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

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Appendix A: interpolation of ages and masses among SEM

Suppose that the effective temperature ([FORMULA]), surface gravity ([FORMULA]) and metallicity ([FORMULA]) of a star are known parameters. As described in Sect.  2, we can then determine a new set of SEM at [FORMULA]. Let [FORMULA], [FORMULA], [FORMULA] and [FORMULA] be the log  [FORMULA], [FORMULA], age and mass, respectively, of the j th EEP of a track with initial mass [FORMULA] when using this new set of SEM, and suppose that our star is contained in the HR region limited by tracks i and [FORMULA], and EEPs j and [FORMULA] (Fig. 6).

[FIGURE] Fig. 6. Model points considered in the interpolation of the age and mass

As a first approximation, it can be assumed that log  [FORMULA], [FORMULA], [FORMULA] and [FORMULA] vary linearly with [FORMULA] inside a given EEP (j). Then, we can construct a new track with initial mass [FORMULA],

[EQUATION]

where

[EQUATION]

We also assume that log  [FORMULA], [FORMULA], and [FORMULA] vary linearly with the age between two EEP points inside a given track. The log  [FORMULA] and [FORMULA] of our star can then be calculated as

[EQUATION]

[FORMULA] being the interpolation factor along an evolutionary track

[EQUATION]

Equations (A6) and (A1) allow us to express the interpolation factor [FORMULA] as

[EQUATION]

Introducing (A2) and (A10) in (A7), we obtain a second degree equation in [FORMULA]

[EQUATION]

whose coefficients are given by

[EQUATION]

and

[EQUATION]

Since [FORMULA] and c can be calculated using SEM and the [FORMULA] and [FORMULA] of the star, we can easily determine the roots of the quadratic equation, and from them, by means of Eq. (A10), we also determine the [FORMULA] values. We can then choose the proper solutions by just imposing the conditions

[EQUATION]

Once [FORMULA] is known, Eq. (A5) allows us to determine the initial mass of our star and, from Eqs. (A4) and (A8), its current mass. Finally, the age is determined by means of Eqs. (A3), (A5) and (A9).

Appendix B: ages, masses and weighting factors in the Overlap Region

Mean ages and masses have been calculated for stars placed in the Overlap Region following expressions  1. Typical differences between computed values and those we would assign if evolutionary phases were known are shown in Table 2 as a function of the position in the HR diagram.


[TABLE]

Table 2. For different pairs (log  [FORMULA], [FORMULA]) covering the Overlap Region in the A-type range the difference between the assigned values for [FORMULA] ([FORMULA]) and M ([FORMULA]) -first and second columns respect.- and the corresponding values calculated as if the phase (A, B or C) where the star is was known. In the third column the probabilities of being to each phase are given


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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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