\documentclass[journal]{rmaa}

\usepackage{paralist}


\title{Mass and metal ejection efficiency in disk galaxies driven by young stellar clusters of nuclear starburst}

\author{
A. Rodr\'iguez-Gonz\'alez,\altaffilmark{1} 
A. Esquivel,\altaffilmark{1}  
A.~C. Raga,\altaffilmark{1} 
and P. Col\'in\altaffilmark{2}}

\altaffiltext{1}{Instituto de Ciencias Nucleares, Universidad Nacional Aut\'onoma de M\'exico, Mexico.}
\altaffiltext{2}{Centro de Radioastronom\'ia y Astrof\'isica, Universidad Nacional Aut\'onoma de M\'exico, Mexico.}


\shortauthor{Rodr\'iguez-Gonz\'alez et al.}

\shorttitle{Mass and metal ejection efficiencies}


\fulladdresses{
\item Alejandro Esquivel, Alejandro C. Raga, and Ary Rodr\'iguez-Gonz\'alez: Instituto de Ciencias Nucleares, Universidad Nacional Aut\'onoma 
de M\'exico, Apdo. Postal 70-543, 04510 M\'exico, D. F., Mexico (esquivel,~raga,~ary@nucleares.unam.mx).
\item Pedro Col\'in: Centro de Radioastronom\'ia y Astrof\'isica, Universidad Nacional Aut\'onoma de M\'exico, Apdo. Postal 3-72, 58090 Morelia, 
Michoac\'an, Mexico (p.colin@crya.unam.mx).}


\listofauthors{A.~Rodr\'iguez-Gonz\'alez, A.~Esquivel, A.~C.~Raga, \& P.~Col\'in}

\indexauthor{Rodr\'iguez-Gonz\'alez, A.}
\indexauthor{Esquivel, A.}
\indexauthor{Raga, A.~C.}
\indexauthor{Col\'in, P.}

\SetVolume{47} 
\SetFirstPage{111} 
\SetYear{2011}
\ReceivedDate{2010 August 26} 
\AcceptedDate{2011 January 31}


\abstract{We present results from models of galactic winds driven by energy injected by nuclear starbursts. The total energy of the starburst 
is provided by young central stellar clusters and parts of the galactic interstellar medium are pushed out as part of the galactic wind (in some 
cases the galactic wind contains an important part of the metals produced in the new generation of stars). We have performed adiabatic and 
radiative 3D N-Body/Smooth Particle Hydrodynamics simulations of galactic winds using the GADGET-2 code. The numerical models cover a wide range 
of starburst (from $\sim 10^2$ to $\sim 10^7~M_\odot$) and galactic gas masses (from $\sim 6 \times 10^6$ to $\sim 10^{11}~M_\odot$). The concentrated 
central starburst regions are an efficient engine for producing the mass and metal loss in galaxies, and also for driving the metal redistribution 
in galaxies.}


\resumen{Presentamos las eficiencias de p\'erdida de masa y metales obtenidas de modelos num\'ericos de vientos gal\'acticos empujados por la 
energ\'ia depositada en brotes de formaci\'on estelar nucleares. Los brotes de formaci\'on estelar contienen c\'umulos estelares j\'ovenes los 
cuales inyectan la energ\'ia suficiente para empujar parte del medio interestelar fuera de las galaxias. En algunos casos los vientos gal\'acticos 
contienen una importante parte de los metales producidos por las nuevas generaciones estelares. Para estudiar las eficiencias de p\'erdida de 
masa y metales hemos desarrollado simulaciones num\'ericas 3D {\it N-Cuerpos/Smooth Particle Hydrodynamics} de vientos gal\'acticos para los casos: 
adiab\'aticos y con p\'erdidas radiativas. Los modelos num\'ericos cubren una amplio intervalo de masas de los brotes de formaci\'on estelar 
(de $\sim$10$^2$ a $\sim$10$^7~M_\odot$) y de masas en las galaxias anfitrionas (de $\sim$6$\times$10$^6$ a $\sim$10$^{11}~M_\odot$). Las regiones 
de formaci\'on estelar concentradas en el centro del potencial son una maquinaria importante para la p\'erdida y redistribuci\'on  de masa y 
metales en este tipo de galaxias.}


\addkeyword{galaxies: starburst}
\addkeyword{galaxies: star clusters: general}
\addkeyword{ISM: general}
\addkeyword{stars: winds, outflows}


\begin{document}

\maketitle


\section{Introduction}

A great amount of evidence, both observational and theoretical, indicates that galactic winds (GWs) are a necessary and very important ingredient 
of the evolution of galaxies and the inter-galactic medium. Their presence can explain, among other things, the very low metallicities found in 
dwarf galaxies, and the abundance of metals observed in the intra-cluster medium. Such outflows have been observed in galaxies of a variety of masses 
and at different redshifts (e.g., Martin 1999; Pettini et al. 2001; Veilleux, Cecil, \& Bland-Hawthorn 2005). At the same time, different analytical 
and numerical  models have been put forward to explain these outflows (e.g., Chevalier \& Clegg 1985; Tomisaka \& Ikeuchi  1988; Heckman, Armus, 
\& Miley 1990; De Young \& Heckman  1994; MacLow \& Ferrara 1999; Strickland \& Stevens 2000; Cant\'o, Raga, \& Rodr\'iguez 2000; Tenorio-Tagle, 
Silich, \& Mu\~noz-Tu\~n\'on 2003; Fujita et al. 2004; Cooper et al. 2008; Fujita et al. 2009; Rodr\'iguez-Gonz\'alez, Raga, \& Cant\'o 2009).

Long ago, Lynds \& Sandage (1963) found a large outflow in M82 and gave the first definition of a starburst (SB). They proposed that material was 
ejected from the nuclear regions as a consequence of a series of SN explosions. Since then, galaxies with concentrated central starbursts (M82, 
NGC~253, NGC~1569, to name a few) have been used to study the GW phenomenon.

Tenorio-Tagle \& Mu\~noz-Tu\~n\'on (1998) showed the importance of the galactic potential component due to the cold dark matter halo in realistic 
models of GW, and concluded that the fate of the stellar wind and SNe material depends on the galactic properties and on the total energy injected 
by the SB. They showed that dwarf galaxies with an interstellar medium (ISM) with masses of the order of $10^9~M_\odot$, retain their metals unless 
they undergo an extreme burst of star formation, much larger than those presently observed. On the other hand, MacLow \& Ferrara (1999) explored 
both, analytically and numerically, the effects of stellar winds and SN explosions, in SBs with mechanical luminosities in the $\sim$10$^{37}-10^{39}$~erg~s$^{-1}$ 
range on the ISM of dwarf galaxies with gas masses in the $\sim$10$^6$ to $10^9~M_\odot$ range. They distinguished between two possibilities of gas 
ejection: (a) A ``blowout'', which requires material be accelerated above the escape speed at a distance of about three times the scale height. (b) 
A ``blow-away'', in which virtually all the ISM is accelerated and lost to the galaxy. The blowout would happen preferably for flatter galaxies; 
rounder galaxies are more likely to be completely disrupted (blow-away) if enough energy is injected. Their conclusion was that dwarf galaxies in 
that mass range undergo blowout for moderate-to-high luminosity starbursts values ($L \sim 10^{38}-10^{39}$~erg~s$^{-1}$), whereas for galaxies with 
gas masses $M_g > 10^{8}~M_\odot$ and mechanical luminosity $L \sim 10^{37}$~erg~s$^{-1}$ the blowout was inhibited. They also found that in the lowest 
mass objects, blow-away occurs virtually independently of $L$.

Many authors have shown that in the nearest starburst galaxies GWs could form through the collective effect of many individual stellar cluster winds, 
which in turn  are formed by the collective effect of many individual stellar winds. Perhaps the best studied starburst galaxy is M82, which has an 
extended, biconical filamentary structure in the optical (Shopbell \& Bland-Hawthorn 1998; Ohyama et al. 2002). In M82, the super stellar clusters 
(SSCs) have H${\alpha}$ luminosities in the $(0.01-23) \times 10^{38}$~erg~s$^{-1}$ range (Melo et al. 2005). These authors compiled a catalog of 197 
SSCs in M82 and 48 SSCs in NGC~253 (Melo 2005), with masses in the $10^4< M/M_{\odot}<10^6$ range (more recently, additional young star clusters in M82 
have been reported by Smith et al. 2007 and Westmoquette et al. 2007). Other examples are NGC~1569 (Anders et al. 2004; Westmoquette, Smith, \& Gallagher 
2008) and  M83 (Harris et al. 2001), with SSCs densities and masses similar to those of M82. Such an exceptional density of massive clusters (i.e. 
$\sim$620~kpc$^{-2}$, see also Anders et al. 2004; Melo et al. 2005; Melo 2005) is the best scenario to study powerful GWs driven by SSC winds originated 
in nuclear starburst regions. 

Yet another interesting effect is when the material ejected from the SSCs winds does not abandon the galaxy. Tenorio-Tagle (1996) explored the effect 
of a non-uniform interstellar medium (density and temperature) in the cooling efficiencies within the giant cavities. In high density regions the cooling 
would be more effective and would produce a series of metal-rich cloudlets. Such cloudlets would fall back to the disk and produce local enhancements 
of metallicity. However, Recchi, Matteucci, \& D'Ercole (2001) and Recchi et al. (2008) presented arguments against this scenario, pointing out that 
thermal conduction and hydrodynamical drag are likely to disrupt the condensations before they reach the disk, and thus the metals would be spread 
more or less uniformly  throughout the galaxy.

In the present paper we study the effects of young stellar clusters in the nuclear starburst (SB) region on the ISM of dwarf disk-galaxies. In particular, 
we calculate the amount of enhanced metallicity material that will end up in the intergalactic medium (IGM) and/or in the outer regions of the galactic 
disk. The paper is organized as follows: in \S~2 we describe the ingredients of the galactic models that we use. In \S~3 we discuss the possible fate of 
the mass and metals in such galaxies. In \S~4 we describe a series of 3D simulations of the interaction of SSCs with the galactic ISM. The results are 
presented in \S~5, and our conclusions in \S~6.


\section{Galactic model ingredients}

Each galaxy in our study consists of a dark matter halo, and a rotationally  supported disk of gas and stars. The models are constructed following the 
approach described in Springel, Di Matteo, \& Hernquist (2005, see also Hernquist 1993 and Springel 2000). A near-equilibrium galaxy model is constructed, 
with a dark matter halo that follows a Hernquist (1990) profile, and a disk with exponential surface gas density. Then, using an iterative procedure, 
the vertical gas profile is  determined self-consistently for a particular effective equation of state.


\subsection{The halos}

We modeled the dark matter mass distribution with the Hernquist profile (Hernquist 1990),
\begin{equation}
\label{rhoh}
\rho_{h}(r)=\frac{M_\mathrm{dm}}{2\pi}\frac{a}{r(r+a)^3}\,,
\end{equation}
where $r$ is the radial distance, and $M_\mathrm{dm}$ is the dark matter mass. Two remarks can be made about the shape of this profile: (i) In the inner 
parts of the galaxy it agrees with the profile given by the Navarro, Frenk, \& White (1996, henceforth NFW) fitting formula. (ii) Its faster than $r^{-3}$ 
decline causes the total mass to converge, allowing the construction of isolated halos without the need of an {\it ad hoc} truncation. The $a$ parameter is 
related to the scale radius $r_s$ of the NFW profile by:
\begin{equation}
\label{a}
a=r_s\sqrt{2[\ln(1+c)-c/(1+c)]}\,,
\end{equation}
where $c=r_{200}/r_s$ is the concentration parameter, $r_{200}$ is the radius at which the enclosed dark matter mean density is 200 times the critical 
value (where the critical density is $\rho_\mathrm{crit}=3H^2/8 \pi G$). We have set $c=9$.


\subsection{The disk}

We modeled gas and star disk components with an exponential surface density profile (in the radial direction of the galactic disk, $R$) the scale of 
length being $R_0$,
\begin{equation}
\label{sigmag}
\Sigma_{g}(R)=\frac{M_{g}}{2\pi R_0^2}\exp(-R/R_0)\,,
\end{equation}
\begin{equation}
\label{sigmas}
\Sigma_{*}(R)=\frac{M_{*}}{2\pi R_0^2}\exp(-R/R_0)\,,
\end{equation}
so that the total mass of the disk is $M_d=(M_g+M_*)=m_d M_\mathrm{tot}$, where $m_d$ is a dimensionless parameter (fixed in this work to $m_d=0.041$), 
and $M_\mathrm{tot}$ is the total mass of the galaxy, incuding the dark matter halo (i.e. $M_\mathrm{tot}=M_d+M_h$, where $M_h$ is the mass of the halo). 
The vertical mass distribution of the stars in the disk is specified by the profile of an isothermal slab with a constant scale height $H$. The 3D 
stellar density in the disk is hence given by,
\begin{equation}
\label{denstar}
\rho_* (R,z)=\frac{M_*}{4 \pi\,H\,R_0^2}~ \mathrm{sech}^2\left(\frac{z}{2H}\right) \exp\left(-\frac{R}{R_0}\right)\,.
\end{equation}
In all models, $H/R_0$ was fixed to $0.2$ whereas the relative content of gas in the disk (the rest of the disk mass is in stars) was set to 0.35. 
On the other hand, the disk spin fraction (the fraction of the total angular momentum in the disk), was set, as is customary, to the value of $m_d$. 
These values are typical of low-mass galaxies (e.g., Springel et al. 2005).


\subsection{Vertical structure of the gas disk}

The vertical structure of the disk at the beginning of the simulation is computed self-consistently with the halo and stellar disk components as follows. 
At $t=0$ the gaseous disk has uniform temperature (fixed to 1000~K). Consider the vertical structure of the gas disk in hydrostatic equilibrium:
\begin{equation}
\label{hidros}
\frac{\partial \rho_g}{\partial z}=-\frac{\rho^2_g}{\gamma P} \frac{\partial \Phi_T}{\partial z}\,,
\end{equation}
where, $\Phi_T=\Phi_h+\Phi_d$ is the total gravitational potential ($\Phi_h$ and $\Phi_d$ are the halo and disk gravitational potentials, respectively). 
For a given $\Phi_T$, the solution of this equation is constrained
by the condition
\begin{equation}
\label{sigmag2}
\Sigma_g(R)=\int \rho_g(R,z)dz\,,
\end{equation}
where $\Sigma_g(R)$ is the gaseous surface mass density (see also equation~\ref{sigmag}). One can obtain the vertical distribution by integrating 
equation~(\ref{hidros}), with a central density value at the midplane ($z=0$) and an effective equation of state of the form $P=P(\rho)$, see Springel 
et al. (2005). The starting value for the density is guessed, then adjusted in an iterative scheme until the desired surface density (equation~\ref{sigmag}) 
is recovered. This process is repeated for different radii to obtain an axisymmetric  gas density distribution.

\subsubsection{Cooling}

For the non-adiabatic cases, we adopt the cooling functions computed in collisional ionization equilibrium (CIE) by Sutherland \& Dopita (1993). The 
energy loss per unit mass of the gas is given by,
\begin{equation}
\label{cool}
\left(\frac{du}{dt}\right)_\mathrm{cool}=-\frac{\Lambda_\mathrm{net}(\rho,T)}{\rho}\,.
\end{equation}

We must note that the cooling function we used in our models assumes solar abundances. However, the kind of galaxies treated in this paper should be less 
metallic and thus the cooling should be lower. For the less metallic galaxies the results would be shifted from our radiative models to the adiabatic ones.


\section{The fate of the gas injected by young stellar clusters}
\label{sec:fate}

Depending on the total amount of kinetic energy injected by the SSCs two outcomes are possible: if the kinetic energy is larger than the gravitational 
potential of the galaxy, all material will escape and become part of the IGM; otherwise, the gas will return to the disk. If the material returns and 
reincorporates to the galaxy two things can happen: it can fall back close to the galactic center or far from it. Considering that GWs are confined 
to a cone, material that does not reach a high altitude will fall close to the galactic center, while gas that reaches a high altitude can fall back 
at large galactocentric radii.


\subsection{Mass loss measures}

In order to quantify the mass loss in numerical models one has to determine which material will be considered lost to the IGM. A natural estimation can 
be derived by comparing the velocity of a parcel of gas and the escape velocity at its current position. This has been done, for instance, in previous 
work by MacLow \& Ferrara (1999) and D'Ercole \& Brighenti (1999). This work, however, used Eulerian codes in which material that leaves the computational 
domain cannot be further analyzed, and therefore was automatically considered unbound. In reality, some of this material would return to the galaxy. Our 
models, that use the SPH (Smooth Particle Hydrodynamics) integration scheme, are not limited by the extent of the computational domain; thus, they were 
able to trace all the material for the duration of the simulation. We will consider the following more stringent criterion: only material that has left 
a cylindrical region of $R_\mathrm{max}=20$~kpc and $|z_\mathrm{max}|=20$~kpc (see also  D'Ercole \& Brighenti 1999) {\it and} has a velocity larger than the escape 
velocity at its current position will be considered effectively lost.

Mass loss in our models is thus carried out by first computing the total mass of unbound (ejected from the galaxy) gas $M_\mathrm{ej}$. To do this, we 
consider the total galaxy  gravitational potential given by,
\begin{equation}
\label{phit}
\Phi_T(r)=\Phi_h(r)+\Phi_d(r)\,,
\end{equation}  
where $r$ is the distance to the center, and $\Phi_h$ and $\Phi_d$ are the dark matter halo and disk components, respectively. Gas particles outside the 
cilinder with velocity larger than the escape velocity
\begin{equation}
v_\mathrm{esc}=\sqrt{-2\Phi_T(r)}
\label{vesc}
\end{equation}
are considered to be unbound. We define a `total mass ejection efficiency' as the ratio of the mass ejected (unbound) to the total gas mass,
\begin{equation}
\label{efmass}
\xi_m \equiv \frac{M_\mathrm{ej}}{M_g}\,.
\end{equation}
The mass ejection efficiency has two limiting cases: $\xi_m \to 0$ when all the galactic ISM is bound, and $\xi_m \to 1$ when all the galactic gas is 
expelled from the galaxy.

One can also define a ``metal ejection efficiency'' as 
\begin{equation}
\label{ej}
\xi_z\equiv\frac{M_{c,\mathrm{ej}}}{M_c},
\end{equation}
where $M_{c}$ is the total mass injected by the SSCs, and $M_{c,\mathrm{ej}}$ the mass, also originated in the SSCs, that is unbound. The mass injected by 
a star cluster contains the metals produced in the lifetime of the massive stars, $t \sim$40~Myr. For that time-span, using the metal yields calculated 
by Meynet \& Maeder (2002), Tenorio-Tagle et al. (2005) showed that the metallicity of the combined stellar winds remains at a value of $\sim$0.5~$Z_\odot$ 
from the start of the SB, up to a $t=3$~Myr evolutionary time, and then rapidly grows to  $\sim$15~$Z_\odot$ at $t\approx 7$~Myr, decreasing gradually 
to $\sim$3~$Z_\odot$ at $t\approx 20$~Myr.

The metal ejection efficiency has two limiting cases: $\xi_z \to 0$, when all the metallic gas injected by the nuclear starburst is kept in the galaxy 
disk, and $\xi_z \to 1$, when all the metallic gas processed in the burst is expelled from the galaxy. Galaxies with $\xi_z \to 1$ would be metal poor, 
while the IGM around them should be metal rich. On the other hand, metallic gas injected from the SBs region with $v_g < v_\mathrm{esc}$ is going to return 
to the galactic plane, and the ISM of such galaxies will be contaminated with the metals produced in the nuclear  starburst. 


%%%%% TABLA 1

\begin{table*}[!t]
\vspace{0.1cm}
\centering
\caption{The galactic models}
\begin{tabular}{lcccccc}
%\label{ini15}
\toprule
Galaxy & $M_{g}$       & $M_h$        & $v_c(r_{200})$ & $H$   & $N^a_\mathrm{ep}$ & $N^r_\mathrm{ep}$ \\
model  & $M_{\odot}$    & $M_{\odot}$   & km~s$^{-1}$    & kpc   &           & \\
\midrule
G1 & $6.2 \times 10^6$ & $4.3 \times 10^8$     & 12      & 0.044 & 335    & 478\\
G2 & $3.0 \times 10^7$ & $2.1 \times 10^9$     & 20.8    & 0.071 & 530    & 766\\
G3 & $7.0 \times 10^7$ & $4.9 \times 10^9$     & 27.7    & 0.094 & 473    & 573\\
G4 & $1.4 \times 10^8$ & $9.4 \times 10^9$     & 34.3  & 0.118 & 458    & 646\\
G5 & $2.1 \times 10^8$ & $1.4 \times 10^{10}$  & 39.4   & 0.135 & 453    & 681\\
G6 & $3.0 \times 10^8$ & $2.1 \times 10^{10}$  & 44.5   & 0.153 & 429    & 666\\
G7 & $4.7 \times 10^8$ & $3.2 \times 10^{10}$  & 51.9   & 0.178 & 379    & 655\\
G8 & $7.0 \times 10^8$ & $4.8 \times 10^{10}$  & 59.2   & 0.203 & 317    & 585\\
G9 & $1.1 \times 10^9$ & $7.3 \times 10^{10}$  & 68     & 0.233 & 255    & 434\\
G10& $1.5 \times 10^{10}$& $1.1 \times 10^{11}$ & 76.8  & 0.263 & 191    & 317\\
\bottomrule
\end{tabular}
\vspace{0.3cm}
\end{table*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Disk contamination measures}

As mentioned above, bound gas ejected by the SSCs will return to the galactic disk, either close to the center if it never reached a high altitude, or 
spread to large galactocentric radii if it did. This mechanism is of course a continuum, and ``high'' or ``low'' altitude are just relative terms. To 
quantify the contamination of the disk by material originated in the SSCs we will consider as `high altitude' a distance of at least 3 times the scale 
height of the disk, $H$.

One can define a new parameter, the fraction of bound gas (that is inside a cylinder of radius $R_\mathrm{max}$ and with velocity less than the escape 
velocity) that reaches at least an altitude of $3H$:
\begin{equation} 
\label{reins}
\chi_b\equiv\frac{M_{\mathrm{ej},3H}}{M_g}\,,
\end{equation}
where $M_{\mathrm{ej},3H}$ is the total bound mass (including the ISM and the material from the SSCs) that reaches an altitude $\geq 3H$. This mass will 
be reinserted into the galactic disk at different radii. However, since it includes material both from the SSCs (with high metallicity) and from the ISM 
around the SSCs (lower metallicity) it is not clear how much the metallicity of the disk will be enhanced by this mixture. For that reason we define the 
fraction of metallic bound gas that leaves the galactic disk and reaches an altitude between $3H$ and $z_\mathrm{max}$ as
\begin{equation} 
\label{reinsz}
\chi_{b,z}\equiv\frac{M_{c,3H}}{M_c}\,,
\end{equation}
where $M_{c,3H}$ is the bound mass injected by the stellar cluster that reaches an altitude $3H \leq z \leq z_\mathrm{max}$, and $M_{c}$ the total mass 
injected by the SSC wind. This gives a better measure of the amount of high metallicity material available to spread throughout the disk. Nevertheless, 
a detailed study of how it will be distributed in the disk is necessary for a complete disk contamination model, which is beyond the scope of this paper.


\section{Numerical simulations}

We have performed adiabatic and radiative 3D $N$-Body/SPH simulations that model the effect of a central compact starburst, in which mechanical energy 
is injected by stellar cluster winds, using GADGET-2\footnote{GADGET-2 is a code for cosmological $N$-Body/SPH-simulation on serial workstations or 
massively parallel computers with distributed memory (Springel, Yoshida, \& White 2001; Springel 2005).}. All simulations were done with three types of 
particles: disk, halo, and gas particles, and,  unless otherwise stated, each consist of 30000, 40000, and 30000 particles, respectively. Two separate 
simulations were run for each model, one adiabatic and a second one including radiative losses.

Most simulations (low-resolution simulations) use a timestep $\sim$1--3$\times 10^5$~yr and a Plummer softening of 0.25~kpc, 0.5~kpc, and 0.25~kpc, for 
gas, halo, and disk particles, respectively. We ran a grid of models using different galaxy masses and starburst energies. The parameters of the models 
are described below.


\subsection{The host galaxies}

We constructed ten isolated galaxies as described in Springel (2000) and in \S~2 of this paper. The galaxies are named G1 to G10; they have ISM 
masses in the range of $\sim$5$\times 10^6-10^{10}~M_\odot$, similar to those used by MacLow \& Ferrara (1999); the total masses are in the 
$\sim$4$\times 10^{8}-1 \times 10^{12}~M_\odot$ range. The disk masses of the galaxies are given by $M_d=0.041~M_\mathrm{tot}$, and the total mass 
of the gas is 35\% of the galactic visible mass in all of  the models.

Table~1 shows the mass of the gas and the halo, the circular velocity at $r_{200}$ and $R_0$  for all the galactic models.


\subsection{The starburst regions}

In order to reproduce the observed properties of starburst galaxies with young stellar clusters, including M82, NGC~253 and NGC~1569, we use a wide 
range of SBs masses. We should note that the numbers of these energetic stellar asociations in the host galaxies were detected at optical wavelengths 
(e.g., for M82 see Melo et al. 2005; for  NGC~1569 see Westmoquette et al. 2008). Since there is a high level of obscuration in the nuclear regions 
of M82, NGC~1569, and NGC~253, the real number of clusters would be larger. Melo et al. (2005) and Smith et al. (2007) have shown that in the nuclear 
SBs region of M82 the SSCs are confined to the central $\sim$200~pc of the galaxy.


%%%%% TABLA 2

\begin{table}[!t]
\vspace{0.1cm}
\centering
\caption{The starburst models}
\begin{tabular}{ccr@{$\times$}l}
%\label{ini15}
\toprule
Starburst Model & $E_{c}$ (erg) & \multicolumn{2}{c}{$M_\mathrm{SB}$ ($M_\odot$)}  \\
\midrule
S1~  & $1 \times 10^{52}$  & $\sim$5.5  & $10^2$\\
S2~  & $5 \times 10^{52}$  & $\sim$2.75 & $10^3$\\
S3~  & $1 \times 10^{53}$  & $\sim$5.5  & $10^3$\\
S4~  & $5 \times 10^{53}$  & $\sim$2.75 & $10^4$\\
S5~  & $1 \times 10^{54}$  & $\sim$5.5  & $10^4$\\
S6~  & $5 \times 10^{54}$  & $\sim$2.75 & $10^5$\\
S7~  & $1 \times 10^{55}$  & $\sim$5.5  & $10^5$\\
S8~  & $5 \times 10^{55}$  & $\sim$2.75 & $10^6$\\
S9~  & $1 \times 10^{56}$  & $\sim$5.5  & $10^6$\\
S10  & $5 \times 10^{56}$  & $\sim$2.75 & $10^7$\\
S11  & $1 \times 10^{57}$  & $\sim$5.5  & $10^7$\\
\bottomrule
\end{tabular}
\vspace{0.3cm}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We estimated the mass of the starburst using Starburst 99 models (Leitherer \& Heckman  1995 and Leitherer et al. 
1999\footnote{http://www.stsci.edu/science/starburst99/.}) with the appropriate mechanical energy and a Salpeter initial mass function (IMF) 
that includes stars between 1 and $100~M_\odot$. In order to obtain a better estimation of the total starburst masses, we have calculated the 
mass of the stars between 0.1 and $1~M_\odot$ using the Kroupa, Tout, \& Gilmore (1993) IMF. We assumed that all of the energy produced in the 
SB lifetime is injected instantaneously, at a given time $t_c$, inside a spherical region of radius $R_c$ at the center of the disk galaxy as 
thermal energy. At this time (corresponding to $\sim$10 orbital periods of the galactic disk at a radius of 150~pc), the gas in the central 
regions of the galaxy has already developed a clumpy structure. Actually, in the present version of the code we cannot consider the case of 
an injection that extends over a finite time period.

The total injected energy is obtained from
\begin{equation}
\label{totalenergy}
E_{c}= \int^{t_f}_0 L_m(t) dt\,,
\end{equation}
where $t_f$ is the typical SBs lifetime, and $L_m(t)$ the mechanical luminosity as a function of time. The most important contribution to the 
mechanical luminosity happens at $\sim$40~Myr (see also Leitherer et al. 1999) and after that time the mechanical luminosity decreases drastically, 
due to the fact that the most massive stars have died by then. Table~2 shows the total energy and the respective mass of all the starburst models 
(S1--S11). 


%%%%%% FIGURA 1

\begin{figure*}[!t]
\centering 
\includegraphics[width=14cm]{arodriguezfig1.eps}
\caption{Time sequence of cuts of  number density in the $xz$-midplane ($y=0$) for the model 
S8+G6 (see Tables~1 and 2). The integration times in years are given in the upper left corner 
of the frames. The top panels (a, b and c) are the adiabatic models, and the bottom panels 
are the radiative ones.}
\label{s8g6}
\vspace{0.3cm}
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{The grid of models}

We computed a total of 220 numerical simulations of GWs produced by the nuclear starburst events, with different total masses of the 
galaxies G1--G10 (see Table~1), combined with different starburst energies S1--S11 (see Table~2), all adiabatic as well radiative (110 
models each). The total energy injected by the starburst of the models S1--S11 was distributed in the gas particles inside a sphere of 
$R_c=150$~pc at the center of each of the galactic models G1--G10.

We choose this size for the region in which the energy of the starburst is injected, because it is representative of the sizes of observed 
starbursts. For example, in M82 the starburst region has a $R_c \sim 175$~pc radius (Moorwood 1996), and in NGC~1569A the starburst hole has 
a $R_c\sim 100$~pc radius (Fujita et al. 2009). An exploration of the effects on our results of varying $R_c$ is shown in \S~6.
 
The number of energetic particles ($N^a_\mathrm{ep}$ and $N^r_\mathrm{ep}$, for the adiabatic and radiative cases, respectively) is related to 
the total number of particles of gas in the initial conditions. The energy per particle in the starburst region is $E_i=E_c/N_\mathrm{ep}$, 
the number of energetic points used in the galactic models, is shown in Table~1. All the cluster particles (the energetic particles) are tagged 
in order to estimate the fate of the metals produced during the starburst lifetime.
                                                                                                    

\section{Results}

We have let the galaxy models evolve on their own for $t_c=1.5\times10^7$~yr. At the time $t_c$, the total energy of the SB is injected in the 
center of the galaxy. MacLow \& Ferrara (1999) injected the energy from the cluster in a continuous way over a period of 40~Myr. Given the fact 
that the flow induced by the cluster takes $\sim$0.5~Gyr to travel a distance of $\sim$10~kpc (i.e., the size of the galaxy), the differences 
that might result from releasing the cluster energy over $\sim$40~Myr=0.04~Gyr (rather than instantaneously) cannot be very important. 

The ejection efficiencies were computed at $t=1$~Gyr in all our models. The values of the efficiencies reach a constant value of integration 
times of $\sim$0.2--0.5~Gyr and $\sim$0.4--0.7~Gyr for the adiabatic and radiative models (respectively).


%%%%% FIGURA 2

\begin{figure}[!t]
\centering
\includegraphics[width=0.85\columnwidth]{arodriguezfig2}
\caption[]{Mass ($\xi_m$, top) and metal ejection efficiency ($\xi_z$, bottom) contours as a function 
of total mass of the gas and total starburst masses, for adiabatic models (top and bottom panel, respectively).}
\label{fig:efa}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Mass and metal ejection efficiencies}
\label{sec:ejec}

Total mass and metal ejection efficiencies have been calculated using the definitions given in \S~\ref{sec:fate}. In particular, the kinetic 
and gravitational potential energies as a function of the radial distance $r_i$ (with respect to the center of the potential) were computed 
for each gas particle. A particle $i$ would escape the galactic gravitational potential if $v_i \geq v_\mathrm{esc}=\sqrt{-2 \Phi_i}$, where 
$v_i$  is the velocity and $\Phi_i$ the total potential at the position of the particle.

We also traced the high metallicity gas particles (injected by the SBs) and the low metallicity gas (galactic ISM gas). The high metallicity gas 
was tagged with a negative passive scalar, the ISM with a positive one. Therefore, $\xi_z$  was calculated using the gas particles  with a 
negative passive scalar, while $\xi_m$ was calculated with all the gas particles (regardless of the sign of the passive scalar). 


%%%%% FIGURA 3

\begin{figure}[!t]
\centering
\includegraphics[width=0.85\columnwidth]{arodriguezfig3}
\caption[]{Same as Figure~2, but for the radiative models.}
\label{fig:efr}
\vspace{0.5cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Figure~\ref{s8g6} shows the temporal evolution of the ejected gas from a central starburst with properties that correspond to starburst model 
S8 at the center of  galaxy model G6 (see Tables~1 and 2). The three timesteps shown are $2 \times10^7$, $6\times10^7$ and $10^8$~yr, for the 
adiabatic case --panels (a), (b) and (c), respectively--, and the radiative one --(d), (e) and (f), respectively. We can see the ejection of 
gas from the plane of the dwarf galaxy with a moderate ISM mass ($3\times10^8~M_\odot$), produced by a central starburst  that ejects a mass of 
$1.5\times10^6~M_\odot$. Such a SB corresponds to a region with $\sim$9000 massive stars, injecting a total energy during its lifetime of around 
$5\times 10^{55}$~erg (see the starburst models of Leitherer \& Heckman 1995 and Leitherer et al. 1999). From the figure we see right away that 
the amount of gravitationally unbound gas is larger in the adiabatic than in the radiative case. However, in both cases an important 
part of the disk gas is pushed above the galactic plane, and in the radiative case a significant part of this gas is returned to the galactic disk.

For this model (S8+G6) the totality of the galaxy gas is pushed above $3H$; however, only 43\% of the gas acquires enough energy to leave the 
galactic potential and become part of the IGM, while the remaining 57\% would eventually return to the disk of the galaxy, since this ISM 
material is gravitationally bound. For the radiative case, the percentage of the gas that is 
gravitationally unbound is of 14\%, while 4.7\% of the ejected gas gets above $3H$ and returns to the galactic disk. On the other hand, in the 
adiabatic case 97\% of the metallic gas ejected by the nuclear starburst escapes from the galactic gravitational potential. In other words, a 
galaxy with those parameters would be contaminated only with 5\% of the metals produced in the massive SB. For the radiative S8+G6 model, 79\% 
of the metals escape from the galaxy.

The mass ejection efficiencies are presented in Figures~\ref{fig:efa} and \ref{fig:efr}, for the adiabatic and the radiative model grids, respectively. 
In these figures, the abscissa sweeps the galactic models (G1--G10), and the ordinate sweeps the SB models (S1--S11).

Figure~\ref{fig:efa} shows isocontour maps of the mass ejection efficiency $\xi_m$ (top panel) and the metal ejection efficiency $\xi_z$ (bottom panel) 
for the adiabatic models as a function of the SB mass and the total gas mass of the galaxy. For the mass ejection efficiency (top panel) we plotted 
isocontours for $\xi_m=$ 0.01, 0.10, 0.3, 0.5, 0.7, 0.9 and 0.99. In this figure, we can see that for the more massive galaxies (in which the gas 
mass is larger than $10^9~M_\odot$) the fraction of unbound gas is less than $\sim$0.25, even for a rather large nuclear starburst. However, galaxies 
with an ISM of mass lower than $10^8~M_\odot$ will lose virtually all of their gas if they undergo a nuclear starburst that injects more than $10^5~M_\odot$. 


%%%%% FIGURA 4

\begin{figure}[!t]
\centering
\includegraphics[width=0.85\columnwidth]{arodriguezfig4}
\caption[]{Fractions of bound gas ($\chi_b$, top) and bound metallic gas ($\chi_{b,z}$, bottom) that reach 
an altitude of at least $3H$, for the adiabatic models.}
\label{fig:reta}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


From the $\xi_z=$ 0.01, 0.10, 0.3, 0.5, 0.7, 0.8, 0.9 and 0.99 isocontours in the bottom panel one can see that most of the gas ejected from the 
cluster escapes the galaxy if the starburst mass is $>10^{6.5}~M_\odot$. Also, most of the ejected gas escapes if the galaxy has a total gas mass 
$<10^7~M_\odot$. On the other hand, for starburst masses $<10^{4.5}~M_\odot$ and galactic gas masses $>10^8~M_\odot$ most of the ejected metals are 
retained by the galaxy. From the bottom panel of Figure~\ref{fig:efa} we also see that a starburst with a mass $>10^{5.5}~M_\odot$ will eject more 
than 50\% of its wind into the IGM.

Figure~\ref{fig:efr} shows the $\xi_m$ (mass ejection) and $\xi_z$ (metal ejection) efficiencies as a function of galactic gas mass and starburst 
mass computed from the radiative models. Both the $\xi_m$ and $\xi_z$ contour maps (top and bottom panels of Figure~\ref{fig:efr}) are qualitatively 
similar to the ones computed from the adiabatic models (see Figure~\ref{fig:efa}). Even though the detailed shapes of the contours vary, the main 
difference between the radiative and adiabatic cases is the following: in order to obtain similar ejection efficiencies, one needs a starburst mass 
larger by a factor of $\sim$2 in the radiative case.


%%%%% FIGURA 5
 
\begin{figure}[!t] 
\centering
\includegraphics[width=0.85\columnwidth]{arodriguezfig5}
\caption[]{Same as Figure~4, but for the radiative models.}
\label{fig:retr}
\vspace{0.3cm}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%% FIGURA 6

\begin{figure*}[!t] 
\centering
\includegraphics[width=12cm]{arodriguezfig6}
\caption[]{Distribution of the SPH particles ejected by the central starburst for model S7+G9 after 1~Gyr. 
The inner region of the domain is shown, and the location of the SPH particles gives the spatial distribution 
of the metals that have been deposited back on the galactic disk. The left plot shows the distribution of 
positions projected onto the galactic disk, and the right plot shows the positions of the SPH particles projected 
onto a plane which includes the rotation axis of the galactic disk. The greyscale shows the distribution of 
the mass density on these two planes.}
\label{fig:f6}
\vspace{0.4cm} 
\end{figure*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Material returned to the galactic disk}
\label{sec:ret}

Figure~\ref{fig:reta} shows the efficiencies of gas ($\chi_b$, top panel) and metal mass ($\chi_{b,z}$, bottom panel) which reach at least 
3 scale heights ($H$) above the disk but remain gravitationally bound, computed from the adiabatic models. Both panels show two low efficiency 
regions divided by a high efficiency ridge (which runs diagonally from low to high SB and galactic gas masses). In the high galactic mass/low 
SB mass region, the SB ejection mostly remains close to the galactic plane, and does not reach the required $3H$ height above the disk plane 
(in order to be counted in the $\chi_b$ and $\chi_{b,z}$ efficiencies). In the low galactic mass/high SB mass region, the SB ejection mostly 
escapes from the galaxy. From the $\chi_{b,z}$ distribution (bottom panel), we see that models with $M_g\sim 10^8 \to 10^9~M_\odot$ and 
$M_\mathrm{SB}\sim 10^3 \to 10^5~M_\odot$ are able to redistribute more than $\sim$50\% of the ejected metals over the galactic ISM. 

Figure~\ref{fig:retr} shows the $\chi_b$ (top panel) and $\chi_{b,z}$ (bottom panel) efficiencies computed from the radiative models. The 
efficiencies show the same qualitative behaviour (with two low efficiency regions separated by a high efficiency ridge) as the ones calculated 
from the adiabatic models. However, the maximum values of the efficiencies are lower. In the adiabatic case, the maximum values of the gas mass 
redistribution efficiency is of $\sim$20\%, while a maximum value of only $\sim$10\% is found for the radiative models. For the metal redistribution 
efficiency, values above $\sim$70\% are obtained for the adiabatic models, while maximum values of only $\sim$20\% are obtained from the radiative 
models.

In Figure~6, we show the distribution of metals that have fallen back onto the galactic disk from the starburst ejection of model S7+G9, after a 
1~Gyr time evolution. The SPH particles corresponding to the collapsed material form a distribution with a height of $\sim$300~pc (about the initial 
scale height of the galactic disk), and with a radial extent of $\sim$2~kpc (see Figure~6), which is about twice the size of the initial scale of 
length ($R_0$) of the G9 galaxy (see Table~1). This is an illustration of the fact that in all of our models the infalling metals are redistributed 
over a large region of the galactic plane.


\section{The effects of varying the radius of the injection region and the resolution of the initial starburst}

In all of the models presented above, we have assumed that the energy from the starburst is injected in a spherical region of radius $R_c=150$~pc 
(which is a representative value of the observed sizes of starburst regions, see \S~4.3). In this section, we explore the effects on the ejection 
efficiencies of varying the value of $R_c$.

We focus on three of the starburst+host galaxy combinations which we have studied above (see Tables~1 and 2). We choose models S5+G3 (a small mass 
galaxy with an intermediate energy starburst, S10+G7 (an intermediate mass galaxy with a strong starburst) and S11+G9 (a high mass galaxy with a 
strong starburst). For these three models, we compute radiative simulations in which the starburst energies are injected in spheres of radii 
$R_c=50$ to 150~pc (in 20~pc steps). The number of SPH particles within the spheres scales as the volume of the initial starburst (45 particles for 
$R_c=50$~pc to 550 particles for $R_c=150$~pc).


%%%%% FIGURA 7

\begin{figure}[!t] 
\includegraphics[width=0.99\columnwidth]{arodriguezfig7}
\caption{The mass ejection efficiency $\xi_m$ computed for models S5+G3, S10+G7 and S11+G9 as 
a function of the initial radius $R_c$ of the starburst.}
\label{fig:f7}
\vspace{0.4cm} 
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In Figure~7, we show the mass ejection efficiency $\xi_m$ (see equation~\ref{efmass}) computed for models S5+G3, S10+G7 and S11+G9 as a function of 
the initial radius $R_c$ of the starburst. We see that there is clear dependence of $\xi_m$ on $R_c$, with differences by factors of $\sim$1.7 (for 
models S11+G9 and S10+G7), and over $\sim$3.5 for model S5+G3.

We should note that the models with the smaller energy injection regions (with $R_c=50$~pc) have smaller mass ejection efficiencies than the models 
with larger energy injection regions (with $R_c=70$~pc, see Figure~7). This is probably due to the fact that in the smaller volumes (covering the denser, 
inner region of the galaxy) the radiative cooling is stronger. In the models with even larger energy injection regions (with $R_c\geq 90$~pc, see 
Figure~7), the momentum per unit mass given to the galactic gas is lower, and therefore less material succeeds in escaping from the galactic potential. 
From these results, we conclude that in our intermediate and high mass galaxy models, variations of factors of $\sim$2 in the efficiencies are to be 
expected for starbursts (with the same energies) with initial radii in the $R_c=50$ to 150~pc range.

We have also explored the effect of changing the total number of SPH particles in the simulations. We again choose models S5+G3, S10+G7 and S11+G9 
(see above and Tables~1 and 2) as representative of our model grid, keeping a constant $R_c=150$~pc initial radius for the starbursts, and we compute 
radiative simulations with  more than one order of magintud more of gas, halo and disk particles (1500000, 2000000 and 1500000, respectively) than 
in the previous models (see \S~4). In these new runs we have also changed the softening length to 50~pc, for all the components. In all the 
new models the mass ejection efficiency increases by a factor of the order of 2. For model S5+G3 this change is a factor of $\sim$1.5, while S10+G7 
and S11+G9 suffer changes of factors of $\sim$2.21 and $\sim$1.6, respectively.

Therefore, we seem to obtain at least a partial convergence for the values of the ejection efficiencies in simulations with $\sim$10$^6$ gas particles. 
However, the computing time necessary for these simulations is quite large ($\sim$30 times longer than for the 30000 gas particle models), so that our 
parameter study has been done at lower resolution.


\section{Discussion}

MacLow \& Ferrara (1999) computed radiative, axisymmetric simulations of the ejection of gas from galaxies due to central starbursts. Their models 
differ from ours in that the starburst energy is injected as a continuous source term during the 50~Myr of their simulations, while in our models the 
energy of the starburst is injected instantaneously at the beginning of the simulations.

If we convert their starburst mechanical luminosities into total energies (by multiplying the luminosities by 50~Myr), and use the relation between 
starburst energy and mass of Table~2, we can find correspondences between their models and ours. We then find that both the mass and metal ejection 
efficiencies computed by MacLow \& Ferrara (1999) are generally consistent with the ones that we have computed, with typical differences of factors 
$\sim$3. We judge this to be a satisfying agreement given the differences between our models and the ones of MacLow \&  Ferrara (1999).

Interestingly, the highest SB mechanical energy/lower galactic mass model of MacLow \& Ferrara (1999) approximately corresponds to our S5+G2 model 
(with $M_\mathrm{SB}=3\times 10^4~M_\odot$ and $M_g=3\times 10^7~M_\odot$). As can be seen in Figure~3, this model lies within the transition from low 
to high efficiency regimes. Therefore, the models of MacLow \& Ferrara (1999) mostly missed the transition between low and high mass loss efficiencies 
which we are describing in the present paper.

D'Ercole \& Brighenti (1999) computed a set of 5 models of galaxies with $M_g<1.4\times 10^8~M_\odot$, similar to our galaxy models G1--G5, in which 
they inject energies in the range of our S6--S8 starbursts. Their ``PEXT'' model (see Table~2 of D'Ercole \& Brighenti 1999) has parameters similar to 
our S8+G4 model (see Tables~1 and 2). The PEXT model has a $\xi_m=0.31$ mass ejection efficiency, which is consistent with the $\xi_m=0.45$ efficiency 
obtained from our S8+G5 model. D'Ercole \& Brighenti (1999) have studied different possibilities (e.g., having galaxies with no rotation, or having 
fixed dark matter mass distributions) which have not been explored in our present work. 

Fragile, Murray, \& Lin (2004) computed mass and metal ejection efficiencies produced in galaxies with $10^9~M_\odot$ (similar to our galaxy model G9) 
in which they inject energies in the range of our model S3-S7 starbursts. Their model~5 (see Table~1 of Fragile et al. 2004) has parameters similar to 
our S7+G9 model. Model~5 has a $\xi_m=0.016$ mass ejection efficiency and $\xi_Z=0.99$ metal ejection efficiency (see Table~2 of Fragile et al. 2004), 
which is consistent with the $\xi_m=0.006$ and $\xi_Z=0.805$ efficiencies of our model S7+G9. Fragile et al. (2004) have explored the metal ejection 
efficiencies for supernova events at different radial positions in the galactic disk. They find that a substantial fraction of the metals is retained 
in their off-center starburst models. These result shows that if we consider off-center galactic starbursts, the mass and metal ejection efficiencies 
will be substantially lower than the ones that we have computed in our present paper.

It is worth mentioning that the strong winds produced by the combination of individual stellar winds and SNe explosions not only produce the GWs, but 
have important consequences on the star formation history of their host galaxy as well. They are the most efficient way to redistribute metals to 
the ISM (and/or the IGM). This produces a feedback mechanism that facilitates the gravitational collapse of the ISM to form stars. The material with 
higher metallicity cools more efficiently, and it therefore forms high density condensations on shorter timescales than the metal poor gas (Scannapiecco 
et al. 2005; Sutherland \&  Dopita 1993). At the same time, the energy feedback from these strong winds can heat and disrupt cold gas clouds, thus 
inhibiting star formation.

Observations and theory suggest (e.g., Larson 1974; White \& Rees 1978; White \& Frenk 1991) that large outflows should be able to develop in small 
systems because of their shallower potential wells. This is because, according to hierarchical galaxy formation scenarios, large systems are formed 
by the aggregation of smaller ones. Thus, the energy feedback from strong winds should have a more important effect for systems in the early stages 
of formation. 

Our models provide quantitative predictions of the amount of metals which are ejected into the IGM and which are redistributed within the host galaxy. 
These predictions could be used in a direct way for computing models of the star formation history of the host galaxy, and could also be included in 
models of the history of galaxy formation. 


\section{Conclusions}

We computed 220 numerical simulations of GWs produced by nuclear starburst energy injected inside a spherical region (of radius $R_c$) at the center of 
the disk galaxy. We used different total masses of the galaxies G1--G10, combined with different starburst energies S1--S11, all adiabatic as well as 
radiative (110 models each). We have performed adiabatic and radiative 3D N-Body/SPH simulations of GW models using GADGET-2. The radiative models have 
been calculated using a solar metallicity cooling function. Finally, the adiabatic models were calculated in order to obtain the maximum possible value 
of the mass and metal ejection effciency in our models of galactic winds, and do not necessarily correspond to real cases of starbursts in galaxies. 

From the grid of models we obtained the mass and metal ejection efficiencies as a function of the mass of the gas in the galaxy and the total mass of the 
nuclear starburst.

In the same way, we computed the efficiencies of gas and metal mass that reaches at least $3H$ above the disk but remains gravitationally bound for both 
the adiabatic and radiative regimes. We present these efficiencies as a function of the mass of interstellar medium in the galaxies versus total mass of the 
nuclear starburst.

We show that for compact starburst regions (e.g., inside the SSC) with masses larger than $10^6~M_\odot$ in the adiabatic case, the metal injected by the 
nuclear starburst is expelled to the IGM (at least 90\% of this gas). We show that in galaxies with gas masses between $10^7$ to $10^{10}~M_\odot$ and with 
a massive nuclear starbursts the available metals are freed from the gravitational potential of the galaxy. In the radiative case the critical SB mass 
at which total loss of metals occurs is around 2 or 3 times larger than in the adiabiatic one.

In addition, we have calculated the efficiency of the metal redistribution in the disk of galaxies. We find that the metal redistribution is more efficient 
for intermediate and low mass starburst masses ($<10^{4.5}~M_\odot$) and galactic gas masses $>10^8~M_\odot$ (for both the adiabatic and radiative cases).

In this paper we presented models in  which we did not consider many elements that are likely to be important in starburst galaxies; among these are:
\begin{itemize}
\item clumpiness in the medium,
\item the generation of new stars,
\item a different fraction of gas in the galaxy disk,
\item the effect of galactic mass loading (see Fujita et al. 2009).
\end{itemize}

Therefore, there is a large range of possibilities for future studies including differential and/or selective galactic winds (also, Pilyugin 1993 and 
Recchi et al. 2001, 2002).

Finally, in our paper we have presented a limited resolution study (in \S~6). Our evaluation of the dependence of the computed efficiencies on resolution 
is limited to a range of $\sim$3 in resolution. This range is of course not very impressive, and questions remain as to the results that would be obtained 
at considerably higher resolutions (presently not possible, at least for calculating relatively extended grids of models). When such higher resolution 
models do become available, one will probably find different numerical values for the efficiencies, but one would expect that the transition between low 
and high ejection efficiency systems (found in the present work) will be preserved.


\vspace{0.25cm}

\acknowledgements

We thank the anonymous referee for very relevant comments that resulted in a substantial revision of the original version of this paper. We thank 
Volker Springel (from the Univ. of Heidelberg) for helpful suggestions which we have been included in this paper. We acknowledge support from the 
Conacyt grants 61547, 101356 and 101975.


\vspace{0.35cm}

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