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Astron. Astrophys. 322, 147-154 (1997)

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2. Determination of stellar age and mass

The evolution of a stellar model of initial mass M is normally presented as a collection of [FORMULA] fixed stages, regardless of the mass considered. We will refer to each of these stages of evolution as Equivalent Evolutionary Phase, or EEP (Praether 1976).

Let us suppose that the surface gravity ([FORMULA]), effective temperature ([FORMULA]) and metallicity ([FORMULA]) of a star are available. The first step in the procedure to determine its age and mass consists in calculating a new set of stellar evolutionary models (SEM hereafter) at a metallicity [FORMULA] through a linear interpolation between two sets of SEM at different metallicities. In Figure 1 the correctness of this procedure has been evaluated. A set of interpolated models at Z =0.008 is in reasonably good agreement with the theoretical ones up to EEP point number 16 -transition towards the beginning of the helium burning phase-, while major differences appear near the Red Giant Branch for massive stars. So, the main uncertainty in this interpolation comes from the error in the observational determination of the stellar metallicity. When the test is repeated with a linear interpolation in log Z the differences are slightly larger than before, so we will adopt our first choice.

[FIGURE] Fig. 1. Differences in [FORMULA] [FORMULA] (upper part), [FORMULA] (middle part) and the relative differences in age (bottom part) between the theoretical stellar evolutionary models of SSMM (Table 1) at [FORMULA]  0.008 and the interpolated ones from models at Z=0.02 and Z =0.004. Calculations have been performed for the initial stellar masses [FORMULA] 2 [FORMULA] (solid line), 5 [FORMULA] (dashed line) and 12 [FORMULA] (long dashed line). The first EEP has been eliminated in the representation of [FORMULA] since in this case the age is almost as small as the differences and, therefore, relative errors are too big

Once the set of evolutionary tracks at [FORMULA] is computed, it is necessary to locate the HR diagram position of the star between two EEP and two evolutionary tracks. Details about the linear interpolation procedure -at a constant Z - to obtain stellar age, present mass and the mass at the moment of its birth ([FORMULA]), are developed in Asiain (1993) and briefly described in the appendix A. As above, the procedure correctness is checked in Figure  2-EEP points in the Overlap Region have been eliminated, since a special treatment has been devised for this region (see next subsection). The small differences in age ([FORMULA] 5 %) and mass ([FORMULA] 2 %) between theoretical and interpolated values prove the proper working of the procedure before the beginning of the Red Giant Branch.

[FIGURE] Fig. 2. Relative differences in age (upper part) and mass (bottom part) between original evolutionary tracks and interpolated ones. Calculations have been performed for the initial stellar masses [FORMULA] 2 (solid line), 5 (dashed line) and 12 [FORMULA] (long dashed line) using Schaller et al. (1992) models. The first EEP is not considered in the representation of [FORMULA] for the same reason than in Figure  1

The algorithm devised is applicable in all regions of the HR diagram where only one evolutionary phase is possible. For this reason, in the following subsection we study the specific assignation of ages and masses to stars in the Overlap Region, easily identifiable as a loop of the evolutionary tracks at the end of the Main Sequence. For stars at the Giant Branch or further evolutionary phases we face a problem that is analogous, though much more complicated, to that of the Overlap Region. Although the analytical method could provide a result, it would be highly unreliable because of the complexity of these regions. Stars that are close to the ZAMS have a big relative error in age, due to their small value of this parameter. The region below ZAMS is discussed in 2.2.

It is worthy to note that the determination of age is not very accurate for late type stars, for which a small uncertainty in [FORMULA] [FORMULA] or [FORMULA] translates into a high error in age. On the other hand, since both atmospheric and stellar models commonly used pertain to normal stars, it is not possible to obtain accurate results either for fast rotating (very common among early type stars) or chemically peculiar stars, among others. Nevertheless, from an analysis of the Hyades cluster age we showed (Figueras et al., 1993) that the whole algorithm can be applied to metallic A type stars without introducing significant variations.

2.1. Overlap Region

At the very end of the Main Sequence, when hydrogen in the core of stars is almost consumed, the star initiates a contraction that produces an increase of its surface gravity and its effective temperature (Fig. 6). The contraction stops when a hydrogen burning shell appears around the star core. In this region of the HR diagram there is no bijective relationship between [ [FORMULA], [FORMULA] ] and [ [FORMULA], M ], and, as a consequence, a pair [ [FORMULA], [FORMULA] ] can correspond to three different phases, namely:

A the Main Sequence phase. Stars spend much more time in this phase than in the other two;

B the contraction phase. Changes in the stellar structure are accelerated;

C the hydrogen burning shell phase. The evolution in this phase is even faster than in the previous case.

If we call [FORMULA]   and [FORMULA]   the ages and masses of a star with atmospheric parameters [FORMULA] and [FORMULA] and evolutionary states A, B and C respectively, then:

[EQUATION]

A solution to this problem consists in adopting the age and mass that corresponds to the most probable state (Grosbol 1978). This state is defined taking into account the relative density of stars in each one of the three possible evolutionary phases. Since stars spend much more time in phase A than in B or C, they can be assumed to be in the Main Sequence, so for most of the stars in the Overlap Region (80-95 %) we would calculate the proper age and mass, while for the rest of them these values would deviated slightly but systematically from the real ones.

Instead, we prefer to weight the three possible results in order to obtain an unbiased mean age and mass for the whole sample. The weights for each of the three possible stages have been assumed to be proportional to the relative density of stars in them. The number of stars [FORMULA] born in (t, t+dt) with masses ranging from M to M +dM can be calculated from:

[EQUATION]

where [FORMULA] is the Initial Mass Function (IMF), and [FORMULA] is the Star Formation Rate (SFR).

We suppose that the IMF profile is independent of t, at least during the short period that a star spends in one of the evolutionary states, and that its shape is given by:

[EQUATION]

where the values of [FORMULA] and x depend on the mass M considered (we adopt the values given by Miller & Scalo 1979). We also assume that the SFR is constant from [FORMULA] to [FORMULA]:

[EQUATION]

Then, the number of stars between two subsequent EEP points of the SEM of mass [FORMULA] can be described as:

[EQUATION]

[FORMULA] being the age difference between these two points. Introducing the number of stars per unit of length in the HR diagram, [FORMULA], where [FORMULA] is the distance in this diagram between the EEP points above defined, and a normalizing factor [FORMULA], we can define the weights [FORMULA] as

[EQUATION]

The age and mass of any star located in the Overlap Region are derived from the following expressions:

[EQUATION]

The ages and masses determined in this way are never equal to the values we would find if the evolutionary state of the star was known. However, since typical values of weights are much higher in the Main Sequence than in the other phases (appendix  5), the age assigned to a star in the Main Sequence state will not change appreciably, and only those stars in states B or C will have differences in age slightly higher than the observational errors.

2.2. Below the ZAMS

When photometry is used to obtain atmospheric parameters, the resulting [FORMULA] and [FORMULA] of a star sometimes place it below the ZAMS. This may be due either to errors on the photometric observations, to the existence of problems associated with the dereddening procedure or the computation of the atmospheric models, or to the presence of certain physical peculiarities (sometimes very difficult to detect). How can we determine the age or mass of a star that is outside the zone covered by the evolutionary tracks in the HR diagram? Actually, we cannot. However, we guess that all these stars are generally young ones, so we shift their position on the diagram -keeping their [FORMULA] constant and varying their [FORMULA] - until they cross the ZAMS. Then, their masses are interpolated between the first SEM points. Since in the first part of the evolution of a star its temperature does not change very much, there is a tight correlation between M and [FORMULA], so the M calculated in this way is presumably a good estimation. Nothing can be said about the age except that these stars may still be near the ZAMS.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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