Issue 
A&A
Volume 545, September 2012



Article Number  A14  
Number of page(s)  10  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201219698  
Published online  29 August 2012 
Theory of stellar population synthesis with an application to Nbody simulations
^{1} University College LondonDepartment of Space & Climate Physics, Mullard Space Science Laboratory, Holmbury St. Mary, Dorking Surrey, RH5 6NT UK
email: sp2@mssl.ucl.ac.uk
^{2} Department of Physics and Astronomy “Galileo Galilei”, University of Padova, 35122 Padova, Italy
Received: 29 May 2012
Accepted: 13 July 2012
Aims. We present here a new theoretical approach to population synthesis. The aim is to predict colour magnitude diagrams (CMDs) for huge numbers of stars. With this method we generate synthetic CMDs for Nbody simulations of galaxies. Sophisticated hydrodynamic Nbody models of galaxies require equal quality simulations of the photometric properties of their stellar content. The only prerequisite for the method to work is very little information on the star formation and chemical enrichment histories, i.e. the age and metallicity of all starparticles as a function of time. The method takes into account the gap between the mass of real stars and that of the starparticles in Nbody simulations, which best correspond to the mass of star clusters with different age and metallicity, i.e. a manifold of single stellar sopulations (SSP).
Methods. The theory extends the concept of SSP to include the phasespace (position and velocity) of each star. Furthermore, it accelerates the building up of simulated CMD by using a database of theoretical SSPs that extends to all ages and metallicities of interest. Finally, it uses the concept of distribution functions to build up the CMD. The technique is independent of the mass resolution and the way the Nbody simulation has been calculated. This allows us to generate CMDs for simulated stellar systems of any kind: from open clusters to globular clusters, dwarf galaxies, or spiral and elliptical galaxies.
Results. The new theory is applied to an Nbody simulation of a disc galaxy to test its performance and highlight its flexibility.
Key words: HertzsprungRussell and CM diagrams / methods: statistical / methods: numerical / stars: kinematics and dynamics
© ESO, 2012
1. Introduction
In the past few decades the unprecedented development of the computational facilities drastically increased the number of Nbody based astrophysical simulations. In some areas of astrophysics, this powerful approach represents the only viable laboratory experiment because gravitational interaction on an astrophysical scale is clearly impossible to reproduce in any laboratory.
Since the pioneering works of von Hoerner (1960) and Aarseth (1963), the orbit integration techniques for the Nbody problem (with N the number of gravitationally interacting bodies) greatly improved (see, e.g., Hockney & Eastwood 1988; Aarseth 2003) both theoretically, with the developments of the treecodes, particlemesh (PM), particleparticleparticle mesh (P^{3}M) or see, e.g., Dehnen & Read (2011) for a recent review, and technically, by exploiting the messagepassinginterface protocol for the use of massively parallel machines. Although the Nbody simulations were originally conceived as a tool to follow the integration of the orbits, the importance of including energy dissipative processes soon became evident. The treatment of baryonic interactions in Nbody simulations is nowadays standardised by popular protocols such as smoothed particle hydrodynamics (SPH) or adaptivemeshrefinement^{1} (AMR) implemented in several codes, e.g., Algodoo (e.g., Koreš 2012), DualSPHysics (e.g., GomezGesteira et al. 2012), SPHflow (e.g., Oger et al. 2006), ENZO (e.g., O’Shea et al. 2004), EvoL (e.g., Merlin et al. 2010), GCD+ (e.g., Kawata & Gibson 2003), GADGET (e.g., Springel et al. 2001), Gasoline (e.g., Wadsley et al. 2004), FLASH Dubey et al. (e.g. 2008) or RAMSES (e.g., Teyssier 2010; Few et al. 2012), which also includes (in astrophysical context) a stickyparticle approach (Bournaud & Combes 2002; Martig & Bournaud 2010). Thus, even though the recipes for implementing the physics of dissipative phenomena are still controversial, several Nbody codes are currently able to evolve with time the baryonic component, i.e. gas and stars, in their mutual interaction. Therefore, it is nowadays possible and mandatory to develop the right tools to compare the results of dynamical Nbody simulations with observational data for the stellar content of galaxies.
Tantalo et al. (2010) recently developed a technique to derive the integrated spectra, magnitudes and colours of the stellar content of simulated galaxies. These authors focused on the evolution of nonresolved stellar populations and exploited the concept of the spectral energy distribution (SED) of an SSP in the context of discrete resolvedmass points of an Nbody simulation. There, the integrated monochromatic flux in a given photometric system (set of discrete passbands Δλ) of a galactic stellar component at the time t and metallicity Z, , is taken to be the convolution of the star formation rate (SFR) (which is assumed to depend only on age and metal content), and the integrated monochromatic flux of constituent SSPs, , i.e. ^{2}.
In this study we present a new theory of population synthesis that is particularly designed to manage very many stars, and is based on the concept of a distribution function (DF). We focus then on generating the CMDs for the stellar system which are simulated with advanced hydrodynamical Nbody simulation techniques. The subject is particularly timely in view of the modern capabilities of data acquisition of starbystar photometry in nearby galaxies. This is thanks to modern instrumentation already in place or in progress for the coming years, and also in view of the Nbody simulations of model galaxies, in which sampling of either real or fake stars with many millions of objects is possible.
The only prerequisite for the method to work is that minimal information about the history of star formation and chemical enrichment is implemented in and read off the Nbody simulations. For each starparticle of the Nbody simulation we must know the age at which it was born and the chemical composition (metallicity) of the parental medium. The new formulation of population synthesis takes naturally into account the gap between the mass of the real stars and the mass of the starparticles in Nbody simulations if these latter are closer to the mass of a star cluster than the mass of a real star. Therefore the starcontent of an Nbody simulation is better described as a manifold of star clusters with different age and metallicity, i.e. a manifold of SSP whose photometric properties, primarily the CMD, are well known. Second, CMDs containing an arbitrary number of stars can be easily simulated by defining and making use of the concept of DF of stars in the CMD.
This novel technique is applicable to the stellar content of single open/globular clusters, to our own Galaxy, to that of nearby galaxies within the Local Group, and to all galaxies of the local Universe whose stellar content can be resolved into stars. Moreover, the algorithm can be easily interfaced with other codes, e.g. Galaxia (e.g., Sharma et al. 2011), or the Galaxy Simulators HRDGST (e.g., Ng et al. 1995, 1996, 2002a,b; Ng & Bertelli 1996; Vallenari et al. 2006), and Besancon model (e.g., Robin et al. 2003), thus extending their performances with the advantages that will become soon clear in what follows.
Setting up the new technique, we focused our attention on a few key requirements that should be met:

1.
The algorithm has to be independent of the particular Nbody simulation it is applied to. Thus, it is designed to accept the output of a simulation as input, without interfering with the Nbody integration.

2.
The algorithm must not depend on the specific prescriptions used to generate the stellar models that are adopted to describe the composite stellar populations and associated CMDs.

3.
The technique needs to handle CMDs populated by several billions of stars, i.e. which are potentially representative of the largest systems of stars known (e.g., giant elliptical galaxies). Indeed, our algorithm should be applicable to every resolved stellar population no matter what the nature of the stellar system under consideration. Therefore CMDs for giant elliptical or spiral galaxies that contain up to 10^{12} stars have to be synthesized with agility.

4.
Finally, we require the code to only require little computational resources. The typical CMD of point 3, has to be obtained on a serial machine, i.e. its realization has to avoid more complicated MPI/OpenMP programming protocols.
To achieve those goals, we have developed a method for synthesising CMDs that takes all the above requirements into account. The plan of the paper is as follows: in Sect. 2 we extend some of the theory concepts of the stellar populations within the framework of the DFs. In Sect. 3 we particularise the general theory to the case of the Nbody simulations. In Sect. 4 we deal with the simulations of CMDs that contains huge numbers of stars. In Sect. 5 we present an example of a synthetic CMD realization. In Sect. 6 we show an example of the technique. Finally we draw some concluding remarks in Sect. 7 and briefly mention some possible applications of the new method to the synthesis of stellar populations.
2. Theory of stellar populations
We extend here the theory of stellar populations to systems (assemblies of stars) with an assigned distribution in the phasespace. This completes the standard theory of stellar populations (e.g., Salaris & Cassisi 2005; Greggio & Renzini 2011) by assigning a position and velocity to each star.
Every real (or realistically simulated) stellar system is a set of stars born at different times and positions, and with different velocities, masses and chemical compositions. We call this assembly of stars compositestellarpopulation (CSP). The position x of each star of a CSP is a point in the space X of coordinates (or space of configurations), and its linear momentum p is geometrically referred to as a fiber of the cotangent space of X at x (Jose & Saletan 1998). Therefore, the phasespace of a CSP is the cotangent bundle of its configuration space X. It follows that T^{ ∗ }X is a manifold, with N the total number of stars in the CSP. Each point in it defines the phasespace of our CSP. We indicate this manifold with the symbol Γ, i.e. Γ ≡ T^{ ∗ }X where . Moreover, to completely define the CSP parameters at a generic time t, we need to specify the distribution in the space of the masses, M, and metallicities, Z of all its members. We call this extended phasespace the existencespace of the CSP, i.e. E ≡ M × Z × Γ of the CSP at the time t and E × R the extendedexistencespace with time t. In E × R, the stars evolve with time, i.e. they continuously move in space, lose mass, enrich in metals, and move in the phasespace. Hence we can safely define in E the distribution function (DF) for the CSPs under the assumption of continuity and differentiability. Let us consider a sample of identical systems whose initial conditions span a certain volume of the space E. We refer to this sample simply as an ensemble, borrowing the nameroot from the grand microcanonical ensemble adopted in Statistical Mechanics. The number of systems dN at the time t with mass within dM, metallicity within dZ and phasespace within dΓ = (dx,dv) is given by (1)with the DF in E, f_{CSP}:R^{+} × R^{+} × R^{6N} × R → R^{ + } continuous and with partial derivative continuous (where R is the set of all real numbers, and ) and the total number of systems in the ensemble is fixed by normalising to one the DF in the space E, i.e. (2)We proceed to formally define an SSP as follows. At a given time t, we divide E into a grid in terms of discrete intervals of metallicity dZ and phasespace dΓ. Then every infinitesimal unit (i.e. every elementary cell) of this subspace of E defines an SSP.
We assume that every CSP originates at time t_{0} from an episode of single star formation in a chemically homogeneous medium. The stars of the CSP have the mass spectrum where is the initial mass function (IMF). Their metallicity distribution is . Finally, their phasespace distribution function is . The DF of the CSP can be written as (3)where n is the number of SSP, with eventually n → ∞, and is the DF of SSP of which Z = Z_{0}, Γ = Γ_{0}, t = t_{0} are the metallicity, SSP’s phasespace and age at the instant t_{0}, respectively.
As time elapses, this stellar population ages. According to their mass, the stars eventually leave the mainsequence (MS) after a time t_{MS} = t_{MS}(M) and soon after die (supernovae) or enter a quiescence stage (white dwarfs), injecting metals into the interstellar medium in form of supernova explosions or quiet winds. The interstellar medium becomes richer in metals due both to selfenrichment by the SSP under consideration and the contribution from all other stellar SSPs. At the generic time t > t_{0}, i.e. after an elapsed time τ ≡ t − t_{0} otherwise known as the age of the stellar population, the SSP has processed stellar material. As a result of this activity, an agemetallicity relationship is built up for the CSP.
In addition to this, the number of trajectories in the phasespace entering a volume dΓ, will in general be different from the number of those leaving the same volume. Thus f_{Γ} evolves following Liouville’s equation of the form (4)which is independent of the nature of the equation of motion (e.g., its correctness in the form of Eq. (4)does not require the existence of a Hamiltonian). ℒ is the Liouvillean operator, eventually extended to account for a creation function C(f_{CSP};t), and i the imaginary unit. The formal solution of this equation is given by the Taylor expansion series of the time dependence of around (5)and defines the infinite series of operators applied on any function on its right side. In this way, the initial SSP has generated a CSP whose distribution in the existence space of the CSP, E, has a DF .
Now that we have stated these fundamental concepts, it is easy to proceed with the following formal definitions to recover the usual concepts of the classical synthesis of population theory:

Presentdaymassfunction: the integral of the DF over themetallicity Z and phasespace Γ (from Eq. (1)) (6)yields the “present”daymassfunction (PDMF), i.e. the total number of stars per mass interval at the time t. This can be expressed by the approximate relation (7)where is IMF of the MS stars. More finetuned approaches based on the evolutionary flux, total luminosity and fuelconsumption theorem can be easily worked out upon necessity.

Agemetallicity function: integrating the DF over the mass M and phasespace Γ(8)we obtain the agemetallicity relation, i.e. the number of stars formed per metallicity interval at the time t.

The phasespace distributionfunction: integrating upon the mass M and metallicity Z(9)where we made use of Eq. (5), yields the the phasespace DF.
In a natural way, we can extend the approach used in defining Eqs. (6), (8)and (9)by performing the following monodimensional integration:
Metallicity phasespace relationship: we consider the following integration (10)to define the metallicityphasespace relationship. Very often the projection of this function onto the configuration space is used (11)where d^{3N}v is the 3N velocity element of the Γ space. This plays an important role, e.g., in relation to radial metallicity gradients of resolved populations in external galaxies and the vertical metallicity gradients in the Milky Way, i.e. ∀t can implicitly define a function that relates the metallicity Z and the radius of a galaxy R once cylindrical coordinates are in use.
Massphasespace relationship: in the same way we define (12)i.e. the massphasespace relation. More often its projection onto the configuration space (13)is used, which is related, e.g., to the evolution of the globular clusters and the masssegregation effects, i.e. we can use it to express the higher concentration of massive (or binary stars) through the centre of the cluster with respect to the less massive stars (e.g., Spitzer 1987).
Massmetallicity relationship: Finally, we define (14)i.e. the massmetallicity relation for a resolved stellar population. This relation is of paramount importance e.g., when the masses of the stars are obtained by parallaxes, transits, and asteroseismology. For nearby galaxies this relation is relevant only in studies of integrated photometry.
3. Stellar populations in Nbody simulations
The concepts we have just developed can be straightforwardly applied to Nbody simulations of any kind, from those for globular clusters with single or multiple stellar populations (e.g., Milone et al. 2012; Norris 2004), to those for the Milky Way stellar fields observed along any line of sight (e.g., Ng et al. 1995; Vallenari et al. 2006), to the dwarf galaxies of the Local Group (e.g., Grebel 1997; Mateo 1998; Tolstoy et al. 2009), or even external galaxies once they are resolved into individual stars (e.g., Harris & Harris 2002; Rejkuba et al. 2011; Crnojević et al. 2011). Indeed, in the adopted formulation, all the dynamical aspects of the processes governing the formation of the astronomical object under investigation are considered through the most general Liouvillean operator, whose form can be specified case by case. In this context, it is worth recalling that a mutual dependence of star formation, which is responsible for the building up of the stellar population of a stellar system, and the dynamical processes that govern the large scale aggregation of it, is in general supposed to occur in extended manybody interactions (e.g. Pasetto et al. 2011, 2012).
Starparticles in Nbody simulations: Before proceeding further, we must suitably link the definition and properties of CSPs to the elemental building blocks of Nbody simulations, i.e. the “starparticles”.
We define an Nbody simulation as a stochastic realization of in the existence space E of the CSPs. We focus our attention on the evolution of a single starparticle in an Nbody simulation. This particle p is born at the instant t_{0,p} ≡ t_{p} with metallicity , at a position in the single particle phasespace γ (a submanifold of T^{ ∗ }X) given by , with a certain metallicity Z = Z(t_{p}) (the metal content of the parent gas component at time t_{p}), and with a mass .
The mass of the starparticles and the smaller mass resolution of the Nbody simulations is one of the greatest problems of modern computational Nbody simulations and companion mathematical techniques. Apart from a few recent Nbody simulations that were specifically tailored for star clusters (e.g., Hut et al. 2003; Zonoozi et al. 2011; Pelupessy & Portegies Zwart 2012), the particle mass of the modern Nbody simulations dedicated galaxies and galaxy clusters can vary within a wide range, typically (e.g., Saitoh et al. 2009, 2008; Guedes et al. 2011; Okamoto & Frenk 2009). This means that the average mass of the starparticles in the Nbody simulations is larger than the average mass of the stars in a SSP. Therefore each starparticle is representative (in mass) of many hundreds stars (or more). However, since all stars in a starparticle of a simulation are assumed to be born at the same time with the same metallicity, the stellar content of the starparticle itself is well described by an SSP (see SSP definition in Sect. 2). Nevertheless, Nbody simulations cannot easily describe the phasespace distribution of individual stars within a starparticle. In this paper we assume that all stars within each starparticle share the same position in the phasespace. As a consequence of this, the volume of the existence space E occupied by elemental cells is bounded by the mass resolution, say dE_{min} ≡ dM_{p} × dZ × dΓ, and it is generally larger than the volume necessary to properly map a CSP, .
On one hand, if the problem clearly shows that attempts to map the complete IMF are at present out of reach, this indicates on the other hand how a possible solution has to be searched for in the concept of SSP itself.
In an Nbody simulation, at the time t > t_{p} a starparticle is located in , has metallicity and mass . Therefore, mass, metallicity and position in the singleparticle phasespace univocally identify the Nbody particle all along its evolutionary history from t_{p} = t_{0,p} to t. At the initial time t_{p} for each particle p we can associate an SSP to the starparticle in a natural way, provided that some rescaling of the SSP mass, M_{SSP}, is made in relation to the starparticle mass, M_{p}. From Eq. (2), the total mass of the SSP can be written as follows: (15)where δ is the ordinary multidimensional Dirac function, and in the last row we used Eq. (7)with t_{p} = t_{0,p} = 0 < t_{MS}∀M. In other words, the SSP mass written in terms of the DF f_{CSP} or the IMF.
In the same way, we can cast the DF of the CSP as a function of the starparticle mass M_{p} and vice versa. The following identities can be written: (16)While associating the SSP to a generic starparticle born at time t_{p} with metallicity Z_{p}(t_{p}) = Z_{p} and mass M_{p}(t_{p}) = M_{p} is straightforward, at a generic time t > t_{p} it is no longer so simple. But if the mass and metallicity of a starparticle do not change with time, the SSP associated to a generic starparticle will simply evolve and age, losing a certain amount of mass in form of gas, changing a fraction of its living stars into remnants, and changing its spectral and photometric properties in a way that is fairly well known from the theory of stellar evolution and population synthesis (e.g. Tosi et al. 1991; Bertelli et al. 1994; Tostoy & Saha 1996; Aparicio et al. 1997; Harris & Zaritsky 2001; Gallart et al. 2005; Bertelli et al. 2008, 2009).
4. Huge CMDs at no cost
With the formalism we developed on the basis of the CMD of CSP, we now tackle the key question: how can we build CMDs for billions of stars in practice at less computational cost? Because real CPS are the result of the past history of star formation in clusters and fields at the complexity levels of galaxies, how can we easily achieve the same level of information in numerical simulations?
4.1. Associating an SSP to a starparticle
The SSP corresponding to the pth starparticle is obtained by twodimensional interpolation (in age and metallicity). The number of stars for the interval of luminosity L and effective temperature T_{eff}, dLdT_{eff} results from
(17)where we referred, by extension of the Jacobeanmatrix formalism, with to the transformation applied on two of the dimensions in which SSP is defined: from agemetallicity space to the effective temperatureluminosity space. Or analogously, (18)for the same transformation to the space of the colour C and magnitude m. In general there is no analytic formulation for the two matrixes and . They are derived numerically from the tabulations of bolometric corrections, and colours of, e.g., the JohnsonCousinGlass system.
The key idea of Eq. (17)or (18)is to describe a CMD as a matrix, i.e. a projected DF, whose elements are the relative frequency (or percentage) of stars of different colour and magnitude in some photometric system, per elemental area of the CMD. For a given photometric system with passbands Δλ_{α} (for instance α stands for U, B, V, ...) for which magnitudes m_{α} and colours C_{αβ} ≡ m_{α} − m_{β} can be computed (magnitudes and colours can be either absolute and intrinsic or apparent and reddened^{3}), Eq. (18)represents a 2D grid of elemental square cells , identified by the coordinates of their centres and (m_{α} − m_{β})^{c}. The path drawn in the CMD by a single SSP of assigned age and chemical composition, see Sect. 2 and Eq. (18), will occupy a number of cells from the main sequence to the last observable stage. Each cell is populated by a certain number of stars according to the underlying luminosity function of the SSP, which in turn is related to the evolutionary rate and IMF (see Eqs. (6)and (7)). If many SSPs are present, as in the case of a composite CSP, each cell contains a total number of stars given by the sum of all contributions by different SSPs passing through this cell. Using Eqs. (3)together with (18), we obtain (19)Therefore, a relative percentage of stars is associated to each cell (matrix element) with respect to the total. The number of stars per cell can be easily normalised according to the problem under investigation. The observational CMDs that contain an arbitrary number of stars, from a few thousands to billions, is reduced to a matrix of number frequencies (relative percentages, Eq. (18)). The prescription can be easily extended to include any history of star formation of any intensity, ψ, as well as kinematical effects (f_{Γ} as in Eq. (5)).
Fig. 1
Left panel: the HRD of SSPs with assigned age and metallicity. The size of the circles indicates the current star mass along the isochrones and/or SSP. The stellar models in use are taken from the library of Bertelli et al. (2008, 2009). The colour code refers to the size of the dots: the smaller is the dot, the more yellow it is and the smaller the mass of the plotted stars. The bigger the dot, the bluer its colour and the bigger the mass of the stars. The mass considered in the models of this example ranges from 0.15 to 20 M_{⊙}. Right panel: the same SSPs displayed in the left panel but in the observational CMD M_{V} vs. M_{V} − M_{I}. The CMD is subdivided to a discrete grid of elemental cells, in each of which a certain number of stars fall. The stars in each cell may belong to different SSPs. The size of the cell is large clearly show evidence the “outoffocus” effect intrinsic to this tessellation technique. Colourcode spans account for the number of stars in the cells from 0 (the bluer colour) to 10^{5} (the light yellow colour). 

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Finally we are left to specify the initial distribution of mass (IMF) and the adopted database of stellar models:

The initial mass function: to calculate an SSP, we need an IMF.The most popular IMF is given by (20)with (21)with ξ_{0} normalisation factor of the IMF, , with lower (upper) limit for the ithmass interval and ΔM_{i} ∩ ΔM_{j} = ∅ for i ≠ j. The case k = 1 in Eq. (20)is often referred to as Salpeter’s IMF with powerlaw slope x_{i} = 2.35 ∀i, the case k = 3 is referred as Kroupa’s IMF with its specific slope and intervals (see e.g., Kroupa 2001) or in general any IMF function can be approximated by a sum as in Eq. (20)for a suitable choice of k. Therefore, in the general case, the total number of stars plotted in an SSP is (22)or for x_{i} = 1. If we choose to sample all starphases of higher luminosity (e.g., asymptotic giant branch (AGB) and planetary nebula (PNs)) with N_{SSP} ≅ 10^{5} stars it becomes immediately evident how the graphical realization of a CMD for an Nbody simulation encounters serious problems once the number of starparticles involved, N, is high, say for N ~ 10^{6 ÷ 8}.

The database of SSPs: we briefly report here on the data base of SSPs that was calculated for the purposes of this study. The stellar models used are those by Bertelli et al. (2008, 2009), which cover a wide grid of helium Y, metallicity Z, and enrichment ratio ΔY/ΔZ. The associated isochrones include the effect of mass loss by stellar wind and the thermally pulsing AGB phase according to the models calculated by Marigo & Girardi (2007). The code used is the last version of YZVAR developed over the years by the Padova group used in many studies (for instance Chiosi & Greggio 1981; Chiosi et al. 1986b,a, 1989; Bertelli et al. 1995; Ng et al. 1995; Aparicio et al. 1996; Bertelli & Nasi 2001; Bertelli et al. 2003) and was recently extended to obtain isochrones and SSPs in a large region of the Z − Y plane. The details on the interpolation scheme at a given ΔY/ΔZ are given in Bertelli et al. (2008, 2009). The present isochrones and SSPs are in the JohnsonCousinsGlass system as defined by Bessell (1990) and Bessell & Brett (1988). The formalism adopted to derive the bolometric corrections is described in Girardi et al. (2002), while the definition and values of the zeropoints are described in Marigo & Girardi (2007) and Girardi et al. (2007) and will not be repeated here. Suffice it to recall that the bolometric corrections stand on an updated and extended library of stellar spectral fluxes. The core of the library now consists of the ODFNEW ATLAS9 spectral fluxes from Castelli & Kurucz (2003), for T_{eff} ∈ [3500,50 000] K, log _{10}g ∈ [− 2,5] (with g the surface gravity), and scaled solar metallicities [M/H] ∈ [− 2.5, + 0.5]. This library is extended at the intervals of high T_{eff} with pure blackbody spectra. For lower T_{eff}, the library is completed with the spectral fluxes for M, L and T dwarfs from Allard et al. (2000), M giants from Fluks et al. (1994), and finally the C star spectra from Loidl et al. (2001). Details about the implementation of this library, and in particular about the C star spectra, are provided in Marigo & Girardi (2007). It is also worth mentioning that in the isochrones we applied the bolometric corrections derived from this library without making any correction for the enhanced He content which has been proved by Girardi et al. (2007) to be low in most common cases. The database of SSP covers the space of existence E of a generic CPS. The number of ages N_{τ} of the SSPs are sampled according to a law of the type τ = i × 10^{j} for i = 1,...,9 and j = 7,...,9, and for N_{Z} metallicities are Z = {0.0001,0.0004,0.0040,0.0080,0.0200,0.0300,0.0400}. The helium content associated to each choice of metallicity is according to the enrichment law ΔYΔZ = 2.5. Each SSP was calculated allowing a small age range around the current value of age given by Δτ = 0.002 × 10^{j} with j = 7,...,9. In total, the data base contains N_{τ} × N_{Z} ≅ 150 SSP. This grid is fully sufficient to illustrate the method. For future practical application of it, finer grids of SSPs can be calculated and made available. Having done this, the normalization constant ξ_{0} of the SSPs remains defined. Finally, for each SSP we computed the “projected” DF, i.e. the number of stars per elemental cell of the CMD, using the value of ξ_{0}. Note that the method for generating the cumulative DFs of Eq. (19)from SSPs does not depend on the particular choice for the data base of stellar tracks, isochrones, or photometric system. Other libraries of stellar models and isochrones can be used to generate the database of SSP, the building blocks of our method.
From the procedure outlined above, it is easy to understand how the use of the stellar DF is able to accelerate the construction of the CMD of a CSP. The reason is as follows: instead of calculating an SSP made by N_{SSP} stars for every star particle of the Nbody simulation (i.e., for a total of N star particles) and counting the stars inside a given bin of magnitude and colour δCδm, i.e. summing over stars, with the above procedure, we need to calculate only stars. However, the number frequencies per cell of the CMD of our reference SSPs are calculated once for all, whereas their combinations can be freely changed according to the underlying star formation history of the Nbody simulation to investigate and the number of calculation required in this method is then just .
The left panel of Fig. 1 shows a few SSPs for the solar metallicity Z = 0.02 and helium content Y = 0.28 in the theoretical HertzsprungRussell diagram (HRD). The size of the dots is proportional to the stellar mass running along the SSPs. The same SSPs are translated into the M_{V} vs. M_{V} − M_{I} plane displayed in the right panel of Fig. 1 in which the cell tessellation is evident. The “outoffocus” effect is due to grouping the stars of different SSPs into the same cell. No photometric errors are applied.
4.2. Simulation of photometric errors and completeness
We finally mention that real data on the magnitudes (and colours) of the stars are affected by photometric errors, whose amplitude in general increases at decreasing luminosities (increasing magnitude). The photometric errors come together with the data themselves provided they are suitably reduced and calibrated. Photometric errors can be easily simulated in our theoretical CMDs. Suppose we know the errors affecting our magnitudes in the two passbands used to build the CMD, and that these are a function of the magnitude itself. Let us indicate the errors by δm_{α}(m), and δm_{β}(m), with α and β the two passbands. When plotting each point of the SSPs on the observational CMDs, the magnitudes are changed by the quantity representing the errors, e.g.: (23)where r and s are two randomly drawn numbers (e.g., from uniform or Gaussian distribution) comprised between 0 and 1. The star frequencies per elemental cell of the CMD are calculated after applying the correction for photometric errors to the reference SSPs.
To compare the real observational data with the theoretical model we have to know the completeness of the former as a function of the magnitudes and passband (Stetson & Harris 1988; Aparicio & Gallart 1995). This is a long known problem that does not require any particular discussion in the context of this paper, and tabulations of the completeness factors must be supplied in advance together with the correction for photometric errors. We mention here is that correcting for completeness will alter the DF of stars in the cells of the observational CMD we aim to analyse (e.g., Crnojević et al. 2011). These completeness factors must be supplied by the user of our method in connection with the specified problem.
Fig. 2
Left panel: the CSP for an Nbody simulation of a disc galaxy. Colour code refers to the frequency of star per bin shown in the companion right panel. Right panel: the histogram of DF for the CSP shown in the left panel. In this figure, one immediately captures the concept of DF described in the text. The characteristic peaks corresponding to MS and RGB stars are shown in yellow. The CMDs have the coordinates of the absolute magnitudes M_{V} and the colours M_{V} − M_{I}. 

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Fig. 3
Snapshot of the simulated disc galaxy whose faceon views of the stellar and gas discs are shown in the left and right panels, respectively. Our assumed location of the Sun is highlighted by a star symbol. The white lines indicate the selected region for constructing the CMD shown in Fig. 5. Colour code ranges from blue (the lowest density) to red (the highest density). 

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5. Generating a CMD with tessellation
To explain the concept of CMD tessellation, we simulated a CMD using the CSP of Nbody TreeSPH simulation (to be described in more detail below). The analysed field contains 8.8 × 10^{4} starparticles of the same mass and known age and metallicity (to each of which an SSP can be associated with the same mass, age and metallicity, see Sect. 3). For the purpose of this example we show only the absolute luminosity and magnitude, i.e. no distance dependence. The results of applying Eq. (19)are shown in Fig. 2. The left panel displays the frequency distribution of CSP in the (M_{V} − M_{I}) vs. M_{V} CMD, while the right panel shows the histogram. The regions occupied by stars in the main sequence are clearly evident: longlived phases display a higher number of stars, on the other hand, red giant, red clump, and asymptotic giant phases are seen in low effective temperatures (red colours) and shows lower frequency due to their short lifetime. These features are similar to a typical observed CMD of the stellar populations in nearby galaxies, e.g. the Magellanic Clouds, M 31 and others, where for all stars in a given galaxy the distance is nearly constant.
6. Application to Nbody simulations
To demonstrate the method, we applied it to an Nbody simulation. The simulation was carried out with an updated version of our original NbodySPH code, GCD+ (Kawata & Gibson 2003; Rahimi & Kawata 2012). We initially set up an isolated Milky Waysized disc galaxy that consists of gas and stellar discs with no bulge component in a static dark matter halo potential, following Rahimi & Kawata (2012). Note that this simulation was used for demonstration purpose, and was not meant to reproduce the Milky Way. We used the standard NavarroFrenkWhite dark matter halo profile (Navarro et al. 1997) with the total mass of M_{tot} = 1.5 × 10^{12} M_{⊙} and the concentration parameter of c = 12. The mass, scale length and scale high of the stellar disc were assumed to be M_{d,s} = 4.0 × 10^{10} M_{⊙}, R_{d,s} = 2.5 kpc and z_{d,s} = 350 pc. The mass and scale length of the gas disc was M_{d,g} = 1.0 × 10^{10} M_{⊙}, R_{d,g} = 4.0 kpc. We initially set 400 000 particles to the gas disc and 1 600 000 particles to the stellar disc. Therefore, our baryon particle mass was M_{p} = 2.5 × 10^{4} M_{⊙}. We applied the threshold density of n_{H} = 1.0 cm^{3} for star formation. Rahimi & Kawata (2012) demonstrated that the star formation in a disc is quite sensitive to the parameters of star formation efficiency, C_{∗}, energy for supernova, E_{SN}, and stellar wind feedback energy, E_{SW}. We here applied C_{∗} = 0.1, E_{SN} = 10^{51} erg and E_{SW} = 10^{37} erg s^{1}. Figure 3 shows a snapshot of the simulation. Strong feedback creates many bubbles in the gas disc.
The simulated galaxy shows a small bar and several spiral arms. We set the Sun at (x,y) = (−8,0) kpc (star symbol in Fig. 3), and selected starparticles in the region of the simulated galaxy “equivalent longitude” 300 < l < 320 and “latitude” − 10 < b < 10 (with implicit reference to the Milky Way galaxy coordinates system l,b), which is enclosed by white lines in Figs. 3 and 4. For each starparticle we measured the distance and extinction from the simulation. To measure the extinction, first the column density for each starparticle was calculated by summing up the line of sight column density of all gas particles between the starparticle and the position of the Sun, using the SPH weighting scheme (Kawata & Rauch 2007). We then converted the column density to the extinction, using N_{H} = 1.9 × 10^{21} cm^{2} × A_{V} (magnitude).
The simulation of the CMD for the selected starparticles is shown in Fig. 2 where with a magnitude cut for true stars at above V > 20 mag about 8.8 × 10^{4} starparticles within the galaxy coordinates defined above are retained. We recall that Fig. 2 shows absolute magnitude and luminosity, which are independent of the distance. To build the observational CMD we need to take into account distance and extinction for each starparticle, which are measured for each particle as described above. The resulting “observational” CMD is a shown in Fig. 5. With the IMF we assumed that the resulting CMD is representative of about 1.08 × 10^{10} true stars. The difference in distance and extinction for each starparticle leads to a more smoothly distributed CMD. As a result, the groups of main sequence stars and red giant stars are hardly recognizable in Fig. 5. This is often observed in the photometric data of Galactic stars.
Fig. 4
Gas distribution in a Galactic coordinate of the simulated galaxy shown in Fig. 3. The region enclosed by the white lines are the selected region for constructing the CMD shown in Fig. 5. The colour coding is by density, so that bluer and more yellow points represent higher and lower density particles respectively. 

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Fig. 5
Observational CMD in apparent magnitudes and colours. The colourcoding is by density, so that bluer and more yellow points represent higher and lower density particles, respectively. 

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7. Concluding remarks
We first extended the theory of the stellar populations to include the phasespace description. We described the concepts of composite stellar population, simple stellar population, star formation rate, initial mass function, etc. using the language of Statistical Mechanics. Now it is fair to ask what we have gained from this rather formal approach. These techniques have proven to be extremely powerful in other branches of physics e.g., for the study of relaxation phenomena, electrical conduction, irreversible processes (see e.g., Landau & Lifshitz 2000) or in general for the development of the thermodynamics of nonequilibrium system (see e.g., de Groot & Mazur 1961) while have been more limited in the treatment of longrange forces e.g., the statistical mechanics of gravitational systems (see, e.g., LyndenBell & Wood 1968; Katz 2003), therefore it is important to understand to what extent they can be applied to the stellar populations. The mutual benefit between Statistical Mechanics and stellar populations theory consists of the temporal evolution of the existence space we introduced, the projection of which onto the CMD is governed by the fuel consumption theorem that drives the relative number of stars in different evolutionary stages, hence cells of the CMD. Moreover, studying the coupling between dynamics and theory of the stellar populations imprinted in the existence space, may reveal unexplored connections.
The second achievement of this paper is indeed the application of this theory to develop a method for handling the synthetic CMDs that one would generate from Nbody simulations, which are to be compared with the CMDs of real galaxies when they are resolvable into stars. The method based on the concept of star frequencies per elemental cell of the CMD can easily simulate observational CMDs that contain huge numbers of stars. This study extends and completes other similar studies in literature that dealt with synthetic CMDs such as those presented by in Carraro et al. (2001); Pasetto et al. (2012). This technique aims to interface Nbody simulations to photometric observations and is developed with the perspective of the ever improving capabilities of the Nbody simulations in the future. Finally, thanks to its agility, the CMD tessellation method can be suitably interfaced with galaxy models based on star counts (see for instance Vallenari et al. 2006) and other hybrid techniques based on the different approaches already present in literature such as the Galaxia model (see e.g., Sharma et al. 2011) or the Besancon model (see e.g. Robin et al. 2003) and it represents a natural extension of previous work (e.g. Tantalo et al. 2010).
The tessellation code for generating the CMD for Nbody simulations is available upon request from the authors.
Where f ∗ g is the standard convolution operator between the generic functions f and g. In general, we can indeed always write F_{λ} = L^{1}[L[ψ]L[φ_{λ}]], with L[ψ] the discrete Laplace transform of the SFR we obtain directly from the Nbody simulations and L[φ_{λ}] the Laplace transform of the monochromatic integrated flux of the a simple stellar population.
Acknowledgments
S.P. acknowledges D. Crnojević and J. Hunt for careful reading of an early version of this manuscript. The authors thank the anonymous referee for the constructive report.
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All Figures
Fig. 1
Left panel: the HRD of SSPs with assigned age and metallicity. The size of the circles indicates the current star mass along the isochrones and/or SSP. The stellar models in use are taken from the library of Bertelli et al. (2008, 2009). The colour code refers to the size of the dots: the smaller is the dot, the more yellow it is and the smaller the mass of the plotted stars. The bigger the dot, the bluer its colour and the bigger the mass of the stars. The mass considered in the models of this example ranges from 0.15 to 20 M_{⊙}. Right panel: the same SSPs displayed in the left panel but in the observational CMD M_{V} vs. M_{V} − M_{I}. The CMD is subdivided to a discrete grid of elemental cells, in each of which a certain number of stars fall. The stars in each cell may belong to different SSPs. The size of the cell is large clearly show evidence the “outoffocus” effect intrinsic to this tessellation technique. Colourcode spans account for the number of stars in the cells from 0 (the bluer colour) to 10^{5} (the light yellow colour). 

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In the text 
Fig. 2
Left panel: the CSP for an Nbody simulation of a disc galaxy. Colour code refers to the frequency of star per bin shown in the companion right panel. Right panel: the histogram of DF for the CSP shown in the left panel. In this figure, one immediately captures the concept of DF described in the text. The characteristic peaks corresponding to MS and RGB stars are shown in yellow. The CMDs have the coordinates of the absolute magnitudes M_{V} and the colours M_{V} − M_{I}. 

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In the text 
Fig. 3
Snapshot of the simulated disc galaxy whose faceon views of the stellar and gas discs are shown in the left and right panels, respectively. Our assumed location of the Sun is highlighted by a star symbol. The white lines indicate the selected region for constructing the CMD shown in Fig. 5. Colour code ranges from blue (the lowest density) to red (the highest density). 

Open with DEXTER  
In the text 
Fig. 4
Gas distribution in a Galactic coordinate of the simulated galaxy shown in Fig. 3. The region enclosed by the white lines are the selected region for constructing the CMD shown in Fig. 5. The colour coding is by density, so that bluer and more yellow points represent higher and lower density particles respectively. 

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In the text 
Fig. 5
Observational CMD in apparent magnitudes and colours. The colourcoding is by density, so that bluer and more yellow points represent higher and lower density particles, respectively. 

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In the text 
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