spiral arm kinematics for milky way stellar populations

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5Mon. Not. R. Astron. Soc. 000, 1–?? (2014) Printed 18 December 2015 (MN LATEX style file v2.2)

Spiral arm kinematics for Milky Way stellar populations

S. Pasetto 1⋆, G. Natale 2, D. Kawata1, C. Chiosi3 & J. A. S. Hunt11University College London, Department of Space & Climate Physics, Mullard Space Science Laboratory, Holmbury St. Mary,

Dorking, Surrey, United Kingdom2Jeremiah Horrocks Institute, University of Central Lancashire, Preston, PR1 2HE, UK3Department of Physics & Astronomy,“Galileo Galilei”, University of Padua, Padova, Italy

Accepted . Received

ABSTRACTWe present a new theoretical population synthesis model (the Galaxy Model) to ex-amine and deal with huge amounts of data from surveys of the Milky Way and todecipher the present and past structure and history of our own Galaxy.

We assume the Galaxy to be made up of the superposition of many compositestellar populations belonging to the thin and thick disks, the stellar halo and thebulge, and to be surrounded by a single dark matter halo component. A global modelfor the Milky Way’s gravitational potential is built up to secure consistency with thedensity profiles from the Poisson equation. In turn, these density profiles are usedto generate synthetic probability distribution functions (PDFs) for the distributionof stars in colour-magnitude diagrams (CMDs). Finally, the gravitational potentialis used to constrain the stellar kinematics by means of the moment method on a(perturbed)-distribution function.

The Galaxy Model contains also a star-count like description of non-axisymmetricfeatures of the Galaxy such as the spiral arms, thus removing the axisymmetric crudeassumptions commonly made in most of the analytical descriptions. Spiral arms per-turb the disk distribution functions in the linear response framework of density-wavetheory where we present an analytical formula of the so-called “reduction factor” interms of Hypergeometric functions. Moreover, we consider a non-axisymmetric modelof extinction to build CMDs in the presence of arbitrary non-axisymmetric features

This galaxy model represents the natural framework to investigate surveys such asGaia-ESO, SEGUE, APOGEE as RAVE as well as the upcoming Gaia data releases.We extend here the model to include a non-axisymmetric treatment of large fieldsof view where the gradients of the underlying stellar density distribution plays a keyrole. At the same time, we introduced a number of critical improvements to find thebest parameter values to represent mock data-surveys, including a recently developedalgorithm based on the concept of probability distribution function to handle colourmagnitude diagrams with a large number of stars and a genetic algorithm to investigatethe parameter space.

Key words: Milky Way kinematics, stellar populations, Gaia

1 INTRODUCTION

The Milky Way (MW) provides and unique environment inwhich to study the origin and evolution of galaxies on astar-by-star basis, with a precision that is simply impossi-ble to reach for any other galaxy in the Universe. The Eu-ropean Space Agency’s cornerstone mission Gaia, togetherwith complementary ground-based spectroscopic follow-upssuch as the Gaia-ESO Survey (e.g., Gilmore et al. 2012),

⋆ E-mail: [email protected]

will map the stellar distribution of the MW with unprece-dented accuracy by providing high-precision phase-space in-formation, physical parameters, and chemical compositions,for roughly one billion of the stars in our Galaxy. The ex-ploitation of this huge amount of data cannot be made us-ing the methods and tools that have been used for manydecades to study much less numerous samples of stars; itrequires the development of, and experience with, cutting-edge multi-dimensional data mining tools, as well as sophis-ticated methodologies to transfer the models from the spaceof “simulations” to the “plane of observers”.

c© 2014 RAS

http://arxiv.org/abs/1512.05367v1

2 S. Pasetto, G. Natale, D. Kawata, C. Chiosi & J. A. S. Hunt

Star-count techniques are born with the aim to answera simple astronomical question: why do we see a given distri-bution of stars in the sky? Since the oldest approach to thestar-count equation(e.g., Trumpler & Weaver 1953), thesetechniques have represented the most natural way to inves-tigate the closest distribution of stars to us, i.e. the MilkyWay. A major advancement of these techniques was achievedby Bahcall & Soneira (1984) who applied the concept ofstellar populations to the solar neighbourhood (see alsoBahcall 1984a,b) and nowadays, more theoretically sophis-ticated star-count models are the standard tools to investi-gate the MW stellar distribution (e.g., the Besanon model,Robin et al. 2003). The ultimate step toward the under-standing of our Galaxy is thus represented by the extensionof the concept of stellar populations to include kinematics,dynamics, photometric and chemical properties together ina global MW modelling approach (e.g., Mendez et al. 2000;Vallenari et al. 2006).

The star-count techniques have the goal to syntheticallyreproduce the observables obtained from an (unknown)-stellar distribution function (DF), i.e. the number of starsin a given range of, e.g., temperature, velocities, densities,proper motions etc., by considering the data distributionin the space of the observable quantities (e.g., photometry,proper motions, radial velocities, etc.). To achieve this goal,a number of founding pillars must be assumed to exist ona global scale, e.g., density-profile laws, star-formation his-tories, age-metallicity relations, age-velocity dispersion re-lations etc. All these relations will ultimately represent away of deciphering and constraining the MW history andevolution.

The more independent constraints a model can repro-duce, the closer these underlying relations are to the trueproperties of the system analysed (the MW in our case).The star-count techniques are a Monte-Carlo type solutionto a multidimensional integration problem of the star-countequation. Historically, in classical textbooks of statistical as-tronomy (Trumpler & Weaver 1953) the star-count equationis generalized to include the kinematics as follows:

dNjdΓdm∆λdCλλ′

= Njfj (Γ) , (1)

where Nj is the number of stars for each given stellar popu-lation, j, with distribution function fj (Γ) in the elementalvolume of the phase space dΓ = dx, dv = dΩdrhel, dv.Here rhel is the heliocentric distance of the stars in an in-finitesimal interval of magnitude dm∆λ in the band ∆λ andcolour dC = m∆λ−m∆λ′ . In Section 2 we will review a gen-eralized framework for Eq.(1) introduced in Pasetto et al.(2012) to recover Eq.(1) as a special case of a multidimen-sional marginalization process.

Two of the major limitations underlying many theo-retical works based on analytical expressions for the DFfj (x,v) are the time independence of fj and its axisym-metry properties in the configuration space. Related to thefirst assumption is the problem of self-consistency: the DFsare not obtained by sampling the phase-space of a systemevolved in time under the effect of self-gravity. In this ap-proach, the DF is not numerical but a parametric function.The second assumption of axisymmetry is led by the neces-sity to keep the treatment of the dynamical evolutions assimple as possible: the corresponding Hamiltonian is cyclic

in some variables and hence more suited for analytical ma-nipulation.

The literature is full of alternatives to overcome thesetwo limitations, e.g., N-body simulations, the Schwarzschildmethod, the Made-to-Measure method, full theoreticalmethods (e.g, Hunt & Kawata 2013; Hunt et al. 2015;Cubarsi 2007; Bienayme et al. 2015; Bienayme & Traven2013) etc. whose review is beyond the goal of this paper.In this work, we will relax the axisymmetry assumption forthe sole thin disk components by implementing a pertur-bative approach carried out to the linear order on suitablesmall parameters to the equation of motion following two dif-ferent works by Lin et al. (1969) and Amendt & Cuddeford(1991). These perturbative linear response frameworks arethe only analytical treatment available up to now that canclaim observational validation.

The perturbative treatment of Amendt & Cuddeford(1991) deals with mirror symmetries about the plane ofthe Galaxy. It has been introduced in the technique we areadopting from (Pasetto PhD Thesis 2005, Vallenari et al.2006) where more detail has been given as to its implemen-tation and to comment about its observational validation.Nowadays, this work represents a good balance between sim-plicity and robustness. More recent formulations can be in-vestigated in the future (e.g., Bienayme 2009).

The treatment of Lin et al. (1969) is referred as DensityWave theory (DWT) and we will review in what follows theliterature that attempts to validate it from the observationalpoint of view.

The history of the attempts to find an explanation ofthe spiral features of the MW and external galaxies is longstanding and still in debate. We recall here (without thepresumption to be complete) a few works of observationalnature that inspired our star-count implementation of thisDWT. The interested reader can look at books such as Shu(1991) or Bertin (2014).

1.1 Observational studies of Density wave theory

The theoretical foundation for the treatment of hydrody-namics of collisionless rotating systems are present in theliterature since the middle sixties (e.g., Marochnik 1964;Lin & Shu 1964; Marochnik 1966, 1967). Nevertheless, thelinear-response-theory for the perturbation of an axisym-metric stellar systems to the perturbation in the presence ofspiral patterns in a flat disk started in 1968/69. In just twoyears the basis for the theoretical interpretation of the MWspiral features in the form of density waves had been settledby the works of Marochnik & Ptitsina (1968) and Lin et al.(1969) (see also Marochnik & Suchkov 1969b,a).

The existence of a theory interpreting the spiralarm phenomenon spurred many research groups to findobservational evidence that could either support or denysuch a theory. The first attempt to interpret the meanproperties of observational velocity fields of young starsin terms of the DWT was by Creze & Mennessier (1973).Creze & Mennessier (1973) set up a method to interpretthe observations in terms of the DWT based on two simpleingredients: a multidimensional parametric fit and anasymptotic expansion on small parameters of the basicequations governing the kinematics of the DWT. Thisseminal study spurred many other studies in which different

c© 2014 RAS, MNRAS 000, 1–??

Milky Way galaxy model 3

results were obtained mainly due to either the adoptedmultidimensional fitting procedure or the large numberof involved parameters or the different data sets in usageand their local/non-local nature in the configuration space.Local models of the velocity space have been consideredwith asymptotic expansions on different small param-eters (e.g., Nelson & Matsuda 1977; Brosche & Schwan1981; Byl & Ovenden 1981; Comeron & Torra 1991;Mishurov et al. 1997; Mishurov & Zenina 1999;Fernandez et al. 2001; Garcıa-Sanchez et al. 2001).

Nowadays this research is still far from being com-plete (see, e.g., Junqueira et al. 2013; Griv et al. 2013, 2014;Vallee 2014; Roca-Fabrega et al. 2014). Recent studies con-sider more complex models based on four spiral arms (e.g.,Lepine et al. 2001) and their connections with the patternof chemical properties of the MW (e.g., Lepine et al. 2003;Andrievsky et al. 2004) and the not monotonic features ofthe MW rotation curve (e.g., Barros et al. 2013). Finally,this research field has been recently boosted by numeri-cal simulations. N-body solvers are achieving higher andhigher resolution and although they are still missing a com-plete self-consistent understanding of the spiral arm dy-namics, several numerical techniques (e.g., the tree-code,Barnes & Hut 1986) allow us to simulate the gravitationalinteractions among millions of particles with masses of theorder of a few thousand solar masses or less (D’Onghia et al.2013; Grand et al. 2012b,a).

The logic flux of the paper is as follows. We first want topresent (Section 2) the concept of stellar population takenfrom a theory developed in its general form in Pasetto et al.(2012) and here adapted to the specific case of the MWstellar populations. This will allow us to generalize the pre-viously introduced concept of star-counts in a larger theo-retical framework, to set a few assumptions, and to empha-size the goals of this novel Galaxy Model. We present thenormalization of the star-count equation for a field of viewof arbitrary size in Section 3 and this allows us to definethe density profiles and the consequent MW potential shape(Section 4). This axysimmetric potential represents the ba-sis for the development of a self-consistent spiral treatmentpresented in the following section, but as explained above,the formulation adopted in our approach is fully analytical,hence parametric, and so are the density-potential coupleintroduced in Section 4. This leaves us with a large numberof parameters to deal with in order to model the MW. Insection 5 a genetic algorithm is introduced for the study ofthese parameters which are used to study the MW data sur-veys. This leads us to the setting of the MW axisymmetricpotential (Table 1) that represents the axysimmetric basisused to develop the spiral arms perturbation theory. Hence,in Section 6 the spiral arms formalism is presented withits implication for the density (Section 6.1) and the CMDs(Section 6.2) once an ad-hoc extinction model is considered(Section 6.2.1). The velocity field description is presented inSection 7. A direct comparison with the most popular Be-sancon model is detailed in Section 8 and the conclusionsare presented in Section 9.

2 THEORY OF STELLAR POPULATIONS

Robust mathematical foundations for the concept of stel-lar populations are still missing, but recently Pasetto et al.(2012) proposed a new formulation for it. We briefly sum-marize here the analysis of Pasetto et al. (2012) because itis the backbone of the population synthesis model we aregoing to describe here. This approach extends the classi-cal concepts presented in books as Salaris & Cassisi (2005)or Greggio & Renzini (2011) to include a phase-space treat-ment for the stellar populations. These definitions will becrucial for the modelling approach and to formally defineour goals. Moreover they will allow us to fix some assump-tions we exploited during our work. Hence, we proceed topin down here the more specific points that in the theoryproposed in Pasetto et al. (2012) are introduced in completegenerality.

2.1 Theoretical framework: EMW

We define every assembly of stars born at different time, po-sitions, with different velocities, masses and chemical com-position a composite stellar population (CSP). The space ofexistence for the Milky Way CSP, EMW is considered asthe Cartesian product of the phase-space Γ = (x,v) =(x1, x2, ..., x3N , v1, v2, ..., v3N ) (N number of stars of theCSP), the mass space M , and the chemical compositionspace Z, EMW ≡M ×Z ×Γ. The inclusion of the time t in-troduces the “extended”-existence space EMW × R. A moreformal geometrical definition of this space and its dimen-sionality for the interested reader is given in Pasetto et al.(2012). Because in the extended existence space the MWstars move continuously; losing mass, enriching in metalsand travelling orbits in the phase-space, we can safely de-fine a distribution for the CSP in EMW, say fMW

CSP ∈ R+ real

always positive function, under the assumption of continu-ity and differentiability, fMW

CSP ∈ C1(

R+)

. We consider nowa sample of identical MW-systems whose initial conditionspans a sub-volume of EMW, let us refer to it as the “MW-ensemble”. The number of these systems, dN , spanning amass range, dM , a metallicity range, dZ, and phase-spaceinterval, dΓ, at the instant t, is given by

dN = NfMWCSPdMdZdΓ, (2)

and the total number of systems in the ensemble is fixed,finite, and subject to the important normalization relation

∫

fMWCSPdMdZdΓ = 1. (3)

Pasetto et al. (2012) proceeded with a foliation of EMW inorthogonal subspaces of metallicity, dZ, and phase-spacealone, dΓ, to define a simple stellar population (SSP) asthese elemental units. A cartoon of the concept of SSPs ispresented in Fig. 1.

As evident, we can assume that the DF of a CSP canbe written as the sum of disjoined DF of SSP,

fMWCSP =

n∑

i=1

fSSP, (4)

where fSSP = fSSP (M,Z0,Γ0; t0) is the DF of a single stel-lar population born at time t0. In this framework we cangive a rigorous geometrical interpretation and definition of

c© 2014 RAS, MNRAS 000, 1–??


Figure 1. Cartoon representing the concept of a foliation ofa CSP over SSPs. SSPs result as an intersection (red rectan-gular area in the Figure) of planes of constant metallicity dZ

(light-brown-colour) and constant phase-space dΓ plane (light-green colour) with the CSP (blue). The SSPs are the fundamental“atoms” to build up the CSPs. Axes are in arbitrary units.

CSPs and SSPs (see Pasetto et al. 2012). As the time passes,the stellar population evolves. According to their masses thestars leave the main sequence and soon after die (supernovaephase) or enter into quiescent stages (white dwarfs phase)injecting chemically processed material into the interstellarmedium in form of supernovae remnants or winds. In thesame way, the evolution of a SSP in the phase-space obeysthe Liouville equation ∂fΓ

∂t= −ιL [fΓ] with L [∗] Liouville

operator and ι imaginary units. It is of interest for us to re-call a few tools which are useful in studying the MW. Withinthis framework we will make use of the following concepts:

• Present-day mass function. This is the result of themarginalization of fMW

CSP over the metallicity Z and phasespace Γ:

∫

NfMWCSPdZdΓ = ξ (M ; t) . (5)

This can be expressed, e.g., by the approximate relation

ξ (M ; t) =

ξ (M) tMS

τtMS < τ

ξ (M) tMS > τ,(6)

where τ = t − t0 is the age of the stellar population,tMS = tMS (M) is the age at which a star exits the mainsequence (MS), and ξ (M) is the initial mass function (IMF)of the MS stars. In our model all the IMFs commonly usedin literature can be considered, among which we recall theIMFs of Kroupa et al. (1993) and/or Kroupa (2001) as thenowadays most popular ones. Even if the present formalismallows us to investigate free-form IMFs with the multi-scalepower law method, we will limit ourselves to use the IMF ofKroupa (2001).

• Age-metallicity relation. By integration of the DF

over the mass, M , and the phase-space, Γ, we can define therelation,

∫

NfMWCSPdΓdM = χ (Z; t) , (7)

which gives the number of stars formed per metallic-ity interval at the time t. Although several studies havebeen devoted to investigate the age-metallicity relation(Rocha-Pinto et al. 2000, 2006; Pilyugin & Edmunds 1996),the small volume of the Galaxy covered by the data doesnot allow us to apply these age-metallicity relationships toa global scale model (Bergemann et al. 2014). The prob-lem becomes even more puzzling for specific stellar com-ponents such as the stellar halo (see, e.g., Leaman et al.2013, for the globular cluster case). Even though the age-metallicity relation of Rocha-Pinto et al. (2006) is includedin our model, we will not use it as a standard assumption.

• Phase-space DF and age-velocity dispersion re-lation. By marginalizing fMW

CSP over the mass and metallicitysub-space we can write the formal relation,

∫

MfMWCSPdMdZ = e−ιLtfΓ (Γ; t0) , (8)

whose analysis within the framework of a perturbative ap-proach of the DWT will be subject of this paper in the fol-lowing sections (ι is the imaginary unit). Here we anticipateonly that by taking the moments on the velocities of Eq.(8)we can obtain the age-velocity dispersion relation:

σv (x; t) ≡∫

d3Nv(v − v)⊗2

∫

MfMWCSPdMdZ

=

∫

(v − v)⊗2e−ιLtfΓ (Γ; t0)d3Nv,

(9)

with a⊗n a standard tensor n-power of the generic vectora accounted for its symmetries. A simplified version of thisrelation for a collisionless stellar system (i.e. where the Liou-ville operator introduced above is simplified with the Boltz-mann operator for collisionless stellar dynamics) are imple-mented in our model with data interpolated from the valuesof the work of Pasetto et al. (2012a) and Rocha-Pinto et al.(2004) (see Eq.(44)).

Finally, by extension of the previous integral formalism ofEq. (5), (7) and (8) we introduce the following relations ofinterest to us:

• Metallicity/phase-space relationship. This rela-tion is formally defined by η (Z,Γ; t) ≡

∫

fMWCSPdM and more

interestingly we can project it onto the configuration space:

η (Z,x; t) =

∫

fMWCSPd

3NvdM. (10)

There is indeed evidence of the presence of chemical radialgradients in the configuration space of the MW thin diskcomponent (see, e.g., Boeche et al. 2014, 2013) and it canbe eventually implemented on the thick disk (Curir et al.2014).

For completeness we remind the reader that the stellar-mass/metallicity relation can easily be defined and imple-mented in our model as presented in Pasetto et al. (2012)once a larger sample of asteroseismology data becomes avail-able (see references in Section 1).

The goal of our research is to develop a technique to

c© 2014 RAS, MNRAS 000, 1–??


investigate EMW, the existence space of the MW, through

the relations that result from the projection of the unknown

fMWCSP in the mentioned subspaces of Eqs.(6), (7), (8) and

(10).To this aim, we need to relate EMW to the space of ob-

servations. This is made possible by the star-count equation,Eq.(1), in combination with Eq.(2) in the space of the ob-servational data. To this aim, one has also to solve a crucialpoint of difficulty with Eq.(1), i.e. the large fields of viewthat are often involved.

3 STAR-COUNT EQUATION FOR LARGESKY COVERAGE

Nowadays and increasingly in the future, we face the chal-lenge of of large sky coverage surveys where the gradientsof the underlying MW stellar density distributions sensiblyvary across the covered survey area. Already large surveys(SDSS, RAVE, SDSS, etc.) present these characteristics, andthe ongoing whole sky survey by Gaia, due to the depth ofthe magnitude limit and the amplitude of the solid angleconsidered (dΩ = 4π), will provide us an enormous amountof data in order to be considered. If the survey spans a largesolid angle dΩ and has a very deep magnitude limit, then thenumber of stars per f.o.v. becomes large and its realizationon a star-by-star basis becomes unpractical. For example,the marginalization of a fMW

CSP over dΓ for large-scale surveydata produces a section over dZ × dM , i.e. a Hertzsprung-Russel or a colour magnitude diagram (CMD) that can beover-dense: to realize it graphically we should draw dots-over-dots and count them. This process should be repeatedevery time we change a single parameter to see the effectof the variation until suitable fitting is achieved. To surpassthese CMD realization problems, Pasetto et al. (2012) pre-sented a novel technique able to substitute the generationof synthetic stars with the computing of a PDF. The con-volution of several SSPs along a line-of-sight (l.o.s.), thanksto Eq.(4), was then substituting the Monte-Carlo genera-tion of stars for a f.o.v., de-facto changing the concept of a“star-count” model with a probability distribution function(PDF) model.

We adopted here the same technique to speed up thegeneration of the fMW

CSP , eventually walking back to a star-count type of model by populating the PDF obtained forfMWCSP only if required. The stellar SSP database used to

build the fMWCSP is the same adopted in Pasetto et al. (2012),

though any other SSP database can be easily implementedvirtually making the modelling approach independent of anyparticular stellar physics recipes adopted by one or anotherresearch group (rotation, overshooting, α-enhancement, he-lium enrichment etc.). More details of this is described forthe interested reader in Pasetto et al. (2012).

Nevertheless, this process of populating the PDF foreach f.o.v. (that can be as large as the full sky) has to betreated with attention because of the normalization relationEq.(3). In particular, the number of stars generated alongthe l.o.s. and appearing in the final CMD has to correctlyaccount for the underlying mass fraction of each stellar com-ponent j of the Galaxy.

Historically, to deal with Eq.(1), or its generalized formin Eq.(2), the approach was based on the sum of several

close f.o.v. of negotiable opening angle. It was required fordΩ to be very small as well as the number of stars perpopulation Nj . The result of these assumptions was thatthe underlying density distributions within dΩ were to agood approximation constant (if the survey was not toodeep in magnitude and hence rhel not too large). To solvethe star-count equation under these approximations wasa trivial exercise and in the past decades it has been in-deed employed by several works in this research area (e.g.,Robin et al. 2003; Mendez et al. 2000; Vallenari et al. 2006;Ng et al. 2002; Girardi et al. 2005, , and references therein).If the hypothesis of small dΩ = dldb cos b (with l and bGalac-tic longitude and latitude) is to be relaxed, the computingof this number has to be performed numerically as follows:

N =

∫

d3Nx

∫

d3Nv

∫

MfMWCSP (M,Z,Γ) dMdZ

=

∫

d3Nx

∫

d3Nve−ιLtfΓ (Γ)

=

∫

dΩ

∫

drhelJρ (x; t) ,

(11)

where J = r2hel |cos b| is the Jacobian of the transformationT between the system of galactocentric coordinates (O,x)to standard galactic coordinates (⊙, rhel, l, b):

T :

x = R⊙ − rhel cos b cos ly = rhel cos b sin lz = z⊙ + rhel sin b,

(12)

where R⊙ =√

x2⊙ + y2⊙ (with y⊙ = 0). After this inte-

gral is evaluated, the relative number of stars within a givenf.o.v. is obtained as a function of observable quantities (e.g.,the galactic coordinates) no matter how large the f.o.v. is(Fig. 2). Although rhel can be unbounded, in practice itis limited by the survey magnitude limits with Pogsons lawand the dust extinction by taking into account an extinctionmodel (Section 6.2.1). We point out how the cone-geometryof Fig. 2 for the volume

∫

dΩdrhel is of exemplificative na-ture. In practice, because every observed star has muchlarger uncertainty in distance rhel + δrhel than in angular

positionl ± δl, b± δb, i.e.∣

∣

∣

δrhelrhel

∣

∣

∣≫∣

∣

δll

∣

∣ and∣

∣

∣

δrhelrhel

∣

∣

∣≫∣

∣

δbb

∣

∣,

the mapping of the synthetically generated fMWCSP of every

survey has a different nature. l, b are not randomly gen-erated but assumed from the data that we want to anal-yse without errors while rhel is randomly generated withinrhel ± δrhel depending on the particular selection function.

4 MILKY WAY DENSITY DISTRIBUTIONSAND GRAVITATIONAL POTENTIALS

In the context of the theory of stellar population introducedabove we can simplify Eq.(8) and obtain the mass densityρ(x) as follows:

ρ =

∫

MfMWCSPdMdZd3Nv

=

∫

e−ιLtfΓ (Γ; t0) d3Nv

≃∫

e−ιBtfγ (γ; t0) d3v,

(13)

c© 2014 RAS, MNRAS 000, 1–??


Figure 2. True spiral arm stellar isocontour. The intersection ofthe yellow surface with the f.o.v. (e.g., violet or red) is the integralperformed in Eq.(11). Two arbitrary solid angles, in blue or red,intersect a single spiral arm SSP over its complicated densityprofile (orange), from a common solar position slightly outside

the plane of the galaxy. The relative contribution to the numberof stars in a given l.o.s. is the result of the intersection of theglobal CSP with the arbitrary cone of the l.o.s..

where in the last row of the equation we reduced the di-mensionality of the phase-space by remembering that it ispossible to show that the two-body relaxation time t2b (con-sidered in the approximation of independent-hyperbolic en-counters) is long enough to allow us to treat the Galaxy toa good approximation as a “collisionless” system. Hence wecan substitute the discrete stellar distribution with a con-tinuous density profile, and the Liouville operator L canbe substituted with the more simple Boltzmann operatorB [fγ ] ≡ ι H,fγ, with H one-particle Hamiltonian and∗, ∗ the Poisson brackets (ι is the imaginary unit).

Unfortunately the explicit form of fMWCSP is unknown (the

blue manifold in Fig. 1) or has to be inferred just from sim-ple theoretical considerations. For this reason, we decidedto base our modelling technique on the density distributionsof stars and dark matter. From the density profiles the po-tential and hence the kinematics is computed. Furthermore,from the same density profiles the relative number of starsper bin of colour and magnitude along a l.o.s. in the CMDis computed. This approach is not the only possible way toproceed in analytical modelling, but we are guided by theexplicit intention to present a model focused on the inter-pretation of the data, where the data are the protagonist inleading our understanding of the phenomenon “Galaxy”.

Therefore, it is of paramount importance to assign toeach component of the MW a plausible density profile ρ(x)to derive a correct global gravitational potential. In the fol-lowing, we present our treatment of the Poisson equationand hence the global gravitational potential of the MW.These results will represent the axisymmetric foundations ofour description the DWT of spiral arms. The gravitational

potential is derived for all components of the MW even ifwe will focus only on the disk components for which, thanksto their proximity, data of good quality can be acquired andaccurate descriptions are possible.

In our model, the location of the Sun is assumed to beat x⊙ = R⊙, φ⊙, z⊙ = 8.00, 0.00, 0.02 kpc in a referenceframe centred on the (yet unknown) mass barycentre of theaxisymmetric model of the MW we are going to build up.

As mentioned above in Eq.(4) we consider a multi-component model of stellar populations. For the ith com-ponent of the thin or thick disk, we implemented a doubleexponential form of the density profiles, that is, with an ex-ponential profile decreasing with Galactocentric radius andvertical distance from the plane. Alternative vertical profiles(power-law and secant-square) are available for investigationbut not breaks of the exponential profiles has been imple-mented (e.g., Pohlen & Trujillo 2006). Because we are goingto develop a kinematical model, no time dependence of thedensity profiles is assumed. Written in cylindrical coordi-nates to exploit the φ-symmetry the profile reads:

ρd (R,φ, z) = ρ⊙e−

R−R⊙

hR−

z−z⊙hz . (14)

This parametric formalism depends on the density at the so-lar neighbourhood ρ⊙ and two scale parameters: scale lengthhR and scale height hz for each stellar population consid-ered. It does not contain an explicit dependence on φ. Thepotential is conveniently expressed as function of one singleintegral with integrand depending on the Bessel function(e.g., Bienayme et al. 1987) being hence extremely rapid tocompute:

Φd (R,φ, z) = −4πGρ0h−1R

∫ ∞

0

J0 (kR)(

h−2R + k2

)−3/2

× h−1z e−k|z| − ke−h

−1z |z|

h−2z − k2

dk

(15)where J0 is the Bessel function of the first kind (e.g.,Abramowitz & Stegun 1972) and the scale parameter foreach component ith should be taken into account but omit-ted for the sake of simplicity. For the ith stellar halo compo-nent of the MW we follow the model proposed in Robin et al.(2003) because it is fine-tuned on the observational con-straints, i.e. it is simple in its form but phenomenologicallyjustified. In the original form Robin’s profile reads:

ρH∗ (r) =ρ0,H∗

r⊙

rα r > hrH∗

hαrH∗ r 6 hrH∗,(16)

where ρ0,H∗ is the central stellar halo density, and hrH∗ thescale length parameter. Because we are interested in the po-tential formalism of this density model, we compute its cor-responding potential solving Poissons equation in sphericalcoordinates and guaranteeing continuity (but not differen-tiability) to the formulation as follows:

ΦH∗ (r) =

4πGρ0,H∗

r−α⊙

r

(α+ 2)hα+3r,H∗ + rα+3

(α+ 2) (α+ 3)∧ r > hr,H∗

−2πGρ0,H∗3h2

r,H∗ − r2

3

(

hr,H∗

r⊙

)α

∧ r 6 hr,H∗,

(17)where the scale parameters dependence of the ith-componentof stellar halo is omitted. The only component that we can

c© 2014 RAS, MNRAS 000, 1–??


start from the potential shape is the dark matter compo-nent, because its presence is indirectly manifest but it is notdirectly observed. We select a simple balance between grav-ity and centrifugal forces for circular orbits to obtain thelogarithmic potential:

ΦDM (R,φ, z) ≡ v20,DM2

log(

h2R,DM +R2 + q−2z2

)

, (18)

where v0 is the scale velocity, hR,DM the scale length andq the flattening factor. The density profile can again be ob-tained by use of Poissons equation as:

ρDM =v20

4πG

q2(

h2r,DM

(

2q2 + 1)

+R2 + 2z2)

− z2

(

q2(

h2r,DM +R2

)

+ z2)2 . (19)

No compelling reasons exist so far to split Dark Matter inmore components. The central regions of the MW are likelypopulated by a non-phase-mixed, fragmented and dynam-ically young component of Dark Matter. The remote pos-sibility exists that streams of Dark Matter, the analog ofstellar streams, are also present. The issue is still a matterof debate (e.g., Yoon et al. 2011). In any case, we expect notto detect granularity in the Dark Matter distribution fromthe kinematic anomalies of the closest stars kinematics. Inview of this, the presence of granularity in the Dark Matterdistribution, mimicking Dark Matter streams, are neglectedin our model.

We complete the review by mentioning that optionallywe can include axisymmetric components adding a hot coro-nal gas (see, e.g., Pasetto et al. 2012, for a model includingit). This does not influence the closest stellar dynamics ofthe MW stars but in mass it is thought to contribute upto ∼ 5 × 1010M⊙ within ∼ 200 kpc from the MW galaxycentre.

A separate work is in preparation on the kinematicaltreatment of the central part of the Galaxy which is of coursevery important. Unfortunately, up to now the modelling ofthe bulge is still imprecise and a subject of debate. A recentfinding of Dekany et al. (2015) shows an example of the on-going research and constantly changing knowledge that wehave about the central regions of the MW. Nevertheless, inthe total potential a bulge component has to be accountedfor and we adopt the following spherical density-potential“couple” (Hernquist 1990):

ρb (r) =Mbhr,b

2πr(r + hr,b)3

Φb (r) = − GMb

r + hr,b,

(20)

where again the dependence of the scale parametersMb, hr,b, bulge mass and scale radius respectively, fromthe stellar bulge component is understood even though thesubscript is omitted. This series of equations represent thebasic potential in asysimmetric approximation. The chosendensity parameters that we are going to assume for theseprofiles are presented in Table 1.

These parameters are chosen in such a way that theynicely reproduce some important observational constraints(see also Appendix A of Pasetto (2005), and Pasetto et al.(2012b)):

4.1 Circular velocity

To date several studies have covered the most im-portant dynamical constraints on the MW potential,i.e. its rotation curve, from several data sets andwith different techniques, both for the total stellarrotation, for gas rotation or for single MW stel-lar populations (e.g., Deason et al. 2012; Xin & Zheng2013; Fermani & Schonrich 2013; Bhattacharjee et al. 2014;Lopez-Corredoira 2014; Deason et al. 2011; Levine et al.2008; Xue et al. 2008, to quote a few). From the potentialadopted here we obtained the rotation curve analytically as

vc =√

r ∂Φ∂r

, where the individual components are not diffi-

cult to evaluate. Using Eq.(18) we get for the dark mattercomponent:

v2c,DM =R2v0

2

R2 + h2r,DM

; (21)

from Eq.(17) for the stellar halo profile we have

v2c,H∗ =

4πGr2(α+3)

rrα⊙

∑

H∗

ρ0,H∗

(

rα+3 − hα+3r,H∗

)

hr,H∗ < r

4πG3

rrα⊙

∑

H∗

ρ0,H∗hαr,H∗ r < hr,H∗ ,

(22)where we assumed the same α∀H∗, H∗ ∈ N indexing thestellar populations, i.e. with a simple abuse of notation we

wrote v2c,H∗ =NH∗∑

H∗=1

v2c,H∗ with NH∗ number of stellar halo

populations implemented in Tab.1 (one in this case). For thebulge population from Eq.(20) we get

v2c,b = GR∑

b

Mb

(R + hr,b)2, (23)

with b ∈ N indexing the populations as above. The disk com-ponents are only slightly more complicated by the presenceof the Bessel function that can be nevertheless handled nu-merically (e.g., Abramowitz & Stegun 1972) from Eq.(15) inthe form:

v2c,d = 4πGR∑

d

ρ0,d

∫ ∞

0

dk1

hR,d(

h−2R,d + k2

)3/2

kJ1(kR)

h−1z,d + k

,

(24)where d ∈ N indexes the disk populations.

Finally in Figure 3, we present the velocity curves of thevarious components of the Galaxy according to the corre-sponding density profiles already discussed above and sum-marized in Table 1.

4.2 Oort functions

The slope of the rotation curve, locally related to the Oortsconstants, has long been known to depend on the local gascontent, which does not monotonically vary with the radiusand contributes significantly to the local gradient of the ro-tation curve (Olling & Merrifield 1998; Minchev & Quillen2007; Olling & Dehnen 2003). The profile of these functionsoutside the solar neighbourhood is what we refer to as “Oortfunctions”. We will present in the next section a map dis-tribution of the gas content in relation to dust distributionand extinction (Fig.8). In the future the estimation of theOort function will represent a challenge for large kinematic

c© 2014 RAS, MNRAS 000, 1–??


Table 1. Kinematic and dynamical properties of the MW components. The first two thin disk stellar components implement the spiral

arm treatment described in the text. Because a map of the metallicity gradients ∇x

[

FeH

]

is still uncertain, no standard default values

are assumed and they are used as free parameters.

Components Scale parameters ∆t[

FeH

]

σii⊙

[Gyr] [dex] [km s−1]

ρd, hR, hz⊙[

M⊙ kpc - 3,kpc,kpc]

Thin disk pop 1 (sp) 1.29 × 107, 2.57, 0.06 [0.1, 0.5[ [-0.70, 0.05[ 27.0,15.0,10.0Thin disk pop 2 (sp) 1.93 × 107, 2.59, 0.06 [0.5, 0.9[ [-0.70, 0.05[ 30.0,19.0,13.0Thin disk pop 3 4.96 × 107, 2.96, 0.07 [0.9, 3.0[ [-0.70, 0.05[ 41.0,24.0,22.0Thin disk pop 4 3.38 × 107, 2.99, 0.09 [3.0, 7.5[ [-0.70, 0.05[ 48.0,25.0,22.0Thin disk pop 5 3.34 × 107, 3.41, 0.25 [7.5, 10.0[ [-0.70, 0.05[ 52.0,32.0,23.0Thick disk 2.40 × 106, 2.23, 1.35 [10.0,12.0[ [-1.90,-0.60[ 51.0,36.0,30.0ISM 2.26 × 107, 4.51, 0.20

ρ0,H∗, d0,H∗, hrH∗ , α[

M⊙ kpc - 3,kpc,kpc]

Stellar halo 2.18 × 104, 1.00, 1.00,−2.44 [12.0,13.0[ < −1.90

Mb, hr,b[

M⊙ kpc - 3,kpc]

bulge 3.4 × 1010, 0.7 [6.0,12.0[ [-0.40,+0.30[

v0, hr,DM , q[

km s−1,kpc]

Dark matter 139.04, 6.70, 0.89

Figure 3. Circular velocities as a function of the radius andcontribution from each stellar component of the model of thegalaxy. See text for the definitions of the equation for the rotationcurve of each sub-population.

surveys such as Gaia. The Oort functions are defined asO± (R) ≡ ± 1

2

(

vcR

∓ dvcdR

)

. Perhaps the greatest difficulty inestimating the Oort functions derives from the presence ofthe derivatives in their definition. Unfortunately, current ob-servations of the rotational motion of the Milky Way arenot good enough to allow a calculation of the derivativesin O± (R) directly from the data. It is also possible to de-termine O+ − O− = vc

Rin an independent way from the

individual values of O+ and O− from proper motion surveysin the direction l = 90 or l = 270. Because along thesedirections the stars have a small dependence on the Galac-tocentric radius so the estimations are less affected by theradial dependence of the Oort functions. Finally, the combi-nation − O−

O+−O−can be estimated from the velocity ellipsoid

of random stellar motions. For the first function O+ ≡ A (R)a compact formulation can be obtained as follows. For the

dark matter component,

ADM (R) =1

4vc,

2R3v20(

R2 + h2r,DM

)2 , (25)

for the stellar halo components of Robin’s density profiles

AH∗ (R) = −πGvc,

r−α⊙

(α+ 3) r2

∑

H∗

ρ0,H∗

(

αrα+3 + 3hα+3r,H∗

)

,

(26)for hr,H∗ < r, while it is clearly null inside the scale radius.For the bulge components contribution we can write

Ab (R) =GMb

4vc,

∑

b

3R + hr,b

(R + hr,b)3 , (27)

and finally for the stellar disks contribution we can write:

Ad (R) =πGR

vc,

∑

d

ρ0,dhR,d

∫

dkk2J2(kR)

h−1z,d + k

1(

h−2R,d + k2

)3/2.

(28)Analogously for the O− ≡ B (R) function we can write: forthe dark matter component

BDM (R) =v20

2vc,

R3 + 2Rh2r,DM

(

R2 + h2r,DM

)2 , (29)

for the stellar halo it reads

BH∗ (R) =πG

vc,

r−α⊙

(α+ 3)r2

∑

H∗

ρ0,H∗

(

(α+ 4)rα+3 − hα+3r,H∗

)

(30)Differently from O+ the contribution from the stellar haloin the central zones for Robin’s profile is not null, but

BH∗ (R) =16πGr

3

∑

H∗

ρ0,H∗

(

hr,H∗

r⊙

)α

, (31)

c© 2014 RAS, MNRAS 000, 1–??


for r 6 hr,H∗ . The level of this contribution is neverthelessextremely weak and added here only for completeness. Ef-fectively it is null compared with the dominant contributionof the bulge component:

Bb (R) =1

4vc,

∑

b

GMbR + 3hr,b

(R + hr,b)3. (32)

Finally, the most significant contributions that account forstar and the local gas distributions are given by:

Bd (R) = 4πG∑

d

ρ0,dhR,d

∫

dkk

h−1z,d + k

kRJ0 (kR) + 2J1 (kR)(

h−2R,d + k2

)3/2.

(33)With this equation and the parameters of Table 1 weobtain the following values at the solar position: O+ =15.1 km s - 1 kpc - 1 and O− = −13.1 km s - 1 kpc - 1.

4.3 Vertical force

The last significant constraint that we consider in the deter-mination of the MW potential is the force acting verticallyon the plane. This constraint is of paramount importance totune the vertical profiles of the disk and the vertical epicyclicoscillations of the orbits, thus several studies have investi-gated the vertical structure of the Milky way on the ba-sis of different observations (Ja locha et al. 2014; Chen et al.2001; Haywood et al. 1997; Bovy et al. 2012; Levine et al.2008; Ja locha et al. 2014; Soubiran et al. 2003). We deter-mine the vertical force at any location within the galaxy asfollows.

For the dark matter component, we evaluate the verticalgradient of the potential at any radial and vertical locationas

Fz,DM =v0

2z

q2(

R2 + h2r,DM

)

+ z2, (34)

that retains information of the flattening parameter q ofthe DM halo. Unfortunately, the alignment of the DM halocomponent with the principal axis of symmetry of the grav-itational potential is far from clear. The triaxiality and thedirections of the eigenvectors of the DM mass distribution isat present unknown and the problem of the stability of rotat-ing disks inside triaxial halos is weakly understood from thetheoretical point of view and still a matter of debate (e.g.,Debattista et al. 2013, and reference therein). We will takeq into account only for completeness and eventually add aflattening of the DM profiles while moving inward in theGalaxy. For the stellar halo components, the same compu-tation yields:

Fz,H∗ (R, z) =4πGzr−α⊙

α+ 3

∑

H∗

ρ0,H∗

(

rα −hα+3r,H∗

r3

)

, (35)

for√R2 + z2 = r > hr,H∗and

Fz,H∗ (R, z) = 4πG∑

H∗

ρ0,H∗z

3r−α⊙ hαr,H∗ , (36)

otherwise. Analogously for the bulge components we get

Fz,b (R, z) =z√

R2 + z2

∑

b

GMb(√R2 + z2 + hr,b

)2 . (37)

Finally, for the disk component we get:

Fz,d (R, z) = 4πG∑

d

ρ0,dhR,dhz,d

×

×∫

dkek|z| − e

h−1

z,d|z|

k2 − h−2z,d

kJ0(kR)e−|z|

(

h−1

z,d+k

)

(

h−2R,d + k2

)3/2.

(38)Our MW potential model with the values of Table 1 presentsa value of |Fz(1.1kpc)|

2πG= 70.0 for the total vertical force on

the plane that match exactly the standard literature valuesof Kuijken & Gilmore (1989a) (see also Kuijken & Gilmore1989c,b). We consider these as the major contributors tothe shape of the underlying MW potential. Adding otherconstraints will not significantly change the distribution ofthe stars in the CMDs and their kinematics.

4.4 Further constraints

By integrating the density profiles of Eqs.(14), (16), (19) and(20) we obtain the total mass as a direct sum of the mass ofthe components. For the dark matter this integration reads:

MDM =v0

2

G

r3max

r2max + h2r,DM

, (39)

for the stellar halo components

MH∗ =4πr⊙

−α

3 (α+ 3)

∑

H∗

ρ0,H∗

d0,H∗

(

3rα+3max + αhα+3

r,H∗

)

, (40)

for the bulge

Mb = 4πGr2max

∑

b

Mb

(hr,b + rmax)2, (41)

and for the disks, by proceeding arbitrarily with an integra-tion in cylindrical coordinates, we can write

Md = 4π∑

d

ρ0,de−Rmaxh

−1R,d

h−2R,dh

−1z,d

(

−Rmaxh−1R,d + e

Rmaxh−1R,d − 1

)

.

(42)With the parameters in Table 1 we obtain a total mass ofM = 1.12 × 1012M⊙ for rmax = 100 kpc.

This constraint has several implications on the orbits ofthe dwarf galaxy satellites of the MW and on the granular-ity of the phase space of the stellar halo, and several studiesare focused on its determination and on the escape speedfrom the MW (e.g., Bhattacharjee et al. 2014; Smith et al.2007; Licquia & Newman 2013; Dehnen & Binney 1998;Xue et al. 2008; Wilkinson & Evans 1999; McMillan 2011).Finally, the determination of the local surface mass densityis tightly related to the integration presented for the totalmass. This is computed in our modelling only for the disk

components Σd = 2∑

d

ρc,iβie−Rαi . This is a relevant con-

straint especially in relation to the disk modelling of thespiral arms that we are going to present here. With the pa-rameters of Table 1 we estimate a value of Σd = 41 M⊙pc - 2

at R = R⊙

We do not consider here a few other issues of minor im-portance that a standard axisymmetric model should takeinto account such as, e.g., the terminal velocities for the in-ner Galaxy (Vallenari et al. 2006). Although it may provide

c© 2014 RAS, MNRAS 000, 1–??


a better constraint than Oort’s constants for an axisymmet-ric galaxy, it it severely affected by non-circular motions ofthe ISM. Its interpretation needs much more precise map-ping of the galactic gas distribution (see also Chemin et al.2015; Golubov & Just 2013). In Section 6.2.1 we will presentour new non axisymmetric distribution of gas.

5 MACHINE LEARNING AND THE PIKAIAGENETIC ALGORITHM

To obtain values representative of the MW stellar, gas andDM components (Table 1) we tune the free parameters of thedensity-potential couples formulated in the previous sectionon theoretical and observational constraints. A genetic algo-rithm is an adaptive stochastic optimization algorithm in-volving search and optimization, and it was first introducedby Holland (1975). Holland created an electronic organismas a binary string (“chromosome”), and then used the ge-netic and evolutionary principles of fitness-proportionate se-lection for reproduction, random crossover, and mutation toexplore the space of solutions. The so-called “genetic pro-gramming languages” apply the same principles using anexpression tree instead of a bit string as a “chromosome”.In astronomy, the Pikaia genetic algorithm has been alreadyconsidered in the galactic kinematics in (see, 2005 Pasetto,PhD thesis, Charbonneau 1995; Metcalfe & Charbonneau2003). We can consider the following quite generic task tomodel a given dataset with a set of adjustable parameters.This task consists of finding the single parameter set thatminimizes the difference between the model’s predictionsand the data. A “top-level” view of the canonical geneticalgorithm for this task can be read as follows: we start bygenerating a set (“population”) of trial solutions, usually bychoosing random values for all model parameters; then eval-uate the goodness of fit (“fitness”) of each member of thecurrent population (through a chi-square measure with thedata). Then the algorithm selects pairs of solutions (“par-ents”) from the current populations, with the probability ofa given solution being selected made proportional to thatsolution’s fitness. It breeds the two solutions selected andproduces two new solutions (“off-spring”). It repeats the se-lection of the population and its progeny until the number ofoff-springs equals the number of individuals in the currentpopulation by replacing the new population of off-springsover the old one. It then repeats the whole sequence untilsome termination criterion is satisfied (i.e. the best solutionof the current population reaches a fit goodness exceedingsome pre-set value).

A genetic-algorithm based approach to a given opti-mization task, as defined above, resembles a kind of forward-modelling: no derivatives of the fit function goodness withrespect to model parameters is needed to be computed.Nothing in the procedure outlined above depends criticallyon using a least-squares statistical estimator; any other ro-bust estimator could be used, with little or no change tothe overall procedure. In the kinematical applications, themodel needs to be evaluated (i.e., given a parameter set,compute a synthetic dataset and the associated goodness offit).

The genetic algorithm has a long history in the pro-cedure of CMD fitting in the Padua group starting from

the works of Ng et al. (2002) and has been implemented inthe kinematic fitting of observational data in Vallenari et al.(2006). The algorithm has been run on true data to re-produce radial velocities (Gilmore et al. 2002), the GSC-II proper motion catalogue Vallenari et al. (2006) and theRAVE dataset equipped with 2MASS proper motions inPasetto et al. (2012a), and Pasetto et al. (2012b). The de-tailed study of the MW potential is beyond the goal ofthe present paper (and maybe meaningless at the sunriseof the Gaia-era), we limit ourselves to present in Table 1the guest parameters for the MW potential just introducedand achieved so far. They will represent the starting valuesof the founding potential that we are going to perturb inthe next section to obtain the spiral arms description whichrepresents the core of this work.

6 DENSITY DESCRIPTION OFNON-AXISYMMETRIC FEATURES

The axisymmetric potential that we have introduced aboverepresents the starting point for the perturbative approachthat we introduce hereafter.

The first framework that we are introducing is the so-called density wave theory. It deals in its original form withthe description of the in-plane motion of the stars in a spiralgalaxy. It is a linear response theory for an unperturbedgeneralized Schwarzschild distribution function (SDF):

fSch ≡ e−12Q(x)+η

Q ≡ (v − v)Tσv (x; t) (v − v) ,(43)

where Q is a quadratic positive definite form, σv (x; t) asecond rank symmetric tensor defined in Eq.(9), η (x) acontinuous and differentiable scalar function and with thesuperscript (∗)T we refer to the transpose of an array.

It is normalized accordingly with (2π)−3/2|σ|−1/2e−η/2 ≡(2π)−3/2|σ|−1/2Σ0 (R). We will recall in what follows thebasis of this theoretical framework without explicit proof,but we will present a new hypergeometric form for the ex-pression of the first moments of the perturbed DF that werepreviously known only in an integral form. We will highlightthe advantages of our formulation.

The second perturbative framework adopted here hasbeen developed by Amendt & Cuddeford (1991) and it hasbeen previously adopted in our modelling technique byPasetto, PhD thesis 2005 in Vallenari et al. (2006). Herewe will only recall the theoretical basis of this second per-turbative framework dealing with the vertical behaviour ofthe kinematics above and below the disk plane, and we willcompare it with the DWT. In the axisymmetric case, thematrix σv (x; t) acquires an especially simple diagonal formand the dependence of the three non-null diagonal termsσRR, σφφ, σzz on the configuration space will be writtenin cylindrical coordinates as:

σii,j (R, z) = ∇zσii,j (R, z) (|z| − zj−1) + σii,j−1 (R, zj−1) ,(44)

for i = R,φ and j = I, II, III , while the vertical profiles ofthe thin disk stellar population will be introduced in Section

7 and where the underlying assumption ofσ2RRσ2zz

= const.is

assumed in agreement with the DWT for spiral arms intro-duced below. On the plane Eq.(44) will be forced to match

c© 2014 RAS, MNRAS 000, 1–??


Figure 4. (left panel) On-the-plane section of the perturbed density profile from Eq.(55) at φ = 0. The blue line is the profile withsingularity at the resonance location R = Rres, the red line is the analytical continuation from Rres ± εR with εres = 1.4kpc and theunderlying green profile is the overall profile continued over the resonance (see text for details). (Right panel) Same green perturbeddensity profile of left panel but for a random set of varying φ angles (and random colour). The purpose is just to show how smooth thepassage is between one line and another at different φ-s.

Table 2. Meridional plane profile of the velocity ellipsoid. Foreach fixed radius the vertical gradient of the two velocity disper-sion tensors is indicated in the fourth and fifth columns. The thirdcolumn gives the vertical ranges.

j zj ∆z∂σRR∂z

, ∂σzz∂z

[kpc] [kpc][

km s−1 kpc−1]

0 z0 = 0.0 |z| = 0 0.0,0.0I z1 = 0.5 |z| ∈ ]0.0, 0.5] 27.2,17.4II z2 = 1.0 |z| ∈ ]0.5, 1.0] 9.7,5.4III |z| > 1.0 0.0,0.0

the profiles

σ2RR (R, 0) = σRR,⊙e

−R−R⊙

hR

σ2φφ (R, 0) = σzz,⊙e

−R−R⊙

hR

σ2zz (R, 0) =

(

1 + ∂ ln vc∂R

) σ2RR(R,0)

2

(45)

and the gradients for the three vertical profiles I, II, III area smooth interpolation of the values in Table 2.

6.1 Linear response theory to a spiralperturbation pattern

The most popular self-consistent fully analytical treat-ment available in literature to study the spiral arms isbased so far on a DWT proposed by Lin et al. (1969),Marochnik & Suchkov (1969b), Marochnik & Suchkov(1969a). In these works, a sinusoidal perturbation to theaxisymmetric potential for a discoidal stellar distribution isconsidered. If we perturb an axisymmetric potential witha sinusoidal wave, we need to search for the self-consistentcondition for a potential of a spiral drawn by a shape

function ψ (R) = −2 cot (p) log(

RR0

)

for a given pitch angle

p ∼ 8, a given starting radius of the spiral perturbationR0 ∼ 2.6kpc, and with m = 2 the number of spiral arms

that we assume. The variable Ωp is the rotation pattern ofthe spiral structure; the theory so far is developed for a con-stant Ωp, even though no strong observational constraintsare available to justify this assumption. Different modellingtechniques (see e.g. N-body simulations from Grand et al.2012b) require e.g. Ωp = Ωp (R). Nevertheless, no strongobservational evidence is available to date to formalize adependence of Ωp from the phase space coordinates. In whatfollows we will simply assume Ωp ∼ 35.0 km s−1 kpc−1.

Φa (R) ≡ −Φa0R e−Rhs is the amplitude of the spiral

arm potential profile, where an indicative scale lengthhs ∼ 2.5 kpc is assumed and Φa0 ∼ 887.0 km2s−2kpc−1

(Roca-Fabrega et al. 2014). These numerical values aretaken form the literature reviewed in Section 1.1 and areof exemplificative nature to the present capability of ourmodelling approach alone. They are not meant to be bestfitting values to any particular survey.

The DWT is developed in epicycle approximation. Theepicycle approximation is probably the weakest of the as-sumptions adopted in our model. In the next section, wewill review some observational evidence of the failure of thisapproximation in the solar neighbourhood. Here we proceedsimply by adopting the modification that the perturbativelinear approach induces on this approximation, without pre-senting a critical review, even though improved tools arealready available (Dehnen 1999a). The main role of this ap-proximation is to decouple in Eq.(43) the radial/azimuthalfrom the vertical direction thus simplifying them. From thepotential introduced in the previous section we can intro-duce the rotation frequency as Ω (R) ≡ vc

Rtogether with its

derivative ∂Ω∂R

= 1R∂vc∂R

− vcR2 . The radial epicycle frequency

is then given by κ = 2Ω√

1 + R2Ω

∂Ω∂R

. Finally, recalling that

the wave-number is the derivative of the shape function in-troduced above, k = ∂ψ

∂R, we can compute Toomres num-

ber as X ≡ kκσRR. In our approach, Toomres number can

eventually acquire a vertical dependence trough the veloc-ity dispersion profiles introduced above (Eq.(44)). Becausea self-consistent theory for the vertical motion of the stars

c© 2014 RAS, MNRAS 000, 1–??


in the presence of spiral arms is missing, a large freedom isleft to the researcher to investigate different approaches.

After the introduction of these quantities, we are in theposition to make use of the results of Lin et al. (1969). Asolution of the evolution equation (i.e. the linearized Boltz-mann equation) ι ∂f1

∂t−B0 [f1]−B1 [Φ1] = 0 is considered by

the authors in the form:

f1 (x,v; t) =

∫ t

−∞

⟨

∇x′Φ1,∂fSch (x′,v′)

∂v′

⟩

dt′, (46)

with natural boundary conditions f1 → 0 as t → +∞.Here 〈∗, ∗〉 represents the standard inner product. Underthe assumption that the perturbations take the form of spi-

ral waves Φ1 (x; t) = Φa (R) eι(mφ−ωt+∫R kdR), in a “tightly

wound” approximation, i.e. |kR| ≫ 1, the authors rapidlyreach the form for the perturbed DF on the plane as:

f1 = − Φ1

σ2RR

fSch

1 − sinc−1 (νπ)×

× 1

2π

∫ π

−π

eι(ντ+X(u sin τ+v(1+cos τ)))dτ

,

(47)

with sinc−1 (νπ) ≡ sin(νπ)νπ

the “sinc” function, where we sethere for simplicity the frequency ratio ν ≡ ω−mΩ

κ, and 1

γ2≡

σ2φφ

σ2RR

= κ2

(2Ω)2in agreement with the hypothesis underlying

Eq.(45). Finally we simplified the notation writing the pe-

culiar velocities as v− v =

vRσRR

, γvφ−vcσRR

, vzσzz

= u, v, w.

We are ready now to proceed to compute the first ordermoments of this DF that we adopted in our kinematic model.

The moment of order zero and one was already car-ried out in numerical form by the authors in AppendixA of Lin et al. (1969) to the first order, and the sec-ond order central moments were recently proposed byRoca-Fabrega et al. (2014) in a work focused on the ver-tex deviation and the bracketing of the resonances. Never-theless, in the original work by Lin et al. (1969) and in thework Roca-Fabrega et al. (2014) the numerical integral waspassed over in favour of a more compact analytical formal-ism, and the divergences due to the resonances were notconsidered.

We present here a different solution for these momentsin the form of Hypergeometrical functions instead of nu-merical integrals. We will underline later the advantages ofour formulation in the context of the present modelling ap-proach. We will also offer a necessary solution to cover theresonances and to make the model suitable for the star-countapproach that we are developing here.

6.1.1 Zero order moments of the perturbed DF

The family of the perturbed density profiles result as thezero order moment of the total DFs given by f = fSch + f1with fSch defined by Eq.(43) and f1 by Eq.(47). We write:

Σ (R,φ; t) ≡∫

fdvRdvφdvz =

∫

(

fSch + f1)

dvRdvφdvz.

(48)

By exploiting the notation introduced above, we can write

Σ1

Σ0=

∫

R3 f1dvRdvφdvz∫

R3 fSchdvRdvφdvz

= − Φ1

σ2RR

1

Σ0

∫

R2

(

1 − sinc−1 (νπ)×

× 1

2π

∫ π

−π


)

fSchdudvdw,

(49)and in particular, we reach the form

Σ = Σ0 + Σ1

= Σ0 − Σ0Φ1

σ2RR

(

1

Σ0

1

2π

∫

R3

(

1 − sinc−1(νπ)×

×∫ π

−π


)

fSchdudvdw

)

,

(50)which is the obvious generalization of the work of Lin et al.(1969) to the case of vertical velocity DFs. If we remember

that∫

R3

1

(2π)3/2e−(u2+v2+w2)dudvdw = 1, the terms inside

the external brackets reads simply

= 1 − sinc−1(νπ)

2π

∫

R3

dudvdwe−u2+v2+w2

2 ×

× 1

(2π)3/2

∫ π

−π

eι(ντ+X(u sin τ+v(1+cos τ)))dτ.

(51)

At this point, by changing the integration order, we canobtain

= − sinc−1 (νπ)

2π

1

(2π)3/2

∫ π

−π

dτ×

×∫

R3

dudvdwe−u2+v2+w2

2 eι(sτ+X(u sin τ+v(1+cos τ)))


2π

∫ π

−π

dτ(

eιντ−X2(1+cos τ)

)


2π

∫ π

−π

dτe−X2(1+cos τ) (cos (ντ) + ι sin (ντ))


2π

∫ π

−π

dτe−X2(1+cos τ) cos (ντ)

= −( 12,1)F(1−ν,1+ν)

(

−2X2)

≡ −( 12,1)F(1−ν,1+ν)

(52)Here we introduced the generalized Hypergeometric func-tion:

2F2 (a1, a2; b1, b2; z) ≡ 2F2 (a1, a2; b1, b2; z)

Γ (b1) Γ (b2)

=1

Γ (b1) Γ (b2)

∞∑

k=0

(a1)k(a2)k(b1)k(b2)k

zk

k!,

(53)

with (a)n ≡ a(a+1) . . . (a+n−1) = Γ(a+n)Γ(a)

the Pochhammersymbol and Γ the Eulero Gamma function. In particular, weadvance the notation of the Hypergeometric function to

(a1,a2)F(b1,b2) ≡ 2F2

(

a1, a2; b1, b2;−2X2) . (54)

c© 2014 RAS, MNRAS 000, 1–??


Figure 5. (left panel): the integral of the continued density profile Σsp (green) and the unperturbed axisymmetric (orange). (rightpanel): Integration profile at different azimuthal directions φ, each for each colour, versus εR.

We can then recollect the terms to express the density in acompact way as:

Σ = Σ0 + Σ1

= Σ0

(

1 − Φ1

σ2RR

(

1−( 12,1)F(1−ν,1+ν)

)

)

= Σ0

(

1 − Φ1

σ2RR

X2

1 − νℜ)

.

(55)

This represents the formula for the density profile perturbedby the spiral arms that we are going to implement.

As a corollary of this result, it is evident that we areable for the first time to propose a closed form for the“reduction-factor” ℜ. This was historically introduced inLin et al. (1969) as the factor to which we have to reducethe response of a stellar disk below the value of a cold disk(this is presented by direct integration in Appendix A too).This compact formulation of the density perturbation dueto spiral perturbations presents extremely rapid computa-tion benefices because of the presence of the hypergeometricfunction 2F2. This will turn out to be especially useful for atechnique that wants to be able to realize mock catalogues,where these integrals have to be computed a larger numberof times to span a huge parameter space or to realize a highnumber of stars by populating PDFs. The plot of the den-sity profiles for the values of the potential of Table 1 and theparameters assumed above are in Fig. 4. As evident from theplot, the previous equation Eq.(55) presents a singularity atthe resonances that we are going to treat in the next section.

6.1.2 Interpolation schemes over the resonances

As evident from Fig. 4, at the radius where the resonancesare located (i.e. wherever1 − ν = 0) a divergence in thedensity profile of Σ is present. To satisfy the normalizationequation Eq.(3) we need to cover this divergence. From Fig.4 it is evident how the closure required by the stellar pop-ulation theory in Eq.(3) (in the special case of Eq.(13) and(43)) leads to a failure of the normalization condition. ThePDF generated by Eq.(55) cannot be populated in a star-count dot-by-dot fashion because of the infinite number ofstars necessary to fill the locality of the radius R = Rres.

Resonances are not eliminable discontinuities. Never-theless, a few viable options are available to cover thesediscontinuities. We can apply a bilinear interpolation incylindrical coordinates. We solve the condition of continu-

ity Σ|p1 = a10 and differentiability

∂Σ∂R, ∂Σ∂φ, ∂Σ∂R∂φ

p1

=

a11, a12, a13 in each of the 4 points of the grid where thepotential scheme introduced above has been valued. The lin-ear matrix for the system of 16 equations in 16 unknownsis invertible and can be solved for two radii, one internal tothe Lindblad resonance Rres − εR, and one external to it,at Rres + εR. Finally the function Eq.(55) is extended (redcurve in Fig. 4).

Alternatively, we can develop the function Eq.(55) ona orthogonal set of basis in cylindrical coordinates (e.g.,the Bessel function Jα introduced in Section 4). Then we

can mimic the behaviour of the DF Σ (R) ∼∞∑

n=1

cnJα,n (R)

where Jα,n (R) ≡ Jα(

zα,nR

Rmax

)

and zα,n is the zero of the

Bessel function Ja, with coefficients cn =〈Σ,Jα,n〉

〈Jα,n,Jα,n〉 . This

approach passes through a long computing of inner prod-ucts and hence is very slow. It does not respect preciselythe values of the original Σ and while it can be worked outefficiently once eigenfunctions of the Laplacian operator areconsidered, it loses efficiency when the purpose is to coverthe resonances on the velocity space.

We want to proceed here with a general scheme thatworks rapidly both for the treatment of resonances on thedensities (i.e. first order moments of fSch) as well as for res-onances for the moments of higher order (mean, dispersionetc.).

To achieve this goal we proceed to investigate here amethod that only extends the function along the radial di-rection R with polynomial P of degree deg (P ) = 4, i.e.

P =4∑

i=0

cixi. The methodology can of course work equally

well with deg (P ) = 3 to match the number of constraintsat the points P (Rres − εR) and P (Rres + εR) where it isvalued together with its derivative. Nevertheless, the samescheme with deg (P ) = 4, allows to impose to the first andsecond order moments closer values to the corresponding un-

c© 2014 RAS, MNRAS 000, 1–??


Figure 6. (left panel) Density profile in the plane of the Milky Way. The Sun is located at at R, φ, z⊙ = 8.0, 0.0[kpc]. A faint bluecircle suggests the solar radius, despite a fixed position for φ⊙ not being investigated here, it is assumed φ⊙ = 0 [deg] for simplicity. Thefigure shows the plane at z = 0. (right panel) out of plane density distribution for two slices symmetric below and above the plane.

perturbed functions, thus allowing us to gently reduce theperturbations to the density and velocity fields to zero ifdesired.

The reason for this polynomial solution to work isthe azimuthal symmetry of the underlying unperturbedmodel. As evident from the condition of resonances, 1 −m(Ωp−Ω(R))

κ(R)= 0, the divergences have no dependence on

the azimuthal angle φ. In the framework of the DWT we canindividuate the resonances location only by analysis of theradial direction R. The results of this interpolation schemeare presented in Fig. 4 (right panel), where a very close az-imuthal spanning is operated to check the validity of thecontinuation polynomial scheme presented, with evidentlysatisfactory results.

The choice of the exact value that the scheme induces atthe resonance is by itself a free parameter that we investigatehere below.

6.1.3 The choice of the interpolating radius

The only parameter left unspecified in this interpolationscheme is the radius at which the scheme has to take overthe DWT predictions. This is a single parameter, one condi-tion is sufficient to fix it and the most natural one is basedon the continuity equation. We require that the differencein mass between the continued Σsp and the unperturbed Σ0

axisymmetric density distributions are the same (see Fig. 5)

Θ (εR) =

∫

R2+

(Σ0 − Σsp (εR))RdR. (56)

This condition is equivalent to minimize the impact of thearbitrary shape that we chose to use to cover the resonances.If we convolve the integrals over all the angular directions weobtain Fig. 5 (right panel). As is evident in the figure, theminimal difference between the integrated mass predictedby Σ0 and Σsp is achieved for εR ∼ 1.5 kpc. Finally in Fig.6 the plot of the density profiles in the plane and aboveand below the plane are shown. A blue line marks the solarradius: the solar location is assumed to be at φ⊙ = 0 but itis not a result of an investigation of any dataset. Hence inFig. 6 a blue circle marks the possible solar radius R⊙. Sofar all the values obtained in Table 1 are the results fromstudies in the axisymmetric formalism of Section 4. A non-axisymmetric investigation of the solar position in the MWplane is within the DWT framework is, to our knowledge,not available (and beyond the goal of this paper).

The vertical density profile of the spiral arms is not di-rectly obtained from the DWT, which is developed only inthe plane. Here we are not searching for a self-consistentdetermination of the density profile, instead we assume de-coupling of the vertical and radial profile in the configura-tion space assigning the asysimmetric density profile of thedisk stellar population to the spiral arms profile too (Fig.7). As evidenced in the Figure, the effect of the spiral armsis a tiny contraction of the vertical profile with respect to

c© 2014 RAS, MNRAS 000, 1–??


Figure 7. Vertical profile of the spiral arm component at R = 6,R = 8 and R = 10 kpc (solid lines). The unperturbed exponentialis added for comparison with a dashed line.

the corresponding unperturbed one. This is in response tothe dependence of the density profiles to the velocity dis-persions. Because of Eq.(9), i.e. the so-called “age-velocitydispersion” relation evident in the MW, the older the SSPof the spiral arm is, the smaller is this contraction.

6.2 Colour magnitude diagrams of spiral features

Once the density is computed, we know the relative con-tribution of all the stellar populations that we want to im-plement in our model (Table 1). At a given distance wecompute the synthetic photometry of an observed f.o.v. bydistributing the SSPs, or the stars, along the density profilesaccording to their relative contribution. The new approachpresented in Section 2 allows us to use virtually any databaseavailable in literature (and this part of the software is freelyavailable upon request to the authors). If we want to includethe treatment of the spiral arm density distribution on thephotometry, the major problem is the extinction along thel.o.s. It has to be accounted for accordingly with the spiralarm distribution of the stars and the gas. In particular, ifwe want to populate a PDF representative of a CMD forthe stellar density distribution computed with the densityprofiles introduced above, we need an extinction model ac-counting for a gas distribution following the spiral arms dis-tribution too.

To account for this extinction, we developed a modelof gas distribution based on the spiral density profiles intro-duced above, but for cold disks (ℜ = 1 in Eq.(55)).

6.2.1 Extinction model

While propagating throughout a galaxy, the intensity of thestar light decreases because of absorption and scattering dueto the presence of interstellar dust. The combined effect,called extinction, has to be taken into account in order toderive the stars intrinsic luminosity from its observed flux. Inorder to predict the effect of interstellar dust on the observedCMDs, we calculated the extinction towards each SSP oreach star in our model galaxy as follows.

We assumed that the dust is traced by the gas in our

Figure 8. ISM gas distribution from Eq.55. Note that this kindof density plots map a linear (i.e. R) and an angular (i.e. φ)quantity over square. This is causing strong distortions over therange of R > 8 kpc and careful attention has to be paid in itsinterpretation.

galaxy model and that its density, relative to the gas den-sity (shown in Fig. 8), as well as its optical properties, arewell described by the dust model of Draine & Li (2007). Thisdust model has been calibrated for the dust extinction curve,metal abundance depletion and dust emission measurementsin the local Milky Way. From this dust model we consider theextinction coefficient kλ,ext per unit gas mass. From kλ,extand the gas density distribution, ρISM, we derive the opti-cal depth crossed by the star light along the path betweeneach star and the observer (located at the sun position):τλ =

∫ ∗

⊙kλ,extρISMdrhel. Then, the extinction in magnitudes

is derived as Aλ = 2.5τλ log e.The determination of the predicted observed flux of a

star taking into account dust extinction in a galaxy modelis affected by several caveats. First, the optical properties ofthe dust are known to change substantially for different l.o.s.within the Milky Way (e.g., Fitzpatrick 1999). Therefore,any Milky Way dust model can only be interpreted as an av-erage model for many directions within the Galaxy. Further-more, the amount of obscuration due to the dust is known tochange significantly between the “diffuse” and “dense” ISM.In particular for very young stars, still embedded in theirparent molecular cloud, our approach is likely to underesti-mate their extinction (since the Draine & Li (2007) model iscalibrated for the diffuse ISM and molecular clouds are notresolved in our model for the gas distribution). In this work,we assumed that the dust follows the gas distribution withinour galaxy model and that the dust optical properties are

c© 2014 RAS, MNRAS 000, 1–??


Figure 9. CMD in V and I band for a field l ∈ [88, 92]

b ∈ [−2, 2] and magnitude limit V < 20 mag. Field stars arerepresented with PDF and only spiral arms star population is ina scatter-type CMD.

uniform. Although quite simple, this approach is sufficientto show the general effect of the presence of spiral arms onthe predicted CMD (Fig. 10).

6.2.2 Sources of stellar tracks, isochrones, SSPs in

different photometric systems

We adopt the stellar models and companion isochrones andSSPs with magnitudes and colours in various photometricsystems of the Padua data-base because they have beenwidely tested and used over the years in many areas of ob-servational stellar astrophysics going from the CMDs of stel-lar clusters, to populations synthesis either star-by-star orintegral photometry (magnitudes and colours) or spectralenergy distributions, and others.

• Stellar tracks. We will not review the physics of thesestellar tracks here but we just mention that over the years,these models were calculated including semi-convectionin massive stars (e.g., Chiosi & Summa 1970), ballistic-convective overshooting from the core (Bressan et al. 1981),overshooting from the the bottom of the convective enve-lope (Alongi et al. 1991), turbulent diffusion from the con-vective core and convective shells Deng et al. (1996a,b);Salasnich et al. (1999), plus several additional improvementsand revisions (Alongi et al. 1993; Fagotto et al. 1994b;Bertelli et al. 1994; Fagotto et al. 1994a,a; Bertelli & Nasi2001; Bertelli et al. 2003, 2008, 2009). The stellar mod-els in use are those by Bertelli et al. (2008, 2009), whichcover a wide grid of helium Y , metallicity Z, and en-richment ratio ∆Y/∆Z. The associated isochrones includethe effect of mass loss by stellar wind and the thermallypulsing AGB phase according to the models calculated byMarigo & Girardi (2007).

• The database of SSPs. We briefly report here onthe data base of isochrones and SSPs that has been cal-culated for the purposes of this study. The code in useis the last version of YZVAR developed over the years

by the Padova group and already used in many stud-ies (for instance Chiosi & Greggio 1981; Chiosi et al. 1986,1989; Bertelli et al. 1995; Ng et al. 1995; Aparicio et al.1996; Bertelli & Nasi 2001; Bertelli et al. 2003) and recentlyextended to obtain isochrones and SSPs in a large re-gion of the Z − Y plane. The details on the interpola-tion scheme at given ∆Y/∆Z are given in Bertelli et al.(2008, 2009). The present isochrones and SSPs are in theJohnson-Cousins-Glass system as defined by Bessell (1990)and Bessell & Brett (1988). The formalism adopted to de-rive the bolometric corrections is described in Girardi et al.(2002), whereas the definition and values of the zero-pointsare as in Marigo & Girardi (2007) and Girardi et al. (2007)and will not be repeated here. Suffice it to recall thatthe bolometric corrections stand on an updated and ex-tended library of stellar spectral fluxes. The core of the li-brary now consists of the DFNEWATLAS9 spectral fluxesfrom Castelli & Kurucz (2003), for Teff ∈ [3500, 50000] K,log10g ∈ [−2, 5] (with g the surface gravity), and scaled-solarmetallicities [M/H] ∈ [−2.5,+0.5]. This library is extendedat the intervals of high Teff with pure black-body spectra.For lower Teff, the library is completed with the spectralfluxes for M, L and T dwarfs from Allard et al. (2000), Mgiants from Fluks et al. (1994), and finally the C star spec-tra from Loidl et al. (2001). Details about the implemen-tation of this library, and in particular about the C starspectra, are provided in Marigo & Girardi (2007). It is alsoworth mentioning that in the isochrones we apply the bolo-metric corrections derived from this library without makingany correction for the enhanced He content which has beenproved by Girardi et al. (2007) to be small in most commoncases. The database of SSP cover the photometric projec-tion of any reasonable EMW. The number of ages Nτ of theSSPs are sampled according to a law of the type τ = i× 10j

for i = 1, ..., 9 and j = 7, ..., 9, and for NZ metallicities areZ = 0.0001, 0.0004, 0.0040, 0.0080, 0.0200, 0.0300, 0.0400.The helium content associated to each choice of metallic-ity is according to the enrichment law ∆Y/∆Z = 2.5. EachSSP has been calculated allowing a small age range aroundthe current value of age given by ∆τ = 0.002 × 10j withj = 7, ...9. In total, the data base contains Nτ ×NZ ∼= 150SSP. This grid is fully sufficient for our purposes. For futurepractical application of it, finer grids of SSPs can be calcu-lated and made available. To calculate SSPs one needs theinitial mass function (see comments after Eq.(6)) of stars ofwhich the are many formulations in the literature. Care mustpaid that the IMF of the SSPs is the same of the Galaxymodel to guarantee self-consistency of the results. By con-struction, the IMF contains a normalization factor whichdepends on the IMF itself and the type of constraint one isusing, e.g. the total number of stars in a certain volume, thetotal mass of stars in a certain galaxy component etc. In thecase of a SSP, the normalization constant is usually definedimposing the total mass of the SSP to be MSSP = 1M⊙, sothat it can immediately be used to find the total luminosity(magnitude) of a stellar assembly with a certain total mass(See section 3 and Eq.(11)). Needless to say that other li-braries of stellar models and isochrones can be used to gener-ate the database of SSPs, the building blocks of our method.The same is true about the code generating the SSPs: wehave adopted our code YZVAR, of course other similar codesin literature can be used provided they may reach the same

c© 2014 RAS, MNRAS 000, 1–??


level of performance. Equally for the photometric systems.So the matrix method for generating the DFs for startingSSPs does not depend on a particular choice for the database of stellar tracks, isochrones, and photometric system.

• Simulation of photometric errors and complete-ness. Real data on the magnitudes (and colours) of thestars are affected by photometric errors, whose amplitudein general increases at decreasing luminosities (increasingmagnitude). The photometric errors come together with thedata itself provided they are suitably reduced and calibrated.Photometric errors can be easily simulated in theoreticalCMDs. The procedure is simple and straightforward (seefor instance Pasetto et al. 2012, for all details). To comparedata acquired along a given line of sight with theory onehas to know the completeness of the former as a functionof the magnitudes and pass-band (Stetson & Harris 1988;Aparicio & Gallart 1995). This is long known problem thatdoes not require any particular discussion in the context ofthis paper, and tabulations of the completeness factors mustbe supplied in advance. The only thing to mention here isthat correcting for completeness will alter the DF of starsin the cells of the observational CMD we want to analyze.These tabulations of completeness factors must be suppliedby the user of our method.

• CMDs rasterisation. Modern, large surveys of stel-lar populations easily generate CMDs containing millions ofstars or more, of different age, chemical composition, po-sition in the host galaxy, suffering different reddening andextinction etc. so that even plotting the CMD can be a prob-lem not to speak about deciphering it for the underlying starformation and chemical enrichment histories, mass and spa-tial distribution of the stellar component under examination.We have already introduced the concept PDF for a stellarpopulation, we want now to particularize it to the case ofthe CMD and introduce the concept of a tesselated CMD.Given two photometric pass-bands δλ and δλ′ and associ-ated magnitudes and colors, one can soon build two CMDsmδλ vs mδλ′ - mδλ and mδλ′ vs [mδλ′ -mδλ] and divide thisin elementary cells of size ∆mδλ and ∆[mδλ-mδλ] To thepopulation of each cell is contributed stars from all SSPswhose evolutionary path crosses the cell. The regions occu-pied by stars in the main sequence, red giant, red clump,and asymptotic giant phase this latter stretching to verylow effective temperatures (red colors) are well evident andno particular remark has to be made but the well knownfact that long lived phases display a higher number of starscompared to the short lived ones. CMDs of this are typicalof the stellar populations in nearby galaxies, e.g. the Mag-ellanic Clouds, M31 and others, where for all the stars in agiven galaxy the distance is nearly constant. Some blurringof the CMD can be caused by varying extinction across thegalaxy under consideration. In the case of f.o.v., the effect ofdifferent distances for the stars has to be included, addinga further dimension to the problem.

6.2.3 Color-Magnitude Diagrams

In Fig. 9 we show two examples of tesselated CMDs forfield stars population of stars observed in the V and I pass-bands and in which two different extinction laws have beenadopted. The left panel shows the case with extinction rep-resented by a simple exponential law, whereas in the right

panel two exponential laws are used for the extinction. Thecolour code is proportional to the value of the PDF in eachcell. In Fig.10 we present the PDF of the CMD in V and Ipass-bands for an ideal field l, bcen ∈ 90, 0 with open-ing angle ∼ 2 and a limiting magnitude of V < 20 mag. Wenote how the PDF is not normalized to [0, 1] but by com-puting the integral Eq.(11) is normalized to the number ofstars effectively predicted by the IMF and density profilesintroduced above. First we have plotted the stars of the spi-ral arms alone (yellow dots) and then added also the starsof the field star populations. The stars in the spiral armspiral almost overlap with the main sequence stars of thefield. The faint evolved sequence is visible at the right of themain sequence. This technique (presented in Pasetto et al.2012) is particularly fast, and because of this it is particu-larly suited to deal with large parameter spaces and there-fore the upcoming era of large surveys. Once the stars aredistributed on the CMD, we can test the effect of the newextinction model. The large impact of the gas distributionalong the spiral arms in the shaping the composite CMD issoon evident. Analysing this particular field in more detailis beyond the goal of the present study. However, it is worthmentioning that the combined effects of the large viewingangle and the density gradient in mass density due to spiralarms have combined to stretch the distribution of stars tothe red side of the CMD. This is because of the more concen-trated gas/dust distribution rising up rapidly at about 1.3kpc from the solar position along the l.o.s. In a near future,surveys like Gaia will be able to provide great insight on thedistribution of gas and dust across the MW. In this context,the technique we have developed can soon be adapted to astar-by-star approach to directly determine the extinction.

7 VELOCITY FIELD DESCRIPTION OFNON-AXISYMMETRIC FEATURES:AZIMUTHAL AND VERTICAL TILT OFTHE VELOCITY ELLIPSOID

The tilt of the velocity ellipsoid with respect to the config-uration space axis generates non-null diagonal terms in thematrix σv (x) introduced in Eq.(43). We derive here thesenon-null terms in the context of two independent theories.

Following what was done for the zero-order moment offSch, we rely on the DWT to derive the moments of order oneor more for the velocity field on the plane, and on the studyby Amendt & Cuddeford (1991) for the velocity momentson the meridional plane. No cross term σzφ is consideredin our model even if spiral arms are expected to couple theazimuthal and vertical directions. In the linear regime, thiscoupling is null and so we strictly assume σzφ = 0.

Finally once the moments of the velocity DF are ob-tained, the velocity of the field is derived from the diagonal-ization of σv (x) by simply solving the eigen-system:

det (σ − λI) = 0, (57)

with I the unit matrix and λ the eigenvalues, and populatingthe corresponding tilted PDF.

c© 2014 RAS, MNRAS 000, 1–??


Figure 10. (left panel). CMD for the sole field stellar population derived by using the extinction model introduced in Section 6.2.1.(right panel) CMD realized assuming a double exponential extinction profile on the sole field stars.

7.1 Radial-azimuthal velocity field

7.1.1 Radial mean stream velocity

The computation of the velocity field proceeds exactly asabove for the moment of order zero. We start with the radialmoment defined as:

vR =1

Σ

∫

fvRdvRdvφdvz

=Σ0

Σv0,R +

Σ0

Σ

1

Σ0

∫

f1vRdvRdvφdvz

=1

Σ

∫

f1vRdvRdvφdvz,

(58)

where v0,R = 1Σ0

∫

fSchvRdvRdvφdvz = 0 and rememberingEq.(47) we evidently need to compute the following

vR = −Σ0

Σ

Φ1

σ2RR

1

Σ0

∫

R3

fSch

(

1 − sinc−1 (νπ)

2π×

×∫ π

−π


)

uσ3RR

γσzzdudvdw.

(59)But the RHS of the previous equation reads

vR = −Σ0

Σ

Φ1

σ2RR

1

Σ0

σ3RR

γσzz

(∫

R3

fSchududvdw −∫

R2

fSch×

× sinc−1 (νπ)

2π

∫ π

−π


)

ududvdw

(60)where the first term in the round brackets on the RHS sum isidentically null because no average radial motion is expectedon an axisymmetric disk. We simplify further this equation

by introducing explicitly Eq. (43) as

vR =ΣSch

Σ

Φ1

σRR

sinc−1 (νπ)

(2π)2

∫ π

−π

dτ×

×∫

R2

dudvdweι(ντ+X(u sin τ+v(1+cos τ)))ue−u2+v2+w2

2 ,

(61)

so that carrying out explicitly the integral on the bottomrow of the previous equation we get

vR =Σ0

Σ

Φ1

σRR

X

2πsinc−1 (νπ)

∫ π

−π

dτ ι sin τeιντ−X2(1+cos τ),

(62)and finally, making use of the Eq.(53), we are able to writethe first order moment in the radial direction as:

vR =Σ0

Σ

Φ1

σRR

X

2sinc−1 (νπ)×

×(

(1/2,1)F(−ν,2+ν) − (1/2,1)F(ν,2−ν)

)

.

(63)

We plot an example of the computing of Eq.(63) in Fig. 11.As evidenced in the moment of order zero, the amplitudeof the response in the velocity field at the resonances growsbeyond the limits permitted by the linear response theoryof Lin et al. (1969) and the theory breaks down. Analyticalcontinuation is applied also in this case within the sameframework developed for the first order moment above.

For a spiral pattern with trailing spiral arms, i > 0, itis easy to prove that the wave number k = ∂ψ

∂R< 0 and so ψ

is a decreasing function of R. Hence, for example, if we areconsidering the regions where Ωp > Ω (R) (see also Fig. 6)we have that ν > 0. Inside the spiral arm it is Σ1 > 0 so thatwe must have Φ1 < 0 . Hence, with the centre of the spiralbeing given by the phase ϕ = 0, we recover the results ex-pected by the density spiral wave theory that presents meanradial motion toward the galactic centre inside the spiralarm and a motion outward in the inter-arm regions ϕ = ±π

c© 2014 RAS, MNRAS 000, 1–??


Figure 11. (left pane) Mean radial velocity vR for a spiral stellar population. The Sun is located at R, φ, z⊙ = 8.0, 0.0, 0.02[kpc,deg, kpc]. The Fig. is on the plane at z = 0. (right panel) Density contrast Σsp − Σ0 at one contour level. Density contrast is negativein blue-colour.

depending on the location where Ωp = Ω (R) as shown inFig.11 just above the ∼ 6.5 kpc (Faure et al. 2014). As evi-denced by the one contour style of this Fig. where Σsp −Σ0

has been plotted, the mean radial velocity field of Fig. 11is in-phase with the density as expected from DWT. Thistest is not only performed to graphically validate the com-putation of the velocity moments through hypergeometricfunctions, but more importantly, to evidence the goodnessof the continuation scheme of Section 6.2.1 over the reso-nance on the velocity space. As evident from the plot theexpectation holds even above and very close to the reso-nances, R ∈ [Rres − εres, Rres + εres], a result which is notobvious to prove analytically.

7.1.2 Azimuthal mean velocity

It is even more interesting to describe the influence of thespiral arms in the mean azimuthal velocity for its implica-tions regarding the location of the Sun relative to the LocalStandard of Rest. Because of the similarity of the integra-tions performed previously in the radial direction, we reporthere simply the results. We obtain for the azimuthal direc-tion

vφ =ΣSch

Σvc−

ΣSch

ΣΦ1

vcσ2RR

(

1 − νπ

sin (νπ) ( 12,1)F(1−ν,1+ν)

)

,

(64)that we plot on Fig. 12. As shown in this figure, the aver-age azimuthal perturbation on the circular velocity is of the

order of 5 to 10 km s−1, i.e. compatible with the motion ofthe sun relative to the LSR. This result is extremely inter-esting no matter where the resonance is located. At everyradius, the spiral arm presence affects the mean motion andcan severely bias the works aimed to determine the motionof the Sun with respect to the circular velocity (the LocalStandard of Rest). This is in line with what is already evi-denced by numerical simulations (e.g., Quillen & Minchev2005; Faure et al. 2014; Kawata et al. 2014). Unfortunately,up to now the result has only a theoretical value because itis affected by the uncertainties on the resonances’ locations,on the validity of the DWT, and on the uncertainty of theSun’s location.

7.1.3 Dispersion velocity tensor

The moments of order two can be calculated by direct inte-gration, but the procedure is more cumbersome and the in-tegrals not easily tractable analytically. A simpler and fullyalgebraic procedure is followed here bypassing the direct in-tegration in favour of the second order not central moments.The desired results will be then achieved with the help of thepreviously obtained Eq. (63) and (64). We present here thecomputation for the first of these moments; the results forthe following orders are obtainable following a similar pro-cedure. From the definition of non-central radial moment of

c© 2014 RAS, MNRAS 000, 1–??


Figure 12. (left panel), Mean stream azimuthal velocity over all the disk. (right panel) Mean stream azimuthal velocity for spiral armdistributed SSP (green). Its evident the strong influence of the spiral arms in the measure circular velocity (see text for detail). Forcomparison the rotation curve of Fig. 3 is added in the range of interest of the spiral arms.

order two we write

v2R =1

Σ

∫

R3

fv2RdvRdvφdvz

=1

Σ

∫

R3

(

fSch + f1)

v2RdvRdvφdvz

=Σ0

Σσ2RR +

1

Σ

∫

R2

f1v2RdvRdvφdvz,

(65)

and remembering the definition of f1 from Eq.(47) we ob-tain:

v2R =Σ0

Σσ2RR − 1

Σ

Φ1

γ

σzz

σ−2RR

∫

R3

Σ0

(2π)3/2σRRσφφσzz×

×e−u2+v2+w2

2

1 − sinc−1(νπ)

2π×

∫ π

−π


u2dudvdw

=Σ0

Σσ2RR − Σ0

ΣΦ1

(

1 − sinc−1(νπ)(

( 12,1)F(1−ν,1+ν)−

−2X2(

( 32,2)F(2−ν,2+ν) − 3( 5

2,3)F(3−ν,3+ν)

)))

.

(66)In the same way, we obtain for the azimuthal term (thecomputation is tedious but straightforward):

v2φ =Σ0

Σv2c

(

1 − Φ1

σ2RR

(

1 − νπ

sin(νπ) ( 12,1)F1−ν,1+ν

))

,

(67)

and for the mixed term:

vRvφ =Φ1

σRR

Σ0

Σ

sinc−1 (νπ)

2

X

γ×

(

γvc(

( 12,1)F(−ν,ν+2) − ( 1

2,1)F(ν,2−ν)

)

+

+ιXσRR(

( 32,2)F(1−ν,ν+3) − ( 3

2,2)F(ν+1,3−ν)

))

.

(68)

The computing of these terms results from a simple appli-cation of the Hypergeometric formalism introduced above.The last step to achieve the dispersion velocity terms forthe velocity ellipsoid perturbed by spiral arms comes as asimple collection of the previous results as:

σRR = v2R − v2R,

σφφ = v2φ − v2φ,

σRφ = vRvφ − vφvR,

(69)

and with Eq.(68), (67), (66), (64), and (63) we conclude thecomputation of the second order moments. Of particular in-terest is for example the plot of the mixed term σRφ becauseof its connection with the azimuthal tilt (of an angle lv )with respect the configuration space cylindrical coordinatesof the velocity ellipsoid on the plane:

lv (R,φ, z) =1

2arctan

(

2σ2Rφ

σ2RR − σ2

φφ

)

, (70)

i.e. the vertex deviation that we plot in Fig. 13. There is

c© 2014 RAS, MNRAS 000, 1–??


Figure 13. Vertex deviation of the velocity ellipsoid at the plane.No graphical smoothing is applied in the contour plot to evidencethe grid resolution adopted.

much observational evidence for the dependence of the tilt-ing of the velocity ellipsoid on the plane (since Lindblad1958; Woolley 1970) and recently it has been studied inPasetto et al. (2012a). In the latter work, a detailed plotof this trend has been shown not only on the plane butalso above and below the plane. These data based on theRAVE survey highlight a decrease of the vertex deviationabove and below the plane. This observational trend is im-portant to validate the vertical treatment of the vertex de-viation outside the plane. Because the theory is not self-consistently validated outside the plane, we point out thatthe dependence on “z” of Eq.(70) comes from the verticaldependence of σRφ, σRR and σφφ. While the behaviour ofσRR and σφφ outside the plane are given by Eq.(44), whichfind observational constraints in the values in Table 2, thetrend of σRφ = σRφ (z) is entirely a simplified assumptionwe adopted in Eq.(69). The results of the convolution of justtwo SSPs of pop 1 and pop 2 of Table 1 (Fig. 14) treatedwith DWT seem to qualitatively reproduce the observationaltrend of Pasetto et al. (2012a) (their left panel in Fig 12).

7.1.4 Limits of the adopted approach

Even if the tilt of the velocity ellipsoid presents a regulartrend, the present theory accounts only for the contributionof the spiral arms which is well known to be incomplete. Thestream motions are expected to have major impact on thevelocity distribution in the Solar Neighbourhood as provenby several authors (e.g., Soubiran et al. 2003; Dehnen1998, 1999b; Seabroke & Gilmore 2007; De Simone et al.

Figure 14. Vertex deviation outside the plane on the solar radialposition lv = lv (R⊙, 0, z).

2004; Bassino et al. 1986; Hilton & Bash 1982; Mayor 1972;Woolley 1970). In order to correctly account for the tilt, acomplete map of the distribution of the molecular clouds en-countered by the stars along their past orbits is needed. Todate, this target is out of reach thus weakening any study ofthe kinematics and dynamics of MW based on direct orbit-integration. This is one of the main reasons spurring us toapply the method of moments in our model of the Galaxy.A limitation of our approach is surely the lack of a self-consistent treatment of the vertical DF. The DWT is lim-ited to in-plane stellar motions (the vertical and planar mo-tions are uncoupled because of the epicyclical approxima-tion). This represents a serious drawback of the theory thatmakes it not fully coherent with the assumptions we aregoing to male on the vertical tilt of the velocity ellipsoid.Nonetheless, we will include the vertical tilt in the GalaxyModel because, as shown in Fig. 14, our approach seems tocapture, at least qualitatively, the information hidden in theobservational data acquired by Pasetto et al. (2012a).

7.2 Vertical velocity field

The study of the local vertical profiles of the MW disks haslong tradition and is still pushed mostly under axisymmet-ric assumption (e.g., Just & Jahreiß 2010; Just et al. 2011;Soubiran et al. 2003; Siebert et al. 2003; Bienayme et al.2006; Soubiran et al. 2008). The study of the vertical kine-matics of spiral arms is still an open research field (e.g.,Williams et al. 2013; Widrow et al. 2012). The kinematicaldescription of the implemented model outside the plane isformally obtained here for the axisymmetric case. Neverthe-less, the tilt of the velocity outside the plane and the non-isothermality for single SSP reduce to zero thus matchingthe in-plane description presented in the previous section.

7.2.1 Vertical tilt of the velocity ellipsoid

The last non null cross term considered in Eq.(43) is σ2Rz =

σ2Rz (R,φ, z). It represents the tilt of the principal axis of the

velocity ellipsoid with respect to cylindrical coordinates out-side the plane. By symmetry in any axisymmetric model weexpect that σ2

Rz (R, 0) = 0, so that the principal axes of the

c© 2014 RAS, MNRAS 000, 1–??


velocity ellipsoid are aligned with the cylindrical coordinatesin the plane. This is not generally true in a non-axisymmetricmodel or in a model with non null radial average velocity(Cubarsi 2014a,b). Out of the plane the alignment is poorlyknown. For small z we can write

σ2Rz (R⊙, z) ≈ σ2

Rz (R⊙, 0) + z∂σ2

Rz (R⊙, 0)

∂z+O

(

z2)

, (71)

where σ2Rz (R⊙, 0) = 0 if and only if the model is axisymmet-

ric. The z-derivative of σ2Rz (the last term of Eq. (71)) eval-

uated on the plane gives the orientation of the velocity el-lipsoid just above or below the plane. Eddington (1915) andDejonghe & de Zeeuw (1988) have shown that the velocitytensor is diagonal in coordinates (if they exist) in which thepotential is separable. In the case where the MW potentialis separable in cylindrical or spherical coordinates, corre-sponding to mass distributions which are highly flattenedand dominated by the disk or spherical halo, we recoverthe limiting case of the vertical titling of the velocity el-lipsoid (van de Ven et al. 2003; Famaey & Dejonghe 2003;Helmi et al. 2003; Verolme & de Zeeuw 2002; Bienayme1999; Mathieu & Dejonghe 1996; Arnold 1995; Osipkov1994; Evans & de Zeeuw 1992; Dejonghe & Laurent 1991;Hunter et al. 1990; Merritt & Stiavelli 1990; Evans et al.1990; Ghosh et al. 1989).

These two cases correspond to the upper and lowerboundaries of the tilt term. They are usually written as

∂σ2Rz (R, 0)

∂z= λ (R)

σ2RR (R, 0) − σ2

zz (R, 0)

R, (72)

whereλ (R) ∈ [0, 1]. The factor λ(R) can be derived ei-ther analytically from orbit integration or is assumedλ = 0 for simplicity (e.g., van der Kruit & Freeman 1986;Lewis & Freeman 1989; Sackett & Sparke 1990). Numericalsimulations (e.g., Carlberg & Innanen 1987; Bienayme 2000)are performed to calculate explicitly the moments for a givengravitational potential. The result of the above studies isthat, at the Sun’s position λ ≃ 0.5. In our model for theGalactic kinematics, we prefer to adopt the analytical for-mulation of λ (R) given by Amendt & Cuddeford (1991):

λ (R) =R2∂R,z,zΦ

3∂RΦ +R∂R,RΦ − 4R∂z,zΦ

∣

∣

∣

∣

(R,z=0)

. (73)

The expression (73) is null for a potential separable in cylin-drical coordinates because the term ∂Φtot

∂R∂z2= 0. In spherical

coordinates λ = 1. The relation (73) will be used in the fol-lowing to describe the tilt of the velocity ellipsoid, obtainingfor the DM halo

λDM (R) = − R2

R2 (qΦ2 − 2) + 2hr,DM2 (qΦ2 − 1)

, (74)

and similarly, a unitary constant value for the bulge and stel-lar halo components is estimated. Finally, for the importantcontribution of the disk we will simplify Eq.(73) as

λd (R) = −∫ ∞

0

kR2J1(kR)h−1z dk

R(

k − 4hz

)

J0(kR) + 2J1(kR)(75)

that has to be included in Eq.(72) with a sum over all thedisks components.

7.2.2 Non-isothermal profile of the Galactic Disks

It is common to assume that the velocity distributionis isothermal in the vertical direction, or more precisely

that∂σ2

zz∂z

= 0 as we have already done in Section 7.1.This is certainly a reasonable assumption for small z, al-though there is no reason why the galactic disk should beisothermal at all. Bahcall (1984b) (but see also Bahcall1984a; Bahcall & Soneira 1984) treated the problem as-suming that non isothermality can be simulated by thesuperposition of more isothermal components. The obser-vations of Kuijken & Gilmore (1989b) (Kuijken & Gilmore1989a,c) show significant departures from isothermality atlarge z. One can prove (Amendt & Cuddeford 1991) that

in a cool disk σ2Rz = −σ2

zz∂σ2

zz∂z

(

∂σ2zz

∂R

)−1

. That tells us

immediately that if the tilt term of the ellipsoid is zero,then the velocity dispersion is constant in the vertical direc-tion. Therefore the assumption of an isothermal structurefor the Galactic disk is true only in the case of a gravi-tational potential which is separable in cylindrical coordi-nates. In the case of no strict isothermality, this approx-imation is valid within 1 kpc from the plane, where thefractional change in σ2

zz is expected to be less than 3%(Amendt & Cuddeford 1991; Cuddeford & Amendt 1991).For all reasonable gravitational potentials, σ2

RR > σ2zz

and one can prove that σ2zz has an extremum (min-

imum) on the plane, i.e.∂σ2

zz∂z

= 0 (Hill et al. 1979;Wainscoat et al. 1989; Fuchs & Wielen 1987; van der Kruit1988). For small z and fixed R = R⊙, one can per-form the Taylor expansion σ2

zz (R⊙, z) ≃ σ2zz (R⊙, 0) +

12z2

∂2σ2zz(R⊙)∂z2

+ o(

z3)

. Assuming now that σ2RR (R⊙, 0) =

α · σ2zz (R⊙, 0) as for Section 7.1, one obtains

σ2zz(R⊙,z)σ2zz(R⊙,0)

≃

1 +λ(R⊙)2R⊙

(α− 1)∣

∣

∣

∂ lnσ2zz

∂R

∣

∣

∣

(R⊙,0)z2. Supposing that σ2

zz ∝ ρ

in the plane, as in van der Kruit & Searle (1982), and thatρ follows an exponential law with constant scale length hR

we haveσ2zz(R⊙,z)σ2zz(R⊙,0)

≃ 1 +λ(R⊙)2R⊙

(

σ2RR(R⊙,0)σ2zz(R⊙,0)

− 1

)

z2

hRwhich

describes the non-isothermal case as,

σ2zz (R⊙, z) ≃ σ2

zz (R⊙, 0) +

+

(

λ (R⊙)

2R⊙

(

σ2RR (R⊙, 0)

σ2zz (R⊙, 0)

− 1

)

z2

hR

)

σ2zz (R⊙, 0)

(76)or

σ2zz (R, z) ≃ σ2

zz (R, 0) +λ (Φ)

(

σ2RR − σ2

zz

)

2 · hR ·R

∣

∣

∣

∣

∣

(R,0)

z2 (77)

which is clearly not constant. We have implemented thislast formulation in our models to take the non-isothermalstructure of the thin disks into account. This SSP non-isothermality based on Amendt & Cuddeford (1991) andCuddeford & Amendt (1991) hydro-dynamical model, ap-plies to single stellar population (SSP) alone. However, itis observationally very difficult to identify a truly homo-geneous population, i.e. in chemistry but also in age (e.g.,Bovy et al. 2012).

c© 2014 RAS, MNRAS 000, 1–??


8 COMPARISON WITH THE LITERATURE

Star count techniques have a long history and a comparisonof all the different flavours of this approach is a complextask. Mainly two kinds of kinematical models and associ-ated star counts are available in literature. The first ones donot depend on the underlying gravitational potential. Theycontain a large number of constants treated as free parame-ters and therefore can quickly fit large samples of data sim-ply because they do not integrate the Poisson equation. Aprototype of this modelling approach is Ratnatunga et al.(1987) (see e.g., Ratnatunga et al. 1989; Casertano et al.1990), who incidentally first applied the kinematical de-convolution of a DF in the phase space. However, due tothe complexity of their approach, the model cannot simplydeal with the tilt of the velocity ellipsoid out of the plane.Another model of this kind, with substantially no differ-ences, is by Gilmore (1984). Mendez & van Altena (1996)and Mendez et al. (2000) further refined the star-counts ap-proach to kinematical problems. Their work is based on theepicycle theory of disk kinematics and presents an originaltreatment of the differential rotation based on the Jeansequations that allow studying of the asymmetric drift foreach disk population. The limitation of these models is thatthe epicycle approximation limits the analysis of the diskkinematics only to regions near the plane and does not con-sider the vertical tilt of the velocity ellipsoid. Models ofthis kind, which are not based on a constrained potentialmay lead to somewhat non-physical solutions. Nevertheless,thanks to their simplification, they are useful to investigatemore difficult problems which require formulations that aremore sophisticated. For instance, the problem of the ver-tex deviation requires the axisymmetric hypothesis in thePoisson-solver to be relaxed. These difficulties are the sub-ject of debate and strictly linked with the problems of mod-elling the bulge too.

The most popular global Galaxy model available in theliterature and to which a finer comparison is due, is theBesancon model. This model has roots in the works byRobin & Creze (1986) and Bienayme et al. (1987), later im-proved by Robin et al. (2003). This model not only simu-lates CMDs, taking into account all evolutionary phases ofa star down to the white dwarf stages, but makes use of akinematical description linked to the gravitational potential.

However, this model is not fully consistent from a dy-namical point of view, equipped with much weaker dy-namical constraints than what we are presenting here andmissing a non-axisymmetric treatment of spiral arms. Themethod developed by Robin et al. (2003) is summarized inBienayme et al. (1987). Assuming suitable density profilesfor the components of a galaxy, their method calculates thetotal density profile using the Poisson-solver in axisymmetricstationary conditions. The authors derive the gravitationalpotential from

∆Φ(I)tot (R, z) = 4πGρtot

(

R, z; R(I)C , ρ

(I)C , M

(I)B , H

(I)z,i

)

, (78)

where R(I)C , ρ

(I)C , M

(I)B , H

(I)z,i are respectively the core ra-

dius of the halo, the central density of the disk, the to-tal mass of the bulge, and the different scale heightsof the disk at their first (I) guess input. This equa-

tion yields the first guess of the potential ⇒ Φ(I)tot (R, z)

where R(I)C , ρ

(I)C , M

(I)B have been considered as param-

eters. Obviously, some standard constraints are imposed

such as the rotation curve v(I)c (R) =

√

R∂RΦ(I)tot(R, z = 0)

which can be compared with the observational data.The parameters are varied to match the rotation curve

vc,bestfit(R) =√

R∂RΦ(I)tot(R, z = 0;R

(II)C , ρ

(II)C ,M

(II)B , H

(I)z,i )

and the new parameters R(II)C , ρ

(II)C , M

(II)B are used to

obtain a second guess for the potential ∆Φ(II)tot (R, z) =

4πGρtot(R, z; R(II)C , ρ

(II)C , M

(II)B , H

(I)z,i ) satisfying the first

dynamical constraint set by the rotation curve. The modelseeks then to satisfy the Boltzmann equation. Using theJeans equation their model assumes, in contrast with ourmodel, that σRz = (vR − vR) (vz − vz) = vRvz = 0 i.e.∂ρvRvz∂R

− ρvRvzR

= 0. With this simplification the Jeans equa-

tion reduces to∂ρv2z∂z

+ ρ ∂Φtot

∂z= 0 and assuming isothermal

behaviour σ2zz∂ρ∂z

+ ρ ∂Φtot

∂z= 0, where σ2

zz = (vz − vz)2 =

v2z . This equation can immediately be solved Φtot(R, z) =c1−σ2

zz ln (ρ(R, z)). This means that another solution of thisequation is Φtot(R, z = 0) = c1 − σ2

zz ln (ρ(R, z = 0)). Sub-tracting the two solutions we obtain another solution and

the implemented equation σ2zz ln

(

ρ(R,z)ρ(R,0)

)

= −Φtot(R, z) +

Φtot(R, 0), that is already present in Mihalas & Routly(1968). The model makes use of this equation to obtainthe best fit of the disk scale length with an iterative pro-cedure. The dynamical consistency is clearly poorer thanthat in our kinematical model, where formulation for themixed terms σRφ = σRφ (Φtot) and σRz = σRz (Φtot) areboth obtained with consistency from the potential as well asthe non-isothermality of σzz (z). This helps to decrease thelarge number of parameters, thus strengthening the consis-tency of our model. Our approximation implicitly assumes akinematically cold disk. Moreover, in the case of the Galac-tic potential no analytical, realistic formulation of the thirdintegral of motion (which is likely responsible for the smallirregularities present in the Galaxy structure) is available.Therefore, it is not yet analytically possible to derive a cor-rect DF that, thanks to the Jeans Theorem, could satisfy theself-consistency requirements. A hypothesis in common toour kinematical model and the one by Robin et al. (2003) isthe stationary state of the DF. The tri-shifted Gaussian rep-resents the only analytical solution of the Boltzmann equa-tion in a steady state and it is commonly adopted in severalmodels.

As it is not easy to satisfy the dynamical consistency,many kinematical models are possible and their solution isdegenerate. Based on these considerations, instead of theiterative procedure adopted in Robin et al. (2003) for whichno unique solution is guaranteed, it is perhaps better to letall parameters remain free to converge to the best fit solutionwith no ad hoc limits. Clearly the best way to proceed isto simultaneously constrain as many parameters as possiblewith the suitable minimization algorithms that we describedin Section 5.

Finally, we point out how in our model, the Poissonsolver is exactly the same as in Robin et al. (2003), but ouranalytical treatment allows us to consider the coupled poten-tial along the vertical-radial direction. This is not possiblewith the Robin et al. (2003) model that neither follows thevertical variation of the vertical tilt of the velocity ellipsoidnor the variations of the velocity ellipsoid vertical axis with

c© 2014 RAS, MNRAS 000, 1–??


the stellar populations nor the radial velocity coupling withthe presence of spiral arms. Both the Robin et al. (2003) andour models allow for a gradient in the vertical component of

the temperature profiles, i.e. σ2zz (R) =

n∑

i=1

σ2zz,i. However,

in the Robin et al. (2003) model all the σ2zz,i are consid-

ered constant for all stellar populations, i.e. σ2zz,i = const.∀i.

Therefore, the temperature gradient is a consequence of thedifferent scale height of each population. In contrast eachstellar population in our model has its own vertical profile,i.e. σ2

zz,i = σ2zz,i (R)∀i. This makes it possible to examine

separately different types of SSP. The task is nowadays fea-sible thanks to the wealth of data and even more in thenear future with Gaia, whose data will probably give defini-tive answers to long lasting problems such as the verticalisothermality, presence of Dark Matter and the origin of theGalactic disks.

Our model has the significant advantage of reproducingseveral observational constraints, such as (Section 4) the ro-tation curve, the outer rotation curve, the Oort functionsand constants, the mass inside 100 kpc, the vertical force,the surface density, and the parameter λ in the solar neigh-bourhood.

Clearly the space of parameter dimensions grows withthe square of the number of parameters; however, the num-ber of important parameters is rather small. The inter-play between a gravitational potential satisfying all the con-straints and parameter adjustment to fit the observationalCMDs secures the kinematical consistency of the model, asthe kinematics are simultaneously derived from the poten-tial and the properties of the stellar populations generatingthe potential. One has to remember that the DF has differ-ent dispersion axes for each population and that the angularmomentum for the orbits is linked to the rotational velocityvia the Boltzmann equation moments.

Finally, we plot the PDF of the proper motion µl,b andradial velocities vr populated for the same field of Fig. 9 inFig. 15.

9 CONCLUSIONS

We presented a Galaxy model which can be used to inves-tigate large datasets focused from MW surveys in great de-tail, of the Milky way with particular attention to the kine-matical modelling. This model gathered the heritage of thePadua model that stems from the early studies of the stel-lar content of the Palomar-Groeningen survey towards theGalactic Centre by Ng et al. (1995); Bertelli et al. (1995),followed by studies of specific groups of stars and interstellarextinction by Bertelli et al. (1995, 1996); Ng & Schultheis(1996); Ng & Bertelli (1996) to mention a few, the studiesof the stellar content towards the Galactic Pole (Ng et al.1997), the development of a new minimization technique forthe diagnostics of stellar population synthesis (Ng 1998b),the study of the Galactic Disc Age-Metallicity relation(Carraro et al. 1998), the possible relationship between thebulge C-stars and the Sagittarius dwarf galaxy (Ng 1998a),the developments of AMORE (Automatic Observation Ren-dering) of a synthetic stellar population’s colour-magnitudediagram based on the genetic algorithm (Ng et al. 2002),the study of 3-D structure of the Galaxy from star-counts in

view of the Gaia mission (Vallenari et al. 2003) and of thekinematics of the Galactic populations towards the NorthPole with mock Gaia data (Vallenari et al. 2004; Pasetto2005; Vallenari et al. 2006).

The building blocks of the Padua Galactic model area synthetic Hertzsprung-Russell Diagram generator a kine-matical model, and a MW gravitational potential model.This tool has now been updated following a novel approachto the theory of population synthesis that borrows andadapts to the present aims a few concepts from StatisticalMechanics:

(i) The model is grounded on the concept of theProbability Distribution Function for stellar populations.The system-Galaxy is framed in a theoretical ExistenceSpace where it is characterized by a number of key re-lations (mass function, age-metallicity, phase-space, andmetallicity/phase-space).

(ii) The model is able to analyze and reproduce observ-able quantities regardless of size and amounts of the data toa analyse: this is achieved thanks to the use of PDF insteadof star-counts.

(iii) The distribution of mass and mass density and asso-ciated gravitational potential are thoroughly discussed andformulated for each component of the MW together with afew other important issues such as the rotation curve, thevertical force acting on the plane, the presence of spiral armsand their effects on dynamics and kinematics, the presenceof non axisymmetric features etc.

(iv) Particular effort is paid to include spiral arms forwhich we develop a completely new treatment of the massdensity, kinematics and extinction. Several treatments of theresonance areas to deal with the star-count technique havebeen explored and implemented.

(v) A novel formulation for the extinction has been imple-mented to account for the new non-axisymmetric features ofthe model and in preparation for a forthcoming star-countmodel of the bulge.

(vi) A genetic algorithm has been included to deal simul-taneously with photometric information, as well as kinemat-ical and gravitational information.

(vii) Particular care is paid to the photometric popula-tion synthesis to simulate the photometric properties, mag-nitudes and colors for samples of stars of unprecedented sizetaking advantage of the concept of PDF to populate CMDsand luminosity functions bearing in mind the data that willsoon be acquired by space observatories like Gaia.

(viii) The model has been compared with similar modelsin literature, for instance the popular Besancon model, tohighlight differences and similarities.

The range of applicability of our Galaxy Model is verylarge. It can already be applied to existing MW surveys onwhich the model has already been tested. None of the exist-ing surveys are actually comparable with Gaia for precisionand amounts of data, but all of them already investigatedifferent aspects of the Galaxy. To mention a few, we re-call the Radial Velocity Experiment (Steinmetz et al. 2006;Zwitter et al. 2008; Siebert et al. 2011) of which the fourthdata release (Kordopatis et al. 2013) has been used to testthe model (Pasetto et al. 2012b,a). The Galaxy Model couldbe applied to the data coming from the Apache Point Obser-vatory Galactic Evolution Experiment (e.g. Majewski et al.

c© 2014 RAS, MNRAS 000, 1–??


Figure 15. Proper motion distribution (left panel) for a field of about 10000 stars with spiral arms. The contribution of the spiral armspopulation is evidenced with subscript“sp”. (Right panel) same as left panel but for radial velocity. The CMD of this example is in Fig.9.

2015), a companion program of the Sloan Digital Sky Sur-vey (Ahn et al. 2014) that makes available an infrared cat-alogue of several hundred thousands radial velocities (fromhigh resolution spectra) that are suitable to stellar popula-tion studies within the plane. This database is especially in-teresting when used in combination with the data from theKepler/K2 mission, because this would allow us to inves-tigate stellar populations also with photometry. The ongo-ing GALAH survey (De Silva et al. 2015), a large Australianproject that will measure the abundances of thirty elementstogether with HERMES providing spectrographic measure-ments of radial velocities is another example to which theGalaxy Model could be applied to investigate the galacticarchaeology and archeochemistry.

Finally, our model can be applied also to entirely differ-ent astrophysical scales, such as in asteroseismology. Whenspectroscopic analysis is combined with seismic information,precise constraints on distances, masses, extinction and fi-nally ages can be obtained. The CoRoT red giant field (e.g.Chiappini et al. 2015) analysis is an example, even thoughthe statistical samples are still very small.

A golden age for systematic studies of MW is imminent.Existing photometric and spectroscopic surveys, as well asfuture ones such as Gaia, will be crucial to obtain the ulti-mate model of our own Galaxy, a fundamental local step tointerpret the Universe in a cosmological framework. In thiscontext, the Galaxy Model we have developed is awaitingcomplete validation by more precise kinematical data, agesand metallicities. In the meantime, we have presented herethe first kinematical model that can simultaneously dealwith the age-velocity and dispersion-metallicity relations ina robust dynamical-kinematical framework over the wholespace of variables defined by the proper motions, radial ve-locities, and multi-band photometric data (magnitudes andcolours). Future developments will include an upgrading ofthe bulge to include non-axisymmetric descriptions of thebar and chemical enrichment. The Galaxy Model can be ac-cessed from the internet interface at www.galmod.org.

ACKNOWLEDGEMENTS

SP thanks Mark Cropper for the constant support in therealization of this work. G.N. would like to acknowledge

support from the Leverhulme Trust research project grantRPG-2013-418.

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Xue X. X., Rix H. W., Zhao G., Re Fiorentin P., Naab T., Stein-metz M., van den Bosch F. C., Beers T. C., Lee Y. S., BellE. F., Rockosi C., Yanny B., Newberg H., Wilhelm R., KangX., Smith M. C., Schneider D. P., 2008, ApJ, 684, 1143

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c© 2014 RAS, MNRAS 000, 1–??


APPENDIX A: HYPERGEOMETRIC FORMULATION OF THE “REDUCTION FACTOR”

There is quite a number of works in the literature, see section 1, that based the interpretation of the reduction factor on thework of Lin et al. (1969) through the computation of the “q”-factor, i.e. the integral of eq. B8 in Appendix B of the mentionedpaper. In this appendix, we express for the first time that integral as a function of the well-known Hypergeometric function.To achieve this result, we make use of the following theorem:

Theorem: from Erdlyi et al. (1954) (Vol 2, pg. 400, Eq.(8)) we have that for any z, α, β, γ, δ ∈ C with Reγ > 0,Reρ > 0 and Re (γ + ρ− α− β) > 0 the following relation holds:

2F2 (ρ, γ + ρ− α− β; γ + ρ− α, γ + ρ− β; z) =Γ (γ + ρ− α) Γ (γ + ρ− β)

Γ (γ) Γ (ρ) Γ (γ + ρ− α− β)ez∫ 1

0

xγ−1(1 − x)ρ−1e−xz2F1 (α, β; γ;x) dx,

(A1)with 2F2 (∗) the hypergeometric function and Γ the Eulero-gamma function.

We want to prove the following corollary to the previous theorem.Corollary: For any z, ν ∈ R the following relation holds:

1

2π

∫ π

−π

cos(νs)e−z(cos(s)+1)ds = 2F2

(

1

2, 1; 1 − ν, ν + 1;−2z

)

, (A2)

where 2F2 is the Hypergeometric-regularized function.Proof. We start relating the Hypergeometric-regularized function, 2F2, to the Hypergeometric function, 2F2, by writing

2F2

(

1

2, 1; 1 − ν, ν + 1;−2z

)

=sin(πν)

πν2F2

(

1

2, 1; 1 − ν, ν + 1;−2z

)

. (A3)

This equation, because of the Theorem Eq.(A1) reduces to:

sin(πν)

πν2F2

(

1

2, 1; 1 − ν, ν + 1;−2z

)

=sin(πν)

πν

Γ(1 − ν)Γ(ν + 1)

πe−2z

∫ 1

0

e2xz 2F1

(

ν,−ν; 12;x)

√1 − x

√x

dx

=1

π

sin(πν)

ν

ν

sin (πν)e−2z

∫ 1

0

e2xz cos (2ν arcsin√x)√

1 − x√x

dx

=e−2z

π

∫ 1

0

2e2y2z cos (2ν arcsin y)√

1 − y2dy

=e−2z

π

∫ π/2

0

2e2zsin2q cos (2qν) dq,

(A4)

where in the second line we made use of 2F1 and Eulero-gamma function properties, in the third and fourth rows we changevariables, x = y2 and y = sin q, accounting for the dominion of integration. The last relation is clearly an even function, thus:

2F2

(

1

2, 1; 1 − ν, ν + 1;−2z

)

=1

π

∫ π

0

e−2z+2zsin2 s2 cos (νs) ds

=1

2π

∫ π

−π

e−z(1+cos s) cos (νs) ds,

(A5)

which concludes our proof. The expression for the Reduction factor comes then easily.Needless to say that the advantage of having this formulation for the reduction factor stands not only in the compact

elegant formalism, but probably more on the rapidity of performing its evaluation. A test on a commercial processor availableto date (Intel core-I7, 3.0 GHrz) shows that the integral evaluation against the Hyper-geometrical formulation is ∼ 10−4 persecond. This translates as the possibility (to date) to generate mock catalogues for a small survey of data with about ∼ 104

stars each second, against a generation of the same kinematics catalogue by numerical integration (with adaptive Runge-Kuttascheme) in about 3 hrs! This result is even more striking if you distribute the computation on several processors.

c© 2014 RAS, MNRAS 000, 1–??

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