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Astron. Astrophys. 361, 73-84 (2000)

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5. Modeling the Galaxy

The description of the Galaxy is done with the code already described by Bertelli et al. (1995) and revised as described in the following sections. First a synthetic population is generated at varying the parameters age, metallicity range, star formation law and initial mass function. Second, the stars are distributed along the line of side following a model of the Galaxy, where all the components are taken into account. Finally, the photometric completeness of the data is taken into account dividing the simulated CMD in magnitude-colour bins and then subtracting from each bin having Nth stars, (1-[FORMULA])Nth, where [FORMULA] is the smallest of the V and I completeness factors given in Fig. 2.

The generation of the synthetic population makes use of the set of stellar tracks by Girardi et al. (1996) for Z=0.0001, Bertelli et al. (1990) for Z=0.001, Bressan et al. (1993) for Z=0.020, Fagotto et al. (1994a,b,c) for Z=0.0004, 0.004, 0.008, 0.05, 0.10.

5.1. The star formation rate

The history of the star formation in the solar neighborhood has been derived using various methods. Several authors suggest that a constant star formation rate can be appropriate for the disk (see among the others Twarog 1980, Haywood et al. 1997). On the basis of the Hipparcos data a star formation rate increasing towards younger ages is derived (Bertelli et al. 1999). In the following, several star formation rates going from constant to increasing or decreasing in time are adopted and the corresponding CMDs and luminosity functions are compared with the data (see Sect. 6).

5.2. The position of the sun

The position of the sun above the Galactic disk mid-plane is found to range from 10 to 42 pc, the upper limit being obtained by star count method (Stobie & Ishida 1987). On the basis of star-counts in 12 fields in the North and South hemispheres, Humphreys & Larsen (1995) suggest that values as low as 20.5[FORMULA]3.5 pc are more appropriate. Even lower values of 15-14 pc are found by Binney et al. (1997), Haywood et al. (1997), Cohen (1995), Hammersley et al. (1995). However, as pointed out by Haywood et al. (1997) the expected offset of the sun deduced from star-counts is found to depend on the scale height of the disk, in the sense that small scale height value favors small offsets: a scale height of 350 pc is compatible with an offset of 20 pc, while a scale height of 200 pc suggests an offset of 15 pc. In the following we adopt an offset of 15 pc.

The distance of the sun from the Galactic center is discussed from 7 to 8.5 Kpc. We adopt 8 Kpc, as a mean value.

5.3. Thin disk

5.3.1. Mass distribution

Two kind of mass distributions are usually adopted:

1) a double exponential law of the form:


where [FORMULA] and [FORMULA] are the scale length and scale height of the disk, respectively;

2) an exponential distribution on the plane and sech2 perpendicularly to the plane:


For z [FORMULA] [FORMULA] the two functions are often considered equivalent. However, in our case, this condition is not always met. In particular, using DIRBE data, Freundenreich (1996) find that the sech2 distribution is greatly superior to the exponential at low latitudes ([FORMULA]), as in our case. Both distributions will be considered and discussed. In our formulation, the constant [FORMULA] can be calculated for every model imposing that the total number of stars in a selected region of the observational CMD is reproduced by the simulations. From [FORMULA] the local mass density distribution will be derived (see following sections).

5.3.2. Scale height and scale length

The scale height [FORMULA] of the thin disk varies from 325 pc (Gilmore & Reid 1983, Reid & Majewski 1993, among others) to 200 pc (Haywood et al. 1997). In our model, this value is assumed as a free parameter.

The scale length [FORMULA] is found to vary from 3.0 Kpc (Eaton et al. 1984, Freudenreich 1997) to 2.0 Kpc (Jones et al. 1981). Intermediate values are derived by Ruelas-Mayorga (1991), Robin et al. (1992) who give [FORMULA]=2.5 Kpc.

In our description, [FORMULA] and [FORMULA] are assumed to be constant with the Galactic radius. This may not be valid in the outer Galaxy, where the dark matter might dominate the mass. In fact radio observations suggest that the thickness of the HI layer reaches about 400 pc at 13 Kpc, while the H2 layer is considerably flatter, reaching 200 pc at 12.5 Kpc of galactocentric distance (see Combes 1991 for a review). However the angle of maximum displacement above the plane is believed to range between 800 (Burton 1988) and 1100 (Diplas & Savage 1991), measured counterclockwise from the direction l=0. The effect in the fields under discussion is negligible.

5.3.3. The metallicity

Suggestions have arisen in literature that no age-metallicity relation is present in the thin disk. Edvardsson et al. (1993) find a spread of about 0.6 dex in [FORMULA] among main sequence stars of similar age in the solar neighborhood: the abundance spread for stars born at roughly the same galactocentric distance is similar in magnitude to the increase in metallicity during the lifetime of the disk. Studies of Galactic open clusters as well as of B stars in the solar neighborhood came to the same conclusion (Friel & Janes 1993, Carraro & Chiosi 1994, Cunha & Lambert 1992). In our simulations, a stochastic age metallicity relation for the disk has been adopted, with Z going from 0.008 to 0.03.

5.3.4. Age components

Up to now the main source of information about the age of this component comes from the open cluster system. If all the open clusters are member of the thin disk, a limiting value might be given by NGC 6791 which is the oldest open cluster with well-determined age. Its age is going from 7 to 10 Gyr (Tripicco et al. 1995). However, Scott et al. (1995) find a somewhat peculiar kinematics for this object. The age of Berkeley 17, which is believed to be one of the oldest disk clusters with 12 Gyr (Phelps 1997) has recently been substantially revised using near-IR photometry to 8-9 Gyr (Carraro et al. 1999). A lower limit to the age of the disk is given by the white dwarfs: recent determinations suggest an age of 6-8 Gyr (Ruiz et al. 1996). This value agrees quite well with the oldest age of field population based on Hipparcos data which is 11[FORMULA] (Jimenez et al. 1998). While the youngest age [FORMULA] of the thin disk component is constrained by the CMD, the oldest age is assumed to be 10 Gyr. [FORMULA] turns out to be in the range 1-5 [FORMULA] yr in all the fields.

The velocity dispersion of the disk stars suggest the presence of more age components, having different scale heights (Bessel & Stringfellow 1993). In the following in addition to the old component with large scale height, a second one is taken into account, having scale height [FORMULA] pc and ages ranging from [FORMULA] to the final age [FORMULA]. [FORMULA] is assumed to be 2 Gyr. This assumption has been verified on the luminosity functions of the observed fields. When [FORMULA] is 3 Gyr, the luminosity functions are difficultly reproduced unless a quite high percentage of thick disk ([FORMULA] 8%) is assumed.

5.4. Thick disc

5.4.1. The age and the metallicity of the thick disk

The mean metallicity is believed to be compatible with the one of the disk globular clusters (-0.6 to -0.7), even if a metal rich tail is expected up to -0.5 together with a metal poor component down to -1.5 dex (Morrison et al. 1990). Gilmore et al. (1995) find a peak of the iron abundance distribution of F/G thick disk stars at -0.7 dex, with no vertical gradient.

Edvardson et al. (1993) estimate an old age for the thick disk component of the solar neighborhood (about 12 Gyr), with no age gap between the thin-thick disk stars. Gilmore et al. (1995) confirm this result. Since the abundance ratios of the thick disk stars reflect incorporation of iron from type I supernovae, these authors suggest a star formation time scale larger than the time scale of SNI which is about 1 Gyr. The kinematics of the thin and thick disks suggest an evolutionary connection between the two components (Wyse & Gilmore 1992, Twarog & Antony-Twarog 1994). It cannot be ruled out that the thick disk is the chemical precursor of the thin disk and is formed through a dissipational collapse after the halo formation and before the end of the thin disk collapse.

However, other scenarios are presented in literature, suggesting that the thick disk formed after the thin disk, either for secular kinematic diffusion of the thin disk stars, or as the result of a violent thin disk heating due to the accretion of a satellite galaxy (Quinn et al. 1993, Robin et al. 1995). At present, it is quite difficult to discriminate between these two models. In the following, we assume the thick disk has an age range from 12 to 8 Gyr, with constant star formation and a metallicity ranging from Z=0.0006 to 0.008.

5.4.2. The mass distribution

As in the case of the thin disk, two density laws are adopted: either double exponential, or exponential in the plane and following sech([FORMULA]) perpendicularly to the plane.

The local density of the thick disk population is poorly known, in spite of the attempt made to measure it. From high proper motion stars, Sandage & Fouts (1987), Casertano et al. (1990) find values ranging from 2% to 10% of the total disk density. Large uncertainties are found also from star-counts, since such estimates are correlated with the determination of the scale height. Small local densities are correlated with large scale heights. The values are going from 1% to 10% of the total density. Robin et al. (1995), Haywood et al. (1997) suggest an intermediate value of 5.6% of the local disk population is due to thick disk. In the following we accept only solutions compatible with a thick disk percentage between 2% and 5%.

5.4.3. The scale height and scale length

Reid & Majewski (1993) summarized the results on the scale height of the thick disk. Gilmore (1984) find a scale height of 1300 pc. Norris (1987), Chen (1996), Gilmore & Reid (1983) found values of 1100 pc, 1170 pc, and 1450 pc respectively. Lower values are suggested by Gould et al. (1995) on the basis of HST counts, by Robin et al. (1995), Haywood et al. (1997) using ground based star counts: 760[FORMULA] pc.

Concerning the scale length of the thick disk Robin et al. (1992), Fux & Martinet (1994), Robin et al. (1995) favor a value of 2.5-2.8 kpc. In our simulations we consider scale height and scale length of the thick disk as free parameters.

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Online publication: September 5, 2000