Online material

Appendix A: PGPUC online

PGPUC online (http://www2.astro.puc.cl/pgpuc) is our web page containing all the PGPUC evolutionary tracks and ZAHB loci. In this web page the Hermite interpolation algorithm for constructing reasonable analytic curves through discrete data points presented by Hill (1982) is used, which produces evolutionary tracks from the MS to the RGB tip for any mass, helium, and metallicities in the ranges 0.5 M_⊙ ≤ M ≤ 1.1 M_⊙, 0.230 ≤ Y ≤ 0.370, and 0.00016 ≤ Z ≤ 0.01570, respectively, for an [α/Fe] = 0.3 and a SC05 mass loss rate with η = 1.0. More specifically, to interpolate an evolutionary track of a given mass M_x, helium abundance Y_x, and metallicity Z_x within a range of masses ({ M_i } , with i = 1...7), helium abundances ({ Y_j } , with j = 1...7), and metallicities ({ Z_k } , with k = 1...9) we follow the next steps:

• Using the first EEP of each evolutionary track with mass M = M₁, helium abundance Y = Y₁, and all the 9 metallicities Z_k, the first EEP of the track with M = M₁, Y = Y₁, and Z = Z_x is interpolated. This process is followed for each EEP of the evolutionary track with M = M₁ and Y = Y₁ with the end result that an interpolated evolutionary track with M = M₁, Y = Y₁, and Z = Z_x is created.

• Following the previous process for each of the seven masses M_i, we have an interpolated set of evolutionary tracks with seven masses M_i, a helium abundance Y = Y₁, and a metallicity Z = Z_x.

• Doing the previous two steps for each helium abundance Y_j, one has a set of evolutionary tracks with seven masses M_i, for each of the seven helium abundances Y_j, and with a given Z = Z_x.

• Now, using the first EEP of each evolutionary track with mass M = M₁, metallicity Z_x, and all the 7 helium abundances Y_j, the first EEP of the track with Y = Y_x, that obviously has a Z = Z_x, is interpolated. This process is followed for each EEP of the evolutionary track with M = M₁ and Z = Z_x, with the end result that an interpolated evolutionary track with M = M₁, Y = Y_x, and Z = Z_x is created.

• When the previous step is repeated for each mass M_i a set of evolutionary tracks with 7 masses M_i, Y = Y_x, and Z = Z_x is obtained.

• Finally, using the first EEP of each mass M_i one can interpolate the first EEP of the mass M_x. Repeating this process for each EEP, one creates the final interpolated evolutionary track with M = M_x, Y = Y_x, and Z = Z_x (see Fig. A.1).

Fig. A.1 Interpolated evolutionary tracks (red lines) from the original theoretical set of evolutionary tracks (blue lines). Upper panel: metallicity interpolation (metallicity increases from left to right) with constant helium abundance Y = 0.245 and mass of 0.9 M⊙. Middle panel: helium interpolation (helium increases from right to left) with constant metallicity Z = 0.0016 and masses of 0.9 (hotter) and 0.5 M⊙ (cooler). Bottom panel: mass interpolation (mass increases from right to left) with constant metallicity Z = 0.0016 and helium abundance Y = 0.245.

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Fig. A.2 Interpolated ZAHB loci (red lines) from the original theoretical set of ZAHB loci (blue lines). Open triangles show the locus where the convective envelope dissapears. Upper panel: Z interpolation (Z increases from top to bottom) with constant Y = 0.245 and Mi = 0.9 M⊙. Middle panel: Y interpolation (Y increases from bottom to top at the red ZAHB and inversely at the blue ZAHB) with constant Z = 0.0016 and Mi = 0.9 M⊙. Bottom panel: Mi (Mi increases from bottom to top at the blue ZAHB) with two constant Z (Z = 0.0016 and Z = 0.00016) and Y = 0.245. Dots are the initial points for each ZAHB locus, with the mass increasing from bottom to top.

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In the case that we have the same set of evolutionary tracks, but for different α-element enhancements [α/Fe] _l, one can follow the previous procedure for each [α/Fe] _l, ending with the interpolation of the EEPs for each evolutionary track with M = M_x, Y = Y_x, Z = Z_x, and [α/Fe] _l to create ones with a given α-element enhancement. Of course, this procedure can be done for each initial property that can be parameterized (e.g., mass loss, mixing length parameter, overshooting, C+N+O abundance, etc.), which is our intention with the PGPUC online web page: that the user can select the initial property that he or she wants.

Moreover, since any evolutionary track can be created, in this web page it is also possible to create any isochrones within the previous ranges of helium and metallicity. However, due to the fact that the maximum mass for each chemical composition is 1.1 M_⊙, the minimum age limit is not constant, with a value equal to the required time for the star with 1.1 M_⊙ of the given chemical composition to evolve from the MS to the RGB tip. In the near future, we plan to increase the maximum mass value.

In the case of the ZAHB, they can also be interpolated for any chemical composition within the range of chemical compositions. This is done using the so-called equivalent ZAHB points (EZAHBPs), which are defined using two criteria, depending on whether the ZAHB star has a convective envelope or not. In Fig. A.2 the open triangles show the points which divide stars with and without a convective envelope, cooler stars being the ones with convective envelopes. For stars with a convective envelope, 34 EZAHBPs are defined depending on the percentage of convective envelope mass, from the coolest point of the ZAHB (the star with higher convective envelope mass) until the point without a convective envelope. For ZAHB stars without a convective envelope, 100 EZAHBPs are defined, depending on the percentage of envelope mass that ZAHB stars have, from the point without a convective envelope (100% of envelope mass) until the hottest point at the given ZAHB (0% of envelope mass). Using the EZAHBPs it is possible to interpolate directly in metallicity and helium abundance (see Fig. A.2). However, the method used to interpolate between progenitor masses is more complex. First, the EZAHBPs can be used only from the hottest ZAHB point until the coolest point of the minimum progenitor mass (in this case, 0.7 M_⊙). The cool part of the ZAHB is extrapolated from the ZAHB with the nearest higher progenitor mass weighted according to the difference in the stellar parameters between the three progenitor masses. As can be seen from Fig. A.2, we note that the interpolated ZAHB loci are quite satisfactory.

© ESO, 2012