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Astron. Astrophys. 320, 428-439 (1997)

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2. Galactic disc model

The model of stellar population synthesis, from which the dynamical model utilized here originates, has been described at length in Robin & Crézé (1986), Bienaymé, Robin & Crézé (1987) (hereafter BRC) and Haywood (1994). A summary of the input parameters for the three stellar populations is given in Table 1 of Paper II. We give here a brief description of the model relevant to the disc population.

2.1. General constraints

The characteristics of the disc stars are represented by the combination of two functions which describe their intrinsic properties and spatial distribution: [FORMULA]. The [FORMULA] function is the frequency distribution of the stars in the HR diagram, assuming a SFR history and IMF, which are the basic parameters that characterize a given disc model. A description of how the [FORMULA] function is calculated is given in Haywood (1994), which also contains the detail of the various inputs (evolutionary tracks, bolometric corrections, B-V calibration etc...). The density distribution of the disc stars is given by the sum of 7 sub-populations, which analytic formula is given in BRC. It is considered that the 6 components older than 0.15 Gyr are isothermal, each component having its own velocity ellipsoid and age range. These are given in Sect. 3.1 (see also BRC).

The combination of the [FORMULA] and [FORMULA] functions gives a consistent description of the galactic disc, provided that the age parameter in [FORMULA] and in [FORMULA] refers to the same clock, linking dynamical and star formation histories. There are two observationally independent local constraints acting respectively on [FORMULA] and [FORMULA]. The first one is the observed LF. The loose constraints on the evolution of the galactic disc provided by the local LF have been thoroughly reviewed by Scalo (1986) and are reconsidered here.

The second constraint is the observed age- [FORMULA] relation, which is not used as a constraint in classical models. An intermediate stage on the way to dynamical self-consistency was reached by BRC, which permitted to use this relation. That is the velocity ellipsoids for the various populations are imposed in the model (instead of being deduced from the potential), but the vertical distribution of disc stars is calculated by integrating the Jeans vertical equation in the plane parallel approximation. The procedure, which leads to the local normalization and parameters for the vertical density laws of the disc, for a given SFR and IMF, is described in the flow-chart of Fig. 1. Using approximate values for the scale height of the disc, we calculate the volume density in the galactic plane for each subpopulation, by requiring that the relative value for each surface density complies with the prescription of the SFR (that is the relative importance of the surface densities in each age interval should correspond to the SFR intensities in these intervals). The second requirement is that the sum of the densities [FORMULA] matchs the value of the local stellar density. This is obtained by forcing the number of stars, calculated as a theoretical LF, to fit the observed LF, in the range of magnitudes where it is best determined, between [FORMULA] =5 and 10. This condition implies that the SFR in our simulation after 10 Gyrs (the age adopted for the galactic disc in our study), is not bound to satisfy the present SFR (as given by the LF at [FORMULA] [FORMULA] 5). However, it is poorly determined and may not be representative of the SFR on the last Gyr, that is the time scale of interest here. Once this first estimate for the [FORMULA] is obtained, the calculation of the dynamical consistency described in Fig. 1 is applied.

[FIGURE] Fig. 1. Determination of a mass model for a given set of SFR, IMF, and age- [FORMULA] relation using dynamical and observational constraints. Condition (1) establishes that at all times, the surface density of a given stellar disc age group stays proportional to the intensity of the star formation rate at the corresponding epoch (SFR([FORMULA])). Condition (2) forces the theoretical LF to have the same number of stars as the observed LF in the absolute magnitude range 5 to 10 and gives the model normalization. These two conditions permit to determine the [FORMULA]. In the box below, a first guess of the potential is obtained by solving the Poisson equation and fitting the observed rotation curve, through the determination of dark corona parameters ([FORMULA]). Below, the Jeans vertical equations is used to determine the new ellipticities for the stellar discs. With these new determinations, we return to conditions (1) and (2), etc. The process converges within a few loops.

The calculation that gives the scale height or disc ellipticities is described in detail in BRC. These parameters, which describe the scale heights of the disc density laws, are obtained for the values of the [FORMULA] and [FORMULA] for each subpopulation. Once these scale heights are obtained, new values of [FORMULA] must be calculated, to insure that they satisfy Eq. (1) and (2). Then new parameters for the scale height and new potential must also be calculated. The process converges within 2 to 3 loops.

For all the models presented below and in Paper II, the potential has been calculated assuming that there is no dynamically significant disc of dark matter in the solar neighbourhood. A surface density dark mass of 8 [FORMULA].pc-3 has been assumed, and taken as an upper limit of the quantity of dark remnants in the solar neighbourhood. This is also within the error bars of the surface density found by Kuijken & Gilmore (1989).

2.2. SFR, IMF and age for galactic disc models

We have adopted the classical parametrization of the SFR as a simple exponential law exp(-t/ [FORMULA]), with [FORMULA] = [FORMULA]  5, [FORMULA]  10, and [FORMULA]. The only concern here being the overall variation over the galactic disc lifetime, and not the details of the evolution. A value of [FORMULA] =5 Gyrs represents a variation by a factor 7 of the star formation intensity throughout galactic disc history, for a disc age of 10 Gyrs. Justifications for adopting this upper limit are given in Sect. 4.

The IMF is represented by a power-law dN/dM [FORMULA]  M [FORMULA], with 3 different slopes on 3 different mass intervals. The details of HR diagram computation are given in Haywood (1994). Let us just mention that we have used evolutionary tracks from Schaller et al. (1992) for stars with mass greater than 1  [FORMULA], and from Vandenberg (private communication) for smaller masses. Evolutionary tracks for Helium-burning stars at masses between 1 and 1.7 [FORMULA] are from Castellani et al. (1992). The age adopted for the galactic disc is 10 Gyrs.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998