UNIVERSITE LIBRE DE BRUXELLES

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Facult´e des Sciences

KINEMATICS AND DYNAMICS OF GIANT STARS IN THE SOLAR NEIGHBOURHOOD

by Benoit FAMAEY

Ph.D. thesis submitted

for the degree of Docteur en Sciences

Institut d’Astronomie et d’Astrophysique Academic year 2003-2004

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savent.

Paul Val´ery, 1937 (L’Homme et la coquille)

For true and false are attributes of speech, not of things. And where speech is not, there is neither truth nor falsehood.

Thomas Hobbes, 1651 (Leviathan, chapter 4)

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Acknowledgements

Je voudrais avant tout remercier le professeur Marcel Arnould, Directeur de l’Institut d’Astronomie et d’Astrophysique, qui m’a accueilli à l’Institut en tant que mathématicien et qui m’a permis de passer ces années de thèse dans des conditions idéales en m’accordant une grande confiance et une grande liberté de travail et en se montrant d’une aide attentive face à mes interrogations et sollicitations.

Je remercie ensuite avec beaucoup de gratitude mon directeur de thèse, le professeur Alain Jorissen, pour sa disponibilité constante durant ces années de doctorat et pour les nombreuses discussions stimulantes que nous avons eues.

Au cours de celles-ci j’ai ressenti une véritable émulation, son esprit scientifique aiguisé en faisant non seulement un chercheur hors pair mais surtout un inter- locuteur extrêmement intéressant et motivant. Je le remercie en outre, par delà ces considérations professionnelles, pour ses qualités humaines hors du commun.

Je voudrais également exprimer ma gratitude à tous les autres membres de l’Institut, en particulier à Dimitri et Marc pour leur soutien face aux aléas de l’informatique, à Carine et Laurent pour m’avoir fourni quelques références intéressantes, à Sylvie, Sophie, Ana, Claire et Viviane pour leur sourire et l’agréable atmosphère de travail qu’elles contribuent à créer à l’Institut, à Yves, Abdel, Stéphane, Lionel et Matthieu pour leur bonne humeur. Merci aussi

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a Samir Keroudj pour la réalisation de l’animation en trois dimensions qui a permis d’illustrer certains résultats de cette thèse.

Il est très important pour moi de remercier ici le professeur Herwig De- jonghe, de l’Université de Gand, véritable instigateur de ce projet, sans lequel rien n’aurait été possible. Ses explications, toujours claires et précises, m’ont permis de me constituer au fil du temps une expertise en dynamique galactique, domaine qui était entièrement neuf pour moi au moment d’aborder ces années de thèse. Je remercie aussi Kathrien Van Caelenberg qui avait entamé une partie de ce travail à l’Université de Gand.

Une grande partie des résultats présentés dans cette thèse dépendent de données obtenues grâce à l’Observatoire de Genève et à la coopération de nombreux observateurs anonymes que je remercie pour leur importante contribu- tion à ce travail. Je remercie tout particulièrement le professeur Michel Mayor ainsi que Catherine Turon d’avoir accepté de me fournir ces données. Je remercie également Stéphane Udry pour son hospitalité lors de mon séjour à l’observatoire de Genève et sa collaboration active lors du débroussaillement

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des donn´ees CORAVEL.

Je suis également extrêmement reconnaissant à Xavier Luri, de l’Université de Barcelone, qui m’a accueilli chaleureusement lors de mon séjour dans son service et qui fut toujours d’une aide radicalement efficace et d’une disponibilité sans faille dans l’analyse des données cinématiques qui a conduit aux principaux résultats de ce travail.

Je voudrais aussi remercier mes parents pour leur soutien constant tout au long de mes études et leurs nombreux conseils judicieux sur le plan humain ou logistique. C’est par ailleurs un plaisir de remercier ici mes amis de longue date Maxime, Benjamin, Stéphane, Bernard, Benoˆıt, Jules, Hugo et Mohamed pour leurs commentaires de non spécialistes de la discipline et leur soutien moral.

Merci enfin `a ma douce Marie-Laure, merci de m’avoir soutenu et encourag´e, et surtout merci d’ensoleiller ma vie chaque jour.

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List of Figures

1.1 Schematic view of the Galaxy . . . . 16

1.2 Lindblad diagram . . . . 23

1.3 Tightly wound and loosely wound spiral patterns . . . . 28

1.4 The swing-amplification . . . . 31

2.1 Difference between the Hipparcos and Tycho-2 proper-motion moduli . . . . 44

2.2 Crude Hertzsprung-Russell diagram for the Hipparcos M stars with positive parallaxes. . . . . 47

2.3 Crude Hertzsprung-Russell diagram for the Hipparcos K stars with a relative error on the parallax less than 20%. . . . . 48

2.4 The concept of line-width parameter . . . . 49

2.5 The (Sb, σ0(vr0)) diagram . . . . 50

2.6 Distribution of the sample on the sky . . . . 52

3.1 Density of stars with precise parallaxes (σπ/π ≤ 20%) in the U V-plane . . . . 57

3.2 Comparison of the distances obtained from a simple inversion of the parallax and the maximum-likelihood distances . . . . 66

3.3 All the stars plotted in theU V-plane with their values ofU and V deduced from the LM method . . . . 67

3.4 HR diagram of group Y . . . . 68

3.5 HR diagram of group HV . . . . 70

3.6 HR diagram of group HyPl . . . . 72

3.7 HR diagram of group Si . . . . 74

3.8 HR diagram of group He . . . . 75

3.9 Histogram of the metallicity in groups B, HyPl and Si for the stars present in the analysis of McWilliam (1990) . . . . 76

3.10 HR diagram of group B. . . . . 77

4.1 Integral space . . . . 87

4.2 Parameter space . . . . 92

4.3 Profile of the logarithm of vertical density at R = 8 kpc for Kuzmin-Kutuzov potentials with different axis ratios . . . . 94

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4.4 Mass isodensity curves in a meridional plane for the five potentials of Table 4.5 . . . . 99 4.5 The effective bulge . . . 100 4.6 The rotation curves of the five selected potentials of Table 4.5 . 101 5.1 Integration area in the (E, I3)-plane . . . 107 5.2 Integration area in the (x, y)-plane . . . 109 5.3 Integration limits inLz . . . 111 5.4 Contour plots of the mass density in a meridional plane, for com-

ponents withz0 equal to 4 kpc and 2 kpc . . . 113 5.5 Logarithm of the galactic plane mass density of different compo-

nents for varyingα1 . . . 114 5.6 Logarithm of the configuration space density of different compo-

nents for varyingα2. . . . 115 5.7 Contour plots of the configuration space density in a meridional

plane, for components with varyingβ . . . 116 5.8 Values of a component distribution function as a function of E

for varyingη . . . 118 5.9 Contour plots of the configuration space density in a meridional

plane, for components with varyings . . . 119 5.10 The ratio _σ^σ^z

R of several components for varying s. . . . 120 5.11 Logarithm of the configuration space density as a function of the

height above the Galactic plane atR= 1 kpc for varyingδ. . . 121 5.12 Fit of a van der Kruit disk . . . 123 5.13 Values of the distribution function (corresponding to the fit of

the van der Kruit disk) as a function ofE for varyingI3 . . . . 124 5.14 Velocity dispersions for the fit of the van der Kruit disk . . . 124 5.15 Axisymmetric substructure in velocity space, for a model based

on real data (Dejonghe & Van Caelenberg 1999) . . . 127

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Summary

We study the motion of giant stars in the Solar neighbourhood and what they tell us about the dynamics of the Galaxy: we thus contribute to the huge project of understanding the structure and evolution of the Galaxy as a whole.

We present a kinematic analysis of 5952 K and 739 M giant stars which in- cludes for the first time radial velocity data from an important survey performed with the CORAVEL spectrovelocimeter at theObservatoire de Haute Provence.

Parallaxes from the Hipparcos catalogue and proper motions from the Tycho-2 catalogue are also used.

A maximum-likelihood method, based on a bayesian approach, is applied to the data, in order to make full use of all the available stars, and to derive the kinematic properties of the subgroups forming a rich small-scale structure in velocity space. Isochrones in the Hertzsprung-Russell diagram reveal a very wide range of ages for stars belonging to these subgroups, which are thus most probably related to the dynamical perturbation by transient spiral waves rather than to cluster remnants. A possible explanation for the presence of young group/clusters in the same area of velocity space is that they have been put there by the spiral wave associated with their formation, while the kinematics of the older stars of our sample has also been disturbed by the same wave. The emerging picture is thus one of dynamical streams pervading the Solar neighbourhood and travelling in the Galaxy with a similar spatial velocity. The term dynamical stream is more appropriate than the traditional term supercluster since it involves stars of different ages, not born at the same place nor at the same time. We then discuss, in the light of our results, the validity of older evaluations of the Solar motion in the Galaxy.

We finally argue that dynamical modeling is essential for a better understanding of the physics hiding behind the observed kinematics. An accurate axisymmetric model of the Galaxy is a necessary starting point in order to understand the true effects of non-axisymmetric perturbations such as spiral waves. To establish such a model, we develop new galactic potentials that fit some fundamental parameters of the Milky Way. We also develop new component distribution functions that depend on three analytic integrals of the motion and that can represent realistic stellar disks.

This thesis has led to the publication of three papers inMonthly Notices of the Royal Astronomy Societyand inAstronomy and Astrophysics:

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Famaey B., Van Caelenberg K., Dejonghe H., 2002, MNRAS, 335, 201 (15 pages). Three-integral models for axisymmetric galactic discs. (Chapter 5 of this thesis)

Famaey B., Dejonghe H., 2003, MNRAS, 340, 752 (11pages). Three-component St¨ackel potentials satisfying recent estimates of Milky Way parameters. (Chap- ter 4 of this thesis)

Famaey B., Jorissen A., Luri X., Mayor M., Udry S., Dejonghe H., Turon C., 2004, A&A, accepted with minor revisions (22 pages). Local kinematics of K and M giants from CORAVEL/Hipparcos/Tycho-2 data. Revisiting the concept of superclusters. (Chapters 2 and 3 of this thesis)

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Chapter 1

Introduction

A better understanding of the Universe as a whole starts with a better understanding of the Galaxy, i.e. the galaxy that we live in and commonly call the Milky Way (the name given to the luminous band crossing the sky). The Galaxy has remained a mystery for a surprisingly long time: Kant (1755) was the first to propose in his “Universal Natural History and the Theory of Heavens” that, by analogy with the planets of the Solar system, the Milky Way could be composed of stars orbiting in a finite flat system, and was maybe not the only finite stellar system of this type. Nevertheless, his hypothesis that some nebulae (like the Andromeda M31 nebula) could be similar stellar systems was confirmed only 80 years ago by Hubble (1925). Today, we know that the Milky Way is just an ordinary galaxy among billions of galaxies in the Universe.

1.1 Components of the Galaxy

Even today, controversy exists on the exact constituency of the Galaxy: roughly, its mass is of the order of several hundred billions Solar masses (1M = 1.99× 10³⁰kg), and it is composed of interstellar gas (representing a mass of 10¹⁰M), dark matter (about 90% of the total mass), and about 10¹¹ stars. The Hubble type (Hubble 1936, Sandage 1961) of the Galaxy seems to be something like SABbc (de Vaucouleurs & Pence 1978), i.e. a spiral barred galaxy with four spiral arms and a weak bar. However, this is still very uncertain and is the subject of a debate. Anyway, the Galaxy can be subdivided into five components, which are described hereafter.

1.1.1 The luminous halo

A luminous stellar halo surrounds the Galaxy. It is spheroidal with a radius of the order of 50 kpc (1 pc = 3.086×10¹⁶m = 3.26 light years), and represents only a small fraction of the total mass of the Galaxy (about 1%). It may have formed before all the other components (Eggen et al. 1962), or may have been

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accreted much later (Searle & Zinn 1978), but most probably results from both processes (Chiba & Beers 2000, 2001). It contains the most metal-poor stars in the Galaxy (as metal-poor as [Fe/H]= log_(Fe/H)^Fe/H

= −3.5, see Ryan et al.

1996, and even some with [Fe/H]= −5, see Christlieb et al. 2002) and has no metallicity gradient. Most halo stars are concentrated in globular clusters of 10² to 10⁵ stars (see Fig. 1.1).

1.1.2 The dark halo

The luminous halo is surrounded by a dark halo or corona, with a radius of the order of 200 kpc. It must dominate the mass of the Galaxy (about 90% of the total mass) in order to explain the constant circular velocity (the “flat” rotation curve) of the stars far from the center of the Galaxy. The determination of the mass distribution of this dark halo, and the determination of its composition (the famous and mysterious dark matter) is a major challenge of modern astronomy, and of modern physics in general. Dark matter is probably mainly non-baryonic, although a small part of it could be composed of “missed” stars (brown dwarfs).

The motion of galaxies in the Local Group suggests that the dark halo ultimately touches that of the Andromeda M31 galaxy.

1.1.3 The bulge

The central bulge (see Fig. 1.1) is spheroidal and could be the inner extension of the luminous halo (Carney et al. 1990). Much of what we know about it is based on the properties of stars in Baade’s windows (small areas in the sky which are almost free of obscuring dust), but considerable progress on the determination of its structural properties has been achieved through observations in the infrared.

Mainly, its triaxial structure is now an evidence (Binney et al. 1997) and its elongated component is called the bar, with a semi-major axis of the order of 3 kpc. The bulge abundance distribution is rather broad, with a mean [Fe/H]=

−0.25 and a spread that goes from [Fe/H]=−1.25 to [Fe/H]= +0.5 (Mc William

& Rich 1994). In the direction of the center of the Galaxy, there is a compact radio source called SgrA*, related to a high concentration of mass at the very center of the system: a careful study of the motion of stars in the dense star cluster near the center by Genzel et al. (2000) has enabled them to identify this high concentration of mass as a supermassive black hole of 2.6±0.2×10⁶M

(Sch¨odel et al. 2002).

1.1.4 The thin disk

Most stars are concentrated in a roughly axisymmetric disk with a radius of the order of 15 kpc, separated into a thin and a thick component. The thin disk is the major stellar component of the Galaxy. It rotates rapidly with a circular velocity at the Solar radius of 220 km s⁻¹, and contains stars of a wide range of ages. Most of the thin disk stars in the Solar neighbourhood have Solar metallicities of [Fe/H]> −0.2 (Edvardsson et al. 1993) and there is a

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metallicity gradient of the order of −0.07 kpc⁻¹ (which means that the inner Galaxy is more metal-rich than the Solar neighbourhood). The link between age, velocity and chemical composition of stars is one of the keys to understand the chemico-dynamical evolution and enrichment history of the disk, which are still poorly known. The thin disk contains the youngest stellar populations, which are located in the famous spiral arms (non-axisymmetric features, see Fig. 1.1).

They emit intensive blue light but represent only a very small fraction of the axisymmetric disk. The number of spiral arms in the disk is still not known with certainty (2 or 4 or even more, see Drimmel 2000). Moreover, the origin, the dynamics and the true nature (stationary or transient) of those spiral arms, as well as their role in star formation process is still the subject of active research in Galactic astronomy.

1.1.5 The thick disk

The existence of the thick disk as a separate stellar component, with about ten times less stars than the thin disk, is well documented today (see e.g. Ohja et al. 1994, Chen et al. 2001). It is not only thicker than the thin disk but is also composed of older and more metal-poor stars, with a mean [Fe/H]=−0.8 (see Robin et al. 2003, Gilmore et al. 1995), moving with a wider dispersion of velocities. It is generally thought to have been formed by dynamical heating of the early thin disk via a merger event with a satellite galaxy (Quinn et al.

1993). However, some authors claim that it has been a step in the formation of the disk (Samland et al. 1997), and that it is connected with the luminous halo (Norris 1996), or that it is related to the bulge (de Grijs & Peletier 1997). In fact, there could be two different thick disks: a very thick one (Chiba & Beers 2000, Gilmore et al. 2002) and a somewhat thinner one (Soubiran et al. 2003), maybe formed by different phenomena.

1.2 Stellar dynamics

A better understanding of the structure of all these components and of the manner they have been formed needs theoretical investigations in the field of

“stellar dynamics”, i.e. the theory of the motion of stars in any gravitationally bound system. Generally, the behavior of a stellar system is solely determined by Newton’s laws of motion and Newton’s law of gravity. General relativistic effects are unimportant, unless we study the motion very close to the horizon of a black hole (like the central supermassive blackhole in the Galaxy).

1.2.1 Relaxation time

The number of stars in the Galaxy is so large that a statistical treatment of the dynamics can prove very useful: stellar dynamics considers the Galaxy as a

“gas” of stars, but without collisions since gravity is a long-range force. Indeed, in a galaxy of 10¹¹ stars, Chandrasekhar (1942) showed that the time for a

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Figure 1.1: Schematic view of the Galaxy (p. 390 of Zeilik 2002, Cambridge University press), showing its main features: halo, bulge and disk. The Sun lies in the disk, on the inner edge of a spiral arm. The realm of the globular clusters defines the luminous halo, shown here only in part. The nuclear bulge in the center surrounds the core, recently identified as a central supermassive black hole

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stellar orbit to be deflected by a tenth of a right angle (i.e. 9^◦) is of the order of 100 rotations of the galaxy. This “relaxation time” represents about the limit of time during which collisional effects can be regarded as negligible. It is also of the order of the estimated age of the Universe: for this reason, the Galaxy is described as a collisionless system, in which stars can be approximated as statistically independent particles moving under the influence of a global gravitational potential.

1.2.2 Hierarchical formation scenario

The collisionless description of the Galaxy is valid at present time, but it was not at the time of Galaxy formation, when gas dynamics, collisions and encounters were very important: for this reason, Galaxy formation is very hard to model, and moreover we are not even sure about the prevailing physical conditions.

Today, there is no single widely accepted theory of Galaxy formation, but the most plausible scenario is the hierarchical “bottom-up” model: the fluctuations observed in the cosmological microwave background (emitted by the baryonic matter 3×10⁵ yr after the “Big Bang”) could be linked with more important fluctuations of the density of non-baryonic dark matter (that is thought to be

“cold”, i.e. composed of particles more massive than 1GeV/c²). These inho- mogeneities of the non-baryonic matter would be responsible for the formation of dark matter halos of about 10⁶M. These halos then merged together and gained angular momentum by tidal forces: the baryonic matter into them cooled down by radiation and formed disks by conservation of the angular momentum (for a more detailed description of the hierarchical bottom-up scenario, refer to Devriendt & Guiderdoni 2003). Once this gaseous disk was formed in our Galaxy, mergers with other galaxies still continued to happen regularly but with a decreasing rate. It is known since the discovery of the absorption of the Sagittarius dwarf galaxy by the Milky Way (Ibata et al. 1994) that some star streams in the Galaxy are remnants of a merger with a satellite galaxy. Helmi et al. (1999) showed that some debris streams are also present in the galactic halo near the position of the Sun. However, if we do not consider specific episodes of the Galactic evolution such as the climax of a major merger, the dynamical collisionless description of the Galaxy is valid.

1.2.3 Boltzmann and Poisson equations

The Galaxy is completely described by its distribution function F(~x, ~v, t), i.e.

the density in 6-dimensional phase space (~x, ~v) as a function of time t. In a collisionless system, bodies do not jump from one point to another in phase space, but rather move continuously into it. As a consequence,F(~x, ~v, t) must be a solution of a continuity equation, the so-called “collisionless Boltzmann equation”:

dF(~x, ~v, t)

dt = 0. (1.1)

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The Newton’s equations of motion write:

d~x dt =~v

d~v

dt =−^∂Φ(~_∂~^x,t)_x (1.2)

where Φ(~x, t) is the gravitational potential. Its opposite ψ(~x, t) = −Φ(~x, t) is also often used for more simplicity in the equations. The introduction of these laws of motion in the collisionless Boltzmann equation (1.1) yields the equation determining the evolution ofF in a given potential Φ:

∂F

∂t +

~ v .∂F

∂~x

− ∂Φ

∂~x.∂F

∂~v

= 0 (1.3)

The Poisson equation (derived from Newton’s law of gravity) relates the gravitational potential Φ to the mass distributionρthat generates it:

∆Φ(~x, t) = 4πGρ(~x, t) (1.4) whereGdenotes the gravitational constant.

If the distribution function describes the mass distribution in phase space, it is related to the mass density ρ(~x, t) in configuration space by the integral equation:

Z Z Z

Fmass(~x, ~v, t) d³~v=ρ(~x, t) (1.5) When we deal with such a distribution function representing the whole system, Equations (1.3), (1.4) and (1.5) form the set of Vlasov equations defining a self-consistent model.

Nevertheless, each separate stellar component can also be described by a distribution function related to the number of stars density g(~x, t) of the component by:

Z Z Z

F_stars(~x, ~v, t) d³~v=g(~x, t) (1.6) In that case, the model is not self-consistent since the set of stars described by F does not generate the potential in which it evolves (it generates only a part of it).

All the observables such as the number density, the mean velocity and the velocity dispersions of the stellar component can be expressed in terms of the moments of this distribution function. Eq. (1.6) corresponds to the 0^th order moment of the distribution function F. Multiplying the integrand of the left- hand side of (1.6) by the velocity or the square velocity in any direction yields respectively a first or a second order moment corresponding to the mean velocity or the velocity dispersion of the component in that direction.

1.2.4 Integrals of the motion

One can derive from the equations of the motion (1.2) that certain quantities depending on (~x, ~v, t) must be conserved along the orbit of any star, depending

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on the adopted potential Φ (the initial conditions (~x₀, ~v₀) att= 0 as a function of the position~xand velocity~vat timet, for example). Such a quantity is called a constant of the motion; if it depends only on the phase space coordinates (~x, ~v) and not ont, it is an integral of the motion. For any orbit, there are six independent constants, but not all are integrals of the motion. The integrals of the motion, in turn, come in two flavours: non-isolating and isolating. An integral of the motionIn is called isolating if it confines the orbit to a (6−n)- dimensional region on the (6−n+ 1)-dimensional surface in phase space defined by the isolating integralsI1, ..., I_n−1.

The Jeans theorem (Jeans 1915) states thatany time-independent function of the integrals of the motion F(I1, ..., In) is automatically a valid distribution function to describe a stationary stellar component of the Galaxy, since it is a solution of the collisionless Boltzmann equation (1.1) by definition of the integrals of the motion:

dF(I1(~x, ~v), ..., In(~x, ~v))

dt = ∂F

∂I1

.dI1(~x, ~v)

dt +...+ ∂F

∂In

.dIn(~x, ~v)

dt = 0 (1.7) Integrals of the motion are associated with symmetries of the system (following the Noether theorem). The two classical independent isolating integrals of the motion in a Galactic model are the vertical component of the angular momentum and the binding energy (the opposite of the energy, i.e. the Hamil- tonian, which is itself an integral of the motion) if we make the assumption that the Galaxy is axisymmetricandstationary(time-symmetric). Indeed, the non- axisymmetric part of the system (the bar and the spiral arms) represents less than 5 % of the total mass and, moreover, throughout the visible Galaxy, the dynamical time ('an orbital period '10⁸ yr) is orders of magnitude shorter than the Hubble time (' order of the age of the Universe ' 10¹⁰ yr), leading to the steady-state approximation since we have no reason to suppose the present day is a particularly exciting moment in the Galaxy’s life, such as the climax of a major merger. In a stationary and axisymmetric gravitational potential Φ(R, z)=−ψ(R, z) (where (R, φ, z) are the cylindrical coordinates), the two classical isolating integrals of the motion are thus

E=−H =ψ(R, z)−1

2(v²_R+v_φ²+v_z²) (1.8) and

L_z=Rv_φ=R²φ.˙ (1.9)

These two integrals of the motion have been used to construct distribution functionsF(E, Lz) that are stationary solutions of the Boltzmann equation (1.1) and that can describe thin axisymmetric disks (Shu 1969, Batsleer & Dejonghe 1995, Bienaym´e & S´echaud 1997) or bulge-disk systems (Jarvis & Freeman 1985, Kent 1991). This two-integral approximation is nevertheless not sufficient to adequately describe the stellar disk: it is a fundamental property of all two- integral distribution functions that the dispersion of the velocity in the radial

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directionσ_R equals the dispersion in the vertical directionσ_z. Indeed, Z Z Z

F(E, L_z)v_Rd³~v= Z Z Z

F(E, L_z)v_zd³~v= 0 (1.10) becauseE andLz are quadratic in vR andvz. And moreover,

gσ²_R= Z Z Z

F(E, Lz)v_R² d³~v= Z Z Z

F(E, Lz)v_z²d³~v=gσ²_z (1.11) because E depends on v²_R and v²_z in the same manner (see Eq. (1.8)), and becauseL_zdoes not depend on them. We know that in the galactic diskσ_R> σ_z (see e.g. Binney & Merrifield 1998; see also Eqs (3.8) and (3.18) of the present thesis), which is in contradiction with the two-integral approximation.

1.2.5 The third integral

Fortunately, in realistic axisymmetric potentials (obtained by solving the Pois- son Eq. (1.4) for realistic mass distributions), it is usually found that a third isolating integral is conserved along the orbits, that are thus “regular”¹: in fact, in realistic galactic potentials, most chaotic orbits are “quasi-regular” and are confined close to regular orbits for a time comparable to the age of the Universe (Contopoulos 1960, Ollongren 1962, Innanen & Papp 1977, Richstone 1982).

This additional quantity conserved along the orbits is called thethird integral of galactic dynamics: when there is an analytic formulation for it, it can be interpreted as the scalar product of the angular momenta about two fixed foci (Lynden-Bell 2003).

Nevertheless, in most Galactic potentials there exists no analytic formulation for the third integral, and it is thus a numerical quantity. It can be taken into account numerically in models of the Galaxy by integrating the equations of motion (1.2) and by using an orbit superposition technique to fit the observations (Schwarzschild 1979, Cretton et al. 1999, Zhao 1999, H¨afner et al. 2000).

But if one wants to model the Galaxy with analytic distribution functions, it is possible to define an approximate third integral for nearly-circular orbits (see Section 1.2.6), or specific to some particular orbital families (de Zeeuw et al.

1996, Evans et al. 1997). It is also possible to foliate phase space with tori on which numerical action-angle variables can be constructed (Mc Gill & Binney 1990, Kaasalainen & Binney 1994, Binney 2002), or to use specific potentials (Lynden-Bell 1962) in which an exact analytic third integral exists for all orbits (see Chapter 4). The strong Jeans theorem then states that when almost all orbits (except a set of null volume in phase space) are regular with incommen- surable frequencies, the steady-state distribution function is a function only of three independent isolating integrals. The proof of this theorem can be found in Appendix 4.A of Binney & Tremaine (1987). Any stellar component of the

1A “regular” orbit is an orbit that has as many isolating integrals as degrees of freedom.

The Fourier spectrum of such an orbit is discrete. A non-regular orbit is called “chaotic”, but if the Fourier spectrum of a chaotic orbit is quasi-discrete, the orbit is called “quasi-regular”.

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steady-state Galaxy can thus be represented by a distribution function of the form F(I₁, I₂, I₃), and more precisely, in the axisymmetric case, by a distribution function of the form F(E, L_z, I₃). Such a stationary and collisionless configuration of the system is called anequilibriumconfiguration. It is in fact a quasi-equilibrium configuration, since there is no maximum-entropy state in a purely gravitational system (see Section 4.7 of Binney & Tremaine 1987).

1.2.6 Galactic orbits

Thanks to the strong Jeans theorem, any stellar component of the Galaxy can be represented by a set of fixed points in integral space rather than by a set of moving points in real space. Indeed, every triple (E, Lz, I3) in integral space represents an orbit in the Galaxy. As noticed in the previous section, the numerical nature of the third integral in most potentials makes it difficult to deal with, but the structure of the (E, Lz)-plane can be studied analytically in every potential.

With the axisymmetric assumption, we can focus on the motion in a meridional plane (i.e. a rotating (R, z)-plane for a givenLz). For a position (R0, z0) in the meridional plane, the expressions (1.8) and (1.9) forE andLzimply that all families of orbits that visit this position have isolating integrals of the motion (E, L_z) that meet the requirement

E≤ψ(R₀, z₀)− L²_z

2R²₀ (1.12)

since we know that

v²_R+v_z²≥0. (1.13)

For givenE andLz, this relation (1.12) restricts possible motion for the corresponding family of orbits to a toroidal volume in configuration space. We define the right-hand side of the inequality (1.12) as the effective potential for a given angular momentum Lz:

ψeff(R, z) =ψ(R, z)− L²_z

2R² (1.14)

With this definition, Newton’s equations of motion describing the evolution ofR andzdo not depend on the azimuthal angleφ:

d²R dt² = ^∂ψ_∂R^eff

d²z dt² = ^∂ψ_∂z^eff

(1.15)

Lindblad diagram

In the (E, L²_z)-space (called the Lindblad diagram, Fig. 1.2), Eq. (1.12) defines the region in which the points correspond to families of orbits passing through (R₀, z₀). The boundary line (equality in Eq. (1.12)) contains orbits that reach the given position with zero velocity (1.13) in the meridional plane. Keeping

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z=z₀ fixed while allowing R to vary then gives us a family of such boundary lines, of which we denote the envelope by

E=S_z₀(L_z). (1.16)

This envelope is defined by the following parametric equations (with parameter R, Batsleer & Dejonghe 1995, Eq. 4 & 5)

E=ψ(R, z0)− L²_z

2R² (1.17)

L²_z=−R³∂ψ

∂R(R, z0). (1.18)

Eq. (1.17) defines the family of boundary lines (equality in Eq. (1.12)), and for every L²_z, the highest binding energyE of all the boundary lines (1.17) has been taken (^∂E_∂R = 0), leading to Eq. (1.18). This envelope Sz₀(Lz) in the (E, L²_z)-plane is plotted on Fig. 1.2.

All points in integral space with E < Sz₀(Lz) represent families of orbits that will pass throughz=z0at a certainR. Orbits for whichE=Sz₀(Lz) also do reach the heightz0, but can never go any higher. All points in integral space with E > Sz0(Lz) represent families of orbits that cannot reach z = z0. Sz0

is thus the minimal binding energy of an orbit that cannot bring a star higher than z₀ above the galactic plane. For every height z₁ > z₀ we find a similar curveE=S_z₁(L_z), withS_z₁(L_z)< S_z₀(L_z) for every value ofL_z.

Similarly, the envelope for the orbits that are confined in the galactic plane, i.e. that cannot go higher thanz= 0, is given byE=S₀(L_z). This curve gives us all circular orbits in the galactic plane, since Eq. (1.18) then becomes

L²_z

R³ =−∂ψ

∂R(R,0). (1.19)

Eq. (1.19) is the condition for a circular orbit of angular momentumLz, since the radial force (the only component of the force in the galactic plane since this plane is a plane of symmetry) is then equal to the centripetal acceleration ^L_R²^z3. Epicyclic motion

Orbits belonging to a disk with a maximum heightz0are thus given by (E, Lz) for whichSz₀(Lz)≤E ≤S0(Lz) (the shaded area in Fig. 1.2), and the thinner the disk is, the closer to circular the orbits are.

For such nearly circular orbits, which are representative of the thin disk, approximate solutions of the equations of motion (1.15) can be found. The solution of Eq. (1.19), i.e. the radius of the circular orbit of angular momentum L_z, is called the guiding radiusR_gand we definex=R−Rg. Then, if we expand the effective potential ψ_eff defined by Eq. (1.14) in a Taylor series about the

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Figure 1.2: Lindblad diagram. The shaded area is the area in the (E, L²_z)-plane for which the corresponding families of orbits cannot go higher than z₀ above the galactic plane. The distribution function of a stellar disk with maximum height z₀ is null outside of this area. The thinner the disk is, the thinner the allowed area in the (E, L²_z)-plane is, and the closer to circular the orbits are.

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point (R_g,0) and suppose thatxandzare small (negligible at the third order), the equations of motion (1.15) become

d²x

dt² =−κ²x

d²z

dt² =−ν²z (1.20)

i.e. the equations of two harmonic oscillators with the epicycle frequency κ²(R_g)≡ −∂²ψeff

∂R² _(R

g,0)

(1.21) and the vertical frequency

ν²(R_g)≡ −∂²ψ

∂z² _(R

g,0)

(1.22) The stellar motion is thus decomposed into a vertical oscillation and a motion parallel to the Galactic plane, itself divided into azimuthal streaming and epicyclic libration (the star moves on an ellipse called the epicycle around the guiding center). The separation of the vertical motion from the motion parallel to the plane is called the Oort-Lindblad approximation, but it is not valid very far outside of the plane. In the plane, the azimuthal streaming dominates over the radial epicyclic libration, so that the disk is said to be dynamically cold.

The application of this epicyclic approximation to the analytic expression of a third integral is straightforward. Indeed, if the star’s orbit is sufficiently nearly circular that the truncation of the Taylor series forψeff at the third order is justified, then the orbit admits three independent isolating integrals of the motion (R, z, Lz) where the epicycle energyR and the vertical energy z are the energies of the oscillators (1.20):

R=1

2(v²_R+κ²(R−Rg)²), (1.23) z=1

2(v²_z+ν²z²). (1.24) Nevertheless, the epicyclic motion is only an approximation and more precise approaches of the third integral have been cited in Section 1.2.5, and will be developed in Chapters 4 and 5 of this thesis.

1.2.7 Non-axisymmetric perturbations

The axisymmetric assumption is of course only an approximation. Indeed, it is well known that the bulge has a non-axisymmetric component called the bar (Section 1.1.3), and that there are spiral arms in the Milky Way disk (Section 1.1.4).

The potential remains stationary if we refer everything to its rotating frame:

the planar orbits in a stationary non-axisymmetric potential are either “box orbits” passing close to every point inside a rectangular box, with no particular

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sense of circulation about the center, or “loop orbits”, i.e. elliptic orbits rotating about the center while oscillating in radius (as the nearly circular orbits in an axisymmetric potential).

A perturbed galaxy is a galaxy with a weak non-axisymmetric component (i.e. when the amplitude of the non-axisymmetric part of the potential is less than 5% of the axisymmetric part), which is the case of the Milky Way. In that case, a linear perturbation analysis can be performed, i.e. we consider that

Φ(R, φ, z, t) = Φ0(R, z) +Φ1(R, φ, z, t) (1.25) with << 1. The axisymmetric approximation is thus a necessary starting point for this kind of analysis, and is a prerequisite if one wants to analyse the bar and the spiral arms on a theoretical basis.

The perturber is said to havem-fold symmetry if its pattern is invariant for rotation over 2π/m. Hence the bar has 2-fold symmetry and is also said to be an “m= 2 mode”, while spiral arms arem= 2 orm= 4 modes (the number of arms in the Milky way is still rather uncertain).

Resonances

In any system, resonances can occur between the driving frequency ωp of a perturber (subscript p denotes the perturber) and the natural frequencies of the system. The usual effect of a resonance is growth in the amplitude of the oscillation of the system, because the driving force acts in phase with the natural oscillation.

In the Galaxy, the standard natural frequencies of stars are the epicycle frequency κ and the vertical frequency ν defined in Eqs. (1.21) and (1.22), while the driving frequency of a perturber of m-fold symmetry is

ωp≡m(Ωp−Ω) (1.26)

where Ωp is the rotational frequency of the perturber and Ω = ^v_R^c the angular velocity of circular orbits (of velocityvc) in the axisymmetric Galaxy. We speak of “resonances” when the stars encounter successive density peaks at a frequency that coincides with the frequency of their natural radial or vertical oscillations, i.e. when

ωp(R) = 0 (corotation) ωp(R) =±κ(R) (radial Lindblad) ωp(R) =±ν(R) (vertical Lindblad)

(1.27) At corotation, the guiding center of the stars corotates with the perturbed potential. The minus signs in Eq. (1.27) correspond to radii inside corotation, called inner Lindblad radii, while the plus signs correspond to radii outside corotation, called outer Lindblad radii. In practice, the name “Lindblad resonances” is used to refer to them= 2 radial Lindblad resonances (ILR and OLR of the bar or of a 2-armed spiral). Depending on the exact shape of the unper- turbed potential, there can be zero, one or two solutions for the inner Lindblad