UNIVERSITE LIBRE DE BRUXELLESFaculte des Sciences

KINEMATICS AND DYNAMICS OF GIANT STARS IN THESOLAR NEIGHBOURHOOD

byBenoit FAMAEY

Ph.D. thesis submittedfor the degree of Docteur en SciencesInstitut d’Astronomie et d’Astrophysique Academic year 2003-2004

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Notre savoir consiste en grande partie a croire savoir, et a croire que d’autressavent.

Paul Valery, 1937 (L’Homme et la coquille)

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

Thomas Hobbes, 1651 (Leviathan, chapter 4)

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Acknowledgements

Je voudrais avant tout remercier le professeur Marcel Arnould, Directeur del’Institut d’Astronomie et d’Astrophysique, qui m’a accueilli a l’Institut en tantque mathematicien et qui m’a permis de passer ces annees de these dans desconditions ideales en m’accordant une grande confiance et une grande libertede travail et en se montrant d’une aide attentive face a mes interrogations etsollicitations.

Je remercie ensuite avec beaucoup de gratitude mon directeur de these, leprofesseur Alain Jorissen, pour sa disponibilite constante durant ces annees dedoctorat et pour les nombreuses discussions stimulantes que nous avons eues.Au cours de celles-ci j’ai ressenti une veritable emulation, son esprit scientifiqueaiguise en faisant non seulement un chercheur hors pair mais surtout un inter-locuteur extremement interessant et motivant. Je le remercie en outre, par delaces considerations professionnelles, pour ses qualites humaines hors du commun.

Je voudrais egalement exprimer ma gratitude a tous les autres membresde l’Institut, en particulier a Dimitri et Marc pour leur soutien face aux aleasde l’informatique, a Carine et Laurent pour m’avoir fourni quelques referencesinteressantes, a Sylvie, Sophie, Ana, Claire et Viviane pour leur sourire etl’agreable atmosphere de travail qu’elles contribuent a creer a l’Institut, a Yves,Abdel, Stephane, Lionel et Matthieu pour leur bonne humeur. Merci aussia Samir Keroudj pour la realisation de l’animation en trois dimensions qui apermis d’illustrer certains resultats de cette these.

Il est tres important pour moi de remercier ici le professeur Herwig De-jonghe, de l’Universite de Gand, veritable instigateur de ce projet, sans lequelrien n’aurait ete possible. Ses explications, toujours claires et precises, m’ontpermis de me constituer au fil du temps une expertise en dynamique galactique,domaine qui etait entierement neuf pour moi au moment d’aborder ces anneesde these. Je remercie aussi Kathrien Van Caelenberg qui avait entame une partiede ce travail a l’Universite de Gand.

Une grande partie des resultats presentes dans cette these dependent dedonnees obtenues grace a l’Observatoire de Geneve et a la cooperation de nom-breux observateurs anonymes que je remercie pour leur importante contribu-tion a ce travail. Je remercie tout particulierement le professeur Michel Mayorainsi que Catherine Turon d’avoir accepte de me fournir ces donnees. Je re-mercie egalement Stephane Udry pour son hospitalite lors de mon sejour al’observatoire de Geneve et sa collaboration active lors du debroussaillement

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des donnees CORAVEL.Je suis egalement extremement reconnaissant a Xavier Luri, de l’Universite

de Barcelone, qui m’a accueilli chaleureusement lors de mon sejour dans sonservice et qui fut toujours d’une aide radicalement efficace et d’une disponibilitesans faille dans l’analyse des donnees cinematiques qui a conduit aux principauxresultats de ce travail.

Je voudrais aussi remercier mes parents pour leur soutien constant tout aulong de mes etudes et leurs nombreux conseils judicieux sur le plan humain oulogistique. C’est par ailleurs un plaisir de remercier ici mes amis de longue dateMaxime, Benjamin, Stephane, Bernard, Benoıt, Jules, Hugo et Mohamed pourleurs commentaires de non specialistes de la discipline et leur soutien moral.Merci enfin a ma douce Marie-Laure, merci de m’avoir soutenu et encourage, etsurtout merci d’ensoleiller ma vie chaque jour.

Contents

1 Introduction 171.1 Components of the Galaxy . . . . . . . . . . . . . . . . . . . . . 17

1.1.1 The luminous halo . . . . . . . . . . . . . . . . . . . . . . 171.1.2 The dark halo . . . . . . . . . . . . . . . . . . . . . . . . . 181.1.3 The bulge . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.1.4 The thin disk . . . . . . . . . . . . . . . . . . . . . . . . . 181.1.5 The thick disk . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 Stellar dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.1 Relaxation time . . . . . . . . . . . . . . . . . . . . . . . . 191.2.2 Hierarchical formation scenario . . . . . . . . . . . . . . . 211.2.3 Boltzmann and Poisson equations . . . . . . . . . . . . . 211.2.4 Integrals of the motion . . . . . . . . . . . . . . . . . . . . 221.2.5 The third integral . . . . . . . . . . . . . . . . . . . . . . 241.2.6 Galactic orbits . . . . . . . . . . . . . . . . . . . . . . . . 251.2.7 Non-axisymmetric perturbations . . . . . . . . . . . . . . 28

1.3 The Solar neighbourhood . . . . . . . . . . . . . . . . . . . . . . 361.3.1 Galactocentric radius of the Sun . . . . . . . . . . . . . . 371.3.2 Rotation curve and Oort constants . . . . . . . . . . . . . 371.3.3 Local dynamical mass . . . . . . . . . . . . . . . . . . . . 391.3.4 LSR, Solar motion and velocity ellipsoid . . . . . . . . . . 411.3.5 Vertex deviation and substructure of velocity space . . . . 42

1.4 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 44

2 Stellar sample 472.1 Selection criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.2 Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Kinematic analysis 573.1 Analysis restricted to stars with the most precise parallaxes . . . 583.2 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . 603.3 Bayesian approach . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3.1 Phenomenological model . . . . . . . . . . . . . . . . . . . 643.3.2 Observational selection and errors . . . . . . . . . . . . . 653.3.3 Maximum likelihood . . . . . . . . . . . . . . . . . . . . . 66

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3.3.4 Group assignment . . . . . . . . . . . . . . . . . . . . . . 673.3.5 Individual distance estimates . . . . . . . . . . . . . . . . 693.3.6 The kinematic groups present in the stellar sample . . . . 693.3.7 Physical interpretation of the groups . . . . . . . . . . . . 82

4 Stackel potentials 894.1 Coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2 Three-component Stackel potentials . . . . . . . . . . . . . . . . 904.3 Selection criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.4 The “winding staircase” . . . . . . . . . . . . . . . . . . . . . . . 954.5 Constraints on the scale height of the thick disk . . . . . . . . . . 954.6 The final selection . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5 Three-integral distribution functions 1075.1 Construction of three-integral components . . . . . . . . . . . . . 1085.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2.1 The case where a = 0 and m is an even integer . . . . . . 1095.2.2 The general case . . . . . . . . . . . . . . . . . . . . . . . 116

5.3 Physical properties of the components . . . . . . . . . . . . . . . 1165.3.1 The parameter z0 . . . . . . . . . . . . . . . . . . . . . . . 1175.3.2 The parameter α1 . . . . . . . . . . . . . . . . . . . . . . 1175.3.3 The parameter α2 . . . . . . . . . . . . . . . . . . . . . . 1175.3.4 The parameter β . . . . . . . . . . . . . . . . . . . . . . . 1205.3.5 The parameter η . . . . . . . . . . . . . . . . . . . . . . . 1205.3.6 The parameter s . . . . . . . . . . . . . . . . . . . . . . . 1205.3.7 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 . . . . . . . . . . . . . . . . . . . 161.2 Lindblad diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 231.3 Tightly wound and loosely wound spiral patterns . . . . . . . . . 281.4 The swing-amplification . . . . . . . . . . . . . . . . . . . . . . . 31

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

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

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

2.4 The concept of line-width parameter . . . . . . . . . . . . . . . . 492.5 The (Sb, σ0(vr0)) diagram . . . . . . . . . . . . . . . . . . . . . . 502.6 Distribution of the sample on the sky . . . . . . . . . . . . . . . 52

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

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

3.3 All the stars plotted in the UV -plane with their values of U andV deduced from the LM method . . . . . . . . . . . . . . . . . . 67

3.4 HR diagram of group Y . . . . . . . . . . . . . . . . . . . . . . . 683.5 HR diagram of group HV . . . . . . . . . . . . . . . . . . . . . . 703.6 HR diagram of group HyPl . . . . . . . . . . . . . . . . . . . . . 723.7 HR diagram of group Si . . . . . . . . . . . . . . . . . . . . . . . 743.8 HR diagram of group He . . . . . . . . . . . . . . . . . . . . . . 753.9 Histogram of the metallicity in groups B, HyPl and Si for the

stars present in the analysis of McWilliam (1990) . . . . . . . . . 763.10 HR diagram of group B. . . . . . . . . . . . . . . . . . . . . . . 77

4.1 Integral space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.3 Profile of the logarithm of vertical density at R = 8 kpc for

Kuzmin-Kutuzov potentials with different axis ratios . . . . . . 94

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4.4 Mass isodensity curves in a meridional plane for the five potentialsof Table 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.5 The effective bulge . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.6 The rotation curves of the five selected potentials of Table 4.5 . 101

5.1 Integration area in the (E, I3)-plane . . . . . . . . . . . . . . . . 1075.2 Integration area in the (x, y)-plane . . . . . . . . . . . . . . . . . 1095.3 Integration limits in Lz . . . . . . . . . . . . . . . . . . . . . . . 1115.4 Contour plots of the mass density in a meridional plane, for com-

ponents with z0 equal to 4 kpc and 2 kpc . . . . . . . . . . . . . 1135.5 Logarithm of the galactic plane mass density of different compo-

nents for varying α1 . . . . . . . . . . . . . . . . . . . . . . . . . 1145.6 Logarithm of the configuration space density of different compo-

nents for varying α2. . . . . . . . . . . . . . . . . . . . . . . . . 1155.7 Contour plots of the configuration space density in a meridional

plane, for components with varying β . . . . . . . . . . . . . . . 1165.8 Values of a component distribution function as a function of E

for varying η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.9 Contour plots of the configuration space density in a meridional

plane, for components with varying s . . . . . . . . . . . . . . . 1195.10 The ratio σz

σRof several components for varying s. . . . . . . . . 120

5.11 Logarithm of the configuration space density as a function of theheight above the Galactic plane at R = 1 kpc for varying δ. . . 121

5.12 Fit of a van der Kruit disk . . . . . . . . . . . . . . . . . . . . . 1235.13 Values of the distribution function (corresponding to the fit of

the van der Kruit disk) as a function of E for varying I3 . . . . 1245.14 Velocity dispersions for the fit of the van der Kruit disk . . . . . 1245.15 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 theytell us about the dynamics of the Galaxy: we thus contribute to the huge projectof 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 performedwith the CORAVEL spectrovelocimeter at the Observatoire de Haute Provence.Parallaxes from the Hipparcos catalogue and proper motions from the Tycho-2catalogue are also used.

A maximum-likelihood method, based on a bayesian approach, is appliedto the data, in order to make full use of all the available stars, and to derivethe kinematic properties of the subgroups forming a rich small-scale structurein velocity space. Isochrones in the Hertzsprung-Russell diagram reveal a verywide range of ages for stars belonging to these subgroups, which are thus mostprobably related to the dynamical perturbation by transient spiral waves ratherthan to cluster remnants. A possible explanation for the presence of younggroup/clusters in the same area of velocity space is that they have been putthere by the spiral wave associated with their formation, while the kinematicsof the older stars of our sample has also been disturbed by the same wave. Theemerging picture is thus one of dynamical streams pervading the Solar neigh-bourhood and travelling in the Galaxy with a similar spatial velocity. The termdynamical stream is more appropriate than the traditional term superclustersince it involves stars of different ages, not born at the same place nor at thesame time. We then discuss, in the light of our results, the validity of olderevaluations 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 accurateaxisymmetric model of the Galaxy is a necessary starting point in order tounderstand the true effects of non-axisymmetric perturbations such as spiralwaves. To establish such a model, we develop new galactic potentials that fitsome fundamental parameters of the Milky Way. We also develop new compo-nent distribution functions that depend on three analytic integrals of the motionand that can represent realistic stellar disks.

This thesis has led to the publication of three papers in Monthly Notices ofthe Royal Astronomy Society and in Astronomy and Astrophysics:

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

Famaey B., Dejonghe H., 2003, MNRAS, 340, 752 (11pages). Three-componentStackel 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 Kand M giants from CORAVEL/Hipparcos/Tycho-2 data. Revisiting the conceptof 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 theMilky Way (the name given to the luminous band crossing the sky). The Galaxyhas remained a mystery for a surprisingly long time: Kant (1755) was the first topropose in his “Universal Natural History and the Theory of Heavens” that, byanalogy with the planets of the Solar system, the Milky Way could be composedof stars orbiting in a finite flat system, and was maybe not the only finite stellarsystem of this type. Nevertheless, his hypothesis that some nebulae (like theAndromeda M31 nebula) could be similar stellar systems was confirmed only80 years ago by Hubble (1925). Today, we know that the Milky Way is just anordinary 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 = 1.99×1030 kg), and it is composed of interstellar gas (representing a mass of 1010M),dark matter (about 90% of the total mass), and about 1011 stars. The Hubbletype (Hubble 1936, Sandage 1961) of the Galaxy seems to be something likeSABbc (de Vaucouleurs & Pence 1978), i.e. a spiral barred galaxy with fourspiral 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 ofthe order of 50 kpc (1 pc = 3.086 × 1016m = 3.26 light years), and representsonly a small fraction of the total mass of the Galaxy (about 1%). It may haveformed before all the other components (Eggen et al. 1962), or may have been

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accreted much later (Searle & Zinn 1978), but most probably results from bothprocesses (Chiba & Beers 2000, 2001). It contains the most metal-poor starsin the Galaxy (as metal-poor as [Fe/H]= log Fe/H

(Fe/H)= −3.5, see Ryan et al.

1996, and even some with [Fe/H]= −5, see Christlieb et al. 2002) and has nometallicity gradient. Most halo stars are concentrated in globular clusters of102 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 theorder of 200 kpc. It must dominate the mass of the Galaxy (about 90% of thetotal mass) in order to explain the constant circular velocity (the “flat” rotationcurve) of the stars far from the center of the Galaxy. The determination of themass distribution of this dark halo, and the determination of its composition (thefamous 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 ultimatelytouches 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 ofthe luminous halo (Carney et al. 1990). Much of what we know about it is basedon the properties of stars in Baade’s windows (small areas in the sky which arealmost free of obscuring dust), but considerable progress on the determination ofits structural properties has been achieved through observations in the infrared.Mainly, its triaxial structure is now an evidence (Binney et al. 1997) and itselongated component is called the bar, with a semi-major axis of the order of 3kpc. 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 compactradio source called SgrA*, related to a high concentration of mass at the verycenter of the system: a careful study of the motion of stars in the dense starcluster near the center by Genzel et al. (2000) has enabled them to identify thishigh concentration of mass as a supermassive black hole of 2.6 ± 0.2 × 106M(Schodel et al. 2002).

1.1.4 The thin disk

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

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metallicity gradient of the order of −0.07 kpc−1 (which means that the innerGalaxy 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 thechemico-dynamical evolution and enrichment history of the disk, which are stillpoorly known. The thin disk contains the youngest stellar populations, whichare 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 theaxisymmetric disk. The number of spiral arms in the disk is still not known withcertainty (2 or 4 or even more, see Drimmel 2000). Moreover, the origin, thedynamics and the true nature (stationary or transient) of those spiral arms, aswell as their role in star formation process is still the subject of active researchin Galactic astronomy.

1.1.5 The thick disk

The existence of the thick disk as a separate stellar component, with about tentimes less stars than the thin disk, is well documented today (see e.g. Ohja etal. 1994, Chen et al. 2001). It is not only thicker than the thin disk but isalso 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 dispersionof velocities. It is generally thought to have been formed by dynamical heatingof 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 ofthe 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). Infact, there could be two different thick disks: a very thick one (Chiba & Beers2000, 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 themanner they have been formed needs theoretical investigations in the field of“stellar dynamics”, i.e. the theory of the motion of stars in any gravitationallybound system. Generally, the behavior of a stellar system is solely determinedby Newton’s laws of motion and Newton’s law of gravity. General relativisticeffects are unimportant, unless we study the motion very close to the horizonof 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 ofthe dynamics can prove very useful: stellar dynamics considers the Galaxy as a“gas” of stars, but without collisions since gravity is a long-range force. Indeed,in a galaxy of 1011 stars, Chandrasekhar (1942) showed that the time for a

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Figure 1.1: Schematic view of the Galaxy (p. 390 of Zeilik 2002, CambridgeUniversity press), showing its main features: halo, bulge and disk. The Sun liesin the disk, on the inner edge of a spiral arm. The realm of the globular clustersdefines the luminous halo, shown here only in part. The nuclear bulge in thecenter surrounds the core, recently identified as a central supermassive blackhole

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

1.2.2 Hierarchical formation scenario

The collisionless description of the Galaxy is valid at present time, but it was notat the time of Galaxy formation, when gas dynamics, collisions and encounterswere 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 themost plausible scenario is the hierarchical “bottom-up” model: the fluctuationsobserved in the cosmological microwave background (emitted by the baryonicmatter 3 × 105 yr after the “Big Bang”) could be linked with more importantfluctuations 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 formationof dark matter halos of about 106M. These halos then merged together andgained angular momentum by tidal forces: the baryonic matter into them cooleddown by radiation and formed disks by conservation of the angular momentum(for a more detailed description of the hierarchical bottom-up scenario, referto Devriendt & Guiderdoni 2003). Once this gaseous disk was formed in ourGalaxy, mergers with other galaxies still continued to happen regularly butwith a decreasing rate. It is known since the discovery of the absorption of theSagittarius dwarf galaxy by the Milky Way (Ibata et al. 1994) that some starstreams in the Galaxy are remnants of a merger with a satellite galaxy. Helmi etal. (1999) showed that some debris streams are also present in the galactic halonear the position of the Sun. However, if we do not consider specific episodesof the Galactic evolution such as the climax of a major merger, the dynamicalcollisionless 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 acollisionless system, bodies do not jump from one point to another in phasespace, but rather move continuously into it. As a consequence, F (~x,~v, t) mustbe a solution of a continuity equation, the so-called “collisionless Boltzmannequation”:

dF (~x,~v, t)dt

= 0. (1.1)

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The Newton’s equations of motion write:

d~xdt = ~v

d~vdt = −∂Φ(~x,t)

∂~x

(1.2)

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

∂F

∂t+(~v .∂F

∂~x

)−(∂Φ∂~x

.∂F

∂~v

)= 0 (1.3)

The Poisson equation (derived from Newton’s law of gravity) relates thegravitational 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 integralequation: ∫ ∫ ∫

Fmass(~x,~v, t) d3~v = ρ(~x, t) (1.5)

When we deal with such a distribution function representing the whole system,Equations (1.3), (1.4) and (1.5) form the set of Vlasov equations defining aself-consistent model.

Nevertheless, each separate stellar component can also be described by adistribution function related to the number of stars density g(~x, t) of the com-ponent by: ∫ ∫ ∫

Fstars(~x,~v, t) d3~v = g(~x, t) (1.6)

In that case, the model is not self-consistent since the set of stars described byF does not generate the potential in which it evolves (it generates only a partof it).

All the observables such as the number density, the mean velocity and thevelocity dispersions of the stellar component can be expressed in terms of themoments of this distribution function. Eq. (1.6) corresponds to the 0th ordermoment 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 yieldsrespectively a first or a second order moment corresponding to the mean velocityor 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 quantitiesdepending 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 functionof the position ~x and velocity ~v at time t, for example). Such a quantity is calleda 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 sixindependent constants, but not all are integrals of the motion. The integralsof the motion, in turn, come in two flavours: non-isolating and isolating. Anintegral 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 definedby the isolating integrals I1, ..., In−1.

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

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

=∂F

∂I1.dI1(~x,~v)

dt+ ...+

∂F

∂In.dIn(~x,~v)

dt= 0 (1.7)

Integrals of the motion are associated with symmetries of the system (fol-lowing the Noether theorem). The two classical independent isolating integralsof the motion in a Galactic model are the vertical component of the angularmomentum 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 thatthe Galaxy is axisymmetric and stationary (time-symmetric). Indeed, the non-axisymmetric part of the system (the bar and the spiral arms) represents lessthan 5 % of the total mass and, moreover, throughout the visible Galaxy, thedynamical time (' an orbital period ' 108 yr) is orders of magnitude shorterthan 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 thepresent day is a particularly exciting moment in the Galaxy’s life, such as theclimax 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), thetwo classical isolating integrals of the motion are thus

E = −H = ψ(R, z)− 12(v2R + v2

φ + v2z) (1.8)

andLz = Rvφ = R2φ. (1.9)

These two integrals of the motion have been used to construct distributionfunctions F (E,Lz) that are stationary solutions of the Boltzmann equation (1.1)and that can describe thin axisymmetric disks (Shu 1969, Batsleer & Dejonghe1995, Bienayme & Sechaud 1997) or bulge-disk systems (Jarvis & Freeman1985, Kent 1991). This two-integral approximation is nevertheless not sufficientto 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,∫ ∫ ∫F (E,Lz) vR d3~v =

∫ ∫ ∫F (E,Lz) vz d3~v = 0 (1.10)

because E and Lz are quadratic in vR and vz. And moreover,

gσ2R =

∫ ∫ ∫F (E,Lz) v2

R d3~v =∫ ∫ ∫

F (E,Lz) v2z d3~v = gσ2

z (1.11)

because E depends on v2R and v2

z in the same manner (see Eq. (1.8)), andbecause 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 presentthesis), 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 thirdisolating integral is conserved along the orbits, that are thus “regular”1: in fact,in realistic galactic potentials, most chaotic orbits are “quasi-regular” and areconfined 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 integralof galactic dynamics: when there is an analytic formulation for it, it can beinterpreted 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 takeninto account numerically in models of the Galaxy by integrating the equationsof motion (1.2) and by using an orbit superposition technique to fit the observa-tions (Schwarzschild 1979, Cretton et al. 1999, Zhao 1999, Hafner et al. 2000).But if one wants to model the Galaxy with analytic distribution functions, it ispossible to define an approximate third integral for nearly-circular orbits (seeSection 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 onwhich numerical action-angle variables can be constructed (Mc Gill & Binney1990, 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 allorbits (except a set of null volume in phase space) are regular with incommen-surable frequencies, the steady-state distribution function is a function only ofthree independent isolating integrals. The proof of this theorem can be foundin 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”, butif 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 theform 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 collisionlessconfiguration of the system is called an equilibrium configuration. It is in fact aquasi-equilibrium configuration, since there is no maximum-entropy state in apurely 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 canbe represented by a set of fixed points in integral space rather than by a set ofmoving points in real space. Indeed, every triple (E,Lz, I3) in integral spacerepresents 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 dealwith, but the structure of the (E,Lz)-plane can be studied analytically in everypotential.

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 thatall families of orbits that visit this position have isolating integrals of the motion(E,Lz) that meet the requirement

E ≤ ψ(R0, z0)−L2z

2R20

(1.12)

since we know thatv2R + v2

z ≥ 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 definethe right-hand side of the inequality (1.12) as the effective potential for a givenangular momentum Lz:

ψeff(R, z) = ψ(R, z)− L2z

2R2(1.14)

With this definition, Newton’s equations of motion describing the evolutionof R and z do not depend on the azimuthal angle φ:

d2Rdt2 = ∂ψeff

∂R

d2zdt2 = ∂ψeff

∂z

(1.15)

Lindblad diagram

In the (E,L2z)-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 reachthe 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 boundarylines, of which we denote the envelope by

E = Sz0(Lz). (1.16)

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

E = ψ(R, z0)−L2z

2R2(1.17)

L2z = −R3 ∂ψ

∂R(R, z0). (1.18)

Eq. (1.17) defines the family of boundary lines (equality in Eq. (1.12)), andfor every L2

z, the highest binding energy E of all the boundary lines (1.17) hasbeen taken (∂E∂R = 0), leading to Eq. (1.18). This envelope Sz0(Lz) in the(E,L2

z)-plane is plotted on Fig. 1.2.All points in integral space with E < Sz0(Lz) represent families of orbits

that will pass through z = z0 at a certain R. Orbits for which E = Sz0(Lz) alsodo reach the height z0, but can never go any higher. All points in integral spacewith E > Sz0(Lz) represent families of orbits that cannot reach z = z0. Sz0is thus the minimal binding energy of an orbit that cannot bring a star higherthan z0 above the galactic plane. For every height z1 > z0 we find a similarcurve 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 givesus all circular orbits in the galactic plane, since Eq. (1.18) then becomes

L2z

R3= − ∂ψ

∂R(R, 0). (1.19)

Eq. (1.19) is the condition for a circular orbit of angular momentum Lz, sincethe radial force (the only component of the force in the galactic plane since thisplane is a plane of symmetry) is then equal to the centripetal acceleration L2

z

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 thinnerthe 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. Thesolution of Eq. (1.19), i.e. the radius of the circular orbit of angular momentumLz, is called the guiding radius Rg and we define x = R−Rg. Then, if we expandthe effective potential ψeff defined by Eq. (1.14) in a Taylor series about the

23

Figure 1.2: Lindblad diagram. The shaded area is the area in the (E,L2z)-plane

for which the corresponding families of orbits cannot go higher than z0 abovethe galactic plane. The distribution function of a stellar disk with maximumheight z0 is null outside of this area. The thinner the disk is, the thinner theallowed area in the (E,L2

z)-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

d2xdt2 = −κ2 xd2zdt2 = −ν2 z

(1.20)

i.e. the equations of two harmonic oscillators with the epicycle frequency

κ2(Rg) ≡ −∂2ψeff

∂R2

∣∣∣(Rg,0)

(1.21)

and the vertical frequency

ν2(Rg) ≡ −∂2ψ

∂z2

∣∣∣(Rg,0)

(1.22)

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

The application of this epicyclic approximation to the analytic expressionof a third integral is straightforward. Indeed, if the star’s orbit is sufficientlynearly circular that the truncation of the Taylor series for ψeff at the third orderis justified, then the orbit admits three independent isolating integrals of themotion (εR, εz, Lz) where the epicycle energy εR and the vertical energy εz arethe energies of the oscillators (1.20):

εR =12(v2R + κ2 (R−Rg)2), (1.23)

εz =12(v2z + ν2 z2). (1.24)

Nevertheless, the epicyclic motion is only an approximation and more preciseapproaches of the third integral have been cited in Section 1.2.5, and will bedeveloped in Chapters 4 and 5 of this thesis.

1.2.7 Non-axisymmetric perturbations

The axisymmetric assumption is of course only an approximation. Indeed, itis 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 (Section1.1.4).

The potential remains stationary if we refer everything to its rotating frame:the planar orbits in a stationary non-axisymmetric potential are either “boxorbits” 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 rotatingabout the center while oscillating in radius (as the nearly circular orbits in anaxisymmetric 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 lessthan 5% of the axisymmetric part), which is the case of the Milky Way. In thatcase, a linear perturbation analysis can be performed, i.e. we consider that

Φ(R,φ, z, t) = Φ0(R, z) + εΦ1(R,φ, z, t) (1.25)

with ε << 1. The axisymmetric approximation is thus a necessary startingpoint for this kind of analysis, and is a prerequisite if one wants to analyse thebar and the spiral arms on a theoretical basis.

The perturber is said to have m-fold symmetry if its pattern is invariant forrotation over 2π/m. Hence the bar has 2-fold symmetry and is also said to bean “m = 2 mode”, while spiral arms are m = 2 or m = 4 modes (the number ofarms in the Milky way is still rather uncertain).

Resonances

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

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

ωp ≡ m (Ωp − Ω) (1.26)

where Ωp is the rotational frequency of the perturber and Ω = vc

R the angularvelocity of circular orbits (of velocity vc) in the axisymmetric Galaxy. We speakof “resonances” when the stars encounter successive density peaks at a frequencythat coincides with the frequency of their natural radial or vertical oscillations,i.e. when

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

(1.27)

At corotation, the guiding center of the stars corotates with the perturbed 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 outsidecorotation, called outer Lindblad radii. In practice, the name “Lindblad reso-nances” is used to refer to the m = 2 radial Lindblad resonances (ILR and OLRof 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 perturbedGalaxy. Indeed, if we refer everything to the frame of a bar rotating like a solidbody (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 looporbit 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 coefficientC(R0) (see Eq. (3-120b) of Binney & Tremaine 1987) becomes infinite at theradii of the ILR, OLR and corotation, and across these resonances the sign ofC changes. Depending on the sign of C the loop orbits will be elongated alongor perpendicular to the major axis of the potential. Resonances thus have aspecial status: the shape of the orbits change at those radii, and moreover, thelinear analysis is no more valid since they are singularities. Therefore, even aweak perturbation can produce a strong response at resonances. This responseaffects the features of the velocity distribution function that can have a bimodalcharacter since both types of orbits (elongated along and perpendicular to thebar) can coexist at resonances (see e.g. Vauterin & Dejonghe 1997, Dehnen1999, 2000).

The bar and its spiral response

The process of formation of the galactic bar is still rather uncertain: in thefirst 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 diskinstability at least 5 Gyr ago. The presence of a supermassive black hole and itsassociated dense star cluster at the center of the Galaxy was thought to causethe destruction of the bar (e.g. Hasan et al. 1993), but recently, high-qualityN -body simulations (Shen & Sellwood 2004) have shown that the bar is morerobust than previously thought, and that it is in theoretical accordance withthe presence of a supermassive black hole at the center.

Bars always rotate like solid bodies, and their morphological properties inexternal galaxies are found to be related to the spiral Hubble type of theirhost galaxy (Elmegreen & Elmegreen 1985): bars of Sa and Sb galaxies haveflat density profiles and end close to their corotation radius, while bars of Sbcand Sc galaxies have exponential density profiles and end well inside (some-times halfway) their corotation radius. The bar’s steep density profile in ourGalaxy (Binney et al. 1997) indicates a Sbc or Sc type for the spiral pattern, inaccordance with the Hubble type proposed by de Vaucouleurs & Pence (1978).

This relation between the spiral pattern and the bar’s morphology, togetherwith the fact that, in barred spiral galaxies, the spiral arms usually start at theend 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 arotating bar and a dissipative interstellar medium (which is not important forthe 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 ofthe interstellar gas has the form of a spiral pattern in such simulations, but thereponse of stars does not (Sanders 1977): the Sanders-Huntley model is thusnot suitable to explain that a spiral pattern is present in the old disk stars aswell 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 structureconsists of a self-sustained quasi-stationary density wave, that is, a density wavethat is maintained in a steady state (in its associated rotating frame) over manyrotation 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) + εΣ1(R,φ, t) (1.30)

with ε << 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 asa “normal mode” Σa(R)eim(φ−Ωpt), and this density can be seen as a densitywave, 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

Σ1(R,φ, t) = Σa(R)eim(φ−Ωpt), (1.31)

i.e.Φ1(R,φ, t) = Φa(R)eim(φ−Ωpt) = A(R)ei f(R)eim(φ−Ωpt) (1.32)

They stated that the responding density to this perturbing potential Φ1 mustbe equal to Σ1, in such a way that the spiral pattern is self-sustained. For thispurpose, they injected the potential Φ(R,φ, t) of Eq. (1.25) and the distributionfunction 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 Section1.3.4) in the linearized Boltzmann equation (Eq.(6A-1) of Binney & Tremaine1987) and hence found the local “dispersion relation for galactic stellar disks”:

ω2p = κ2 − 2πGΣ0

∣∣∣ dfdR

∣∣∣F(ωp, κ,dfdR

, σR) (1.33)

28

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

where the function F is the “reduction factor” (always between 0 and 1), ωp isthe driving frequency of the perturber defined in Eq. (1.26), κ is the epicyclicfrequency 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 ofthe spiral pattern from Eqs. (1.29) and (1.32), and σR is the velocity dispersionin the radial direction. This is the key equation for the local analytic study ofquasi-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 perturbationis not growing exponentially) if

Q ≡ σRκ

3.36GΣ0> 1 (1.34)

The dispersion relation of Eq. (1.33) is also the condition for the existence ofstable non-axisymmetric disturbances of an initially axisymmetric disk. Nu-merical experiments suggest that Q > 1 implies stability to all axisymmetricperturbations, but not to non-axisymmetric ones (where Q must be even higherto ensure stability).

Eqs. (1.27) and (1.33) imply that stable m = 2 or m = 4 perturbations ofa purely stellar disk can only occur between the ILR and the OLR or betweenthe IUHR and OUHR of the perturber (because F ≥ 0). However, in gaseousdisks, 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 pressuresupport, which makes it more stable (because there is an additional positiveterm 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 andconglomerate at certain radii even in the absence of perturbing forces. Thestable behaviour of the gaseous component is thus mainly responsible for theobservation of grand-design spiral patterns in the Milky Way and in externalgalaxies. This different behaviour of gas and stars could explain the observationof different spiral patterns in the Galaxy, depending on the age of the stars(young freshly formed stars are situated near the gaseous spiral arms wherethey 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 Galaxyand in external galaxies. However, there is little evidence for the hypothesis thatthe spiral pattern is (quasi-)stationary, since the anti-spiral theorem (Lynden-Bell & Ostriker 1967) states that if a steady-state solution of a time reversibleset of equations has the form of a trailing spiral, then there must be an identicalsolution in the form of a leading spiral. The prevalence of trailing spirals inexternal galaxies implies that non-stationary effects are involved in the expla-nation of the spiral structure (for example, they may be growing in amplitudein non-stable periods). In fact, spiral arms could be transient phenomena, con-stantly renewed by self-regulating instabilities. Thus, ironically, the Lin-Shuhypothesis, which has proved to be fruitful in advancing our understanding ofdisk 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 impliesthat the waves propagate radially (Toomre 1969), and this has consequences onthe 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 describedabove) 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 theresults of such a numerical simulation (Toomre 1981). We see on Fig. 1.4 thatan 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 theresulting trailing wave is about twenty times larger than the amplitude of theinitial leading wave, and moreover, an even stronger transient spiral pattern isformed at intermediate stages. This effect is called the swing-amplification (i.e.the resulting trailing wave is stronger than the initial leading wave, as clearlyseen in panels 4 to 9 on Fig.1.4), and it is not captured by the tight-windingapproximation. 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 astellar 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 densitywaves whose wavecrests swing around from leading to trailing. The amplitudes

30

of those waves are amplified transitorily and then die away. Their duration is ofthe order of an orbital period (108 years in the Solar neighbourhood). Toomre(1981) has argued vigorously that this swing-amplification mechanism is theprincipal process of spiral arms formation in galactic disks and that the spiralarms are thus of transient nature.

The biggest question about the theory of transient spiral waves is wether itis able to explain the observed grand-design patterns in most spiral galaxies.Sellwood & Carlberg (1984) demonstrated that a self-regulating instability inthe stellar disk (producing transient spirals) may be able to produce grand-design, open spiral structure similar to that seen in most Sc galaxies. Gasinstabilities lead to the formation of transient spiral arms (similar to those ofJulian & Toomre 1966 and Toomre 1981) in which new stars are created. Thesenew stars have low velocity dispersion. This lowering of the velocity dispersionreduces Q (see Eq. (1.34)), induces instability in the stellar disk, boosts theswing-amplification and increases the strength of the spiral arms while boostingstar formation. These spiral waves impart angular momentum to stars andincrease 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 thesusceptibility of further spiral-making. Gas is re-ejected by supernovae and coolsdown rapidly: this cooling leads to gravitational instability of the gas, in sucha way that the transient spiral arms are constantly renewed.

Numerical simulations by De Simone et al. (2004) have shown that suchtransient density waves could cause the distribution function to become clumpy,and the velocity dispersions to increase, especially far from corotation. Sincetransient 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 infavour of the transient nature of spiral waves, since the quasi-stationary wavesof Lin-Shu cannot heat the disk anywhere else than at Lindblad resonances,and cannot explain the observed age-velocity dispersion relation. Moreover, theentire disk can also be affected by the radial mixing of orbits around corotation(Sellwood & Binney 2002). At corotation, the perturbation force acts on starswith a constant direction for a long time, and this enables large changes inthe angular momentum of stars. Indeed, Lz and E are no longer integrals ofthe motion in the presence of a non-axisymmetric perturbation. Stars insidecorotation overtake the wave and gain angular momentum as they fall into thespiral arm, which causes them to move to larger radii and slow their drift relativeto the wave. Some stars close to corotation then reverse their speed relative tothe wave, and subsequently slip backwards relative to the wave, fall backwardsinto the other arm, lose angular momentum and move to smaller radii. Theseorbits are called “horseshoe orbits” (Goldreich & Tremaine 1982). A spiral wavecan cause a radial migration of stars of 2 or 3 kpc around corotation in muchless than 1 Gyr (Lepine et al. 2003). An important feature of radial migrationnear corotation is that random motions caused by the spiral perturbation arequite moderate, and that stars stay on quasi-circular epicyclic orbits. Indeed, inthe 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 theswing-amplification. The time interval between panels is one-half of a rotationperiod at corotation. An initially leading wave (panel 1) first unwinds into arelatively open pattern (panel 3), and then into a stronger trailing pattern thatbecomes more and more tightly wound (panel 9). At intermediate stages (panels4, 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 starare related by

∆E = −Ωp∆Lz (1.36)

If we define JR as the radial action, which is simply εR/κ (see Eq. (1.23)) inthe epicyclic approximation, we can write the following equality between thedifferential applications:

dE =∂E

∂JRdJR +

∂E

∂LzdLz = −κ dJR − Ω dLz (1.37)

Eqs. (1.36) and (1.37) lead to the expression of ∆JR, which is a measure of thedistance from the original quasi-circular orbit:

∆JR =Ωp − Ωκ

∆Lz (1.38)

Hence, large changes in Lz at corotation produce no significant heating (no in-crease of v2

R), and the radial mixing does not increase the random motion ofstars, 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 migrationover 2 or 3 kpc without causing radical changes in the observed velocity distri-bution. This mechanism could explain the fact that no age-metallicity relationis 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 bya 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 Orionarm could be an example of such a transient spiral arm corotating with the Sunin 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 providingvery precise data on stars (and in particular on their positions and velocitiesrelative to the Sun), and is thus the benchmark test for models of the Galacticstructure and evolution. In this section, most results are based on the axisym-metric assumption, which is the 0th order analysis, and which is a necessarystarting 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 heliocentriccartesian coordinate system associated with galactic coordinates is:

x = d cosl cosby = d sinl cosbz = d sinb

(1.39)

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 toknow where the Sun is located in the Galaxy. The view of the Galactic planefrom earth as a luminous band crossing the sky implies that we are very closeto that plane (in fact 12 ± 8 pc above the plane; see e.g. Batsleer & Dejonghe1994), between the Perseus (in the outer Galaxy) and Sagittarius (in the innerGalaxy) spiral arms, and inside the Orion arm. On the other hand, the distanceof the Sun to the Galactic center is very difficult to estimate: the direct methodis to compare the average radial velocities with the average proper motions (theangular velocities on the sky) of maser spots in star forming regions near theGalactic center (see Section 2.2.4 of Binney & Merrifield 1998), but this methodneeds very accurate observations. If the density of the objects of the luminoushalo peaks at the galactic center, the galactocentric radius of the Sun can thenalso 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 thestellar candles (for example cepheids for which the period-luminosity relation isknown) will reverberate in the estimation of the galactocentric distance of theSun. 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 of8 kpc for the galactocentic Solar radius, with an uncertainty of the order of 0.5kpc.

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 rotationcurve) vc(R), where R is the galactocentric radius, and in particular vc(R)(whereR denotes the galactocentric radius of the Sun everywhere in this thesis)is one of the hardest problems in Galactic astronomy. The rotation curve isdetermined by observations of the kinematics of the gas, and in particular ofthe neutral hydrogen 21 cm line, but the rotation curve is not well established forgalactic radii R > R. Observations of many outer spiral galaxies indicate thatthe rotation curve remains more or less flat after attaining its maximum (e.g.Casertano & van Gorkom 1991): the rotation curve of the Milky Way thereforeis thought to behave similarly. This flatness of the rotation curve implies thepresence of a dark matter halo in addition to all the observed matter in orderto 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 betterknowledge of local Galactic structure. Lindblad (1925) and Oort (1927a,b) de-veloped the model of differential axisymmetric rotation with Ω = vc

R dependingonly on the distance R to the galactic center. Oort (1927a) introduced twoconstants (the Oort constants A and B), that can be determined from propermotions (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 theSolar neighbourhood on a circular orbit, Taylor expanding Ω(R) to first orderin (R − R) yields for the line-of-sight radial velocity vr and the transversevelocity vt relative to the Sun:

vr = Ad sin2l (1.40)

vt = Ad cos2l +B d (1.41)

withA =

12(vcR− dvc

dR)∣∣∣R, (1.42)

B = −12(vcR

+dvcdR

)∣∣∣R

(1.43)

Kuijken & Tremaine (1991) showed that the Taylor expansion terms arisingfrom non-circularity of the orbits are negligible, just as they should be if theMilky 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 onthe 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 propermotions, 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 propermotions of the Cepheids that were measured by Hipparcos has been derivedby Feast & Whitelock (1997) who found A = 14.82 ± 0.84 km s−1kpc−1 andB = −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 curveis slightly declining in the Solar neighbourhood.

The determination of vc(R) follows from the determination of the Oortconstants: using the values of Feast & Whitelock (1997), we find vc(R) =(218±8km s−1)(R/8kpc), a value which is consistent with vc(R) = 220 km s−1

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 sourceSgr 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 leavesus 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 forany mass model of the Galaxy. It can be surmised that this parameter is notdirectly observed: it is deduced from the positions and velocities of tracer starsin the direction perpendicular to the galactic plane (the z-direction), using theBoltzmann equation (1.3) and Poisson equation (1.4) in various forms.

Jeans equations (first order moments of the Boltzmann equation (1.3)) implythat the vertical acceleration Kz is related to the vertical velocity dispersion σ2

z

and to the vertical number density g(z) of a population of stars by:

Kz(z) = σ2z

d ln(g(z)/g(0))dz

, (1.44)

if the vertical motion can be separated from the radial and azimuthal motionof the stars (Oort-Lindblad approximation), if the velocity ellipsoid is alignedwith the cylindrical coordinate axes (i.e 〈vRvz〉 = 0), and if the population isisothermal (i.e. σ2

z 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 becomedynamically well mixed in the z-direction. Then he derived ρ using Poissonequation for nearly circular orbits in the Solar neighbourhood (derived from Eq.(1.4), (1.42) and (1.43) if we assume that KR(R) = −v2c

R ):

4πGρ = −dKz

dz− 2(A2 −B2) (1.45)

Oort found ρ = 0.09Mpc−3 in 1932 and ρ = 0.15Mpc−3 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 about0.07Mpc−3, see Creze et al. 1998).

Afterwards, many other similar studies (Yasuda 1961; Eelsalu 1961; Woolley& Stewart 1967; Turon 1971; Gould & Vandervoort 1972; Jones 1972; Balakirev1976; Hill et al. 1979) gave discrepant results, suffering from inhomogeneitiesin the data, systematic errors due to the use of photometric distances, andundersampling near the galactic plane. Because of this undersampling, somestudies were even based on young O and B stars, assuming that the gas anddust 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 ofisothermal components (i.e. components described by distribution functionsFz(z, vz) leading to a constant velocity dispersion σz, see Spitzer 1942) andanalyzed the nonlinear self-consistent equations in which the matter producesthe potential (Poisson equation) and is also affected by the potential (Jeansequations for each isothermal component). Bahcall et al. (1992) applied thismethod to a sample of K giants and found ρ = 0.26Mpc−3, a result leadingonce again to the presence of dark matter in the disk.

36

At about the same time, Kuijken & Gilmore (1989a,b,c, 1991) used anothermethod which does not use the Jeans equations, inspired by the method of vonHoerner (1960). This method is based on the assumption that the phase spacedistribution function Fz(z, vz) of any tracer population depends only on thevertical energy

εz = Φz(z) +12v2z (1.46)

where

−dΦz(z)dz

= Kz(z). (1.47)

This assumption follows from the classical separability of the vertical motionof the stars (Oort-Lindblad approximation) and from the Jeans theorem in onedimension. The potential Φz(z) is quadratic in z near the plane, in accordancewith Eq. (1.24). The density in configuration space g(z) of a tracer populationis then related to its density in phase space by the integral equation:

g(z) =∫ ∞

−∞Fz(z, vz)dvz = 2

∫ ∞

Φz

Fz(εz)√2(εz − Φz)

dεz = f(Φz(z)) (1.48)

So, there is a unique relation between g(z) and Fz(εz): Kuijken & Gilmore(1989c, 1991) inverted the Abel transform (1.48) and used the space densityprofile g(z) of distant K stars to predict their velocity distribution at differentheights for different Φz(z). Then they compared these predictions to the ve-locity data and used a maximum likelihood technique to select the best-fittingpotential. The data they used were too far from the plane to constrain ρ andthey found a surface mass density between z = ±1.1 kpc of 71Mpc−2, a resultrejecting the presence of dark matter in the disk.

So, all the investigations between 1932 and the ESA Hipparcos mission hadfailed to converge to a reliable determination of ρ: they were all very uncertain,essentially because of inhomogeneities in the tracer samples, undersampling nearthe Galactic plane and the use of photometric distances. Hipparcos data solvedall these problems: the dense probe near the plane eliminated the inhomogeneityand the undersampling, while the accurate parallaxes2 eliminated the use ofphotometric distances3.

Creze et al. (1998) and Holmberg & Flynn (2000) used Eq. (1.48) to de-termine ρ from complete samples of nearby A-F stars. Given the functionsg(z) and Fz(z = 0, v2

z) = Fz(εz) from the observed vertical density and velocitydistribution, the function Φz(z) can be derived. Then, to determine ρ, theOort constants have to be used in the Poisson equation (1.45): these are alsomuch better known since the Hipparcos mission (see Section 1.3.2). Creze et al.(1998) estimated ρ = 0.076 ± 0.015Mpc−3 while Holmberg & Flynn (2000)

2The parallax of a star is the angle formed by the two lines going from the center of thestar 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

estimated ρ = 0.102± 0.010Mpc−3. These differences between local densityestimates using almost the same Hipparcos data are due to different a priorihypothesis on the vertical potential Φz. Recent investigations at StrasbourgUniversity using a local Hipparcos sample combined with two samples at thegalactic 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 thedirection “Sun - Galactic center”, its direction is the opposite of the radial veloc-ity in galactocentric coordinates vR), the y-velocity V (parallel to the directionof 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 relativeto the Local Standard of Rest (the LSR), i.e. a reference frame that follows theclosed 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 starrelative to the LSR is called the peculiar velocity of the star, denoted u, v andw in the three directions of the coordinate system (1.39). The peculiar velocityof the Sun is called the Solar motion and is denoted U, V and W

4 in thethree directions of the coordinate system (1.39).

Clearly, since the Galactic plane is a plane of symmetry, there should beno net vertical motion in the Galaxy. Therefore, the vertical Solar motion Wis simply the opposite of the mean vertical motion 〈W 〉 of an homogeneoussample of stars in the Solar neighbourhood. Dehnen & Binney (1998a) foundon the basis of Hipparcos proper motions of main sequence stars W = 7.17±0.38 km s−1. The determination of the azimuthal motion V is less simple,since the “asymmetric drift” (Eq. (10.12) of Binney & Merrifield 1998, see alsoEq. (3.10) of the present thesis) predicts a linear dependence of 〈v〉 with theradial velocity dispersion : V has been estimated at V = 5.25 ± 0.62 km s−1

(Dehnen & Binney 1998a) by linearly extrapolating the mean motion 〈V 〉 tozero velocity dispersion. Following the axisymmetric assumption, there shouldbe no net radial motion at the position of the Sun in the Galaxy: following thathypothesis, Dehnen & Binney (1998a) found U = −〈U〉 = 10.00±0.36 km s−1.Nevertheless, those values for the Solar motion will be revisited and discussedin Chapter 3.

If we get back to the epicyclic approximation of Section 1.2.6, a relationbetween the peculiar velocity v of an orbit at Solar radius and the position ofits guiding center can be derived (Eq. (3-74) of Binney & Tremaine 1987):

v = −2Bxg =κ2

2Ωxg (1.49)

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

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):

εR =12(u2 + κ2 x2

g) =12(u2 + (2Ω/κ)2v2) (1.50)

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

F (u, v, w) = (2Π)−3/2(σuσvσw)−1e−(εR/σ2u)−(εz/σ

2w), (1.51)

we have by the Jeans theorem (see Section 1.2.4) a time-independent solutionof the collisionless Boltzmann equation (1.1) since F depends on the velocitiesonly through the integrals εR and εz. Moreover, since σ2

v/σ2u = κ2/4Ω2 (Eq.

(3-72) and Eq.(4-52) of Binney & Tremaine 1987), Eq. (1.50) yields

εR/σ2u =

12(u2/σ2

u + v2/σ2v), (1.52)

and Eq. (1.51) becomes

F (u, v, w) = (2Π)−3/2(σuσvσw)−1e−(u2/2σ2u)−(v2/2σ2

v)−(w2/2σ2w). (1.53)

This 3-dimensional gaussian distribution is called the Schwarzschild distributionfunction. Schwarzschild (1907) pointed out that the distribution of stellar ve-locities in the Solar neighbourhood was similar to that of, say, a population ofoxygen molecules at room temperature. The main difference between the caseof molecules in air and stars is that in the former case the velocity dispersionis 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 constanton ellipsoids with axis lengths in the ratio σu : σv : σw. The ellipsoid withsemi-axes of length σu, σv and σw is called the Schwarzschild velocity ellipsoid.

1.3.5 Vertex deviation and substructure of velocity space

The covariance σ2uv of any population of stars having a velocity distribution

function of the type of Eq. (1.53) is null. However, when estimating thiscovariance for samples of stars in the Solar neighbourhood, one often finds asignificantly non-zero value for this covariance (especially for young stars). Thuswe must rewrite a phenomenological distribution function in order to accountfor this phenomenon:

F (u, v, w) = (2Π)−3/2(σu′σv′σw)−1e−(u′2/2σ2u′ )−(v′2/2σ2

v′ )−(w2/2σ2w) (1.54)

whereu′ = u cos lv − v sin lvv′ = u sin lv + v cos lv

(1.55)

39

The angle lv in velocity space is called the vertex deviation. If we define thevelocity dispersion tensor of a population of stars as 〈(~v− 〈~v〉)⊗ (~v− 〈~v〉)〉, thevertex deviation lv is the angle one has to rotate the (u, v) coordinate systemin 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〈u′v′〉 = 0 yields:

lv = 1/2 arctan (2σ2uv/(σ

2u − σ2

v)). (1.56)

We state that σu′ > σv′ in order to raise the ambiguity introduced by the arctanfunction.

Nevertheless, if we insert the velocity distribution (1.54) (multiplied by anarbitrary 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 backthe distribution (1.53). The possible causes of this vertex deviation is one ofthe main topics of this thesis.

The classical hypothesis is that the observed vertex deviation is due to thefact that many samples of stars are not in dynamical equilibrium, implying thatthe number of truly independent velocities employed to derive the distributionfunction is strongly reduced. In fact, many objects in samples of young starscould be members of moving groups, generally thought to be vestiges of theclusters and associations in which most stars form. According to the commonlyaccepted theory, early-type, young stars still carry the kinematic signature oftheir place of birth. As a consequence, their distribution in velocity space isclumpy (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 lotof substructure that appears in velocity space. These inhomogeneities can bespatially 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 thesame kinematics and may occupy extended regions in the Galaxy, and a movinggroup as the part of the supercluster that enters the Solar neighbourhood andcan 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). Ithas 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 openclusters (Eggen 1958). The best documented groups (see Montes et al. 2001and references therein) are the Hyades supercluster associated with the Hyadescluster (600 Myr) and the Ursa Major group (also known as the Sirius super-cluster) associated with the UMa cluster of stars (300 Myr). Another kinematicgroup called the Local Association or Pleiades moving group is a reasonablycoherent kinematic group with embedded young clusters and associations likethe Pleiades, α Persei, NGC 2516, IC 2602 and Scorpius-Centaurus, with agesranging from about 20 to 150 Myr. Two other young moving groups are the IC2391 supercluster (35-55 Myr) and the Castor moving group (200 Myr). Thekinematic 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 hypothesisthat supercluster stars share a common origin: Chereul et al. (1998) proposethat the supercluster-like velocity structure is just a chance juxtaposition ofseveral cluster remnants.

Nevertheless, significant clumpiness in velocity space has also been recentlyreported for late-type stars (Dehnen 1998), thus raising the questions of theage of those late-type stars and of the exact origin of this clumpiness. Forexample, Montes et al. (2001) suppose that, if a late-type star belongs to amoving group or supercluster, it may be considered as a sign that the star isyoung, 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 velocityspace and for the vertex deviation. The first one is the absorption of satel-lite galaxies that could generate streams everywhere in the Galaxy (see Section1.2.2). The second one is the perturbation of the distribution function by anon-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 neighbourhoodand what they tell us about the dynamics of the Galaxy: it is thus a smallcontribution to the huge and ambitious project of understanding the structureand evolution of the Galaxy as a whole.

We restrict our study of stellar kinematics to late-type giants (of spectraltypes 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 Solarneighbourhood, which makes them good tracers for the study of kinematics anddynamics. The next two Chapters are dedicated to the kinematic analysis ofthe data, while Chapters 4 and 5 are dedicated to the development of new toolsto establish dynamical models.

In Chapter 2, we describe a stellar sample of 5952 K and 739 M giants. Forthe 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 thisthesis). Indeed, their radial velocities have been measured at the Observatoire deHaute 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 thedata (Section 3.3). This allows to make full use of all the available stars, and toderive 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 withthe three main hypotheses proposed to account for the substructure in velocity

41

space (the merger remnants described in Section 1.2.2, the non-axisymmetricperturbations described in Section 1.2.7, and the vestiges of clusters describedin Section 1.3.5). Our results suggest a dynamical non-axisymmetric origin, inthe 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 Solarmotion in the Galaxy, which will be more difficult to evaluate than previouslythought.

Nevertheless the weak point of all the recent simulations of dynamical per-turbations is the simplicity of their initial conditions, based on two-dimensionalSchwarzschild 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 integralsof the motion exist for all orbits, defined by five parameters and designed tomodel the Galaxy. We study the parameter ranges of the presented potentialsmatching the fundamental parameters of the Milky Way, especially in the Solarneighbourhood. Five different valid potentials are presented and analyzed indetail.

In Chapter 5, we present new equilibrium component distribution functionsthat 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 2and 3. These models could give ideal initial conditions (more complex than asimple two-dimensional Schwarzschild velocity ellipsoid) for three-dimensionalN -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 Galaxywhere we can retrieve the most precise data on star positions and velocities. Thestory of the efforts to obtain more and more accurate data is standard textbookfare, and nicely illustrates how theoretical progress and data acquisition haveto 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 theHipparcos satellite mission (ESA 1997), that provided accurate parallaxes andproper 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 previewof how Hipparcos data could improve our knowledge of stellar motions in theGalaxy was given by Kovalevsky (1998); then, many studies (e.g. Creze etal. 1998, Dehnen & Binney 1998a, Holmberg & Flynn 2000) used Hipparcosproper motions to derive some fundamental kinematic parameters of the Solarneighbourhood. They did not use radial velocities from the literature because,at that time, radial velocities had been measured preferentially for high propermotions stars and their use would have introduced a kinematic bias. Hipparcosdata were also used by Chereul et al. (1998) to derive the small scale structureof the velocity distribution of early-type stars in the Solar neighbourhood.

The situation has now dramatically improved thanks to the efforts of a largeEuropean 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 fromother specific programmes. This unique database, comprising about 45000 starsmeasured with the CORAVEL spectrovelocimeter (Baranne et al. 1979) at atypical accuracy of 0.3 km s−1, combines a high precision and the absence ofkinematic bias. It thus represents an unprecedented data set to test the resultsobtained by previous kinematic studies based solely on Hipparcos data. An-other feature of the present thesis is the use of Tycho-2 proper motions, whichcombine 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 εµ. The solid line refers to the 859spectroscopic binaries (SB) present in our sample, whereas the dashed line cor-responds to the 5832 non-SB stars. Note that the sample of binaries containsmore 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 propermotions themselves, which are based on very accurate positions, but extendingover a limited 3-year time span. Results from the Geneva-Copenhagen surveyfor about 14000 F and G dwarfs present in the CORAVEL database have beenrecently published by Nordstrom et al. (2004). The present thesis, on the otherhand, 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 HipparcosCatalogue (ESA 1997) have been selected, and the Hipparcos parallaxes havebeen 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 thanthe Hipparcos ones. Fig. 2.1 compares the proper motions from Tycho-2 andHipparcos, and reveals that the Tycho-2 proper motions are sufficiently differ-ent from the Hipparcos ones (especially for binaries) to warrant the use of theformer in the present study; (iii) Radial-velocity data for stars belonging to thisfirst list (stars with spectral type K and M) have then been retrieved from theCORAVEL database. Only stars from the northern hemisphere (δ > 0), ob-served with the 1-m Swiss telescope at the Haute-Provence Observatory, havebeen 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 spectraltypes later than G5, where b is the galactic latitude; those stars are flagged ‘S’ infield H68 of the Hipparcos Catalogue) are present in the CORAVEL database,thanks to a large observing campaign dedicated to those stars. The CORAVELdatabase contains in addition stars monitored for other purposes (e.g., binarityor 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 theM-type stars of the Hipparcos Catalogue has been constructed from distancesestimated from a simple inversion of the parallax (Fig. 2.2), and reveals a clearseparation between dwarfs and giants. Clearly, all M stars with MHp < 4 mustbe giants. For K-type stars with a relative parallax error less than 20%, theHertzsprung-Russell diagram (Fig. 2.3) shows that the giant branch connectsto the main sequence around (V − I = 0.75,MHp = 4). Therefore, in order toselect 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 avoidcontamination 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 singleprofile: the cross-correlation profile. The lines are contained in a template or mask, consistingof slits representing a simplified synthetic spectrum. For each velocity, the template is shiftedover the observed absorption spectrum, and at the velocity of the object, the light transmittedthrough the slits is minimal, see Fig. 2.4

46

searches, allowed us to further screen out stars with peculiar spectra, such asT Tau stars, Mira variables and S stars. T Tau stars have been eliminatedbecause they belong to a specific young population; S stars because they are amixture of extrinsic and intrinsic stars, which have different population charac-teristics (Van Eck & Jorissen 2000); Mira variables because their center-of-massradial velocity is difficult to derive from the optical spectrum, where confusionis 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 andM giants brighter than V = 7.3 + 1.1| sin b|. To fix the ideas, for a typical giantstar with MV = 0, this magnitude threshold translates into distances of 290 pcin the galactic plane and 480 pc in the direction of the galactic pole (these valuesare, however, very sensitive upon the adopted absolute magnitude, and become45 pc for a subgiant star with MV = +4 in the galactic plane, and 2900 pc fora supergiant star with MV = −5, corresponding to the range of luminositiespresent in our sample; see Chapter 3).

The final step in the preparation and cleaning of the sample involves theidentification 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 studiesshould make use of the center-of-mass velocity. In order to identify those, theobserving strategy was to obtain at least two radial-velocity measurements perstar, 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 & Jorissen2000), the identification of binaries requires a specific strategy, which makes useof 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-correlationprofile, i.e., corrected from the instrumental width (see Fig. 2.4 for more precisedefinitions). The CORAVEL instrumental profile is obtained from the observa-tion of the cross-correlation dip of minor planets reflecting the sun light, aftercorrection for the Solar rotational velocity and photospheric turbulence (seeBenz & Mayor 1981 for more details). The Sb parameter is directly relatedto the average spectral line width, which is in turn a function of spectral typeand luminosity, the later-type and more luminous stars having larger Sb values.On the other hand, the measurement error ε has been quadratically subtractedfrom the radial velocity standard deviation σ(vr0) to yield the effective standarddeviation σ0(vr0).

The identification of binaries among K or M giants follows different steps.The standard χ2 variability test, comparing the standard deviation σ(vr0) of themeasurements to their uncertainty ε, cannot be applied to M giants, becauseintrinsic radial-velocity variations (“jitter”) associated with envelope pulsations

47

0 1 2 3 4 5

10

5

0

V-I

Figure 2.2: Crude Hertzsprung-Russell diagram for the Hipparcos M stars withpositive parallaxes.

48

0 0.5 1 1.5 2

10

5

0

V-I

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

49

Figure 2.4: The concept of line-width parameter Sb = (s2− s20)1/2 is illustratedhere by comparing the CORAVEL cross-correlation (smoothed) profiles (blackdots) of 3 M giants or supergiants with the gaussian instrumental profile (ofsigma s0 = 7 km s−1; solid line). Each profile corresponds to a single radial-velocity measurement for the given stars. The s value listed in the above panelsrefers to the sigma parameter of a gaussian fitted to the observed correlationprofile.

50

Figure 2.5: The (Sb, σ0(vr0)) diagram (see text) for K (upper panel, and zoominside) and M giants (lower panel). Star symbols denote supergiant stars, filledsymbols 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 firststep, M giants having σ0(vr0) ≥ 1 km s−1have been monitored with the ELODIEspectrograph (Baranne et al. 1996) at the Haute-Provence Observatory (France)since August 2000 (Jorissen et al. 2004). These supplementary data pointsmade it possible to distinguish orbital variations from radial-velocity jitter bya 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 ofthe sample. Stars located below the dashed line in Fig. 2.5 may be supposedto be single (although some very long-period binaries, with P > 10 yr, maystill hide among those). Their radial velocity dispersion suffers from a jitterwhich clearly increases with increasing spectral line-width, as represented bythe 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 inthe lower panel of Fig. 2.5). Many semi-regular and irregular variables arelocated in the intermediate region, with 5 < Sb < 9 km s−1. Strangely enough,spectroscopic binaries seem to be lacking in this region. Nevertheless a detaileddiscussion of the properties of the binaries found among M giants would deservean entire thesis and is out of the scope of the present one: it could be thesubject 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(N−1)[σ(vr0)/ε]2 axis (where N is the number of measurements for a given star)follows a χ2 distribution, as expected (e.g., Jorissen & Mayor 1988). This holdstrue 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 bya straight χ2 test. A 1% threshold for the first kind risk (of rejecting the nullhypothesis that the star is not a binary while actually true, i.e., of consideringthe 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 spectroscopicbinaries (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 thepresent work. The new binaries are identified in Table A.1 (flags 0, 1, 5, 6 and 9in 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 numerousenough to derive an orbit), estimated (in the case of low-amplitude orbits) ortaken from the literature for 216 systems only, thus leaving 643 binaries whichhad to be discarded from the kinematic study, because no reliable center-of-massvelocity could be estimated. After excluding those large-amplitude binaries aswell as the dubious cases, 5311 K giants and 719 M giants remain in the finalsample. Fig. 2.6 shows the distribution of our final sample on the sky. Amongthis 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 theprevious Chapter. Although the velocity distribution is a function of position inthe Galaxy, it will be assumed in this Chapter that our local stellar sample maybe used to derive the properties of the velocity distribution at the location ofthe 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 starswith parallaxes accurate to better than 20%. In a second approach, designed tomake full use of the 6030 available stars but without being affected by the biasesappearing when dealing with low-precision parallaxes, the kinematic parametersare evaluated with a Monte-Carlo method. Although easy to implement, theMonte Carlo method faces some limitations (the parallaxes were drawn froma Gaussian distribution centered on the observed parallax, not the true one asit 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 simultaneouslymaximum likelihood estimators of luminosity and kinematic parameters, andwhich can identify possible groups present in the sample by performing a clusteranalysis.

In all cases, we correct for differential Galactic rotation in order to dealwith the true velocity of a star relative to its own standard of rest. Given theobserved values of the line-of-sight radial velocity vr0 and the proper motionsµl0 and µb0, the corrected values are (Trumpler & Weaver 1953):

vr = vr0 −Ad cos2b sin2l (3.1)

κµl = κµl0 −A cos2l −B (3.2)

κµb = κµb0 + 1/2A sin2b sin2l, (3.3)

where κ is the factor to convert proper motions into space velocities, andd is the distance to the Sun. We used the values of the Oort constants Aand 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 velocityof the star and that of the Sun relative to the LSR.

3.1 Analysis restricted to stars with the mostprecise parallaxes

The components of the velocity of a star with respect to the Sun in the cartesiancoordinate 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 velocityW . As we decide not to use any a priori value for the Solar motion, we use thevelocities U , V , and W instead of the peculiar velocities of the stars relative tothe 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 theproper motions and radial velocities as given by Eqs. (3.1), (3.2) and (3.3). Thissimple procedure faces two major difficulties: (i) the inverse parallax is a biasedestimator of the distance (especially when the relative error on the parallax islarge), and (ii) the individual errors on the velocities cannot be derived from asimple 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 welimit 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 generalbehaviour of the K and M giants in the Solar neighbourhood. We choose insteadto restrict the sample to the stars with a relative parallax error smaller than20%, in such a way that 2774 stars (2524 K and 250 M) remain. In that casethe classical first order approximation for the calculation of the errors may stillbe applied (and the bias is very small, see Brown et al. 1997), which yields fromthe errors on the proper motions, radial velocities and parallaxes

〈εU 〉 = 4.03 km s−1

〈εV 〉 = 3.22 km s−1

〈εW 〉 = 2.54 km s−1.(3.4)

The contribution of the measurement errors to the uncertainty on the samplemean velocity is N−1/2 times the values given by (3.4), where N(= 2774) isthe 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%, weobtain

〈U〉 = −10.24± 0.66 km s−1

〈V 〉 = −20.51± 0.43 km s−1

〈W 〉 = −7.77± 0.34 km s−1.(3.5)

If there is no net radial and vertical motion at the Solar position in theGalaxy, we have hence estimated U = −〈U〉 and W = −〈W 〉 (see Section1.3.4, but see also Section 3.3.6).

55

Knowing that 〈V 〉 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 interestingto compare the mean value of V for the K and M giants separately. For the 250M giants, we obtain

〈V 〉 = −23.42± 1.48 km s−1 (3.6)

while for the 2524 K giants we obtain

〈V 〉 = −20.22± 0.44 km s−1. (3.7)

This difference (at the 2σ level) between the two subsamples can be understoodin terms of the age-velocity dispersion relation. Indeed, the M giants must bea little older than the K giants on average because only the low-mass stars canreach the spectral type M on the Red Giant Branch and because the lifetimeon the main sequence is longer for lower mass stars. This implies that thesubsample of M giants has a larger velocity dispersion and rotates more slowlyabout the Galactic Center than the subsample of K giants, in agreement withthe asymmetric drift relation.

The velocity dispersion tensor is defined as 〈(~v−〈~v〉)⊗(~v−〈~v〉)〉 (see Section1.3.5). The diagonal components are the square of the velocity dispersions whilethe mixed components correspond to the covariances. For the stars with arelative parallax error smaller than 20%, we obtain the classical ordering for thediagonal components, i.e. σU > σV > σW :

σU = 34.46± 0.46 km s−1

σV = 22.54± 0.30 km s−1

σW = 17.96± 0.24 km s−1.(3.8)

The asymmetric drift relation predicts a linear dependence of 〈v〉 = 〈V 〉 + Vwith the radial velocity dispersion σ2

U . If we adopt the peculiar velocity of theSun from Dehnen & Binney (1998a), V = 5.25 km s−1, we find for the fullsample (K and M together)

〈v〉 = −15.26 km s−1. (3.9)

On the other hand, if we adopt the parameter k = 80 ± 5 km s−1 from theasymmetric drift equation of Dehnen & Binney (1998a) and Binney & Merrifield(1998), we find

〈v〉 = −σ2U/k = −14.9± 1.3 km s−1. (3.10)

This independent estimate of 〈v〉 is in accordance with (3.9) and our values of〈V 〉 and σ2

U 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 Mgiants to be higher. We find

σU (Mgiants) = 35.95± 1.6 km s−1, (3.11)

56

andσU (K giants) = 34.32± 0.48 km s−1. (3.12)

As expected, the difference is not very large, and the radial velocity dispersionsfor the two subsamples satisfy the asymmetric drift equation (3.10).

The mixed components of the velocity dispersion tensor involving verticalmotions vanish within their errors. Nevertheless, the mixed component in theplane, that we denote σ2

UV , is non-zero: this is not allowed in an axisymmetricGalaxy (see Section 1.3.5) and is further discussed in Section 3.3. We obtain

σ2UV = 134.26± 13.28 km2 s−2 (3.13)

where the error corresponds to the 15 and 85 percentiles of the correlationcoefficient, 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 usefulquantity is the vertex deviation lv (see Eq. (1.56):

lv ≡ 1/2 arctan (2σ2UV /(σ

2U − σ2

V )) = 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 theSolar 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 vertexdeviation 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 〈U〉 of the starswith V < −35 km s−1, we find that it is largely negative (〈U〉 = −22 km s−1). Itdenotes a global outward radial motion of the stars that lag behind the galacticrotation. We see on Fig. 3.1 that this anomaly is due to a clump located atU ' −35 km s−1, V ' −45 km s−1, the already known “Hercules” stream (Fux2001). On the other hand, Fig. 3.1 reveals in fact a rich small-scale structure inthe 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 (Pleiadessupercluster). More precise values for these peaks will be given in Section 3.3where 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%. Inthis Section, we use all the available stars, without any parallax truncation, butperform a Monte-Carlo simulation to properly evaluate the errors and biases onthe 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 contoursindicate the bins with 3, 4, 7, 12, 17, 20, 25, 30, 35 and 40 stars respectively. Theconcentration of stars around U ' −35 km s−1, V ' −45 km s−1 contributeslargely 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 agaussian distribution centered on the observed parallax value for the correspond-ing star, and with a dispersion corresponding to the uncertainty on the observedparallax. Note that this procedure is not strictly correct, as the gaussian distri-bution should in fact be centered on the (unknown) true parallax. To overcomethis difficulty, a fully bayesian approach will be applied in yet another analysisof the data, as described next in Section 3.3.

Some a priori information has nevertheless been included to truncate thegaussian parallax distribution in the present Monte Carlo approach, since thegiant 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) forthe stars flagged as supergiants. This prescription thus corresponds to assigninga minimum admissible parallax πmin to any given star. It prevents that verysmall parallaxes drawn from the gaussian distribution yield unrealistically largespace velocities. The parallax distribution used in the Monte Carlo simulationthus writes

P (π) =1

(2Π)1/2σπobs

exp

(−1

2

(π − πobsσπobs

)2)

if π ≥ πmin (3.15)

P (π) = 0 if π < πmin (3.16)

The mean space velocities and the velocity dispersions are calculated foreach simulated sample, and finally we adopt the average of these values (of themean velocities and of the velocity dispersions) over 4000 simulated samples asthe best estimate of the true kinematic parameters. We thus obtain:

〈U〉 = −10.25± 0.15 km s−1

〈V 〉 = −22.81± 0.15 km s−1

〈W 〉 = −7.98± 0.09 km s−1.(3.17)

If there is no net radial and vertical motion in the Solar neighbourhood(see however Section 3.3.6), we may write U = −〈U〉 and W = −〈W 〉. Theresults given by (3.17) are very close to those estimated from stars with relativeerrors on the parallax smaller than 20% (see (3.5)). They are in agreementwith the values derived by Dehnen & Binney (1998a) on the basis of Hipparcosproper motions of main sequence stars (U = 10.00±0.36 km s−1, W = 7.17±0.38 km s−1). The value of U is not in perfect accordance with the one derivedby Brosche et al. (2001), who found U = 9.0 ± 0.5 km s−1 from photometricdistances and Hipparcos proper motions of K0-5 giants, nor with the one derivedby Zhu (2000) who found U = 9.6±0.3 km s−1 with the same stars as ours butwithout the radial velocity data. The value of W contradicts slightly the onederived by Bienayme (1999), who found W = 6.7± 0.2 km s−1 from Hipparcosproper motions. Nevertheless, these considerations on the Solar motion are notvery useful since we stress in Section 3.3.6 that the motion of the Sun is difficultto derive anyway because there are conceptual uncertainties on the mean motionof stars in the Solar neighbourhood.

59

For the velocity dispersions, we find

σU = 40.72± 0.58 km s−1

σV = 32.23± 1.41 km s−1

σW = 22.55± 0.95 km s−1.(3.18)

These velocity dispersions are somewhat larger than those found for therestricted sample (3.8), which is not surprising since 60% of the high-velocitystars (as identified in Section 3.3.6) are not present in the sample restricted tothe most precise parallaxes. It could also be a slight effect of the larger volumeof the Galaxy probed by the complete sample. Regarding the asymmetric driftand the slightly larger age of the M giants, the Monte Carlo method and theanalysis of the restricted sample reach the same conclusion.

Concerning the mixed component of the velocity dispersion tensor in theplane, we find

σ2UV = 188.23± 40.9 km2 s−2 (3.19)

which leads to a clearly non-zero vertex deviation of

lv ≡ 1/2 arctan (2σ2UV /(σ

2U − σ2

V )) = 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 anaxisymmetric Galaxy: this is absolutely not the case for our sample of late-typegiants. Furthermore, the value we find for the vertex deviation is larger thanthe one derived for late-type stars by Dehnen & Binney (1998a), who showedthat the vertex deviation drops from 30 for young stellar populations (maybedue to young groups concentrated near the origin of the UV -plane, see alsoSections 1.3.5 and 3.3.7) to a constant value of 10 for older populations. Bi-enayme (1999) also found from Hipparcos proper motions a vertex deviation of9.2 for the giant stars. We conclude from our sample that the vertex deviationis 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 youngerpopulations) 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 modeldescribing the basic morphological characteristics we can safely expect from thesample (spatial distribution, kinematics and absolute magnitudes), and a modelof the selection criteria used to define the sample. These models are used to builda distribution function intended to describe the observational characteristics ofthe sample. The a priori distribution function adopted is a linear combinationof 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) witha gaussian magnitude distribution, and an exponential height distribution uncor-related with the velocities. The phenomenological model adopted is obviouslynot completely rigorous, but has the advantage of being able to identify andquantify 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 determinedfrom the sample using a bayesian approach: the model is adjusted to the sampleby a maximum likelihood fit of the parameters. The values of the parameters soobtained provide the best representation of the sample given the a priori modelsassumed.

In the following Sects. 3.3.1 and 3.3.2 we describe the ingredients of themodels used in this Chapter. Thereafter, in Sects. 3.3.3 to 3.3.6, we presentthe results of the maximum likelihood fit, and we discuss in Section 3.3.7 thedifferent 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 mixtureof stars coming from several “base groups”. A given group represents a fractionwi of the total sample and its characteristics are described by the followingcomponents:

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

ϕe(d, l, b) = exp(−|d sin b|

Z0

)d2 cos b (3.21)

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 thestars with respect to their Standard of Rest:

ϕv(U ′, V ′,W ) = e− 1

2

“U′

σU′

”2− 1

2

“V ′

σV′

”2− 1

2

“W−W0

σW

”2

(3.22)

whereU ′ = (U − U0) cos lv − (V − V0) sin lvV ′ = (U − U0) sin lv + (V − V0) cos lv

(3.23)

and where lv is the vertex deviation.

• Galactic rotation: An Oort-Lindblad rotation model at first order (seeSection 1.3.2), where the rotation velocity is added to the mean LSRvelocity (given by the Schwarzschild ellipsoid above). The same valueswere adopted for the Oort constants as in Sects. 3.1 and 3.2

61

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

ϕM(M) = exp

(−1

2

(M −M0

σM

)2). (3.24)

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

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

The distribution function of each base group is simply the product of thespace, velocity and magnitude distributions presented here, with different valuesfor the model parameters. The total distribution function D of the samplein phase-magnitude space is a linear combination of the partial distributionfunctions 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 inSection 3.3.3. However, the number of groups (ng) composing our sample isnot known a priori. A likelihood test – like Wilk’s test, Soubiran et al. (1990)– will be used to determine it: maximum likelihood estimations are performedwith ng = 1, 2, 3, . . . and the maximum likelihoods obtained for each case arecompared 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 characteristicsof the sample requires that its selection criteria be included in the model. Inour case, our sample of giants (like the Hipparcos Catalogue as a whole) iscomposed primarily of stars belonging to the Hipparcos survey plus stars addedon the basis of several heterogeneous criteria.

In the case of the Hipparcos survey, the selection criteria are just based onthe 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” inHp 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, butleaves 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 zeroat Hp = 11.5

This condition together with the completeness up to Hp = 7.5 + 1.1| sin b|, andwith the cutoff in declination, defines the selection of the complete sample.

The individual errors on the astrometric and photometric data are also takeninto account in the model. We assume that the observed values are producedby gaussian distributions of observational errors around the “true” values, withstandard deviations given by the errors quoted in the Hipparcos Catalogue, theTycho-2 catalogue or the CORAVEL database.

In the end, we can define for each star a joint distribution function of thetrue and observed values:

µ(~x, ~z‖~θ) = D(~x‖~θ) ε(~z‖~x)S(~z) (3.27)

where ~x are the true values (position in the LSR, velocities with respect to theLSR, absolute magnitude), ~z the observed values (position on the sky, parallax,proper motions, radial velocity, apparent magnitude), ~θ the set of parametersof the model, S the selection function taking into account the selection criteriaof the sample, and ε the gaussian distribution of observational errors.

For each star, we can then easily deduce the distribution function of theobserved values:

O(~z‖~θ) =∫µ(~x, ~z‖~θ) d~x (3.28)

The fact that the selection function and the error distribution are taken intoaccount 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 lawgiven by O(~z‖~θ0), where~θ0 = (w1,M01 , σM1 , U01 , σU ′1 , V01 , σV ′1 ,W01 , σW1 , Z01 , lv1 , w2, . . . , lvng

) is the setof unknown parameters on which it depends.

The Likelihood Function is defined as

L(~θ) =n∗∏i=1

O(~zi‖~θ). (3.29)

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 beshown that ~θML is asymptotically non-biased, asymptotically gaussian and thatfor 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 usedto 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 possiblebiases: once a ML estimation ~θML has been obtained, several samples corre-sponding to the parameters ~θML are simulated and the method is applied tothem. The comparison of the results (which could be called ”the estimation ofthe estimation”) with the original ~θML allows us to detect possible biases. Thedispersion 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 witha single base group indicates that the kinematic properties of giant stars in theSolar neighbourhood cannot be fitted by a single Schwarzschild ellipsoid. Thefirst acceptable solution requires three base groups: one of bright giants andsupergiants with “young” kinematics (further discussed in Section 3.3.6), oneof high-velocity stars and finally one group of “normal” stars. This third groupexhibits plenty of small-scale structure, and this small-scale structure can besuccessfully modelled by 3 more base groups (leading to a total of 6 groupsin the sample, called groups Y, HV, HyPl, Si, He, and B, further discussedin Section 3.3.6). These supplementary groups are statistically significant, asrevealed by a Wilks test. Statistically, solutions with 7 or 8 groups are evenbetter, but these solutions are not stable anymore and are thus not useful (theydepend too much on the observed values of some individual stars). Table 3.1lists the values of the ML parameters for a model using just the 5177 (out of5397) survey stars which comply with the completeness law (Eq. (3.26)). Table3.2 lists those values for a model using the whole sample of 6030 stars with ourmodified selection law. The models for the survey stars alone and for the wholesample 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 gofar enough above the galactic plane), thus providing a strong indication that theresults for the whole sample are reliable. The errors on the parameters listed inTables 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 providedby the LM method, and the most probable group along with the correspondingprobability 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 astar belongs to the jth group and Oj(~z | ~θj) the distribution of the observedquantities ~z in this group (deduced from the phenomenological model adoptedand depending on the parameters ~θj of this model for the group). Then, using

64

Table 3.1: Maximum-Likelihood parameters obtained using 5177 survey starsonly. The velocities are expressed in km s−1, the distances for Z0 in pc, and thevertex deviation in degrees. The number n of stars belonging to each group (aslisted in column 34 of Table A.1) is given in the last row. Although obtainedfrom 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 %) ofthe whole sample belonging to the considered group, as listed in the previousrow. Therefore, the observed fraction of stars in each group could be slightlydifferent from the true fraction wi of that group in the entire population.

group Y group HV group HyPl group Si group He group BM0 0.6 ± 0.9 2.2 ± 0.2 1.1 ± 0.4 1.1 ± 0.6 1.3 ± 0.4 1.2 ± 0.2σM 1.8 ± 0.2 1.3 ± 0.1 0.9 ± 0.2 1.2 ± 0.2 0.9 ± 0.1 1.1 ± 0.1U0 -11.6 ± 2.3-17.6 ± 3.5 -31.2 ± 1.0 5.2 ± 3.0 -42.1 ± 4.8 -2.9 ± 1.5σU ′ 15.6 ± 1.2 53.4 ± 2.1 11.5 ± 1.5 13.5 ± 2.7 26.1 ± 13.1 31.8 ± 1.6V0 -11.6 ± 2.1-43.9 ± 3.2 -20.0 ± 0.7 4.2 ± 1.9 -50.6 ± 4.7 -15.1 ± 2.4σV ′ 9.4 ± 1.4 36.1 ± 1.8 4.9 ± 1.1 4.6 ± 4.2 8.6 ± 3.0 17.6 ± 0.8W0 -7.8 ± 0.9 -7.8 ± 2.3 -4.8 ± 1.8 -5.6 ± 2.1 -6.9 ± 3.6 -8.2 ± 0.9σW 6.9 ± 0.8 32.5 ± 1.5 8.8 ± 1.1 9.4 ± 2.2 16.7 ± 1.6 16.3 ± 0.8Z0 73.4 ± 6.3 901 ± 440 128.7 ± 19.2149.4 ± 23.7201.8 ± 55.2196.1 ± 12.3lv 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 ± 4.7n 345 505 334 204 372 3417

Table 3.2: Same as Table 3.1 for the full sample of 6030 starsgroup 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 ± 0.1σM 1.8 ± 0.1 1.4 ± 0.1 0.7 ± 0.1 1.1 ± 0.1 0.9 ± 0.1 1.1 ± 0.1U0 -10.4 ± 0.9 -18.5 ± 2.8 -30.3 ± 1.5 6.5 ± 1.9 -42.1 ± 1.9 -2.8 ± 1.1σU ′ 15.4 ± 1.1 58.0 ± 1.9 11.8 ± 1.3 14.4 ± 2.0 28.3 ± 1.7 33.3 ± 0.7V0 -12.4 ± 0.9 -53.3 ± 3.1 -20.3 ± 0.6 4.0 ± 0.7 -51.6 ± 1.1 -15.4 ± 0.8σV ′ 9.9 ± 0.7 41.4 ± 1.7 5.1 ± 0.8 4.6 ± 0.7 9.3 ± 1.2 17.9 ± 0.8W0 -7.7 ± 0.6 -6.6 ± 1.8 -4.8 ± 0.8 -5.8 ± 1.1 -8.1 ± 1.3 -8.3 ± 0.4σW 6.7 ± 0.6 39.1 ± 1.7 8.7 ± 0.7 9.7 ± 0.8 17.1 ± 1.6 17.6 ± 0.3Z0 80.3 ± 6.2 208.1 ± 27.9106.3 ± 14.4128.0 ± 19.6132.9 ± 9.4141.2 ± 5.2lv 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 ± 1.5n 413 401 392 268 529 4027

65

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

P (∗ ∈ Gj | ~z∗) =wjO(~z∗|~θj)∑ng

k=1 wkOk(~z∗|~θk). (3.30)

Using this formula the a posteriori probabilities that the star belongs to a givengroup 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, willhave a certain percentage of misclassifications. However, the reliability of eachassignment 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 thegroup parameters and the observed values for the star ~z∗, one can obtain themarginal probability density lawR(d) for the distance of the star from the globalprobability density function.

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

d =∫ ∞

0

dR(d) dd (3.31)

and its dispersion

ε2d =∫ ∞

0

(d− d)2 R(d) dd. (3.32)

The first can be used as a distance estimator free from biases and the secondas its error (Columns 28 and 29 of Table A.1). Fig. 3.2 reveals that the biasesin the distance derived from the inverse parallax are a combination of thoseresulting from truncations in parallax and apparent magnitude as discussed inLuri & 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 expectedvalues of U and V deduced from the LM method. The 6 different groups arerepresented on this figure. The structure of the UV -plane is similar to the onealready 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 theLM method are not artifacts of this method. Several groups can be identifiedwith known kinematic features of the Solar neighbourhood.

66

0 200 400 600 800 10000

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 theLM method.

67

-200 -100 0 100

-200

-100

0

100

U (km/s)

Figure 3.3: All the stars plotted in the UV -plane with their values of U and Vdeduced from the LM method. The 6 different groups are represented on thisfigure: group Y in yellow, group HV in blue, group HyPl in red, group Si inmagenta, 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

0 1 2 3 44

2

0

-2

-4

-6

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 colourtransformation 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 〈MHp〉 for this group is 0.7 (see Table 3.2), thebrightest among all 6 groups. An important property of this group is thus itsyoung kinematics coupled with its large average luminosity: the most luminousgiants and supergiants 1 in our sample thus appear to have a small dispersion inthe UV -plane, in agreement with the general idea that younger, more massivegiants are predominantly found at larger luminosities in the Hertzsprung-Russelldiagram 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 acorrelation between velocities and luminosity artificially. This result must thusbe considered as a robust result, even more so since the group Y of young giantswas present in all solutions, irrespective of the number of groups imposed. Thisgroup is centered on the usual antisolar motion (Dehnen & Binney 1998a) in Uand W as seen in Table 3.2.

Fig. 3.4 locates stars from group Y in the HR diagram, constructed from theV − I colour index as provided by the colour transformation of Platais et al.(2003), based on the measured Hp−VT2 colour index (and is thus more accuratethan the Hipparcos V −I index; see Appendix for more details). The isochronesof Lejeune & Schaerer (2001), for a typical metallicity Z = 0.008, indicate thatthe 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 absolutemagnitude for the group listed in Tables 3.1 and 3.2 (〈MHp〉 = 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 trueaverage 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 valueof the scale height of this population is poorly constrained (hence the largedifference between that parameter derived from the full sample – Table 3.2–and from survey stars only – Table 3.1) because our sampling distance is toosmall: indeed Fig. 3.2 shows that the distance of most stars in our sample issmaller 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 consistentwith the mass fraction of the thick disk relative to the thin disk in the Solarneighbourhood.

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

70

0 1 2 36

4

2

0

-2

-4

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 inEqs. (3.8) and (3.18), notice that 247 stars (out of 401) of this group haverelative errors on their Hipparcos parallax higher than 20%.

Clearly we see on Fig. 3.5 that stars of this group are old, mostly older than1 Gyr and it seems clear that some stars at the bottom of the HR diagram are asold 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 centeredon a zero-velocity frame with respect to the galactic center, and with a radiusof the order of 280 km s−1. We refrain, however, from providing numericalvalues for the position of this circular envelope in the UV -plane, because of therather limited number of stars defining this limit and the absence of antirotatinghalo stars in our sample. Note that a lower limit to the local escape velocity isprovided by the velocity (U2 +(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 andallows 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 theHyades and Pleiades superclusters. The large spatial dispersion of the HyPlstars clearly hints at their supercluster rather than cluster nature (see Section1.3.5), since they are spread all over the sky with a wide range of distances (upto 500 pc). The Hyades (〈U〉 = −40 km s−1, 〈V 〉 = −25 km s−1, see Dehnen1998) and Pleiades (〈U〉 = −15 km s−1, 〈V 〉 = −25 km s−1, see Dehnen 1998)superclusters are known since Eggen (1958, 1975). It was also noticed (Eggen1983) that several young clusters (NGC 2516, IC 2602, α Persei) have the sameV -component as those superclusters. We obtain 〈V 〉 = −20.3 km s−1for oursupercluster structure, and thus refine the old value 〈V 〉 = −25 km s−1, within addition the fact that the group is tilted in the anti-diagonal direction of theUV -plane, i.e., it has a negative vertex deviation (lv = −8.7). The concept ofbranches in the UV -plane (Skuljan et al. 1999), of quasi-constant V but slightlytilted in the anti-diagonal direction, will be further discussed below in relationwith 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 closeto 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 Bfor 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 Hegroups discussed below, especially because the HyPl group is lacking young su-pergiants). Although a precise determination of the ages of HyPl stars wouldrequire the knowledge of their metallicities, it seems nevertheless clear that thereare a lot of clump stars with ages reaching 1 Gyr, and that there are clearly stars

72

0 0.5 1 1.56

4

2

0

-2

-4

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). Thestars at the bottom-right of the diagram could be very old metal rich stars (incontradiction with the age-metallicity relation which is obviously not correct ifno other factors are taken into account).

73

older than 2 Gyr, in sharp contrast with the ages of about 80 and 600 Myr forthe Pleiades and Hyades clusters themselves. This result of age heterogeneity,already noticed by Chereul & Grenon (2001) for the Hyades supercluster, is animportant clue to identify the origin of supercluster-like structures, as discussedin Section 3.3.7.

Finally, the value 〈W 〉 = −4.8 ± 0.8 km s−1 differs significantly from thevertical Solar motion (W = 7 to 8 km s−1; see Sects. 3.1, 3.2, and group B inTables 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 sinceEggen (1958, 1960) and is traditionally located at 〈U〉 = 10 km s−1, 〈V 〉 =−5 km s−1 (Dehnen 1998). The LM method refines those values and locatesit at 〈U〉 = 6.5 km s−1, 〈V 〉 = 3.9 km s−1. The spatial distribution onceagain indicates that this group has a supercluster-like nature. Metallicity fromMcWilliam (1990) is available for 12 stars of the Si group (Fig. 3.9). Contrarilyto the situation prevailing for the HyPl group, the metallicity distribution withinthe Si group appears similar to that for the bulk of the giants in the Solarneighbourhood, as represented by group B. Isochrones for a typical value ofZ = 0.008 (Fig. 3.7) indicate that the ages are widely spread, even more sothan for the HyPl group.

Group He: Hercules stream

The “Hercules stream” (represented in green in Fig. 3.3), located by the LMmethod at 〈U〉 = −42 km s−1, 〈V 〉 = −51 km s−1has been named by Raboudet al. (1998) the u-anomaly. It corresponds to a global outward radial motionof the stars which lag behind the galactic rotation: known since Blaauw (1970),it is traditionally centered on 〈U〉 = −35 km s−1, 〈V 〉 = −45 km s−1(see Fux2001). It is strongly believed since several years that its origin is of dynamicalnature, 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 atypical value of Z = 0.008).

Note that the three groups HyPl, Si and He could be interpreted in term ofextended branches crossing the UV -plane (Skuljan et al. 1999, Nordstrom etal. 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 theanti-diagonal direction. These properties could be understood in the classicaltheory of moving groups as discussed in next Section but also in the context ofa dynamical origin for the substructure.

Group B: Smooth Background

Most stars are part of an “axisymmetric”, “smooth” background representedin black in Fig. 3.3. The average metallicity of this group seems to be slightly

74

0 0.5 1 1.56

4

2

0

-2

-4

V-I

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

0 0.5 1 1.56

4

2

0

-2

-4

V-I

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

-0.6 -0.4 -0.2 0 0.20

10

20

30

40

[Fe/H]

group Bgroup HyPlgroup Si

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

77

0 1 2 36

4

2

0

-2

-4

V-I

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 theirsample of F and G dwarfs, Nordstrom et 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 thatthere is a large spread in age. This is typical of the mixed population of thegalactic disk, composed of stars born at many different epochs since the birthof the Galaxy.

In the UV -plane, the velocity ellipsoid of this group is not centered on thevalue commonly accepted for the antisolar motion: it is centered instead on〈U〉 = −2.78 ± 1.07 km s−1. However, the full data set (including the varioussuperclusters) 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 Solarmotion in the presence of streams in the Solar neighbourhood: does there existin the Solar neighbourhood a subset of stars having no net radial motion? If thesmooth background is indeed an axisymmetric background with no net radialmotion, 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, especiallyif the “superclusters” have a dynamical origin, as proposed in the next section.Moreover, the group Y of young giants is centered on the commonly acceptedvalue of 〈U〉 = −10.41 ± 0.94 km s−1 and this difference between groups Yand 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 ona circular orbit, then the 〈U〉 value of group Y is the acceptable one for theantisolar motion, but nothing proves that the GMC’s are not locally movingoutward in the Galaxy under the effect of the spiral pattern. No value can thusat 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 natureof the substructures observed in velocity space. Theoretical investigations anddynamical simulations thus appear to be the only ways to solve this problem.On the other hand, the value of 〈W 〉 = −8.26 ± 0.38 km s−1 for group B is inaccordance 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 asmaller 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 confrontthem 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 galacticrotations, the cluster will evaporate and form a tube called “supercluster”. Starsin the “supercluster” still share common V velocities when located in the sameregion of the tube (for example in the Solar neighbourhood) for the followingreason, first put forward by Woolley (1961): if the present galactocentric radiusof 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, fromEq. (1.49), its guiding-center radius Rg writes

Rg = R − xg = R +v

2B(3.33)

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 whichstayed together in the Galaxy after a few galactic rotations (since they are allcurrently observed in the Solar neighbourhood) must necessarily have the sameperiod 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). Thistheory has thus the great advantage of predicting extended horizontal branchescrossing the UV -plane (similar to those observed in our sample on Fig. 3.3 andalready identified by Skuljan et al. 1999, and by Nordstrom et al. 2004 in theirsample of F and G dwarfs). Moreover, it is easy to understand in this frameworkthat 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 branchesthat 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 branchand another with the Hyades-Pleiades branch). Chereul et al. (1998) suggestedthat the supercluster-like velocity structure is just a chance juxtaposition of sev-eral cluster remnants, but this hypothesis requires extraordinary long survivaltimes 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 havinga mass of m × 104 M can survive up to m Gyr in the Galaxy: indeed heavyclusters of more than 2× 104 M are probably very rare in the disk.

An explanation for the wide range of ages observed in the superclusterscould be the capture of some older stars by the high concentration of mass ina molecular cloud, at the time of the formation of a new group of stars (whileother stars would be scattered by the molecular cloud) . Though this hypothesiscould be tested in N -body simulations involving gas, it would imply that thelocal clumps of the potential not only perturb the motion of stars but dominateit, which is in contradiction with the high predominance of the global potentialon the local clumps. This explanation is thus highly unlikely too.

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Merger remnants

The second class of mechanisms involves the theory of hierarchic formation ofthe galaxies (see Section 1.2.2): following this theory, galaxies were built up bythe merging of smaller precursor structures. It is known since the discovery ofthe 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 satellitegalaxy. Helmi et al. (1999) showed that some debris streams are also present inthe galactic halo near the position of the Sun. Such streams could also be presentin the velocity substructure of the disk: it seems to be the case of the Arcturusgroup at U ' 0 km s−1, V ' −115 km s−1, as recently proposed by Navarroet al. (2004). In this scenario, the streams observed in our kinematic studycould also be the remnants of merger events between our Galaxy and a satellitegalaxy. The merger would have triggered star formation whereas the oldestgiant stars would be stars accreted from the companion galaxy. A merger witha satellite galaxy would moreover induce a perturbation in W , as we observe ingroup HyPl in which 〈W 〉 = −4.8 ± 0.8 km s−1. If the hierarchical (“bottom-up”) cosmological model is correct, the Milky Way system should have accretedand subsequently tidally destroyed approximately 100 low-mass galaxies in thepast 12 Gyr (see Bullock & Johnston 2004), which leads to one merger every120 Myr, but the chance that two of them (leading to the Hyades-Pleiades andSirius superclusters) have left such important signatures in the disk near theposition of the Sun is statistically unlikely (although not impossible).

Dynamical streams

The third class of mechanisms is the class of purely dynamical ones. A dynamicalmechanism that could cause substructure in the local velocity distribution is thedisturbing effect of a non-axisymmetric component of the gravitational potential(Section 1.2.7), like the rotating galactic bar. The Hercules stream was recentlyidentified with the bimodal character of the local velocity distribution (Dehnen1999, 2000) due to the rotation of the bar if the Sun is located at the bar’s outerLindblad resonance (OLR). Indeed, stars in the Galaxy will have their orbitselongated along or perpendicular to the major axis of the bar (orbits respectivelycalled the LSR and OLR modes in the terminology of Dehnen 2000), dependingupon their position relative to the resonances, and both types of orbits coexistat the OLR radius (see Section 1.2.7). Moreover, all orbits are regular in a2-dimensional (2D) axisymmetric potential, but the perturbation of the triaxialbar will induce some chaos. Fux (2001) showed that in the region of the Herculesstream in velocity space, the chaotic regions, decoupled from the regular regions,are more heavily crowded. The Hercules stream has thus also been interpretedas 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 tothat of the bar, the “chaotic” Hercules stream remains a strong feature of thelocal 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 toinhomogeneous star formation. Following Muhlbauer & Dehnen (2003), theGalactic 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 relatedto the Hercules stream. Nevertheless, the other streams present in the data arealso partly responsible for the vertex deviation. The likely dynamical origin ofthe Hercules stream has been the first example of a non-axisymmetric originfor a stream in velocity space: the other streams could thus be related to othernon-axisymmetric effects. Chereul & Grenon (2001) proposed that the Hyadessupercluster represents in fact an extension of the Hercules stream, but the LMmethod has shown in this thesis that the two features are clearly separated inthe 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 oftransient spiral waves (Julian & Toomre 1966; see Section 1.2.7 of this thesisfor a detailed description of the phenomenon) in the shearing sheet (i.e. a smallportion of an infinitesimally thin disk that can be associated with the Solarneighbourhood, see Section 1.2.7). These transient density waves tend to putstars in some specific regions of the UV -plane in the simulations of De Simoneet al. (2004), thus creating streams as observed in our sample. These spiralwaves cause radial migration in the galactic disk near their corotation radius,while not increasing the random motions and preserving the overall angularmomentum distribution (Sellwood & Binney 2002; see also Section 1.2.7). Theseemingly peculiar chemical composition of the group HyPl (i.e., a metallicityhigher 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 originin the inner Galaxy (where the interstellar medium is more metal-rich than inthe Solar neighbourhood) and that it was perturbed by a spiral wave at a certainmoment. This specific scenario (Pont et al., in preparation) would explain whythis group is composed of stars sharing a common metallicity but not a commonage. The group Si could also be a clump recently formed by the passage of atransient spiral.

The simulations of De Simone et al. (2004) can create streams with a rangeof 3 Gyr or more in age, and this is thus a mechanism that can explain thatmain result of our study. Another characteristic of the simulations is that theytend to reproduce the observed branches, and their origin ultimately lies in thesame mechanism as that elucidated by Woolley (1961). However, the tilt of thebranches in the UV -plane is not reproduced by the simulations.

A question which arises in the framework of this dynamical scenario is thefollowing: 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 halfof the OB associations listed in Table A.1 of de Zeeuw 1999 are situated in theregions of the HyPl and Si groups on Fig. 3.3)? Probably not, which suggeststhat the same transient spiral which gave their peculiar velocity to the HyPland Si groups has put these young clusters and associations along the same

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UV branches. For example, it is quite striking that the Hyades cluster itself ismetal-rich with a mean [Fe/H]= 0.13 (Boesgaard 1989), thus pointing towardsthe same galactocentric origin as the HyPl group. The entire cluster could havebeen shifted in radius while remaining bound since the effect of a spiral wave onstars depends on the stars’ phase with respect to the spiral, and the phase doesnot vary much across the cluster. Moreover, since most clusters and associationsare young, they should not have crossed many transients, which suggests thatone 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 timehave given them their peculiar velocity in the UV -plane. The relation betweenspiral 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 ofperiod 500 Myr related to the spiral pattern). As a corollary, we conclude thatthe dynamical streams observed among K and M giants are young kinematicfeatures: integrating backwards (in a smooth stationnary axisymmetric poten-tial) the orbits of the stars belonging to the streams makes thus absolutely nosense, and reconstructing the history of the local disk from the present data ofstars in the Solar neighbourhood becomes tricky. In this dynamical scenario, thedeviation from dynamical equilibrium that is present among samples of youngstars 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 bethe result of chance (Chereul et al. 1998).

If this dynamical scenario is correct, the term dynamical stream for thebranches in velocity space seems more appropriate than the term superclustersince they are not caused by contemporaneous star formation but rather involvestars that do not share a common place of birth: stars in the streams just shareat present time a common velocity vector.

It should be noted that those non-axisymmetric perturbations, as well asthe minor mergers, could lead to some asymmetries in the spatial distributionof stars in the galactic disk on a large scale (see Parker et al. 2004). Sinceour sample does not cover the whole sky, and is anyway restricted to the Solarneighbourhood, 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 velocityspace is largely preferred over the theory which views it as remnants of clustersof stars sharing a common initial origin, essentially because of the wide range ofages of the stars composing the identified subgroups. In fact, we stress that bothphenomena could be closely related to each other. The presence of dynamicalstreams in our sample of K and M giants is clearly responsible for the vertexdeviation found for late-type giants, but we even suggest that the same origincould hold as well for early-type stars. It is indeed quite striking on Fig. 3.3that the group Y extends just far enough in the UV -plane to touch both theHyPl 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 ofthe peculiar velocity imparted by the spiral wave that created them) imposes a

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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 populationscould in fact have the same dynamical origin as the vertex deviation for oldones was already proposed by Mayor (1972, 1974). Here, we also argue thateven the specific initial conditions of young groups of stars could be due to thesame phenomenon.

However, nature is of course not so simple and the features of the distributionfunction are presumably related to a mixture of several phenomena. Notably, theinitial conditions in the simulations should be more complex than the simple 2DSchwarzschild velocity ellipsoid used by De Simone et al. (2004): in fact, purelyaxisymmetric 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 thestructure 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 clearlybe pursued, and in particular dynamical modeling. We have shown that the finestructure of phase space in the Solar neighbourhood cannot be interpreted interms of an axisymmetric steady-state model. Nevertheless, an axisymmetricmodel revealing all the fine structure of the axisymmetric distribution functionin the Solar neighbourhood (Dejonghe & Van Caelenberg 1999) is a necessarystarting point in order to understand the true effects of the non-axisymmetricperturbations.

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 collisionlessBoltzmann equation (1.1) (or (1.3)), taking into account the velocity dispersionsanisotropy (contrarily to two-integral models, see Section 1.2.4) observed in theSolar 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 2DSchwarzschild 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).

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Chapter 4

Stackel potentials

An axisymmetric equilibrium model representing a stellar component of theGalaxy is a pair (Φ, F ), where Φ(R, z) is the steady axisymmetric gravitationalpotential, 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 findan acceptable potential Φ. This Chapter is dedicated to the development ofaxisymmetric Galactic potentials Φ(R, z), satisfying recent estimates of MilkyWay 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 ofthe gravitational force field: they concluded that a wide variety of models canemerge from this fitting process and that the mass distribution of the Galaxyis still ill-determined. As previously explained in Section 1.2.5, the numericalnature of the third integral in most potentials makes it difficult to deal with. Toavoid this problem, we choose to use specific potentials in which an exact ana-lytic third integral exists for all orbits, Stackel potentials (Stackel 1890). Thesepotentials were introduced into stellar dynamics by Eddington (1915) and havesince been used in a number of papers (e.g. Lynden-Bell 1962; de Zeeuw 1985;Dejonghe 1993; Sevenster, Dejonghe & Habing 1995; Durand, Dejonghe & Acker1996; Bienayme 1999) : in fact, the regularity of typical galactic potentials couldbe understood in terms of their proximity to Stackel potentials (e.g. Gerhard1985).

The Stackel potentials are non-rotating potentials for which the Hamilton-Jacobi equation H(~x, ~p) = −E is separable (where ~x are the positions, ~p thegeneralized impulses, E the binding energy). This means that there exists arelation

pi = pi(xi, I1, I2, I3) (4.1)Thus, all orbits admit three analytic integrals of the motion. They form themost general set of potentials that contain one free function, for which threeexact integrals of the motion are known, and which can be relevant as modelsfor 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 Stackelpotentials can fit most known parameters of the Milky Way (including Hipparcoslatest findings). In order to do this, we continue the work of Batsleer & Dejonghe(1994, hereafter BD) who presented a set of simple Stackel potentials with twomass components (halo and disk) and a flat rotation curve, that we generalize byadding a thick disk to them since its existence as a separate stellar componentis now well documented (Ojha et al. 1994; Chen et al. 2001; see also Section1.1.5 and 3.3.6). These new potentials are described by five parameters and wewill show that many different combinations of these parameters are consistentwith fundamental constraints for a mass model of the Milky Way.

4.1 Coordinate system

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

R2

τ + α+

z2

τ + γ= 1 α < γ < 0, (4.2)

and (R,φ, z) cylindrical coordinates. The parameters α and γ are both constantand we assume them smaller than zero. The coordinate surfaces λ =constant areprolate ellipsoids, while the coordinate surfaces ν =constant are hyperboloids.It is convenient to define the axis ratio of the coordinate surfaces as ε = a

c withα = −a2 and γ = −c2. Together with the focal distance ∆ =

√γ − α, the axis

ratio defines the coordinate system, since

γ = ∆2/(1− ε2)α = ε2 γ

(4.3)

4.2 Three-component Stackel potentials

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

Φ(λ, ν) = −f(λ)− f(ν)λ− ν

, (4.4)

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 inspheroidal coordinates, and therefore the orbits admit three analytic isolatingintegrals of the motion: the two classical integrals E and Lz (see Section 1.2.4)and the third integral of galactic dynamics:

I3 =12(L2

x + L2y) + (γ − α)

[12v2z − z2G(λ)−G(ν)

λ− ν

](4.5)

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Figure 4.1: Each point inside the volume is an orbit (clockwise or counterclock-wise with I2 = L2

z/2). Outside of the volume, the values are not allowed orcorrespond to unbound orbits (Dejonghe & de Zeeuw 1988)

All the orbits in such a potential are short axis tubes, i.e. orbits delimited bya minimum and maximum λ (λ− and λ+), and by a minimum and maximumν (ν− and ν+). A detailed orbit analysis can be found in de Zeeuw (1985). Inintegral space, the volume corresponding to allowed bound orbits is delimitedby the planes E = 0, L2

z = 0, I3 = 0 and by a curved surface for whichλ− = λ+ = λ0 (the infinitesimally thin tubes) and whose parametric equations(of parameter λ0) are

E = G(λ0) + (λ0 + α)dGdλ

(λ0)− I3(γ − α)/(λ0 + γ)2 (4.6)

L2z = −2(λ0 + α)2

(dGdλ

(λ0) + I3/(λ0 + γ)2)

(4.7)

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 moveoutside of the Galactic plane). Fig. 4.1 illustrates the structure of the fullintegral space when a Stackel potential is used.

The Milky Way is composed of several mass components: the bulge, thethin 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

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each of these components. For example, BD presented a set of two-component(halo-disk) Stackel potentials with a flat rotation curve.

Our goal is to show that different Stackel potentials are able to fit the latestestimates for the fundamental parameters of the Galaxy. We first generalize thepotentials of BD by adding a thick disk to them since its existence as a separatestellar component is now well established (Ojha et al. 1994; Chen et al. 2001).Our potentials have thus three mass components: two “flat” components andone spheroidal. The spheroidal component accounts for the stellar and darkhalo, and we shall see in Section 4.6 that our potentials turn out to have aneffective bulge, which enables us to avoid the explicit introduction of a bulge-component.

We assume that all three components of our potentials generate a Stackelpotential, with three different coordinate systems but the same focal distance.It is straightforward to show that the superposition of three Stackel potentialsis still a Stackel potential when all three coordinate systems have the same focaldistance. Although the functions fthin, fthick and fhalo are arbitrary, we assumethat they each generate a Kuzmin-Kutuzov potential, defined by

G(τ) =GM√τ + c

(4.8)

with M the total mass of the system. Such a potential becomes a point masspotential (Φ = −GM

R ) in the Galactic Plane when λ→∞. We use this potentialessentially because it is an extremely simple but representative Stackel potential:we will show that it is not necessary to use complicated Stackel potentials inorder to match all the known and most recently determined parameters of theMilky Way.

Near the center, in a meridional plane, the lines of constant mass densitycorresponding to a Stackel 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 oblatespheroids (contrarily to the spheroids of constant λ, which are prolate), andincreasing ε = a

c produces more flattening. So, the ratio ε has to be high for thethin 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 andthe central value of the potentials equal to −1. Each of these potentials is asuperposition of three Kuzmin-Kutuzov potentials in three different coordinatesystems:

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

λthin−νthin

−kthickfthick(λthick)−fthick(νthick)

λthick−νthick− (1− kthin − kthick)

fhalo(λhalo)−fhalo(νhalo)λhalo−νhalo

(4.9)This new class of potentials is thus defined by five parameters (the three axisratios 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 reasonableaugmentation with respect to the BD potentials that were defined by threeparameters.

We denote

αthin − αthick = γthin − γthick = q1 ≥ 0αthin − αhalo = γthin − γhalo = q2 ≥ q1 ≥ 0. (4.10)

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

Φp(λthin, νthin) = −GM( kthin√λthin+

√νthin

+ kthick√λthin+q1+

√νthin+q1

+ 1−kthin−kthick√λthin+q2+

√νthin+q2

)(4.11)

The dimensionless rotation curve corresponding to such potentials is givenby:

v2c (Rp) = Rp∂Φp/∂Rp(Rp, 0) = GMR2

p(kthin√

λthin(√λthin+cthin)2

+ kthick√λthin+q1(

√λthin+q1+

√c2thin+q1)2

+ 1−kthin−kthick√λthin+q2(

√λthin+q2+

√c2thin+q2)2

)(4.12)

where Rp denotes the dimensionless galactocentric radius. We shall imposeconstraints on the shape and flatness of the rotation curve in Section 4.3.

In order to transform these dimensionless potentials into dimensional onesfor the Milky Way, we denote the dimensionless radius where the rotation curveattains its first maximum as Rp,M : since the Milky Way attains its globalamplitude of 220 km/s for the first time at a radius of about 1.5 kpc (Fich &Tremaine 1991), we define a distance scale factor rS = 1.5kpc

Rp,M. The conversion

between dimensionless and dimensional distances is then given by:

R(kpc) = rSRpz(kpc) = rSzp

(4.13)

Then, the total mass M of the Galaxy is adjusted in such a way that thedimensional circular velocity at the Solar radius (R = 8± 0.5 kpc, see Section1.3.1) is equal to 220 km s−1: we obtain thus a minimum and a maximum valuefor M , for the two extreme values of the galactocentric distance of the sun.That adjustment also fixes the local mass density in the Solar neighbourhoodρ (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 thelight of the observational constraints reviewed in Section 1.3.

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By definition of the dimensional potentials (see Section 4.2), the local circularspeed vc(R) is equal to 220 km s−1 for all the potentials. The first fundamentalselection criterion is the flatness of the rotation curve (Section 1.3.2): this featurecan be examined in the dimensionless frame-work. Even the BD potentials, withonly two mass components, could produce many different shapes of rotationcurves. So, we adopt the same simple diagnostic as BD: we denote Rp,M thedimensionless radius where the circular speed attains its first maximum and welook for a range in R where vc(R) remains larger than 80% of the maximumvelocity and is thus more or less constant. We denote Rp,F the dimensionlessradius where vc(Rp,F ) = 0.8vc(Rp,M ). A rotation curve is considered sufficientlyflat if:

EF =Rp,F −Rp,M

Rp,M> 8 (4.14)

This is a minimum requirement.The second selection criterion is based on the latest determinations of the

Oort constants (Section 1.3.2). Since we impose vc(R) = 220 km s−1 andR = 8± 0.5 kpc for all our potentials,

vc(R)R

= 27.6± 1.7km s−1kpc−1 (4.15)

which is in accordance with the value determined by Feast & Whitelock (1997),vc(R)R

= 27.2± 0.9km s−1kpc−1. The first derivative of the circular velocity inthe Solar neighbourhood corresponding to the Oort constants found by Feast& Whitelock (1997) is dvc

dR (R) = −2.4± 1.2km s−1kpc−1. Our potentials havetwo extreme values for this derivative, depending on the position of the sun; inorder to fit the above interval, we select the potentials such that:

max(dvc

dR (R)) > −3.6km s−1kpc−1

min(dvc

dR (R)) < −1.2km s−1kpc−1 (4.16)

This feature is not as essential as the flatness of the rotation curve, because ofthe 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 forits 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 inSection 1.3.3, we select the potentials such that:

max(ρ) > 0.06Mpc−3

min(ρ) < 0.12Mpc−3 (4.17)

This feature is more important than the Oort constants: we give the priority tothat constraint, contrarily to Sevenster et al. (2000) who studied the structure ofthe inner Galaxy, and therefore constructed Stackel potentials with reasonablevalues 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 observationallyand are not considered as fundamental constraints for the potential.

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4.4 The “winding staircase”

Our goal is now to find and select, among the class of potentials defined inSection 4.2, some representative potentials that differ with respect to their formand features, and that satisfy the selection criteria of Section 4.3. As it isdifficult to visualize a five-dimensional parameter space, we shall first visualizeits structure when εthin and εhalo are fixed. Then we shall restrict the parameterspace by imposing additional constraints on the flatness of the thick disk. Weshall finally select five representative potentials in Section 4.6.

As an example of the consequences of the choice of the selection criteriaof Section 4.3, we look for all the values of εthick, kthin and kthick that yieldpotentials satisfying the selection criteria for εthin = 75 and εhalo = 1.02 (withthe thick disk always thicker than the thin disk, i.e. εthick < 75). The accordancewith the selection criteria results from a precise mixing of the two disks and ofthe halo. Fig. 4.2 illustrates the volume in parameter space correspondingto these satisfactory potentials: the volume looks like a “winding staircaise”.When kthick = 0, we see that all the values of εthick 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 zeroexcept when εthick is very close to 1, i.e. when the thick disk is a pretty roundcomponent (similar to the halo) and helps to keep the rotation curve flat. Whenkthin is smaller than 15%, the possibilities for kthick are more numerous, i.e. thethick disk can take a part of the mass. When kthin attains the critical valueof 12%, the volume is inflected and the thick disk has to be thin and non-zeroin order to counter-balance the lack of mass in the thin disk. Decreasing themass 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 inparameter space for εthin > 75 have the same form, and are bigger essentiallybecause there is more freedom for εthick.

4.5 Constraints on the scale height of the thickdisk

In Fig. 4.2, there are some solutions with a thick disk more massive than thethin disk. Even though the mass fractions of the disks and the flattening of thecomponents 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 thethin disk (and represent at most 13% of the local thin disk density in the Solarneighbourhood) and the latest determination of the thick disk scale height basedon 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 definitiveanswer: some other studies indicate that the scale height could be of the orderof 1 kpc (Gilmore 1984; Ojha et al. 1996). We shall reject the potentials thatare 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

010

2030

4050

6070

80

a_thick/c_thick

00.02

0.040.06

0.080.1

0.120.14

0.16

k_thin

00.020.040.060.08

0.10.120.140.16

k_thick

Figure 4.2: This “winding staircase” figure displays the possible values of thecoordinate axis ratio ε = a

c for the thick disk and the possible values of thecontributions k of the disks to the total mass in order to satisfy the selectioncriteria of Section 4.3 (for fixed values εthin = 75 and εhalo = 1.02, and withthe thick disk always thicker than the thin disk, i.e. εthick < 75). It gives arough vision of the region of parameter space that satisfies the criteria. Onlythe region 1.3 ≤ εthick ≤ 2 is in fact relevant for a model of the Milky Way (seeSection 4.5).

93

Table 4.1: Column 1 contains the axis ratio of the coordinate surfaces for thethick disk. Column 2 gives the corresponding scale height for a Kuzmin-Kutuzovpotential with rS = 1.ε hz(pc)1.3 9141.4 8341.5 7821.8 6962 663

In order to determine the interval of axis ratios that should be considered forthe thick disk, we have fitted the vertical mass distribution corresponding to asimple Kuzmin-Kutuzov potential with rS = 1 to an exponential law e−z/hz . Weconclude that the potentials with 1.3 ≤ εthick ≤ 2 have a scale height between665 pc and 1 kpc and are the ones we should examine in detail. For each axisratio, Table 4.1 gives the corresponding scale height. However, Fig. 4.3 revealsthat the exponential fit is not valid for the axis ratios used to model the thindisk, which is not surprising since a thin disk could be better understood asa superposition of isothermal sheets (i.e. ρ(z) = ρ0 sech2(z/2hz); see Spitzer1942) than by a simple exponential law.

4.6 The final selection

In this section, we look for some three-component potentials with different formsand 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 newselection criteria: they are listed in Table 4.2. All the two-component potentialswith εdisk = 50, 75, 130, 200 and εhalo = 1.005, 1.01, 1.02, 1.03 are examined.We do not consider disks with a/c > 200 because then the uncertainty on ρbecomes too large. We see in Table 4.2 that, in order to reproduce the Oortconstants in the two-component framework, the shape of the halo cannot vary(εhalo = 1.02).

If we take the two-component potentials as a starting point, there are twoways to add a thick disk. The first way is to decrease the contribution of thehalo and to put the remaining mass into the thick disk: the local density in theSolar neighbourhood is then slightly larger while the rotation curve is decreasingfaster. If we take the third potential of Table 4.2 as a starting point, the firstpotential of Table 4.5 (potential I) illustrates this first case. The other way is todecrease the contribution of the thin disk and to put the remaining mass intothe thick disk: the local density is then slightly decreasing while the rotationcurve is more flat. If we take the fourth potential of Table 4.2 as a startingpoint, 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 ε = 1.3 (top left), ε = 2 (top right), ε = 75(bottom left) and ε = 200 (bottom right). Only the two first cases resembleexponentials (axes: ln(ρ) (no dimension) and z (kpc)).

95

the halo, so we look for three-component potentials with a halo rounder thanεhalo = 1.02. In order to keep the rotation curve flat and retain the localdensity as well as the Oort constants in the allowed interval, we need to couplea very thin disk with the rounder halo: indeed, our investigations show thatno solution can be found for εthin = 50 and εhalo = 1.01. However, if we takeεthin = 200, Table 4.3 gives solutions for a halo with εhalo = 1.01: we select thesolution 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 εhalo = 1.02 wouldcontain 292 entries and is omitted here. There are much less solutions whenεthin = 75, as can be seen in Table 4.4. However, as stated in Section 4.3, we donot assign high priority to the Oort constants, and we select a potential withεthin = 75, εhalo = 1.01 and a relative mass of the thick disk relative to the thindisk of 13% (i.e. a smaller fraction than any of the solutions of Table 4.4), butwith a quite large local radial derivative of the circular speed (potential IV ofTable 4.5).

For εhalo = 1.005, it is totally impossible to find a potential satisfying theOort constants criterion: the radial derivative of the circular speed in the Solarneighbourhood is always positive. Nevertheless, if one is willing to ignore theestimates of A and B, one could select a potential with εhalo = 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, forwhich the interval in ρ is precisely [0.06Mpc−3, 0.12Mpc−3], and which isclose to the Chen et al. (2001) findings , i.e. εthin = 200 which is the thinnestthin disk that we consider in order not to have a too large interval for ρ, arelative mass of the thick disk to the thin disk of 10%, a scale height of the thickdisk of 612.5 pc and a relatively large EF (EF = 13.44).

Table 4.5 summarizes the main features of the selected Stackel potentialswith different forms and features and that we shall use for dynamical modelingof the Milky Way: potentials III and V have a very thin disk associated with aquite massive thick disk and a pretty round halo, while potentials I and IV havea thicker thin disk with a quasi-negligible thick disk (Fig. 4.4 shows the massisodensity 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 verydifferent and become larger with a rounder halo and that a rounder halo impliesthat this halo is much more extended. A closer look to the mass density inthe equatorial plane indicates that, for each potential, the density grows fasterthan 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 theintroduction of an explicit bulge component. We have fitted the mass densityin the plane to an exponential law down to R = 3 kpc in order to check thatthe scale length of the disk is realistic: the last column of Table 4.5 gives thescale length corresponding to each selected potential and we conclude that theyare realistic but do not distinguish the different potentials. For the potentialswith the biggest and smallest scale length, Fig. 4.5 illustrates the shape of thelogarithm 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 eachof the five selected potentials: the rotation curve of potential II is more flat thanthe one of potential I in the vicinity of the sun, while the rotation curves of thepotentials with εhalo = 1.01 (potentials III and IV) are even more flat and theone of potential V is slightly increasing.

The five potentials Φ of Table 4.5 can be used as potential for the Milky Wayin an equilibrium model (Φ, F ) representing a stellar population such as the giantstars of Chapter 3. The next step in the establishment of the equilibrium modelis 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 εdisk =50, 75, 130, 200 and εhalo = 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 of0.01 for the relative contribution of the disk). Columns 1 and 2 contain the axisratios of the coordinate surfaces for the disk and the halo. Column 3 containsthe relative contribution of the disk to the total mass. Column 4 contains theextent of the flat part of the rotation curve (see Eq. 4.14). Column 5 containsthe minimum and maximum local spatial density, while column 6 contains theminimal and maximal local radial derivative of the circular velocity, each timefor the two extreme positions of the sun.εdisk εhalo kdisk EF ρ in Mpc−3 dvc

dR (R) in km s−1kpc−1

75 1.02 0.11 13.07 0.04, 0.06 −1.90, −1.1875 1.02 0.12 11.74 0.04, 0.06 −2.03, −1.4275 1.02 0.13 10.43 0.04, 0.07 −2.25, −1.7475 1.02 0.14 9.25 0.04, 0.07 −2.46, −2.0575 1.02 0.15 8.20 0.05, 0.08 −2.68, −2.37130 1.02 0.08 17.16 0.04, 0.06 −2.07,−1.12130 1.02 0.09 16.03 0.04, 0.07 −1.75, −0.85130 1.02 0.10 14.56 0.05, 0.08 −1.73, −0.92130 1.02 0.11 13.03 0.05, 0.09 −1.86, −1.15130 1.02 0.12 11.70 0.06, 0.10 −2.00, −1.39130 1.02 0.13 10.38 0.06, 0.11 −2.22, −1.72130 1.02 0.14 9.20 0.07, 0.11 −2.44, −2.04130 1.02 0.15 8.08 0.07, 0.12 −2.69, −2.38200 1.02 0.08 17.21 0.04, 0.09 −2.02, −1.07200 1.02 0.09 16.07 0.05, 0.10 −1.71, −0.81200 1.02 0.10 14.59 0.06, 0.12 −1.69, −0.88200 1.02 0.11 13.05 0.07, 0.13 −1.82, −1.12200 1.02 0.12 11.64 0.07, 0.14 −2.00, −1.40200 1.02 0.13 10.32 0.08, 0.16 −2.22, −1.72200 1.02 0.14 9.14 0.08, 0.17 −2.44, −2.04200 1.02 0.15 8.08 0.09, 0.18 −2.66, −2.36

98

Table 4.3: The three-component potentials with εthin = 200 and εhalo = 1.01that satisfy the selection criteria are listed in this table (with a step of 0.01 forthe relative contribution of the two disks). Columns 1, 2, 3 contain the axisratios of the coordinate surfaces for the two disks and the halo. Columns 4 and5 contain the relative contribution of the repectively thin and thick disk to thetotal 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 spatialdensity in Mpc−3, while column 8 contains the minimal and maximal localradial derivative of the circular velocity, each time for the two extreme positionsof the sun in km s−1kpc−1.εthin εthick εhalo kthin kthick EF ρ

dvc

dR (R)200 1.3 1.01 0.09 0.07 10.69 0.06, 0.12 −1.34, −1.13200 1.3 1.01 0.09 0.08 10.06 0.06, 0.12 −1.64, −1.45200 1.3 1.01 0.10 0.06 9.51 0.07, 0.15 −1.35, −1.29200 1.3 1.01 0.10 0.07 8.95 0.07, 0.14 −1.64, −1.59200 1.3 1.01 0.10 0.08 8.45 0.07, 0.13 −1.92, −1.88200 1.3 1.01 0.11 0.04 8.51 0.09, 0.18 −1.22, −1.17200 1.3 1.01 0.11 0.05 8.02 0.08, 0.17 −1.51, −1.45200 1.4 1.01 0.09 0.07 9.81 0.06, 0.13 −1.46, −1.32200 1.4 1.01 0.09 0.08 9.11 0.06, 0.12 −1.77, −1.65200 1.4 1.01 0.10 0.06 8.63 0.07, 0.14 −1.50, −1.47200 1.4 1.01 0.10 0.07 8.07 0.07, 0.14 −1.78, −1.78200 1.5 1.01 0.08 0.07 10.86 0.06, 0.11 −1.25, −1.00200 1.5 1.01 0.09 0.06 9.93 0.07, 0.13 −1.21, −1.09200 1.5 1.01 0.09 0.07 9.10 0.06, 0.13 −1.54, −1.45200 1.5 1.01 0.09 0.08 8.34 0.06, 0.12 −1.86, −1.80200 1.5 1.01 0.10 0.06 8.01 0.07, 0.15 −1.59, −1.57200 1.8 1.01 0.08 0.07 9.42 0.06, 0.12 −1.34, −1.22200 1.8 1.01 0.09 0.06 8.55 0.07, 0.13 −1.34, −1.32200 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 εthin = 75 and εhalo = 1.01 thatsatisfy the selection criteria are listed in this table (with a step of 0.01 for therelative contribution of the two disks). Columns have the same meaning as intable 4.3.εthin εthick εhalo kthin kthick EF ρ

dvc

dR (R)75 1.3 1.01 0.11 0.05 8.21 0.05, 0.07 −1.49, −1.4575 1.4 1.01 0.11 0.04 8.06 0.05, 0.07 −1.32, −1.2775 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 ofTable 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. Rightpanel: large scale view of the halo, axes: position (kpc) from 0 to 15 in theplane and from 0 to 15 in the vertical direction).

100

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

101

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

102

Table 4.5: Among the class of potentials defined in Section 4.2, five differentpotentials regarding form and features have been selected. Columns 1, 2, 3contain 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 andthick disk to the total mass. Column 6 contains the extent of the flat part ofthe rotation curve (see Eq. 4.14). Column 7 contains the scale factor whichcorresponds to the focal distance of the coordinate system of the dimensionalpotential. Column 8 contains the minimum and maximum local spatial densityin Mpc−3, while column 9 contains the minimum and maximum local radialderivative of the circular velocity in km s−1kpc−1 and column 10 the minimumand maximum total mass of the Galaxy in 1011M, each time for the twoextreme positions of the sun. The scale length hR in the equatorial plane downto 3 kpc has been calculated and is presented in column 11 (in kpc).

εthin εthick εhalo kthin kthick EF rS ρdvc

dR (R) M hRI 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 2.73II 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 2.63III 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 2.65IV 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 2.78V 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 distributionfunctions

Now that we have developed in Chapter 4 acceptable Stackel potentials for theGalaxy, we need to know which form to give to the distribution function F ofthe stellar population we want to model. We express F as a linear combinationof component distribution functions FΛ:

F (E,Lz, I3) =∑Λ

cΛFΛ(E,Lz, I3) (5.1)

This Chapter will be dedicated to the development of new component distribu-tion functions FΛ. It is not quite obvious to define suitable distribution functionsFΛ(E,Lz, I3) that depend on three exact analytic integrals in a Stackel poten-tial, and that can somewhat realistically represent our ideas of a real stellardisk. For example, Bienayme (1999) made three-integral extensions of the two-integral parametric distribution functions described in Bienayme & Sechaud(1997), but these ones were built to model the kinematics of stars in only asmall portion of the Milky Way. Dejonghe & Laurent (1991) also defined thethree-integral Abel distribution functions, but these ones could not provide verythin disks in the two-integral approximation. Robijn & de Zeeuw (1996) con-structed three-integral distribution functions for oblate galaxy models, but theyalso had problems to recover the two-integral approximation.

In this Chapter we continue the work of Batsleer & Dejonghe (1995), whoconstructed component distribution functions that are two-integral, but thatcan represent (very) thin disks when a judicious linear combination of them ischosen. We use these components as a basis for new component distributionfunctions that are three-integral, of which the Batsleer & Dejonghe componentsare 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 massdensity 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 verythin disk systems (e.g. Jarvis & Freeman 1985), with the dramatic restrictionthat both vertical and radial dispersions are equal. We know that this is nottrue in the Milky Way disk (see especially Eqs. (3.8) and (3.18) of the presentthesis). So we want to create three-integral distribution functions that cancreate anisotropy in the velocity dispersions but that can also describe very thindisks in the two-integral approximation, unlike the Abel distribution functionsof Dejonghe & Laurent (1991).

The Fricke components (Fricke 1952) of the form EηLβz favour that partof phase space where stars populate circular orbits, so they could be taken asa starting point. However, they cannot be used in their basic form to modeldisks with a finite extent because they populate orbits which can reach arbitrarylarge heights: therefore, we will take as a starting point the components definedin Batsleer & Dejonghe (1995, Eq. 19), who constructed disk-like componentdistribution functions with a finite extent in vertical direction by setting themequal 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, weintroduce the factor (p + qE + rL2

z + sI3)δ in which the parameter s (and δ)will be responsible for the three-integral character of the components. Thecoefficients 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

F (E,Lz, I3) = f(Lz)(

E − Sz0(Lz)S0(Lz)− Sz0(Lz)

)η(p+ qE + rL2

z + sI3)δ (5.2)

if E − Sz0(Lz) ≥ 0p+ qE + rL2

z + sI3 ≥ 0 . (5.3)

The distribution function is identically zero in all other cases. The functionf(Lz) is defined as

f(Lz) =1

1 + e−aLz(2L2

zSz0(Lz))β(Sz0(Lz))

α1e− α2

Sz0 (Lz) . (5.4)

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 diskcomponent. The parameter η is responsible for the favouring of nearly circularorbits, i.e. orbits with a binding energy E as close as possible to that of circularorbits 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 anincreasing function of E: this forces q to be positive. Since a large I3 impliesthat the orbit can reach a large height above the galactic plane (see de Zeeuw1985 for a complete analysis of the orbits in a Stackel potential), the orbitswith small I3’s have to be favoured in order to describe thin disks: this forcess to be negative. The parameter a is the rotation parameter (the value of ainfluences only the odd moments of F , see Section 5.2). If a = 0, there is norotation for the component, and if a = +∞, F represents a maximum streamingcomponent with no counter-rotating stars. Finally, The requested exponentialdecline 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 spheroidalcoordinate system are defined as

µl,m,n(λ, ν) =∫ ∫ ∫

F (E,Lz, I3)vlλvmφ v

nν dvλdvφdvν , (5.5)

with vλ, vφ, vν the components of the velocity in the λ, the φ and the ν directionof the spheroidal coordinate system, and l, m, n integers.The mass density, the mean velocity and the velocity dispersions of the stellarsystem represented by F can easily be expressed in terms of the moments (5.5)by

ρ(λ, ν) = µ0,0,0(λ, ν)ρ〈vφ〉(λ, ν) = µ0,1,0(λ, ν)ρσ2

λ(λ, ν) = µ2,0,0(λ, ν)ρ〈v2

φ〉(λ, ν) = µ0,2,0(λ, ν)ρσ2

ν(λ, ν) = µ0,0,2(λ, ν)

(5.6)

To obtain the value of one of the moments, we have to integrate over thevolume in velocity-space corresponding to all orbits that pass through the point(λ, φ, ν) in the spheroidal coordinate system of the Stackel 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 anodd 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 norotation and if m is odd, the moment is zero) and that m is an even integer (thegeneral case will easily be derived from this one in the next subsection). Underthese assumptions, Eq. (5.5) becomes (see Dejonghe & de Zeeuw 1988):

106

µl,m,n =2

l+n2 +1

R(λ− ν)l+n2

∫f(L2

z)√L2z

vmφ dL2z

∫ ∫dEdI3

(E − Sz0S0 − Sz0

)η×(p+ qE + rL2

z + sI3)δ(I+3 − I3)

l−12 (I3 − I−3 )

n−12 (5.7)

where I+3 and I−3 are given by

I+3 (E,L2

z) = (λ+ γ)[G(λ)− E]− λ+γ2(λ+α)L

2z

I−3 (E,L2z) = (ν + γ)[G(ν)− E]− ν+γ

2(ν+α)L2z.

(5.8)

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

I−3 (E,L2z) ≤ I3 ≤ I+

3 (E,L2z)

Sz0(Lz) ≤ E ≤ ψ(λ, ν)− L2z

2R2

p+ qE + rL2z + sI3 ≥ 0.

(5.9)

For this double integral to be analytically solved, however, we will have tomake use of those combinations of p, q, r and s for which the integration areais transformed into the triangle bounded by

I3 = I+3 (E,L2

z)I3 = I−3 (E,L2

z)p+ qE + rL2

z + sI3 = 0,(5.10)

as shown in Fig. 5.1. We will be in this situation (for all the points (λ, φ, ν) ofconfiguration space, where the moments are calculated) whenever p+qE+rL2

z+sI3 = 0 does intersect the E-axis for E ≥ Sz0(Lz). If we take p = −Sz0 andr ≤ 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 theother three factors in the integrandum (corresponding to the bounding lines ofthe integration surface):

E − Sz0S0 − Sz0

= t(I+3 − I3) + u(I3 − I−3 ) + v(p+ qE + rL2

z + sI3). (5.11)

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

η∑i=0

η−i∑j=0

(ηi

)(η − ij

)vi tη−i−j uj

∫ ∫dE dI3 (p+ qE + rL2

z + sI3)δ+i

×(I+3 − I3)η+

l−12 −i−j (I3 − I−3 )

n−12 +j(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 whosezero-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 integrationvariables x and y, defined by (for a fixed L2

z)

x(E, I3) = I+3 (E)− I3

y(E, I3) = I3 − I−3 (E).(5.13)

The line in the (E, I3)-plane p+ qE + rL2z + 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

xmax = −p+ rL2

z − h(ν)ψ(λ, ν) + s(ν + γ)G(ν)− L2z

2R2(q − sz2)

(λ− ν)h(ν)

≡ − (λ− ν)h(ν)

x′max (5.15)

and

ymax(x) =h(ν)h(λ)

(xmax − x) (5.16)

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

1(λ−ν)δ+i+1

∫ xmax

0xη+

l−12 −i−jdx

∫ ymax(x)

0y

n−12 +j [h(λ)(y − ymax(x))]

δ+idy

= (−h(λ))δ+i

(λ−ν)δ+i+1B(n−12 + j + 1, δ + i+ 1)

×∫ xmax

0xη+

l−12 −i−jymax(x)

n−12 +i+j+δ+1dx

(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 thewhole summation (5.12)

η∑i=0

η−i∑j=0

(ηi

)(η − ij

)vi tη−i−j uj

(−h(ν))δ+ n+12 +i+j

(λ− ν)δ+i+1(−h(λ))n+1

2 +j

×(xmax)δ+η+l+n2 +1 Γ(η + l+1

2 − i− j) Γ(n+12 + j) Γ(δ + i+ 1)

Γ(δ + η + l+n2 + 2)

(5.18)

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 coefficientst, u and v are calculated by equalizing term by term in equation (5.11)

1S0−Sz0

= −t(λ+ γ) + u(ν + γ) + vq

0 = −t+ u+ vs

− Sz0S0−Sz0

= t(λ+ γ)[G(λ)− L2

z

2(λ+α)

]− u(ν + γ)

[G(ν)− L2

z

2(ν+α)

]+ vrL2

z + vp

(5.19)We find for the coefficients

v =v′

x′max

, u =u′

(λ− ν)x′max

, t =t′

(λ− ν)x′max

(5.20)

with

v′ =ψ(λ, ν)− Sz0 −

L2z

2R2

S0 − Sz0, u′ = −h(λ)v′ − x′max

S0 − Sz0, t′ = u′ + s(λ− ν)v′ (5.21)

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

Sz0(Lz) ≤ E ≤ ψ − L2z

2R2, (5.22)

the integration limits for the integral in L2z are, in a first time, determined by

the intersections of Sz0(Lz) and the line E = ψ − L2z

2R2 in the (E,L2z)-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

xmax ≥ 0 ⇔ x′max ≥ 0. (5.23)

Since the condition ψ − L2z

2R2 − Sz0 ≥ 0 is automatically verified when condition(5.23) is verified, the equality in (5.23) fixes the minimal and maximal angularmomentum to take into account in the integration.

The extremum of x′max(L2z) is calculated: if it is smaller than zero, the

integral is null, else the zero’s of x′max(L2z) are calculated and are the bounds of

the integration.Knowing that L2

z = R2v2φ, the resulting expression for the moment (with

a = 0 and m even) becomes

111

Figure 5.3: The (E,L2z)-plane. The integrations limits in L2

z, (Lz)2min and(Lz)2max, must be between the intersections of the curve E = Sz0(Lz) and the

line E = ψ − v2φ2 . Condition (5.23) then definitively fixes these limits. E0 and

E1 are the minimal and maximal energy taken into account in the integration.

112

µl,m,n = 2l+n2 +β+1R2β Γ(δ + 1)Γ(n+1

2 )Γ(η + l+12 )

Γ(δ + η + l+n2 + 2)

∫ (v2φ)max

(v2φ)min

(v2φ)β+ m−1

2

×(Sz0)β+α1+α3

e− α2

Sz0

2(x′max)

δ+ l+n2 +1

(−h(λ))n+1

2 (−h(ν))η+ l+12

η∑i=0

η−i∑j=0

(ηi

)(η − ij

)v′i

×t′η−i−j u′j Γ(δ + i+ 1)Γ(δ + 1)

Γ(η + l+12 − i− j) Γ(n+1

2 + j)Γ(η + l+1

2 ) Γ(n+12 )

(−h(ν))i+j

(−h(λ))jdv2φ (5.24)

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 :

P =1− e−aR|vφ|

1 + e−aR|vφ|(5.25)

because∫ (v2φ)1/2max

(v2φ)1/2min

f(v2φ)v

2k+1φ

11 + e−aRvφ

dvφ +∫ −(v2φ)

1/2min

−(v2φ)1/2max

f(v2φ)v

2k+1φ

11 + e−aRvφ

dvφ

=12

∫ (v2φ)max

(v2φ)min

f(v2φ)(v

2φ)k 1− e−aR|vφ|

1 + e−aR|vφ|dv2φ (5.26)

with k = m−12 and the factor 1

2 already present in Eq. (5.24) (division by 2 ofthe 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 distributionfunctions: their mass density has a finite extent in the vertical direction and anexponential decline in the galactic plane, they favour almost circular orbits andtheir velocity dispersions are different in the vertical and radial direction. Byvarying 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 themoments of many component distribution functions with different values for theparameters.

As galactic potential, we use potential II of Table 4.5. In the implementationof 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 ameridional 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 (rightpanel) respectively (note the very different scale for R and z). The disks becomethinner with smaller z0, while the mass density is zero above z = z0 (and evenso 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 thenext lower contour.

5.3.1 The parameter z0

This parameter was introduced in order to impose a maximum height above thegalactic plane for the disk-like component (Fig. 5.4): indeed, when E ≥ Sz0 , anorbit cannot go higher than z = z0, and the distribution function (5.2) is nullfor E ≤ Sz0(Lz). In order to model samples of stars belonging to populationswith different characteristic heights above the galactic plane, we can use a setof 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-negativevalues of α1, the factor where it appears will behave as a declining function ofLz, 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 alarge fraction of bound orbits. When it is increasing, this parameter helps toproduce mass close to the center.

When a given exponential decline is requested, α1 will be a function of theother 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 massdistribution 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 planemass density (in arbitrary units and scaling) of different components for varyingα1. A rising α1 helps to produce mass close to the center. The other parametershave 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 planeand 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) exceptthat α1 is adjusted to built-in a given exponential decline. We see that thecomponents have an exponential decline in the galactic plane and that α2 isroughly the reciprocal of the component’s scale length.

116

radii: this implies that, in the galactic plane,

L2z ∼ R (5.27)

and that (Batsleer & Dejonghe 1995)

Sz0(Lz) ∼1L2z

∼ 1R

(5.28)

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 muchwith 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 determinedby 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-constantbehaviour of the other factors at large radii, making the global contribution ofall 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 firstcase, the density is maximum in the center and falls off smoothly. In the lattercase, the density is null in the center since Lz = 0 for R = 0. In order to modelreal stellar systems, we need components with β = 0 to have some mass in thecenter. However, in a real galaxy, the maximum number of stars occurs in theintermediate region where the bulge meets the disk: this justifies the utilizationof components with β > 0 when modeling real stellar systems. We see themaximum 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 ina smaller region of configuration space.

5.3.5 The parameter η

For a given Lz, the largest value of the factor (E − Sz0)η 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 almostcircular 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 negatives, the orbits cannot reach the height z0 above the galactic plane. We see onFig. (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(E − Sz0 + sI3)δ 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 massdensity) 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 themaximum number of stars moves away from the center and that the mass ismore 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 toE = S0) is much higher for η = 8.

119

Figure 5.9: Contour plots of the configuration space density (the massdensity) 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 z0is reached only in the case s = 0. The more negative s, the more the mass isconcentrated 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 thedispersion 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 anynonzero s will produce a ratio σz

σRless than 1 (Fig. 5.10). The ratio is closer

to unity in the center than in the outer regions: this indicates the physicallyrealistic character of our components. For s = 0, we find σz = σR since we aredealing 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 thestars to stay close to the plane by favouring low I3-values. As we can see onFig. (5.11), a component with a larger δ has more stars in the galactic planeand shows a steeper decline with respect to z.

120

Figure 5.10: This figure displays the ratio σz

σRof several components (in the

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 (inarbitrary units) as a function of the height above the Galactic plane at R = 1kpc for varying δ. A rising δ implies a steeper decline. The other parametershave the same values as in Fig. 5.4 (with z0 equal to 2 kpc).

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5.4 Modeling

The component distribution functions described in this Chapter are very usefulas basis functions in the method described by Dejonghe (1989), in order to find,in a given potential, a three-integral distribution function that reproduces anyobservable quantities of a population of stars (spatial mass density, velocitydispersions, average radial velocities on a sky grid,...).

As an illustration, we present the application of the method to fit a givenspatial density ρ0(R, z) (see Batsleer & Dejonghe 1995 for a similar applicationin the two-integral approximation). We look for a linear combination of ourcomponents ∑

Λ

cΛFΛ (5.29)

that fits ρ0(R, z), with Λ = (α1, α2, β, δ, η, z0, s, a) and cΛ the coefficients thatare 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:

χ2 =∑i

[(∑Λ

cΛµ(Λ)0,0,0(Ri, zi)

)− ρ0(Ri, zi)

]2

(5.30)

This minimization, together with the constraint that the distribution functionmust 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 resemblesthat of a real disk, i.e. a van der Kruit law, for which the vertical disributionis a good compromise between an exponential and an isothermal sheet (Spitzer1942, van der Kruit 1988).

ρ0(R, z) ∝ exp(− R

hR

)sech

(z

hz

)(5.31)

In order to have a zero derivative with respect to R on the rotation axis, weadopt a mass density that follows closely the van der Kruit law, without a cuspin the center (see also Batsleer & Dejonghe 1995):

ρ0(R, z) =1 + 2R/hR1 +R/hR

exp(− R

hR

)sech

(z

hz

), (5.32)

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 thatcontains the stellar disk we want to model. Here, we adopt potential II of Table4.5.

The first step in the actual modeling consists in the selection of a subsetof components out of the (infinite) set of possible components. This subset ischosen so that certain features, that we suppose to be present in the stellar disk,

123

0 2 4 6 8 10-3

-2

-1

0

1

radial position

Figure 5.12: Data for a van der Kruit disk (hR = 3kpc and hz = 0.25kpc) weregenerated in the region 0 kpc ≤ R ≤ 10 kpc and 0 kpc ≤ z ≤ 1 kpc. The initialsubset 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 massdensity of the components linear combination at these heights. The only regionof our disk where the fit deviates a little from the data is the 2 kpc centralregion: our components are primarily intended to describe the outer regionsrather than the central region of the Galaxy since there is in reality a strongdeviation from axisymmetry in the central region (bar, see Section 1.2.7), anda supermassive black hole (cusp).

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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 thefit 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 tubeorbits and is smaller than S0 (see Eqs. (4.6) and (4.7)).

Figure 5.14: The left panel displays the ratio σz

σRin the galactic plane for the

fit obtained in Fig. 5.12. The right panel gives the shapes of the individualvelocity dispersions curve σR (solid line) and σz (dotted line) in the galacticplane.

125

such as circular orbits, are included. For example, we expect the mass densitycorresponding to a component to have an exponential behaviour close to themass 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 subsetthat produces the lowest minimum for that function (5.30). Then the programiterates, selecting and adding at each iteration the component which, togetherwith the components already chosen in a previous run, produces the best fit.Once the minimum of the χ2-variable does not change significantly any morewith the addition of extra components, the program is halted because too lowa value for χ2 could imply that the QP program starts producing a distributionfunction featuring unnecessary oscillations.

As an example, we model a modified van der Kruit disk with hR = 3kpcand hz = 0.25kpc. Batsleer & Dejonghe (1995) already showed that a linearcombination of two-integral components (with s = 0 and δ = 0) could fit such adisk, but with σR = σz. In order to model real anisotropic velocity data in thefuture, the dependence on the third integral will be needed. We show that, bychoosing 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 fitwith 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 Stackel dynamics to model the galactic disk, we have constructeda completely explicit and analytic distribution function, with an explicit depen-dence on the third integral. Fig. 5.13 displays the distribution function obtainedby QP as a function of E, for Lz = 0.1 and for two values of I3 (I3 = 0 andI3 = 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 (seeEqs. (4.6) and (4.7)). We see on Fig. 5.13 that the distribution function isdecreasing with increasing I3 (particularly near E = Sz0), and that it has someclumps. These clumps at I3 = 0.05 are not discontinuities since the distributionfunction is a linear combination of continuous components.

Many different three-integral distribution functions correspond to a givenspatial density, and there is no guarantee that they will yield realistic velocitydispersions. It is a major result of this Chapter to show that it is possible to finda linear combination of our components yielding realistic velocity dispersions.Fig. 5.14 displays the ratio σz

σRin the galactic plane: at the radius corresponding

to the Solar position in the Milky Way (7.5-8.5 kpc), the classical value ofσz

σR' 0.4 is obtained. The local maximum in the σz

σRcurve is due to the

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 Stackelpotentials 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 〈vr〉 on thesky, 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 constructeda model with basis functions of Abel type (Dejonghe & Laurent 1991), lessadapted 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 (withrespect to Toomre criterion, Eq. (1.34)) when modeling real data. Once thedistribution function of the population will be found in integral space, we couldcompare 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, seeSection 3.3.6) to refine the solution.

We could iterate the processus with the five potentials of Table 4.5 and seewhich one leads to the best fit. We would thus have indirect constraints on themass distribution of the Milky Way. Moreover, the solution would yield idealinitial conditions (more complex than a simple two-dimensional Schwarzschildvelocity ellipsoid, see Fig. 5.15) for three-dimensional N -body simulations in theSolar neighbourhood, in order to study the formation of the dynamical streamsobserved in Chapter 3.

127

Figure 5.15: The velocity distribution at Solar position in the Galactic planefor the three-integral model of Dejonghe & Van Caelenberg (1999). We seethat some axisymmetric substructure is present in velocity space, even if it doesnot correspond to the regions of the dynamical streams of Chapter 3 (see Fig.3.3 with U ' −vR and V ' −vφ − 220 km s−1, without correcting for theunknown Solar motion). Nevertheless, it is clear that the initial conditions fora perturbation analysis should be more complex than a Schwarzschild velocityellipsoid.

128

Chapter 6

Conclusions andperspectives

This thesis presented a kinematic analysis of 5952 K and 739 M giants in theSolar neighbourhood which included for the first time radial velocity data froman important survey performed with the CORAVEL spectrovelocimeter. Wealso used proper motions from the Tycho-2 catalogue, and parallaxes from theHipparcos catalogue.

First, we analyzed the kinematics of the sample restricted to the 2774 starswith parallaxes accurate to better than 20%, and then we made full use of the6030 available stars and evaluated the kinematic parameters with a Monte-Carlomethod. We found that the asymmetric drift is larger for M giants than for Kgiants due to the fact that the M giants must be a little older than the K giantson average. We also found the usual value for the Solar motion when assumingthat the whole sample has no net radial and vertical motion.

Then a maximum-likelihood method, based on a bayesian approach (Luri etal. 1996, LM method), has been applied to the data and allowed us to derivesimultaneously maximum likelihood estimators of luminosity and kinematic pa-rameters, and to identify subgroups present in the sample. Several subgroupshave 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 verywide range of ages for stars belonging to these subgroups. This result excludedthe classical hypothesis which views them as cluster remnants. Moreover, sinceit is unlikely that two or three mergers (leading to the Hyades-Pleiades andSirius superclusters) have left such important signatures in the disk near theposition of the Sun, we concluded that the substructure in velocity space ismost 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 “dynamicalstreams”.

129

130

A possible explanation for the presence of open clusters and associations inthe same area of the UV -plane as those streams is that they have been put thereby a spiral wave, while kinematics of the older stars of our sample have also beendisturbed by the same wave. It would be interesting to use N -body simulationsto check wether a cluster could be displaced by a spiral wave without beingdisrupted. The slightly supersolar metallicity of the Hyades-Pleiades stream(also reported by Chereul & Grenon 2001) suggests that this stream originatesfrom a specific galactocentric distance and that it was perturbed by a spiralwave at a certain moment, radially pushed by the wave, and sent in the Solarneighbourhood (see Sellwood & Binney 2002). This would explain why thisstream is composed of stars sharing a common metallicity but not a commonage. A careful metallicity analysis of this stream would be of great interestin order to confirm this scenario. If this scenario is correct, the radial mixingshould be taken into account in any future model of the chemical evolution of theGalaxy (see e.g. Koppen 2003 for older models). The Sirius moving group couldalso be a feature recently formed by the passage of a transient spiral, while theHercules stream would be related to the bar’s outer Lindblad resonance (Dehnen1999, 2000, Fux 2001). The position of all these streams in the UV -plane isresponsible for the vertex deviation of 16.2 ± 5.6 for the whole sample. Weeven argue that the vertex deviation observed among large samples of early-typestars (see Dehnen & Binney 1998a) and the specific kinematic initial conditionsof some young open clusters and OB associations could in fact have the samedynamical origin as those streams of giants. A better understanding of thestreams should start with a chemical analysis of the stars belonging to them.As a first step, their photometric indices could be investigated. An importantconsequence of the dynamical origin of the streams is that it makes no sense tointegrate backwards over Gyrs in a smooth stationary axisymmetric potentialthe orbits of the stars belonging to them. This warning may even apply to mostold disk stars, as they may be expected to have been dynamically disturbed bya transient spiral wave at least once within their lifetime (which could explainthe absence of age-metallicity relation in the Solar neighbourhood; Edvardssonet al. 1993).

We then discussed, in the light of our results, the validity of older estimationsof the Solar motion in the Galaxy. Indeed, the group of background stars (groupB, see Section 3.3.6) has a distribution in velocity space close to a Schwarzschildellipsoid but is not centered on the classical value found for −U in Section 3.2when considering the full sample, including the streams. Instead we find 〈U〉 =−2.78±1.07 km s−1. This discrepancy clearly raises the essential question of howto derive the Solar motion in the presence of dynamical perturbations alteringthe kinematics of the Solar neighbourhood (the net radial motion of stars in theSolar neighbourhood can be of the order of 10 km s−1 in the simulation of DeSimone et al. 2004): does there exist in the Solar neighbourhood a subset ofstars having no net radial motion which can be used as a reference against whichto measure the Solar motion? We do not have the answer to that question, butwe have shown that the reliability of the older estimations of the Solar motion(Dehnen & Binney 1998a, Bienayme 1999, 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 ofphase space in the Solar neighbourhood cannot be interpreted in terms of anaxisymmetric steady-state model. Nevertheless, an axisymmetric model reveal-ing all the fine structure of the axisymmetric distribution function in the Solarneighbourhood would give ideal initial conditions (more complex than a simpletwo-dimensional Schwarzschild velocity ellipsoid) for three-dimensional N -bodysimulations that could afterwards reproduce some non-axisymmetric featuresobserved in the Solar neighbourhood (see Fux 1997, De Simone et al. 2004 fortwo-dimensional simulations). Indeed, the weak point of all the recent simu-lations is the choice of the initial conditions, especially for three-dimensionalsimulations 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 HyPlgroup in Section 3.3.6). We developed new tools to establish such a model inthe last two Chapters.

To make an analytic model possible, we chose to deal with a special kind ofpotential (Stackel potentials) for which all orbits admit three analytic integralsof the motion. We have shown that some different simple Stackel potentialscan fit most known parameters of the Milky Way (especially Hipparcos latestfindings). We have generalized the two-component potentials of Batsleer & De-jonghe (1994) by adding a thick disk, and we have studied how the parameterscan vary in order to satisfy selection criteria based on the latest observationalconstraints. 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 andwe have selected five different valid potentials listed in Table 4.5. It should benoted that, in fact, there could be two different thick disks in the Galaxy, a verythick one (Chiba & Beers 2000; Gilmore, Wyse & Norris 2002) and a thinnerone (Soubiran, Bienayme & Siebert 2002): in that case, the three-componentmodeling presented in this thesis could be easily extended, but this would implya growth of parameter space. A major result of Chapter 4 is that, even thoughStackel potentials are negligible in the set of all potentials, many of them are stillable 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 ofTable 4.5) were already used by Vlemmings et al. (2004) to infer the Galacticorbit of two pulsars over a few Myrs.

Then, we have constructed new analytic three-integral stellar distributionfunctions F (E,Lz, I3) yielding σR 6= σz: they are generalizations of two-integralones that can describe thin disks with the restriction that σR = σz (Batsleer &Dejonghe 1995). We first reduced the triple integral defining their moments toa simple one, by making some assumptions on the parameters. Then we lookedfor the effects of the different parameters and showed the disk-like (physicallyrealistic) features of our distribution functions: they have a finite extent in ver-tical direction and an exponential decline in the galactic plane, while favouringalmost 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, byvarying the parameters responsible for this dependence (s and δ). We showedthat 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 realisticanisotropy in velocity dispersions. This implies that they are very promisingtools to model real data with σR 6= σz by using the quadratic programmingalgorithm 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 innerGalaxy (already used by Dejonghe & Van Caelenberg 1999). The axisymmetricvelocity distribution in the Solar neighbourhood deduced from such a modelcould be compared with the velocity distribution in Figs 3.1 and 3.3. Then wecould use this model to create realistic initial conditions for N -body simula-tions in order to understand the exact mechanisms that lead to the observeddynamical streams in the Solar neighbourhood. Moreover, we could see whichof the five potentials of Table 4.5 yields the best fit when modeling those realdata, and thus have indirect constraints on the overall mass distribution of theGalaxy, which is still ill-determined.

In conclusion, the results of our kinematic analysis of giant stars in theSolar neighbourhood led to a profound revision of our understanding of “su-perclusters” (or “moving groups”), renamed “dynamical streams” since theyare most probably related to the dynamical perturbation by non-axisymmetriccomponents of the Galaxy. A better understanding of those streams needs moretheoretical investigations in the field of galactic dynamics. Our contribution totheoretical research was to develop new tools (potentials and distribution func-tions) for axisymmetric dynamical modeling: indeed, an accurate axisymmetricmodel of the Galaxy is a necessary starting point in order to understand thetrue effects of non-axisymmetric perturbations, such as spiral waves, that arethought to lead to the formation of the “dynamical streams” exhibited in thisthesis.

Appendix A

Contents of the data table

Table A.11 (on the CD-ROM attached to this thesis) contains 6691 lines, andcontains the following information in the successive columns (note that missingdata 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 H9of 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 foundin columns 9 to 12; the kinematic study made then use of the Hipparcosproper motions instead of the Tycho-2 ones. In most cases, the absenceof a Tycho-2 proper motion is caused by the fact that the star has a closevisual 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-2proper motions: ∆µ = |µHip−µTyc2|/εµ, where µ = (µ2

α cos2 δ+µ2δ)

1/2 andεµ = (ε2µHip

+ ε2µTyc−2)1/2, where εi denotes the standard error of quantity

i. The quantity ∆µ may be used as a diagnostic tool to identify long-period binaries (Kaplan & Makarov 2003). The basic idea behind thistool 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 recordedby Hipparcos is in fact the vector addition of the true proper motion and ofthe 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 (beyondthe combined error bar encapsulated by εµ) very likely hints at the binarynature of the star. This diagnostic will be fully exploited in a forthcomingpaper devoted to the binary stars present in our sample. Note that Fig. 1presents the histogram of ∆µ, which indeed reveals the presence of anabnormally large tail at ∆µ ≥ 1.5.

18-19 Hipparcos Hp magnitude and associated standard error.

20 Tycho-2 VT2 magnitude.

21 Hp−VT2. For visual binaries (flag 4 in column 24), this colour index has notbeen listed and the value 0.0 is given instead, because the Hp magnitudeappears to be a composite value for the two visual components. In mostof these cases, the VT2 magnitude of the visual companion is given in thelast column of the table. For large amplitude variable stars, the Hp−VT2

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 H40field of the Hipparcos catalogue does not provide a directly measuredquantity. It has instead been computed from various colour transforma-tions based on the B−V index from field H37, which neither is a directlymeasured quantity.

23 V − I colour index in Cousins’ system as provided by the colour trans-formation based on the measured Hp − VT2 colour index (Platais et al.2003). This value is thus in principle more reliable than the HipparcosH40 value (note, however, that the colour transformation provided byPlatais et al. (2003) had to be extrapolated somewhat, since it is pro-vided in the range −2.5 ≤ Hp − VT2 ≤ −0.20, whereas our data set goesup to Hp − VT2 = 0.1). It has been used to draw the HR diagram of thesample. For large-amplitude variable stars, the median V − I index hasbeen taken directly from Platais et al. (2003), instead of being computedfrom the colour transformation based on Hp − VT2 (set to 0. in thosecases).

24 Binarity flag:*: no evidence for radial-velocity variations;0: spectroscopic binary (SB), with no orbit available. The star had to bediscarded 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’scenter-of-mass velocity. The reference with the orbit used to derive thecenter-of-mass velocity is listed in the last column. If no reference isgiven, the center-of-mass velocity has been estimated from the availableCORAVEL 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 6′′ away (as listed by theTycho-2 catalogue);5,6,7,8: as 0,1,2,3 but for a visual binary. It should be stressed herethat visual binaries have not been searched for exhaustively among ourtarget stars. Whenever the Tycho-2 catalogue lists a companion star lessthan 6′′ away from the target star, the binarity flag has been set to 4.In most of these cases, the Hipparcos Hp magnitude corresponds to thecomposite magnitude and the Tycho-2 proper motions are identical forthe two components. The VT2 magnitude of the companion is listed in thelast 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 codeof the reference providing the orbit used), and standard error (set to0.3 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 distancelisted 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 modelof Arenou et al. (1992).

31-33 The U, V and W components of the heliocentric space velocity deducedfrom the LM method (corrected for the galactic differential rotation tofirst 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: Bibliographiccode (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 code2005A&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. Anasterisk 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, asthey were derived by the LM method. They depend upon, e.g., the particularchoice for the values of the Oort constants, the interstellar extinction model,the a priori choice of the various distribution functions,...

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