Free Access
Volume 541, May 2012
Article Number A41
Number of page(s) 21
Section Stellar structure and evolution
Published online 26 April 2012

Online material

Appendix A: Transformation between Z and [Fe/H]

We establish in this appendix a relation between the mass fraction Zinit = 1 − Xinit − Yinit of all elements heavier than helium and the metallicity  [Fe/H] init = log [X(Fe)/Xinit − log [X(Fe)/X, where the subscript init refers to the abundances of the interstellar medium from which stars are born and refers to the photospheric abundances of the Sun today. The explicit reference to initial abundances of stars is important in order to distinguish them from photospheric abundances of running models, the latter having possibly been modified in the course of stellar evolution by atomic diffusion. A simple relation between Zinit and  [Fe/H] init can be established if we make two assumptions on the chemical evolution of the Galaxy. The first assumption concerns the galactic evolution of the helium mass fraction Yinit with metallicity Zinit. A linear relation is assumed of the form (A.1) where Y0 is the primordial helium mass fraction of the universe resulting from Big Bang nucleosynthesis and ΔYZ reflects the chemical enrichment of the Galaxy whereby H is globally transformed into He and heavier elements. Equation A.1 gives (A.2)Second, we assume a constant X(Fe)/Z ratio throughout the evolution of the Galaxy; i.e., we neglect enhancement of α elements. We then have, for all Z, (A.3)Transformation relations between  [Fe/H] init and Zinit result from assumptions (A.2) and (A.3): (A.4)With the values adopted in this paper (Y0 = 0.248, ΔYZ = 1.2857 and Z/X = 0.0174), Eqs. (A.4) become (A.5)It is important to keep in mind that Eqs. (A.4) and (A.5) relate to abundances at the time of star formation. In our grids, these relations are also valid during the MS and post-MS models for stars equal to or more massive than 1.1 M. For stars less massive than 1.1 M, however, these relations are no longer valid because they do not take the alteration with time of the surface abundances due to atomic diffusion into account. For example, the solar model in our grids, which is obtained when the Zinit = 0.014, 1.0 M model has reached the solar luminosity and radius at the age of 4.57 Gyr, predicts surface abundances equal to X = 0.7524, Y = 0.2346, Z = 0.0131, and Z/X = 0.0174, in agreement with observations. With this value for Z at the surface of the Sun today, however, Eq. (A.5) predicts a value for  [Fe/H]  of 0.018. This is obviously wrong, as by definition,  [Fe/H]  = 0.0. The error comes from the use of Eq. (A.5), which does not include the effect of atomic diffusion during the past 4.57 Gyr evolution of the solar model. The effect is seen to amount to  ~2% on  [Fe/H]  for the Sun, and is expected to amount to at most  ~5% for the other low-mass models (see Fig. 14). Within this precision, Eqs. (A.4) and (A.5) can be used for all cases.

Appendix B: Construction of the basic tracks

B.1. Main sequence

B.1.1. Reference points in the HR diagram

The tracks of several representative masses in the HR diagram are shown in Fig. B.1. They reveal a morphology of the MS that depends on the stellar mass. In the low-mass regime (M <  0.9 M), the morphology is rather simple: both the effective temperature and the luminosity monotonically increase with time during the whole MS phase. At higher masses, several complexities appear in the MS morphology:

  • 1.

    For M ≳ 0.9 M, Teff starts to decrease during the MS. The point where it starts to decrease is located at the end of the MS for the lower masses, and occurs earlier when we consider more massive stars. Teff decreases right from the ZAMS for M ≥  1.6 M.

  • 2.

    Two turn off points appear in the HR diagram for M ≳ 1.2 M.

  • 3.

    The surface luminosity is a strictly increasing function of time during all the MS for M ≲  1.4 M. For masses M ≳  1.4 M, however, the surface luminosity decreases after the second turn off point, reaches a minimum and increases again as the star evolves off the MS.

To better cope with these different morphologies on the MS, we shift from the HR diagram to the log R − log L plane. This removes the first complexity because the stellar radius is always a strictly increasing function of time from the ZAMS up to the first loop, if it exists, as illustrated in Fig. B.2. For constructing the basic tracks we use the morphologies in the log R − log L plane.

thumbnail Fig. B.1

Selected Zinit = 0.014 tracks in the HR diagram covering the MS and part of the post-MS to illustrate the different morphologies. The main sequence is represented in thick lines, the filled circle locating its end.

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B.1.2. Reference points in the log R – log L diagram

thumbnail Fig. B.2

Same as Fig. B.1, but in the log R − log L diagram.

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thumbnail Fig. B.3

Upper panel: MS track of the Zinit = 0.014, 2 M models in the log R − log L plane. Second to lowest panels: functions used to identify reference points on the MS track (see Sect. B.1.2).

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thumbnail Fig. B.4

Same as Fig. B.3, but for the 1.2 M models.

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thumbnail Fig. B.5

log L as a function of decreasing core hydrogen mass fraction, in logarithm, for Zinit = 0.014 models of, from bottom to top, 1.0, 1.1, 1.16, 1.18, 1.2, 1.22, 1.24, 1.26, 1.28, 1.3, 1.4 and 1.5 M. The locations of the three reference points within the MS are shown by filled circles on each track. The dashed lines denote tracks for which the reference points are defined by fixed values of Xc(H) (see Sect. B.1.2).

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The tracks in the log R − log L plane are shown in Fig. B.2. They display the hooks already identified in the HR diagram for stellar masses above  ~1.2 M (Sect. B.1.1). The basic tracks are constructed in three steps:

  • 1.

    We first identify the second turn-off point in thelog R − log L plane. To this aim, we define the line connecting the locations of the ZAMS and TAMS in that plane (dotted line in panel a of Fig. B.3), and compute the geometrical distance of every model on the track to that line. This distance, called function 1, is shown in panel b of the figure. The first minimum of function 1, searched backward from the end of the MS, defines the second turning point. It is shown by a filled circle in panel b of the figure.

  • 2.

    We then identify the first-turn off point of the track. We proceed in a similar way to the above, but work in the log R −  function 1 plane rather than in the log R − log L plane. We define the line connecting the ZAMS to the second turning point (dotted line in panel b of Fig. B.3), and compute the geometrical distance of function 1 to that line. This new distance, called function 2, is shown in panel c. The location of the first turning point is defined as the minimum of that function. It is shown by a filled circle in the panel.

  • 3.

    Finally, we define a third reference point on the MS track, between the ZAMS and the first turning point, based on the distance of function 2 to the line connecting the ZAMS and the first turning point in the log R −  function 2 plane (dotted line in panel c of Fig. B.3). That new distance is called function 3, and the reference point is defined by the minimum of that function. It is shown by a filled circle in panel d.

The three reference points so defined are summarized in panel a of Fig. B.3. The procedure is applicable to masses as low as 1.2 M, even if no hook is visible in the log R − log L plane at that mass, as illustrated in Fig. B.4 for the 1.2 M case.

For masses lower than 1.2 M, the technique fails because there is no deflection point on the MS tracks in the log R − log L due to the absence of a convective core. For those low mass stars, the three reference points are taken at fixed values of Xc(H) based on the values found at the higher mass stars. Figure B.5 displays log L as a function of the core hydrogen mass fraction for stars in the mass range 1.2 to 1.4 M, and locates the three reference points on the tracks. From these results, the reference points are fixed at Xc(H) = 0.26, 0.065 and 0.008 for all the stars with M < 1.2 M at Zinit = 0.014. The same values of Xc(H) are adopted for the tracks at higher metallicities. At lower metallicities, different values are adopted: Xc(H) = 0.29, 0.070 and 0.010 at Zinit = 0.010, and Xc(H) = 0.33, 0.090 and 0.015 at Zinit = 0.006. At Zinit = 0.006, these fixed values of Xc(H) are used to define the reference points of the 1.2 M track as well.

B.1.3. Basic MS tracks

Once the reference points are defined on the MS, basic tracks are constructed by distributing a fixed number of models between the reference points. The distribution is done regularly in age in the first two intervals (i.e. between ZAMS and the first reference point and between the first and second reference points) and regularly in log Xc(H) between the two remaining intervals (i.e. between the second and third reference points and between the third reference point and the TAMS). A regular distribution in log Xc(H) cannot be done before the first turning point because the hydrogen mass fraction in the core is not a monotonically decreasing function of time for all masses.

B.2. Post-main sequence

The construction of the basic tracks during the post-MS phase is easier to perform than that of the MS. We locate the base of the RGB with a technique similar to the one used to locate the turn off points on the MS. For that purpose, we use the angle between the X-axis in the log R − log L plane and the line connecting the TAMS and the running point on the post-MS track. The base of the RGB is located at the point where this angle starts to increase. We then distribute a fixed number of models between the TAMS and that reference point at the base of the RGB, regularly spaced in age.

Appendix C: Isochrone construction

thumbnail Fig. C.1

Logarithm of the age, in 109 yr, of the models in the basic tracks as a function of model index for stellar masses between 1.0 and 1.4 M as labeled next to the curves. The models are taken at Zinit = 0.014.

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thumbnail Fig. C.2

Isochrone at 1 Gyr, Zinit = 0.014, in the HR diagram, illustrating the isochrone computation procedure. Basic tracks are shown in gray for different stellar masses, continuous lines for the MS and dashed lines for the post-MS phase. The isochrone is plotted in black, each filled circle representing a stellar model.

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thumbnail Fig. C.3

Same as Fig. C.2, but for isochrones at 3.7, 4.0, and 4.3 Gyr as labeled next to the isochrones.

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Isochrones are computed from the basic tracks by taking care to reproduce adequately the hooks at the end of the MS. The simplest procedure would consist in considering, in turn, each evolutionary stage available in the basic tracks (identified by the model indexes in those tracks as described in Sect. 4.1), and interpolating in the set of models with the same index from the different tracks to get a model at the requested age. The resulting isochrone would then consist of stellar models distributed according to the evolutionary stages defined in the basic tracks, which are, by construction, well sampled around the hooks in the HR diagram.

This procedure is, however, not applicable owing to the appearance of a convective core during the MS at stellar masses around 1.20 M. Figure C.1 plots the age of the models in the Z = 0.014 basic tracks as a function of model index for masses between 1.0 and 1.4 M. The age at the end of the MS (indexes above 400) is seen to monotonically decrease with increasing stellar masses, as expected. But this is not the case in the middle of the MS. The age of models at index 200, for example, decreases with increasing mass for all masses except when passing from 1.18 to 1.20 M and again when passing from 1.24 to 1.26 M. The former case corresponds to the mass range at which a convective core appears in the middle-to-end of the MS, and the latter case to the mass range at which the convective core settles during the whole MS phase (see Fig. 7).

We therefore proceed differently. At the given metallicity, we compute a very dense grid of models by interpolating in the basic tracks with a mass step that depends on the evolutionary stage. We take mass steps of 0.01 M between the ZAMS and the second MS reference point, 0.005 M between the second and third reference points, 0.0005 M between the third and fourth, 0.0001 M between the fourth reference point and the TAMS, and 0.0005 M in the post-MS phase. We then interpolate in each track of this dense grid to obtain a model at the requested age, and add it to the isochrone. The resulting isochrone obtained at 1 Gyr is displayed in Fig. C.2.

This issue related to the development of the convective core during the MS does impact the morphology of isochrones that have their MS hooks at stellar masses between 1.18 and 1.26 M. At Zinit = 0.014, it occurs for isochrones around 4 Gyr, shown in Fig. C.3. The 4 Gyr isochrone clearly displays the signature of the appearance of the convective core in the middle of the MS at stellar masses between 1.18 and 1.20 M and of the propagation of the convective core to the whole MS at stellar masses between 1.24 and 1.26 M. These effects of the development of the convective core on the morphology of the isochrones is limited to ages between 3.8 and 4.2 Gyr. The isochrones at 3.7 and 4.3 Gyr, for example, are seen in Fig. C.3 to normally behave, the former with the two hooks characteristic of more massive stars and the latter with no hook characteristic of lower mass stars.

Appendix D: Data access

Table D.1

a to f: Basic tracks at Zinit = 0.006, 0.01, 0.014, 0.02, 0.03, 0.04, respectively. Only a portion of table c is shown. See text for a description of the column.

Basic and interpolated tracks, as well as iso-Zsurf lines, will be made available for download on our web site3. Each downloaded file corresponds to either a given metallicity (for basic and interpolated tracks and for isochrones) or a given surface metallicity (for iso-Zsurf lines), and contains as many tracks as the number of stellar masses requested by the user. An extract of the file is given in Table D.1. It contains a file header that indicates the number of tracks included in the file, and then lists the different tracks one after the other. For each track, a track header first summarizes general information on the track. Those are

  • the type of track (in the example given in Table D.1,it is a basic track);

  • the metallicity, either Zinit or Zsurf depending on the type of track;

  • the age or the stellar mass of the track, depending on whether it is an isochrone or not;

  • the number of models in the track;

  • the model indexes of the reference points in the track. The reference points define phases in the following way:

    • phase 20: starts at the ZAMS,

    • phase 21: starts at the second (after ZAMS) reference point on the MS,

    • phase 22: starts at the third reference point, which corresponds to the occurrence of the first turn-off point on the MS or to the equivalent point if there is no hook,

    • phase 23: starts at the fourth reference point, which corresponds to the occurrence of the second turn-off point on the MS, or to the equivalent point if there is no hook,

    • phase 30: starts at the TAMS,

    • phase 31: base of the red giant branch.

The data of the models are then listed, one line per model. For each model, we give, ordered by column:

  • 1.

    the model index;

  • 2.

    the phase of the model;

  • 3.

    t (yr): the age;

  • 4.

    M/M: the stellar mass;

  • 5.

    log L/L: the logarithm of the surface luminosity;

  • 6.

    log Teff: the logarithm of the effective temperature;

  • 7.

    log ρc (g/cm3): the logarithm of the central density;

  • 8.

    log Tc (K): the logarithm of the central temperature;

  • 9.

    Qcc: the mass of the convective core relative to the stellar mass;

  • 10.

    Xc(H): the hydrogen mass fraction at the center;

  • 11.


  • 12.


  • 13.


  • 14.


  • 15.


  • 16.


  • 17.


  • 18.


  • 19.


  • 20.


  • 21.

    Xs(H): the hydrogen mass fraction at the surface;

  • 22.


  • 23.


  • 24.


  • 25.


  • 26.


  • 27.


  • 28.


  • 29.


  • 30.


  • 31.


  • 32.

    Minit/M: the stellar mass of the initial model of the evolutionary track from which this model has been computed;

  • 33.

    Xinit: the hydrogen mass fraction of the initial model of the evolutionary track from which this model has been computed;

  • 34.

    Yinit: same as Xinit, but for the initial helium mass fraction;

  • 35.

    Zinit: same as Xinit, but for the initial metallicity;

  • 36.

    Zsurf: the surface metallicity;

  • 37.

    R/R: the stellar radius;

  • 38.

    : the logarithm of the mean stellar density relative to the solar mean density, with  = 1.411 g/cm3;

  • 39.

    νmax: the frequency of stellar oscillation at maximum amplitude, given by Eq. (3);

  • 40.

    Amax/Amax, ⊙: the maximum oscillation amplitude relative to that of the Sun given by Eq. (4);

  • 41.

    Δν: the large frequency separation, given by Eq. (2).

© ESO, 2012

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