^{45}$\\
|
|
g1\_b & $1914$ & $4.89\times10^{14}$ & $3.1\times10^7$ &
|
|
$4.1\times10^7$ & $1.2\times10^{44}$\\
|
|
g1\_c & $1448$ & $2.12\times10^{14}$ & $1.5\times10^7$ &
|
|
$2.5\times10^7$ & $3.0\times10^{43}$\\
|
|
g1\_d & $1258$ & $1.39\times10^{14}$ & $1.6\times10^7$ &
|
|
$1.9\times10^7$ & $1.8\times10^{43}$\\
|
|
g1\_e & $1085$ & $8.92\times10^{13}$ & $1.1\times10^7$ &
|
|
$1.6\times10^7$ & $7.0\times10^{42}$\\
|
|
\hline
|
|
\hline
|
|
\end{tabular}
|
|
\caption{Physical properties of our sample of simulated galaxy
|
|
clusters at $z=0$ and at $200\rho_c$. For different galaxy clusters,
|
|
labeled in the
|
|
first column, cluster radius, total mass, mass-- and
|
|
emission--weighted gas temperature and X--ray luminosity are listed,
|
|
respectively. Note that the values refer to the simulations with
|
|
cooling and star formation, without bubble heating included.
|
|
\label{tab_GCpar}}
|
|
\ec
|
|
\end{table*}
|
|
|
|
Unlike simulations of isolated clusters, cosmological simulations require a
|
|
special algorithmic method for placing bubbles, since the position of the
|
|
cluster centre and properties like virial mass are not known a priori, and
|
|
change with time. To address this problem, we run for every AGN duty cycle a
|
|
fast parallel FOF group finder on the fly as a part of the simulation code,
|
|
obtaining a list of all halos with their basic properties. We then adopt two
|
|
different schemes for injecting bubbles. We either consider only the most
|
|
massive halo found in the high-resolution zone, which can be identified with
|
|
the most massive progenitor of the final cluster, or we introduce AGN-driven
|
|
bubbles in all halos above a given fixed mass threshold value. The injection
|
|
of bubbles in all large halos is motivated by the observational indications
|
|
that probably most if not all of the spheroidal galaxies harbour a
|
|
supermassive black hole at their centres. Note that the larger number of
|
|
bubbles in this second scenario can also cause additional effects during
|
|
merger events, where bubble material can be torn apart and mixed into outer
|
|
regions of the cluster.
|
|
%%
|
|
|
|
\subsection{Global gas properties of simulated galaxy clusters}\label{Gas prop}
|
|
|
|
Before analysing the properties of simulated galaxy clusters with and
|
|
without AGN bubble heating, we briefly discuss issues of numerical
|
|
convergence. For this purpose we consider the S1 and S2 runs, and
|
|
compare their spherically averaged radial profiles at two epochs,
|
|
namely at $z=3$ and $z=0$\footnote{These two epochs delimit the time
|
|
interval during which the bubble heating is active, and hence the
|
|
period of time where our analysis is performed.}. The dark
|
|
matter and stellar density profiles of the S1 and S2 galaxy clusters are
|
|
in excellent agreement at both epochs, as well as the gas density
|
|
profiles, with the residual differences at early times being
|
|
consistent with what is expected from the increased noise. Both the
|
|
emission-weighted and the mass-weighted temperature profiles do not
|
|
noticeably differ at low redshifts, while there is a hint a of
|
|
slightly higher gas temperature for the S2 cluster at early
|
|
times. Thus, we conclude that for radii larger than the gravitational
|
|
softening length the properties of our simulated galaxy clusters are
|
|
numerically robust and have converged quite well.
|
|
|
|
|
|
|
|
%%
|
|
In Figure~\ref{Shot_S1_z}, we compare the gas entropy profiles with
|
|
and without bubble heating at three different epochs, $z=1.64$,
|
|
$z=0.44$ and $z=0$. For this comparison, we use both of our AGN
|
|
heating models, the one based on the Magorrian relationship and the
|
|
``BHAR model''. The entropy has been estimated by calculating the
|
|
ratio of the emission-weighted temperature to the gas density
|
|
elevated to the $2/3$ power, where the temperature is measured in
|
|
Kelvin and the gas density is given in $h^2 {\rm M}_{\odot}{\rm
|
|
kpc}^{-3}$. We selected only the hot gas component to compute these
|
|
profiles, i.e.~we avoided the cold, star-forming gas by imposing a cut
|
|
in density and ionisation level. With this choice, the gas profiles
|
|
are smoother because they have no contributions from cool
|
|
substructures at various radii. Nonetheless, it is also important to
|
|
investigate the fate of the cold gas in the central cluster region, an
|
|
issue we will address separately in Section~\ref{Stellar prop}. The
|
|
blue continuous lines are for the run without AGN feedback, the red
|
|
dot-dashed lines correspond to the ``Magorrian model'', while the
|
|
green dashed lines are for the ``BHAR model''. The vertical dotted lines
|
|
denote the softening length and the virial radius at the different
|
|
epochs, respectively. When $E_{\rm bub}$ is computed from the ``BHAR
|
|
model'', the effect of bubbles is less prominent at low redshifts than
|
|
in our other AGN heating scenario. However, at early times the
|
|
situation is opposite, as expected. Here the ``BHAR model'' heats the
|
|
ICM gas more prominently, right from the initial injection epoch
|
|
($z=3$) until $z \sim 0.4$. It is interesting to note that at $z \sim
|
|
0.4$ the bubble energy content is already much lower than in the
|
|
``Magorrian model'', indicating that the efficient heating at early
|
|
times has a prolonged effect on the thermodynamic state of the galaxy
|
|
cluster. We find that the ``Magorrian model'' starts to affect the gas
|
|
entropy from $z \approx 1.6$ in a noticeable way and it becomes very
|
|
important at late times, where it suppresses the cooling flow
|
|
completely at $z=0$.
|
|
\begin{figure*}
|
|
\centerline{
|
|
\hbox{
|
|
\psfig{file=figs/Shot_S1_z1.eps,width=6.2truecm,height=6.2truecm}
|
|
\hspace{-0.3truecm}
|
|
\psfig{file=figs/Shot_S1_z2.eps,width=6.2truecm,height=6.2truecm}
|
|
\hspace{-0.3truecm}
|
|
\psfig{file=figs/Shot_S1_z3.eps,width=6.2truecm,height=6.2truecm}
|
|
}}
|
|
\caption{Radial profiles of gas entropy of the S1 galaxy cluster
|
|
simulation. The blue continuous lines are for the run with cooling and star
|
|
formation only, the red dot-dashed lines refer to the case when AGN heating
|
|
based on our ``Magorrian-like scheme'' is included as well, while the green
|
|
dashed lines are for the BHAR-based model. The dotted vertical lines
|
|
denote the
|
|
gravitational softening and the virial radius at the given redshift; the
|
|
latter is indicated in the upper-left corner. The profiles do not extent
|
|
down to vanishingly small radii because they have been calculated
|
|
exclusively from the hot gas component (basically the gas above $1\,{\rm
|
|
keV}$), excluding the cold dense gas in the centre. Note that both the
|
|
spatial and the entropy scale vary between the three different panels.}
|
|
\label{Shot_S1_z}
|
|
\end{figure*}
|
|
%%
|
|
%%
|
|
As a further comparison, we show in Figure~\ref{Prof_S1} radial
|
|
profiles of gas density, temperature, X--ray luminosity, and
|
|
local cooling time for the S1 galaxy cluster at $z=0$. AGN heating
|
|
alters the gas properties out to a radius of $\approx 300\,h^{-1}{\rm
|
|
kpc}$, reducing the central gas density and increasing its
|
|
temperature. The X--ray emissivity, being more sensitive to the gas
|
|
density, is substantially lower when bubble feedback is active. In the
|
|
lower-right panel of Figure~\ref{Prof_S1}, we show the cooling time of
|
|
all ICM gas, estimated isobarically as \citep{Sarazin}
|
|
%
|
|
\be \label{tcool_eq} t_{\rm cool} = 8.5
|
|
\times 10^{10} \bigg( \frac{n_p}{10^{-3}\,{\rm cm}^{-3}} \bigg)^{-1} \bigg(
|
|
\frac{T}{10^8\,{\rm K}} \bigg)^{1/2}\, [{\rm yrs}],
|
|
\ee
|
|
%
|
|
where $n_p$ is the number density of hydrogen. When AGN heating is
|
|
not included, the cooling radius, i.e. the radius where $t_{\rm cool}
|
|
= t_{\rm H}$, lies at $\sim 60\,h^{-1}{\rm kpc}$, while it gets
|
|
reduced to $\sim 25\,h^{-1}{\rm kpc}$ in our ``BHAR model''. However,
|
|
the cooling radius vanishes for the ``Magorrian model'', where the
|
|
bubbles injection heats the gas above $1\,{\rm keV}$.
|
|
|
|
\begin{figure*}
|
|
\centerline{\vbox{
|
|
\hbox{
|
|
\psfig{file=figs/Rhohot_S1.eps,width=8.truecm,height=8truecm}
|
|
\hspace{0.truecm}
|
|
\psfig{file=figs/Thot_S1.eps,width=8.truecm,height=8truecm}
|
|
}
|
|
\hbox{
|
|
\psfig{file=figs/Lhot_S1.eps,width=8.truecm,height=8truecm}
|
|
\hspace{0.truecm}
|
|
\psfig{file=figs/tcool_S1.eps,width=8.truecm,height=8truecm}
|
|
}
|
|
}}
|
|
\caption{Radial profiles at $z=0$ of gas density (upper-left panel),
|
|
emission-weighted temperature (upper-right panel), and X--ray
|
|
luminosity (lower-left panel) estimated with the bremsstrahlung
|
|
approximation given by eq.~(\ref{Lx_eq}). The lower-right panel shows
|
|
the cooling time of all gas particles as a function of radius,
|
|
computed using eq.~(\ref{tcool_eq}). The continuous horizontal line
|
|
indicates the Hubble time at $z=0$. The blue continuous lines are
|
|
for the case without AGN feedback, the red dot-dashed lines are for
|
|
the model where $E_{\rm bub} \propto M_{200}^{4/3}(z)$, while the
|
|
green dashed lines are for the scenario where the bubble energy
|
|
depends on the BHAR given by eq.~(\ref{Ebub_BHAR_eq}).}
|
|
\label{Prof_S1}
|
|
\end{figure*}
|
|
%%
|
|
|
|
|
|
%%
|
|
Even though the spatial extent of the bubble particles reaches out to the
|
|
virial radius of the cluster, they are not capable of heating the gas in outer
|
|
regions, simply because their entropy content becomes comparable to the
|
|
entropy of the surrounding ICM gas at intermediate radii. This finding is
|
|
analogous to the previously discussed case of isolated halos, but the
|
|
dynamical evolution of the cluster with its associated merger processes makes
|
|
spreading of bubble material towards the outskirts more efficient.
|
|
|
|
In Figure~\ref{MM_g676}, we show emission-weighted temperature maps of
|
|
the g676 galaxy cluster at four different epochs, in order to more
|
|
closely discuss the spatial distribution of bubbles during merger
|
|
events. The over-plotted dots represent the particles that at least
|
|
once belonged to a bubble, and they are colour-coded according to their
|
|
temperature, the darkest ones are particles with $T > 10^8\,{\rm K}$,
|
|
while the lightest have $T < 10^4\,{\rm K}$. For this analysis, we
|
|
have introduced AGN feedback in all halos above $5 \times 10^{10}
|
|
h^{-1}{\rm M}_\odot$ and we scaled the energy content of the bubbles
|
|
with $M_{200}^{4/3}(z)$ of the host galaxy cluster.
|
|
|
|
In the first panel at redshift $z=0.86$, there is a smaller halo on
|
|
the lower right corner which enters the most massive cluster
|
|
progenitor at that epoch (which roughly has three times larger mass),
|
|
which is located at the centre of the panel. Both the massive halo and
|
|
the smaller one are AGN heated, but the bubbles are less energetic for
|
|
the infalling halo due to our assumed mass dependence. Moreover, it
|
|
can be noticed that the bubbles occupying the central regions are
|
|
hotter, both because they are more recent and thus have had less time
|
|
to lose their energy content, and also because at later times bubbles
|
|
are intrinsically more energetic in our ``Magorrian scenario''. The
|
|
second panel of Figure~\ref{MM_g676} (at $z=0.62$) illustrates what
|
|
happens to the bubble distribution when the smaller halo is crossing
|
|
the central region of the massive cluster. The bubbles are literally
|
|
pushed out of the way, upwards and to the right, and they are also
|
|
heated. The next panel (at $z=0.48$) shows that the bubble
|
|
distribution is still quite asymmetric, but at the same time, bubbles
|
|
have spread efficiently into outer regions and also have
|
|
cooled. Finally, the last panel shows how the cluster appears at
|
|
$z=0.20$, where it starts to be fairly relaxed with a quite symmetric
|
|
distribution of bubbles.
|
|
|
|
\begin{figure*}
|
|
\centerline{\vbox{
|
|
\hbox{
|
|
\psfig{file=figs/MM_bubbles-1.1-45.eps,width=8.5truecm,height=8.truecm}
|
|
\hspace{-0.truecm}
|
|
\psfig{file=figs/MM_bubbles-1.1-53.eps,width=8.5truecm,height=8.truecm}
|
|
}
|
|
\hbox{
|
|
\psfig{file=figs/MM_bubbles-1.1-59.eps,width=8.5truecm,height=8.truecm}
|
|
\hspace{-0.truecm}
|
|
\psfig{file=figs/MM_bubbles-1.1-75.eps,width=8.5truecm,height=8.truecm}
|
|
}
|
|
}}
|
|
\caption{Emission-weighted temperature maps of the g676 galaxy cluster
|
|
simulation during a major merger event at $z=0.86$, $0.62$, $0.48$ and
|
|
$0.20$, respectively. Over-plotted dots represent gas particles that
|
|
have been at least once part of a bubble, and they are colour-coded
|
|
according to their temperature, the darkest ones being the hottest. It
|
|
can be noticed that both the spatial distribution of bubbles and their
|
|
energy content are drastically influenced by the passage of the
|
|
smaller halo through the central region of the massive cluster.}
|
|
\label{MM_g676}
|
|
\end{figure*}
|
|
%%
|
|
|
|
\subsection{Stellar properties of galaxy clusters}\label{Stellar prop}
|
|
|
|
In this section, we analyse the effects of AGN heating on the
|
|
properties of the stellar components of galaxy clusters. We
|
|
concentrate on the properties of the central cluster galaxy, which is
|
|
the one affected most by the bubble heating. From the initial
|
|
injection epoch ($z=3$) until $z=0$, we compute for this purpose the
|
|
stellar and gaseous mass, star formation rates, stellar ages, and
|
|
colours of the cD galaxy that sits in the main progenitor of our final
|
|
halo.
|
|
|
|
%%
|
|
In Figure \ref{CD_zf}, we show a histogram of the formation times of stars
|
|
belonging to the cD galaxy at $z=0$ of our S1 galaxy cluster. The histograms
|
|
have been computed by binning the expansion factors that correspond to each
|
|
stellar formation time, and they have been normalised to the maximum bin. The
|
|
white histogram is for the run without AGN heating, the grey coloured one for
|
|
the ``Magorrian model'', while the hatched histogram gives the result for
|
|
``BHAR model''. When AGN heating is absent, the histogram of stellar formation
|
|
redshifts clearly shows an extended tail at low redshifts; together with the
|
|
considerable SFR of order of $100\,{\rm M}_{\odot}{\rm yr}^{-1}$, this implies
|
|
that stars are formed until $z=0$ in situ. Nevertheless, there is a
|
|
possibility that some of these stars have been formed elsewhere, e.g.~in
|
|
merging substructures, and that they only ended up later in the cD galaxy by
|
|
merging. For redshifts less than $0.3$ this is certainly not an important
|
|
mechanism, because in our ``Magorrian model'', the central SFR is completely
|
|
suppressed for $z < 0.3$ and at the same time the $z_{\rm sf}$ histogram is
|
|
truncated. Thus, the difference of $ \sim 6 \times
|
|
10^{11} h^{-1}{\rm M}_{\odot}$ in the stellar mass of the final cD galaxy in
|
|
these two cases gives an indication on how many stars have formed in the cD
|
|
galaxy from $z=0.3$ until today, when AGN feedback is not present. At the same
|
|
time, also the mass of the cold gas (below $1\,{\rm keV}$) is reduced in the
|
|
``Magorrian model'', from $\sim 2 \times 10^{10}\,h^{-1}{\rm M}_{\odot}$ to
|
|
zero. Moreover, the suppression of star formation at late times has an
|
|
immediate effect on colours of the cD galaxy, which becomes redder. To estimate
|
|
the colours we used Bruzual \& Charlot's stellar population synthesis models
|
|
\citep{Charlot}, computing rest-frame magnitudes in the SDSS bands, assuming
|
|
Solar metallicity and a Chabrier initial mass function. The $u$-band magnitude
|
|
is changed from $-23.8$ to $-22.8$ and the $u-r$ colour is increased from
|
|
$2.0$ to $2.6$.
|
|
|
|
In contrast, the star formation of the ``BHAR model'' proceeds in a quite
|
|
different manner, as can been seen from the hatched histogram, which is
|
|
substantially lower for $0.4 < z < 2.0$, but quite similar to the case without
|
|
AGN feedback at very low redshifts, $z < 0.3$. The first feature arises due to
|
|
two different processes happening at the same time. The second peak present in
|
|
the histograms is due to a major merger event that happens roughly at $z=1$.
|
|
One consequence of this merger event is a central burst of star formation,
|
|
which happens to be absent in the ``BHAR model'' due to its very efficient
|
|
bubble heating at this epoch. Nonetheless, there are still some stars becoming
|
|
part of the cD galaxy at $z=0$ that have formation times in the corresponding
|
|
time interval. Tracing back these stars in time and considering that the SFR
|
|
of the cD galaxy in the ``BHAR model'' during this epoch is practically
|
|
zero, we see that these stars have been produced in other small galaxies and
|
|
were indeed accreted onto the central cD galaxy at later times. Assuming that a
|
|
similar amount of stars accreted onto the cD galaxy also in the case without
|
|
AGN feedback, a total mass of $\sim 10^{12} \, h^{-1}{\rm M}_{\odot}$ in stars
|
|
has formed in situ between $z=2$ and $z=0.4$, with a considerable part created
|
|
as part of the central starburst induced by the major merger event.
|
|
|
|
The tail at low redshifts present in the hatched histogram can be explained by
|
|
the reduced energy content of the bubbles, which are not efficient any more in
|
|
suppressing the cooling flow, and consequently the star formation in the central
|
|
cD galaxy. The total SFR within $R_{200}$ follows a similar trend as the SFR
|
|
of the cD galaxy, with comparable systematic differences for the different runs,
|
|
implying that the bubble heating mainly affects the central stellar properties
|
|
and not the residual star formation in the cluster volume.
|
|
|
|
\begin{figure}
|
|
\bc
|
|
\centerline{\includegraphics[width=8.5truecm,height=8.5truecm]{figs/CD_stage_z.0_S1.eps}}
|
|
\caption{Distribution of formation redshifts of the stars belonging to
|
|
the cD galaxy of the S1 galaxy cluster at $z=0$. The white histogram
|
|
corresponds to the case with cooling and star formation only; here it can be
|
|
seen that stars continue to be formed until $z=0$. The grey coloured
|
|
histogram shows how the stellar formation times change when our ``Magorrian
|
|
model'' of bubble heating is switched on, which essentially suppresses any
|
|
central star formation for $z < 0.25$ completely. Finally, the result for
|
|
the ``BHAR model'' of AGN feedback is illustrated with the hatched
|
|
histogram, which shows a considerable suppression of central star formation
|
|
at intermediate redshifts, $0.3<z<2.0$.}
|
|
\label{CD_zf}
|
|
\ec
|
|
\end{figure}
|
|
%%
|
|
|
|
\subsection{The metallicity distribution in the simulated clusters}
|
|
|
|
In this section, we analyse the metal distribution in our simulated galaxy
|
|
clusters. It is a well known problem that numerical simulations which include
|
|
cooling and star formation processes in general fail to reproduce the observed
|
|
shallow metallicity gradients, especially so if efficient feedback mechanisms
|
|
that help spreading the metals are absent. In particular, metals produced by
|
|
stars remain locked in the dense, star forming regions, even though supernovae
|
|
feedback is included which regulates the star formation process itself. As a
|
|
result, the metallicity distribution remains lumpy and exhibits a rather step
|
|
gradient. Moreover, most of the metals are produced in the central cD galaxy
|
|
due to its excessive star formation, which is a manifestation of the cooling
|
|
flow problem.
|
|
|
|
This motivates the search for physical feedback processes that can spread and
|
|
mix metals more efficiently, acting both in the central region and on the
|
|
scale of the whole galaxy cluster. While the galactic wind model suggested by
|
|
\cite{SH03} helps in reducing this discrepancy and also diminishes the total
|
|
SFR over cosmological time, the model fails to qualitatively change the gas
|
|
and stellar properties of galaxy clusters discussed in Sections~\ref{Gas prop}
|
|
and \ref{Stellar prop}. Especially at late times, mixing of metals due to
|
|
winds from the cD galaxy proves inefficient; here the cluster potential well
|
|
is simply to deep and the ram pressure of the ICM too high to allow winds to
|
|
travel far. Bubble heating may fare considerably better in this respect. In
|
|
the following we therefore analyse the effect of AGN on the metal
|
|
distribution in our simulations, and we also compare simulations
|
|
with or without galactic winds of velocity $\sim 480\,{\rm km\,s^{-1}}$.
|
|
|
|
In Figure~\ref{Zhot_S1}, we show radial profiles of the gas
|
|
metallicity of the S1 galaxy cluster, and in Figure~\ref{Zmaps} we
|
|
illustrate the corresponding mass-weighted gas metallicity maps. When
|
|
additional feedback mechanisms are taken into account, the amount of
|
|
metals in the hot gas component is increased with respect to runs with
|
|
cooling and star formation only (continuous blue line on
|
|
Figure~\ref{Zhot_S1} and left panel of Figure~\ref{Zmaps}). Also, the
|
|
metallicity distribution becomes less lumpy because the metals are
|
|
more efficiently transported out of the dense regions, increasing the
|
|
fraction of enriched gas. It can be noticed that already bubble
|
|
heating without winds (red dot-dashed line in Figure~\ref{Zhot_S1},
|
|
and middle panel of Figure~\ref{Zmaps}) is capable of producing a more
|
|
homogeneous metallicity distribution, while the AGN heating coupled
|
|
with the galactic winds (green dashed line in Figure~\ref{Zhot_S1} and
|
|
right panel of Figure~\ref{Zmaps}) slightly decreases the radial
|
|
metallicity gradient and makes the spreading of the metals even more
|
|
efficient.
|
|
|
|
Nevertheless, the total amount of metals in the three different
|
|
ICM phases -- hot ICM gas, cold star-forming gas, and stars --
|
|
remains very similar in all runs, implying that there are no
|
|
substantial metal-enriched gas outflows from the galaxy cluster
|
|
itself. The situation can be rather different if winds with similar
|
|
velocities are present in less massive systems, as demonstrated by
|
|
\cite{SH03}. When the galactic wind velocities become comparable to
|
|
the escape velocity from the system in consideration, then the winds
|
|
may lead to gaseous outflows, eventually polluting the surrounding
|
|
medium with metals produced by cluster stars. The heating provided by
|
|
bubbles might help the metal spreading even more. Considering our
|
|
``BHAR model'' at early times where it is very efficient and
|
|
when the corresponding halo mass is considerably smaller, there is
|
|
evidence that not only many metals are transfered into the ``hot
|
|
phase'', but also that metal enriched gas is pushed towards the
|
|
cluster outskirts, and in part beyond the virial radius. However, this
|
|
effect is transitory, because $E_{\rm bub}$ decreases with time,
|
|
and more importantly, because the forming cluster is so massive that
|
|
it behaves effectively like a closed box at late times. Consequently,
|
|
at $z=0$, the distribution of metals looks quite similar to the
|
|
``Magorrian model'' discussed above. Finally, note
|
|
that even though bubble heating improves metal spreading in the
|
|
ICM, at the same time it does not disrupt the central metallicity
|
|
gradient of the galaxy clusters, as can be clearly seen from
|
|
Figure~\ref{Zhot_S1}. Therefore, our AGN heating prescription is
|
|
qualitatively in good agreement with the metallicity gradients observed in cool
|
|
core clusters \citep[e.g.][]{Boehringer2002,DeGrandi2004,Boehringer04}.
|
|
|
|
\begin{figure}
|
|
\bc
|
|
\centerline{\includegraphics[width=8.5truecm,height=8.5truecm]{figs/Zhot_S1.eps}}
|
|
\caption{Radial profiles of the gas metallicity of the S1 galaxy
|
|
cluster. Only the diffuse hot gas component has been used to estimate the
|
|
metallicity, which is given in Solar units. The blue solid line is for the
|
|
run without AGN heating, the red dot-dashed line is for bubble injection
|
|
based on our ``Magorrian model'', while the green dashed line is also for
|
|
the ``Magorrian model'' but with additional inclusion of feedback by
|
|
galactic winds. We can see that the feedback processes manage to better
|
|
spread the metals into the diffuse ICM gas, and the increased mixing leads
|
|
to a less clumpy metallicity distribution.}
|
|
\label{Zhot_S1}
|
|
\ec
|
|
\end{figure}
|
|
|
|
|
|
\begin{figure*}
|
|
\centerline{
|
|
\psfig{file=figs/met_csfS1_central.eps,width=6.1truecm,height=5.5truecm}
|
|
\hspace{-0.truecm}
|
|
\psfig{file=figs/met_bcsfS1_central.eps,width=6.1truecm,height=5.5truecm}
|
|
\hspace{-0.truecm}
|
|
\psfig{file=figs/met_bcsfwS1_central.eps,width=5.9truecm,height=5.5truecm}
|
|
}
|
|
\caption{Mass-weighted gas metallicity maps of the S1 galaxy cluster at
|
|
$z=0$. The left panel corresponds to the case with cooling and star
|
|
formation only. The middle panel shows how the metallicity of the hot gas
|
|
component changes when AGN heating (here the ``Magorrian model'' was
|
|
assumed) is included, while the right panel illustrates the case where in
|
|
addition galactic winds with a velocity of $ \sim 480\,{\rm km \, s^{-1}}$
|
|
were included, making the metallicity distribution more homogeneous.}
|
|
\label{Zmaps}
|
|
\end{figure*}
|
|
|
|
\subsection{Sound waves or merger induced weak shocks?} \label{S2_24}
|
|
|
|
While the unsharp mask technique is very useful in detecting X-ray
|
|
cavities and associated sound waves, it is potentially easy to confuse
|
|
these ripples with shock waves stemming from merger events. To
|
|
demonstrate this danger in interpreting observations, we here analyse
|
|
a specific case where the presence of smaller systems passing trough
|
|
the cluster might induce such a misleading conclusion.
|
|
|
|
To this end, we computed projected maps of the S2 galaxy cluster
|
|
without AGN heating at $z=0.13$. At this time, two substructures at
|
|
radial distances of about half the virial radius are moving towards
|
|
the centre. These substructures are still visible in the gas density
|
|
map (upper right panel of Figure~\ref{S2_24maps}), but almost
|
|
completely vanish when the X--ray luminosity map is
|
|
computed. Nevertheless, when the unsharp masking procedure is applied
|
|
to the $L_{\rm X}$ map, a two-lobed feature is clearly visible (upper left
|
|
panel of Figure~\ref{S2_24maps}). Even though this feature at
|
|
first glance looks strikingly similar to the ripples produced by
|
|
bubble heating events, it is exclusively a product of the specific
|
|
spatial geometry of the substructures in the cluster, and is also
|
|
enhanced due to projection effects.
|
|
|
|
|
|
In the mass-weighted temperature map (lower left panel of
|
|
Figure~\ref{S2_24maps}), two spherical regions can be noticed that are
|
|
slightly cooler than their surroundings. They correspond to the two
|
|
substructures. Especially around the substructure in the lower right
|
|
part of the map, hotter surrounding gas can be seen. To distinguish
|
|
between a cold front or a shock, we can analyse the pressure of the
|
|
surrounding gas and compute the Mach number map\footnote{We made a
|
|
crude estimate of the Mach number by calculating for every gas
|
|
particle its velocity in the galaxy cluster reference frame and
|
|
dividing it by the local sound speed.} (lower right panel of
|
|
Figure~\ref{S2_24maps}). A clear jump of pressure in the region
|
|
adjacent to the hot gas around the substructure together with the
|
|
mildly supersonic motion of the substructure indicates the presence of
|
|
weak shocks. Note that these shocks cannot be easily identified
|
|
observationally as being caused by infalling substructures. Thus, if a
|
|
careful analysis is not performed, these features could be
|
|
associated mistakingly with sound waves caused by AGN bubbles, especially if at
|
|
the same time some fossil but unrelated bubble is detected in the
|
|
cluster by its radio emission.
|
|
|
|
|
|
\begin{figure*}
|
|
\centerline{\vbox{
|
|
\hbox{
|
|
\psfig{file=figs/Lx_S2_24.eps,width=9.truecm,height=7.6truecm}
|
|
\hspace{-0.truecm}
|
|
\psfig{file=figs/rho_S2_24.eps,width=9.truecm,height=7.6truecm}
|
|
}
|
|
\hbox{
|
|
\psfig{file=figs/Tm_S2_24.eps,width=9.truecm,height=7.6truecm}
|
|
\hspace{-0.truecm}
|
|
\psfig{file=figs/Mach_S2_24_xy.eps,width=9.truecm,height=7.6truecm}
|
|
}
|
|
}}
|
|
\caption{Projected maps of different gas properties of the
|
|
S2 galaxy cluster at $z=0.13$, where a merger with two smaller subclumps is
|
|
in progress. AGN heating has not been included in this simulation. The
|
|
upper left panel shows the unsharped masked image of the X--ray luminosity
|
|
on a scale of $160\,h^{-1}{\rm kpc}$, and the upper right panel illustrates
|
|
the gas density map. On the lower left panel we show a map of mass-weighted
|
|
cluster gas temperature, while the lower right panel gives a crude estimate
|
|
of the mass-weighted Mach number.}
|
|
\label{S2_24maps}
|
|
\end{figure*}
|
|
|
|
|
|
|
|
\subsection{AGN heating in galaxy clusters of different mass}
|
|
|
|
Here we discuss the effect of AGN heating in clusters spanning a wide range in
|
|
mass, following their cosmological evolution up to $z=0$. Their main
|
|
properties are summarized in Tables~\ref{tab_simpar} and~\ref{tab_GCpar}. We
|
|
analyze how the gas properties of clusters of different mass and at various
|
|
radii, normalized to $R_{200}$, change due to the AGN-driven bubbles.
|
|
|
|
In Figure~\ref{L_T}, we show for all clusters in our sample at $z=0$ the
|
|
cumulative X--ray luminosity and mean gas temperature at four different radii,
|
|
$r\,=\,0.03\,R_{200}$, $0.1\,R_{200}$, $0.5\,R_{200}$ and $R_{200}$, marked
|
|
with different colours and symbols. In this $L_{\rm X}-T$ plane, clusters are
|
|
found at different places according to their mass, with less massive systems
|
|
being located in the lower-left part of the panel and more massive ones in the
|
|
upper-right part, as indicated in the figure. Also, cluster luminosity and
|
|
temperature vary systematically with increasing radius $r$, with values at the
|
|
virial radius occupying the upper-left part of the plot. The arrows attached
|
|
to each cluster model show how $L_{\rm X}$ and $T$ change when bubble heating
|
|
is present. Clearly, there is a systematic trend of decreasing X--ray
|
|
luminosity for all clusters and at all considered radii. This effect is more
|
|
pronounced in the most inner regions and it is important both for massive
|
|
clusters and smaller systems. Thus, in the cosmological simulations the bubble
|
|
heating efficiency is not as clearly related to the mass of the host cluster
|
|
as is the case for the isolated halos. From Figure~\ref{L_T}, it can be seen
|
|
that the two low mass clusters exhibit prominent imprints caused by AGN
|
|
activity, highlighting that the bubble heating is more complex in cosmological
|
|
simulations due to the hierarchical merging histories of clusters.
|
|
|
|
The gas temperature does not follow an equally clear trend as $L_{\rm
|
|
X}$ for all the clusters in our sample, but in most cases, and
|
|
especially for central clusters regions, it is boosted towards higher
|
|
values, implying that bubble injection leads to en effective heating
|
|
of the ICM. These trends in $L_{\rm X}$ and $T$ confirm our previous
|
|
findings for the S1/S2 clusters (presented in Figure~\ref{Prof_S1}),
|
|
and show that they are general features of our AGN heating model.
|
|
Interestingly, recent observational work by \cite{Croston05} indicates
|
|
that radio-loud elliptical-dominated groups have $L_{\rm X}-T$ scaling
|
|
relations systematically different from those of elliptical-dominated
|
|
radio-quiet groups. Their observed trends in the $L_{\rm X}-T$ plane
|
|
are qualitatively similar with what we find, showing that for a given
|
|
X--ray luminosity radio-loud groups have systematically higher gas
|
|
temperature values. Nevertheless, since they are probing smaller mass
|
|
systems, simulated galaxy groups with a matching range in mass are
|
|
needed in order to make a more detailed comparison. Recent
|
|
observational works \citep{McNamara05, Nulsen05} revealed the presence
|
|
of very powerful outbursts due to the AGN activity, which are also
|
|
accompanied with large-scale shocks. These clusters are located above
|
|
the mean $L_{\rm X}-T$ relation, probably because we are witnessing
|
|
the very early stages of bubble feedback, where pressure
|
|
equilibrium with the surrounding ICM has not yet been established.
|
|
|
|
|
|
|
|
|
|
\begin{figure}
|
|
\bc
|
|
\centerline{\includegraphics[width=8.5truecm,height=8.5truecm]{figs/L_g1.eps}}
|
|
\caption{Cumulative X--ray luminosity of seven galaxy clusters as
|
|
a function of their mass-weighted gas temperature. We give results for
|
|
redshift $z=0$, and at four different radii. Different symbols and colours
|
|
denote estimates of $L_{\rm X}$ and $T$ at different radii normalized to
|
|
$R_{200}$: blue crosses correspond to $0.03\,R_{200}$, green stars to
|
|
$0.1\,R_{200}$, yellow diamonds to $0.5\,R_{200}$, and red triangles to
|
|
$R_{200}$. The arrows indicate how the cluster luminosity and temperature
|
|
change when AGN heating is included, i.e.~systematically decreasing $L_{\rm
|
|
X}$ for all the considered radii. More massive clusters are located in the
|
|
upper-right part of the panel, while less massive systems are found in the
|
|
lower-left corner of the figure, as indicated.}
|
|
\label{L_T}
|
|
\ec
|
|
\end{figure}
|
|
|
|
|
|
|
|
\section{Discussion and Conclusions} \label{DIS}
|
|
|
|
In this work, we discussed a simulation model for AGN heating in the
|
|
form of hot, buoyant bubbles, which are inflated by active phases of
|
|
supermassive black holes at the centres of massive halos. The
|
|
motivation for such a mode of feedback stems from the rich
|
|
phenomenology of X-ray cavities and radio bubbles observed in clusters
|
|
of galaxies, and the suggestion that this AGN activity may represent
|
|
the solution of the `cooling flow puzzle' posed by clusters of
|
|
galaxies.
|
|
|
|
|
|
Several previous studies in the literature \citep[e.g.][]{Churazov01,Quilis01,
|
|
Brueggen03,Hoeft04,DVecchia04} have analysed bubble feedback in
|
|
isolated galaxy clusters using hydrodynamical mesh codes. We here
|
|
present the first implementation of this feedback in a SPH code, so an
|
|
important goal was to test whether our results are consistent
|
|
with these previous studies based on very different hydrodynamical
|
|
techniques. Reassuringly, this is the case, both qualitatively and
|
|
quantitatively. In particular, for galaxy clusters of
|
|
similar masses as considered by \citet{Quilis01, DVecchia04}, and with
|
|
bubbles parametrised in an analogous way, we find changes induced by
|
|
AGN-heating in gas properties, e.g. density and temperature radial
|
|
profiles, central mass deposition rates, that are in excellent
|
|
agreement. Also, the morphology of the bubbles and their time
|
|
evolution are very similar. This is important because it implies that
|
|
the SPH technique, which is more easily applicable to full blown
|
|
cosmological simulations of cluster formation, can be reliably used to
|
|
study bubble feedback.
|
|
|
|
|
|
In our simulations, we considered both, isolated halos of different mass, and
|
|
cosmological simulations of the $\Lambda$CDM model that follow galaxy cluster
|
|
assembly from high redshift to the present. The isolated simulations served
|
|
as a laboratory to study the dynamics of bubbles in detail. By considering
|
|
halos with masses ranging from $10^{12}\,h^{-1}{\rm M}_{\odot}$ to
|
|
$10^{15}\,h^{-1}{\rm M}_{\odot}$, they also allowed us to gain some insight in
|
|
how the coupling of radiative cooling to the AGN heating varies with cluster
|
|
mass. An important conclusion from these experiments is that for systems of
|
|
mass lower than $\sim 10^{13}\,h^{-1}{\rm M}_{\odot}$ bubbles with reasonable
|
|
energy content are not capable of preventing excessive gas cooling. If one
|
|
nevertheless allows for very large energy in these systems, a stable
|
|
suppression of the cooling requires a delicate fine tuning of the bubbles.
|
|
However, the inefficiency of bubble heating in lower mass systems is also
|
|
caused in part by our injection prescription, which is based on global
|
|
properties of the host galaxy cluster without accounting for the actual amount
|
|
of cooling gas present in the very centre.
|
|
|
|
|
|
In our cosmological simulations, we find that bubble injection can
|
|
substantially affect galaxy cluster properties, especially in massive,
|
|
relaxed clusters and at late cosmological times. Central cluster gas
|
|
is efficiently heated, and thus both the amount of cold baryons and
|
|
the star formation in the central cD galaxy is reduced. Also, an
|
|
excessive mass deposition rate by cooling flows is
|
|
prevented. AGN-driven bubbles not only modify the properties
|
|
of the most central cluster parts, but alter the whole
|
|
inner region of massive clusters, out to radii of order $\sim
|
|
300\,h^{-1}{\rm kpc}$, where the gas density is reduced and the
|
|
temperature is increased. As a result, the X--ray luminosity is
|
|
considerably reduced, while the gas entropy exhibits a flat core in
|
|
the central region. These trends are all in the direction required to
|
|
reconcile hydrodynamical simulations of cluster formation in the
|
|
$\Lambda$CDM model with observations of real galaxy clusters.
|
|
|
|
|
|
We tried several variants of our bubble model in order to explore the
|
|
dependence of obtained results on the detailed assumptions made about
|
|
how the energy is released in the bubbles. In particular, we compared
|
|
an instantaneous injection of energy into the bubbles with a scheme
|
|
where the energy is released over a certain period of time (from
|
|
$5\times10^7$yrs to $5\times10^8$yrs), we imposed that the bubble
|
|
particles should not cool during a given time interval (e.g. $\sim
|
|
10^8$ yrs), we tried different spatial patterns for the bubble
|
|
placement, and we also varied the initial epoch where our AGN heating
|
|
started (from $z=6$ to $z=3$). None of these changes was really
|
|
capable of modifying our results considerably. However, it appears
|
|
that our findings are much more sensitive to the adopted model for the
|
|
time evolution of the bubble energy. We explicitly demonstrated this
|
|
by changing the rate at which the energy is released with time, under
|
|
the constraint that the total energy injected from $z=3$ to $z=0$ was
|
|
kept constant. When we linked the bubble energy content in this way
|
|
to a model for the BH accretion rate, the fraction of cold gas and the
|
|
star formation rate can be noticeably reduced even at early times, but
|
|
this is compensated by a reduced efficiency of bubble heating at late
|
|
times, such that cooling flows are not suppressed sufficiently at
|
|
$z=0$. The ``Magorrian model'', where the feedback occurs primarily at
|
|
low redshift fares better in this respect, and gives therefore a
|
|
better match to the properties of observed rich clusters of
|
|
galaxies. In addition, we explored yet another scaling between
|
|
the energy of the bubbles and the mass of the host galaxy cluster,
|
|
namely $E_{\rm bub} \propto M_{200}^{5/3}$, which can be motivated by the
|
|
observed $M_{\rm BH}-\sigma$ relation as well. Also, \cite{Ferrarese2005}
|
|
pointed out a relationship between the mass of the black hole and that
|
|
of the hosting dark matter halo, in the form of $M_{\rm BH} \propto
|
|
M_{\rm DM}^{1.65}$. Thus, if the energy content of the bubbles is a
|
|
linear function of the black hole mass, the above scaling is
|
|
obtained. One also arrives at $E_{\rm bub} \propto M_{200}^{5/3}$ if
|
|
one assumes that the energy in the bubbles is some small fraction of
|
|
the total thermal cluster energy, which roughly scales as
|
|
$M_{200}^{5/3}$. In performing the analysis with the modified slope,
|
|
we fixed the normalization of the $E_{\rm bub} \propto M_{200}^{5/3}$
|
|
relation at redshift $z=0$ to be equal to the value in our ordinary
|
|
``Magorrian model''. The results of this analysis showed that the
|
|
change in the slope from $4/3$ to $5/3$ produces qualitatively very
|
|
similar results, and hence it follows that the properties of our
|
|
simulated galaxy clusters are not very sensitive to such a modest
|
|
change of the AGN heating prescription.
|
|
|
|
|
|
|
|
In the newly emerging picture for the joint evolution of galaxies and
|
|
supermassive black holes, the interplay between AGN and their host galaxies
|
|
may be composed of two modes. One mode is caused by the quiescent accretion of
|
|
intracluster gas onto the central BH, provoking periodic AGN activity which
|
|
manifests itself in jets and radio bubbles. This mode plays a more important
|
|
role at late cosmological epochs, in massive and more relaxed systems, and it
|
|
is often referred to as a ``radio-mode'' \citep[e.g.][]{Croton05}. The other
|
|
mode occurs in merging pairs of galaxies, where strong tidal forces
|
|
efficiently funnel large amounts of {\em cold} gas towards the nuclei of the
|
|
merging galaxies, where it becomes available for fueling the embedded
|
|
supermassive BHs~\citep{DiMatteo05,Springel04}. The associated intense
|
|
accretion triggers quasar activity, which is more frequent at higher redshift
|
|
due to the larger merger rates there. If a small fraction of the bolometric
|
|
luminosity of the quasar couples thermally to the surrounding gas, a prominent
|
|
gas outflow can eventually be created during the formation of ellipticals,
|
|
which then shuts off further accretion and star formation~\citep{Springel2005}
|
|
and establishes the $M_{\rm BH}-\sigma$ relationship~\citep{DiMatteo05}.
|
|
|
|
The latter process, the `quasar mode', has already been explored in direct
|
|
simulation models of galaxy mergers, but not yet in cosmological simulations.
|
|
It would therefore be extremely interesting to couple these two modes of AGN
|
|
feedback in a unified simulation model for supermassive black hole growth, and
|
|
to carry out cosmological simulations with it. In such a model, the
|
|
energetics of the bubbles and the periods of AGN activity can then be made
|
|
directly dependent on the current BH mass and on the local physics of
|
|
accreting gas, removing much of the freedom in our present bubble models. We
|
|
will present such a model in forthcoming work. In addition, our work suggests
|
|
that for a more complete picture of the ICM dynamics, additional physical
|
|
processes should be incorporated as well. This includes thermal conduction,
|
|
even though it is probably relevant only in the most massive systems. We also
|
|
suggest that the physical viscosity expected for the ICM gas should be
|
|
considered as well, since the efficiency of non-local AGN heating by viscous
|
|
dissipation of sound waves will depend crucially on this input. Finally, radio
|
|
observations strongly suggest that bubbles are prevalently filled with
|
|
relativistic particles, which appear as radio lobes, and in many cases are
|
|
coincident with X--ray cavities. Therefore, it would be important to address,
|
|
using fully cosmological simulation of cluster formation, the role of
|
|
non-thermal radio plasma in bubbles for heating of the ICM. It appears that
|
|
for some time to come clusters of galaxies will remain one of the most
|
|
interesting places to study complex hydrodynamic phenomena in the Universe.
|
|
|
|
\section*{Acknowledgements}
|
|
|
|
We thank Martin Jubelgas for providing a code to set-up isolated halos in
|
|
equilibrium, Elena Rasia for help in producing mock observations with {\small
|
|
X-MAS}, and Klaus Dolag for providing cluster initial conditions. We are
|
|
indebted to Simon White and Eugene Churazov for very useful comments on the
|
|
manuscript. DS acknowledges the PhD fellowship of the International Max Planck
|
|
Research School in Astrophysics, and received support from Marie Curie Host
|
|
Fellowship for Early Stage Research Training.
|
|
|
|
\bibliographystyle{mnras}
|
|
|
|
\bibliography{paper}
|
|
|
|
\end{document}
|
|
|