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Astron. Astrophys. 359, 103-112 (2000)

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2. The model of population synthesis

We have used a revised version of the Besançon model of population synthesis. Previous versions were described in Bienaymé et al. (1987a; 1987b) and Haywood et al. (1997b).

The model is based on a semi-empirical approach, where physical constraints and current knowledge of the formation and evolution scenario of the Galaxy are used as a first approximation for the population synthesis. The model involves 4 populations (disc, thick disc, halo and bulge) each deserving a specific treatment. The bulge population which is irrelevant for this spheroid analysis will be described elsewhere.

2.1. The disc population

A standard evolution model is used to produce the disc population, based on a set of usual parameters: an initial mass function (IMF), a star formation rate (SFR), a set of evolutionary tracks (see Haywood et al., 1997 and references therein). The disc population is assumed to evolve during 10 Gyr. A set of IMF slopes and SFR's are tentatively assumed and tested against star counts. The tuning of disc parameters against relevant observational data was described in Haywood et al. (1997a; 1997b).

The model fixes the distribution of stars in the space of intrinsic parameters: effective temperature, gravity, absolute magnitude, mass and age. These parameters are converted into colours in various systems through stellar atmosphere models corrected to fit empirical data (Lejeune et al., 1997; Lejeune et al., 1998). While some errors still remain in the resulting colours for some spectral types, the overall agreement is good in the major part of the HR diagram.

Since the Haywood et al. model was based on evolutionary tracks at solar metallicities, inverse blanketing corrections are introduced to give to the disc a metallicity distribution in agreement with Twarog (1980) age/metallicity distribution (mean and dispersion about the mean).

The model returns the present-day distribution of stars as a function of intrinsic parameters in a unit volume column centered at the sun position. Since the evolution model does not account for orbital evolution, stars are redistributed in the reference volume over the z axis. The key for redistributing stars along the z-axis is age: an empirical relation associates z velocity dispersions to ages. Then the Boltzmann equation is used to convert z velocity distributions into z density. The model is dynamically self-consistent in the sense that the potential used in the Boltzmann equation is the one generated by the total mass distribution of stellar populations. The self consistency is established iteratively. We slice the disc populations into seven isothermal populations of different ages, from 0 to 10 Gyr. Each sub-population (except the youngest one, which cannot be considered as relaxed) has its velocity dispersion imposed by the age/velocity dispersion relation. We then deduce the scale height of each sub-population using the Boltzmann equation. The overall scheme is described in Bienaymé et al. (1987a).

Resulting density laws are used to correct the evolution model distribution in and off the plane, then to compute the stellar densities all over the Galaxy.

2.2. The thick disc population

A detailed analysis of the thick disc population from photometric and astrometric star counts has been given elsewhere (Ojha et al., 1994a; Ojha et al., 1994b; Ojha et al., 1996; Robin et al., 1996; Ojha et al., 1999). The kinematics, metallicity, and density law were measured allowing us to constrain the origin for this population. In this series of papers, evidence was given that the majority of thick disc stars should originate from a merging event at the beginning of the life of the thin disc, after the first collapse. One or several satellite galaxies have heated the thin disc, then the gas re-collapsed and reformed a new thin disc (Robin et al., 1996).

In the population synthesis process, the thick disc population is modeled as originating from a single epoch of star formation. We use Bergbusch & Vandenberg (1992) oxygen enhanced evolutionary tracks. No strong constraint exists on the thick disc age until now. We assume an age of 11 Gyr, which is slightly older than the disc and younger than the halo. The initial mass function is modeled by a simple power law with a slope about [FORMULA], referring to the notation [FORMULA].

The thick disc metallicity can be chosen between -0.4 and -1.5 dex in the simulations. The standard value of -0.7 dex is usually adopted, following in situ spectroscopic determination from Gilmore et al. (1995) and photometric star count determinations (Robin et al., 1996; Buser et al., 1999). The low metallicity tail of the thick disc seems to represent a weak contribution to general star counts (Morrison, 1993b). It was neglected here. An internal metallicity dispersion among the thick disc population is allowed. The standard value for this dispersion is 0.25 dex. No evidence has been found for a significant metallicity gradient in the thick disc population (Robin et al., 1996).

The thick disc density law is assumed to be a truncated exponential: at large distances the law is exponential. At short distances it is a parabola. This formula ensures the continuity and derivability of the density law (contrarily to a true exponential) and eases the computation of the potential. The scale height of the exponential can vary between 600 and 2600 pc. The standard value, 760 pc, has been obtained from star count fitting in various directions (Robin et al., 1996). Nevertheless, it can be shown that star counts when restricted to a small number of galactic directions and a small magnitude range do not give a strong constraint on the scale height, but rather on the parameter: (local density)[FORMULA](scale height)2. At present there is no accurate determination of the thick disc density in the solar neighbourhood, independently from the scale height. But reasonable values range between 700 to 1200 pc for the scale height and 1 to 4% for the local density relative to the thin disc.

2.3. The spheroid

We assume a homogeneous population of spheroid stars with a short period of star formation. We thus use the Bergbusch & Vandenberg (1992) oxygen enhanced models, assuming an age of roughly 14 Gyr (until more constraints on the age are available), a mean metallicity of -1.7 dex and a dispersion of 0.25 about this value. No galactocentric gradient is assumed. The IMF has to be constrained either from globular clusters (if they are representative of the spheroid population) or from deep star counts. This point is discussed in Sect. 3.

The density of spheroid stars is modeled by a power law:

[EQUATION]

where [FORMULA] is the local density, n is the power law index and [FORMULA] is the flattening.

The local density can be constrained by local measurements of high velocity stars, or by remote counts of giants (spectroscopically selected) or dwarfs (photometrically selected). The local density cannot be determined independently from the other density parameters with our limited number of data sets. Thus we have used independent constraints from the literature on the local spheroid density.

2.4. The local spheroid density

The local stellar spheroid density, [FORMULA], is bounded by observational data on halo dwarfs and giants. Fig. 1 shows the luminosity function obtained by different authors. We only selected recent results obtained in good conditions from sufficiently large samples. Bahcall & Casertano (1986) and Gizis & Reid (1999) derived their values from high proper motion dwarf samples. Dahn et al. (1995) determined accurate parallaxes for local late-type subdwarfs and deduced the local luminosity function of halo stars in the absolute visual magnitude range 9 to 14. These three results are biased by the kinematic selection. They took the bias correction into account but this correction is model dependent and introduces an unknown uncertainty into the result. We expect that the differences between the three measurements rely upon this correction. On the giant side, Morrison (1993a) used a non kinematically-biased sample of halo giants, selected from their metallicity to estimate the spheroid local density. In Fig. 1a we show the luminosity function from Bergbusch & Vandenberg (1992) for a population of 14 Gyr with a metallicity of -1.75 and an IMF slope [FORMULA] of 2. If we let the local halo density vary from a factor of 0.75 to 1.25 relative to this reference model (dotted lines in Fig. 1a), we get a good agreement with the specified observations given their uncertainties. In the next section we allow the local density to vary within these limits.

[FIGURE] Fig. 1a and b. Luminosity function of the stellar halo. (Gizis & Reid, 1999): diamonds; (Bahcall & Casertano, 1986): squares; (Dahn et al., 1995): plus, (Morrison, 1993a): cross. a  solid line: modeled luminosity function with an IMF slope ([FORMULA]) of 2.0;the dotted lines give the same luminosity function renormalized by a factor 0.75 or 1.25. b  luminosity functions as a function of IMF slope (as indicated in the graph).

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Online publication: June 30, 2000
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