3 Calculations and results

3.1 Details of the calculations

We have calculated two complete sets of models, one (canonical) without and one with particle diffusion (denoted "C'' and "D''). In each set the following values of mass and composition were explored:

1.

Mass: $M=0.6\cdots 1.3\,M_\odot$ in steps of $0.1\,M_\odot$;

2.

Helium: $Y=0.20,\; 0.25,\; 0.30$;

3.

Metallicity: $Z=0.0001,\; 0.0003,\; 0.001,\; 0.003$.

Due to the varying helium abundance, $\rm [Fe/H]$ is not constant for fixed Z. Table 1 lists $\rm [Fe/H]$ for all mixtures.

   

Table 1: $\rm [Fe/H]$ for the twelve initial compositions, for which evolutionary tracks have been computed

$\downarrow\,Y\,\vert\,Z\,\rightarrow$	0.0001	0.0003	0.0010	0.0030
0.20	-2.645	-2.168	-1.644	-1.166
0.25	-2.617	-2.140	-1.616	-1.138
0.30	-2.587	-2.110	-1.586	-1.108

The metallicity range is that for typical globular clusters, but is not covering the most metal-rich ones like 47 Tuc or M 107. The same enhancement of $\alpha$-elements ([Salaris & Weiss 1998]) is always assumed and is the one for which we have the opacity tables available. Z therefore denotes the total metallicity including the $\alpha$-element enhancement in all cases. Because $\alpha$-element enhancement is typical for Pop II stars, no calculation for solar metal ratios has been done. We recall that for very low metallicities, the evolution depends primarily on the total metallicity ([Salaris et al. 1993]) and only slightly on the internal metal distribution. This, however, becomes non-negligible at the upper end of our metallicity range. For example, [Salaris & Weiss 1998] find that already at Z=0.002 the turn-off of isochrones is about 0.05 mag brighter for models which include $\alpha$-enhancment as compared to a solar-scaled mixture with identical Z. Also, the RGB colour is bluer by $\approx$ 0.05 mag.

It is not yet clear whether the amount of oxygen enhancement in metal-poor stars is independent of metallicity (see [Gratton et al. 2000] for a recent result), or whether there are systematic variations of $\rm [O/Fe]$ with $\rm [Fe/H]$ ( see [Israelian et al. 1998] for unevolved metal-poor stars). In both cases, the oxygen enhancement of our metal mixture ( $\rm [O/Fe]=0.5$) is a good representation of the average enhancement in the metallicity range under consideration. The same is true for the magnesium overabundance ([Fuhrmann 1998]; see also [Salaris & Weiss 1998] for the spread of abundances of other $\alpha$-elements). Variations around the mean $\alpha$-enhancement are a second-order effect, which could be considered only in modeling individual objects, provided the availability of appropriate opacity tables.

The helium values of our mixtures were chosen as to certainly cover the possible range, with the central value of Y=0.25being close to a primordial value of $0.244\pm0.002$([Izotov & Thuan 1998]). This value is somewhat higher than the traditionally assumed 0.23, which, however is too low, even for the more generally accepted primordial value of $0.234\pm0.002$ ([Olive et al. 1997]). Different initial helium contents in the calculations allow save interpolation to any prefered value or to keep it as a free parameter. Some additional mixtures were considered for specific mass values in order to be able to compare with published results (see Sect. 4.1).

Figure 1: Evolution (without diffusion) in the HRD for all masses ( $0.6,\,0.7,\ldots1.3\;M_\odot$) with composition Y=0.25, Z=3 10-4

Figure 2: Influence of composition changes on the evolution of the $0.8\,M_\odot$ model (no diffusion)

Figure 3: Influence of diffusion on the evolutionary tracks (left panel) and lifetimes (right panel) for selected masses ( $0.7,\,0.9,\,1.1,\,1.3\,M_\odot$) and the same composition as in Fig. 1

All calculations were started from homogeneous zero-age main-sequence (ZAMS) models with vanishing gravothermal energies. This implies adjustment of isotopes to their equilibrium values in the stellar core; this period lasts for several 10⁷ years. The resulting small loop in the HRD is omitted in all figures and tables and the ages reset to zero for the models with minimum gravothermal energy production. This definition does not necessarily coincide with the minimum luminosity during the initial loop, which would be an alternative choice for the ZAMS position. For the lower masses our definition corresponds to ages of a few 10⁷ yrs, for the higher masses to about 10⁵ yrs or even less. The evolution is followed up to the tip of the red giant branch (RGB), when helium violently ignites in an off-center shell (core helium flash). No shell-shifting or other approximation is done on the RGB; the full evolution is followed. Typically, the calculations need about 200 time-steps until core hydrogen exhaustion, another 300 until the onset of the first dredge-up, 700 to the end of it and a further 8000 to the tip of the RGB.

The spatial resolution of the models is such that on the main sequence of order 600 and on the RGB twice as many grid-points are needed. We verified that increasing the number of grid-points and time-steps does not influence the relation between luminosity and age by more than a per cent.

3.2 Evolutionary tracks

We display in Fig. 1 the evolution without diffusion ("C''-set) of all masses for the case which is close to the centre of our 3 $\times$ 4 composition space, i.e. for $(Y,\,Z) = (0.25,\,0.0003)$, and in Fig. 2 the changes of the evolution due to variations of the composition for the case of the $0.8\,M_\odot$ model. The left panels show the HRD-tracks (top: varying helium content; bottom: varying metallicity) and the right ones the evolutionary speed. The well-known effects, such as a lower effective temperature for higher Z or lower Y or shorter main-sequence (MS) lifetimes for higher Y or lower Z, are recognizable.

Figure 4: Influence of metal diffusion on the evolutionary tracks (left panel) and lifetimes (right panel) for composition $(Y,\,Z)=(0.25,\,0.001)$ and the same selected masses composition as in Fig. 3. Comparison is made with the case of hydrogen-helium diffusion

The influence of diffusion both on the track in the HRD and on the evolutionary speed is displayed in Fig. 3 for the same reference composition. For sake of clarity the evolution of only a few selected masses are shown. The effects - for example, lower effective temperature and brightness during the main sequence - are as known from other investigations (e.g. [Cassisi et al. 1998]). MS-lifetimes get shorter due to the diffusion of helium into the center, which is effectively equivalent to a faster aging of the star. For given MS-luminosity, TO-models with diffusion can be younger by up to 1 Gyr compared to those calculated canonically. We recall that we include only H/He-diffusion in the grid of models of this paper. To verify that the additional metal diffusion has a negligible influence on the evolutionary tracks and in particular on lifetimes, we show in Fig. 4 the comparison between models with H/He- and H/He/Z-diffusion in the case of mixture $(Y,\,Z)=(0.25,\,0.001)$. We chose a higher metallicity than in the previous example because the depletion of the stellar envelopes in metals due to diffusion is expected to have a higher effect for higher initial metallicity. As Fig. 4 demonstrates, the age-luminosity relation is almost identical and the track in the HRD only slightly shifted to the blue because of the decrease in surface metallicity. After the turn-off, the deepening convective envelope is mixing back quickly the diffused elements such that the initial envelope composition is almost restored (cf. [Salaris et al. 2000]). The tracks approach each other therefore during the subgiant evolution. The surface metallicity drops to a minimum of 42% of the initial one for the $M/M_\odot=0.8$ model, which is in good agreement with results by [Salaris et al. 2000].

The evolutionary properties for all cases calculated are given in tables in Appendix A.

Figure 5: $t(\log L)$ (left axis) from the calculation of a (case C) model with $M/M_\odot = 0.90$, Y=0.25, Z=0.001 (solid) and as obtained from Eq. (1) (dotted) or Eq. (2) (dashed). The relative accuracy (absolute value) for both fitting functions is also shown (corresponding thin lines; right axis)

   

Table 2: Parameters for the function $t= a - \exp(-(\log(L/L_\odot)+b)\cdot c)$. t is in 10¹⁰ years. For each combination of mass $M/M_\odot$, metallicity Z and helium content Y the fitting parameters a, b and c are given. Case "C'' (no diffusion)

$M/M_\odot$	Z	Y=0.20			Y=0.25			Y=0.30
		a	b	c	a	b	c	a	b	c
0.60	0.0001	4.68683	0.35586	3.05625	3.53605	0.34283	3.02336	2.60818	0.33038	3.02014
	0.0003	4.77712	0.36200	3.06499	3.59575	0.34984	3.05363	2.65045	0.33699	3.05964
	0.0010	5.10040	0.37619	3.07399	3.82210	0.36438	3.09124	2.81103	0.35093	3.11996
	0.0030	6.06607	0.37026	2.94939	4.52971	0.38835	3.13366	3.24782	0.37750	3.16778
0.70	0.0001	2.75644	0.23948	3.07744	2.04816	0.22465	3.11651	1.50112	0.20670	3.16830
	0.0003	2.80616	0.24708	3.12424	2.08362	0.23132	3.17035	1.52506	0.21224	3.22277
	0.0010	2.98995	0.26352	3.19686	2.21790	0.24602	3.25090	1.61912	0.22531	3.30198
	0.0030	3.49486	0.29429	3.25585	2.58532	0.27583	3.33550	1.88109	0.25322	3.39598
0.80	0.0001	1.68343	0.13525	3.22279	1.27581	0.11091	3.25539	0.93729	0.08773	3.30100
	0.0003	1.74860	0.13805	3.27157	1.29541	0.11599	3.31912	0.94964	0.09107	3.37295
	0.0010	1.86228	0.15350	3.37035	1.37588	0.12939	3.40681	1.00329	0.10161	3.46594
	0.0030	2.18269	0.18574	3.48119	1.60682	0.15929	3.52297	1.16197	0.12978	3.56537
0.90	0.0001	1.13503	0.02607	3.34182	0.84599	0.00074	3.38876	0.62575	-0.02647	3.43726
	0.0003	1.15214	0.03091	3.41167	0.85656	0.00347	3.47214	0.63156	-0.02646	3.53322
	0.0010	1.22304	0.04434	3.51816	0.90404	0.01403	3.58107	0.66144	-0.02009	3.66277
	0.0030	1.43144	0.07579	3.65932	1.04797	0.04264	3.71174	0.75716	0.00348	3.81259
1.00	0.0001	0.78944	-0.07630	3.47280	0.59243	-0.10569	3.53159	0.44517	-0.13915	3.61112
	0.0003	0.79898	-0.07378	3.56344	0.59726	-0.10614	3.64016	0.44737	-0.14257	3.73657
	0.0010	0.84194	-0.06398	3.70478	0.62523	-0.10057	3.79452	0.46323	-0.14090	3.91179
	0.0030	0.97527	-0.03456	3.87128	0.71489	-0.07726	3.99567	0.51923	-0.12586	4.15912
1.10	0.0001	0.57401	-0.17656	3.62913	0.44051	-0.21321	3.73425	0.34106	-0.26244	3.91988
	0.0003	0.57955	-0.17673	3.74518	0.44161	-0.21628	3.87160	0.34008	-0.26682	4.06653
	0.0010	0.60410	-0.17304	3.95072	0.45619	-0.21517	4.07949	0.34790	-0.27014	4.34218
	0.0030	0.69070	-0.14894	4.19878	0.50901	-0.19791	4.33119	0.37785	-0.25579	4.66150
1.20	0.0001	0.44274	-0.27904	3.85851	0.34658	-0.32992	4.05838	0.26632	-0.38467	4.26117
	0.0003	0.44597	-0.28260	4.01666	0.34535	-0.33435	4.22578	0.26563	-0.40273	4.59135
	0.0010	0.45712	-0.28112	4.25898	0.35288	-0.33823	4.54620	0.27248	-0.42019	5.10977
	0.0030	0.50782	-0.26315	4.55547	0.38208	-0.32214	4.91328	0.28638	-0.40417	5.49256
1.30	0.0001	0.35440	-0.38785	4.17826	0.27521	-0.44247	4.38679	0.21011	-0.50132	4.62344
	0.0003	0.35614	-0.39527	4.39311	0.27396	-0.45857	4.71954	0.21034	-0.53311	5.13498
	0.0010	0.36100	-0.39740	4.74103	0.28144	-0.47611	5.28857	0.21569	-0.55507	5.76582
	0.0030	0.39037	-0.38075	5.16706	0.29669	-0.46093	5.76533	0.22379	-0.52555	5.86558

3.3 Age-luminosity relations

It would be desirable to have an analytical formula t(L,M,Y,Z), which returns the age of a star for any given set of observed quantities. However, there is no simple analytical fit to the results of the evolutionary calculations and high-order fitting formulae are not practical. We have attempted to provide fits which are a compromise between accuracy and simplicity and start with providing a fitting formula to obtain t(L) for each individual mass calculated. This formula is

$$t{\rm [10^{10}~yr]} = a - \exp(-(\log(L/L_\odot)+b)\cdot c)$$ (1)

and in general fits the evolutionary ages after the initial 100-200 Myr with an accuracy of a few percent. The coefficients a, band c depend on mass and composition. Tables 2 and 3 contain the values of all of them for all cases. To illustrate the fit quality, both the fit and the relative fitting accuracy are shown for two selected cases in Figs. 5 and 6 (solid vs. dotted lines).

 

 

Table 3: As Table 2, but for case "D'' (with diffusion)

$M/M_\odot$	Z	Y=0.20			Y=0.25			Y=0.30
		a	b	c	a	b	c	a	b	c
0.60	0.0001	4.32887	0.37567	3.02190	3.26232	0.36413	2.99265	2.41195	0.35193	2.99435
	0.0003	4.40146	0.38369	3.04085	3.31250	0.37205	3.02861	2.44809	0.35903	3.03903
	0.0010	4.68208	0.40064	3.06122	3.50928	0.38869	3.07645	2.58894	0.37435	3.10479
	0.0030	5.49931	0.40621	2.96719	4.04219	0.41951	3.11132	2.96640	0.40530	3.16241
0.70	0.0001	2.60728	0.25458	3.07649	1.93855	0.23953	3.11674	1.42432	0.22087	3.16418
	0.0003	2.65059	0.26277	3.12750	1.97077	0.24646	3.16889	1.44621	0.22648	3.22177
	0.0010	2.82116	0.28025	3.20341	2.09415	0.26202	3.25551	1.53329	0.24023	3.30384
	0.0030	3.27308	0.31450	3.26899	2.42215	0.29481	3.34407	1.76776	0.27068	3.41326
0.80	0.0001	1.64775	0.14155	3.22820	1.22635	0.12050	3.26636	0.90293	0.09644	3.31164
	0.0003	1.67685	0.14833	3.28972	1.24557	0.12581	3.32984	0.91498	0.09989	3.38364
	0.0010	1.78747	0.16442	3.38334	1.32252	0.13983	3.41672	0.96556	0.11174	3.47145
	0.0030	2.11192	0.18609	3.33429	1.53360	0.17155	3.54847	1.11111	0.14061	3.58888
0.90	0.0001	1.10076	0.03244	3.36031	0.82064	0.00676	3.40366	0.60561	-0.02245	3.46209
	0.0003	1.11760	0.03745	3.43144	0.83141	0.00965	3.48631	0.61216	-0.02207	3.55489
	0.0010	1.18750	0.05159	3.53355	0.87804	0.02080	3.59323	0.64233	-0.01466	3.68069
	0.0030	1.38137	0.08447	3.66996	1.01169	0.05043	3.73441	0.73241	0.01050	3.82981
1.00	0.0001	0.76908	-0.07288	3.49792	0.57533	-0.10408	3.56717	0.42600	-0.13838	3.63774
	0.0003	0.77900	-0.07044	3.59247	0.58121	-0.10383	3.66964	0.42929	-0.14256	3.77247
	0.0010	0.82251	-0.05927	3.72051	0.61003	-0.09704	3.81815	0.44773	-0.14085	3.95451
	0.0030	0.94850	-0.02913	3.89619	0.69552	-0.07280	4.02281	0.50360	-0.12202	4.19106
1.10	0.0001	0.55857	-0.17708	3.67657	0.41930	-0.21320	3.75835	0.31158	-0.25374	3.86183
	0.0003	0.56415	-0.17722	3.79350	0.42210	-0.21775	3.91291	0.31300	-0.26297	4.04566
	0.0010	0.59210	-0.17067	3.97315	0.44043	-0.21741	4.13788	0.32390	-0.26772	4.30476
	0.0030	0.67508	-0.14611	4.22927	0.49579	-0.19793	4.40999	0.36014	-0.26203	4.79190
1.20	0.0001	0.41916	-0.28013	3.88606	0.31552	-0.32160	3.99623	0.23514	-0.36652	4.11702
	0.0003	0.42181	-0.28496	4.05931	0.31673	-0.33159	4.20871	0.23532	-0.38079	4.35825
	0.0010	0.43987	-0.28553	4.33719	0.32785	-0.33961	4.54623	0.24104	-0.39874	4.84120
	0.0030	0.49553	-0.26471	4.64456	0.36493	-0.33050	5.06159	0.26657	-0.40618	5.60667
1.30	0.0001	0.32287	-0.38193	4.13115	0.24368	-0.42705	4.25755	0.18201	-0.47279	4.36969
	0.0003	0.32393	-0.39226	4.37277	0.24378	-0.44282	4.53974	0.17950	-0.49064	4.69556
	0.0010	0.33549	-0.39965	4.75053	0.24998	-0.45894	5.04221	0.18393	-0.50652	5.21632
	0.0030	0.37394	-0.39030	5.34027	0.27690	-0.46505	5.89274	0.20304	-0.51820	6.03615

Figure 6: As Fig. 5, but for a model with diffusion (case D) and $M/M_\odot = 1.00$, Y=0.30, Z=0.003

 

 

Table 4: Fitting the mass-dependence of the coefficients of Table 2. The three lines of each composition correspond to the three parameters a, b, and c; for example, the first lines contain coefficients a₀, a₁, a₂, and a₃ of Eq. (2)

Y	Z	a, b, c
0.20	0.0001	29.2077	-70.7155	58.7312	-16.4698
		1.4060	-2.4031	1.3708	-0.4489
		0.1911	9.0341	-9.6597	3.8994
0.20	0.0003	29.7043	-71.7262	59.3967	-16.6087
		1.3975	-2.3636	1.3456	-0.4525
		-0.2651	10.6190	-11.4234	4.6238
0.20	0.0010	31.4324	-75.6675	62.4585	-17.4163
		1.3120	-2.0165	0.9768	-0.3362
		-1.5348	14.8288	-15.9140	6.3227
0.20	0.0030	36.8363	-88.4998	72.8976	-20.2998
		0.8201	-0.4514	-0.5362	0.1330
		-4.3950	23.6936	-24.8876	9.4754
0.25	0.0001	22.5447	-55.0764	46.1447	-13.0444
		1.1258	-1.5493	0.4975	-0.1800
		2.7059	1.1295	-1.7242	1.4248
0.25	0.0003	22.8452	-55.7195	46.5951	-13.1477
		1.4040	-2.4222	1.4181	-0.5054
		-1.0202	13.5479	-15.1164	6.2240
0.25	0.0010	24.0297	-58.3325	48.5215	-13.6191
		1.6165	-3.0661	2.1179	-0.7673
		-4.8060	26.6803	-29.9153	11.8184
0.25	0.0030	27.9680	-67.6242	55.9981	-15.6582
		1.4430	-2.3857	1.3930	-0.5264
		-4.6154	26.4036	-29.9896	12.1702
0.30	0.0001	16.8533	-41.4895	35.0619	-9.9976
		0.8082	-0.5057	-0.6429	0.1976
		4.7946	-5.7016	5.4242	-0.8754
0.30	0.0003	17.0790	-41.9770	35.3982	-10.0704
		0.9524	-1.0040	-0.0459	-0.0471
		3.5676	-0.6182	-1.2110	2.0137
0.30	0.0010	18.1267	-44.4785	37.4060	-10.6092
		0.7120	-0.1966	-0.8545	0.1964
		7.8855	-13.4807	10.8945	-1.3620
0.30	0.0030	20.4666	-49.7150	41.3637	-11.6164
		0.3008	1.3422	-2.5921	0.8232
		20.2151	-54.5229	55.3097	-16.8058

 

 

Table 5: As Table 4, but the case D coefficients of Table 3

Y	Z	a, b, c
0.20	0.0001	26.4015	-63.2538	52.1585	-14.5656
		1.2280	-1.6948	0.5350	-0.1415
		2.1368	2.1647	-1.8484	1.0488
0.20	0.0003	26.7886	-64.1169	52.8122	-14.7334
		1.2478	-1.7368	0.5949	-0.1764
		1.6609	3.9792	-4.0180	1.9709
0.20	0.0010	28.3804	-67.7799	55.6983	-15.5053
		1.2038	-1.5540	0.4377	-0.1471
		1.7792	4.1186	-4.7200	2.5472
0.20	0.0030	32.9185	-78.2930	64.0441	-17.7525
		1.1598	-1.4451	0.4589	-0.2035
		-4.9779	25.6408	-27.2815	10.5084
0.25	0.0001	19.9386	-47.9299	39.6462	-11.1021
		1.1975	-1.5798	0.3577	-0.0798
		2.2408	1.7358	-1.2970	0.8889
0.25	0.0003	20.2355	-48.6066	40.1675	-11.2377
		1.1670	-1.4564	0.2420	-0.0571
		2.5421	1.0307	-0.8450	0.9498
0.25	0.0010	21.4126	-51.3506	42.3545	-11.8293
		1.0981	-1.1875	-0.0110	0.0024
		2.0184	3.4628	-4.2511	2.5979
0.25	0.0030	24.4112	-58.2151	47.7292	-13.2546
		1.2151	-1.5061	0.3984	-0.1801
		-1.2354	15.6116	-18.9391	8.5766
0.30	0.0001	14.7503	-35.5174	29.4317	-8.2556
		1.0895	-1.2037	-0.1142	0.0889
		3.4016	-2.1258	2.9171	-0.5447
0.30	0.0003	15.0295	-36.2278	30.0487	-8.4381
		1.0026	-0.8747	-0.4748	0.2030
		3.2613	-1.4881	2.0549	-0.0469
0.30	0.0010	15.8313	-38.0600	31.4652	-8.8065
		0.5327	0.7118	-2.1346	0.7475
		9.0922	-19.5958	20.0909	-5.6196
0.30	0.0030	18.0913	-43.3256	35.6464	-9.9272
		0.0508	2.3708	-3.8357	1.2883
		20.6253	-56.0962	57.0595	-17.3318

   

Table 6: Comparison between our models and those of [Baraffe et al. 1997] with Z=0.0002 (upper group) and 0.001 (lower group). The first line in each case (0.7 and 0.8 $M/M_\odot$) gives the [Baraffe et al. 1997] data at 10 Gyr, the second and third line our results for the two bracketing metallicities resp. the second line our corresponding model. The comparison is made at same age (Cols. 2 and 3) or at same luminosity (Cols. 4 and 5; age in Gyr). All models have Y=0.25

M/Z	$\lg T_{\rm eff}$	$\log L/L_\odot$	$\lg T_{\rm eff}$	age
$M/M_\odot=0.8$	3.825	0.334
Z=0.0001	3.832	0.285	3.835	10.4
Z=0.0003	3.825	0.255	3.828	10.8
$M/M_\odot=0.7$	3.772	-0.265
Z=0.0001	3.784	-0.233	3.781	8.8
Z=0.0003	3.780	-0.250	3.779	9.4
$M/M_\odot=0.8$	3.800	0.199
Z=0.001	3.809	0.165	3.810	10.4
$M/M_\odot=0.7$	3.755	-0.326
Z=0.001	3.771	-0.298	3.768	8.8

As a next step, we tried to model the dependence of the fitting coefficients in Eq. (1) on mass. Globally, a(M) and c(M) appear to be reminiscent of a parabolic function, while b(M) is close to a linear one. However, individual coefficient values lie off the main trend, such that higher order fits are required. We used the cubic polynomial

$$a = a_0 + a_1\cdot (M/M_\odot) + a_2\cdot (M/M_\odot)^2 + a_3 \cdot (M/M_\odot)^3$$ (2)

(and equivalent expressions for b and c). The individual coefficients still depend on Y and Z. We found that with this approximation the coefficients could be modelled again with an accuracy of a few percent. The coefficients $a_0\ldots c_3$ are listed for both sets of calculations in Tables 4 and 5. The fits obtained by using Eq. (2) with the coefficients taken from Tables 4 and 5 and inserting the proper mass value to obtain the coefficients of Eq. (1) are shown in Figs. 5 and 6 (dashed lines) as well. While in the former case the fit quality is not degraded, it is worse in the latter one, but the errors remain within 5% for most of the evolution (see the thin dashed line), except for the very end of the main-sequence, where the luminosity of the more massive stars shows the complicated "kink''-behaviour in the HRD, which cannot be modelled by this fitting function.

It is not useful to continue with finding fitting functions in composition, because we have only a $3\times 4$ composition space and such functions would require at least 3 parameters for each dimension. Further fitting would therefore only increase the total number of coefficients. Rather, we recommend to interpolate between the ages obtained through Eqs. (1) and (2) to the composition of any observed star.