UNIVERSITE LIBRE DE BRUXELLES Facult´edes 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 2 Notre savoir consiste en grande partie `acroire savoir, et `acroire que d’autres 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)

3 4 Acknowledgements

Je voudrais avant tout remercier le professeur Marcel Arnould, Directeur de l’Institut d’Astronomie et d’Astrophysique, qui m’a accueilli `al’Institut en tant que math´ematicien et qui m’a permis de passer ces ann´ees de th`esedans des conditions id´ealesen m’accordant une grande confiance et une grande libert´e de travail et en se montrant d’une aide attentive face `ames interrogations et sollicitations. Je remercie ensuite avec beaucoup de gratitude mon directeur de th`ese, le professeur Alain Jorissen, pour sa disponibilit´econstante durant ces ann´ees de doctorat et pour les nombreuses discussions stimulantes que nous avons eues. Au cours de celles-ci j’ai ressenti une v´eritable ´emulation, son esprit scientifique aiguis´een faisant non seulement un chercheur hors pair mais surtout un inter- locuteur extrˆemement int´eressant et motivant. Je le remercie en outre, par del`a ces consid´erations professionnelles, pour ses qualit´eshumaines hors du commun. Je voudrais ´egalement exprimer ma gratitude `atous les autres membres de l’Institut, en particulier `aDimitri et Marc pour leur soutien face aux al´eas de l’informatique, `aCarine et Laurent pour m’avoir fourni quelques r´ef´erences int´eressantes, `aSylvie, Sophie, Ana, Claire et Viviane pour leur sourire et l’agr´eable atmosph`ere de travail qu’elles contribuent `acr´eer `al’Institut, `aYves, Abdel, St´ephane, Lionel et Matthieu pour leur bonne humeur. Merci aussi `aSamir Keroudj pour la r´ealisation de l’animation en trois dimensions qui a permis d’illustrer certains r´esultats de cette th`ese. Il est tr`es important pour moi de remercier ici le professeur Herwig De- jonghe, de l’Universit´ede Gand, v´eritable instigateur de ce projet, sans lequel rien n’aurait ´et´epossible. Ses explications, toujours claires et pr´ecises, m’ont permis de me constituer au fil du temps une expertise en dynamique galactique, domaine qui ´etaitenti`erement neuf pour moi au moment d’aborder ces ann´ees de th`ese. Je remercie aussi Kathrien Van Caelenberg qui avait entam´eune partie de ce travail `al’Universit´ede Gand. Une grande partie des r´esultatspr´esent´es dans cette th`ese d´ependent de donn´eesobtenues grˆace`al’Observatoire de Gen`eve et `ala coop´erationde nom- breux observateurs anonymes que je remercie pour leur importante contribu- tion `ace travail. Je remercie tout particuli`erement le professeur Michel Mayor ainsi que Catherine Turon d’avoir accept´ede me fournir ces donn´ees. Je re- mercie ´egalement St´ephane Udry pour son hospitalit´elors de mon s´ejour `a l’observatoire de Gen`eve et sa collaboration active lors du d´ebroussaillement

5 6 des donn´ees CORAVEL. Je suis ´egalement extrˆemement reconnaissant `aXavier Luri, de l’Universit´e de Barcelone, qui m’a accueilli chaleureusement lors de mon s´ejour dans son service et qui fut toujours d’une aide radicalement efficace et d’une disponibilit´e sans faille dans l’analyse des donn´ees cin´ematiques qui a conduit aux principaux r´esultats de ce travail. Je voudrais aussi remercier mes parents pour leur soutien constant tout au long de mes ´etudes 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, St´ephane, Bernard, Benoˆıt,Jules, Hugo et Mohamed pour leurs commentaires de non sp´ecialistes de la discipline et leur soutien moral. Merci enfin `ama douce Marie-Laure, merci de m’avoir soutenu et encourag´e,et surtout merci d’ensoleiller ma vie chaque jour. Contents

1 Introduction 17 1.1 Components of the Galaxy ...... 17 1.1.1 The luminous halo ...... 17 1.1.2 The dark halo ...... 18 1.1.3 The bulge ...... 18 1.1.4 The thin disk ...... 18 1.1.5 The thick disk ...... 19 1.2 Stellar dynamics ...... 19 1.2.1 Relaxation time ...... 19 1.2.2 Hierarchical formation scenario ...... 21 1.2.3 Boltzmann and Poisson equations ...... 21 1.2.4 Integrals of the motion ...... 22 1.2.5 The third integral ...... 24 1.2.6 Galactic orbits ...... 25 1.2.7 Non-axisymmetric perturbations ...... 28 1.3 The Solar neighbourhood ...... 36 1.3.1 Galactocentric radius of the Sun ...... 37 1.3.2 Rotation curve and Oort constants ...... 37 1.3.3 Local dynamical mass ...... 39 1.3.4 LSR, Solar motion and velocity ellipsoid ...... 41 1.3.5 Vertex deviation and substructure of velocity space . . . . 42 1.4 Outline of this thesis ...... 44

2 Stellar sample 47 2.1 Selection criteria ...... 49 2.2 Binaries ...... 50

3 Kinematic analysis 57 3.1 Analysis restricted to stars with the most precise parallaxes . . . 58 3.2 Monte Carlo simulation ...... 60 3.3 Bayesian approach ...... 63 3.3.1 Phenomenological model ...... 64 3.3.2 Observational selection and errors ...... 65 3.3.3 Maximum likelihood ...... 66

7 8

3.3.4 Group assignment ...... 67 3.3.5 Individual distance estimates ...... 69 3.3.6 The kinematic groups present in the stellar sample . . . . 69 3.3.7 Physical interpretation of the groups ...... 82

4 St¨ackel potentials 89 4.1 Coordinate system ...... 90 4.2 Three-component St¨ackel potentials ...... 90 4.3 Selection criteria ...... 93 4.4 The “winding staircase” ...... 95 4.5 Constraints on the scale height of the thick disk ...... 95 4.6 The final selection ...... 97

5 Three-integral distribution functions 107 5.1 Construction of three-integral components ...... 108 5.2 Moments ...... 109 5.2.1 The case where a = 0 and m is an even integer ...... 109 5.2.2 The general case ...... 116 5.3 Physical properties of the components ...... 116 5.3.1 The parameter z0 ...... 117 5.3.2 The parameter α1 ...... 117 5.3.3 The parameter α2 ...... 117 5.3.4 The parameter β ...... 120 5.3.5 The parameter η ...... 120 5.3.6 The parameter s ...... 120 5.3.7 The parameter δ ...... 123 5.4 Modeling ...... 126

6 Conclusions and perspectives 133

A Contents of the data table 137 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 UV -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 the UV -plane with their values of U 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

9 10

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 in Lz ...... 111 5.4 Contour plots of the mass density in a meridional plane, for com- ponents with z0 equal to 4 kpc and 2 kpc ...... 113 5.5 Logarithm of the galactic plane mass density of different compo- nents for varying α1 ...... 114 5.6 Logarithm of the configuration space density of different compo- nents for varying α2...... 115 5.7 Contour plots of the configuration space density in a meridional plane, for components with varying β ...... 116 5.8 Values of a component distribution function as a function of E for varying η ...... 118 5.9 Contour plots of the configuration space density in a meridional plane, for components with varying s ...... 119 5.10 The ratio σz of several components for varying s...... 120 σR 5.11 Logarithm of the configuration space density as a function of the height above the Galactic plane at R = 1 kpc for varying δ. . . 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 of E for varying I3 . . . . 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 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 the Observatoire 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 neigh- bourhood 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 under- standing 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 compo- nent 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 in Monthly Notices of the Royal Astronomy Society and in Astronomy and Astrophysics:

11 12

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) Chapter 1

Introduction

A better understanding of the Universe as a whole starts with a better under- standing 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 × 30 10 10 kg), and it is composed of interstellar gas (representing a mass of 10 M ), 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 sub- ject 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

13 14 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 = −3.5, see Ryan et al. (Fe/H) 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 6 high concentration of mass as a supermassive black hole of 2.6 ± 0.2 × 10 M (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 15 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 16

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 17 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 6 of dark matter halos of about 10 M . 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) = 0. (1.1) dt 18

The Newton’s equations of motion write:

d~x = ~v dt (1.2) d~v ∂Φ(~x,t) dt = − ∂~x 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 of F in a given potential Φ: ∂F  ∂F  ∂Φ ∂F  + ~v. − . = 0 (1.3) ∂t ∂~x ∂~x ∂~v The Poisson equation (derived from Newton’s law of gravity) relates the gravitational potential Φ to the mass distribution ρ that generates it:

∆Φ(~x,t) = 4πGρ(~x,t) (1.4) where G denotes 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: ZZZ 3 Fmass(~x,~v, t) d ~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 com- ponent by: ZZZ 3 Fstars(~x,~v, t) d ~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 0th 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 19

on the adopted potential Φ (the initial conditions (~x0,~v0) at t = 0 as a function of the position ~x and velocity ~v at time t, 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 on t, 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 motion In 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 integrals I1, ..., In−1. The Jeans theorem (Jeans 1915) states that any 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 (I (~x,~v), ..., I (~x,~v)) ∂F dI (~x,~v) ∂F dI (~x,~v) 1 n = . 1 + ... + . n = 0 (1.7) dt ∂I1 dt ∂In dt

Integrals of the motion are associated with symmetries of the system (fol- lowing 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 axisymmetric and stationary (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), lead- ing 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 po- tential Φ(R, z)=−ψ(R, z) (where (R, φ, z) are the cylindrical coordinates), the two classical isolating integrals of the motion are thus

1 E = −H = ψ(R, z) − (v2 + v2 + v2) (1.8) 2 R φ z and 2 ˙ Lz = Rvφ = R φ. (1.9) These two integrals of the motion have been used to construct distribution functions F (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 20

direction σR equals the dispersion in the vertical direction σz. Indeed, ZZZ ZZZ 3 3 F (E,Lz) vR d ~v = F (E,Lz) vz d ~v = 0 (1.10) because E and Lz are quadratic in vR and vz. And moreover, ZZZ ZZZ 2 2 3 2 3 2 gσR = F (E,Lz) vR d ~v = F (E,Lz) vz d ~v = gσz (1.11)

2 2 because E depends on vR and vz in the same manner (see Eq. (1.8)), and because Lz does not depend on them. We know that in the galactic disk σR > σz (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 the third 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 formula- tion 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 observa- tions (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”. 21 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 distri- bution function of the form F (E,Lz,I3). Such a stationary and collisionless configuration of the system is called an equilibrium configuration. 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 nu- merical 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 merid- ional plane (i.e. a rotating (R, z)-plane for a given Lz). For a position (R0, z0) in the meridional plane, the expressions (1.8) and (1.9) for E and Lz imply that all families of orbits that visit this position have isolating integrals of the motion (E,Lz) that meet the requirement

2 Lz E ≤ ψ(R0, z0) − 2 (1.12) 2R0 since we know that 2 2 vR + vz ≥ 0. (1.13)

For given E and Lz, this relation (1.12) restricts possible motion for the corre- sponding 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: L2 ψ (R, z) = ψ(R, z) − z (1.14) eff 2R2 With this definition, Newton’s equations of motion describing the evolution of R and z do not depend on the azimuthal angle φ:

2 d R ∂ψeff dt2 = ∂R 2 (1.15) d z ∂ψeff dt2 = ∂z

Lindblad diagram

2 In the (E,Lz)-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 22

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 = Sz0 (Lz). (1.16)

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

L2 E = ψ(R, z ) − z (1.17) 0 2R2

∂ψ L2 = −R3 (R, z ). (1.18) z ∂R 0 Eq. (1.17) defines the family of boundary lines (equality in Eq. (1.12)), and 2 for every Lz, the highest binding energy E of all the boundary lines (1.17) has ∂E been taken ( ∂R = 0), leading to Eq. (1.18). This envelope Sz0 (Lz) in the 2 (E,Lz)-plane is plotted on Fig. 1.2.

All points in integral space with E < Sz0 (Lz) represent families of orbits that will pass through z = z0 at a certain R. Orbits for which E = Sz0 (Lz) also do reach the height z0, 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 curve E = Sz1 (Lz), with Sz1 (Lz) < Sz0 (Lz) for every value of Lz. Similarly, the envelope for the orbits that are confined in the galactic plane, i.e. that cannot go higher than z = 0, is given by E = S0(Lz). This curve gives us all circular orbits in the galactic plane, since Eq. (1.18) then becomes

L2 ∂ψ z = − (R, 0). (1.19) R3 ∂R

Eq. (1.19) is the condition for a circular orbit of angular momentum Lz, since the radial force (the only component of the force in the galactic plane since this 2 Lz plane is a plane of symmetry) is then equal to the centripetal acceleration R3 .

Epicyclic motion

Orbits belonging to a disk with a maximum height z0 are thus given by (E,Lz) for which Sz0 (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 Lz, is called the guiding radius Rg and we define x = R−Rg. Then, if we expand the effective potential ψeff defined by Eq. (1.14) in a Taylor series about the 23

2 Figure 1.2: Lindblad diagram. The shaded area is the area in the (E,Lz)-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 2 allowed area in the (E,Lz)-plane is, and the closer to circular the orbits are. 24

point (Rg, 0) and suppose that x and z are small (negligible at the third order), the equations of motion (1.15) become

d2x 2 dt2 = −κ x d2z 2 (1.20) dt2 = −ν z i.e. the equations of two harmonic oscillators with the epicycle frequency

∂2ψ 2 eff κ (Rg) ≡ − 2 (1.21) ∂R (Rg ,0) and the vertical frequency

∂2ψ 2 ν (Rg) ≡ − 2 (1.22) ∂z (Rg ,0) The stellar motion is thus decomposed into a vertical oscillation and a mo- tion 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 for ψeff at the third order is justified, then the orbit admits three independent isolating integrals of the motion (R, z,Lz) where the epicycle energy R and the vertical energy z are the energies of the oscillators (1.20): 1  = (v2 + κ2 (R − R )2), (1.23) R 2 R g 1  = (v2 + ν2 z2). (1.24) z 2 z 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 25 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 have m-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 are m = 2 or m = 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)

vc where Ωp is the rotational frequency of the perturber and Ω = R the angular velocity of circular orbits (of velocity vc) 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) (1.27) ωp(R) = ±ν(R) (vertical Lindblad) At corotation, the guiding center of the stars corotates with the perturbed po- tential. 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 reso- nances” is used to refer to the m = 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 26 resonance. The m = 4 radial Lindblad resonances are called “ultra-harmonic” resonances (IUHR and OUHR). The resonances play an important role in the shape of orbits in the perturbed Galaxy. Indeed, if we refer everything to the frame of a bar rotating like a solid body (in such a way that the line φ = 0 coincides with the long-axis of the bar), if we set R(t) = R0 + R1(t), where R0 is the radius of an unperturbed orbit, and φ(t) = −ωp t, so that R(φ) = R0 + R1(φ), and if we consider a planar loop orbit closing in the corotating frame, then a linear perturbation analysis yields:

R1(φ) = C cos(mφ). (1.28)

R1 is thus the radial excursion due to the perturber at azimuth φ. The coefficient C(R0) (see Eq. (3-120b) of Binney & Tremaine 1987) becomes infinite at the radii of the ILR, OLR and corotation, and across these resonances the sign of C changes. Depending on the sign of C the loop orbits will be elongated along or perpendicular to the major axis of the potential. Resonances thus have a special status: the shape of the orbits change at those radii, and moreover, the linear analysis is no more valid since they are singularities. Therefore, even a weak perturbation can produce a strong response at resonances. This response affects the features of the velocity distribution function that can have a bimodal character since both types of orbits (elongated along and perpendicular to the bar) can coexist at resonances (see e.g. Vauterin & Dejonghe 1997, Dehnen 1999, 2000).

The bar and its spiral response The process of formation of the galactic bar is still rather uncertain: in the first Gyrs of the life of the Galaxy, the disk was probably unstable (Fuchs & von Linden 1998), and it is well possible that the bar formed via a disk instability at least 5 Gyr ago. The presence of a supermassive black hole and its associated dense star cluster at the center of the Galaxy was thought to cause the destruction of the bar (e.g. Hasan et al. 1993), but recently, high-quality N-body simulations (Shen & Sellwood 2004) have shown that the bar is more robust than previously thought, and that it is in theoretical accordance with the presence of a supermassive black hole at the center. Bars always rotate like solid bodies, and their morphological properties in external galaxies are found to be related to the spiral Hubble type of their host galaxy (Elmegreen & Elmegreen 1985): bars of Sa and Sb galaxies have flat density profiles and end close to their corotation radius, while bars of Sbc and Sc galaxies have exponential density profiles and end well inside (some- times halfway) their corotation radius. The bar’s steep density profile in our Galaxy (Binney et al. 1997) indicates a Sbc or Sc type for the spiral pattern, in accordance with the Hubble type proposed by de Vaucouleurs & Pence (1978). This relation between the spiral pattern and the bar’s morphology, together with the fact that, in barred spiral galaxies, the spiral arms usually start at the end of the bar, suggests that the bar could be the origin of the spiral pattern. Indeed the origin and true nature of the spiral arms is still the subject of great 27 debate in Galactic astronomy. The simulations of Sanders & Huntley (1976) demonstrate that the formation of spiral arms requires only the presence of a rotating bar and a dissipative interstellar medium (which is not important for the global dynamics since it represents only a few percents of the total mass, but which is capital for the formation of such a spiral pattern). The response of the interstellar gas has the form of a spiral pattern in such simulations, but the reponse of stars does not (Sanders 1977): the Sanders-Huntley model is thus not suitable to explain that a spiral pattern is present in the old disk stars as well as in the young stars and gas in the Galaxy.

Quasi-stationary spiral waves On the other hand, the Lin-Shu (1964, 1966) hypothesis is that spiral structure consists of a self-sustained quasi-stationary density wave, that is, a density wave that is maintained in a steady state (in its associated rotating frame) over many rotation periods. A spiral arm can be seen as a curve:

m(φ − Ωpt) + f(R) = constant (modulo 2π) (1.29) where m is the number of arms and f the shape function of the spiral pattern. The linear perturbation analysis considers a small spiral perturbation Σ1 of the total surface mass density of the disk Σ0, i.e.

Σ(R, φ, t) = Σ0(R) +  Σ1(R, φ, t) (1.30) with  << 1. If we separate the rapid radial density variations as one passes between arms from the slower azimuthal variations (this is called the “tight-winding” approximation, see Fig. 1.3), we can write the perturbing spiral density Σ1 as im(φ−Ωpt) a “normal mode” Σa(R)e , and this density can be seen as a density wave, not made up of the same stars at different times. Lin & Shu (1966) calculated the gravitational potential Φ1 (Eq. (1.25)) induced by a normal mode

im(φ−Ωpt) Σ1(R, φ, t) = Σa(R)e , (1.31) i.e. im(φ−Ωpt) i f(R) im(φ−Ωpt) Φ1(R, φ, t) = Φa(R)e = A(R)e e (1.32)

They stated that the responding density to this perturbing potential Φ1 must be equal to Σ1, in such a way that the spiral pattern is self-sustained. For this purpose, they injected the potential Φ(R, φ, t) of Eq. (1.25) and the distribution function F (R, φ, vR, vφ, t) associated to the density Σ(R, φ, t) of Eq. (1.30) (assuming a 2-dimensional gaussian distribution of the velocities, see Section 1.3.4) in the linearized Boltzmann equation (Eq.(6A-1) of Binney & Tremaine 1987) and hence found the local “dispersion relation for galactic stellar disks”: df df ω2 = κ2 − 2πGΣ F(ω , κ, , σ ) (1.33) p 0 dR p dR R 28

Figure 1.3: The left panel represents a m = 2 tightly wound (T.W.) spiral pattern: the tight-winding approximation is valid since the radial density varia- tions are rapid in comparison to the azimuthal ones. The right panel represents a m = 2 loosely wound (L.W.) spiral pattern: in that case the tight-winding approximation is not valid.

where the function F is the “reduction factor” (always between 0 and 1), ωp is the driving frequency of the perturber defined in Eq. (1.26), κ is the epicyclic frequency defined in Eq. (1.21), G is the gravitational constant, Σ0 is the un- perturbed surface density of the disk from Eq. (1.30), f is the shape function of the spiral pattern from Eqs. (1.29) and (1.32), and σR is the velocity dispersion in the radial direction. This is the key equation for the local analytic study of quasi-stationary density waves and of disk stability. Indeed, a m = 0 pertur- bation is an axisymmetric perturbation, and Eq. (1.33) leads to the Toomre’ stability criterion (Toomre 1964) for axisymmetric galactic disks. The disk is lo- cally stable with respect to an axisymmetric perturbation (i.e. the perturbation is not growing exponentially) if σ κ Q ≡ R > 1 (1.34) 3.36GΣ0 The dispersion relation of Eq. (1.33) is also the condition for the existence of stable non-axisymmetric disturbances of an initially axisymmetric disk. Nu- merical experiments suggest that Q > 1 implies stability to all axisymmetric perturbations, but not to non-axisymmetric ones (where Q must be even higher to ensure stability). Eqs. (1.27) and (1.33) imply that stable m = 2 or m = 4 perturbations of a purely stellar disk can only occur between the ILR and the OLR or between the IUHR and OUHR of the perturber (because F ≥ 0). However, in gaseous disks, some density waves can cross the Lindblad resonances: indeed, Eq.(1.33) is slightly different for an ideal gas than for stars since a gas has a pressure support, which makes it more stable (because there is an additional positive term in the right-hand side of Eq. (1.33)). Moreover, Icke (1979) showed that 29 the dissipational nature of gas causes it to flow in the radial direction and conglomerate at certain radii even in the absence of perturbing forces. The stable behaviour of the gaseous component is thus mainly responsible for the observation of grand-design spiral patterns in the Milky Way and in external galaxies. This different behaviour of gas and stars could explain the observation of different spiral patterns in the Galaxy, depending on the age of the stars (young freshly formed stars are situated near the gaseous spiral arms where they were born). The response of gas implies that the Lin-Shu hypothesis could be an accept- able explanation for the presence of a grand-design spiral pattern in the Galaxy and in external galaxies. However, there is little evidence for the hypothesis that the spiral pattern is (quasi-)stationary, since the anti-spiral theorem (Lynden- Bell & Ostriker 1967) states that if a steady-state solution of a time reversible set of equations has the form of a trailing spiral, then there must be an identical solution in the form of a leading spiral. The prevalence of trailing spirals in external galaxies implies that non-stationary effects are involved in the expla- nation of the spiral structure (for example, they may be growing in amplitude in non-stable periods). In fact, spiral arms could be transient phenomena, con- stantly renewed by self-regulating instabilities. Thus, ironically, the Lin-Shu hypothesis, which has proved to be fruitful in advancing our understanding of disk dynamics, may still prove to be largely irrelevant to spiral structure.

Transient spiral waves The stable response of the dissipative gas could explain the presence of a grand- design long-lived spiral pattern. However, the dissipative nature of gas implies that the waves propagate radially (Toomre 1969), and this has consequences on the behaviour of leading spiral waves. Indeed a gaseous, tightly wound, lead- ing disturbance inevitably becomes a loosely wound disturbance, and the tight- winding approximation (basis of the quasi-stationary waves of Lin-Shu described above) is no more valid. Numerical simulations without the tight-winding ap- proximation show that it is also the case for a stellar disk: Fig. 1.4 presents the results of such a numerical simulation (Toomre 1981). We see on Fig. 1.4 that an initially leading wave unwinds into an open pattern and then into a trail- ing pattern that becomes more and more tightly wound. The amplitude of the resulting trailing wave is about twenty times larger than the amplitude of the initial leading wave, and moreover, an even stronger transient spiral pattern is formed at intermediate stages. This effect is called the swing-amplification (i.e. the resulting trailing wave is stronger than the initial leading wave, as clearly seen in panels 4 to 9 on Fig.1.4), and it is not captured by the tight-winding approximation. Julian & Toomre (1966) discovered this phenomenon by ana- lytically studying without the tight-winding approximation, in a small portion (called the “shearing sheet”) of an infinitesimally thin disk, the response of a stellar disk to a point-like concentration of gas (i.e. a non-axisymmetric pertur- bation of the potential). They found that this response is in the form of density waves whose wavecrests swing around from leading to trailing. The amplitudes 30 of those waves are amplified transitorily and then die away. Their duration is of the order of an orbital period (108 years in the Solar neighbourhood). Toomre (1981) has argued vigorously that this swing-amplification mechanism is the principal process of spiral arms formation in galactic disks and that the spiral arms are thus of transient nature. The biggest question about the theory of transient spiral waves is wether it is able to explain the observed grand-design patterns in most spiral galaxies. Sellwood & Carlberg (1984) demonstrated that a self-regulating instability in the stellar disk (producing transient spirals) may be able to produce grand- design, open spiral structure similar to that seen in most Sc galaxies. Gas instabilities lead to the formation of transient spiral arms (similar to those of Julian & Toomre 1966 and Toomre 1981) in which new stars are created. These new stars have low velocity dispersion. This lowering of the velocity dispersion reduces Q (see Eq. (1.34)), induces instability in the stellar disk, boosts the swing-amplification and increases the strength of the spiral arms while boosting star formation. These spiral waves impart angular momentum to stars and increase the velocity dispersion (see e.g. Fuchs 2001, De Simone et al. 2004). The increase of velocity dispersion raises Q, stabilizes the disk and reduces the susceptibility of further spiral-making. Gas is re-ejected by supernovae and cools down rapidly: this cooling leads to gravitational instability of the gas, in such a way that the transient spiral arms are constantly renewed. Numerical simulations by De Simone et al. (2004) have shown that such transient density waves could cause the distribution function to become clumpy, and the velocity dispersions to increase, especially far from corotation. Since transient waves with a wide range of pattern speeds develop in rapid succession, the entire disk can be affected by this phenomenon. This is a strong argument in favour of the transient nature of spiral waves, since the quasi-stationary waves of Lin-Shu cannot heat the disk anywhere else than at Lindblad resonances, and cannot explain the observed age-velocity dispersion relation. Moreover, the entire disk can also be affected by the radial mixing of orbits around corotation (Sellwood & Binney 2002). At corotation, the perturbation force acts on stars with a constant direction for a long time, and this enables large changes in the angular momentum of stars. Indeed, Lz and E are no longer integrals of the motion in the presence of a non-axisymmetric perturbation. Stars inside corotation overtake the wave and gain angular momentum as they fall into the spiral arm, which causes them to move to larger radii and slow their drift relative to the wave. Some stars close to corotation then reverse their speed relative to the wave, and subsequently slip backwards relative to the wave, fall backwards into the other arm, lose angular momentum and move to smaller radii. These orbits are called “horseshoe orbits” (Goldreich & Tremaine 1982). A spiral wave can cause a radial migration of stars of 2 or 3 kpc around corotation in much less than 1 Gyr (L´epine et al. 2003). An important feature of radial migration near corotation is that random motions caused by the spiral perturbation are quite moderate, and that stars stay on quasi-circular epicyclic orbits. Indeed, in the rotating frame of the spiral perturbation, the Jacobi’s integral is an integral 31

Figure 1.4: Numerical simulation (performed by Toomre 1981, Cambridge Uni- versity Press) of a density wave in a stellar disk with Q = 1.5, illustrating the swing-amplification. The time interval between panels is one-half of a rotation period at corotation. An initially leading wave (panel 1) first unwinds into a relatively open pattern (panel 3), and then into a stronger trailing pattern that becomes more and more tightly wound (panel 9). At intermediate stages (panels 4, 5 and 6), an extremely strong transient spiral pattern is formed. 32 of the motion (Binney & Tremaine 1987):

EJ = −E − ΩpLz (1.35)

Hence, changes of the binding energy and of the angular momentum of a star are related by ∆E = −Ωp∆Lz (1.36)

If we define JR as the radial action, which is simply R/κ (see Eq. (1.23)) in the epicyclic approximation, we can write the following equality between the differential applications: ∂E ∂E dE = dJR + dLz = −κ dJR − Ω dLz (1.37) ∂JR ∂Lz

Eqs. (1.36) and (1.37) lead to the expression of ∆JR, which is a measure of the distance from the original quasi-circular orbit: Ω − Ω ∆J = p ∆L (1.38) R κ z

Hence, large changes in Lz at corotation produce no significant heating (no in- 2 crease of vR), and the radial mixing does not increase the random motion of stars, which stay on quasi-circular orbits. Moreover, large changes in individ- ual Lz do not imply a change of the overall angular momentum distribution (Sellwood & Binney 2002). Thus a single transient can cause radial migration over 2 or 3 kpc without causing radical changes in the observed velocity distri- bution. This mechanism could explain the fact that no age-metallicity relation is observed for old stars in the Solar neighbourhood (Edvardsson et al. 1993) since most old disk stars may be expected to have been radially displaced by a transient spiral wave at least once within their lifetime. On a longer term, this mechanism could cause the age-velocity dispersion relation all over the disk, because of heating far from corotation by many successive waves. The Orion arm could be an example of such a transient spiral arm corotating with the Sun in the Galaxy (Mayor 1972).

1.3 The Solar neighbourhood

A better understanding of the Galaxy does not only need theoretical investiga- tions: it also needs very precise observational data. The Solar neighbourhood, i.e. a sample volume of a few hundreds pc around the Sun, is ideal for providing very precise data on stars (and in particular on their positions and velocities relative to the Sun), and is thus the benchmark test for models of the Galactic structure and evolution. In this section, most results are based on the axisym- metric assumption, which is the 0th order analysis, and which is a necessary starting point for any subsequent perturbation analysis. In order to locate a star relative to the Sun in the Galaxy, the “galactic co- ordinates” (d, l, b) are used, where d denotes the distance of the star to the Sun, 33 l is the “galactic longitude”, and b is the “galactic latitude”. The heliocentric cartesian coordinate system associated with galactic coordinates is:

x = d cosl cosb y = d sinl cosb (1.39) z = d sinb

1.3.1 Galactocentric radius of the Sun In order to know what implications the kinematics of stars in the Solar neigh- bourhood could have on the global structure of the Galaxy, it is important to know where the Sun is located in the Galaxy. The view of the Galactic plane from earth as a luminous band crossing the sky implies that we are very close to that plane (in fact 12 ± 8 pc above the plane; see e.g. Batsleer & Dejonghe 1994), between the Perseus (in the outer Galaxy) and Sagittarius (in the inner Galaxy) spiral arms, and inside the Orion arm. On the other hand, the distance of the Sun to the Galactic center is very difficult to estimate: the direct method is to compare the average radial velocities with the average proper motions (the angular velocities on the sky) of maser spots in star forming regions near the Galactic center (see Section 2.2.4 of Binney & Merrifield 1998), but this method needs very accurate observations. If the density of the objects of the luminous halo peaks at the galactic center, the galactocentric radius of the Sun can then also be measured by determining the distance of this density peak: the prob- lem of this method is that any uncertainty in the absolute magnitude of the stellar candles (for example cepheids for which the period-luminosity relation is known) will reverberate in the estimation of the galactocentric distance of the Sun. The measurements of this distance have been reviewed by Reid (1993), and they tend to approach 8 kpc. In this thesis, we will assume this estimate of 8 kpc for the galactocentic Solar radius, with an uncertainty of the order of 0.5 kpc.

1.3.2 Rotation curve and Oort constants The stars of the disk travel in nearly circular orbits around the galactic center (see Section 1.2.6). The determination of the circular velocity (the rotation curve) vc(R), where R is the galactocentric radius, and in particular vc(R ) (where R denotes the galactocentric radius of the Sun everywhere in this thesis) is one of the hardest problems in Galactic astronomy. The rotation curve is determined by observations of the kinematics of the gas, and in particular of the neutral hydrogen 21 cm line, but the rotation curve is not well established for galactic radii R > R . Observations of many outer spiral galaxies indicate that the rotation curve remains more or less flat after attaining its maximum (e.g. Casertano & van Gorkom 1991): the rotation curve of the Milky Way therefore is thought to behave similarly. This flatness of the rotation curve implies the presence of a dark matter halo in addition to all the observed matter in order to maintain the stars in the Galaxy. 34

As the global shape of the rotation curve is not known precisely, the determi- nation of its local shape in the Solar neighbourhood is fundamental for a better knowledge of local Galactic structure. Lindblad (1925) and Oort (1927a,b) de- vc veloped the model of differential axisymmetric rotation with Ω = R depending only on the distance R to the galactic center. Oort (1927a) introduced two constants (the Oort constants A and B), that can be determined from proper motions (angular velocities on the sky) of neighbouring stars and that are di- rectly related to the local shape of the rotation curve. Indeed, for a star of the Solar neighbourhood on a circular orbit, Taylor expanding Ω(R) to first order in (R − R ) yields for the line-of-sight radial velocity vr and the transverse velocity vt relative to the Sun:

vr = A d sin2l (1.40)

vt = A d cos2l + B d (1.41) with 1 v dv A = ( c − c ) , (1.42) 2 R dR R 1 v dv B = − ( c + c ) (1.43) 2 R dR R Kuijken & Tremaine (1991) showed that the Taylor expansion terms arising from non-circularity of the orbits are negligible, just as they should be if the Milky Way is a perturbed axisymmetric galaxy and not a triaxial one. Oort (1927b) showed for the first time this sinusoidal effect of the galactic rotation on the radial velocities and on the proper motions. He found A = 19 km s−1kpc−1 and B = −24 km s−1kpc−1. Indeed, by fitting Eq.(1.41) to observed proper motions, one can determine A and B if one is sure that the frame is not rotating. This last requirement was pretty unsure before the ESA Hipparcos mission. The most reliable determination of the Oort constants based on the proper motions of the Cepheids that were measured by Hipparcos has been derived by Feast & Whitelock (1997) who found A = 14.82 ± 0.84 km s−1kpc−1 and B = −12.37 ± 0.64 km s−1kpc−1. These values were confirmed by Mignard (2000) who found A = 14.5±1.0 km s−1kpc−1 and B = −11.5±1.0 km s−1kpc−1 by using proper motions of distant giants. This indicates that the rotation curve is slightly declining in the Solar neighbourhood. The determination of vc(R ) follows from the determination of the Oort constants: using the values of Feast & Whitelock (1997), we find vc(R ) = −1 −1 (218±8km s )(R /8kpc), a value which is consistent with vc(R ) = 220 km s that we choose to adopt in this thesis. However, although Hipparcos data have improved our knowledge of the Oort constants, the measurement of the proper motion of the compact radio source Sgr A* (Backer 1996), seems to indicate that A − B = 30.1 ± 0.8km s−1kpc−1 when assuming that Sgr A* is stationary with respect to the galactic center. This is inconsistent with the determination based on Hipparcos data, and leaves us with an uncertainty which still awards a resolution. 35

1.3.3 Local dynamical mass

The mass density in the Solar neighbourhood ρ is an essential constraint for any mass model of the Galaxy. It can be surmised that this parameter is not directly observed: it is deduced from the positions and velocities of tracer stars in the direction perpendicular to the galactic plane (the z-direction), using the Boltzmann equation (1.3) and Poisson equation (1.4) in various forms. Jeans equations (first order moments of the Boltzmann equation (1.3)) imply 2 that the vertical acceleration Kz is related to the vertical velocity dispersion σz and to the vertical number density g(z) of a population of stars by:

d ln(g(z)/g(0)) K (z) = σ2 , (1.44) z z dz if the vertical motion can be separated from the radial and azimuthal motion of the stars (Oort-Lindblad approximation), if the velocity ellipsoid is aligned with the cylindrical coordinate axes (i.e hvRvzi = 0), and if the population is 2 isothermal (i.e. σz is constant as a function of z). Oort (1932; 1960) applied this now classical formula to stars of spectral- type A to M, with the assumption that they were old enough to have become dynamically well mixed in the z-direction. Then he derived ρ using Poisson equation for nearly circular orbits in the Solar neighbourhood (derived from Eq. 2 vc (1.4), (1.42) and (1.43) if we assume that KR(R) = − R ): dK 4πGρ = − z − 2(A2 − B2) (1.45) dz −3 −3 Oort found ρ = 0.09M pc in 1932 and ρ = 0.15M pc in 1960: this last result indicated that there might be a lot of dark matter in the disk, by comparison with star counts (today the density of known matter is about −3 0.07M pc , see Cr´ez´eet al. 1998). Afterwards, many other similar studies (Yasuda 1961; Eelsalu 1961; Woolley & Stewart 1967; Turon 1971; Gould & Vandervoort 1972; Jones 1972; Balakirev 1976; Hill et al. 1979) gave discrepant results, suffering from inhomogeneities in the data, systematic errors due to the use of photometric distances, and undersampling near the galactic plane. Because of this undersampling, some studies were even based on young O and B stars, assuming that the gas and dust out of which they recently formed was already roughly relaxed (Stothers & Tech 1964; von Hoerner 1966). More recently, Bahcall (1984a,b,c) described the disk matter as a series of isothermal components (i.e. components described by distribution functions Fz(z, vz) leading to a constant velocity dispersion σz, see Spitzer 1942) and analyzed the nonlinear self-consistent equations in which the matter produces the potential (Poisson equation) and is also affected by the potential (Jeans equations for each isothermal component). Bahcall et al. (1992) applied this −3 method to a sample of K giants and found ρ = 0.26M pc , a result leading once again to the presence of dark matter in the disk. 36

At about the same time, Kuijken & Gilmore (1989a,b,c, 1991) used another method which does not use the Jeans equations, inspired by the method of von Hoerner (1960). This method is based on the assumption that the phase space distribution function Fz(z, vz) of any tracer population depends only on the vertical energy 1  = Φ (z) + v2 (1.46) z z 2 z where dΦ (z) − z = K (z). (1.47) dz z This assumption follows from the classical separability of the vertical motion of the stars (Oort-Lindblad approximation) and from the Jeans theorem in one dimension. The potential Φz(z) is quadratic in z near the plane, in accordance with Eq. (1.24). The density in configuration space g(z) of a tracer population is then related to its density in phase space by the integral equation: Z ∞ Z ∞ Fz(z) g(z) = Fz(z, vz)dvz = 2 p dz = f(Φz(z)) (1.48) −∞ Φz 2(z − Φz)

So, there is a unique relation between g(z) and Fz(z): Kuijken & Gilmore (1989c, 1991) inverted the Abel transform (1.48) and used the space density profile g(z) of distant K stars to predict their velocity distribution at different heights for different Φz(z). Then they compared these predictions to the ve- locity data and used a maximum likelihood technique to select the best-fitting potential. The data they used were too far from the plane to constrain ρ and −2 they found a surface mass density between z = ±1.1 kpc of 71M pc , a result rejecting the presence of dark matter in the disk. So, all the investigations between 1932 and the ESA Hipparcos mission had failed to converge to a reliable determination of ρ : they were all very uncertain, essentially because of inhomogeneities in the tracer samples, undersampling near the Galactic plane and the use of photometric distances. Hipparcos data solved all these problems: the dense probe near the plane eliminated the inhomogeneity and the undersampling, while the accurate parallaxes2 eliminated the use of photometric distances3. Cr´ez´eet al. (1998) and Holmberg & Flynn (2000) used Eq. (1.48) to de- termine ρ from complete samples of nearby A-F stars. Given the functions 2 g(z) and Fz(z = 0, vz ) = Fz(z) from the observed vertical density and velocity distribution, the function Φz(z) can be derived. Then, to determine ρ , the Oort constants have to be used in the Poisson equation (1.45): these are also much better known since the Hipparcos mission (see Section 1.3.2). Cr´ez´eet al. −3 (1998) estimated ρ = 0.076 ± 0.015M pc while Holmberg & Flynn (2000)

2The parallax of a star is the angle formed by the two lines going from the center of the star to the Sun and to the Earth 3The photometric distance of an object is an estimation of its distance based on a com- parison of the apparent magnitude with the expected true luminosity of the star, known e.g. thanks to the period-luminosity relation for some pulsating stars 37

−3 estimated ρ = 0.102 ± 0.010M pc . These differences between local density estimates using almost the same Hipparcos data are due to different a priori hypothesis on the vertical potential Φz. Recent investigations at Strasbourg University using a local Hipparcos sample combined with two samples at the galactic poles (Siebert et al. 2003) confirm the values obtained by Holmberg & Flynn (2000).

1.3.4 LSR, Solar motion and velocity ellipsoid The components of the velocity of a star with respect to the Sun in the helio- centric cartesian coordinate system (1.39) are the x-velocity U (parallel to the direction “Sun - Galactic center”, its direction is the opposite of the radial veloc- ity in galactocentric coordinates vR), the y-velocity V (parallel to the direction of Galactic rotation at the azimuth of the Sun), and the z-velocity W = vz (pos- itive towards the North Galactic pole). Velocities are often expressed relative to the Local Standard of Rest (the LSR), i.e. a reference frame that follows the closed orbit in the plane that passes through the present location of the Sun. In the axisymmetric approximation, this orbit is circular. The velocity of a star relative to the LSR is called the peculiar velocity of the star, denoted u, v and w in the three directions of the coordinate system (1.39). The peculiar velocity 4 of the Sun is called the Solar motion and is denoted U , V and W in the three directions of the coordinate system (1.39). Clearly, since the Galactic plane is a plane of symmetry, there should be no net vertical motion in the Galaxy. Therefore, the vertical Solar motion W is simply the opposite of the mean vertical motion hW i of an homogeneous sample of stars in the Solar neighbourhood. Dehnen & Binney (1998a) found on the basis of Hipparcos proper motions of main sequence stars W = 7.17 ± −1 0.38 km s . The determination of the azimuthal motion V is less simple, since the “asymmetric drift” (Eq. (10.12) of Binney & Merrifield 1998, see also Eq. (3.10) of the present thesis) predicts a linear dependence of hvi with the −1 radial velocity dispersion : V has been estimated at V = 5.25 ± 0.62 km s (Dehnen & Binney 1998a) by linearly extrapolating the mean motion hV i to zero velocity dispersion. Following the axisymmetric assumption, there should be no net radial motion at the position of the Sun in the Galaxy: following that −1 hypothesis, Dehnen & Binney (1998a) found U = −hUi = 10.00 ± 0.36 km s . Nevertheless, those values for the Solar motion will be revisited and discussed in Chapter 3. If we get back to the epicyclic approximation of Section 1.2.6, a relation between the peculiar velocity v of an orbit at Solar radius and the position of its guiding center can be derived (Eq. (3-74) of Binney & Tremaine 1987):

κ2 v = −2Bx = x (1.49) g 2Ω g

4Even if the Solar motion is a peculiar velocity, it is traditionnaly denoted by upper-case letters 38

where xg is the the x-coordinate of the guiding center in the heliocentric frame (1.39). In the Solar neighbourhood approximation, yg << Rg so that xg ' R −Rg, and we can rewrite the epicycle energy (1.23): 1 1  = (u2 + κ2 x2) = (u2 + (2Ω/κ)2v2) (1.50) R 2 g 2

If we define a distribution function in velocity space (with parameters σu, σv and σw representing the velocity dispersions in the radial, azimuthal and vertical directions) at the position of the Sun in the Galactic plane (z = 0)

2 2 −3/2 −1 −(R/σ )−(z /σ ) F (u, v, w) = (2Π) (σuσvσw) e u w , (1.51) we have by the Jeans theorem (see Section 1.2.4) a time-independent solution of the collisionless Boltzmann equation (1.1) since F depends on the velocities 2 2 2 2 only through the integrals R and z. Moreover, since σv/σu = κ /4Ω (Eq. (3-72) and Eq.(4-52) of Binney & Tremaine 1987), Eq. (1.50) yields 1  /σ2 = (u2/σ2 + v2/σ2), (1.52) R u 2 u v and Eq. (1.51) becomes

−3/2 −1 −(u2/2σ2 )−(v2/2σ2)−(w2/2σ2 ) F (u, v, w) = (2Π) (σuσvσw) e u v w . (1.53) This 3-dimensional gaussian distribution is called the Schwarzschild distribution function. Schwarzschild (1907) pointed out that the distribution of stellar ve- locities in the Solar neighbourhood was similar to that of, say, a population of oxygen molecules at room temperature. The main difference between the case of molecules in air and stars is that in the former case the velocity dispersion is independent of direction. In the stellar case, the 3-dimensional gaussian dis- tribution (1.53) implies that the density of stars in velocity space is constant on ellipsoids with axis lengths in the ratio σu : σv : σw. The ellipsoid with semi-axes of length σu, σv and σw is called the Schwarzschild velocity ellipsoid.

1.3.5 Vertex deviation and substructure of velocity space 2 The covariance σuv of any population of stars having a velocity distribution function of the type of Eq. (1.53) is null. However, when estimating this covariance for samples of stars in the Solar neighbourhood, one often finds a significantly non-zero value for this covariance (especially for young stars). Thus we must rewrite a phenomenological distribution function in order to account for this phenomenon:

02 2 02 2 2 2 −3/2 −1 −(u /2σ 0 )−(v /2σ 0 )−(w /2σ ) F (u, v, w) = (2Π) (σu0 σv0 σw) e u v w (1.54) where 0 u = u cos lv − v sin lv 0 (1.55) v = u sin lv + v cos lv 39

The angle lv in velocity space is called the vertex deviation. If we define the velocity dispersion tensor of a population of stars as h(~v − h~vi) ⊗ (~v − h~vi)i, the vertex deviation lv is the angle one has to rotate the (u, v) coordinate system in velocity space in order to diagonalize this tensor. Multiplying lines in Eq. (1.55) by each other, averaging the result for a sample of stars, and stating that hu0v0i = 0 yields: 2 2 2 lv = 1/2 arctan (2σuv/(σu − σv)). (1.56)

We state that σu0 > σv0 in order to raise the ambiguity introduced by the arctan function. Nevertheless, if we insert the velocity distribution (1.54) (multiplied by an arbitrary distribution of the positions) in the collisionless Boltzmann equation (1.3), we find that lv must be zero if the Galaxy is axisymmetric, and find back the distribution (1.53). The possible causes of this vertex deviation is one of the main topics of this thesis. The classical hypothesis is that the observed vertex deviation is due to the fact that many samples of stars are not in dynamical equilibrium, implying that the number of truly independent velocities employed to derive the distribution function is strongly reduced. In fact, many objects in samples of young stars could be members of moving groups, generally thought to be vestiges of the clusters and associations in which most stars form. According to the commonly accepted theory, early-type, young stars still carry the kinematic signature of their place of birth. As a consequence, their distribution in velocity space is clumpy (e.g. de Bruijne et al. 1997, Figueras et al. 1997, de Zeeuw et al. 1999): the isodensity ellipsoids of Eqs. (1.53) and (1.54) are distorted by a lot of substructure that appears in velocity space. These inhomogeneities can be spatially confined groups of young stars (OB associations in the Gould’s belt, young clusters) but can also be spatially extended groups. There is, however, some confusion in the literature about the related terminology. Eggen (1994) defines a supercluster as a group of stars gravitationally unbound that share the same kinematics and may occupy extended regions in the Galaxy, and a moving group as the part of the supercluster that enters the Solar neighbourhood and can be observed all over the sky. Unfortunately, the same term “moving group” is sometimes also applied to OB associations (e.g. de Zeeuw et al. 1999). It has long been known that, in the Solar vicinity, there are several superclus- ters and moving groups that share the same space motions as well-known open clusters (Eggen 1958). The best documented groups (see Montes et al. 2001 and references therein) are the Hyades supercluster associated with the Hyades cluster (600 Myr) and the Ursa Major group (also known as the Sirius super- cluster) associated with the UMa cluster of stars (300 Myr). Another kinematic group called the Local Association or Pleiades moving group is a reasonably coherent kinematic group with embedded young clusters and associations like the Pleiades, α Persei, NGC 2516, IC 2602 and Scorpius-Centaurus, with ages ranging from about 20 to 150 Myr. Two other young moving groups are the IC 2391 supercluster (35-55 Myr) and the Castor moving group (200 Myr). The kinematic properties of all these moving groups and superclusters are listed by 40

Montes et al. (2001) and Chereul et al. (1999). However, the recent observation (e.g. Chereul et al. 1998, 1999) that some of those superclusters involve early- type stars spanning a wide range of ages contradicts the classical hypothesis that supercluster stars share a common origin: Chereul et al. (1998) propose that the supercluster-like velocity structure is just a chance juxtaposition of several cluster remnants. Nevertheless, significant clumpiness in velocity space has also been recently reported for late-type stars (Dehnen 1998), thus raising the questions of the age of those late-type stars and of the exact origin of this clumpiness. For example, Montes et al. (2001) suppose that, if a late-type star belongs to a moving group or supercluster, it may be considered as a sign that the star is young, an assumption that will be largely challenged by the results of this thesis. In fact two other mechanisms (in addition to the classical theory of clus- ter remnants) could be responsible for the existence of clumpiness in velocity space and for the vertex deviation. The first one is the absorption of satel- lite galaxies that could generate streams everywhere in the Galaxy (see Section 1.2.2). The second one is the perturbation of the distribution function by a non-axisymmetric component of the potential such as transient spiral waves (see Section 1.2.7).

1.4 Outline of this thesis

In this thesis, we study the motion of giant stars in the Solar neighbourhood and what they tell us about the dynamics of the Galaxy: it is thus a small contribution to the huge and ambitious project of understanding the structure and evolution of the Galaxy as a whole. We restrict our study of stellar kinematics to late-type giants (of spectral types K and M, without any a priori hypothesis on their motion or age). Indeed, the giants are intrinsically bright, and thus probe a large area in the Solar neighbourhood, which makes them good tracers for the study of kinematics and dynamics. The next two Chapters are dedicated to the kinematic analysis of the data, while Chapters 4 and 5 are dedicated to the development of new tools to establish dynamical models. In Chapter 2, we describe a stellar sample of 5952 K and 739 M giants. For the first time, the three-dimensional velocities of those stars are made available (they are listed in Table A.1, to be found on the CD-ROM attached to this thesis). Indeed, their radial velocities have been measured at the Observatoire de Haute Provence with the CORAVEL spectrovelocimeter, while their parallaxes (Hipparcos) and proper motions (Tycho-2) were already available. In Chapter 3, after a first crude kinematic analysis (Sections 3.1 and 3.2), a maximum-likelihood method, based on a bayesian approach, is applied to the data (Section 3.3). This allows to make full use of all the available stars, and to derive the kinematic properties of the subgroups forming a rich small-scale struc- ture in velocity space. Then we confront the results of our kinematic study with the three main hypotheses proposed to account for the substructure in velocity 41 space (the merger remnants described in Section 1.2.2, the non-axisymmetric perturbations described in Section 1.2.7, and the vestiges of clusters described in Section 1.3.5). Our results suggest a dynamical non-axisymmetric origin, in the light of recent simulations discussed in Section 1.2.7. We discuss the conse- quences of these non-axisymmetric perturbations for the derivation of the Solar motion in the Galaxy, which will be more difficult to evaluate than previously thought. Nevertheless the weak point of all the recent simulations of dynamical per- turbations is the simplicity of their initial conditions, based on two-dimensional Schwarzschild velocity ellipsoids in the Galactic plane (see Section 1.3.4). There- fore, we develop in Chapters 4 and 5 new tools to establish three-integral models (see Section 1.2.5), exact solutions of the collisionless Boltzmann equation (1.1), that could yield more complex initial conditions for the simulations. In Chapter 4, we present a set of potentials, in which three exact integrals of the motion exist for all orbits, defined by five parameters and designed to model the Galaxy. We study the parameter ranges of the presented potentials matching the fundamental parameters of the Milky Way, especially in the Solar neighbourhood. Five different valid potentials are presented and analyzed in detail. In Chapter 5, we present new equilibrium component distribution functions that depend on three analytic integrals, and that can be used to model the pop- ulations of the galactic disk, such as the late-type giants studied in Chapters 2 and 3. These models could give ideal initial conditions (more complex than a simple two-dimensional Schwarzschild velocity ellipsoid) for three-dimensional N-body simulations, that could afterwards reproduce the non-axisymmetric fea- tures observed in our stellar sample (Chapter 3). 42 Chapter 2

Stellar sample

As we stressed in Section 1.3, the Solar neighbourhood is the place in the Galaxy where we can retrieve the most precise data on star positions and velocities. The story of the efforts to obtain more and more accurate data is standard textbook fare, and nicely illustrates how theoretical progress and data acquisition have to go hand in hand if one wants to gain insight in the structure and evolu- tion of the Galaxy. Clearly, another chapter in this story has begun with the Hipparcos satellite mission (ESA 1997), that provided accurate parallaxes and proper motions for a large number of stars (about 118000): with accurate par- allaxes available, it was no longer necessary to resort to photometric distances. The Hipparcos data enabled several kinematic studies of the Solar neighbour- hood, but those studies generally lacked radial velocity data. A first preview of how Hipparcos data could improve our knowledge of stellar motions in the Galaxy was given by Kovalevsky (1998); then, many studies (e.g. Cr´ez´eet al. 1998, Dehnen & Binney 1998a, Holmberg & Flynn 2000) used Hipparcos proper motions to derive some fundamental kinematic parameters of the Solar neighbourhood. They did not use radial velocities from the literature because, at that time, radial velocities had been measured preferentially for high proper motions stars and their use would have introduced a kinematic bias. Hipparcos data were also used by Chereul et al. (1998) to derive the small scale structure of the velocity distribution of early-type stars in the Solar neighbourhood. The situation has now dramatically improved thanks to the efforts of a large European consortium to obtain radial velocities of Hipparcos stars with a spec- tral type later than about F5 (Udry et al. 1997). The sample includes Hipparcos “survey” stars (flag S in field H68 of the Hipparcos Catalogue), and stars from other specific programmes. This unique database, comprising about 45000 stars measured with the CORAVEL spectrovelocimeter (Baranne et al. 1979) at a typical accuracy of 0.3 km s−1, combines a high precision and the absence of kinematic bias. It thus represents an unprecedented data set to test the results obtained by previous kinematic studies based solely on Hipparcos data. An- other feature of the present thesis is the use of Tycho-2 proper motions, which combine Hipparcos positions with positions from much older catalogues (Høg et

43 44

Figure 2.1: Difference between the Hipparcos and Tycho-2 proper-motion mod- uli, normalized by the root mean square of the standard errors on the Hippar- cos and Tycho-2 proper motions, denoted µ. The solid line refers to the 859 spectroscopic binaries (SB) present in our sample, whereas the dashed line cor- responds to the 5832 non-SB stars. Note that the sample of binaries contains more cases where the Hipparcos and Tycho-2 proper motions differ significantly, in agreement with the argument of Kaplan & Makarov (2003). 45 al. 2000). They represent a subsantial improvement over the Hipparcos proper motions themselves, which are based on very accurate positions, but extending over a limited 3-year time span. Results from the Geneva-Copenhagen survey for about 14000 F and G dwarfs present in the CORAVEL database have been recently published by Nordstr¨omet al. (2004). The present thesis, on the other hand, concentrates on the K and M giants from the CORAVEL database.

2.1 Selection criteria

Our stellar sample is the intersection of several data sets: (i) in a first step, all stars with spectral types K and M appearing in field H76 of the Hipparcos Catalogue (ESA 1997) have been selected, and the Hipparcos parallaxes have been used in the present study; (ii) Proper motions were taken from the Tycho- 2 catalogue (Høg et al. 2000). These proper motions are more accurate than the Hipparcos ones. Fig. 2.1 compares the proper motions from Tycho-2 and Hipparcos, and reveals that the Tycho-2 proper motions are sufficiently differ- ent from the Hipparcos ones (especially for binaries) to warrant the use of the former in the present study; (iii) Radial-velocity data for stars belonging to this first list (stars with spectral type K and M) have then been retrieved from the CORAVEL database. Only stars from the northern hemisphere (δ > 0◦), ob- served with the 1-m Swiss telescope at the Haute-Provence Observatory, have been considered. As described by Udry et al. (1997), all stars from the Hip- parcos survey (including all stars brighter than V = 7.3 + 1.1| sin b| for spectral types later than G5, where b is the galactic latitude; those stars are flagged ‘S’ in field H68 of the Hipparcos Catalogue) are present in the CORAVEL database, thanks to a large observing campaign dedicated to those stars. The CORAVEL database contains in addition stars monitored for other purposes (e.g., binarity or rotation), but they represent a small fraction of the Hipparcos survey stars. The sample resulting from the intersection of these 3 data sets was fur- ther cleaned as follows. First, a crude Hertzsprung-Russell diagram for all the M-type stars of the Hipparcos Catalogue has been constructed from distances estimated from a simple inversion of the parallax (Fig. 2.2), and reveals a clear separation between dwarfs and giants. Clearly, all M stars with MHp < 4 must be giants. For K-type stars with a relative parallax error less than 20%, the Hertzsprung-Russell diagram (Fig. 2.3) shows that the giant branch connects to the main sequence around (V − I = 0.75,MHp = 4). Therefore, in order to select only K and M giants, only stars with MHp < 4 were kept in the sample. For K stars, the stars with V − I < 0.75 and MHp > 2 were eliminated to avoid contamination by K dwarfs. Diagnostics based on the radial-velocity variabil- ity and on the CORAVEL cross-correlation profiles1, combined with literature

1The Doppler-information of a large number of lines in the spectrum is joined in one single profile: the cross-correlation profile. The lines are contained in a template or mask, consisting of slits representing a simplified synthetic spectrum. For each velocity, the template is shifted over the observed absorption spectrum, and at the velocity of the object, the light transmitted through the slits is minimal, see Fig. 2.4 46 searches, allowed us to further screen out stars with peculiar spectra, such as T Tau stars, Mira variables and S stars. T Tau stars have been eliminated because they belong to a specific young population; S stars because they are a mixture of extrinsic and intrinsic stars, which have different population charac- teristics (Van Eck & Jorissen 2000); Mira variables because their center-of-mass radial velocity is difficult to derive from the optical spectrum, where confusion is introduced by the pulsation (Alvarez et al. 2001). The primary sample includes 5952 K giants and 739 M giants (6691 stars). 86% of those stars are “survey” stars: our sample is thus complete for the K and M giants brighter than V = 7.3 + 1.1| sin b|. To fix the ideas, for a typical giant star with MV = 0, this magnitude threshold translates into distances of 290 pc in the galactic plane and 480 pc in the direction of the galactic pole (these values are, however, very sensitive upon the adopted absolute magnitude, and become 45 pc for a subgiant star with MV = +4 in the galactic plane, and 2900 pc for a supergiant star with MV = −5, corresponding to the range of luminosities present in our sample; see Chapter 3). The final step in the preparation and cleaning of the sample involves the identification of the spectroscopic binaries, as described in Section 2.2.

2.2 Binaries

The identification of the binaries, especially those with large velocity ampli- tudes, is an important step in the selection process, because kinematic studies should make use of the center-of-mass velocity. In order to identify those, the observing strategy was to obtain at least two radial-velocity measurements per star, spanning 2 to 3 yr. Monte-Carlo simulations reveal that, with such a strat- egy, binaries are detected with an efficiency better than 50 % (Udry et al. 1997). Since late-type giants exhibit intrinsic radial velocity jitter (Van Eck & Jorissen 2000), the identification of binaries requires a specific strategy, which makes use of the (Sb, σ0(vr0)) diagram (Fig. 2.5), where vr0 is the measured radial velocity. The parameter Sb is a measure of the intrinsic width of the cross-correlation profile, i.e., corrected from the instrumental width (see Fig. 2.4 for more precise definitions). The CORAVEL instrumental profile is obtained from the observa- tion of the cross-correlation dip of minor planets reflecting the sun light, after correction for the Solar rotational velocity and photospheric turbulence (see Benz & Mayor 1981 for more details). The Sb parameter is directly related to the average spectral line width, which is in turn a function of spectral type and luminosity, the later-type and more luminous stars having larger Sb values. On the other hand, the measurement error ¯ has been quadratically subtracted from the radial velocity standard deviation σ(vr0) to yield the effective standard deviation σ0(vr0). The identification of binaries among K or M giants follows different steps. 2 The standard χ variability test, comparing the standard deviation σ(vr0) of the measurements to their uncertainty ¯, cannot be applied to M giants, because intrinsic radial-velocity variations (“jitter”) associated with envelope pulsations 47

0

5

10

0 1 2 3 4 5 V-I

Figure 2.2: Crude Hertzsprung-Russell diagram for the Hipparcos M stars with positive parallaxes. 48

0

5

10

0 0.5 1 1.5 2 V-I

Figure 2.3: Crude Hertzsprung-Russell diagram for the Hipparcos K stars with a relative error on the parallax less than 20%. 49

2 2 1/2 Figure 2.4: The concept of line-width parameter Sb = (s − s0) is illustrated here by comparing the CORAVEL cross-correlation (smoothed) profiles (black dots) of 3 M giants or supergiants with the gaussian instrumental profile (of −1 sigma s0 = 7 km s ; solid line). Each profile corresponds to a single radial- velocity measurement for the given stars. The s value listed in the above panels refers to the sigma parameter of a gaussian fitted to the observed correlation profile. 50

Figure 2.5: The (Sb, σ0(vr0)) diagram (see text) for K (upper panel, and zoom inside) and M giants (lower panel). Star symbols denote supergiant stars, filled symbols denote spectroscopic binaries with (squares) or without (circles) center- of-mass velocity available. 51 would flag them as velocity variables in almost all cases. Therefore, in a first −1 step, M giants having σ0(vr0) ≥ 1 km s have been monitored with the ELODIE spectrograph (Baranne et al. 1996) at the Haute-Provence Observatory (France) since August 2000 (Jorissen et al. 2004). These supplementary data points made it possible to distinguish orbital variations from radial-velocity jitter by a simple visual inspection of the data. It turns out that, in the (Sb, σ0(vr0)) diagram, all the confirmed binaries (filled symbols in the lower panel of Fig. 2.5) are located in the upper left corner, and are clearly separated from the bulk of the sample. Stars located below the dashed line in Fig. 2.5 may be supposed to be single (although some very long-period binaries, with P > 10 yr, may still hide among those). Their radial velocity dispersion suffers from a jitter which clearly increases with increasing spectral line-width, as represented by the parameter Sb. For very large values of Sb (in excess of about 9 km s−1), the diagram is populated almost exclusively by supergiants (star symbols in the lower panel of Fig. 2.5). Many semi-regular and irregular variables are located in the intermediate region, with 5 < Sb < 9 km s−1. Strangely enough, spectroscopic binaries seem to be lacking in this region. Nevertheless a detailed discussion of the properties of the binaries found among M giants would deserve an entire thesis and is out of the scope of the present one: it could be the subject of further studies. For K giants, no such structure is apparent in the (Sb, σ0(vr0)) diagram. It has been checked that the distribution of stars along a 2 (N −1)[σ(vr0)/] axis (where N is the number of measurements for a given star) follows a χ2 distribution, as expected (e.g., Jorissen & Mayor 1988). This holds true irrespective of the Sb value, thus confirming the absence of structure in the (Sb, σ0(vr0)) diagram. The binaries among K giants may thus be identified by a straight χ2 test. A 1% threshold for the first kind risk (of rejecting the null hypothesis that the star is not a binary while actually true, i.e., of considering the star as binary while actually single) has been chosen in the present study. Among M giants, 42 binaries are found (corresponding to an observed fre- quency of spectroscopic binaries of 42/739, or 5.7%). Among the 5952 K giants, 817 spectroscopic binaries are found, corresponding to a frequency of 13.7%. The large difference between these two frequencies could be the subject of fur- ther studies and is out of the scope of this thesis. Most of the spectroscopic binaries (SBs) detected among our samples of K and M giants are first detec- tions. This large list of new SBs constitutes an important by-product of the present work. The new binaries are identified in Table A.1 (flags 0, 1, 5, 6 and 9 in column 24 of Table A.1), to be found on the CD-ROM attached to this thesis. Among this total sample of 859 spectroscopic binaries, the center-of-mass ve- locity could be computed (whenever the available measurements were numerous enough to derive an orbit), estimated (in the case of low-amplitude orbits) or taken from the literature for 216 systems only, thus leaving 643 binaries which had to be discarded from the kinematic study, because no reliable center-of-mass velocity could be estimated. After excluding those large-amplitude binaries as well as the dubious cases, 5311 K giants and 719 M giants remain in the final sample. Fig. 2.6 shows the distribution of our final sample on the sky. Among this final sample of 6030 stars, 5397 belong to the Hipparcos “survey”. 52

Figure 2.6: Distribution of the final sample on the sky, in galactic coordinates, the galactic center being at the center of the map. The selection criterion δ > 0◦ is clearly apparent on this map. Chapter 3

Kinematic analysis

In this Chapter, we proceed to the kinematic analysis of the data presented in the previous Chapter. Although the velocity distribution is a function of position in the Galaxy, it will be assumed in this Chapter that our local stellar sample may be used to derive the properties of the velocity distribution at the location of the Sun if we correct the data from the effects of differential galactic rotation. First, we analyze the kinematics of the sample restricted to the 2774 stars with parallaxes accurate to better than 20%. In a second approach, designed to make full use of the 6030 available stars but without being affected by the biases appearing when dealing with low-precision parallaxes, the kinematic parameters are evaluated with a Monte-Carlo method. Although easy to implement, the Monte Carlo method faces some limitations (the parallaxes were drawn from a Gaussian distribution centered on the observed parallax, not the true one as it should). Therefore, in a third approach, we make use of a bayesian method (the LM method; Luri et al. 1996), which allows us to derive simultaneously maximum likelihood estimators of luminosity and kinematic parameters, and which can identify possible groups present in the sample by performing a cluster analysis. In all cases, we correct for differential Galactic rotation in order to deal with the true velocity of a star relative to its own standard of rest. Given the observed values of the line-of-sight radial velocity vr0 and the proper motions µl0 and µb0, the corrected values are (Trumpler & Weaver 1953):

2 vr = vr0 − A d cos b sin2l (3.1)

κ µl = κ µl0 − A cos2l − B (3.2)

κ µb = κ µb0 + 1/2 A sin2b sin2l, (3.3) where κ is the factor to convert proper motions into space velocities, and d is the distance to the Sun. We used the values of the Oort constants A and B derived by Feast & Whitelock (1997) from Hipparcos Cepheids, i.e. A = 14.82 km s−1kpc−1 and B = −12.37 km s−1kpc−1 (see Section 1.3.2). The

53 54 observed velocity is, after that correction, a combination of the peculiar velocity of the star and that of the Sun relative to the LSR.

3.1 Analysis restricted to stars with the most precise parallaxes

The components of the velocity of a star with respect to the Sun in the cartesian coordinate system of Eq. (1.39) are the velocity towards the galactic center U, the velocity in the direction of Galactic rotation V , and the vertical velocity W . As we decide not to use any a priori value for the Solar motion, we use the velocities U, V , and W instead of the peculiar velocities of the stars relative to the LSR u, v, and w (see Section 1.3.4). The basic technique to calculate U, V and W is to invert the parallax to estimate the distance and then use the proper motions and radial velocities as given by Eqs. (3.1), (3.2) and (3.3). This simple procedure faces two major difficulties: (i) the inverse parallax is a biased estimator of the distance (especially when the relative error on the parallax is large), and (ii) the individual errors on the velocities cannot be derived from a simple first-order linear propagation of the individual errors on the parallaxes. It is thus extremely hard to estimate an error on the average velocities. If we limit the sample to the stars with a relative parallax error smaller than 10%, we are left with 786 stars, which is too small a sample to analyze the general behaviour of the K and M giants in the Solar neighbourhood. We choose instead to restrict the sample to the stars with a relative parallax error smaller than 20%, in such a way that 2774 stars (2524 K and 250 M) remain. In that case the classical first order approximation for the calculation of the errors may still be applied (and the bias is very small, see Brown et al. 1997), which yields from the errors on the proper motions, radial velocities and parallaxes

−1 hU i = 4.03 km s −1 hV i = 3.22 km s (3.4) −1 hW i = 2.54 km s . The contribution of the measurement errors to the uncertainty on the sample mean velocity is N −1/2 times the values given by (3.4), where N(= 2774) is the sample size. This contribution is in fact negligible with respect to the “Poisson noise” (obtained from the intrinsic velocity dispersion of the sample, see Eq. (3.8)). For the stars with a relative parallax error smaller than 20%, we obtain

hUi = −10.24 ± 0.66 km s−1 hV i = −20.51 ± 0.43 km s−1 (3.5) hW i = −7.77 ± 0.34 km s−1. If there is no net radial and vertical motion at the Solar position in the Galaxy, we have hence estimated U = −hUi and W = −hW i (see Section 1.3.4, but see also Section 3.3.6). 55

Knowing that hV i is affected by the asymmetric drift (Eq. (10.12) of Binney & Merrifield 1998), which implies that the larger a stellar sample’s velocity dis- persion is, the more it lags behind the circular Galactic rotation, it is interesting to compare the mean value of V for the K and M giants separately. For the 250 M giants, we obtain hV i = −23.42 ± 1.48 km s−1 (3.6) while for the 2524 K giants we obtain

hV i = −20.22 ± 0.44 km s−1. (3.7)

This difference (at the 2σ level) between the two subsamples can be understood in terms of the age-velocity dispersion relation. Indeed, the M giants must be a little older than the K giants on average because only the low-mass stars can reach the spectral type M on the Red Giant Branch and because the lifetime on the main sequence is longer for lower mass stars. This implies that the subsample of M giants has a larger velocity dispersion and rotates more slowly about the Galactic Center than the subsample of K giants, in agreement with the asymmetric drift relation. The velocity dispersion tensor is defined as h(~v −h~vi)⊗(~v −h~vi)i (see Section 1.3.5). The diagonal components are the square of the velocity dispersions while the mixed components correspond to the covariances. For the stars with a relative parallax error smaller than 20%, we obtain the classical ordering for the diagonal components, i.e. σU > σV > σW :

−1 σU = 34.46 ± 0.46 km s −1 σV = 22.54 ± 0.30 km s (3.8) −1 σW = 17.96 ± 0.24 km s .

The asymmetric drift relation predicts a linear dependence of hvi = hV i + V 2 with the radial velocity dispersion σU . If we adopt the peculiar velocity of the −1 Sun from Dehnen & Binney (1998a), V = 5.25 km s , we find for the full sample (K and M together)

hvi = −15.26 km s−1. (3.9)

On the other hand, if we adopt the parameter k = 80 ± 5 km s−1 from the asymmetric drift equation of Dehnen & Binney (1998a) and Binney & Merrifield (1998), we find 2 −1 hvi = −σU /k = −14.9 ± 1.3 km s . (3.10) This independent estimate of hvi is in accordance with (3.9) and our values of 2 hV i and σU are thus in good agreement with this value of k. As we already noticed, the M giants are a little bit older than the K giants on average and we thus expect the velocity dispersions of the subsample of M giants to be higher. We find

−1 σU (M giants) = 35.95 ± 1.6 km s , (3.11) 56 and −1 σU (K giants) = 34.32 ± 0.48 km s . (3.12) As expected, the difference is not very large, and the radial velocity dispersions for the two subsamples satisfy the asymmetric drift equation (3.10). The mixed components of the velocity dispersion tensor involving vertical motions vanish within their errors. Nevertheless, the mixed component in the 2 plane, that we denote σUV , is non-zero: this is not allowed in an axisymmetric Galaxy (see Section 1.3.5) and is further discussed in Section 3.3. We obtain

2 2 −2 σUV = 134.26 ± 13.28 km s (3.13) where the error corresponds to the 15 and 85 percentiles of the correlation coefficient, assuming that the sample of U, V velocities is drawn from a two- dimensional gaussian distribution. In order to parametrize the deviation from dynamical axisymmetry, a useful quantity is the vertex deviation lv (see Eq. (1.56):

2 2 2 ◦ ◦ lv ≡ 1/2 arctan (2σUV /(σU − σV )) = 10.85 ± 1.62 . (3.14) This vertex deviation for giant stars is not in accordance with a perfectly axisym- metric Galaxy and could be caused by non-axisymmetric perturbations in the Solar neighbourhood (see Section 1.2.7), or by a deviation from equilibrium (i.e. moving groups due to inhomogeneous star formation, following the classical the- ory of moving groups; see Section 1.3.5). A hint to the true nature of this vertex deviation could be the local anomaly in the UV -plane, the so-called u-anomaly (e.g. Raboud et al. 1998). If we calculate the mean velocity hUi of the stars with V < −35 km s−1, we find that it is largely negative (hUi = −22 km s−1). It denotes a global outward radial motion of the stars that lag behind the galactic rotation. We see on Fig. 3.1 that this anomaly is due to a clump located at U ' −35 km s−1, V ' −45 km s−1, the already known “Hercules” stream (Fux 2001). On the other hand, Fig. 3.1 reveals in fact a rich small-scale structure in the UV -plane, with several clumps which can be associated with known kine- matic features: we clearly see small peaks at U = −40 km s−1, V = −25 km s−1 (corresponding to the Hyades supercluster), at U = 10 km s−1, V = −5 km s−1 (Sirius moving group), and at U = −15 km s−1, V = −25 km s−1 (Pleiades supercluster). More precise values for these peaks will be given in Section 3.3 where their origin will also be discussed.

3.2 Monte Carlo simulation

To reduce the bias introduced by the non-linearity of the parallax-distance trans- formation, the results discussed in the previous Section were obtained by re- stricting the sample to stars with a relative parallax error smaller than 20%. In this Section, we use all the available stars, without any parallax truncation, but perform a Monte-Carlo simulation to properly evaluate the errors and biases on the kinematic parameters. 57

Figure 3.1: Density of stars with precise parallaxes (σπ/π ≤ 20%) in the UV - plane. The colours indicate the number of stars in each bin. The contours indicate the bins with 3, 4, 7, 12, 17, 20, 25, 30, 35 and 40 stars respectively. The concentration of stars around U ' −35 km s−1, V ' −45 km s−1 contributes largely to the vertex deviation, while other peaks already identified by Dehnen (1998) at U = −40 km s−1, V = −25 km s−1 (Hyades supercluster), at U = 10 km s−1, V = −5 km s−1 (Sirius moving group), and at U = −15 km s−1, V = −25 km s−1 (Pleiades supercluster) are also present. 58

We constructed synthetic samples by drawing the stellar parallax from a gaussian distribution centered on the observed parallax value for the correspond- ing star, and with a dispersion corresponding to the uncertainty on the observed parallax. Note that this procedure is not strictly correct, as the gaussian distri- bution should in fact be centered on the (unknown) true parallax. To overcome this difficulty, a fully bayesian approach will be applied in yet another analysis of the data, as described next in Section 3.3. Some a priori information has nevertheless been included to truncate the gaussian parallax distribution in the present Monte Carlo approach, since the giant stars in our sample should not be brighter than MHp = −2.5 (see Figs. 3.6, 3.7, 3.8, 3.10). That threshold has been increased to MHp = −5 (see Fig.3.4) for the stars flagged as supergiants. This prescription thus corresponds to assigning a minimum admissible parallax πmin to any given star. It prevents that very small parallaxes drawn from the gaussian distribution yield unrealistically large space velocities. The parallax distribution used in the Monte Carlo simulation thus writes

! 1 1 π − π 2 P (π) = exp − obs if π ≥ π (3.15) 1/2 min (2Π) σπobs 2 σπobs

P (π) = 0 if π < πmin (3.16) The mean space velocities and the velocity dispersions are calculated for each simulated sample, and finally we adopt the average of these values (of the mean velocities and of the velocity dispersions) over 4000 simulated samples as the best estimate of the true kinematic parameters. We thus obtain:

hUi = −10.25 ± 0.15 km s−1 hV i = −22.81 ± 0.15 km s−1 (3.17) hW i = −7.98 ± 0.09 km s−1. If there is no net radial and vertical motion in the Solar neighbourhood (see however Section 3.3.6), we may write U = −hUi and W = −hW i. The results given by (3.17) are very close to those estimated from stars with relative errors on the parallax smaller than 20% (see (3.5)). They are in agreement with the values derived by Dehnen & Binney (1998a) on the basis of Hipparcos −1 proper motions of main sequence stars (U = 10.00 ± 0.36 km s , W = 7.17 ± −1 0.38 km s ). The value of U is not in perfect accordance with the one derived −1 by Brosche et al. (2001), who found U = 9.0 ± 0.5 km s from photometric distances and Hipparcos proper motions of K0-5 giants, nor with the one derived −1 by Zhu (2000) who found U = 9.6 ± 0.3 km s with the same stars as ours but without the radial velocity data. The value of W contradicts slightly the one −1 derived by Bienaym´e(1999), who found W = 6.7 ± 0.2 km s from Hipparcos proper motions. Nevertheless, these considerations on the Solar motion are not very useful since we stress in Section 3.3.6 that the motion of the Sun is difficult to derive anyway because there are conceptual uncertainties on the mean motion of stars in the Solar neighbourhood. 59

For the velocity dispersions, we find

−1 σU = 40.72 ± 0.58 km s −1 σV = 32.23 ± 1.41 km s (3.18) −1 σW = 22.55 ± 0.95 km s . These velocity dispersions are somewhat larger than those found for the restricted sample (3.8), which is not surprising since 60% of the high-velocity stars (as identified in Section 3.3.6) are not present in the sample restricted to the most precise parallaxes. It could also be a slight effect of the larger volume of the Galaxy probed by the complete sample. Regarding the asymmetric drift and the slightly larger age of the M giants, the Monte Carlo method and the analysis of the restricted sample reach the same conclusion. Concerning the mixed component of the velocity dispersion tensor in the plane, we find 2 2 −2 σUV = 188.23 ± 40.9 km s (3.19) which leads to a clearly non-zero vertex deviation of

2 2 2 ◦ ◦ lv ≡ 1/2 arctan (2σUV /(σU − σV )) = 16.2 ± 5.6 . (3.20)

Interestingly, Soubiran et al. (2003) found no vertex deviation for low- metallicity stars in the disk, and concluded that this was consistent with an axisymmetric Galaxy: this is absolutely not the case for our sample of late-type giants. Furthermore, the value we find for the vertex deviation is larger than the one derived for late-type stars by Dehnen & Binney (1998a), who showed that the vertex deviation drops from 30◦ for young stellar populations (maybe due to young groups concentrated near the origin of the UV -plane, see also Sections 1.3.5 and 3.3.7) to a constant value of 10◦ for older populations. Bi- enaym´e(1999) also found from Hipparcos proper motions a vertex deviation of 9.2◦ for the giant stars. We conclude from our sample that the vertex deviation is significantly non-zero: we will suggest a possible origin for this vertex devi- ation (which could be the same origin as for the vertex deviation of younger populations) in Section 3.3.7.

3.3 Bayesian approach

To obtain the kinematic characteristics of our sample in a more rigorous way, we have applied the Luri-Mennessier (LM) method described in detail by Luri (1995) and Luri et al. (1996). The starting point of this method is a model describing the basic morphological characteristics we can safely expect from the sample (spatial distribution, kinematics and absolute magnitudes), and a model of the selection criteria used to define the sample. These models are used to build a distribution function intended to describe the observational characteristics of the sample. The a priori distribution function adopted is a linear combination of partial distribution functions, each of which describes a group of stars (called “base group”). Each partial distribution function combines a kinematic model 60

(the velocity ellipsoid introduced by Schwarzschild 1907; see Section 1.3.4) with a gaussian magnitude distribution, and an exponential height distribution uncor- related with the velocities. The phenomenological model adopted is obviously not completely rigorous, but has the advantage of being able to identify and quantify the different subgroups present in the data and possibly related to ex- tremely complex dynamical phenomema, which cannot be easily parametrized. The values of the parameters of the distribution function can be determined from the sample using a bayesian approach: the model is adjusted to the sample by a maximum likelihood fit of the parameters. The values of the parameters so obtained provide the best representation of the sample given the a priori models assumed. In the following Sects. 3.3.1 and 3.3.2 we describe the ingredients of the models used in this Chapter. Thereafter, in Sects. 3.3.3 to 3.3.6, we present the results of the maximum likelihood fit, and we discuss in Section 3.3.7 the different possible physical interpretations of those results.

3.3.1 Phenomenological model To build a phenomenological model of our sample, we assume that it is a mixture of stars coming from several “base groups”. A given group represents a fraction wi of the total sample and its characteristics are described by the following components:

• Spatial distribution: An exponential disk of scale height Z0 in the direction perpendicular to the galactic plane

  |d sin b| 2 ϕe(d, l, b) = exp − d cos b (3.21) Z0

and a uniform distribution along the galactic plane (a realistic approxi- mation for samples in the Solar neighbourhood like ours).

• Velocity distribution: A Schwarzschild ellipsoid for the velocities of the stars with respect to their Standard of Rest:

“ 0 ”2 “ 0 ”2 “ ”2 − 1 U − 1 V − 1 W −W0 0 0 2 σ 0 2 σ 0 2 σ (3.22) ϕv(U ,V ,W ) = e U V W

where 0 U = (U − U0) cos lv − (V − V0) sin lv 0 (3.23) V = (U − U0) sin lv + (V − V0) cos lv

and where lv is the vertex deviation. • Galactic rotation: An Oort-Lindblad rotation model at first order (see Section 1.3.2), where the rotation velocity is added to the mean LSR velocity (given by the Schwarzschild ellipsoid above). The same values were adopted for the Oort constants as in Sects. 3.1 and 3.2 61

• Luminosity: We have adopted a gaussian distribution for the absolute magnitudes of the stars:

 2! 1 M − M0 ϕM(M) = exp − . (3.24) 2 σM

This is just a first rough approximation. A better model would include a dependence of the absolute magnitude on a colour index but is more complex to implement (we leave it for future studies).

• Interstellar absorption: In the LM method, the correction of interstellar absorption is integrated in the formalism. A model, giving its value as a function of the position (d, l, b), is needed: the Arenou et al. (1992) model has been chosen here.

The distribution function of each base group is simply the product of the space, velocity and magnitude distributions presented here, with different values for the model parameters. The total distribution function D of the sample in phase-magnitude space is a linear combination of the partial distribution functions of the different base groups. Both the relative fractions of the different base groups and the model pa- rameters will be determined using a Maximum Likelihood fit, as described in Section 3.3.3. However, the number of groups (ng) composing our sample is not known a priori. A likelihood test – like Wilk’s test, Soubiran et al. (1990) – will be used to determine it: maximum likelihood estimations are performed with ng = 1, 2, 3,... and the maximum likelihoods obtained for each case are compared using the test to decide on the most likely value of ng.

3.3.2 Observational selection and errors As pointed out above, the correct description of the observed characteristics of the sample requires that its selection criteria be included in the model. In our case, our sample of giants (like the Hipparcos Catalogue as a whole) is composed primarily of stars belonging to the Hipparcos survey plus stars added on the basis of several heterogeneous criteria. In the case of the Hipparcos survey, the selection criteria are just based on the apparent V magnitude. For a given line of sight, with galactic latitude b, the survey is complete up to

V = 7.3 + 1.1| sin b|. (3.25)

In our case, the Hp Hipparcos magnitude has been used instead of V . There- fore the survey stars in our sample will follow a similar “completeness law” in Hp that we have adopted to be approximately

Hp = 7.5 + 1.1| sin b| (3.26) 62

assuming an average Hp − V of 0.2 mag for the stars in our sample. This law, altogether with the cutoff in declination (δ > 0◦), allows us to quite realistically model the selection of the survey stars in the sample, but leaves out 10% of the total sample. In order to be able to use the full sample, we must define the selection of the non-survey stars as well, as (approximately) done by the following completeness law:

Completeness linearly decreasing from 1 at Hp = 7.5 + 1.1| sin b| down to zero at Hp = 11.5

This condition together with the completeness up to Hp = 7.5 + 1.1| sin b|, and with the cutoff in declination, defines the selection of the complete sample. The individual errors on the astrometric and photometric data are also taken into account in the model. We assume that the observed values are produced by gaussian distributions of observational errors around the “true” values, with standard deviations given by the errors quoted in the Hipparcos Catalogue, the Tycho-2 catalogue or the CORAVEL database. In the end, we can define for each star a joint distribution function of the true and observed values:

µ(~x,~zkθ~) = D(~xkθ~) (~zk~x) S(~z) (3.27) where ~x are the true values (position in the LSR, velocities with respect to the LSR, absolute magnitude), ~z the observed values (position on the sky, parallax, proper motions, radial velocity, apparent magnitude), θ~ the set of parameters of the model, S the selection function taking into account the selection criteria of the sample, and  the gaussian distribution of observational errors. For each star, we can then easily deduce the distribution function of the observed values: Z O(~zkθ~) = µ(~x,~zkθ~) d~x (3.28)

The fact that the selection function and the error distribution are taken into account in this distribution function prevents from all the possible biases.

3.3.3 Maximum likelihood The principle of maximum likelihood (ML) can be briefly described as follows: let ~z be the random variable of the observed values following the density law ~ given by O(~zkθ0), where θ~ = (w ,M , σ ,U , σ 0 ,V , σ 0 ,W , σ ,Z , l , w , . . . , l ) is the set 0 1 01 M1 01 U1 01 V1 01 W1 01 v1 2 vng of unknown parameters on which it depends. The Likelihood Function is defined as

n∗ ~ Y ~ L(θ) = O(~zikθ). (3.29) i=1 63

~ ~ The value of θ which maximizes this function is the ML estimator, θML, ~ of the parameters θ0 characterizing the density law of the sample. It can be ~ shown that θML is asymptotically non-biased, asymptotically gaussian and that for large samples, it is the most efficient estimator (see Kendall & Stuart 1979). Once the ML estimator of the parameters has been found, simulated sam- ples (generation of random numbers following the given distribution) are used to check the equations and programs developed for the estimation. They also al- low a good estimation of the errors on the results and the detection of possible ~ biases: once a ML estimation θML has been obtained, several samples corre- ~ sponding to the parameters θML are simulated and the method is applied to them. The comparison of the results (which could be called ”the estimation of ~ the estimation”) with the original θML allows us to detect possible biases. The ~ dispersion of these results σ(θML) can be taken as the error of the estimation. The low maximum likelihood obtained when the full sample is modelled with a single base group indicates that the kinematic properties of giant stars in the Solar neighbourhood cannot be fitted by a single Schwarzschild ellipsoid. The first acceptable solution requires three base groups: one of bright giants and supergiants with “young” kinematics (further discussed in Section 3.3.6), one of high-velocity stars and finally one group of “normal” stars. This third group exhibits plenty of small-scale structure, and this small-scale structure can be successfully modelled by 3 more base groups (leading to a total of 6 groups in the sample, called groups Y, HV, HyPl, Si, He, and B, further discussed in Section 3.3.6). These supplementary groups are statistically significant, as revealed by a Wilks test. Statistically, solutions with 7 or 8 groups are even better, but these solutions are not stable anymore and are thus not useful (they depend too much on the observed values of some individual stars). Table 3.1 lists the values of the ML parameters for a model using just the 5177 (out of 5397) survey stars which comply with the completeness law (Eq. (3.26)). Table 3.2 lists those values for a model using the whole sample of 6030 stars with our modified selection law. The models for the survey stars alone and for the whole sample give very similar results (except for the scale height Z0 of group HV – see Section 3.3.6 – which is not well constrained since our sample does not go far enough above the galactic plane), thus providing a strong indication that the results for the whole sample are reliable. The errors on the parameters listed in Tables 3.1 and 3.2 are the dispersions coming from the simulated samples.

3.3.4 Group assignment The probabilities for a given star to belong to the various groups is provided by the LM method, and the most probable group along with the corresponding probability is listed in Table A.1(on the CD-ROM joined to this thesis). Indeed, let wj be the a priori probability (i.e. the ML parameter of Table 3.2) that a ~ star belongs to the jth group and Oj(~z | θj) the distribution of the observed quantities ~z in this group (deduced from the phenomenological model adopted ~ and depending on the parameters θj of this model for the group). Then, using 64

Table 3.1: Maximum-Likelihood parameters obtained using 5177 survey stars −1 only. The velocities are expressed in km s , the distances for Z0 in pc, and the vertex deviation in degrees. The number n of stars belonging to each group (as listed in column 34 of Table A.1) is given in the last row. Although obtained from the assignment process described in Section 3.3.4, it is not, strictly, a ML- parameter, in contrast to the fraction wi (i = 1, ...6) (expressed here in %) of the whole sample belonging to the considered group, as listed in the previous row. Therefore, the observed fraction of stars in each group could be slightly different from the true fraction wi of that group in the entire population. group Y group HV group HyPl group Si group He group B M0 0.6 ± 0.9 2.2 ± 0.2 1.1 ± 0.4 1.1 ± 0.6 1.3 ± 0.4 1.2 ± 0.2 σM 1.8 ± 0.2 1.3 ± 0.1 0.9 ± 0.2 1.2 ± 0.2 0.9 ± 0.1 1.1 ± 0.1 U0 -11.6 ± 2.3-17.6 ± 3.5 -31.2 ± 1.0 5.2 ± 3.0 -42.1 ± 4.8 -2.9 ± 1.5 σU 0 15.6 ± 1.2 53.4 ± 2.1 11.5 ± 1.5 13.5 ± 2.7 26.1 ± 13.1 31.8 ± 1.6 V0 -11.6 ± 2.1-43.9 ± 3.2 -20.0 ± 0.7 4.2 ± 1.9 -50.6 ± 4.7 -15.1 ± 2.4 σV 0 9.4 ± 1.4 36.1 ± 1.8 4.9 ± 1.1 4.6 ± 4.2 8.6 ± 3.0 17.6 ± 0.8 W0 -7.8 ± 0.9 -7.8 ± 2.3 -4.8 ± 1.8 -5.6 ± 2.1 -6.9 ± 3.6 -8.2 ± 0.9 σW 6.9 ± 0.8 32.5 ± 1.5 8.8 ± 1.1 9.4 ± 2.2 16.7 ± 1.6 16.3 ± 0.8 Z0 73.4 ± 6.3 901 ± 440 128.7 ± 19.2149.4 ± 23.7201.8 ± 55.2196.1 ± 12.3 lv 17.1 ± 5.8 0.2 ± 4.9 -5.6 ± 4.6 -14.2 ± 25.3 -5.7 ± 11.2 -0.2 ± 5.2 % 8.6 ± 1.3 14.9 ± 2.2 7.1 ± 0.7 5.0 ± 3.0 6.5 ± 1.7 57.9 ± 4.7 n 345 505 334 204 372 3417

Table 3.2: Same as Table 3.1 for the full sample of 6030 stars group Y group HV group HyPl group Si group He group B M0 0.7 ± 0.2 2.0 ± 0.2 1.0 ± 0.1 0.9 ± 0.2 1.2 ± 0.2 1.0 ± 0.1 σM 1.8 ± 0.1 1.4 ± 0.1 0.7 ± 0.1 1.1 ± 0.1 0.9 ± 0.1 1.1 ± 0.1 U0 -10.4 ± 0.9 -18.5 ± 2.8 -30.3 ± 1.5 6.5 ± 1.9 -42.1 ± 1.9 -2.8 ± 1.1 σU 0 15.4 ± 1.1 58.0 ± 1.9 11.8 ± 1.3 14.4 ± 2.0 28.3 ± 1.7 33.3 ± 0.7 V0 -12.4 ± 0.9 -53.3 ± 3.1 -20.3 ± 0.6 4.0 ± 0.7 -51.6 ± 1.1 -15.4 ± 0.8 σV 0 9.9 ± 0.7 41.4 ± 1.7 5.1 ± 0.8 4.6 ± 0.7 9.3 ± 1.2 17.9 ± 0.8 W0 -7.7 ± 0.6 -6.6 ± 1.8 -4.8 ± 0.8 -5.8 ± 1.1 -8.1 ± 1.3 -8.3 ± 0.4 σW 6.7 ± 0.6 39.1 ± 1.7 8.7 ± 0.7 9.7 ± 0.8 17.1 ± 1.6 17.6 ± 0.3 Z0 80.3 ± 6.2 208.1 ± 27.9106.3 ± 14.4128.0 ± 19.6132.9 ± 9.4141.2 ± 5.2 lv 16.4 ± 10.3 0.2 ± 4.8 -8.8 ± 4.1 -11.9 ± 3.5 -6.5 ± 2.8 -2.2 ± 1.6 % 9.6 ± 0.8 10.6 ± 1.0 7.0 ± 0.8 5.3 ± 0.9 7.9 ± 0.9 59.6 ± 1.5 n 413 401 392 268 529 4027 65

Bayes formula, the a posteriori probability for a star to belong to the jth group given its measured values ~z∗ is:

~ wjO(~z∗|θj) P (∗ ∈ Gj | ~z∗) = . (3.30) Png ~ k=1 wkOk(~z∗|θk) Using this formula the a posteriori probabilities that the star belongs to a given group can be compared, and the star can be assigned to the most likely one (Table A.1). Note that this procedure, like any method of statistical classification, will have a certain percentage of misclassifications. However, the reliability of each assignment is clearly indicated by the probability given in column 35 of Ta- ble A.1.

3.3.5 Individual distance estimates Once a star has been assigned to a group, and given the ML estimator of the group parameters and the observed values for the star ~z∗, one can obtain the marginal probability density law R(d) for the distance of the star from the global probability density function.

This can then be used to obtain the expected value of the distance

Z ∞ d = d R(d) dd (3.31) 0 and its dispersion

Z ∞ 2 2 d = (d − d) R(d) dd. (3.32) 0 The first can be used as a distance estimator free from biases and the second as its error (Columns 28 and 29 of Table A.1). Fig. 3.2 reveals that the biases in the distance derived from the inverse parallax are a combination of those resulting from truncations in parallax and apparent magnitude as discussed in Luri & Arenou (1997).

3.3.6 The kinematic groups present in the stellar sample We have plotted in Fig. 3.3 the stars in the UV -plane by using the expected values of U and V deduced from the LM method. The 6 different groups are represented on this figure. The structure of the UV -plane is similar to the one already identified (in Fig. 3.1) for the stars with precise parallaxes (σπ/π ≤ 20%). This similarity is a strong indication that the subgroups identified by the LM method are not artifacts of this method. Several groups can be identified with known kinematic features of the Solar neighbourhood. 66

1000

800

600

400

200

0 0 200 400 600 800 1000

Figure 3.2: Comparison of the (biased) distances obtained from a simple inver- sion of the parallax, and the maximum-likelihood distances d obtained from the LM method. 67

100

0

-100

-200

-200 -100 0 100 U (km/s)

Figure 3.3: All the stars plotted in the UV -plane with their values of U and V deduced from the LM method. The 6 different groups are represented on this figure: group Y in yellow, group HV in blue, group HyPl in red, group Si in magenta, group He in green and group B in black. Note that the yellow group (Y) extends just far enough to touch both the red (HyPl) and magenta (Si) groups. 68

-6

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2

4 0 1 2 3 4 V-I

Figure 3.4: HR diagram of group Y. The absolute magnitude used here is MV . Isochrones of Lejeune & Schaerer (2001) for Z = 0.008 and log(age(yr))= 7.4, 7.6, 8, 8.3, 8.55, 8.75, 8.85, 9. V − I indices were computed from the colour transformation of Platais et al. (2003) 69

Group Y: The young giants

The first group is one with “young” kinematics (yellow group on Fig. 3.3), ◦ with a small velocity dispersion and with a vertex deviation of lv = 16.4 . Quite remarkably, the value of hMHpi for this group is 0.7 (see Table 3.2), the brightest among all 6 groups. An important property of this group is thus its young kinematics coupled with its large average luminosity: the most luminous giants and supergiants 1 in our sample thus appear to have a small dispersion in the UV -plane, in agreement with the general idea that younger, more massive giants are predominantly found at larger luminosities in the Hertzsprung-Russell diagram and are, at the same time, concentrated near the origin of the UV - plane. It must be stressed that nothing in the LM method can induce such a correlation between velocities and luminosity artificially. This result must thus be considered as a robust result, even more so since the group Y of young giants was present in all solutions, irrespective of the number of groups imposed. This group is centered on the usual antisolar motion (Dehnen & Binney 1998a) in U and W as seen in Table 3.2. Fig. 3.4 locates stars from group Y in the HR diagram, constructed from the V − I colour index as provided by the colour transformation of Platais et al. (2003), based on the measured Hp −VT 2 colour index (and is thus more accurate than the Hipparcos V −I index; see Appendix for more details). The isochrones of Lejeune & Schaerer (2001), for a typical metallicity Z = 0.008, indicate that the age of stars from group Y is on the order of several 106 to a few 108 yr. It must be remarked at this point that the observed members of group Y, as displayed on Fig. 3.4, appear to be brighter than the true average absolute magnitude for the group listed in Tables 3.1 and 3.2 (hMHpi = 0.7) and as- signed by the LM method. This is a natural consequence of the Malmquist bias (Malmquist 1936) for a magnitude-limited sample. Nevertheless, even the true average absolute magnitude makes group Y the brightest and youngest one.

Group HV: The high-velocity stars

This group is composed of high-velocity stars (represented in blue on Fig. 3.3). Those stars are probably mostly halo or thick-disk stars, even though the value of the scale height of this population is poorly constrained (hence the large difference between that parameter derived from the full sample – Table 3.2– and from survey stars only – Table 3.1) because our sampling distance is too small: indeed Fig. 3.2 shows that the distance of most stars in our sample is smaller than the scale height of the thick disk (665 pc < Z0,thick < 1000 pc). This group represents about 10% of the whole sample (Table 3.2, although, like Z0, this parameter is not well constrained), and this value is consistent with the mass fraction of the thick disk relative to the thin disk in the Solar neighbourhood.

1almost all stars a priori flagged as supergiants (see Sects. 2.2, 3.2 and Fig. 2.5) indeed belong to group Y 70

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6 0 1 2 3 V-I

Figure 3.5: HR diagram of group HV. Isochrones of Lejeune & Schaerer (2001) for Z = 0.008 and log(age(yr))= 8.3, 8.55, 8.75, 8.85, 9, 9.3, 9.45. 9.5, 9.6, 9.7. V − I indices were computed from the colour transformation of Platais et al. (2003) 71

Going back to the different results obtained for the velocity dispersions in Eqs. (3.8) and (3.18), notice that 247 stars (out of 401) of this group have relative errors on their Hipparcos parallax higher than 20%. Clearly we see on Fig. 3.5 that stars of this group are old, mostly older than 1 Gyr and it seems clear that some stars at the bottom of the HR diagram are as old as the Galaxy itself. It is also striking that the envelope of this kinematically “hot” group seems to correspond to a portion of circle approximately centered on a zero-velocity frame with respect to the galactic center, and with a radius of the order of 280 km s−1. We refrain, however, from providing numerical values for the position of this circular envelope in the UV -plane, because of the rather limited number of stars defining this limit and the absence of antirotating halo stars in our sample. Note that a lower limit to the local escape velocity is provided by the velocity (U 2 +(V +220)2 +W 2)1/2 = 281 km s−1 of HIP 89298, the fastest star in group HV. The detection of this group of high-velocity stars “cleans” the sample and allows us to study the fine structure of the velocity distribution of disk stars.

Group HyPl: Hyades-Pleiades supercluster The group represented in red in Fig. 3.3 occupies the well-known region of the Hyades and Pleiades superclusters. The large spatial dispersion of the HyPl stars clearly hints at their supercluster rather than cluster nature (see Section 1.3.5), since they are spread all over the sky with a wide range of distances (up to 500 pc). The Hyades (hUi = −40 km s−1, hV i = −25 km s−1, see Dehnen 1998) and Pleiades (hUi = −15 km s−1, hV i = −25 km s−1, see Dehnen 1998) superclusters are known since Eggen (1958, 1975). It was also noticed (Eggen 1983) that several young clusters (NGC 2516, IC 2602, α Persei) have the same V -component as those superclusters. We obtain hV i = −20.3 km s−1for our supercluster structure, and thus refine the old value hV i = −25 km s−1, with in addition the fact that the group is tilted in the anti-diagonal direction of the ◦ UV -plane, i.e., it has a negative vertex deviation (lv = −8.7 ). The concept of branches in the UV -plane (Skuljan et al. 1999), of quasi-constant V but slightly tilted in the anti-diagonal direction, will be further discussed below in relation with the other groups. Metallicity is available for 17 stars of the HyPl group (McWilliam 1990), and it is interesting to remark that the average metallicity seems to be close to Solar ([Fe/H]= 0; Fig. 3.9) as already noticed by Chereul & Grenon (2001). This is quite unusual for giant stars. The metallicity of the stars from group B for example is centered on [Fe/H]= −0.2. Plotting the HyPl stars in a HR dia- gram (Fig. 3.6) and using the Solar-metallicity isochrones of Lejeune & Schaerer (2001) reveals that the stars forming the HyPl group are by far not coeval (de- spite the fact that HyPl stars have a smaller age spread than the Si and He groups discussed below, especially because the HyPl group is lacking young su- pergiants). Although a precise determination of the ages of HyPl stars would require the knowledge of their metallicities, it seems nevertheless clear that there are a lot of clump stars with ages reaching 1 Gyr, and that there are clearly stars 72

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6 0 0.5 1 1.5 V-I

Figure 3.6: HR diagram of group HyPl. Isochrones of Lejeune & Schaerer (2001) for Z = 0.02 and log(age(yr))= 8.3, 8.55, 8.75, 8.85, 9, 9.3, 9.45. 9.5, 9.6, 9.7. We calculated V-I using the colour transformation of Platais et al. (2003). The stars at the bottom-right of the diagram could be very old metal rich stars (in contradiction with the age-metallicity relation which is obviously not correct if no other factors are taken into account). 73 older than 2 Gyr, in sharp contrast with the ages of about 80 and 600 Myr for the Pleiades and Hyades clusters themselves. This result of age heterogeneity, already noticed by Chereul & Grenon (2001) for the Hyades supercluster, is an important clue to identify the origin of supercluster-like structures, as discussed in Section 3.3.7. Finally, the value hW i = −4.8 ± 0.8 km s−1 differs significantly from the −1 vertical Solar motion (W = 7 to 8 km s ; see Sects. 3.1, 3.2, and group B in Tables 3.1 and 3.2), indicating that the group has a slight net vertical motion.

Group Si: Sirius moving group The Sirius moving group (represented in magenta in Fig. 3.3) is known since Eggen (1958, 1960) and is traditionally located at hUi = 10 km s−1, hV i = −5 km s−1 (Dehnen 1998). The LM method refines those values and locates it at hUi = 6.5 km s−1, hV i = 3.9 km s−1. The spatial distribution once again indicates that this group has a supercluster-like nature. Metallicity from McWilliam (1990) is available for 12 stars of the Si group (Fig. 3.9). Contrarily to the situation prevailing for the HyPl group, the metallicity distribution within the Si group appears similar to that for the bulk of the giants in the Solar neighbourhood, as represented by group B. Isochrones for a typical value of Z = 0.008 (Fig. 3.7) indicate that the ages are widely spread, even more so than for the HyPl group.

Group He: Hercules stream The “Hercules stream” (represented in green in Fig. 3.3), located by the LM method at hUi = −42 km s−1, hV i = −51 km s−1has been named by Raboud et al. (1998) the u-anomaly. It corresponds to a global outward radial motion of the stars which lag behind the galactic rotation: known since Blaauw (1970), it is traditionally centered on hUi = −35 km s−1, hV i = −45 km s−1(see Fux 2001). It is strongly believed since several years that its origin is of dynamical nature, as further discussed in the next section. Once again, the Hertzsprung- Russell diagram of Fig. 3.8 indicates a wide range of ages in this group (for a typical value of Z = 0.008). Note that the three groups HyPl, Si and He could be interpreted in term of extended branches crossing the UV -plane (Skuljan et al. 1999, Nordstr¨omet al. 2004): one is Hercules, another is the combination of Hyades and Pleiades, and the third one is Sirius. The high dispersion of the streams azimuthal (V ) component confirms this view. The branches are somewhat tilted along the anti-diagonal direction. These properties could be understood in the classical theory of moving groups as discussed in next Section but also in the context of a dynamical origin for the substructure.

Group B: Smooth Background Most stars are part of an “axisymmetric”, “smooth” background represented in black in Fig. 3.3. The average metallicity of this group seems to be slightly 74

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Figure 3.7: HR diagram of group Si. Isochrones of Lejeune & Schaerer (2001) for Z = 0.008 and log(age(yr))= 8.3, 8.55, 8.75, 8.85, 9, 9.3, 9.45. 9.5, 9.6, 9.7. 75

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Figure 3.8: HR diagram of group He. Isochrones of Lejeune & Schaerer (2001) for Z = 0.008 and log(age(yr))= 8.3, 8.55, 8.75, 8.85, 9, 9.3, 9.45. 9.5, 9.6, 9.7. 76

40

group B group HyPl group Si

30

20

10

0 -0.6 -0.4 -0.2 0 0.2 [Fe/H]

Figure 3.9: Histogram of the metallicity in groups B, HyPl and Si for the stars present in the analysis of McWilliam (1990) 77

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Figure 3.10: HR diagram of group B. Isochrones of Lejeune & Schaerer (2001) for Z = 0.008 and log(age(yr))= 8.3, 8.55, 8.75, 8.85, 9, 9.3, 9.45. 9.5, 9.6, 9.7. 78 subsolar (Fig. 3.9) as expected for a sample of disk giants: Girardi & Salaris (2001) found an average metallicity of [Fe/H]=−0.12 ± 0.18, while for their sample of F and G dwarfs, Nordstr¨omet al. (2004) found [Fe/H]=−0.14 ± 0.19. Using isochrones for a typical value of Z = 0.008 (see Fig. 3.10), we see that there is a large spread in age. This is typical of the mixed population of the galactic disk, composed of stars born at many different epochs since the birth of the Galaxy. In the UV -plane, the velocity ellipsoid of this group is not centered on the value commonly accepted for the antisolar motion: it is centered instead on hUi = −2.78 ± 1.07 km s−1. However, the full data set (including the various superclusters) does yield the usual value for the Solar motion (see Section 3.2). This discrepancy clearly raises the essential question of how to derive the Solar motion in the presence of streams in the Solar neighbourhood: does there exist in the Solar neighbourhood a subset of stars having no net radial motion? If the smooth background is indeed an axisymmetric background with no net radial motion, we have found a totally different value for the Solar motion. Never- theless, we have no strong argument to assess that this is the case, especially if the “superclusters” have a dynamical origin, as proposed in the next section. Moreover, the group Y of young giants is centered on the commonly accepted value of hUi = −10.41 ± 0.94 km s−1 and this difference between groups Y and B clearly prevents us from deriving without ambiguity the Solar motion. If the giant molecular clouds (GMC’s) from which the young stars arose are on a circular orbit, then the hUi value of group Y is the acceptable one for the antisolar motion, but nothing proves that the GMC’s are not locally moving outward in the Galaxy under the effect of the spiral pattern. No value can thus at the present time be given for the radial Solar motion but only some differ- ent estimates depending on the theoretical hypothesis we make on the nature of the substructures observed in velocity space. Theoretical investigations and dynamical simulations thus appear to be the only ways to solve this problem. On the other hand, the value of hW i = −8.26 ± 0.38 km s−1 for group B is in accordance with the usual motion of the sun perpendicular to the galactic disk (see Section 3.2), and thus seems to be a reliable value because streams have a smaller effect on the vertical motion of stars.

3.3.7 Physical interpretation of the groups Several mechanisms may be responsible for the substructrure observed in ve- locity space in the Solar neighbourhood. Hereafter, we list them and confront them with the results of our kinematic study.

Cluster remnants A first class of mechanisms is that associated with inhomogeneous star forma- tion responsible for a deviation from equilibrium in the Solar neighbourhood (see Section 1.3.5): this theory states that a large number of stars are formed (almost) simultaneously in a certain region of the Galaxy and create a cluster- 79 like structure with a well-defined position and velocity. After several galactic rotations, the cluster will evaporate and form a tube called “supercluster”. Stars in the “supercluster” still share common V velocities when located in the same region of the tube (for example in the Solar neighbourhood) for the following reason, first put forward by Woolley (1961): if the present galactocentric radius of a star on a quasi-circular epicyclic orbit equals that of the sun (denoted R ), and if such a star is observed with a peculiar velocity v = V + V , then, from Eq. (1.49), its guiding-center radius Rg writes v R = R − x = R + (3.33) g g 2B where xg is the position of its guiding-center in the cartesian reference frame (1.39) (in the Solar neighbourhood approximation, the impact of yg is negli- gible), and B is the second Oort constant (see Section 1.3.2). Woolley (1961) pointed out that disk stars (most of which move on quasi-circular epicyclic or- bits, see Section 1.2.6) which formed at the same place and time, and which stayed together in the Galaxy after a few galactic rotations (since they are all currently observed in the Solar neighbourhood) must necessarily have the same period of revolution around the Galactic center, and thus the same guiding- center Rg, and thus the same velocity V = v −V according to Eq. (3.33). This theory has thus the great advantage of predicting extended horizontal branches crossing the UV -plane (similar to those observed in our sample on Fig. 3.3 and already identified by Skuljan et al. 1999, and by Nordstr¨omet al. 2004 in their sample of F and G dwarfs). Moreover, it is easy to understand in this framework that the Group HyPl seems to be more metal-rich than the smooth background (see Fig. 3.9). However, this theory does not explain the tilt of the branches that we observe in our sample. Moreover, to explain the wide range of ages ob- served in those branches (see Figs. 3.6, 3.7, and also Chereul & Grenon 2001), the stars must have formed at different epochs (see e.g. Weidemann et al. 1992) out of only two large molecular clouds (one associated with the Sirius branch and another with the Hyades-Pleiades branch). Chereul et al. (1998) suggested that the supercluster-like velocity structure is just a chance juxtaposition of sev- eral cluster remnants, but this hypothesis requires extraordinary long survival times for the oldest clusters (with ages > 2 Gyr) in the supercluster-like struc- ture. This long survival time is made unlikely by the argument of Boutloukos & Lamers (2003) who found that clusters within 1 kpc from the Sun having 4 a mass of m × 10 M can survive up to m Gyr in the Galaxy: indeed heavy 4 clusters of more than 2 × 10 M are probably very rare in the disk. An explanation for the wide range of ages observed in the superclusters could be the capture of some older stars by the high concentration of mass in a molecular cloud, at the time of the formation of a new group of stars (while other stars would be scattered by the molecular cloud) . Though this hypothesis could be tested in N-body simulations involving gas, it would imply that the local clumps of the potential not only perturb the motion of stars but dominate it, which is in contradiction with the high predominance of the global potential on the local clumps. This explanation is thus highly unlikely too. 80

Merger remnants

The second class of mechanisms involves the theory of hierarchic formation of the galaxies (see Section 1.2.2): following this theory, galaxies were built up by the merging of smaller precursor structures. It is known since the discovery of the absorption of the Sagittarius dwarf galaxy by the Milky Way (Ibata et al. 1994) that some 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. Such streams could also be present in the velocity substructure of the disk: it seems to be the case of the Arcturus group at U ' 0 km s−1, V ' −115 km s−1, as recently proposed by Navarro et al. (2004). In this scenario, the streams observed in our kinematic study could also be the remnants of merger events between our Galaxy and a satellite galaxy. The merger would have triggered star formation whereas the oldest giant stars would be stars accreted from the companion galaxy. A merger with a satellite galaxy would moreover induce a perturbation in W , as we observe in group HyPl in which hW i = −4.8 ± 0.8 km s−1. If the hierarchical (“bottom- up”) cosmological model is correct, the Milky Way system should have accreted and subsequently tidally destroyed approximately 100 low-mass galaxies in the past 12 Gyr (see Bullock & Johnston 2004), which leads to one merger every 120 Myr, but the chance that two of them (leading to the Hyades-Pleiades and Sirius superclusters) have left such important signatures in the disk near the position of the Sun is statistically unlikely (although not impossible).

Dynamical streams

The third class of mechanisms is the class of purely dynamical ones. A dynamical mechanism that could cause substructure in the local velocity distribution is the disturbing effect of a non-axisymmetric component of the gravitational potential (Section 1.2.7), like the rotating galactic bar. The Hercules stream was recently identified with the bimodal character of the local velocity distribution (Dehnen 1999, 2000) due to the rotation of the bar if the Sun is located at the bar’s outer Lindblad resonance (OLR). Indeed, stars in the Galaxy will have their orbits elongated along or perpendicular to the major axis of the bar (orbits respectively called the LSR and OLR modes in the terminology of Dehnen 2000), depending upon their position relative to the resonances, and both types of orbits coexist at the OLR radius (see Section 1.2.7). Moreover, all orbits are regular in a 2-dimensional (2D) axisymmetric potential, but the perturbation of the triaxial bar will induce some chaos. Fux (2001) showed that in the region of the Hercules stream in velocity space, the chaotic regions, decoupled from the regular regions, are more heavily crowded. The Hercules stream has thus also been interpreted as an overdensity of chaotic orbits (Fux 2001) due to the rotating bar. Quillen (2003) has confirmed that when the effect of the spiral structure is added to that of the bar, the “chaotic” Hercules stream remains a strong feature of the local distribution function, whose boundaries are refined by the spiral structure. This stream in the local distribution function seems thus related to a non- 81 axisymmetric perturbation rather than to a deviation from equilibrium due to inhomogeneous star formation. Following M¨uhlbauer & Dehnen (2003), the Galactic bar naturally induces a non-zero vertex deviation on the order of 10◦ and the vertex deviation found in Sects. 3.1 and 3.2 would thus be partly related to the Hercules stream. Nevertheless, the other streams present in the data are also partly responsible for the vertex deviation. The likely dynamical origin of the Hercules stream has been the first example of a non-axisymmetric origin for a stream in velocity space: the other streams could thus be related to other non-axisymmetric effects. Chereul & Grenon (2001) proposed that the Hyades supercluster represents in fact an extension of the Hercules stream, but the LM method has shown in this thesis that the two features are clearly separated in the UV -plane. De Simone et al. (2004) have shown that the structure of the local distri- bution function could be due to a lumpy potential related to the presence of transient spiral waves (Julian & Toomre 1966; see Section 1.2.7 of this thesis for a detailed description of the phenomenon) in the shearing sheet (i.e. a small portion of an infinitesimally thin disk that can be associated with the Solar neighbourhood, see Section 1.2.7). These transient density waves tend to put stars in some specific regions of the UV -plane in the simulations of De Simone et al. (2004), thus creating streams as observed in our sample. These spiral waves cause radial migration in the galactic disk near their corotation radius, while not increasing the random motions and preserving the overall angular momentum distribution (Sellwood & Binney 2002; see also Section 1.2.7). The seemingly peculiar chemical composition of the group HyPl (i.e., a metallicity higher than average for field giants, as suggested by the 17 giant stars ana- lyzed by McWilliam 1990 and displayed in Fig. 3.9, also reported by Chereul & Grenon 2001) thus suggests that the group has a common galactocentric origin in the inner Galaxy (where the interstellar medium is more metal-rich than in the Solar neighbourhood) and that it was perturbed by a spiral wave at a certain moment. This specific scenario (Pont et al., in preparation) would explain why this group is composed of stars sharing a common metallicity but not a common age. The group Si could also be a clump recently formed by the passage of a transient spiral. The simulations of De Simone et al. (2004) can create streams with a range of 3 Gyr or more in age, and this is thus a mechanism that can explain that main result of our study. Another characteristic of the simulations is that they tend to reproduce the observed branches, and their origin ultimately lies in the same mechanism as that elucidated by Woolley (1961). However, the tilt of the branches in the UV -plane is not reproduced by the simulations. A question which arises in the framework of this dynamical scenario is the following: is it by chance that such a large number of young clusters and as- sociations are situated on the same branches in the UV -plane (more than half of the OB associations listed in Table A.1 of de Zeeuw 1999 are situated in the regions of the HyPl and Si groups on Fig. 3.3)? Probably not, which suggests that the same transient spiral which gave their peculiar velocity to the HyPl and Si groups has put these young clusters and associations along the same 82

UV branches. For example, it is quite striking that the Hyades cluster itself is metal-rich with a mean [Fe/H]= 0.13 (Boesgaard 1989), thus pointing towards the same galactocentric origin as the HyPl group. The entire cluster could have been shifted in radius while remaining bound since the effect of a spiral wave on stars depends on the stars’ phase with respect to the spiral, and the phase does not vary much across the cluster. Moreover, since most clusters and associations are young, they should not have crossed many transients, which suggests that one and the same spiral transient could have formed some clusters and associa- tions (by boosting star formation in the gas cloud) and could at the same time have given them their peculiar velocity in the UV -plane. The relation between spiral waves and star formation is indeed well established (e.g. Hernandez et al. 2000 who found a star formation rate SFR(t) with an oscillatory component of period 500 Myr related to the spiral pattern). As a corollary, we conclude that the dynamical streams observed among K and M giants are young kinematic features: integrating backwards (in a smooth stationnary axisymmetric poten- tial) the orbits of the stars belonging to the streams makes thus absolutely no sense, and reconstructing the history of the local disk from the present data of stars in the Solar neighbourhood becomes tricky. In this dynamical scenario, the deviation from dynamical equilibrium that is present among samples of young stars is closely related to the deviation from axisymmetry existing in the Galaxy. Of course, we do not exclude that the position of some clusters and OB associ- ations in the same region of velocity space as the dynamical streams could be the result of chance (Chereul et al. 1998). If this dynamical scenario is correct, the term dynamical stream for the branches in velocity space seems more appropriate than the term supercluster since they are not caused by contemporaneous star formation but rather involve stars that do not share a common place of birth: stars in the streams just share at present time a common velocity vector. It should be noted that those non-axisymmetric perturbations, as well as the minor mergers, could lead to some asymmetries in the spatial distribution of stars in the galactic disk on a large scale (see Parker et al. 2004). Since our sample does not cover the whole sky, and is anyway restricted to the Solar neighbourhood, we are not in a position to detect those asymmetries. To conclude this Section and this Chapter, let us stress that the dynamical, non-axisymmetric theory to explain the substructure observed in the velocity space is largely preferred over the theory which views it as remnants of clusters of stars sharing a common initial origin, essentially because of the wide range of ages of the stars composing the identified subgroups. In fact, we stress that both phenomena could be closely related to each other. The presence of dynamical streams in our sample of K and M giants is clearly responsible for the vertex deviation found for late-type giants, but we even suggest that the same origin could hold as well for early-type stars. It is indeed quite striking on Fig. 3.3 that the group Y extends just far enough in the UV -plane to touch both the HyPl and Si branches. The presence of stars from these two streams in group Y (sent in that region of the UV -plane at the time of their formation because of the peculiar velocity imparted by the spiral wave that created them) imposes a 83 very specific value to the vertex deviation (ranging from 15◦ to 30◦ and more), in agreement with the high values often observed for young stars (see Dehnen & Binney 1998a). This idea that the vertex deviation for younger populations could in fact have the same dynamical origin as the vertex deviation for old ones was already proposed by Mayor (1972, 1974). Here, we also argue that even the specific initial conditions of young groups of stars could be due to the same phenomenon. However, nature is of course not so simple and the features of the distribution function are presumably related to a mixture of several phenomena. Notably, the initial conditions in the simulations should be more complex than the simple 2D Schwarzschild velocity ellipsoid used by De Simone et al. (2004): in fact, purely axisymmetric substructure could already be present in the Solar neighbourhood (see the structure of the UV -plane in Dejonghe & Van Caelenberg 1999; Fig. 5.15 of this thesis). Other phenomena that could have an influence on the structure of velocity space are the following: a triaxial or clumpy dark halo, giant molecular clouds, and close encounters with the Magellanic Clouds (Rocha- Pinto et al. 2000). Theoretical investigations in this area should thus clearly be pursued, and in particular dynamical modeling. We have shown that the fine structure of phase space in the Solar neighbourhood cannot be interpreted in terms of an axisymmetric steady-state model. Nevertheless, an axisymmetric model revealing all the fine structure of the axisymmetric distribution function in the Solar neighbourhood (Dejonghe & Van Caelenberg 1999) is a necessary starting point in order to understand the true effects of the non-axisymmetric perturbations. In the next two Chapters, we develop new tools to establish axisymmet- ric three-integral models (see Section 1.2.5), exact solutions of the collisionless Boltzmann equation (1.1) (or (1.3)), taking into account the velocity dispersions anisotropy (contrarily to two-integral models, see Section 1.2.4) observed in the Solar neigbourhood (Eqs. (3.8) and (3.18), Tables 3.1 and 3.2). These mod- els could prove to be ideal initial conditions (more complex than a simple 2D Schwarzschild velocity ellipsoid) for 3D N-body simulations that could after- wards reproduce some non-axisymmetric features observed in the Solar neigh- bourhood, such as the dynamical streams that we have identified in this Chapter (see Fux 1997, De Simone et al. 2004 for 2D simulations in a perturbed disk). 84 Chapter 4

St¨ackel potentials

An axisymmetric equilibrium model representing a stellar component of the Galaxy is a pair (Φ,F ), where Φ(R, z) is the steady axisymmetric gravitational potential, generated by the whole mass distribution of the Galaxy (Eq.(1.4)), and F (I1,I2,I3) is the steady distribution function of the component in inte- gral space. The first step in the construction of such a model is thus to find an acceptable potential Φ. This Chapter is dedicated to the development of axisymmetric Galactic potentials Φ(R, z), satisfying recent estimates of Milky Way parameters, especially in the Solar neighbourhood. Caldwell & Ostriker (1981), Rohlfs & Kreitschmann (1988) and, recently, Dehnen & Binney (1998b) fitted axisymmetric mass models of the Milky Way to various measurements of the gravitational force field: they concluded that a wide variety of models can emerge from this fitting process and that the mass distribution of the Galaxy is still ill-determined. As previously explained in Section 1.2.5, the numerical nature of the third integral in most potentials makes it difficult to deal with. To avoid this problem, we choose to use specific potentials in which an exact ana- lytic third integral exists for all orbits, St¨ackel potentials (St¨ackel 1890). These potentials were introduced into stellar dynamics by Eddington (1915) and have since been used in a number of papers (e.g. Lynden-Bell 1962; de Zeeuw 1985; Dejonghe 1993; Sevenster, Dejonghe & Habing 1995; Durand, Dejonghe & Acker 1996; Bienaym´e1999) : in fact, the regularity of typical galactic potentials could be understood in terms of their proximity to St¨ackel potentials (e.g. Gerhard 1985). The St¨ackel potentials are non-rotating potentials for which the Hamilton- Jacobi equation H(~x,~p) = −E is separable (where ~x are the positions, ~p the generalized impulses, E the binding energy). This means that there exists a relation pi = pi(xi,I1,I2,I3) (4.1) Thus, all orbits admit three analytic integrals of the motion. They form the most general set of potentials that contain one free function, for which three exact integrals of the motion are known, and which can be relevant as models for a global potential in galactic dynamics (Lynden-Bell 1962).

85 86

In this Chapter, our goal is to show that a wide variety of simple St¨ackel potentials can fit most known parameters of the Milky Way (including Hipparcos latest findings). In order to do this, we continue the work of Batsleer & Dejonghe (1994, hereafter BD) who presented a set of simple St¨ackel potentials with two mass components (halo and disk) and a flat rotation curve, that we generalize by adding a thick disk to them since its existence as a separate stellar component is now well documented (Ojha et al. 1994; Chen et al. 2001; see also Section 1.1.5 and 3.3.6). These new potentials are described by five parameters and we will show that many different combinations of these parameters are consistent with fundamental constraints for a mass model of the Milky Way.

4.1 Coordinate system

Axisymmetric St¨ackel potentials are best expressed in spheroidal coordinates (λ, φ, ν), with λ and ν the roots for τ (λ > ν) of the equation

R2 z2 + = 1 α < γ < 0, (4.2) τ + α τ + γ and (R, φ, z) cylindrical coordinates. The parameters α and γ are both constant and we assume them smaller than zero. The coordinate surfaces λ =constant are prolate ellipsoids, while the coordinate surfaces ν =constant are hyperboloids. It is convenient to define the axis ratio of the coordinate surfaces as  = a with √ c α = −a2 and γ = −c2. Together with the focal distance ∆ = γ − α, the axis ratio defines the coordinate system, since

γ = ∆2/(1 − 2) (4.3) α = 2 γ

4.2 Three-component St¨ackel potentials

An axisymmetric potential is of St¨ackel form if there exists a spheroidal coor- dinate system (λ, φ, ν) in which the potential can be written as

f(λ) − f(ν) Φ(λ, ν) = − , (4.4) λ − ν for an arbitrary function f(τ) = (τ + γ)G(τ), G(τ) ≥ 0, τ = λ, ν. The function −G(λ) then represents the potential in the z = 0 plane. For this kind of potential, the Hamilton-Jacobi equation is separable in spheroidal coordinates, and therefore the orbits admit three analytic isolating integrals of the motion: the two classical integrals E and Lz (see Section 1.2.4) and the third integral of galactic dynamics:

1 1 G(λ) − G(ν) I = (L2 + L2) + (γ − α) v2 − z2 (4.5) 3 2 x y 2 z λ − ν 87

Figure 4.1: Each point inside the volume is an orbit (clockwise or counterclock- 2 wise with I2 = Lz/2). Outside of the volume, the values are not allowed or correspond to unbound orbits (Dejonghe & de Zeeuw 1988)

All the orbits in such a potential are short axis tubes, i.e. orbits delimited by a minimum and maximum λ (λ− and λ+), and by a minimum and maximum ν (ν− and ν+). A detailed orbit analysis can be found in de Zeeuw (1985). In integral space, the volume corresponding to allowed bound orbits is delimited 2 by the planes E = 0, Lz = 0, I3 = 0 and by a curved surface for which λ− = λ+ = λ0 (the infinitesimally thin tubes) and whose parametric equations (of parameter λ0) are

dG E = G(λ ) + (λ + α) (λ ) − I (γ − α)/(λ + γ)2 (4.6) 0 0 dλ 0 3 0 dG L2 = −2(λ + α)2 (λ ) + I /(λ + γ)2
(4.7) z 0 dλ 0 3 0 Eqs (4.6) and (4.7) can be compared with Eqs (1.17) and (1.18) when the (E,Lz)-plane was investigated. We find back the condition for circular orbits (1.19) when I3 = 0 (the plane I3 = 0 corresponds to orbits that cannot move outside of the Galactic plane). Fig. 4.1 illustrates the structure of the full integral space when a St¨ackel potential is used. The Milky Way is composed of several mass components: the bulge, the thin disk, the thick disk, the stellar halo, and the dark halo (see Section 1.1). It is not fundamental that a mass model aknowledges explicitly the existence of 88 each of these components. For example, BD presented a set of two-component (halo-disk) St¨ackel potentials with a flat rotation curve. Our goal is to show that different St¨ackel potentials are able to fit the latest estimates for the fundamental parameters of the Galaxy. We first generalize the potentials of BD by adding a thick disk to them since its existence as a separate stellar component is now well established (Ojha et al. 1994; Chen et al. 2001). Our potentials have thus three mass components: two “flat” components and one spheroidal. The spheroidal component accounts for the stellar and dark halo, and we shall see in Section 4.6 that our potentials turn out to have an effective bulge, which enables us to avoid the explicit introduction of a bulge- component. We assume that all three components of our potentials generate a St¨ackel potential, with three different coordinate systems but the same focal distance. It is straightforward to show that the superposition of three St¨ackel potentials is still a St¨ackel potential when all three coordinate systems have the same focal distance. Although the functions fthin, fthick and fhalo are arbitrary, we assume that they each generate a Kuzmin-Kutuzov potential, defined by

GM G(τ) = √ (4.8) τ + c with M the total mass of the system. Such a potential becomes a point mass GM potential (Φ = − R ) in the Galactic Plane when λ → ∞. We use this potential essentially because it is an extremely simple but representative St¨ackel potential: we will show that it is not necessary to use complicated St¨ackel potentials in order to match all the known and most recently determined parameters of the Milky Way. Near the center, in a meridional plane, the lines of constant mass density corresponding to a St¨ackel potential are approximately ellipsoidal (de Zeeuw, Peletier & Franx 1986). For a Kuzmin-Kutuzov potential (see e.g. Dejonghe & de Zeeuw 1988), when a > c, the isodensity surfaces are flattened oblate spheroids (contrarily to the spheroids of constant λ, which are prolate), and a increasing  = c produces more flattening. So, the ratio  has to be high for the thin disk, intermediate for the thick disk and close to unity for the halo. We first define a class of dimensionless potentials Φp in dimensionless units (Rp, zp), with a focal distance ∆ = 1 for all three coordinate systems and the central value of the potentials equal to −1. Each of these potentials is a superposition of three Kuzmin-Kutuzov potentials in three different coordinate systems:

fthin(λthin)−fthin(νthin) Φp(λthin, λthick, λhalo, νthin, νthick, νhalo) = −kthin λthin−νthin

fthick(λthick)−fthick(νthick) fhalo(λhalo)−fhalo(νhalo) −kthick − (1 − kthin − kthick) λthick−νthick λhalo−νhalo (4.9) This new class of potentials is thus defined by five parameters (the three axis ratios of the coordinate surfaces and the relative contribution of the thin and 89

thick disk masses to the total mass, i.e. kthin and kthick), which is a reasonable augmentation with respect to the BD potentials that were defined by three parameters. We denote α − α = γ − γ = q ≥ 0 thin thick thin thick 1 (4.10) αthin − αhalo = γthin − γhalo = q2 ≥ q1 ≥ 0.

So, we can express the class of potentials Φp as a function of λthin, νthin and the two constants q1 and q2 (we also use Eq. (4.8)) to give the final form of Φp:

kthin kthick Φp(λthin, νthin) = −GM( √ √ + √ √ λthin+ νthin λthin+q1+ νthin+q1 (4.11) + √ 1−kthin−√kthick ) λthin+q2+ νthin+q2 The dimensionless rotation curve corresponding to such potentials is given by:

2 2 √ √kthin v (Rp) = Rp∂Φp/∂Rp(Rp, 0) = GMR ( 2 c p λthin( λthin+cthin) (4.12) k 1−k −k + √ √ thick √ + √ √ thin thick√ ) 2 2 2 2 λthin+q1( λthin+q1+ cthin+q1) λthin+q2( λthin+q2+ cthin+q2) where Rp denotes the dimensionless galactocentric radius. We shall impose constraints on the shape and flatness of the rotation curve in Section 4.3. In order to transform these dimensionless potentials into dimensional ones for the Milky Way, we denote the dimensionless radius where the rotation curve attains its first maximum as Rp,M : since the Milky Way attains its global amplitude of 220 km/s for the first time at a radius of about 1.5 kpc (Fich & 1.5kpc Tremaine 1991), we define a distance scale factor rS = . The conversion Rp,M between dimensionless and dimensional distances is then given by:

R(kpc) = r R S p (4.13) z(kpc) = rSzp

Then, the total mass M of the Galaxy is adjusted in such a way that the dimensional circular velocity at the Solar radius (R = 8 ± 0.5 kpc, see Section 1.3.1) is equal to 220 km s−1: we obtain thus a minimum and a maximum value for M, for the two extreme values of the galactocentric distance of the sun. That adjustment also fixes the local mass density in the Solar neighbourhood ρ (see Section 4.3).

4.3 Selection criteria

We shall now establish the features that a potential (as defined in Section 4.2) must have to be considered as a plausible potential for the Milky Way, in the light of the observational constraints reviewed in Section 1.3. 90

By definition of the dimensional potentials (see Section 4.2), the local circular −1 speed vc(R ) is equal to 220 km s for all the potentials. The first fundamental selection criterion is the flatness of the rotation curve (Section 1.3.2): this feature can be examined in the dimensionless frame-work. Even the BD potentials, with only two mass components, could produce many different shapes of rotation curves. So, we adopt the same simple diagnostic as BD: we denote Rp,M the dimensionless radius where the circular speed attains its first maximum and we look for a range in R where vc(R) remains larger than 80% of the maximum velocity and is thus more or less constant. We denote Rp,F the dimensionless radius where vc(Rp,F ) = 0.8vc(Rp,M ). A rotation curve is considered sufficiently flat if: Rp,F − Rp,M EF = > 8 (4.14) Rp,M This is a minimum requirement. The second selection criterion is based on the latest determinations of the −1 Oort constants (Section 1.3.2). Since we impose vc(R ) = 220 km s and R = 8 ± 0.5 kpc for all our potentials, v (R ) c = 27.6 ± 1.7km s−1kpc−1 (4.15) R which is in accordance with the value determined by Feast & Whitelock (1997), vc(R ) = 27.2 ± 0.9km s−1kpc−1. The first derivative of the circular velocity in R the Solar neighbourhood corresponding to the Oort constants found by Feast dvc −1 −1 & Whitelock (1997) is dR (R ) = −2.4 ± 1.2km s kpc . Our potentials have two extreme values for this derivative, depending on the position of the sun; in order to fit the above interval, we select the potentials such that:

dvc −1 −1 max( dR (R )) > −3.6km s kpc dvc −1 −1 (4.16) min( dR (R )) < −1.2km s kpc This feature is not as essential as the flatness of the rotation curve, because of the intriguing measurement of the proper motion of Sgr A* (see Section 1.3.2). The last selection criterion is the local mass density in the Solar neighbour- hood: this number is determined by the adopted total mass M. We look for its values in both extreme positions for the sun (R , z ) = (7.5, 0.004) kpc and (R , z ) = (8.5, 0.02) kpc. Following the Hipparcos latest findings reviewed in Section 1.3.3, we select the potentials such that: −3 max(ρ ) > 0.06M pc −3 (4.17) min(ρ ) < 0.12M pc This feature is more important than the Oort constants: we give the priority to that constraint, contrarily to Sevenster et al. (2000) who studied the structure of the inner Galaxy, and therefore constructed St¨ackel potentials with reasonable values for the Oort constants but values for ρ that are too low. The total mass of the Galaxy, the mass fractions of the disks and the flatten- ing and scale length of the components are not well established observationally and are not considered as fundamental constraints for the potential. 91

4.4 The “winding staircase”

Our goal is now to find and select, among the class of potentials defined in Section 4.2, some representative potentials that differ with respect to their form and features, and that satisfy the selection criteria of Section 4.3. As it is difficult to visualize a five-dimensional parameter space, we shall first visualize its structure when thin and halo are fixed. Then we shall restrict the parameter space by imposing additional constraints on the flatness of the thick disk. We shall finally select five representative potentials in Section 4.6. As an example of the consequences of the choice of the selection criteria of Section 4.3, we look for all the values of thick, kthin and kthick that yield potentials satisfying the selection criteria for thin = 75 and halo = 1.02 (with the thick disk always thicker than the thin disk, i.e. thick < 75). The accordance with the selection criteria results from a precise mixing of the two disks and of the halo. Fig. 4.2 illustrates the volume in parameter space corresponding to these satisfactory potentials: the volume looks like a “winding staircaise”. When kthick = 0, we see that all the values of thick are allowed when 13% ≤ kthin ≤ 15%, which results from the fact that no thick disk is in fact present. When kthin is close to its maximum possible value (15%), kthick has to be zero except when thick is very close to 1, i.e. when the thick disk is a pretty round component (similar to the halo) and helps to keep the rotation curve flat. When kthin is smaller than 15%, the possibilities for kthick are more numerous, i.e. the thick disk can take a part of the mass. When kthin attains the critical value of 12%, the volume is inflected and the thick disk has to be thin and non-zero in order to counter-balance the lack of mass in the thin disk. Decreasing the mass of the thin disk forces the thick disk to become thinner and more massive, yielding the “winding staircase” volume of Fig. 4.2. The similar volumes in parameter space for thin > 75 have the same form, and are bigger essentially because there is more freedom for thick.

4.5 Constraints on the scale height of the thick disk

In Fig. 4.2, there are some solutions with a thick disk more massive than the thin disk. Even though the mass fractions of the disks and the flattening of the components are not well established and should be tested in a dynamical study, we know that the mass fraction of the thick disk is smaller than that of the thin disk (and represent at most 13% of the local thin disk density in the Solar neighbourhood) and the latest determination of the thick disk scale height based on star count data from the Sloan Digital Sky Survey is 665pc (Chen et al. 2001). However, Chen et al. (2001) insist on the difficulty to converge to a definitive answer: some other studies indicate that the scale height could be of the order of 1 kpc (Gilmore 1984; Ojha et al. 1996). We shall reject the potentials that are completely inconsistent with those characteristics (0.6 kpc < hz < 1 kpc) and in particular those with a thick disk more massive than the thin disk. 92

k_thick

0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0 0.02 0.04 80 0.06 70 60 0.08 50 k_thin 0.1 40 0.12 30 20 a_thick/c_thick 0.14 10 0.16 0

Figure 4.2: This “winding staircase” figure displays the possible values of the a coordinate axis ratio  = c for the thick disk and the possible values of the contributions k of the disks to the total mass in order to satisfy the selection criteria of Section 4.3 (for fixed values thin = 75 and halo = 1.02, and with the thick disk always thicker than the thin disk, i.e. thick < 75). It gives a rough vision of the region of parameter space that satisfies the criteria. Only the region 1.3 ≤ thick ≤ 2 is in fact relevant for a model of the Milky Way (see Section 4.5). 93

Table 4.1: Column 1 contains the axis ratio of the coordinate surfaces for the thick disk. Column 2 gives the corresponding scale height for a Kuzmin-Kutuzov potential with rS = 1.  hz(pc) 1.3 914 1.4 834 1.5 782 1.8 696 2 663

In order to determine the interval of axis ratios that should be considered for the thick disk, we have fitted the vertical mass distribution corresponding to a −z/hz simple Kuzmin-Kutuzov potential with rS = 1 to an exponential law e . We conclude that the potentials with 1.3 ≤ thick ≤ 2 have a scale height between 665 pc and 1 kpc and are the ones we should examine in detail. For each axis ratio, Table 4.1 gives the corresponding scale height. However, Fig. 4.3 reveals that the exponential fit is not valid for the axis ratios used to model the thin disk, which is not surprising since a thin disk could be better understood as 2 a superposition of isothermal sheets (i.e. ρ(z) = ρ0 sech (z/2hz); see Spitzer 1942) than by a simple exponential law.

4.6 The final selection

In this section, we look for some three-component potentials with different forms and features, all satisfying the selection criteria defined in Section 4.3 and con- sistent with what is known about the thick disk. First of all, we look for the two-component BD potentials that satisfy the new selection criteria: they are listed in Table 4.2. All the two-component potentials with disk = 50, 75, 130, 200 and halo = 1.005, 1.01, 1.02, 1.03 are examined. We do not consider disks with a/c > 200 because then the uncertainty on ρ becomes too large. We see in Table 4.2 that, in order to reproduce the Oort constants in the two-component framework, the shape of the halo cannot vary (halo = 1.02). If we take the two-component potentials as a starting point, there are two ways to add a thick disk. The first way is to decrease the contribution of the halo and to put the remaining mass into the thick disk: the local density in the Solar neighbourhood is then slightly larger while the rotation curve is decreasing faster. If we take the third potential of Table 4.2 as a starting point, the first potential of Table 4.5 (potential I) illustrates this first case. The other way is to decrease the contribution of the thin disk and to put the remaining mass into the thick disk: the local density is then slightly decreasing while the rotation curve is more flat. If we take the fourth potential of Table 4.2 as a starting point, the first potential of Table 4.5 illustrates this second case. The presence of a third component allows more freedom for the shape of 94

Figure 4.3: Profile of the logarithm of vertical density at R = 8 kpc for Kuzmin- Kutuzov potentials with rS = 1 and  = 1.3 (top left),  = 2 (top right),  = 75 (bottom left) and  = 200 (bottom right). Only the two first cases resemble exponentials (axes: ln(ρ) (no dimension) and z (kpc)). 95 the halo, so we look for three-component potentials with a halo rounder than halo = 1.02. In order to keep the rotation curve flat and retain the local density as well as the Oort constants in the allowed interval, we need to couple a very thin disk with the rounder halo: indeed, our investigations show that no solution can be found for thin = 50 and halo = 1.01. However, if we take thin = 200, Table 4.3 gives solutions for a halo with halo = 1.01: we select the solution where the mass of the thick disk relative to the thin disk is the smallest (potential III of Table 4.5). Remark that a similar Table for halo = 1.02 would contain 292 entries and is omitted here. There are much less solutions when thin = 75, as can be seen in Table 4.4. However, as stated in Section 4.3, we do not assign high priority to the Oort constants, and we select a potential with thin = 75, halo = 1.01 and a relative mass of the thick disk relative to the thin disk of 13% (i.e. a smaller fraction than any of the solutions of Table 4.4), but with a quite large local radial derivative of the circular speed (potential IV of Table 4.5). For halo = 1.005, it is totally impossible to find a potential satisfying the Oort constants criterion: the radial derivative of the circular speed in the Solar neighbourhood is always positive. Nevertheless, if one is willing to ignore the estimates of A and B, one could select a potential with halo = 1.005 and reason- ably low values for the cirular speed radial derivative in the Solar neighbourhood (potential V of Table 4.5). Finally, we select a potential (potential II) satisfying all the criteria, for −3 −3 which the interval in ρ is precisely [0.06M pc , 0.12M pc ], and which is close to the Chen et al. (2001) findings , i.e. thin = 200 which is the thinnest thin disk that we consider in order not to have a too large interval for ρ , a relative mass of the thick disk to the thin disk of 10%, a scale height of the thick disk of 612.5 pc and a relatively large EF (EF = 13.44). Table 4.5 summarizes the main features of the selected St¨ackel potentials with different forms and features and that we shall use for dynamical modeling of the Milky Way: potentials III and V have a very thin disk associated with a quite massive thick disk and a pretty round halo, while potentials I and IV have a thicker thin disk with a quasi-negligible thick disk (Fig. 4.4 shows the mass isodensity curves of each potential in a meridional plane for two different scales). It should be noted that the total masses associated with those potentials are very different and become larger with a rounder halo and that a rounder halo implies that this halo is much more extended. A closer look to the mass density in the equatorial plane indicates that, for each potential, the density grows faster than an exponential in the central 3 kpc corresponding to the bulge region: the potentials have thus an effective bulge, which did enable us to avoid the introduction of an explicit bulge component. We have fitted the mass density in the plane to an exponential law down to R = 3 kpc in order to check that the scale length of the disk is realistic: the last column of Table 4.5 gives the scale length corresponding to each selected potential and we conclude that they are realistic but do not distinguish the different potentials. For the potentials with the biggest and smallest scale length, Fig. 4.5 illustrates the shape of the logarithm of the density in the plane (the other potentials have a similar shape 96 for that curve). Finally, Fig. 4.6 shows the rotation curve associated with each of the five selected potentials: the rotation curve of potential II is more flat than the one of potential I in the vicinity of the sun, while the rotation curves of the potentials with halo = 1.01 (potentials III and IV) are even more flat and the one of potential V is slightly increasing. The five potentials Φ of Table 4.5 can be used as potential for the Milky Way in an equilibrium model (Φ,F ) representing a stellar population such as the giant stars of Chapter 3. The next step in the establishment of the equilibrium model is to find the form to give to the distribution function F of the population. 97

Table 4.2: The characteristics of the two-component BD potentials with disk = 50, 75, 130, 200 and halo = 1.005, 1.01, 1.02, 1.03 have been examined. The po- tentials satisfying the selection criteria are listed in this table (with a step of 0.01 for the relative contribution of the disk). Columns 1 and 2 contain the axis ratios of the coordinate surfaces for the disk and the halo. Column 3 contains the relative contribution of the disk to the total mass. Column 4 contains the extent of the flat part of the rotation curve (see Eq. 4.14). Column 5 contains the minimum and maximum local spatial density, while column 6 contains the minimal and maximal local radial derivative of the circular velocity, each time for the two extreme positions of the sun. −3 dvc −1 −1 disk halo kdisk EF ρ in M pc dR (R ) in km s kpc 75 1.02 0.11 13.07 0.04, 0.06 −1.90, −1.18 75 1.02 0.12 11.74 0.04, 0.06 −2.03, −1.42 75 1.02 0.13 10.43 0.04, 0.07 −2.25, −1.74 75 1.02 0.14 9.25 0.04, 0.07 −2.46, −2.05 75 1.02 0.15 8.20 0.05, 0.08 −2.68, −2.37 130 1.02 0.08 17.16 0.04, 0.06 −2.07,−1.12 130 1.02 0.09 16.03 0.04, 0.07 −1.75, −0.85 130 1.02 0.10 14.56 0.05, 0.08 −1.73, −0.92 130 1.02 0.11 13.03 0.05, 0.09 −1.86, −1.15 130 1.02 0.12 11.70 0.06, 0.10 −2.00, −1.39 130 1.02 0.13 10.38 0.06, 0.11 −2.22, −1.72 130 1.02 0.14 9.20 0.07, 0.11 −2.44, −2.04 130 1.02 0.15 8.08 0.07, 0.12 −2.69, −2.38 200 1.02 0.08 17.21 0.04, 0.09 −2.02, −1.07 200 1.02 0.09 16.07 0.05, 0.10 −1.71, −0.81 200 1.02 0.10 14.59 0.06, 0.12 −1.69, −0.88 200 1.02 0.11 13.05 0.07, 0.13 −1.82, −1.12 200 1.02 0.12 11.64 0.07, 0.14 −2.00, −1.40 200 1.02 0.13 10.32 0.08, 0.16 −2.22, −1.72 200 1.02 0.14 9.14 0.08, 0.17 −2.44, −2.04 200 1.02 0.15 8.08 0.09, 0.18 −2.66, −2.36 98

Table 4.3: The three-component potentials with thin = 200 and halo = 1.01 that satisfy the selection criteria are listed in this table (with a step of 0.01 for the relative contribution of the two disks). Columns 1, 2, 3 contain the axis ratios of the coordinate surfaces for the two disks and the halo. Columns 4 and 5 contain the relative contribution of the repectively thin and thick disk to the total mass. Column 6 contains the extent of the flat part of the rotation curve (see Eq. 4.14). Column 7 contains the minimum and maximum local spatial −3 density in M pc , while column 8 contains the minimal and maximal local radial derivative of the circular velocity, each time for the two extreme positions of the sun in km s−1kpc−1. dvc thin thick halo kthin kthick EF ρ dR (R ) 200 1.3 1.01 0.09 0.07 10.69 0.06, 0.12 −1.34, −1.13 200 1.3 1.01 0.09 0.08 10.06 0.06, 0.12 −1.64, −1.45 200 1.3 1.01 0.10 0.06 9.51 0.07, 0.15 −1.35, −1.29 200 1.3 1.01 0.10 0.07 8.95 0.07, 0.14 −1.64, −1.59 200 1.3 1.01 0.10 0.08 8.45 0.07, 0.13 −1.92, −1.88 200 1.3 1.01 0.11 0.04 8.51 0.09, 0.18 −1.22, −1.17 200 1.3 1.01 0.11 0.05 8.02 0.08, 0.17 −1.51, −1.45 200 1.4 1.01 0.09 0.07 9.81 0.06, 0.13 −1.46, −1.32 200 1.4 1.01 0.09 0.08 9.11 0.06, 0.12 −1.77, −1.65 200 1.4 1.01 0.10 0.06 8.63 0.07, 0.14 −1.50, −1.47 200 1.4 1.01 0.10 0.07 8.07 0.07, 0.14 −1.78, −1.78 200 1.5 1.01 0.08 0.07 10.86 0.06, 0.11 −1.25, −1.00 200 1.5 1.01 0.09 0.06 9.93 0.07, 0.13 −1.21, −1.09 200 1.5 1.01 0.09 0.07 9.10 0.06, 0.13 −1.54, −1.45 200 1.5 1.01 0.09 0.08 8.34 0.06, 0.12 −1.86, −1.80 200 1.5 1.01 0.10 0.06 8.01 0.07, 0.15 −1.59, −1.57 200 1.8 1.01 0.08 0.07 9.42 0.06, 0.12 −1.34, −1.22 200 1.8 1.01 0.09 0.06 8.55 0.07, 0.13 −1.34, −1.32 200 2 1.01 0.08 0.07 8.75 0.06, 0.12 −1.38, −1.31

Table 4.4: The three-component potentials with thin = 75 and halo = 1.01 that satisfy the selection criteria are listed in this table (with a step of 0.01 for the relative contribution of the two disks). Columns have the same meaning as in table 4.3. dvc thin thick halo kthin kthick EF ρ dR (R ) 75 1.3 1.01 0.11 0.05 8.21 0.05, 0.07 −1.49, −1.45 75 1.4 1.01 0.11 0.04 8.06 0.05, 0.07 −1.32, −1.27 75 1.5 1.01 0.10 0.05 8.87 0.04, 0.07 −1.26, −1.23 99

Figure 4.4: Mass isodensity curves in a meridional plane for the five potentials of Table 4.5, with two different scales (Left panel: zoom on the disk, axes: position (kpc) from 0 to 8 in the plane and from 0 to 0.5 in the vertical direction. Right panel: large scale view of the halo, axes: position (kpc) from 0 to 15 in the plane and from 0 to 15 in the vertical direction). 100

Figure 4.5: The logarithm of the mass density in the equatorial plane for the two potentials with extreme scale lengths.These curves very much resemble each other, and the effective bulge appears clearly. 101

Figure 4.6: The rotation curves of the five selected potentials of Table 4.5. The total mass used to plot thes curves is the mean total mass of the two extreme values of Table 4.5. 102

Table 4.5: Among the class of potentials defined in Section 4.2, five different potentials regarding form and features have been selected. Columns 1, 2, 3 contain the axis ratios of the coordinate surfaces for the two disks and the halo. Columns 4 and 5 contain the relative contribution of the repectively thin and thick disk to the total mass. Column 6 contains the extent of the flat part of the rotation curve (see Eq. 4.14). Column 7 contains the scale factor which corresponds to the focal distance of the coordinate system of the dimensional potential. Column 8 contains the minimum and maximum local spatial density −3 in M pc , while column 9 contains the minimum and maximum local radial derivative of the circular velocity in km s−1kpc−1 and column 10 the minimum 11 and maximum total mass of the Galaxy in 10 M , each time for the two extreme positions of the sun. The scale length hR in the equatorial plane down to 3 kpc has been calculated and is presented in column 11 (in kpc). dvc thin thick halo kthin kthick EF rS ρ dR (R ) M hR I 75 1.5 1.02 0.13 0.01 9.86 0.93 0.04, 0.07 −2.51, −2.05 2.37, 2.41 2.73 II 200 1.8 1.02 0.10 0.01 13.44 0.88 0.06, 0.12 −2.05, −1.31 2.37, 2.41 2.63 III 200 1.3 1.01 0.11 0.04 8.51 0.95 0.09, 0.18 −1.22, −1.17 3.19, 3.22 2.65 IV 75 1.8 1.01 0.11 0.015 9.07 0.98 0.05, 0.08 −0.62, −0.55 3.56, 3.58 2.78 V 200 1.3 1.005 0.07 0.01 18.30 1.01 0.11, 0.23 +0.69, +1.47 6.13, 6.20 2.72 Chapter 5

Three-integral distribution functions

Now that we have developed in Chapter 4 acceptable St¨ackel potentials for the Galaxy, we need to know which form to give to the distribution function F of the stellar population we want to model. We express F as a linear combination of component distribution functions FΛ:

X F (E,Lz,I3) = cΛFΛ(E,Lz,I3) (5.1) Λ

This Chapter will be dedicated to the development of new component distribu- tion functions FΛ. It is not quite obvious to define suitable distribution functions FΛ(E,Lz,I3) that depend on three exact analytic integrals in a St¨ackel poten- tial, and that can somewhat realistically represent our ideas of a real stellar disk. For example, Bienaym´e(1999) made three-integral extensions of the two- integral parametric distribution functions described in Bienaym´e& S´echaud (1997), but these ones were built to model the kinematics of stars in only a small portion of the Milky Way. Dejonghe & Laurent (1991) also defined the three-integral Abel distribution functions, but these ones could not provide very thin disks in the two-integral approximation. Robijn & de Zeeuw (1996) con- structed three-integral distribution functions for oblate galaxy models, but they also had problems to recover the two-integral approximation. In this Chapter we continue the work of Batsleer & Dejonghe (1995), who constructed component distribution functions that are two-integral, but that can represent (very) thin disks when a judicious linear combination of them is chosen. We use these components as a basis for new component distribution functions that are three-integral, of which the Batsleer & Dejonghe components are a special case.

103 104

5.1 Construction of three-integral components

We intend to create three-integral stellar distribution functions, for the con- struction of stellar disks: we want to achieve an exponential decline in the mass density for large radii, while we want to introduce a preference for (nearly) circular orbits (see Section 1.2.6). It has been known for some time that two-integral models can describe very thin disk systems (e.g. Jarvis & Freeman 1985), with the dramatic restriction that both vertical and radial dispersions are equal. We know that this is not true in the Milky Way disk (see especially Eqs. (3.8) and (3.18) of the present thesis). So we want to create three-integral distribution functions that can create anisotropy in the velocity dispersions but that can also describe very thin disks in the two-integral approximation, unlike the Abel distribution functions of Dejonghe & Laurent (1991). η β The Fricke components (Fricke 1952) of the form E Lz favour that part of phase space where stars populate circular orbits, so they could be taken as a starting point. However, they cannot be used in their basic form to model disks with a finite extent because they populate orbits which can reach arbitrary large heights: therefore, we will take as a starting point the components defined in Batsleer & Dejonghe (1995, Eq. 19), who constructed disk-like component distribution functions with a finite extent in vertical direction by setting them equal to zero for E < Sz0 (Lz) (see Section 1.2.6, Eqs. (1.16), (1.17) and (1.18), and Fig. 1.2). In order to make the components depending on the third integral I3, we 2 δ introduce the factor (p + qE + rLz + sI3) in which the parameter s (and δ) will be responsible for the three-integral character of the components. The coefficients p, q, r and s can, in the most general case, be functions of Lz. This leads us towards a general three-integral disk component of the form

 η E − Sz0 (Lz) 2 δ F (E,Lz,I3) = f(Lz) (p + qE + rLz + sI3) (5.2) S0(Lz) − Sz0 (Lz) if  E − Sz0 (Lz) ≥ 0 2 . (5.3) p + qE + rLz + sI3 ≥ 0 The distribution function is identically zero in all other cases. The function f(Lz) is defined as

α2 1 2 β α1 − S (L ) z0 z f(Lz) = (2LzSz0 (Lz)) (Sz0 (Lz)) e . (5.4) 1 + e−aLz

The distribution function (5.2) is thus defined by 11 parameters: z0, η, p, q, r, s, δ, a, β, α1 and α2. The parameter z0 is the maximum height of the disk component. The parameter η is responsible for the favouring of nearly circular orbits, i.e. orbits with a binding energy E as close as possible to that of circular orbits in the galactic plane (see Section 5.3). Furthermore, if we want to favour 105

(nearly) circular orbits, we have to suppose the distribution function to be an increasing function of E: this forces q to be positive. Since a large I3 implies that the orbit can reach a large height above the galactic plane (see de Zeeuw 1985 for a complete analysis of the orbits in a St¨ackel potential), the orbits with small I3’s have to be favoured in order to describe thin disks: this forces s to be negative. The parameter a is the rotation parameter (the value of a influences only the odd moments of F , see Section 5.2). If a = 0, there is no rotation for the component, and if a = +∞, F represents a maximum streaming component with no counter-rotating stars. Finally, The requested exponential decline in the mass density with large radii is controlled by the parameter α2 (and to some extent by the parameter α1, see Section 5.3). Other constraints (on p, r, and η) will be imposed in Section 5.2, in order to enable the analytical calculations of the moments.

5.2 Moments

The moments of a distribution function F at the point (λ, φ, ν) of a spheroidal coordinate system are defined as ZZZ l m n µl,m,n(λ, ν) = F (E,Lz,I3)vλvφ vν dvλdvφdvν , (5.5) with vλ, vφ, vν the components of the velocity in the λ, the φ and the ν direction of the spheroidal coordinate system, and l, m, n integers. The mass density, the mean velocity and the velocity dispersions of the stellar system represented by F can easily be expressed in terms of the moments (5.5) by

ρ(λ, ν) = µ0,0,0(λ, ν) ρhvφi(λ, ν) = µ0,1,0(λ, ν) 2 ρσλ(λ, ν) = µ2,0,0(λ, ν) (5.6) 2 ρhvφi(λ, ν) = µ0,2,0(λ, ν) 2 ρσν (λ, ν) = µ0,0,2(λ, ν) To obtain the value of one of the moments, we have to integrate over the volume in velocity-space corresponding to all orbits that pass through the point (λ, φ, ν) in the spheroidal coordinate system of the St¨ackel potential.

5.2.1 The case where a = 0 and m is an even integer

Since all three integrals of the motion are quadratic in vλ and vν , if l or n is an odd integer, the moment µl,m,n is identically zero. If l and n are even integers, the moment µl,m,n can be written as an integral computed in the integral space. In the development below, we assume that a = 0 (in that case, there is no rotation and if m is odd, the moment is zero) and that m is an even integer (the general case will easily be derived from this one in the next subsection). Under these assumptions, Eq. (5.5) becomes (see Dejonghe & de Zeeuw 1988): 106

l+n η 2 +1 Z 2 ZZ   2 f(Lz) m 2 E − Sz0 µl,m,n = v dL dEdI3 l+n p 2 φ z R(λ − ν) 2 Lz S0 − Sz0 l−1 n−1 2 δ + 2 − 2 ×(p + qE + rLz + sI3) (I3 − I3) (I3 − I3 ) (5.7)

+ − where I3 and I3 are given by

+ 2 λ+γ 2 I3 (E,Lz) = (λ + γ)[G(λ) − E] − 2(λ+α) Lz (5.8) − 2 ν+γ 2 I3 (E,Lz) = (ν + γ)[G(ν) − E] − 2(ν+α) Lz.

We want to reduce the triple integral (5.7) to a simple one by solving the innermost double integral analytically. For our components given by Eq. (5.2), the integration surface in the (E,I3)-plane is defined by

− 2 + 2 I3 (E,Lz) ≤ I3 ≤ I3 (E,Lz) L2 z (5.9) Sz0 (Lz) ≤ E ≤ ψ(λ, ν) − 2R2 2 p + qE + rLz + sI3 ≥ 0. For this double integral to be analytically solved, however, we will have to make use of those combinations of p, q, r and s for which the integration area is transformed into the triangle bounded by

+ 2 I3 = I3 (E,Lz) − 2 I3 = I3 (E,Lz) (5.10) 2 p + qE + rLz + sI3 = 0, as shown in Fig. 5.1. We will be in this situation (for all the points (λ, φ, ν) of 2 configuration space, where the moments are calculated) whenever p+qE+rLz + sI3 = 0 does intersect the E-axis for E ≥ Sz0 (Lz). If we take p = −Sz0 and r ≤ 0, it is the case for all the Lz relevant in the integration. In this situation, we can express the factor (E − Sz0 )/(S0 − Sz0 ) as a linear combination of the other three factors in the integrandum (corresponding to the bounding lines of the integration surface):

E − Sz0 + − 2 = t(I3 − I3) + u(I3 − I3 ) + v(p + qE + rLz + sI3). (5.11) S0 − Sz0

We impose η to be an integer: then the double integral in the (E,I3)-plane transforms into a sum of integrals:

η η−i X X  η   η − i  ZZ vi tη−i−j uj dE dI (p + qE + rL2 + sI )δ+i i j 3 z 3 i=0 j=0 l−1 n−1 + η+ 2 −i−j − 2 +j ×(I3 − I3) (I3 − I3 ) (5.12) 107

Figure 5.1: The integration area for the inner double integral in Eq.(5.7) in the (E,I3)-plane. 108

For each integral in this summation, the integrandum consists of factors whose zero-points define the bounding lines for the integration surface in the (E,I3)- plane. These integrals can be solved analytically. In order to solve the integrals analytically, one uses the new integration 2 variables x and y, defined by (for a fixed Lz)

+ x(E,I3) = I3 (E) − I3 − (5.13) y(E,I3) = I3 − I3 (E).

2 The line in the (E,I3)-plane p + qE + rLz + sI3 = 0 becomes y = ymax(x), and the root of ymax(x) = 0 is xmax (see Fig. 5.2). To make a more compact notation possible, we first define the auxiliary function h(τ) as h(τ) = s(τ + γ) − q. (5.14)

We then have

 L2  (λ − ν) x = − p + rL2 − h(ν)ψ(λ, ν) + s(ν + γ)G(ν) − z (q − sz2) max z 2R2 h(ν) (λ − ν) ≡ − x0 (5.15) h(ν) max and h(ν) y (x) = (x − x) (5.16) max h(λ) max

Solving the integral part of one of the terms in Eq. (5.12) for y then yields

xmax l−1 ymax(x) n−1 δ+i 1 R η+ 2 −i−j R 2 +j (λ−ν)δ+i+1 0 x dx 0 y [h(λ)(y − ymax(x))] dy

(−h(λ))δ+i n−1 = (λ−ν)δ+i+1 B( 2 + j + 1, δ + i + 1)

xmax l−1 n−1 R η+ 2 −i−j 2 +i+j+δ+1 × 0 x ymax(x) dx (5.17) where B is the special B-function (Abramowitz & Stegun 1972). After solving for x (analogous to what we did for y), one obtains for the whole summation (5.12)

η η−i     δ+ n+1 +i+j X X η η − i (−h(ν)) 2 vi tη−i−j uj i j δ+i+1 n+1 +j i=0 j=0 (λ − ν) (−h(λ)) 2 l+1 n+1 l+n Γ(η + − i − j) Γ( + j) Γ(δ + i + 1) δ+η+ 2 +1 2 2 ×(xmax) l+n (5.18) Γ(δ + η + 2 + 2) 109

Figure 5.2: The integration area for the inner double integral in Eq.(5.7) in the (x(E,I3), y(E,I3))-plane (see Eq.(5.13) 110 where Γ is the Euler function (see Abramowitz & Stegun 1972). The coefficients t, u and v are calculated by equalizing term by term in equation (5.11)

1 = −t(λ + γ) + u(ν + γ) + vq S0−Sz0 0 = −t + u + vs h 2 i h 2 i Sz0 Lz Lz 2 − = t(λ + γ) G(λ) − − u(ν + γ) G(ν) − + vrLz + vp S0−Sz0 2(λ+α) 2(ν+α) (5.19) We find for the coefficients

v0 u0 t0 v = 0 , u = 0 , t = 0 (5.20) xmax (λ − ν)xmax (λ − ν)xmax with

2 Lz 0 ψ(λ, ν) − S − 2 x v0 = z0 2R , u0 = −h(λ)v0 − max , t0 = u0 + s(λ − ν)v0 (5.21) S0 − Sz0 S0 − Sz0

So we have a one-dimensional numerical integration to perform. We now have to determine the integration limits of the simple integral in Lz. Since the definition of E and condition (5.3) imply that

L2 S (L ) ≤ E ≤ ψ − z , (5.22) z0 z 2R2

2 the integration limits for the integral in Lz are, in a first time, determined by 2 Lz 2 the intersections of Sz0 (Lz) and the line E = ψ − 2R2 in the (E,Lz)-plane (see Fig. 5.3). Furthermore, in order to have a non-degenerate (i.e. not empty) triangle in Fig. 5.1, we must have

0 xmax ≥ 0 ⇔ xmax ≥ 0. (5.23)

2 Lz Since the condition ψ − 2R2 − Sz0 ≥ 0 is automatically verified when condition (5.23) is verified, the equality in (5.23) fixes the minimal and maximal angular momentum to take into account in the integration. 0 2 The extremum of xmax(Lz) is calculated: if it is smaller than zero, the 0 2 integral is null, else the zero’s of xmax(Lz) are calculated and are the bounds of the integration. 2 2 2 Knowing that Lz = R vφ, the resulting expression for the moment (with a = 0 and m even) becomes 111

2 2 2 Figure 5.3: The (E,Lz)-plane. The integrations limits in Lz,(Lz)min and 2 (Lz)max, must be between the intersections of the curve E = Sz0 (Lz) and the 2 vφ line E = ψ − 2 . Condition (5.23) then definitively fixes these limits. E0 and E1 are the minimal and maximal energy taken into account in the integration. 112

2 n+1 l+1 (v )max l+n Γ(δ + 1)Γ( )Γ(η + ) Z φ m−1 2 +β+1 2β 2 2 2 β+ 2 µl,m,n = 2 R l+n (vφ) Γ(δ + η + + 2) 2 2 (vφ)min α 2 l+n − S 0 δ+ +1 η η−i e z0 (x ) 2     β+α1+α3 max X X η η − i 0i ×(Sz ) v 0 2 n+1 η+ l+1 i j (−h(λ)) 2 (−h(ν)) 2 i=0 j=0

Γ(δ + i + 1) Γ(η + l+1 − i − j) Γ( n+1 + j) (−h(ν))i+j ×t0η−i−j u0j 2 2 dv2 (5.24) Γ(δ + 1) l+1 n+1 (−h(λ))j φ Γ(η + 2 ) Γ( 2 ) This integration can be performed numerically by parts.

5.2.2 The general case In the general case a 6= 0, the expression (5.24) is still valid for the even mo- ments. When m is odd, the integrandum has to be multiplied by a factor P :

1 − e−aR|vφ| P = (5.25) 1 + e−aR|vφ| because

2 1/2 2 1/2 Z (vφ)max Z −(vφ)min 2 2k+1 1 2 2k+1 1 f(vφ)vφ −aRv dvφ + f(vφ)vφ −aRv dvφ 2 1/2 1 + e φ 2 1/2 1 + e φ (vφ)min −(vφ)max

(v2 ) 1 Z φ max 1 − e−aR|vφ| = f(v2 )(v2 )k dv2 (5.26) φ φ −aR|v | φ 2 2 1 + e φ (vφ)min m−1 1 with k = 2 and the factor 2 already present in Eq. (5.24) (division by 2 of the third factor of the integrand).

5.3 Physical properties of the components

In this section, we show the realistic disk-like character of our stellar distribution functions: their mass density has a finite extent in the vertical direction and an exponential decline in the galactic plane, they favour almost circular orbits and their velocity dispersions are different in the vertical and radial direction. By varying the parameters, we can give a wide range of shapes to the components. In order to illustrate the role of the different parameters, we calculate the moments of many component distribution functions with different values for the parameters. As galactic potential, we use potential II of Table 4.5. In the implementation of the theory we choose p = −Sz0 , q = 1, and r = 0. 113

Figure 5.4: Contour plots of the mass density (i.e. the moment µ0,0,0) in a meridional plane, for components with the parameters (α1, α2, β, δ, η, z0, s, a) = (3, 3, 1, 1, 2, z0, −0.5, −5), with z0 equal to 4 kpc (left panel) and 2 kpc (right panel) respectively (note the very different scale for R and z). The disks become thinner with smaller z0, while the mass density is zero above z = z0 (and even so below z = z0 because s 6= 0). In this figure and in the following similar ones, every contour corresponds to a density that is a factor of 10 smaller than the next lower contour.

5.3.1 The parameter z0 This parameter was introduced in order to impose a maximum height above the galactic plane for the disk-like component (Fig. 5.4): indeed, when E ≥ Sz0 , an orbit cannot go higher than z = z0, and the distribution function (5.2) is null for E ≤ Sz0 (Lz). In order to model samples of stars belonging to populations with different characteristic heights above the galactic plane, we can use a set of components with different values for this parameter.

5.3.2 The parameter α1

The parameter α1 enters Eq. (5.4) as the exponent of Sz0 : so, for non-negative values of α1, the factor where it appears will behave as a declining function of

Lz, in the same way as Sz0 (Lz) does, showing a steeper decline for larger α1 (see Fig. 5.5). A large α1 thus results in a distribution function that favours a large fraction of bound orbits. When it is increasing, this parameter helps to produce mass close to the center. When a given exponential decline is requested, α1 will be a function of the other parameters rather than a fixed parameter (see Section 5.3.3).

5.3.3 The parameter α2

The parameter α2 occurs as exponent in the distribution function’s exponen- tial factor . Increasing values of this parameter will contribute to the mass distribution near the center (Fig. 5.6), like in the α1 case (but exponentially). On the other hand, our potential is approximately Keplerian at very large 114

Figure 5.5: This figure displays the decline of the logarithm of the galactic plane mass density (in arbitrary units and scaling) of different components for varying α1. A rising α1 helps to produce mass close to the center. The other parameters have the same values as in Fig. 4 (with z0 equal to 2 kpc) except that α2 = 0. 115

Figure 5.6: Logarithm of the configuration space density (in the galactic plane and in arbitrary units) of different components for varying α2. The other pa- rameters have the same values as in Fig. 5.4 (with z0 equal to 2 kpc) except that α1 is adjusted to built-in a given exponential decline. We see that the components have an exponential decline in the galactic plane and that α2 is roughly the reciprocal of the component’s scale length. 116 radii: this implies that, in the galactic plane,

2 Lz ∼ R (5.27) and that (Batsleer & Dejonghe 1995)

1 1 Sz0 (Lz) ∼ 2 ∼ (5.28) Lz R

So, at very large radii, α2 is the reciprocal of the component’s scale length, if the contribution of the other factors to the mass density does not vary much with respect to R (for very large R). In practice, it is often desirable to use com- ponents for which an exponential decline and a given scale length (as determined by α2) is already built-in between two radii (say R1 and R2). In such cases, the parameter α1 is adjusted in such a way that it corrects for the non-constant behaviour of the other factors at large radii, making the global contribution of all factors (except the one in α2) constant at R1 and R2.

5.3.4 The parameter β For this parameter, there are two distinct cases: β = 0 and β > 0. In the first case, the density is maximum in the center and falls off smoothly. In the latter case, the density is null in the center since Lz = 0 for R = 0. In order to model real stellar systems, we need components with β = 0 to have some mass in the center. However, in a real galaxy, the maximum number of stars occurs in the intermediate region where the bulge meets the disk: this justifies the utilization of components with β > 0 when modeling real stellar systems. We see the maximum density moving away from the center when β is rising (Fig. 5.7). We also see on Fig. (5.7) that an increasing β will concentrate the mass in a smaller region of configuration space.

5.3.5 The parameter η

η For a given Lz, the largest value of the factor (E − Sz0 ) is obtained when the binding energy E = S0 corresponds to the circular orbits in the galactic plane (see Fig. 1.2). So, the parameter η is responsible for the favouring of almost circular orbits: a larger η implies a larger contribution of almost circular orbits (Fig. 5.8) and thus a mass density located closer to the plane.

5.3.6 The parameter s

Condition (5.3) E ≥ Sz0 − sI3 implies that for I3 6= 0 and a strictly negative s, the orbits cannot reach the height z0 above the galactic plane. We see on Fig. (5.9) that the height z0 is reached only in the case s = 0. Furthermore, since a large I3 corresponds to an orbit that can reach a large height, the factor δ (E − Sz0 + sI3) favours orbits that stay low. So, by setting s more negative, we confine the orbits closer to the galactic plane. 117

Figure 5.7: Contour plots of the configuration space density (the mass density) in a meridional plane, for components with the parameters (α1, α2, β, δ, η, z0, s, a) = (3, 3, β, 1, 2, 2, −0.5, −5), with β = 0 (top left), β = 0.5 (top right), β = 4 (bottom left) and β = 6 (bottom right). We see that the maximum number of stars moves away from the center and that the mass is more concentrated in configuration space for an increasing β. 118

Figure 5.8: For I3 = 0 and Lz = 0.1, this figure displays the value of a com- ponent distribution function (in arbitrary units) between E = Sz0 and E = S0 for η = 1 and η = 8. The other parameters have the same values as in Fig. 5.4 (with z0 equal to 2 kpc). We see that the proportion of circular orbits (close to E = S0) is much higher for η = 8. 119

Figure 5.9: Contour plots of the configuration space density (the mass density) in a meridional plane, for components with the parameters (α1, α2, β, δ, η, z0, s, a) = (3, 3, 1, 1, 2, 2, s, −5), with s = 0 (top left), s = −0.5 (top right), s = −1 (bottom left) and s = −4 (bottom right). The height z0 is reached only in the case s = 0. The more negative s, the more the mass is concentrated near the galactic plane.

A very important property of our components is the possibility of introduc- ing a certain amount of anisotropy in the stellar disk: if we denote by σz the dispersion of the velocity in the direction perpendicular to the galactic plane, and by σR the dispersion of the radial velocity in the galactic plane, then any nonzero s will produce a ratio σz less than 1 (Fig. 5.10). The ratio is closer σR to unity in the center than in the outer regions: this indicates the physically realistic character of our components. For s = 0, we find σz = σR since we are dealing with a two-integral component again.

5.3.7 The parameter δ

A large δ has partly the same effects as a large η: it favours circular orbits. Furthermore, a large δ augments the effects of the negative s and forces the stars to stay close to the plane by favouring low I3-values. As we can see on Fig. (5.11), a component with a larger δ has more stars in the galactic plane and shows a steeper decline with respect to z. 120

Figure 5.10: This figure displays the ratio σz of several components (in the σR Plane) for varying s. The other parameters have the same value as in Fig. 5.9. The dependence of the components on the third integral induces anisotropy. 121

Figure 5.11: Decline of the logarithm of the configuration space density (in arbitrary units) as a function of the height above the Galactic plane at R = 1 kpc for varying δ. A rising δ implies a steeper decline. The other parameters have the same values as in Fig. 5.4 (with z0 equal to 2 kpc). 122

5.4 Modeling

The component distribution functions described in this Chapter are very useful as basis functions in the method described by Dejonghe (1989), in order to find, in a given potential, a three-integral distribution function that reproduces any observable quantities of a population of stars (spatial mass density, velocity dispersions, average radial velocities on a sky grid,...). As an illustration, we present the application of the method to fit a given spatial density ρ0(R, z) (see Batsleer & Dejonghe 1995 for a similar application in the two-integral approximation). We look for a linear combination of our components X cΛFΛ (5.29) Λ that fits ρ0(R, z), with Λ = (α1, α2, β, δ, η, z0, s, a) and cΛ the coefficients that are to be determined. In practice, to find this linear combination we must introduce a grid (Ri, zi) in configuration space and minimize the quadratic function in cΛ: " ! #2 2 X X (Λ) χ = cΛµ0,0,0(Ri, zi) − ρ0(Ri, zi) (5.30) i Λ This minimization, together with the constraint that the distribution function must be positive in phase space, is a problem of quadratic programming (here- after QP) described by Dejonghe (1989). Here, we choose to adopt for ρ0 a spatial density which closely resembles that of a real disk, i.e. a van der Kruit law, for which the vertical disribution is a good compromise between an exponential and an isothermal sheet (Spitzer 1942, van der Kruit 1988).

 R   z  ρ0(R, z) ∝ exp − sech (5.31) hR hz In order to have a zero derivative with respect to R on the rotation axis, we adopt a mass density that follows closely the van der Kruit law, without a cusp in the center (see also Batsleer & Dejonghe 1995):     1 + 2R/hR R z ρ0(R, z) = exp − sech , (5.32) 1 + R/hR hR hz with hR and hz denoting the horizontal and vertical scale factor, respectively. Since the moments µ0,0,0 are dependent on the potential of the galaxy (in- cluding the dark matter), we have to choose a potential for the galaxy that contains the stellar disk we want to model. Here, we adopt potential II of Table 4.5. The first step in the actual modeling consists in the selection of a subset of components out of the (infinite) set of possible components. This subset is chosen so that certain features, that we suppose to be present in the stellar disk, 123

1

0

-1

-2

-3 0 2 4 6 8 10 radial position

Figure 5.12: Data for a van der Kruit disk (hR = 3kpc and hz = 0.25kpc) were generated in the region 0 kpc ≤ R ≤ 10 kpc and 0 kpc ≤ z ≤ 1 kpc. The initial subset of components was made of the components with β = 0, 1, 3, 5, 7; α1 = 1; α2 = 0.15, 0.3, 2; z0 = 1, 2, 4; η = 1, 5, 10; s = 0, −0.5, −1; δ = 0.01, 1, 4; a = 0. Components with non-zero s and non-zero δ were selected by the QP program. The crosses indicate the data for z = 0pc, 200pc, 400pc, 600pc, 800pc, 1kpc (from top to bottom), with error bars. The solid lines correspond to the mass density of the components linear combination at these heights. The only region of our disk where the fit deviates a little from the data is the 2 kpc central region: our components are primarily intended to describe the outer regions rather than the central region of the Galaxy since there is in reality a strong deviation from axisymmetry in the central region (bar, see Section 1.2.7), and a supermassive black hole (cusp). 124

Figure 5.13: For Lz = 0.1 and a fixed I3 (left panel:I3 = 0, riht panel:I3 = 0.05), this figure displays the values of the distribution function (corresponding to the fit obtained in Fig. 5.12) as a function of E (for the bound orbits). For I3 = 0.05, the maximum value of E is the one corresponding to infinitesimally thin tube orbits and is smaller than S0 (see Eqs. (4.6) and (4.7)).

Figure 5.14: The left panel displays the ratio σz in the galactic plane for the σR fit obtained in Fig. 5.12. The right panel gives the shapes of the individual velocity dispersions curve σR (solid line) and σz (dotted line) in the galactic plane. 125 such as circular orbits, are included. For example, we expect the mass density corresponding to a component to have an exponential behaviour close to the mass density we want to model. The QP program first minimizes the function (5.30) for one component FΛ and chooses the component of the initial subset that produces the lowest minimum for that function (5.30). Then the program iterates, selecting and adding at each iteration the component which, together with the components already chosen in a previous run, produces the best fit. Once the minimum of the χ2-variable does not change significantly any more with the addition of extra components, the program is halted because too low a value for χ2 could imply that the QP program starts producing a distribution function featuring unnecessary oscillations. As an example, we model a modified van der Kruit disk with hR = 3kpc and hz = 0.25kpc. Batsleer & Dejonghe (1995) already showed that a linear combination of two-integral components (with s = 0 and δ = 0) could fit such a disk, but with σR = σz. In order to model real anisotropic velocity data in the future, the dependence on the third integral will be needed. We show that, by choosing components with β = 0, 1, 3, 5, 7; α1 = 1; α2 = 0.15, 0.3, 2; z0 = 1, 2, 4; η = 1, 5, 10; s = 0, −0.5, −1; δ = 0.01, 1, 4 and a = 0 in the initial subset, a fit with components featuring s 6= 0 and δ 6= 0 can be obtained too (see Fig. 5.12). The fit is obtained for a linear combination of 25 components at 231 configu- ration space points (206 degrees of freedom). If we assume relative errors of 6%, we obtain for our minimum χ2 = 246, and the probability that a value of χ2 larger than 246 should occur by chance is Q(246/2, 206/2) ' 0.1 (Abramowitz & Stegun 1972), which makes the goodness-of-fit believable (Press et al. 1986). By using St¨ackel dynamics to model the galactic disk, we have constructed a completely explicit and analytic distribution function, with an explicit depen- dence on the third integral. Fig. 5.13 displays the distribution function obtained by QP as a function of E, for Lz = 0.1 and for two values of I3 (I3 = 0 and

I3 = 0.05). For I3 = 0, the distribution function is non-zero if Sz0 ≤ E ≤ S0 (with z0 = 4kpc); for I3 > 0, instead, the maximum value of E is the one cor- responding to infinitesimally thin short axis tubes and is smaller than S0 (see Eqs. (4.6) and (4.7)). We see on Fig. 5.13 that the distribution function is decreasing with increasing I3 (particularly near E = Sz0 ), and that it has some clumps. These clumps at I3 = 0.05 are not discontinuities since the distribution function is a linear combination of continuous components. Many different three-integral distribution functions correspond to a given spatial density, and there is no guarantee that they will yield realistic velocity dispersions. It is a major result of this Chapter to show that it is possible to find a linear combination of our components yielding realistic velocity dispersions. Fig. 5.14 displays the ratio σz in the galactic plane: at the radius corresponding σR to the Solar position in the Milky Way (7.5-8.5 kpc), the classical value of σz ' 0.4 is obtained. The local maximum in the σz curve is due to the σR σR individual shapes of the velocity dispersions curves (Fig. 5.14). In the future, we could use the component distribution functions defined in this Chapter to model real data in the inner Galaxy, with one of the St¨ackel potentials defined in Table 4.5. Indeed, Eq. (5.30) can easily be extended to 126

higher order moments, such as the projected mean radial velocity hvri on the sky, or the projected velocity dispersion of those radial velocities. For example, we could use the data of Messineo (2004), or of Dejonghe & Van Caelenberg (1999) (radial velocities of AGB stars in the inner Galaxy) who constructed a model with basis functions of Abel type (Dejonghe & Laurent 1991), less adapted than ours to model a disk population. For such a three-integral model, we can be virtually sure that it will be possible to find a stable solution (with respect to Toomre criterion, Eq. (1.34)) when modeling real data. Once the distribution function of the population will be found in integral space, we could compare it in velocity space to the one found in Chapter 3 (Figs. 3.1 and 3.3). Moreover, we could use the stars of group B (“axisymmetric” background, see Section 3.3.6) to refine the solution. We could iterate the processus with the five potentials of Table 4.5 and see which one leads to the best fit. We would thus have indirect constraints on the mass distribution of the Milky Way. Moreover, the solution would yield ideal initial conditions (more complex than a simple two-dimensional Schwarzschild velocity ellipsoid, see Fig. 5.15) for three-dimensional N-body simulations in the Solar neighbourhood, in order to study the formation of the dynamical streams observed in Chapter 3. 127

Figure 5.15: The velocity distribution at Solar position in the Galactic plane for the three-integral model of Dejonghe & Van Caelenberg (1999). We see that some axisymmetric substructure is present in velocity space, even if it does not correspond to the regions of the dynamical streams of Chapter 3 (see Fig. −1 3.3 with U ' −vR and V ' −vφ − 220 km s , without correcting for the unknown Solar motion). Nevertheless, it is clear that the initial conditions for a perturbation analysis should be more complex than a Schwarzschild velocity ellipsoid. 128 Chapter 6

Conclusions and perspectives

This thesis presented a kinematic analysis of 5952 K and 739 M giants in the Solar neighbourhood which included for the first time radial velocity data from an important survey performed with the CORAVEL spectrovelocimeter. We also used proper motions from the Tycho-2 catalogue, and parallaxes from the Hipparcos catalogue. First, we analyzed the kinematics of the sample restricted to the 2774 stars with parallaxes accurate to better than 20%, and then we made full use of the 6030 available stars and evaluated the kinematic parameters with a Monte-Carlo method. We found that the asymmetric drift is larger for M giants than for K giants due to the fact that the M giants must be a little older than the K giants on average. We also found the usual value for the Solar motion when assuming that the whole sample has no net radial and vertical motion. Then a maximum-likelihood method, based on a bayesian approach (Luri et al. 1996, LM method), has been applied to the data and allowed us to derive simultaneously maximum likelihood estimators of luminosity and kinematic pa- rameters, and to identify subgroups present in the sample. Several subgroups have been identified with known kinematic features of the Solar neighbourhood (namely the Hyades-Pleiades supercluster, the Sirius moving group and the Her- cules stream). Isochrones in the Hertzsprung-Russell diagram revealed a very wide range of ages for stars belonging to these subgroups. This result excluded the classical hypothesis which views them as cluster remnants. Moreover, since it is unlikely that two or three mergers (leading to the Hyades-Pleiades and Sirius superclusters) have left such important signatures in the disk near the position of the Sun, we concluded that the substructure in velocity space is most probably related to the dynamical perturbation by transient spiral waves (as recently modeled by De Simone et al. 2004). Those velocity groups, usu- ally called “superclusters” or “moving groups”, are thus renamed “dynamical streams”.

129 130

A possible explanation for the presence of open clusters and associations in the same area of the UV -plane as those streams is that they have been put there by a spiral wave, while kinematics of the older stars of our sample have also been disturbed by the same wave. It would be interesting to use N-body simulations to check wether a cluster could be displaced by a spiral wave without being disrupted. The slightly supersolar metallicity of the Hyades-Pleiades stream (also reported by Chereul & Grenon 2001) suggests that this stream originates from a specific galactocentric distance and that it was perturbed by a spiral wave at a certain moment, radially pushed by the wave, and sent in the Solar neighbourhood (see Sellwood & Binney 2002). This would explain why this stream is composed of stars sharing a common metallicity but not a common age. A careful metallicity analysis of this stream would be of great interest in order to confirm this scenario. If this scenario is correct, the radial mixing should be taken into account in any future model of the chemical evolution of the Galaxy (see e.g. K¨oppen 2003 for older models). The Sirius moving group could also be a feature recently formed by the passage of a transient spiral, while the Hercules stream would be related to the bar’s outer Lindblad resonance (Dehnen 1999, 2000, Fux 2001). The position of all these streams in the UV -plane is responsible for the vertex deviation of 16.2◦ ± 5.6◦ for the whole sample. We even argue that the vertex deviation observed among large samples of early-type stars (see Dehnen & Binney 1998a) and the specific kinematic initial conditions of some young open clusters and OB associations could in fact have the same dynamical origin as those streams of giants. A better understanding of the streams should start with a chemical analysis of the stars belonging to them. As a first step, their photometric indices could be investigated. An important consequence of the dynamical origin of the streams is that it makes no sense to integrate backwards over Gyrs in a smooth stationary axisymmetric potential the orbits of the stars belonging to them. This warning may even apply to most old disk stars, as they may be expected to have been dynamically disturbed by a transient spiral wave at least once within their lifetime (which could explain the absence of age-metallicity relation in the Solar neighbourhood; Edvardsson et al. 1993). We then discussed, in the light of our results, the validity of older estimations of the Solar motion in the Galaxy. Indeed, the group of background stars (group B, see Section 3.3.6) has a distribution in velocity space close to a Schwarzschild ellipsoid but is not centered on the classical value found for −U in Section 3.2 when considering the full sample, including the streams. Instead we find hUi = −2.78±1.07 km s−1. This discrepancy clearly raises the essential question of how to derive the Solar motion in the presence of dynamical perturbations altering the kinematics of the Solar neighbourhood (the net radial motion of stars in the Solar neighbourhood can be of the order of 10 km s−1 in the simulation of De Simone et al. 2004): does there exist in the Solar neighbourhood a subset of stars having no net radial motion which can be used as a reference against which to measure the Solar motion? We do not have the answer to that question, but we have shown that the reliability of the older estimations of the Solar motion (Dehnen & Binney 1998a, Bienaym´e1999, Zhu 2000, Brosche et al. 2001) is 131 questionable. Theoretical investigations in this area should thus clearly be pursued, and in particular dynamical modeling. We have shown that the fine structure of phase space in the Solar neighbourhood cannot be interpreted in terms of an axisymmetric steady-state model. Nevertheless, an axisymmetric model reveal- ing all the fine structure of the axisymmetric distribution function in the Solar neighbourhood would give ideal initial conditions (more complex than a simple two-dimensional Schwarzschild velocity ellipsoid) for three-dimensional N-body simulations that could afterwards reproduce some non-axisymmetric features observed in the Solar neighbourhood (see Fux 1997, De Simone et al. 2004 for two-dimensional simulations). Indeed, the weak point of all the recent simu- lations is the choice of the initial conditions, especially for three-dimensional simulations where the vertical motion of stars is important (if we want to un- derstand the increase of σW with age, or the peculiar velocity W of the HyPl group in Section 3.3.6). We developed new tools to establish such a model in the last two Chapters. To make an analytic model possible, we chose to deal with a special kind of potential (St¨ackel potentials) for which all orbits admit three analytic integrals of the motion. We have shown that some different simple St¨ackel potentials can fit most known parameters of the Milky Way (especially Hipparcos latest findings). We have generalized the two-component potentials of Batsleer & De- jonghe (1994) by adding a thick disk, and we have studied how the parameters can vary in order to satisfy selection criteria based on the latest observational constraints. We have shown that the presence of a thick disk allows more flex- ibility in the form of the potentials, especially for the shape of the halo and we have selected five different valid potentials listed in Table 4.5. It should be noted that, in fact, there could be two different thick disks in the Galaxy, a very thick one (Chiba & Beers 2000; Gilmore, Wyse & Norris 2002) and a thinner one (Soubiran, Bienaym´e& Siebert 2002): in that case, the three-component modeling presented in this thesis could be easily extended, but this would imply a growth of parameter space. A major result of Chapter 4 is that, even though St¨ackel potentials are negligible in the set of all potentials, many of them are still able to match the most recent estimates for the parameters of the Milky Way, and furthermore very simple ones (superpositions of three Kuzmin-Kutuzov po- tentials) are sufficient to do this. These potentials (especially potential I of Table 4.5) were already used by Vlemmings et al. (2004) to infer the Galactic orbit of two pulsars over a few Myrs. Then, we have constructed new analytic three-integral stellar distribution functions F (E,Lz,I3) yielding σR 6= σz: they are generalizations of two-integral ones that can describe thin disks with the restriction that σR = σz (Batsleer & Dejonghe 1995). We first reduced the triple integral defining their moments to a simple one, by making some assumptions on the parameters. Then we looked for the effects of the different parameters and showed the disk-like (physically realistic) features of our distribution functions: they have a finite extent in ver- tical direction and an exponential decline in the galactic plane, while favouring almost circular orbits. A very important feature induced by the dependence on 132 the third integral is their ability to introduce a certain amount of anisotropy, by varying the parameters responsible for this dependence (s and δ). We showed that a van der Kruit disk can be modeled by a linear combination of such distri- bution functions with an explicit dependence on the third integral and a realistic anisotropy in velocity dispersions. This implies that they are very promising tools to model real data with σR 6= σz by using the quadratic programming algorithm described by Dejonghe (1989). We plan to use our potentials and distribution functions in order to estab- lish an axisymmetric model, based on radial velocities of AGB stars in the inner Galaxy (already used by Dejonghe & Van Caelenberg 1999). The axisymmetric velocity distribution in the Solar neighbourhood deduced from such a model could be compared with the velocity distribution in Figs 3.1 and 3.3. Then we could use this model to create realistic initial conditions for N-body simula- tions in order to understand the exact mechanisms that lead to the observed dynamical streams in the Solar neighbourhood. Moreover, we could see which of the five potentials of Table 4.5 yields the best fit when modeling those real data, and thus have indirect constraints on the overall mass distribution of the Galaxy, which is still ill-determined.

In conclusion, the results of our kinematic analysis of giant stars in the Solar neighbourhood led to a profound revision of our understanding of “su- perclusters” (or “moving groups”), renamed “dynamical streams” since they are most probably related to the dynamical perturbation by non-axisymmetric components of the Galaxy. A better understanding of those streams needs more theoretical investigations in the field of galactic dynamics. Our contribution to theoretical research was to develop new tools (potentials and distribution func- tions) for axisymmetric dynamical modeling: indeed, 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, that are thought to lead to the formation of the “dynamical streams” exhibited in this thesis. Appendix A

Contents of the data table

Table A.11 (on the CD-ROM attached to this thesis) contains 6691 lines, and contains the following information in the successive columns (note that missing data are replaced by null values):

1 HIP number. 2 HD number. 3-4 BD number, only when there is no HD number. 5-6 Right ascension and declination in decimal degrees from fields H8 and H9 of the Hipparcos Catalogue (ICRS, equinox 2000.0; epoch 1991.25). 7-8 Hipparcos parallax and standard error.

9-10 µα∗ ≡ µα cos δ from Tycho-2 and standard error.

9-10 µδ from Tycho-2 and standard error. For 291 stars, null values are found in columns 9 to 12; the kinematic study made then use of the Hipparcos proper motions instead of the Tycho-2 ones. In most cases, the absence of a Tycho-2 proper motion is caused by the fact that the star has a close visual companion or is too bright for the Tycho detection. 13-16 Same as columns 9 to 12 for Hipparcos proper motions. 17 The normalized absolute difference between the Hipparcos and Tycho-2 2 2 2 1/2 proper motions: ∆µ = |µHip−µTyc2|/µ, where µ = (µα cos δ+µδ) and  = (2 + 2 )1/2, where  denotes the standard error of quantity µ µHip µTyc−2 i i. The quantity ∆µ may be used as a diagnostic tool to identify long- period binaries (Kaplan & Makarov 2003). The basic idea behind this tool is the following. For binary stars with orbital periods much longer

1Table A.1 is also available in electronic form at the CDS via anonymous ftp to cdsarc.u- strasbg.fr (130.79.128.5)

133 134

than the duration of the Hipparcos mission, the proper motion recorded by Hipparcos is in fact the vector addition of the true proper motion and of the orbital motion. This orbital motion averages out in the Tycho-2 value, since it is derived from measurements spanning a much longer time base (on the order of a century, as compared to 3 yr for Hipparcos). Therefore, a difference between the Tycho-2 and Hipparcos proper motions (beyond the combined error bar encapsulated by µ) very likely hints at the binary nature of the star. This diagnostic will be fully exploited in a forthcoming paper devoted to the binary stars present in our sample. Note that Fig. 1 presents the histogram of ∆µ, which indeed reveals the presence of an abnormally large tail at ∆µ ≥ 1.5.

18-19 Hipparcos Hp magnitude and associated standard error.

20 Tycho-2 VT 2 magnitude.

21 Hp−VT 2. For visual binaries (flag 4 in column 24), this colour index has not been listed and the value 0.0 is given instead, because the Hp magnitude appears to be a composite value for the two visual components. In most of these cases, the VT 2 magnitude of the visual companion is given in the last column of the table. For large amplitude variable stars, the Hp − VT 2 index is meaningless as well, and as been set to 0. 22 V −I colour index in Cousins’ system as provided by field H40 of the Hip- parcos catalogue. This index is useful for constructing the Hertzsprung- Russell (HR) diagram of the sample. It should be stressed that the H40 field of the Hipparcos catalogue does not provide a directly measured quantity. It has instead been computed from various colour transforma- tions based on the B − V index from field H37, which neither is a directly measured quantity. 23 V − I colour index in Cousins’ system as provided by the colour trans- formation based on the measured Hp − VT 2 colour index (Platais et al. 2003). This value is thus in principle more reliable than the Hipparcos H40 value (note, however, that the colour transformation provided by Platais et al. (2003) had to be extrapolated somewhat, since it is pro- vided in the range −2.5 ≤ Hp − VT 2 ≤ −0.20, whereas our data set goes up to Hp − VT 2 = 0.1). It has been used to draw the HR diagram of the sample. For large-amplitude variable stars, the median V − I index has been taken directly from Platais et al. (2003), instead of being computed from the colour transformation based on Hp − VT 2 (set to 0. in those cases). 24 Binarity flag: *: no evidence for radial-velocity variations; 0: spectroscopic binary (SB), with no orbit available. The star had to be discarded from the kinematic analysis; 1: SB with an orbit available, or with a center-of-mass velocity which 135

can be estimated reasonably well. Column 25 then contains the system’s center-of-mass velocity. The reference with the orbit used to derive the center-of-mass velocity is listed in the last column. If no reference is given, the center-of-mass velocity has been estimated from the available CORAVEL data; 2: supergiant star, with a substantial radial-velocity jitter (see Fig. 5); 3: uncertain case: either SB or supergiant; 4: visual binary with a companion less than 600 away (as listed by the Tycho-2 catalogue); 5,6,7,8: as 0,1,2,3 but for a visual binary. It should be stressed here that visual binaries have not been searched for exhaustively among our target stars. Whenever the Tycho-2 catalogue lists a companion star less than 600 away from the target star, the binarity flag has been set to 4. In most of these cases, the Hipparcos Hp magnitude corresponds to the composite magnitude and the Tycho-2 proper motions are identical for the two components. The VT2 magnitude of the companion is listed in the last column of the table; 9: binary supergiant with an orbit available.

25-26 Average radial velocity (based on CORAVEL observations) or center-of- mass velocity for SBs (the last column provides the bibliographic code of the reference providing the orbit used), and standard error (set to 0.3 km s−1 in the case of center-of-mass velocity).

27 The absolute Hp magnitude, corrected for interstellar reddening (accord- ing to the model of Arenou et al. 1992), and based on the LM distance listed in column 28.

28-29 The maximum-likelihood distance based on the LM estimator (see Sec- tion 3.3), and its associated standard error (see Eq. (3.32)).

30 The interstellar absorption AV , based on the LM distance and the model of Arenou et al. (1992).

31-33 The U, V and W components of the heliocentric space velocity deduced from the LM method (corrected for the galactic differential rotation to first order using A = 14.82 km s−1kpc−1 and B = −12.37 km s−1kpc−1).

34-35 The most likely group to which the star belongs, and the associated prob- ability. The various groups are the following: 1 or Y: stars with a young kinematics; 2 or HV: high-velocity stars; 3 or B: stars defining the smooth background (the most populated group); 4 or HyPl: stars belonging to the Hyades-Pleiades stream; 5 or Si: stars belonging to the Sirius stream; 6 or He : stars belonging to the Hercules stream. 136

36 In the case of a spectroscopic binary with an available orbit: Bibliographic code (according to the standard ADS/CDS coding) of the reference list- ing the orbit used as the source of the center-of-mass velocity (The code 2005A&A...???..???J refers to the forthcoming paper by Jorissen et al. 2005 devoted to the analysis of the binary content of the present sample). In the case of a visual binary, the VT2 magnitude of the companion. An asterisk in that column indicates the presence of a note in the remark file. It must be stressed that columns 27 to 35 contain model-dependent data, as they were derived by the LM method. They depend upon, e.g., the particular choice for the values of the Oort constants, the interstellar extinction model, the a priori choice of the various distribution functions,... Bibliography

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