Theoretical models for the formation of stars

Thesis

Reference

Theoretical models for the formation of stars

HAEMMERLE, Lionel

Abstract

Le présent travail est consacré à la modélisation de la structure interne des étoiles au cours de la pré-séquence principale, à l'aide du Code d'Évolution Stellaire de Genève. On étudie d'abord les liens entre la rotation et la déplétion du lithium à la surface des étoiles de type solaire. On présente ensuite les premiers modèles de pré-séquence principale incluant simultanément l'accrétion et la rotation. Nos modèles montrent que la formation des étoiles massives nécessite un mécanisme de freinage, et suggèrent que l'efficacité de l'accrétion par un disque diminue lorsque la masse de l'étoile augmente. On présente enfin une grille de rotateurs rapides sur la séquence principale, et on propose qu'en moyenne les étoiles du Petit Nuage de Magellan ont un moment cinétique plus élevé que celles de la Voie Lactée.

HAEMMERLE, Lionel. Theoretical models for the formation of stars. Thèse de doctorat : Univ. Genève, 2014, no. Sc. 4753

URN : urn:nbn:ch:unige-483656

DOI : 10.13097/archive-ouverte/unige:48365

Available at:

http://archive-ouverte.unige.ch/unige:48365

Disclaimer: layout of this document may differ from the published version.

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UNIVERSIT´E DE GEN`EVE FACULT´E DES SCIENCES

D´epartement d’Astronomie Professeur G. Meynet

Docteur P. Eggenberger

Theoretical Models

For The Formation Of Stars

TH`ESE

pr´esent´ee `a la Facult´e des sciences de l’Universit´e de Gen`eve

pour obtenir le grade de Docteur `es sciences, mention astronomie et astrophysique par

Lionel Haemmerl´e de

Carouge (GE)

Th`ese N^◦ 4753

GEN`EVE

Centre d’Impression de l’Universit´e de Gen`eve 2015

1

R´esum´e en fran¸cais

Les ´etoiles se forment par l’effondrement gravitationnel de nuages denses et froids.

Au cours de l’effondrement, la pression augmente dans le nuage, jusqu’`a s’´equilibrer avec la gravit´e: l’´equilibre hydrostatique est atteint et l’´etoile entre dans sa phase de pr´e-s´equence principale. Le pr´esent travail est consacr´e `a la mod´elisation de la structure interne des ´etoiles au cours de cette phase, `a l’aide du Code d’´Evolution Stellaire de Gen`eve, incluant les effets de l’accr´etion et de la rotation. On consid`ere des ´etoiles dans un large intervalle de masses, depuis des masses telles que celle du Soleil (M) jusqu’`a des masses sup´erieures `a 100M.

Apr`es une introduction historique sur le d´eveloppement des mod`eles hydrosta- tiques et hydrodynamiques de formation stellaire (Chap. 1), on ´etudie d’abord le cas des ´etoiles de faibles masses (∼1M,Chap. 2). On s’int´eresse `a la d´epl´etion du lithium `a la surface de ces ´etoiles au cours de leur pr´e-s´equence principale, et plus particuli`erement `a ses liens avec l’histoire rotationnelle de l’´etoile. On s’int´eresse en particulier aux effets du disc-locking, le couplage magn´etique entre l’´etoile et son disque d’accr´etion. Nos mod`eles montrent que la rotation favorise la d´epl´etion du lithium `a la fin de la pr´e-s´equence principale, `a travers le m´elange rotationnel. Toute- fois, on obtient paradoxalement que, pour une dispersion dans le temps de vie du disque d’accr´etion, les ´etoiles qui tournent le plus lentement `a la fin de la pr´e-s´equence principale sont les plus d´epl´et´ees en lithium, ce qui est en accord avec les observa- tions. Nos mod`eles apportent aussi une explication `a l’´eventuelle corr´elation entre la pr´esence de plan`etes autour d’une ´etoile et son abondance de surface en lithium.

Pour les ´etoiles de masses plus ´elev´ees (&2M), il est n´ecessaire d’inclure l’accr´etion pour rendre compte de l’´evolution pr´e-s´equence principale. Trois versions du code d’´evolution avec accr´etion ont ´et´e utilis´ees dans ce travail (Chap. 3):

- version 1: sans rotation;

- version 2: avec rotation;

- version 3: avec rotation et une am´elioration dans le calcul des d´eriv´ees temporelles.

Les versions 1 et 2 contiennent une incoh´erence qui produit une perte artificielle d’entropie, et de moment cin´etique lorsque l’on introduit la rotation. La perte arti- ficielle d’entropie conduit `a sous-estimer l’amplitude du gonflement de l’´etoile en cas d’accr´etion lors de la transition entre la structure convective et la structure radiative au cours de la pr´e-s´equence principale.

LeChap. 4pr´esente des mod`eles incluant l’accr´etion et la rotation pour des ´etoiles atteignant des masses comprises entre 2 et 22M ayant la composition chimique du

Soleil, en utilisant le taux d’accr´etion de Churchwell-Henning. Ces mod`eles ont

´

et´e calcul´es avec la version 2 du code, avant la correction du calcul des d´eriv´ees temporelles. Dans ce cas, on obtient qu’il est impossible de former des rotateurs rapides parmi les ´etoiles massives, `a cause de la perte artificielle de moment cin´etique.

Les profils de rotation obtenus `a la fin de la pr´e-s´equence principale sont proches des profils d’´equilibre pour l’´evolution sur la s´equence principale. On montre que l’´evolution sur la s´equence principale est ind´ependante du profil initial de rotation en termes de trac´e ´evolutif, de vitesse de rotation et de composition chimique. Ce r´esultat est ind´ependant de la version du code utilis´ee.

Dans le Chap. 5, on consid`ere le cas des ´etoiles massives (20M.M .100M) sans rotation. Les mod`eles pr´esent´es dans ce chapitre ont ´et´e calcul´es avec la version 1 du code, avant la correction du calcul des d´eriv´ees temporelles, comme dans le chapitre pr´ec´edent. On a utilis´e des taux d’accr´etion inspir´es des r´esultats de simulations hydrodynamiques de formation stellaire. En utilisant un mod`ele simplifi´e d’extinction par l’enveloppe de poussi`ere, on obtient une birthline qualitativement similaire `a celles d’autres auteurs, qui sont en accord avec les observations.

LeChap. 6´etend les conclusions du Chap. 4 au cas des ´etoiles de faibles m´etallicit´es.

On utilise ces r´esultats pour calculer une grille de rotateurs rapides dans l’intervalle des masses interm´ediaires, pour trois m´etallicit´es diff´erentes. On obtient paradoxale- ment qu’au d´ebut de la s´equence principale, `a moment cin´etique donn´e, la vitesse de rotation `a la surface des ´etoiles d´epend peu de leur m´etallicit´e. Ce r´esultat indique qu’en moyenne les ´etoiles du Petit Nuage de Magellan ont un moment cin´etique plus ´elev´e que celles de la Voie Lact´ee. Ceci sugg`ere que l’efficacit´e des m´ecanismes d’´evacuation du moment cin´etique n´ecessaires lors de la formation des ´etoiles d´epend de la composition chimique du nuage pr´e-stellaire.

Enfin, on consid`ere au Chap. 7 la formation des ´etoiles massives par accr´etion, avec et sans rotation, en utilisant la version 3du code. On utilise d’abord des taux d’accr´etion constants, et on montre que dans ce cas nos r´esultats sont en accord avec la litt´erature. On utilise ensuite le taux de Churchwell-Henning, et on en d´eduit que l’efficacit´e de l’accr´etion par un disque d´ecroˆıt lorsque la masse de l’´etoile augmente.

Quand on inclut la rotation, on obtient qu’il est impossible de former des ´etoiles massives dans le sc´enario consid´er´e. On en d´eduit qu’un m´ecanisme de freinage est n´ecessaire pour former des ´etoiles massives, comme dans le cas des ´etoiles de faibles masses. On obtient en outre que le transport de moment cin´etique dans l’int´erieur stellaire est n´egligeable au cours de la phase d’accr´etion, et on propose qu’il existe une loi d’accr´etion du moment cin´etique produisant des profils de rotation d’´equilibre d`es l’entr´ee sur la s´equence principale.

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Introduction

The topic of the present work is the formation of stars, in the context of one- dimensional hydrostatic models of stellar evolution. We consider stars in a broad mass spectrum, from low-mass stars like the Sun to stars of masses higher than 100M. The work is organized as follows.

We start with a historical introduction (Chap. 1), reviewing the developments of the hydrostatic and hydrodynamic models of star formation. Sect. 1.1 summarizes the progressive discovery of the pre-main sequence of low- and intermediate-mass stars (M . 8M), introducing the accretion scenario. Sect. 1.2 is focused on the case of massive stars (M & 8M), recalling the various difficulties that arise when we try to extrapolate the formation scenario of low- and intermediate-mass stars to this higher-mass range (Sect. 1.2.1). In Sect. 1.2.2, we present the progresses accom- plished during the last two decades on the important issue of the multi-dimensional hydrodynamic simulations for multiple star formation. In Sect. 1.2.3 we report the main works that have been done until now in the context of the one-dimensional hy- drostatic modeling of pre-main sequence objects destined to become massive stars.

InChap. 2, we start with the range of low-mass stars, around the mass of the Sun, neglecting accretion. The main issue of this chapter is the surface depletion in lithium during the pre-main sequence, and its relations with the rotational history of pre- main sequence stars. We consider rotation through its effects on the stellar structure (Sect. 2.1.2) and on the mixing of chemical species (Sect. 2.1.3). An important feature in our treatment of rotation is the disc-locking, i.e. the magnetic coupling between the star and its accretion disc (Sect. 2.1.4). We present models illustrating the effects of the various rotational parameters (Sect. 2.2, 2.3, 2.5 and 2.9), of the mass (Sect. 2.4), of the chemical composition (Sect. 2.6, 2.7) and of other parameters (Sect. 2.8). In Sect. 2.10, we compare our results with observations and previous models, focusing on two observational correlations discussed in the literature: the correlation between the rotation velocity and the surface abundance in lithium of pre-main sequence stars (Sect. 2.10.1), and the correlation between the detection of planets around a star and its surface abundance in lithium (Sect. 2.10.2).

In order to compute models of pre-main sequence stars in the intermediate- or high-mass range, we have to consider accretion. In Chap. 3, we present the im- provements made in the Geneva Stellar Evolution Code, used in the present work,

in order to include accretion. Three versions of the code are presented: version 1 (Sect. 3.2), without rotation, version 2 (Sect. 3.3), with rotation, and version 3 (Sect. 3.4), with rotation and an improvement in the computation of the time derivatives. This improvement removes an inconsistency that appeared in versions 1 and 2, responsible for an artificial loss of entropy in both versions, and of angular momentum in version 2. In Sect. 3.5, we show two examples of accreting models in order to illustrate the effects of this inconsistency.

Chap. 4 treats the case of the intermediate-mass range for a solar chemical com- position, including the effects of rotation. The results presented in this chapter have been obtained before the improvement in the computation of the time derivatives, with the version 2 of the code. We present different series of models of various masses and rotation velocities, during the accretion phase and during the contrac- tion at constant mass towards the main sequence (Sect. 4.2, 4.3, 4.4 and 4.5). In particular, we show the internal rotation profiles obtained once the star enters its main sequence phase (Sect. 4.2), and we obtain that with this version of the code it is not possible to form fast rotators among massive stars (Sect. 4.5). Then we look at the consequences of this pre-main sequence scenario on the main sequence evolution (Sect. 4.6).

InChap. 5, we enter the range of massive stars, first in the case without rotation.

As in the previous chapter, the results presented in Chap. 5 have been obtained before the improvement in the computation of the time derivatives, with the version 1 of the code. We use accretion rates inspired by hydrodynamical simulations (Sect. 5.1) to compute models of accreting stars (Sect. 5.2). We use a simplified treatment of the circumstellar extinction in order to compare the observational features of the considered scenario with those of other scenarios (Sect. 5.3).

In Chap. 6, we go back to the intermediate-mass range (with the version 2 of the code), considering this time models at different metallicities. First, we extend the results of Chap. 4 to the case of the metallicity of the Small Magellanic Cloud (Sect. 6.1). Then we use this results to compute a stellar grid of fast rotators in the intermediate-mass range at various metallicities (Sect. 6.2). In Sect. 6.3, we study an interesting effect that appeared in the models of the grid, about the relations between the angular momentum, the surface velocity and the metallicity of main sequence stars.

Finally, inChap. 7, we consider the formation of massive stars by accretion with the version 3of the code, including the improvement in the computation of the time derivatives. We first consider the case without rotation (Sect. 7.1), with a constant mass-accretion rate (Sect. 7.1.1) and with a mass-accretion rate increasing with time (Sect. 7.1.2). In this last case, we try to deduce some hints on the efficiency of accretion through discs. Then we consider the case with rotation (Sect. 7.2). We show that it is not possible to form massive stars in the considered scenario, due

5

to the hydrostatic effects of rotation. We conclude that a braking mechanism is necessary for the formation of massive stars (Sect. 7.2.3). In Sect. 7.2.4, we discuss how the improvements in the version 3 of the code modify the main results of the previous chapters.

Chap. 8 summarizes the main conclusions of the present work, and opens some future perspectives in this promising field.

Contents

1. Star formation: a history of the models 8

1.1. The pre-main sequence 8

1.2. The formation of massive stars 17

2. Low-mass stars: the impact of rotation on the surface lithium depletion

during pre-MS 39

2.1. Introduction 39

2.2. Fiducial model 42

2.3. The effects of the initial rotation velocity 48

2.4. Mass dependence 50

2.5. The effects of the disc lifetime 52

2.6. The effects of metallicity 56

2.7. The effects of the abundance in helium 59

2.8. The mixing length 61

2.9. Magnetic braking 63

2.10. Comparison with previous results and with observations 63 3. Accretion in the Geneva Stellar Evolution Code 74

3.1. Motivations 74

3.2. Version 1 74

3.3. Version 2 80

3.4. Version 3 82

3.5. Examples of models 87

4. Intermediate-mass stars at solar metallicity: rotating models computed

with the version 2 of the code 99

4.1. Scenario 99

4.2. Series I: birthline and contractions at fixed Ωini 100 4.3. Series II: birthlines and contractions at various Ω_ini 111

4.4. Series III: contractions at Ω_ini ∝M_fin 113

4.5. Series IV: birthline and contractions at v_ini=v_crit 116

4.6. Main sequence 117

5. Massive star formation by accretion without rotation 139

5.1. Mass-accretion rates 139

5.2. Models 141

5.3. The dust envelope and the birthline 145

5.4. Discussion: comparison with Behrend & Maeder (2001) 154

6. Star formation at low metallicity 157

6.1. Rotation profiles on the ZAMS and MS evolution at the SMC metallicity 157

6.2. Grid of fast rotators in the B-type range 160

6.3. Surface velocities on the ZAMS 161

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7. Massive star formation by accretion with rotation 186

7.1. Case without rotation 186

7.2. Case with rotation 195

8. Conclusions and perspectives 209

References 214

1. Star formation: a history of the models

1.1. The pre-main sequence. Stars form in the gravitational collapse of cold dense clouds. As the collapse proceeds, the internal pressure increases until the hydrostatic equilibrium is reached: the star enters the pre-Main Sequence (pre-MS) phase.

The determination of the pre-MS evolutionary tracks required several decades of effort. With the improvements brought to the physics included in the models, the shape of this tracks has been significantly modified. The first pre-MS models were based on the assumption of monolithic homologous collapse, in which the whole cloud’s mass reaches the hydrostatic equilibrium at the same time, and forms a single star.

1.1.1. The radiative branch. The first models of pre-MS stars have been developed by Lev´ee (1953) and Henyey et al. (1955). Built on the assumptions of hydrostatic equilibrium and spherical symmetry, without accretion nor rotation, this models con- sider pre-MS stars as fully radiative and chemically homogeneous, with gravitational contraction as the only energy source.

The equations are

(1)

dP

dr =−ρg g = GM_r r² dM_r

dr = 4πr²ρ P = k_B

µm_HρT dL_r

dM_r =_{g g} =−3γ−4 γ−1

k_BT µm_H

˙ r r L_r

4πr² =−4acT³ 3κρ

dT

dr κ=κ₀ρT^−3.5

where we used the law of ideal gas for the equation of state¹, the Kramer’s law for the opacity and the assumption of homology for _g.

The timescale for such a contraction is the thermal adjustment timescale, i.e. the Kelvin-Helmholtz time

(2) t_KH = GM²

RL For the Sun, Lev´ee (1953) obtained the value

(3) tpre−MS() = 5×10⁷yr

1The radiation pressure is not included here. The conclusions obtained in these works apply only to low- and intermediate-mass stars. For massive stars, the radiation pressure cannot be neglect anymore (Sect. 1.2).

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Henyey et al. (1955) computed evolutionary tracks for such models and obtained that evolution proceeds at quasi-constant luminosity: starting from the red, pre-MS stars join the ZAMS corresponding to their mass by following a nearly horizontal track on the HR diagram, the radiative branch, (line CD on the left panel of Fig. 1).

Salpeter (1954, 1955a) added to this results some important qualitative remarks about the isochrones, in particular about the effects of the burning of light elements². Indeed, the temperature needed for the burning of light elements is lower than the temperature for hydrogen burning (H-burning), so that one expects them to be destroyed, at least partially, during pre-MS (Salpeter 1954). In particular, deuterium and lithium are destroyed by proton capture³

(4) D + ¹H 7−→ ³He + γ ⁷Li + ¹H 7−→ ⁴He + ⁴He at temperatures

(5) T_D= 1−2×10⁶K T_Li = 2.5×10⁶K

Depending on the abundances of the light elements, the energy produced by their destruction can stop the gravitational contraction, modifying the evolution time on the pre-MS and so the shape of the isochrones. This effect is not included in the models of Lev´ee (1953) and Henyey et al. (1955), which consider only the energy generated by the gravitational contraction.

1.1.2. The Hayashi line. Using the conclusions of Hayashi & Hoshi (1961) about the surface properties of red giants, which have a convective envelope, to the case of pre-MS stars, Hayashi (1961) showed that the fully radiative initial models used by Henyey et al. (1955) are not valid, because of their boundary conditions (i.e. the properties of the photosphere). With their own boundary conditions, Hayashi (1961) obtained initial models that are fully convective, located higher on the HR diagram than those of Henyey et al. (1955). The first stages of pre-MS evolution are done at quasi-constant color instead of quasi-constant luminosity. The track corresponds to an almost vertical line, the Hayashi line, close to the red giant branch, that pre- MS stars follow downwards while they contract. The regions of the HR diagram located at the right-hand side of this line are forbidden for any star in hydrostatic equilibrium. As for the red-giant branch, the exact location of the Hayashi line depends on the mass of the star. The track is redder and at lower luminosities if the mass of the star is smaller.

The left panel of Fig. 1 shows a schematic picture of pre-MS tracks. The line AB is the Hayashi line, and the line CD the radiative branch, corresponding to the

2Deuterium (D = ²H), lithium (⁶Liand⁷Li), beryllium (⁹Be) and bore (¹⁰B and¹¹B).

3The reactions D(D, n)³HeandD(D,¹H)³H(, e⁻ν)³Healso participate to D-burning, but the proton capture remains the main channel for energy generation as well as variation of chemical abundances.

Figure 1. Left panel: Schematic view of pre-MS tracks. AB is the Hayashi line, CD the radiative branch, and the curve ”M.S.” is the ZAMS. The pre-MS of Hayashi (1961) corresponds to APD, the one of Henyey et al. (1955) to CD.Right panel: Isochrones of Hayashi (1961) (solid lines) and of Henyey et al. (1955) (dashed lines). The hatched regions corresponds to the observations of the cluster NGC 2264. From Hayashi (1961).

track of Henyey et al. (1955). In the models of Hayashi (1961), regions located at the right-hand side of the line AB are forbidden for any object in hydrostatic equi- librium. In particular, the initial conditions of Henyey et al. (1955), corresponding to the point C, are unstable. A pre-stellar object located in C evolves rapidly along the line R=Const and joins the Hayashi line, where hydrostatic equilibrium is estab- lished. The evolution timescale along the line R=Const is the dynamical adjustment timescale, i.e. the free-fall time

(6) t_ff =

r 3π 32Gρ¯ where ¯ρ=M/⁴₃πR³ is the mean density.

Once the star reaches hydrostatic equilibrium on the Hayashi line, the evolution continues at a Kelvin-Helmholtz timescale (Eq. 2). The star is fully convective and follows downwards the Hayashi line, as mentioned above. When it approaches point P, a radiative core appears in the centre and increases in mass. The star joins the radiative branch, following the track of Henyey et al. (1955). So the pre-MS track of Hayashi (1961) corresponds to the line APD.

In order to check his results, Hayashi (1961) compared the isochrones he obtained with the HR diagram of the cluster NGC 2264. The result is shown on the right panel of Fig. 1. The continuous lines are the isochrones of Hayashi (1961), the dashed lines

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are those of Henyey et al. (1955). The hatched region corresponds to the observations of the cluster NGC 2264, which are better reproduced by the isochrones of Hayashi (1961) than by those of Henyey et al. (1955).

1.1.3. Initial conditions. The initial conditions for pre-MS evolution, and in partic- ular the temperature-, density- and entropy-profiles at the top of the Hayashi line, depends in principle on the previous evolution, before the hydrostatic equilibrium is established. This stages correspond to the free-fall collapse of the pre-stellar cloud, at a timescale given by Eq. (6). The collapse starts at densities around ∼10⁻¹⁸g cm⁻³ (Hayashi & Nakano 1965). The cloud is initially transparent, and becomes opaque around ρ∼ 10⁻¹³g cm⁻³ (Hayashi & Nakano 1965). At this stage, the central pres- sure is still too low for hydrostatic equilibrium, and the collapse proceeds in free-fall, adiabatically (Gaustad 1963).

Nakano et al. (1968, 1970) and Narita et al. (1970) studied specifically the optically thick stages of the collapse, by solving numerically the hydrodynamical equations.

They obtained that, when the star reaches hydrostatic equilibrium (i.e. when it enters pre-MS), its internal structure is made of a dense quasi-isothermal core surrounded by an envelope of lower density. The corresponding entropy profile initially increases outwards, and the star is fully radiative. But with their low opacity, the external regions radiate quickly their entropy, while the central regions, with their higher opacity, keep their entropy. In addition, the fact that central regions are quasi- isothermal implies that their contraction is quasi-adiabatic:

(7) ∇T '0 =⇒ F '0 =⇒ L_r '0 =⇒ _g '0 =⇒ dq dt '0 And due to the high central value of T, a small entropy decrease in the centre is able to produce a high g >0 if needed, since g =−T ds/dt. As a consequence, the entropy profile flattens in a time short compared to pre-MS evolution, so that the star becomes soon fully convective.

1.1.4. The accretion scenario and the birthline. Larson (1969, 1972) solved numer- ically the hydrodynamical equations for the entire collapse of the pre-stellar cloud, from the beginning of the transparent stage until the whole mass has reached the hydrostatic equilibrium. The collapse turns out to be highly non-homologous. The density profile adjusts quickly on ρ∝r⁻², leading to a free-fall time t_ff ∝r (Eq. 6).

A small (∼ 0.001M) hydrostatic core forms first in the centre, while the rest of the cloud is still collapsing in free-fall. For the external regions of the cloud, t_ff ∼ 10⁵−10⁶ yr, which is not negligible compared to the Kelvin-Helmholtz time.

As a consequence the hydrostatic core, whose mass increases with time, evolves sig- nificantly in its gravitational contraction before it reaches its final mass. This is the accretion scenario.

Figure 2. Left panel: Birthlines of Larson (1972). The thick solid (resp. dashed) line is the birthline for a cloud’s temperature of 10K (resp. 20K). The empty circles on the solid birthline corresponds to the points where stars of different final masses, indicated in M, become optically visible. The thin lines are the isochrones and the ZAMS. Right panel: Schematic view of the system hydrostatic core + circumstellar envelope during the accretion phase, from Stahler et al.

(1980a).

The consequence of such a scenario is the shortening of the pre-MS tracks. The time the whole mass is accreted by the hydrostatic core, the core has already lost a significant part of its initial entropy in its gravitational contraction (g =−T ds/dt).

The location of the star on the HR diagram corresponds to a more evolved stage than in the case of a homologous collapse. The set of this points on the HR diagram, where stars of different masses become optically visible, is called the birthline. It corresponds to the upper envelope of pre-MS stars on the HR diagram. An example of birthline is shown on the left panel of Fig. 2.

With initial conditions corresponding to the Jeans criterium, i.e.

(8) R∝ρ^−1/2 ∝M^−1/2R^3/2 =⇒ R ∝M

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the free-fall time of the external regions is longer when the mass of the cloud is bigger:

(9) t_ff ⁽⁶⁾=

s 3π 32G

4 3πR³

M = π

2√ 2G

R^3/2

M^1/2 ∝M

Since the Kelvin-Helmholtz time decreases with the mass, the ratio tff/tKH is an increasing function of the mass. When the mass is high enough, one hastff/tKH ∼1 and the pre-MS does not exist anymore: the star⁴ joins the MS before it reaches its final mass. This is the reason why the birthline becomes closer and closer to the ZAMS, finally joining it, when the mass increases (Fig. 2, left panel). It fixes an upper limit for the mass of a pre-MS star. The exact value of this limit depends on the physical conditions of the pre-stellar collapse. Larson (1972) obtained a value of

∼3M (Fig. 2, left panel).

In the models of Larson (1972), only the stars with a final mass smaller than

∼2Mgo through a Hayashi line. Higher masses can only be reached on the radiative branch or the MS. In this case, the star accretes while it is already radiative. When the star becomes radiative, from the centre to the surface, the high efficiency of the energy transport produces an increase in the flux, taken on the gravitational energy of the central regions, i.e. on their entropy (g =−T ds/dt). The external regions, still convective, with their high opacity, absorb this entropy and the stellar radius consequently increases. A luminosity wave travels across the star, from the centre to the surface, producing a violent jump in luminosity (factor 2 in a few days⁵). When the wave reaches the surface, the star contracts again and follows its evolution along a track similar to the radiative branch.

Hayashi (1970) criticized the treatment by Larson (1969) of the shock wave corre- sponding to the accretion front, where the material of the envelope falling on the star reaches the hydrostatic equilibrium. According to Hayashi, the Larson’s treatment underestimates the entropy that is accreted, leading to artificially low luminosities at the end of accretion, and to an excessive shortening of the pre-MS tracks. Shu (1977) criticized also the initial and boundary conditions used by Larson (1969, 1972).

In order to solve this controversy, Stahler et al. (1980a,b) computed the evolution of the whole system hydrostatic core + circumstellar envelope during the accretion phase, from the formation of a hydrostatic core of 0.01M to the end of accretion, at 1M, with a particularly careful treatment of the accretion shock front. The originality of the method adopted by Stahler et al. (1980a,b) is to split the system in a series of regions. In each of this regions, some simplifications in the physics can be done. The difficulties related to the variety of orders of magnitude entering

4In the context of accretion,star means the hydrostatic core.

5In the model of 2M of Larson (1972), the accretion rate is on average 5×10⁻⁶Myr⁻¹.

simultaneously in the problem can then be turned as an advantage. A schematic view of the system can be seen on the right panel of Fig. 2. The different regions are the following:

- In the centre is thehydrostatic core, surrounded by the accretion shock.

- Then, if the gas is still optically thick, comes the radiative precursor, until the gas photosphere.

- After the gas photosphere, the flow is transparent: this is theopacity gap.

- When the temperature becomes low enough to allow dust grains to survive, above the dust destruction front, in the dust envelope, the accretion flow is opaque again.

- Around the dust envelope, density decreases until the flow becomes transpar- ent again, at the dust photosphere, which is the observable photosphere.

- In theouter envelope the gas is transparent and isothermal.

The treatment of the accretion shock is based on jump conditions, that provide the boundary conditions for the hydrostatic core. If the accretion flow is optically thin above the accretion shock (i.e. without a radiative precursor), the boundary conditions are

(10)

P = M˙ 4πR²

r2GM R σT⁴ = 3

4σT_eff⁴ − 1 4F

whereP, T andF are respectively the pressure, the temperature and the flux under the accretion shock. If the accretion flow is optically thick (i.e. in the presence of a radiative precursor), the second condition is replaced by

(11) T =T⁰

where T⁰ is the temperature above the accretion shock. This boundary conditions are based on the assumption of spherical symmetry (scenario of spherical accretion).

In case accretion proceeds through a disc, the boundary conditions of the hydrostatic core have to be modified. In the extreme scenario where the entropy of the accreted material is the same as the entropy of the stellar surface before the material con- sidered is accreted (scenario of cold disc accretion), the accretion flow has no more influence on the stellar surface, and the correct boundary conditions are the usual photospheric conditions (Palla & Stahler 1992):

(12) P = 2

3 GM κR² L= 4πR²σT_eff⁴

15

With the formulation of Stahler et al. (1980a,b), the various regions of the system can be studied separately. For instance, it is only through the boundary conditions and the accretion rate ˙M =dM/dtthat the evolution of the hydrostatic core depends on the evolution of the rest of the cloud. Inversely, the hydrostatic core influences the rest of the cloud only through the gravitation field and the radiatif feedback, i.e.

through M, L and T_eff.

The results of Stahler et al. (1980a,b) essentially confirms those of Larson (1969, 1972) about the shortening of the Hayashi line. For a mass of 1M, the value of the radius obtained at the end of accretion is 4.7R, closer to the 2.1R of Larson (1972) than to the 25−50R of Hayashi & Nakano (1965). For the internal stellar structure, Stahler et al. (1980a,b) obtained that, no matter what the initial conditions are, the entropy profile converges quickly to a profile increasing outwards. So the star is fully radiative in the beginning of the accretion phase (M ' 0.1M). The hydrostatic core is quasi-isothermal, and it follows that the luminosity is close to 0 (L∝ −∇T). The main contribution to the luminosity at the accretion shock comes from the energy liberated by the accreted material when it adjusts to the internal structure of the star. In central regions, L ' 0 and so _g ' 0 which means that the evolution is nearly adiabatic (_g =−dq/dt=−T ds/dt). The central regions are even characterized by a temperature inversion (∇T > 0). The consequence is that D-burning starts in excentred regions, where the temperature has its maximum while the central regions remain inert ('0). D-burning produces a significant amount of entropy and, except in the inert central core, the evolution is not adiabatic anymore.

In addition, the energy produced is too important for the radiative transport, and a convective zone develops from the D-burning layer towards the surface. In parallel, an external convective zone appears. Both zones join rapidly, and the star becomes fully convective, except the central core which is still inert. This structure is conserved until the hydrostatic core reaches a mass of 1M, corresponding to the end of the computations of Stahler et al. (1980a,b). So in the models of Stahler et al. (1980a,b), the fully convective structure of the star is a consequence of D-burning.

Remark: In their models with radiative regions, Stahler et al. (1980a,b) used an explicit method to solve their differential equations, instead of the implicit method used in the convective regions.

The complete birthline, i.e. the track of an accreting star (here at rates in the range M˙ = 10⁻⁵−10⁻⁴Myr⁻¹), until a mass of ∼10M, has been computed by Stahler (1983, 1988, 1989) and Palla & Stahler (1990, 1991, 1992). The upper mass limit for pre-MS stars is ∼10M, which is clearly higher than the ∼3M of Larson (1972).

The comparison with T Tauri and Herbig Ae/Be stars confirm that the theoretical birthline of Stahler et al. (1980a,b) and Palla & Stahler (1991) corresponds to the upper envelope of the observed pre-MS stars (Fig. 3). The observations of outflow

Figure 3. Comparison between the theoretical birthline and the ob- servations. Left panel: Birthline of Stahler et al. (1980a,b) and T Tauri stars from different clusters. From Stahler (1983). Right panel: Birth- line of Palla & Stahler (1991), Herbig Ae/Be stars (empty signs), and mass outflows (filled signs). From Palla & Stahler (1990).

jets fit well with the position of the birthline (Fig. 3, right panel), which is consistent with the idea that such jets are related to accretion processes.

The evolution of the internal structure of the star in the models of Stahler (1983, 1988, 1989) and Palla & Stahler (1990, 1991, 1992) can be split in four stages:

(1) While the star is fully convective, D burns at the same rate as it is accreted (steady-state deuterium burning). D-burning has a thermostatic effect, and the central temperature remains nearly constant, which leads to a radius increasing linearly with the mass of the hydrostatic core (Tc ∼ M/R), i.e.

with the age ( ˙M = cst).

(2) When the central regions become radiative, the centre receive no more deu- terium from the external regions, central D-burning stops, and convection disappears in the centre. The star is then made of a radiative core and a con- vective envelope, and the mass fraction of the radiative core increases with

17

time. D-burning continues at the bottom of the convective envelope (deu- terium shell-burning), while the central radiative regions contract, radiating their entropy. This entropy is then absorbed by the external regions, and the radius increases considerably (by a factor ∼ 2−3) in agreement with the results of Larson (1972). At the surface, the luminosity increases when the luminosity wave reaches the surface and the convective envelope disappears.

(3) Soon after the envelope has disappeared, the radius reaches a maximum and the star returns to gravitational contraction. The star is fully radiative, but deuterium shell-burning continues close to the surface, due to the newly accreted deuterium.

(4) When the radius has decreased by a factor ∼ 2−3, H-burning starts in the centre and a convective core appears. The gravitational contraction slows down and stops, the radius reaches a minimum and the star is on the ZAMS.

The internal structure on the birthline for ˙M = 10⁻⁵Myr⁻¹ is visible on Fig. 4. In this case, the maximum radius and the mass on the ZAMS are

(13) R_max= 8.5R M_ZAMS= 8M

while for ˙M = 10⁻⁴Myr⁻¹, Palla & Stahler (1992) obtained

(14) R_max = 15.8R M_ZAMS= 15M

1.2. The formation of massive stars.

1.2.1. The upper limit for the mass of stars. When one tries to apply to massive stars (M &8M) the same formation scenario as for low- (M .2M) and intermediate- mass stars (2M .M .8M) described in Sect. 1.1.4, some difficulties arise. This difficulties have been first listed by Larson & Starrfield (1971), with the assumption of spherical symmetry.

The first problem concerns the time-scales. In the same way as the accretion scenario leads to an upper mass limit for pre-MS stars (Sect. 1.1.4), it fixes an upper mass limit for MS stars: since the main sequence time t_MS decreases when the mass increases, and since t_ff increases with the mass (Eq. 9), above a certain mass, one has t_MS < t_ff and the hydrostatic core reaches the end of the MS before the mass considered can be accreted. Larson & Starrfield (1971) evaluated this limit around M_max = 60M, emphasizing that the exact value is very sensitive to the initial conditions.

The other limitations are related to the radiation field of the star. When the stellar radiation emerges from the gas photosphere, it crosses freely the opacity gap (Fig. 2, Sect. 1.1.4). Once it reaches the dust destruction front, it is easily absorbed and re- emitted in the infra-red (Fig. 6, upper panel). Above a certain mass, the radiation

Figure 4. Internal structure on the birthline for ˙M = 10⁻⁵Myr⁻¹. The vertical axis indicates the mass, which is a function of time, while the horizontal axis shows the distance from the centre. The thick curve at the right-hand side is the stellar radius, the dashed lines are the iso-mass, the filled areas are convective regions and the hatched layers close to the surface indicate the place of deuterium shell-burning that survives to the receding of the convective envelope. From Palla

& Stahler (1991).

pressure of the direct radiation field (optical and UV) on the dust destruction front can stop the accretion flow. The secondary radiation field (IR) that is re-emitted by the dust from the absorbed direct radiation field, produces also a radiation pressure on the external parts of the dust envelope, that goes against the accretion flow. In this external regions, the absorption of the secondary radiation field leads to an increase in the temperature, and thus in the pressure of the gas, which can potentially reverse the accretion flow. Finally, if the luminosity and the effective temperature of the star become high enough, the direct radiation field ionize the surrounding gas, producing an HII region. The brutal increase in gas pressure resulting from the ionization can also inhibit accretion. For all this mechanisms, Larson & Starrfield (1971) estimated upper mass-limits between M_max = 10 and 90M, with strong uncertainties in the numerical values.

19

Kahn (1974) studied in details the limitation due to the radiation pressure of the secondary radiation field acting on the external regions of the cloud, that he consid- ered as the more restrictive. In such a problem, the treatment of the dust grains is a crucial point. Dust grains are responsible for the absorption of the direct radiation field, for its re-emission as the secondary field, and for the absorption of the sec- ondary field in the external regions of the dust envelope. Using only graphite grains, which are the more resistant to the direct radiation field, Kahn (1974) computed sta- tionary hydrodynamic models for the whole cloud with the assumptions that the gas pressure is negligible and that gas and dust are coupled. He obtained that, for the accretion flow to be reversed, the ratio L/M shall not exceed ∼5500L/M. With the mass-luminosity relation of Papaloizou (1973), he deduced an upper mass-limit of

(15) M_max = 40M

Relaxing the hypothesis of Kahn (1974), and adding the effects of the accretion luminosity (i.e. the luminosity generated at the accretion shock), Yorke & Kruegel (1977) confirmed the results of Kahn (1974).

Wolfire & Cassinelli (1987) computed new mass limits for various accretion rates, using the dust abundances and opacities of Mathis et al. (1977) for the InterStellar Medium (ISM). They obtained that accretion until ∼100M is only possible if the dust abundance is reduced by a factor 4 compared to the abundance of Mathis et al.

(1977). Without this condition, the radiation pressure of the secondary radiation field reverses the accretion flow too early. Moreover, even with such a modified chemical composition, a rate & 10⁻³Myr⁻¹ is necessary to obtain a final mass & 100M. The various limits in the case with the modified chemical composition are shown on Fig. 5. The limit C, due to the radiation pressure of the direct radiation field on the dust destruction front, is the more restrictive. If the accretion rate is lower than this limit, the radiation pressure of the direct radiation field wins over the ram pressure of the accretion flow:

(16) M v <˙ L

c

and accretion stops. If we consider initial conditions corresponding to the Jeans criterium with a purely thermal support, and an accretion rate corresponding to the free-fall (t_ff ∝ ρ^−1/2, Eq. 6), the accretion rate can be related to the cloud’s temperature:

(17) MJ ∝T^3/2ρ^−1/2

(18) M˙ = MJ

t_ff ∝T^3/2

Figure 5. The range of accretion rates needed for the formation of massive stars by Wolfire & Cassinelli (1987).

On Fig. 5, the temperature corresponding to each accretion rate is indicated on the right-hand side vertical axis.

The conclusion of this series of works is that the formation of massive stars (∼

100M) by accretion in spherical symmetry requires extreme conditions in terms of chemical composition and temperature. Nakano (1989) suggested that such extreme conditions are not necessary when we relax the assumption of spherical symmetry, i.e. if we consider that accretion proceeds through a disc. The formation of accretion discs, supported essentially by the centrifugal force (quasi-Keplerian velocities), is a natural consequence of angular momentum conservation during the collapse, which implies high rotation velocities in central regions⁶ (v ∝ r⁻¹). The presence of such discs is considered as a common feature of low-mass stars during their formation (Shu

6Indeed, the contraction is so important during the collapse (∼6−7 orders of magnitude) that the increase in velocity in case the angular momentum is conserved would make it impossible to form stars, the centrifugal force exceeding quickly the gravitational attraction. Efficient mechanisms for the evacuation of angular momentum are thus necessary during star formation (Maeder 2009).

21

Figure 6. Schematic view of the accretion flow and the radiation field in the case of spherical accretion (upper panel) and of disc accretion (lower panel). From Kuiper et al. (2010).

& Adams 1987). In the case of massive stars, the idea of Nakano (1989) was that if the disc is thin enough, the secondary radiation field can escape freely in the polar directions, so that the main part of the disc remains unperturbed. The shielding by the internal disc has to be efficient enough to impose an anisotropy to the secondary radiation field (Fig. 6, lower panel).

The first hydrodynamic simulations of the proto-stellar collapse that relaxed the assumption of spherical symmetry have been published in a series of papers: Bo- denheimer et al. (1990); Yorke et al. (1993, 1995); Laughlin & Bodenheimer (1994);

Yorke & Bodenheimer (1999); Yorke & Sonnhalter (2002). This simulations were based on the assumption of axial- (rotational axis of the cloud) and planar-symmetry (equatorial plane, perpendicular to the rotational axis), and they have been per- formed on two-dimensional grids. They included radiative feedback⁷ through the flux-limited diffusion approximation, and the angular momentum transport was treated through a viscosity tensor that mimics the effects of the shear (α-prescription, Shakura & Sunyaev 1973). The initial angular momentum of the cloud corresponds to j ∼ 10²¹cm²s⁻¹. The evolution of the cloud depends on the properties of the central hydrostatic core only through the gravitation field and the radiative feed- back, i.e. through M,L and T_eff (Sect. 1.1.4). The comprehensive treatment of the hydrostatic core would need a too high resolution, so that in all the hydrodynamic simulations listed above the central cell, in which the star is located, is considered as a sink cell: all the material that enters this cell is considered as instantaneously accreted by the star. The radiative feedback is then included using pre-computed stellar models: L and T_eff are obtained from M, the total mass captured by the sink cell since the beginning of the simulation. A term can be added to Lto include the contribution of the accretion luminosity, i.e. the energy generated by the material that is accreted, when it settles onto the star. If all the kinetic energy gained by the material during the collapse is radiated, and with the assumption of free-fall (for which the kinetic energy is maximal), the accretion luminosity is given by

(19) LM˙ max= dE_cin dM

dM

dt =−dE_pot dM

dM

dt = GMM˙ R

In general, we expect the real accretion luminosity to be only a fraction η < 1 of LM˙ max, i.e.

(20) LM˙ =η×GMM˙

R

In particular, Yorke & Sonnhalter (2002) tested numerically the idea of Nakano (1989) to use the disc geometry to circumvent the mass limitation due to radiation pressure. However, the higher mass reached in their simulations was only 42.9M, a value that exceeds only marginally the limit of Kahn (1974) in the spherical case.

In the simulations of Yorke & Sonnhalter (2002), the disc appears as expected, but it is too short-lived. The collapse proceeds initially nearly in spherical symmetry in the whole cloud. The accretion rate in the sink cell increases rapidly, reaching

∼2×10⁻³Myr⁻¹in a few thousands years, and then remains nearly constant. When

7 However, the ionizing effects of the radiation field were not included.

23

the stellar mass reaches ∼30M, the radiation pressure of the direct radiation field of the central star cleans the polar regions, where cavities form. The accretion on the central star continues only in the equatorial regions, through a disc. The polar cavities grow rapidly, starving the disc from material coming from the envelope. The accretion rate decreases progressively as the disc dissipate. When accretion stops, the stellar mass doesn’t exceed significantly the upper limit of the spherically symmetric case.

This limit of∼40M is incompatible with the observations, that show stars with masses exceeding 100M (e.g. Massey & Hunter 1998).⁸ Krumholz et al. (2007, 2009) tried to circumvent the limitation of Yorke & Sonnhalter (2002) by computing three-dimensional hydrodynamic simulations, including the effects of gravitational instability in the accretion flow. They obtained that such instabilities lead to the fragmentation of the cloud, and that the lower density regions between the fragments allow the radiation to escape, while the accretion of the dense fragments on the disc continues. According to Krumholz et al. (2007, 2009), this continuous contribution from the external regions of the cloud allows the disc to survive to the direct radiation field of the star. But unfortunately, their simulations were stopped before the end of accretion, when the stellar mass was only 46.9M, so that the validity of the argument of the fragmentation was not confirmed numerically.

The first hydrodynamic simulations that overcame the radiation pressure and formed stars with masses exceeding 100Mwere those of Kuiper et al. (2010). These simulations have been performed on two-dimensional grids, in contradiction with the argument of fragmentation of Krumholz et al. (2007, 2009). The critical difference between the simulations of Yorke & Sonnhalter (2002) and those of Kuiper et al.

(2010) is the size of the sink cell. The sink cell of Yorke & Sonnhalter (2002), with its radius of 160 ua, contained the dust sublimation front (Sect. 1.1.4), which was not included in the computational domain. Kuiper et al. (2010) showed that in this case the shielding becomes inefficient, because the radiative feedback acts on regions with an artificially low optical depth, that are not able to impose a significant anisotropy to the secondary radiation field. In their simulations, Kuiper et al. (2010) used a sink cell with a radius of 10 ua, and they obtained that the dust sublimation front is located at a radius around 15-20 ua. With such a sink cell, Kuiper et al. (2010) formed stars with masses above 100M. Their simulations have been computed

8 In order to solve this problem, Bonnell et al. (1998) proposed the coalescence scenario, in which massive stars form form collisions and fusions of intermediate-mass stars. This scenario is not considered in this section, which focuses on the scenario of monolithic collapse. The scenario of multiple star formation, that allows such collisions, is treated in Sect. 1.2.2. However, we can already notice that such collisions require extremely high densities of stars in clusters, so that we need again to invoke extreme conditions, like in the case of spherical accretion.

starting from clouds of different masses M_cloud, and the final mass of the star M_fin is an increasing function of M_cloud:

Mcloud= 60 120 240 480 M

Mfin= 28.2 56.5 92.6 >137.2 M

M_fin/M_cloud= 47% 47% 39% >29%

The simulation with Mcloud = 480M has been stopped before the end of accretion, when the sink cell accreted already 137.2M, with 67.8M still in the computa- tional domain. The final mass is thus < 205M, i.e. M_fin/M_cloud < 43%, which suggests a small decrease of the Star Formation Efficiency (SFE), defined as the ratio M_fin/M_cloud, as a function of the cloud’s mass, due to the effect of the radia- tive feedback. In the case without rotation (i.e. with spherical symmetry), Kuiper et al. (2010) obtained again the same upper mass limit of 40M as Kahn (1974) and Yorke & Kruegel (1977), independently of the cloud’s mass, which demonstrates that it is the change in the geometry of the accretion flow that allows to overcome the radiation pressure problem.

The simulations of Kuiper et al. (2010) provide mass-accretion rates, given by the accretion flow at the boundary of the sink cell. The accretion rates of Kuiper et al.

(2010) are qualitatively similar to those of Yorke & Sonnhalter (2002), and can be divided in two phases (Fig. 7):

(1) Spherical accretion: the accretion flow is close to the free-fall. The accretion rate is high, and constant due to the initial density profile:

(21)

ρ∝r⁻² =⇒ Mr ∝r =⇒ ρ¯= Mr 4

3πr³ ∝r⁻²

=(6)⇒ t_ff ∝r =⇒ M˙_ff = Mr

t_ff ∝ r

r = cst ('10⁻³−10⁻²Myr⁻¹) (2) Disc accretion: the accretion flow is governed by the angular momentum

transport in the disc, given by the viscosity tensor. It decreases with time, as the disc’s mass decreases.

The accretion rates show also strong oscillations, in particular for the more massive cloud. This oscillations are produced by the radiative feedback: when the accretion rate is high, the radiative pressure due to the accretion luminosity becomes high enough to slow down the accretion flow; the accretion luminosity decreases and the accretion flow can accelerate again; thus the accretion luminosity increases again, and so on.

This results have been confirmed by Kuiper et al. (2011) in the three-dimensional case. In the three-dimensional simulations, the transport of angular momentum in the disc is governed by the gravitational instability itself, leading to the formation of

25

Figure 7. Accretion rates of Kuiper et al. (2010).

spiral arms, without any artificial viscosity. During the spherical accretion phase, the accretion rates are strictly identical to the two-dimensional case. In the disc accretion phase, the formation of spiral arms leads to an oscillation in the accretion rate on the central star, around a mean value slightly higher than in the two-dimensional case.

This simulations also confirmed that the disc is in quasi-Keplerian rotation (& 99%

of the Keplerian velocity), i.e. that it is essentially supported by the centrifugal force.

The external regions of the disc show even super-Keplerian motions, in agreement with the observations of accretion discs around massive stars (Beuther & Walsh 2008).

The conclusion of Kuiper et al. (2010, 2011) is thus thatin the case of disc accre- tion, the radiation pressure of the direct and secondary radiation fields is not able to reverse the accretion flow. However, the ionizing effects of the direct radiation field (UV) were not included in the simulations of Kuiper et al. (2010, 2011). Since the development of an HII region with high pressure in the surrounding of the star can potentially reverse the accretion flow (Larson & Starrfield 1971), a comprehensive treatment of the radiative feedback requires to include the ionizing effects.

1.2.2. Multiple star formation. Infra-red observations of star forming regions indicate that the majority of stars form in clusters (e.g. Lada et al. 1991, 1993; Zinnecker et al.

1993; Lada & Lada 2003). The interactions between the stars in a cluster can modify significantly their evolution. Moreover, the observations of MS stars show that the majority of stars belong to a multiple system (e.g. Duquennoy & Mayor 1991). It indicates that a full understanding of the star formation process requires to go beyond

the scenario of monolithic collapse, in which each cloud form one single star, and to consider instead the scenario of multiple star formation, in which several stars are formed in the collapse of the same parental cloud.

The simulations of Kuiper et al. (2010, 2011), as well as those of Yorke & Sonnhal- ter (2002), are based on the assumption of monolithic collapse, and they do not allow multiple formation. The validity of such an assumption can be tested a posteriori by evaluating the stability of the accretion flow using the Jeans criterium. It requires a high enough resolution so that the Jeans length is described by at least four cells in each point of the accretion flow (Truelove et al. 1997). In the simulations of Kuiper et al. (2011), this condition is not satisfied in the spiral arms, so that the assumption of monolithic collapse cannot be justified.

The modeling of multiple star formation with hydrodynamic simulations requires the use of sink particles (Bate et al. 1995), similar to the sink cell in the case of monolithic collapse, except that:

- the sink particles can form at any place in the cloud and at any time during the collapse as long as the physical conditions are satisfied;

- the sink particles are not fixed on the grid, they move on it depending on their initial and accreted momentum and on the gravitational field.

The typical conditions for the formation of a sink particle are:

- the density has to be higher than a given threshold;

- the material considered has to be gravitationally bound.

Once a sink particle has formed, the material included in it is considered as forming a star, described with a single value of mass, momentum and angular momentum.

The initial value of this quantities is determined by the properties of the material from which the sink particle formed, while their evolution is given by the properties of the accreted material (and of the gravitational field in the case of the momentum).

In general, the material entering the sink particle is not necessarily accreted: the condition for the material to be accreted is that it is gravitationally bound to the star in the particle.

Bonnell et al. (1997, 1998, 2001a,b, 2003) and Bonnell & Bate (2002, 2006) per- formed hydrodynamic simulations using the sink particles technique in order to form star clusters.⁹ This simulations revealed the interesting feature of competitive accre- tion: the different stars that form share the same mass reservoir, so that they are in competition to accrete the material it contains. In this conditions, the location of a star at a given time is critical for the subsequent evolution of its mass. Bon- nell et al. (1997, 2001a) obtained that stars that form in the centre of the cloud are favoured for accretion compared to stars formed in the external regions, because they

9 The radiative feedback is not included in their computations.

27

benefit from the whole potential well of the cluster. It follows that once accretion ends, the more massive stars are located in the centre of the cluster. Such a mass segregation has been observed in the majority of young clusters (Zinnecker et al.

1993; Hillenbrand et al. 1995; Testi et al. 1997), but it is generally interpreted as a purely dynamical effect. The simulations of Bonnell et al. (1997, 2001a) suggest an alternative scenario, in which this mass segregation is a direct consequence of the accretion process.

The effects of competitive accretion are crucial in the issue of the Initial Mass Function (IMF) and of its universality. Bonnell et al. (2001b, 2003) and Bonnell &

Bate (2002, 2006) obtained in their simulations a mass distribution that is in good agreement with the IMF of Salpeter (1955b) for the high mass range,

(22) dlogN

dlogM ' −1.35

and proposed competitive accretion as the mechanism responsible for the IMF.

The simulations of Bonnell et al. (2003) revealed also the feature of hierarchical cluster formation, in which several sub-clusters form in the different parts of the cloud, and then merge together in one big cluster (Fig. 8). Star formation occurs essentially before the coalescence. Such a scenario leads to gravitational interactions that are particularly strong between stars in the same sub-cluster, with possible disruptions on their accretion discs or planetary system.¹⁰

Krumholz et al. (2005) criticized the scenario of competitive accretion. According to this authors, the typical gas densities in observed star forming regions are too low, and the velocity dispersions are too high, to allow significant accretion. Low densities give high Jeans masses (Eq. 17), inhibating fragmentation, while high ve- locities inhibit the accretion of the gas by the forming stars. Bonnell & Bate (2006) showed that the estimations of Krumholz et al. (2005) were wrong, because they were based on the global properties of star-forming regions. In the simulations of Bonnell et al. (2003), star formation occurs along high density filaments (Fig. 8 B). In this filaments, the density exceeds by two orders of magnitude the mean density of the cloud, and the velocity dispersion is reduced by a factor ∼5 compared to the mean value. With such local conditions, Bonnell & Bate (2006) obtained densities high enough and velocities dispersions low enough to allow fragmentation and accretion.

10In their simulations, Bonnell & Bate (2002) obtained stellar densities high enough to allow the coalescence of intermediate-mass stars. The hierarchical formation in the simulations of Bonnell et al. (2003) leads to stellar densities even higher. Thus the coalescence scenario for massive star formation cannot be rejected, even if the original motivations of Bonnell et al. (1998, 2003) and Bonnell & Bate (2002) to propose such a scenario are no longer relevant, since the radiation pressure problem has now been solved (Sect. 1.2.1).

Figure 8. Hierarchical cluster formation: the gray regions represent the gas and the white dots are the stars. From Bonnell et al. (2003).

The filamentary structures that appear in the simulations of Bonnell et al. (2003) are due to supersonic turbulences¹¹. We know since Larson (1981) that turbulence plays a crucial role in star formation. At large scales, it gives a support against gravitational collapse in addition to the thermal support, while at smaller scales it can trigger the collapse (Klessen 2011). In the simulations of Bonnell et al. (2003), supersonic turbulences produce shocks in the gas, leading to the over-density fila- ments, gravitationally unstables, along which stars form. This is the scenario of gravoturbulent fragmentation (Klessen 2004). In such a scenario, the properties of turbulence are crucial for the size and the structure of the cluters, as well as for the issue of the IMF. A precise treatment of turbulence is thus necessary in the hydro- dynamic simulations of multiple star formation. Klessen & Burkert (2000, 2001),

11Such filamentary structures have also been observed for instance by the Herschel space tele- scope, and seem to be a common feature of star forming regions (e.g. Andr´e et al. 2012).

29

Klessen (2000, 2001b,a), Klessen et al. (2000) and Schmeja & Klessen (2004) studied in details the properties of turbulence in the pre-stellar clouds in such simulations.

In particular, Klessen (2001a) considered the effects of turbulence on the IMF, while Klessen (2001b) and Schmeja & Klessen (2004) deduced accretion rates from their simulations.

Another important mechanism in multiple star formation is the radiative feed- back (Sect. 1.1.4 and 1.2.1). In the simulations of multiple star formation described above, no radiative feedback was included. In the simulations of monolithic collapse described in Sect. 1.2.1, the radiative feedback was included only through radiation pressure, without the ionizing effects. The simulations showed that the radiation pressure alone cannot halt the accretion flow when accretion proceeds through a disc, but the ionization of the stellar neighbourhood (i.e. the development of an HII region) can in principle have catastrophic consequences on the accretion flow, because of the brutal increase in the gas pressure occuring when the gas is ionized (Sect. 1.2.1). The first simulations of multiple star formation including such ionizing effects have been performed by Peters et al. (2010a,b). The results show that HII regions appear around stars when they exceed ∼ 10M, and that the shape of this HII regions vary with time, going successively through all possible morphologies.

However, the simulations of Peters et al. (2010a) show that the high pressure of this HII regions does not prevent the formation of massive stars. The mechanism that gives an upper mass limit to stars appears to be the fragmentation, which leads to the formation of secondary stars in the accretion flow of massive stars. This mech- anism is called fragmentation-induced starvation (Peters et al. 2010a): secondary stars accrete the majority of the material from the accretion flow before it reaches the central star. Accretion on the central star stops, while the secondary stars be- come massive too. In order to distinguish the effects of fragmentation from those of the radiative feedback, Peters et al. (2010a) computed simulations of monolithic collapse including the effect of the ionizing flux, and obtained a star of 70M at the end of their simulation, while the accretion was not yet finished. As for the radia- tion pressure, the pressure of the ionized gas stops accretion in polar directions, but not in the equatorial plane. In the case of multiple star formation, secondary stars appear when the central star becomes massive (M &8M). Then the accretion on the central star slows down, and ends when its mass is only ∼ 25M (Fig. 9). The decrease in the accretion rate on the central star is due to a decrease in the gas density around the star. The ionizing flux give birth to an HII region. The results of this simulations modify the competitive accretion scenario of Bonnell et al. (1997), in which the central star is favoured for accretion due to its location at the centre of the potential well of the whole cluster.