### The self-regulation of star-forming galaxies : Implementation of the bathtub model and

### study of the metallicity

### Student : Steve Blandin

### Project Supervisor - IRAP : Nicolas Bouché School Supervisor - Supaéro : Sébastien Massenot

### School Supervisor - KTH : Felix Ryde TRITA-FYS 2015:17

### ISSN 0280-316X

### ISRN KTH/FYS/- -15:17—SE

### March 6, 2015

Abstract

The evolution of galaxies is a research topic that has evolved a lot during the last few years. In particular, it appears that the composition of galaxies (in gas and stars) follows some relatively simple scaling relations. Recently, researches have found that the star formation law scales with the growth law of the dark matter, which is the main component of the galactic halo.

The purpose of the master thesis is the analysis of the metal production (every elements except hydrogen and helium) in general within a simplified model that will be solved numerically through a program written in Python. We aim to model some fundamental quantities such as the evolution of metallicity through the cosmological time or determine, at a given redshift, the metallicity as a function of the stellar mass of a galaxy. Then the results will be compared with analytic solutions and observational data.

Acknowledgements

I express my thanks to Nicolas Bouché, my supervisor at IRAP, for his necessary advice during my research as well as the writing of this report. I would also like to thank the members of GAHEC who provided scientific guidance. Finally I express my thanks to Sébastien Massenot and Felix Ryde, my supervisors at Supaéro and KTH respectively, for providing necessary information regarding the organisation of the master thesis.

## Contents

1 Introduction 3

1.1 The IRAP . . . 3

1.2 History of studies of galaxies . . . 5

1.3 Presentation of the subject . . . 6

1.3.1 The metallicity in star forming galaxies . . . 6

1.3.2 The study of the Z − M∗ relation : motivations . . . 6

2 The 3 diffenrential equations model 7 2.1 The bathtub model . . . 7

2.1.1 Model of halo formation . . . 7

2.1.2 Implementation of the bathtub model . . . 9

2.2 Study of Metallicity . . . 15

3 Results of metallicity in star forming galaxies 18 3.1 First results on metallicity . . . 18

3.1.1 The "no min" model . . . 18

3.1.2 The accretion floor model . . . 18

3.1.3 The quasi-steady equilibrium . . . 19

3.1.4 The impact of the accretion floor . . . 19

3.2 The mass/metallicity relation . . . 20

3.2.1 Choice of the model for the outflows . . . 21

3.2.2 The MZR without accretion floor . . . 22

3.3 Discusion and comparison with existing MZR models . . . 24

3.3.1 Lilly et al. (2013) . . . 24

3.3.2 Peng & Maiolino (2014) . . . 25

3.3.3 Zahid et al. (2014c) . . . 26

3.3.4 Yabe et al. (2014) . . . 26

4 Beyond equilibrium : Addition of stochastic events 30 4.1 The Z − M∗− SF R relation . . . 30

4.1.1 Results . . . 30

4.1.2 SFR or relative specific SFR for the fundamental plane ? . . . . 32

4.2 Comparison with Salim et al. (2014) . . . 33

5 Conclusion 35

References 36

## Chapter 1 Introduction

I have done my master thesis at the Institute of Research in Astrophysics and Planetol- ogy (IRAP) in Toulouse (France). In this introduction I will first present the Institute, and then describe briefly the history of studies of galaxies, and finally make a general presentation of the subject studied during this master thesis.

### 1.1 The IRAP

IRAP (logo on Figure 1.1) was created on the 1st January 2011. Martin Giard, Director of Research at CNRS, is its Director since its creation.

Figure 1.1: The logo of IRAP.

It is a Mixed Research Unity (UMR) depending on Paul Sabatier University (Toulouse, France) and the Centre National de la Recherche Scientifique (CNRS).

It is actually a gathering of the employees coming from some laboratories of the Ob- servatory Midi-Pyrénées (OMP).

There are about 300 people at IRAP :

• 110 researchers and lecturers

• 80 permanent staff supporting the research

• 50 post-docs

• 50 doctoral students

• several intern students

Figure 1.2: Organisation at IRAP.

Figure 1.3: Hubble’s classification scheme. Source : http://www.universetoday.com.

There is a Management Team at the head of IRAP organisation. It contains the director, the deputy, technical, administrative and financial directors. Below the institute is divided in four entities around four scientific thematic areas (Figure 1.2).

The researchers are spread among 5 groups, which are :

• Planetary Geophysics and Space Plasma (GPPS)

• Sun and Stars Physics (PSE)

• Interstellar Medium, Cycle of Matter, Astro-Chemistry (MICMAC)

• Galaxies, High Energies Astrophysics and Cosmology (GAHEC)

• Signal and Images in Sciences of the Universe (SISU)

To do this project, I joined the GAHEC group, whose aim is among others the study of formation and evolution of first stars and galaxies.

### 1.2 History of studies of galaxies

Our astronomical heritage is derived from Asian and European cultures, so that the two Magellanic clouds were not record until the 16th century, delaying galaxy catalogue compared to star catalogue. The Andromeda Galaxy had been listed in the catalogues with other nebulae and star clusters. Then Charles Messier published his catalogue in 1781, recording 140 of these non-stellar astronomical objects, followed by William Herschel who discovered 2500 nebulae and star clusters between 1786 and 1802.

However, it is only since the early 20th century that galaxies are known as fully in- dependent objects, equivalent to our Milky Way. In 1924, Hubble succeed in resolving Andromeda “nebula” into stars and thus proved that is was an other galaxy and not a nebula made of gas. The main morphological types are classified within the famous Hubble sequence (Figure 1.3), where galaxies are separated between elliptical, spirals, lenticulars and irregulars.

In this project, the study will be constrained to spiral galaxies surrounded by dark matter (DM) halos accreting gas, which will be the basis of the bathtub model that will be described later, but first, to finish this introduction I will present the main parts of the project.

### 1.3 Presentation of the subject

The first goal of my master thesis was to study some fundamental properties of the metallicity in star forming galaxies, and then the relation existing between the metal- licity, noted Z and the stellar mass, noted M∗.

### 1.3.1 The metallicity in star forming galaxies

The metallicity of galaxies is a research subject that evolves a lot currently. Indeed, it is with the stellar mass the most fundamental property of galaxies. The stellar mass reflects the amount of baryons locked up in stars, and the metallicity the amount of metals (in astrophysics, every elements heavier than helium) that are present in the Interstellar Medium (ISM). They also allow a better understanding about other pertinent parameters such as the amount of metals produced by stars and returned into the ISM, or the star formation and metallicity histories, determining for example the amount of gas accreted. The primary goal is however the understanding of the role played by the outflows and inflows in the evolution of a galaxy.

In literature many models characterise the metallicity, a large part of them being fitting functions or not self-consistent. However, the model that I have used is the first self-consistent model, giving the results presented in chapter 3.

### 1.3.2 The study of the Z − M

_{∗}

### relation : motivations

The existence of this relation has been highlighted since the beginning of the last decade by authors using observational data (Garnett (2002), Tremonti et al. (2004), Cowie & Barger (2008)), its physical origin is still on discussion. Since the observa- tions can only be made at a given redshift, it would be indeed an important result if we could predict the tracks followed by galaxies in the Z − M∗ plane and observe the role played by the inflows and outflows on this relation.

Some authors suggested that the Z − M∗ relation, sometimes referred to as the mass/metallicity relation (MZR), came rather from a more fundamental 3D relation, known as the Fundamental Metallicity Relation (FMR), including the Star Formation Rate (SFR) as a third parameter. Indeed, we expect the shift between the mean MZR and some perturbed solution to be dependent on the SFR. The fundamental character of this relation comes from the fact that it might be redshift invariant.

First this project proposes to look at the mean Z − M∗ relation and compare the results to the others present in literature. Finally I will investigate the FMR by adding some physically motivated scatter, allowing me to investigate the dependence of the perturbed solution on other parameters, leading to a better insight on whether or not the SFR is the best candidate for the third parameter of the FMR.

## Chapter 2

## The 3 diffenrential equations model

### 2.1 The bathtub model

The bathtub (or reservoir) is a relatively simplified model describing the evolution of gas and stellar masses in a galaxy through cosmological time. It is described by a set of differential equations : the continuity equation of gas, balancing the inflows and outflows and the star-formation law. At the beginning of galaxy formation, a DM halo is formed and accretes gas into the reservoir. It is its growth rate that regulates the accretion of gas. The first results that I will present were published in Bouché et al. (2010). I will summarise them in order to provide a better insight on the key ingredients involved in the study of metallicity.

### 2.1.1 Model of halo formation

Cosmological N -body simulations have allowed us to understand better and approx-
imate the average mass growth rate of DM halos of virial mass M_{h} at redshift z,
obtaining the fitting function for the evolution of the mass of halo :

M˙_{h} ' 510M_{h,12}^{s} (1 + z)^{t}_{3.2}Myr^{−1}, (2.1)
where Mh,12= Mh/10^{12}M, z is the redshift, (1 + z)3.2 = (1 + z)/3.2, t ' 2.2 and
s ' 1.

The accretion of gas into the reservoir is regulated by the halo growth rate such that

M˙_{A}= _{in}f_{b}M˙_{h}, (2.2)

where f_{b} = 1/7 is the baryonic fraction and _{in} is the accretion efficiency.

It has been suggested that the accretion efficiency was a function of M_{h}. Indeed
for more massive halos (M_{h} > M_{max}) gas accretion is suppressed because the gas
cannot cool as the cooling time is too long compared to the dynamical time of the
halo. Moreover, for low halo masses M_{h} < M_{min}, evidences have shown a drop in the
efficiency of galaxy formation. These two phenomena induce a floor and a ceiling for
the accretion.

Furthermore, _{in} is expected to decline in time at a given mass, since the fraction
of stars and hot gas in the Inter-Galactic Medium (IGM) grows, so that _{in} = _{in}(z) =

Figure 2.1: Left: Some halos growth histories. The range of Mh where accretion occurs is showed
(between Mmin and Mmax). Right : The maximum baryonic accretion rate. Halos may never reach
the accretion threshold or the ceiling mass. The integration starts at z = 10 with halos with initial
masses Mh(z = 10) ∈ [3.10^{9}; 6.10^{10}]M, with non constant mass step. Source : Bouché et al. (2010).

0.7f (z)^{1}.

To summarise, these three effects on accretion can be modelled by the following relations :

_{in}(M_{h}, z) =

0 if M_{h} > M_{max} = 1.5 × 10^{12}M

0 if M_{h} < M_{min} = 10^{11}M

0.7f (z) if M_{min} ≤ M_{h} ≤ M_{max}

Taking into account these parameters, after numerical integration of the equa- tion (2.1) the halos growth history can be plotted as a function of redshift (Figure 2.1).

Figure 2.1 is very important, indeed it will be used all along the study for the choices in the initial masses of the halos.

Let’s point out some important properties of the halos :

• Low masses halos (M_{h} < 3.10^{9}M) never reach the accretion threshold.

• More massive halos reach the threshold first and reach the ceiling mass Mmax

over a shorter timescale.

• The halo growth rate (and thus the accretion rate) first increases with time, reaches a maximum, and then slowly decreases.

Knowing the main properties of the halos, the bathtub model can now start being implemented.

1f(z) is a function linear with time such that f (2.2) = 1 and f (0) = 0.5.

Figure 2.2: Scheme representing the bathtub model, characterising inflows, outflows and changes in gas mass due to stellar formation.

### 2.1.2 Implementation of the bathtub model

The bathtub model, which has been used all along this project, is built with two
crucial parameters that are the accretion efficiency in and the star formation (SF) ef-
ficiency _{SF R}. It is a useful tool that has been used in previous studies to get a better
knowledge of the influence of some physical parameters involved in the evolution of
star-forming galaxies.

Let’s consider the galactic halo as the reservoir that is filled from a source and emptied into a drain. In this analogy the source is the newly accreted gas and the drain represents the consumption of gas through star formation and outflows, as shown on Figure 2.2.

The model starts with the following simple equation expressing the conservation of gas mass :

M˙_{gas} = ˙M_{A}− (1 − R) ˙M_{∗}− ˙M_{gas,out}, (2.3)
where (1 − R) ˙M∗ is the net SFR, R representing the fraction of recycled gas^{2},
and ˙Mgas,out is the rate of mass outflow. Those outflows are generated by supernovae
that drive gas and dust out of the galaxy in what we call "winds". The observa-
tions show that the outflow rate ˙M_{gas,out} is roughly proportional to the SFR so that
M˙gas,out= η ˙M∗. The mass loading factor η characterises the winds and may vary with
M_{h} or M∗, depending on the model used for the winds. The choice that I made for

2During this study we will take R = 0.52.

the mass loading factor is justified in section 3.1.1.

The amount of gas consumed in star formation is described by an empirical relation called the star formation law (Kennicutt (1998) & Schmidt (1959)):

SF R = _{SF R}M_{gas}/t_{dyn}, (2.4)

where tdyn = R_{1/2}/V_{c} is the galaxy dynamical time and can be written tdyn =
2 × 10^{7}yr(1 + z)^{−1.5}_{3.2} , _{SF R} is the SFR efficiency parameter^{3}.

Finally, the reservoir model is described by the differential equations :

M˙_{gas} = ˙M_{A}− (1 − R + η) ˙M∗

M˙∗ = _{SF R}(1 − R)M_{gas}/t_{dyn} (2.5)
where the accretion rate is calculated from equation (2.2), and with initial condi-
tions M_{gas}(z = 10) = M∗(z = 10) = 0.

This coupled system shows how M_{gas} and M∗ interact with each other, since ˙M_{gas}
depends on ˙M∗, which depends on M_{gas}. We can already assume the behaviour of the
solution : when the mass of gas increases (due to accretion), the star formation starts
and increases as well, leading to a balance between inflows and outflows( ˙M_{gas} = 0).

Then a quasi-steady state is reached and the SFR is mostly set by the accretion rate.

If for any reason the mass of gas came to increase again (reservoir overfilled), then the rate of star formation would increase as well thus the galaxy would go back to the initial quasi-steady state, due to an increase of the outflows.

This system being set up, we can solve it and analyse some properties, depending on whether or not an accretion floor is set. During this thesis the resolution of the system has been performed between the redshifts z = 10 and z = 0.

The "no min" model

First, we will consider the model where there is no minimum of halo mass to start the accretion of gas. It means that the accretion will start at z = 10, regardless on the initial mass of the halo. It is a good first result to understand some properties of the bathtub model, as shown in Figure 2.3 (dashed curves).

We can make some remarks :

• At z = 10 the accretion starts so the masses of gas and stars start increasing, according to equation. (2.5).

• At z = 4 the mass of gas decreases because the mass of the halo M_{h} reaches the
mass ceiling, so the accretion stops.

• As expected from the star formation law, ˙M∗ > 0 so the stellar mass is always increasing.

3Kennicutt(1998) has estimated this value to be close to 0.02.

Figure 2.3: Evolution of gas and stellar masses in both models for the same initial mass of halo.

When the accretion floor is set (solid curves) the accretion starts later (here at z = 8.2) but finally
the masses reach the same values. The initial halo mass taken is Mh(z = 10) = 3.2 × 10^{10}M_{}.

It is a first approach to see how the model works, but the more interesting to study for us is the accretion floor model.

The "accretion floor" model

It has been shown that observational data could be well fitted by using an accretion floor on our model. Figure 2.3 shows the evolution of gas and stellar mass (solid curves) for the same halo as previously. By comparing the results with the model without an accretion floor, we can highlight a few points :

• The integration starts at z = 10 but the accretion (and so M_{gas} and M∗) begins
at z = 8.8, due to the floor.

• Even if the accretion starts later, the values reached by M_{gas} and M∗ when z
tends towards 0 are the same in both model.

The SFR

The study of the SFR is interesting as well. In Figure 2.4 on the left we can see different steps in its evolution :

• The Rising state : since an increasing amount of gas penetrates into the reservoir, more and more stars are created.

• The ’Steady’ state : a quasi-equilibrium is reached and the rate of stars formed is proportional to the rate of gases accreted, as explained in more details below.

Figure 2.4: Left : different states of the SFR : rising state, steady state and the state when no more gas is accreted. Right : Star formation history for 20 halos. In the chosen range two of them don’t reach the ceiling mass. The more massive halos reach the mass floor earlier and form stars on a shorter time scale.

• The decrease : due to the reaching of the mass ceiling, the accretion stops so the amount of af gas in the reservoir decreases. The stars keep being created but at a lower rate.

Since it is also interesting to observe how the SFR may vary according to the halo, Figure 2.4 (right) shows an overview of the evolution of the SFR in several halos. We thus can observe that more massive halos reach the mass floor earlier and form stars on a shorter time scale. This confirms analyses that showed that in the local universe, stars seem to have formed earlier and over a sorter time-span.

The steady state appears when the SFR scales with the accretion rate, which is a
fundamental property of the model. This effect is better shown on Figure 2.5, indeed
during the steady state ( ˙M_{gas} = 0) equation (2.5) gives

M˙_{A}

SF R = 1 − R + η

1 − R , (2.6)

which only depends on the model used for the winds (η). This equation tells us that
when the SFR is high enough to meet the condition ˙M_{gas} = 0, it scales with the gas
accretion. It is what we can observe on Figure 2.5, when the mass of gas become
constant the SFR is fed by the accretion.

Finally in Figure 2.6 the solid curves show the specific SFR (sSFR^{4}) main sequence
i.e. the M∗−sSFR relation at different redshift. These are the isochrones showing
how the sSFR evolves according to the stellar masses. The dashed curves are the
evolutionary tracks for the different halos i.e. the tracks followed by individual halos
between z = 10 and z = 0 in the M∗−sSFR plane.

4sSFR=SFR/M_{∗}.

Figure 2.5: Scaling of SFR with accretion when the mass of gas is constant. To show the scaling I have chosen an halo that doesn’t reach the ceiling mass.

Figure 2.6: Evolution of the main sequence i.e. the M_{∗}−sSFR relation at different redshift. The
dashed curves represent the tracks followed by individual halos between z = 10 and z = 0 in the
M_{∗}−sSFR plane.

Inflows

ZA Accretion metallicity

RZ Rate of restored mass of metal

E˙_{Z} = y_{Z}(1 − R) ˙M∗ Rate of newly produced metals
Outflows

Z ˙M∗ Mass of metals turned into stars
Z ˙W Mass of metals ejected by winds^{7}

Table 2.1: Parameters involved in the inflows and outflows of the model of metallicity.

### 2.2 Study of Metallicity

Once the bathtub model implemented, I extended it to include the evolution of metal-
licity, noted Z. In my notations, Z is the amount of metals present in the ISM. I will
express it in term of solar metallicity Z5. As what was done to build equation (2.5),
we must write down the metallicity continuity equation, i.e. the parameters taking
part of the evolution of Z, assuming outflows made of 100% entrained materials^{6}.
Table 2.1 summarises the inflows and outflows of metals used to describe the rate of
change ˙Z.

Based on these ingredients, we can write in the most general case
dM_{Z,gas}

dt = Z_{A}M˙_{A}− Z ˙M∗+ R_{Z}+ ˙E_{Z}− Z ˙W , (2.7)
where Z_{A} is the accretion metallicity, so Z_{A}M˙_{A}is the accretion rate of mass of metals,
Z ˙M∗ is the rate of metal consumption due to star formation, R_{Z} = RZ ˙M∗ is the rate
of restored mass of metals, ˙E_{Z}(t) = y_{Z}(1 − R) ˙M_{∗} is the rate of newly produced mass
of metals and Z ˙W is the mass of metals ejected by winds^{8}.

It leads to

ZM˙ _{gas}+ ˙M_{gas}Z = Z_{A}M˙_{A}+ [y_{Z} − Z(1 + η

1 − R)](1 − R) ˙M∗ (2.8) Using equation (2.5), equation (2.8) reduces to

Z =˙ 1

τ_{gas}[y + (Z_{A}− Z)M˙_{A}

M˙_{∗}] (2.9)

where τ_{gas} = M_{gas}/ ˙M∗is the gas depletion time scale and y is the metallicity yield^{9}.
It represents in the model the amount of newly produced metals by supernovae. Here

5Z_{}' 0.02.

6It means that the winds have the same metallicity as the interstellar medium.

8W = η ˙˙ M_{∗}

9y = yZ(1 − R) where yZ is the yield reduced per stellar generation.

the assumption that the winds have the ISM metallicity has been made. This is the
most generic result under the instantaneous recycling approximation (IRA)^{10}. Z is
expressed in unit of solar metallicity, with Z_{A}= 0.01Z_{} and y = 1/42/0.02Z_{}.

Finally the complete 3 differential equations are :

M˙_{gas} = ˙M_{A}− (1 − R + η) ˙M∗

M˙_{∗} = _{SF R}(1 − R)M_{gas}/t_{dyn}
Z =˙ _{τ}^{1}

gas[y + (Z_{A}− Z)^{M}^{˙}_{˙}^{A}

M∗]

(2.10)

with the initial conditions Mgas(z = 10) = M∗(z = 10) = Z(z = 10) = 0. In order
to simplify the study of the system, I have restricted myself to the halos that don’t
reach the maximum mass of accretion. Figure 2.1 gives us a range for the initial M_{h}
between 3.10^{9}M and 10^{10}M where this condition is met.

To summarise, the solving process is mostly in 3 stages, as shown in Figure 2.7 : 1. Resolution of the equation determining the mass of halos through time, giving

M_{h}(t) and ˙M_{h}(t).

2. Resolution of the bathtub model giving M_{gas}(t) and M∗(t).

3. Resolution of the metallicity equation to obtain Z(t) for each halo.

10R = R ˙˙ M_{∗}and ˙RZ= RZ ˙M_{∗}. The assumption is that the gas and stars have the same metallicity.

Figure 2.7: Process followed to solve the system : 1) Resolution of the equation determining
the mass of halos through time; 2) Resolution of the bathtub model giving Mgas(t) and M_{∗}(t); 3)
Resolution of Z(t).

## Chapter 3

## Results of metallicity in star forming galaxies

### 3.1 First results on metallicity

Once the equations being set up, we can solve the system and draw some important results such as the evolution of metallicity through time.

### 3.1.1 The "no min" model

Figure 3.1 (left) shows the evolution of metallicity through time for 20 different halos.

Since there is no minimum of accretion the increase of Z starts at z = 10 for every
halos. Even if the production of metals starts at the same time, the fact that the
metallicity is independent of the mass of halos might be disturbing. In fact, as ex-
plained above, Z is not a mass of metals (which should be higher for heavier galaxies)
but the ratio between metals and every elements, and thus follows the same path for
every halos. Indeed, τ_{gas} does not depend on M_{h} and we have seen that the ratio ^{M}^{˙}_{˙}^{A}

M∗

tends to a value that varies slowly with time, and is almost independent of the halo.

Indeed, as soon as the accretion of gas starts (at z = 10 for every halos here) this ratio is almost the same for each halo.

To summarise, when the accretion starts, the metallicity in the galaxy is first fed
by an accreted gas with metallicity Z_{A} = 0.01Z. Then the increasing creation of
stars, producing more metals contribute to the large increase of Z. Finally when the
SFR scales with the accretion the metallicity reaches a quasi-equilibrium that we can
observe for low redshifts, where the yield is offset by the winds and metals turned into
stars.

### 3.1.2 The accretion floor model

I had assumed that setting the accretion floor might have a repercussion on the differ- ences in evolution of Z between halos. Indeed, Figure 3.1 (right) shows that since the inflows don’t start feeding the metallicity at the same times, the evolution is different.

Figure 3.1: Left : Evolution of Z for 20 halos without accretion floor. Right : Evolution of Z for 20 halos with an accretion floor, implying a shift to the right for decreasing halo masses. In both cases a quasi-equilibrium is rapidly reached and then Z approaches the solar metallicity.

Let’s point out 2 things :

• In both cases a quasi-equilibrium is rapidly reached, which will be quantified in the next paragraph.

• For low redshifts, Z approaches the solar metallicity, which is logical since more and more stars are created.

### 3.1.3 The quasi-steady equilibrium

The solution at equilibrium is obtained by solving equation (2.9) for ˙Z = 0. It gives the equilibrium metallicity

Zeq = y
M˙_{A}
M˙∗

+ ZA (3.1)

which, injecting equation (2.6) for the quasi-steady state (when the SFR scales with the accretion), can be expressed as

Z_{eq} = y

1 − R + η + Z_{A} (3.2)

By roughly approximating η ' 0.8 for low redshifts we find Z_{eq}' 0.94Z which is
a good approximation to cross-check the numerical solution.

### 3.1.4 The impact of the accretion floor

One might question the reason of such an impact of the accretion floor on metallic- ity. Indeed when no minimum is set, every halos experience the same evolution of

Figure 3.2: Comparison between τ_{gas} and t_{acc} without minimum of accretion (left) and with mini-
mum of accretion (right). Left : condition τ_{gas}< t_{acc}= ^{M}_{˙}^{h}

M_{h} met around z = 3, the time to meet this
condition is similar for every halos. Right : τ_{gas} and t_{acc} for 2 halos in accretion floor model : one
which meets the condition later (red) and one whose condition is already met (black).

metallicity (Figure 3.1(left)) through time. In fact, when the accretion starts, the
yield dominates in the rate of change of Z (Equation (2.9)), since the second term is
small. Then the impact of the second term (Z_{A}− Z)^{M}^{˙}_{˙}^{A}

M∗ becomes more important but it appears that at a given redshift, it doesn’t vary much from one halo to another.

The condition for the quasi-steady state solution is
τ_{gas}< t_{acc} = M_{h}

M˙_{h},

which is already met when the yield no longer dominates. For our halos, Figure 3.2 (left) shows that it appears around z = 3, and that the time to meet this condition is similar for every halos.

In accretion floor model the time to meet the condition is different for every ha-
los (Figure 3.2 (right)), so that the quasi-steady state is not reached under the same
timescale : when the accretion starts later τ_{gas} is already lower than t_{acc}, so that the
SFR scales directly with the accretion, but when the accretion starts earlier, the ac-
cretion of gas is performed but since the quasi-steady state takes time to be reached,
it will be higher at the end. This explains the differences in metallicity growth history
between both models.

Once these first analyses done I could focus on further studies regarding the evo- lution of metallicity. One of the main goals of this thesis was the study of the mass/metallicity relation (MZR). In the following section I will present the results obtained for the MZR and then compare my model with some others found in litera- ture.

### 3.2 The mass/metallicity relation

As explained in the introduction, the M∗− Z relation is an important result because it can be easily compared with those existing in literature, which is what I will present in this part. I will compare my results with those of Zahid et al. (2014c) shown in

Figure 3.3: (A) shows the data for the M_{∗}− Z relation at different redshifts (legend on (B)). The
solid curves are the fitting model implemented by Zahid et al. (2014c). Data at different redshifts
seem to tend toward the same metallicity. Source : Zahid et al. (2014c).

Figure 3.3, which is a plot from Zahid et al. (2014c), whose model fits the data also.

The following results are made using the accretion floor model, I will explained in the last paragraph the interest of a minimum accretion floor for the MZR.

### 3.2.1 Choice of the model for the outflows

So far the model used to express the mass loading factor η hasn’t been described, because all the quantitative results obtained were not affected by the choice made. I have tested three models that depend on M∗ because they are easier to implement :

• Davé et al. (2011) : η = 2.25 × (^{M}_{10}^{∗}^{(M}9.9^{}^{)})^{−1}

• Hopkins et al. (2012) : η = 3 × (^{M}_{10}^{∗}^{(M}9.9^{}^{)})^{−1.1}

• Zahid et al. (2014a) : η = 0.57 × (^{M}_{10}^{∗}^{(M}9.9^{}^{)})^{−0.41}

Figure 3.4 shows the M∗− Z relation at z = 0 for those three models.

Comparing with Figure 3.3, the model suggested for the winds by Zahid et al.

(2014a) appeared to be a better choice for 3 main reasons, the third one being a consequences of the first two ones:

• A larger range of masses is covered.

• The metallicity reaches higher values.

• The quasi-equilibrium seems to be almost reached.

It is also interesting to observe how this relation looks like without outflows(η = 0).

The comparison is shown on Figure 3.5. We observe that without wind, the metallicity reaches higher values for higher stellar masses. Since the amount of gas in the reser- voir is higher, the SFR is higher as well and thus more stars are created, increasing the amount of newly produced metals. The value reached in the quasi-equilibrium

Figure 3.4: The M_{∗}− Z relation for our models of winds. The last plot (model of winds suggested
by Zahid et al. (2014a)) appears to be the closest to the data shown in Zahid et al. (2014c).

Figure 3.5: The M_{∗}− Z relation compared with and without outflows. Without outflows Z goes
over the solar metallicity, which is higher than the expected value.

is beyond the solar metallicity, which is less close to reality than when we used the model suggested by Zahid et al. (2014a).

The mass loading factor was the last parameter needed to implement the system.

Table 3.1 summarises all the parameters that have been used so far. Those precision for the mass loading factor being made, we can now start comparing our results.

### 3.2.2 The MZR without accretion floor

So far my results were made using the bathtub model with an accretion floor, it is however interesting to see how the mass/metallicity relation would be without min- imum of accretion. Figure 3.1 showed that for different halos the metallicity would follow practically the same evolution if the accretion started at the same time, so that we cannot expect a really good result in the MZR at a given redshift. Indeed, Fig- ure 3.6 shows that it hasn’t a relevant behaviour since Z remains virtually constant for a given redshift.

Halos

z_{initial} 10

zf inal 0

M_{h}(z_{i}) ∈ [3.10^{9}, 10^{11}]M

f_{b} 1/7

_{in}(z) 0.7 if z > 2

0.7f (z) if z ≤ 2

f (z) linear function with time with f (2.2) = 1 and f (0) = 0.5 Bathtub model

Mmin 10^{11}M

M_{max} 1.5 × 10^{12}M

R 0.52

η 0.57 × (^{M}_{10}^{∗}^{(M}9.9^{}^{)})^{−0.41}

_{SF R} 0.02

t_{dyn} 2 × 10^{7}yr(1 + z)^{−1.5}_{3.2}

Metallicity

τ_{gas} M_{gas}/ ˙M∗

y 1/42

Z_{A} 0.01Z

Table 3.1: The ingredients used in the reservoir model.

This shows that our general system of equations (2.10) is more accurate under the assumption of an accretion floor.

Figure 3.6: The MZR at the same redshifts as Zahid et al. (2014c), with no accretion floor. The absence of accretion floor impacts strongly the MZR, too far from the data.

### 3.3 Discusion and comparison with existing MZR mod- els

In the following paragraphs I will briefly describe the models made in different papers and then compare them with my results.

### 3.3.1 Lilly et al. (2013)

In lilly et al. (2013) the chemical evolution of metals in galaxies is described, giving a formula for the metallicity as a function of the specific SFR. On Figure 3.7 we can see a comparison between the model from Lilly et al.(2013) (on the left, plot made with observational data) and similar results using my model. It qualitatively reproduces the observed evolution.

The comparison is not easy since a thinner range of masses is covered in my results.

Indeed, I have restrained the range of galaxies for those that didn’t reach the ceiling mass. Still, the offset between the 2 reshifts and the behaviour of the relation at z = 0 appear to be similar.

Figure 3.7: Left : model implemented by Lilly et al. (2013). Source : Lilly et al. (2013). Right : Our model at the same redshifts. The curves and the ranges of masses covered look similar.

Figure 3.8: The M∗− Z relation at z = 0 for Peng & Maiolino’s model. A quasi-equilibrium is reached and is less high as that in our results.

### 3.3.2 Peng & Maiolino (2014)

In Peng & Maiolino (2014) the authors suggested an analytic solution for equa- tions (2.10), for the special case when the accretion, the star-formation efficiency and the yield are constant.

With the input parameters defined in table 3.2 the analytic formula given for some galaxies properties can be used. Figure 3.8 shows the MZR at z = 0 using this model.

We observe some differences with my plots :

• A quasi-equilibrium is reached and is less high as that in our results.

• Contrary to our solution, high stellar masses (> 10^{10.2}M) are not covered.

• Contrary to our solution, low stellar masses (< 10^{9.5}M) are covered.

Input parameters

Accretion rate M˙_{A}

Star formation efficiency = _{SF R}/t_{d}yn

Mass-loading factor^{1} η

Equilibrium timescale τ_{eq} = _{(1−R+η)}^{1}
Predicted galaxy properties

M_{gas}(t) M˙_{A}τ_{eq}(1 − e^{−}^{t−t0}^{τeq} )
SF R(t) M˙_{A}τ_{eq}(1 − e^{−}^{t−t0}^{τeq} )

M∗(t) (1 − R) ˙M_{A}τ_{eq}(t − t_{0}− τ_{eq}(1 − e^{−}^{t−t0}^{τeq} ))
Zgas(t) ZA+ y^{SF R}_{M}_{˙}

A + [Zgas(t0) − ZA− y^{SF R}_{˙}

MA ]e^{−}

MA˙ Mgas(t−t0)

Table 3.2: Analytic solutions derived in Peng & Maiolino (2014) for the special case when the accretion, the star-formation efficiency and the yield are constant.

### 3.3.3 Zahid et al. (2014c)

In Zahid et al. (2014c), a fitting model is suggested such that
12 + log(O/H) = Z_{0} + log[1 − exp(−[M∗

M0

]^{γ})] (3.3)

The best fits are made by adjusting Z0, M0 and γ. The result is obtained on Figure 3.3.

Figure 3.9 shows our model (right) plotted like Zahid’s (left). In both figures we see that for higher redshifts the quasi-equilibrium is not reached. We also remark that Zahid’s curves seem to converge to the same points for both high and low M∗, which is not the case for our curves, which are almost parallel. Except for z = 1.55 the data seem to be in agreement with Zahid’s model on that point.

By comparing the numerical solution of a physically motivated model with the existing ones we could observe that our MZR had a good behaviour according to what we could expect, and without making too many assumptions. This remark is valid especially at z = 0, where more studies will be performed in the next chapter.

### 3.3.4 Yabe et al. (2014)

In Yabe et al. (2014), it is suggested to study a fitting model according to the Gas Mass Fraction parameter, defined as

γ = M_{gas}
M_{gas}+ M∗

Figure 3.9: Left : Model implemented by Zahid et al. (2014c). Source : Zahid et al. (2014c). Right : Our model at the same redshifts. The comparison shows that the curves look similar but ours don’t tend toward the same value.

Figure 3.10 from Yabe et al. (2014) shows 2 plots that are a good basis for comparison.

As expected, the data tell us that the gas mas fraction is inversely proportional to the stellar mass, which is logical, and that the metallicity in the galaxy increases when the gas mass fraction decreases, since a part of the production of metals is due to the consumption of gas into star formation. Those stars will then create metals. Even if the fitting model is questionable, we can use the data to compare them to our model, with and without outflows.

In Figure 3.11 (η = 0) we see that the plot for γ as a function of the stellar
masses (left) appears to be similar with Yabe’s, despite the fact that in our model a
smaller range of stellar masses is covered. Indeed, since we constrained ourselves to
the halos that don’t reach the ceiling mass, high stellar masses (M∗ > 10^{11}M) at
z = 0 can hardly be reached. Moreover, the accretion floor prevents us from covering
lower stellar masses (M∗ < 4.10^{9}M).

Those remarks are valid for the metallicity plots (right), indeed it seems like there would be the same graduation in the curves if I could extrapolate same. However those curves lead to too high values of metallicity, which is due to the absence of winds to reduce Z.

Figure 3.12 includes the winds in the computation. We see that the results on the left plot are not changed at all : with the winds the gas is ejected, implying a diminution of Mgas, which affects directly the the SFR and thus the stellar mass, so the gas fraction remains unchanged at a given stellar mass. However, a change appears on the right figure, where we observe more reasonable values for the metallicity (in agreement with the data from Yabe et al. (2014)), which is lower now for a given γ, which is not surprising : we have already seen that without winds Z reaches higher values.

Figure 3.10: Fitting model suggested by Yabe et al. (2014). The gas mas fraction is inversely proportional to the stellar mass and the metallicity in the galaxy increases when the gas mass fraction decreases. Source : Yabe et al. (2014).

Figure 3.11: Model without outflows plotted at the same redshifts as Yabe et al. (2014). The curves follow the same trend.

Figure 3.12: Model with outflows plotted at the same redshifts as Yabe et al. (2014). Here Z reaches more reasonable values at a given γ.

Those last results concluded this chapter about the computation of the Z − M∗

relation obtained using our model and its comparison with others found in literature.

The next interesting part is the study of the so called fundamental Z − M∗ − SF R plane, in the model computed beyond equilibrium.

## Chapter 4

## Beyond equilibrium : Addition of stochastic events

### 4.1 The Z − M

_{∗}

### − SF R relation

For the motivations explained in the introduction, studying the Z −M∗−SF R relation is one of the main goals of the project. Indeed, it is currently proposed that this relation may be redshift invariant, which is the reason why some have called it "fundamental"

relation (Mannucci et al. (2010)). In the following paragraphs I will present my results and then discuss them.

### 4.1.1 Results

To highlight the properties of the fundamental plane we need to solve the system in a state out of equilibrium in order to obtain scattered results and see how the SFR varies regarding the scatter. For that purpose, I have randomly added some stochastic events on the time line, which could represent the accretion of small satellites, where the accretion varies such that

M˙_{A,scatter} = ˙M_{A,eq}± 0.3dex

After solving the equation with the scatter term, the result obtained for the MZR
is shown on Figure 4.1. We observe the scatter and we can now study the dispersion
of the FMR to see if there are any correlation with other physical parameters such as
the SFR. Figure 4.2 (left) shows the same relation as previously but now the points
are coloured as a function of the SFR. As expected, the SFR is higher for higher values
of M_{∗}, which is logical. However we can hardly identify any correlation between the
SFR and the scatter. To get a better insight on this, I have plotted the residuals as a
function af the SFR on Figure 4.2 (right). The residuals are defined such that

res(t) = Z_{scatter}(t) − Zequilibrium(t)
Zequilibrium(t)

I thought that the correlation would be more obvious but it appeared that, accord- ing to the model, there is no evidence for using the SFR as a third parameter of the fundamental plane.

Figure 4.1: The Z − M_{∗} relation at z = 0 with the scattered points. The system has been solved
for 2000 halos. The red curve is the equilibrium solution, i.e. the FMR without addition of random
events. The blue points indicate the FMR for each halo.

Figure 4.2: The Z − M∗ relation showing 2 ways of observing the variations in SFR. No evidence of correlation between the Z − M∗ relation and the SFR appears here.

### 4.1.2 SFR or relative specific SFR for the fundamental plane

### ?

Salim et al. (2014) suggested that the MZR residuals might be better characterised by either the sSF R or the relative sSF R defined as

∆ log sSF R = log sSF R − hsSF Ri_{M}_{∗},

where hsSF Ri_{M}_{∗} is the median of logsSFR of galaxies having a mass M∗. This
parameter represents the median shift in the sSFR sequence (Figure 2.6).

Their work is based on observational data^{1} provided by SDSS spectroscopic survey.

Some previous cuts were used to define the sample, the procedure was adopted in
Mannucci et al. (2010)(M10) in order to select the star-forming galaxies : 928,000
galaxies dropping to 212,000 after the application of M10 redshift (i.e. 0.07 < z < 0.30)
and H_{α} signal-to-noise ratio cuts. Then some minor cuts are made, leading to a final
M10-like sample of 141,000 galaxies.

In Salim et al.(2014) the metallicity is the average of 12+log(O/H) estimates for two strong-line methods :

• the ratio of four oxygen lines ([OII]3727 doublet and [OIII]4958, 5007) to Hβ

• the ratio of [NII]6584 to Hα

Our model describes the metallicity in general but the data provided by the au- thors remain a good basis of comparison.

For the purpose of the analysis, the authors sorted the data according to extreme values (highest and lowest 2.5%) of absolute SFR, absolute sSFR and relative sSSFR.

Their results are shown on Figure 4.3.

They made the following remarks :

• High SFRs galaxies : At the highest masses, they show no offset with the overall sample, while at lowest masses they keep an offset of 0.2 dex.

• Low SFRs galaxies : They are found in low masses regions with an offset of 0.05 dex above the overall median. The fact that the SFR scales with stellar masses motivates the study of the sSFR.

• High sSFRs galaxies : They are found among the lower mass galaxies. Median offsets remain large but are smaller as those in high SFR galaxies for low masses.

• Low sSFRs galaxies : They are found among the higer mass galaxies. Median offsets are similar to those for low SFRs galaxies.

• High relative sSFRs galaxies : An offset is present even for high masses.

1The authors suggested a physically-motivated analysis of the FMR considering the relative specific SFR as a third parameter. They performed analyses on the impact of this parameter on metallicity in order to, in future work, get a better insight on the fundamental character of the relation.

Figure 4.3: The Z − M_{∗}relation for different sorting. The green regions give the 90% of the overall
sample, the white line representing the median. The coloured line show the median of the sorted
samples. Source : Salim et al. (2014).

• Low relative sSFRs galaxies : An offset as large as in the previous sorting is present, but now covering a full range of masses.

The conclusion made by the authors is that the relative specific SFR might be a better way to characterise the MZR residuals.

### 4.2 Comparison with Salim et al. (2014)

The sorting made in figure 4.3 can be reproduced in our model and are a good way to compare it to observational data. Figure 4.4 shows the same sorting but with a selection by 10 percentiles since my sample is smaller. We see that the observations made by Salim et al. (2014) regarding the offsets apply for our results, which reinforces the belief in the stochastic model as a good tool to study the FMR.

I can now highlight this conclusion by plotting Figure 4.2, but with the relative sSFR as a third parameter. The result is shown in Figure 4.5 where the correlation between the dispersion and the relative sSFR is clearly highlighted. Indeed we see that when the scatter increases, the relative specific SFR become larger, which confirms our expectations about the role played by this parameter in the FMR.

Figure 4.4: The Z − M∗ relation at z = 0 for different sorting using our model. The red curve is the numerical solution without perturbation. The scattered results (in blue) are sorted according to low and high values of SFR, sSFR and relative sSFR, giving equivalent results as those obtained by Salim et al. (2014).

Figure 4.5: Left : The Z − M∗ relation with a colouring according to the relative sSFR. Right :
Residuals as a function of the relative sSFR. It is clear that the relative specific SFR has an impact
on the Z − M_{∗} relation.

## Chapter 5 Conclusion

These studies show that the physically-motivated model that I have used is a useful tool allowing the understanding of key parameters involved in the evolution of galax- ies, such as the metallicity. The equations that were set have allowed me to perform further studies such as the MZR and investigate the impact of winds on this relation.

Finally, I have been able to show that the relative specific SFR was the more fun- damental third parameter of the FMR as observations suggest (Salim et al. (2014)), which has also confirmed our model as an efficient tool for the study of metallicity.

From a personal point of view, this master thesis has allowed me to get a better insight into research in laboratories. I have learned to be more self-reliant and inde- pendent, but also to synthesize research results. Finally I am glad to contribute to the understanding of a current research topic such as metallicity in star-forming galaxies.

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