[Paczynski 1997] has argued that the achievable accuracy in the observations of detached eclipsing binaries in globular clusters translates into an accuracy of the determined stellar age of order 2%. Very justified, Paczynski states that "the uncertainties in the stellar models are certainly larger than that''. To give an impression of how large these uncertainties might be, we compare our results to some other contemporary calculations of comparable models.

In a first step we compare ZAMS positions. [Tout et al. 1996] have given analytic functions for ZAMS-positions as function of mass and metallicity based on their own calculations. Since they used a fixed Y-Z-relation we cannot straightforwardly compare their results with our low-metallicity models. The composition closest to their relation is that with Y=0.25 and Z=0.0010, which is to be compared with Y=0.2420 (and the same Z). We find that over the mass range of our calculations our ZAMS models are 0.04 dex brighter in L (with very small variation) and slightly hotter (0.04 dex in $T_{\rm eff}$ for the lowest masses to 0.01 dex for the highest ones). Both effects are consistent with the higher helium content of our models and the fact that we are using a more up-to-date EOS. For a solar-like mixture (Y=0.28, Z=0.02), for which we made additional calculations, the differences are below the 0.02 dex level, reflecting the EOS-change only. We have also calculated a set of ZAMS models for all three metallicities and the same helium content as in [Tout et al. 1996], but using our old EOS. In this case differences are below 0.01 dex both in luminosities and effective temperatures with no systematic effect recognizable.

Next, we compared with results by [Baraffe et al. 1997], where the most important difference to our calculations is the use of the Saumon-Chabrier EOS ([Saumon et al. 1995]). [Baraffe et al. 1997] list data for a set of models with Y=0.25 and metallicities of Z=0.001 and 0.0002, the latter one being intermediate between the lower two of our values. The age of these models is 10 Gyr. Table 6 shows how our models compare either at the same age or for the same luminosity. Comparing at the same age, our models are less luminous by about 0.06 dex for $M/M_\odot=0.8$ but brighter by about 0.02 dex for $M/M_\odot=0.7$ . This translates into age differences (if comparison at identical luminosity is made) of about +0.5 resp. -1.0 Gyr for both metallicities. [Baraffe et al. 1997] used solar metal ratios, but at these low absolute metallicities there is almost no influence of the internal metal composition (solar or $\alpha$ -enhanced) as verified by [Baraffe et al. 1997] themselves (see also [Salaris & Weiss 1998]).

As a further test, we compared with results obtained with the FRANEC code, in particular those by [Cassisi et al. 1998], who provide results for several combinations of input physics data. Their case-8 models are very similar to ours with the major exception being the treatment of the EOS outside the OPAL-range. We compare turn-off (TO) data for several cases in Table 7. The helium content is 0.23 for all models; we have made additional calculations with the same metallicity for this purpose. They were done without an explicit network (equilibrium abundances for the participating nuclei assumed). An explicit network increases the TO-age of the $0.8\,M/M_\odot$ model (Z=0.001) by 0.5 Gyr; the inclusion of both network and pre-main sequence phase results in an increase of only 0.3 Gyr. We add that the calculations of [Salaris & Weiss 1998], done with a variant of the FRANEC code, produce practically the same results as those by [Cassisi et al. 1998]; the small differences can be traced back mainly to the slightly higher helium content of 0.233 (at Z=0.001). Overall, our models take longer to finish the MS-phase, with the differences getting smaller for higher metallicity and mass. With one exception, TO-ages are larger by less than 1 Gyr, or 5-10%. The comparison has been extended for RGB-tip data recently by [Castellani et al. 2000], finding similar agreement. In the same table, for the highest metallicity, data from the latest Padua-tracks ([Girardi et al. 1999]) are listed as well. In this case, the agreement with our own results is even better.

We close this part with a few remarks on comparing isochrones with those by [D'Antona et al. 1997] ([D'Antona et al. 1997], Table 2), who provide turn-off data for a large number of metallicities. At Z=0.0002 and Y=0.23, for which we again have made separate calculations, our 12 Gyr isochrone's TO is at $\log L/L_\odot = 0.469$ and $\log T_{\rm eff} = 3.829$ , which is 0.021 dex brighter and 0.03 dex hotter than the corresponding one by [D'Antona et al. 1997] (for mixing-length theory convection). The turn-off mass of $0.820~M_\odot$ is larger by $0.012~M_\odot$ . For Z=0.001, the differences are very similar ( $\delta\log L/L_\odot = 0.031$ and a TO-mass higher by $0.018~M_\odot$ ). In fact, our TO-values are very close to those of the 11 Gyr isochrone of [D'Antona et al. 1997]. Part of the difference can be ascribed to different helium abundances, which is Y=0.235 for their models in this case. A similar comparison with the [Salaris & Weiss 1998] isochrones (for Y=0.233 and Z=0.001) gave an almost identical result: while at 9 Gyr our isochrone is very close to their corresponding one, the TO-brightness of our 13 Gyr isochrone is almost coincident with the 12 Gyr one of [Salaris & Weiss 1998]. This result is naturally to be expected from the comparison of Table 7. To conclude, it appears that the different low-mass star calculations agree with each other rather well, but the remaining differences, which are partly due to physical assumptions and partly due to technical details translate into age differences of up to 10% for any given composition, mass and luminosity. This can be viewed as the inherent uncertainty the evolutionary calculations carry with them.

Table 7: Comparison between turn-off data for our models and those of [Cassisi et al. 1998] ([Cassisi et al. 1998], their Table 2) and [Girardi et al. 1999]. All models have Y=0.23
Z $M/M_\odot$ age $\log L/L_\odot$ $\lg T_{\rm eff}$ age $\log L/L_\odot$ $\lg T_{\rm eff}$ age $\log L/L_\odot$ $\lg T_{\rm eff}$

Cassisi et al. (1998) Girardi et al. (1999) this paper

0.0001 0.80 11.6 0.410 3.826 12.2 0.415 3.829

0.0002 0.80 11.2 0.378 3.824 12.1 0.380 3.824

0.0010 0.70 20.0 0.060 3.777 21.6 0.074 3.784 21.8 0.074 3.783

0.0010 0.80 11.7 0.231 3.799 12.4 0.240 3.807 12.4 0.249 3.805

0.0010 0.90 7.4 0.393 3.822 7.6 0.406 3.829 7.7 0.394 3.828

0.0010 1.00 5.0 0.577 3.852 5.0 0.567 3.855 5.5 0.586 3.857

**Table 7:** Comparison between turn-off data for our models and those of [Cassisi et al. 1998] ([Cassisi et al. 1998], their Table 2) and [Girardi et al. 1999]. All models have Y=0.23
Z	$M/M_\odot$	age	$\log L/L_\odot$	$\lg T_{\rm eff}$	age	$\log L/L_\odot$	$\lg T_{\rm eff}$	age	$\log L/L_\odot$	$\lg T_{\rm eff}$
		Cassisi et al. (1998)	Girardi et al. (1999)	this paper
0.0001	0.80	11.6	0.410	3.826				12.2	0.415	3.829
0.0002	0.80	11.2	0.378	3.824				12.1	0.380	3.824
0.0010	0.70	20.0	0.060	3.777	21.6	0.074	3.784	21.8	0.074	3.783
0.0010	0.80	11.7	0.231	3.799	12.4	0.240	3.807	12.4	0.249	3.805
0.0010	0.90	7.4	0.393	3.822	7.6	0.406	3.829	7.7	0.394	3.828
0.0010	1.00	5.0	0.577	3.852	5.0	0.567	3.855	5.5	0.586	3.857

4.2 A test application

The intended application of our tracks are detached eclipsing binary systems in globular clusters, which have been detected mainly by the OGLE team in several clusters ([Kaluzny et al. 1996]; [Kaluzny et al. 1997a]). Presently, for none of them follow-up spectroscopy needed to determine absolute parameters, has been concluded, although for one system in $\omega$ Cen preliminary data have been obtained (Kaluzny, private communication). Neither is there any other suitable system from another source available. All well-known systems, e.g. CM Dra and YY Gem ([Chabrier & Baraffe 1995]), $\mu$ Cas ([Lebreton et al. 1999]), or Gl570BC ([Forveille et al. 1999]), are too metal-rich ( ${\rm [Fe/H]} > -0.76$ ). We therefore turned to appropriate single stars to apply our relations and tracks. [Fuhrmann 1998] provides a list of nearby disk and halo stars for which absolute parameters have been derived from a careful spectroscopic analysis in conjunction with Hipparcos parallaxes. From this list we have selected the five most metal-poor stars, of which two, however, are slightly beyond the upper boundary of our metallicity range (Table 8). All stars are enriched in Mg ( $0.28 \le {\rm [Mg/Fe]} \le 0.46$ ) and therefore are assumed to be $\alpha$ -enriched in agreement with our model compositions. Errors in $\rm [Fe/H]$ and $M_{\rm bol}$ are given in Table 8 as well, and are usually very small. The largest uncertainty comes from the mass, which [Fuhrmann 1998] estimates to be of order 5%, or generally, less than 10%. The uncertainty in $T_{\rm eff}$ is $\pm 80$ K in all cases. [Fuhrmann 1998] classifies 4 of the selected stars as halo stars, and the fifth one (HD 201891) as belonging to the thick disk.

If atomic diffusion is in operation, the presently observed and spectroscopically determined metallicity depends on both the initial one and on age. While in globular clusters the initial metallicity of main-sequence and turn-off stars can be estimated quite accurately from that of cluster giants ([Salaris et al. 2000]), this is not possible for field stars. The degeneracy mentioned therefore does not allow to determine the age independently of some assumptions about the initial metallicity. We therefore applied only our t(L)-fitting formulae Eqs. (1) and (2) without diffusion (case C) to these objects.

Table 8 contains age estimates in three steps: Col. 6 gives the age derived from the models with Y=0.25 and a metallicity closest to the determined one (Col. 2), i.e. without any interpolation in $\rm [Fe/H]$ (cf. Table 1). In Col. 8, the age obtained from interpolation to the observed $\rm [Fe/H]$ (but the same helium content) is listed, and in Col. 10 that resulting from interpolation to Y=0.235 (the "generic'' Pop II helium abundance).

Table 8: Application of Eq. (2) (for the "C''-case without diffusion) to dwarfs from the sample of [Fuhrmann 1998] (Cols. 1-5). Errors according to the original paper are given in the second line for each object. Ages are derived in three steps (groups (1)-(3)): The first one using a mixture with Y=0.25 and the metallicity closest to that of the object (cf. Table 1). The second step is to interpolate to the observed metallicity, but still assuming Y=0.25. Finally, full interpolation to the object's metallicity and Y=0.235 is done. For this case, the age uncertainties due to the errors in metallicity, bolometric magnitude and mass ( $\triangle t_1-\triangle t_3$ ) are given. All ages (in 10⁹ yrs) were obtained from application of Eq. (2), while theoretical effective temperatures, given next to ages, were obtained from the evolutionary tracks. Explanation of remarks: (1) age taken directly from evolutionary tracks: step (1): 8.79; step (3): 10.43 Gyr; (2) $T_{\rm eff}$ taken from $0.6\,M_\odot$ -tracks only, since $0.7\,M_\odot$ is always brighter than observed $M_{\rm bol}$ ; (3) no $T_{\rm eff}$ derived, [Fe/H] being too high
object stellar parameters (1) (2) (3)

$\rm [Fe/H]$ $M_{\rm bol}$ $M_\odot$ $T_{\rm eff}$ age $T_{\rm eff}$ age $T_{\rm eff}$ age $T_{\rm eff}$ $\triangle t_1$ $\triangle t_2$ $\triangle t_3$ note

HD 19445 -1.95 4.91 0.74 6016 10.8 6289 11.3 6243 13.4 6122 $\pm 0.2$ $\pm 1.0$ 9.9-18.0

0.07 0.11 0.037 80 12.5-14.3

HD 45282 -1.52 1.98 0.90 5282 8.9 5321 9.2 5218 10.2 5267 $\pm 0.2$ $\pm 0.1$ 8.3-12.6 1

0.06 0.31 0.045 80 9.8-10.6

HD 103095 -1.35 6.33 0.64 5110 8.9 5128 5.8 5195 9.7 5184 $\pm 1.5$ $\pm 1.6$ < 18.1 2

0.10 0.05 0.032 80

HD 194598 -1.12 4.45 0.84 6058 9.3 6232 9.4 6230 11.1 6151 $\pm 0.4$ $\pm 1.0$ 7.7-15.0

0.07 0.16 0.042 80 10.3-11.9

HD 201891 -1.05 4.46 0.81 5943 11.9 -- 12.4 -- 14.4 -- +1.4 $\pm 0.5$ 10.5-18.7 3

0.08 0.09 0.041 80

**Table 8:** Application of Eq. (2) (for the "C''-case without diffusion) to dwarfs from the sample of [Fuhrmann 1998] (Cols. 1-5). Errors according to the original paper are given in the second line for each object. Ages are derived in three steps (groups (1)-(3)): The first one using a mixture with Y=0.25 and the metallicity closest to that of the object (cf. Table 1). The second step is to interpolate to the observed metallicity, but still assuming Y=0.25. Finally, full interpolation to the object's metallicity and Y=0.235 is done. For this case, the age uncertainties due to the errors in metallicity, bolometric magnitude and mass ( $\triangle t_1-\triangle t_3$ ) are given. All ages (in 10⁹ yrs) were obtained from application of Eq. (2), while theoretical effective temperatures, given next to ages, were obtained from the evolutionary tracks. Explanation of remarks: (1) age taken directly from evolutionary tracks: step (1): 8.79; step (3): 10.43 Gyr; (2) $T_{\rm eff}$ taken from $0.6\,M_\odot$ -tracks only, since $0.7\,M_\odot$ is always brighter than observed $M_{\rm bol}$ ; (3) no $T_{\rm eff}$ derived, [Fe/H] being too high
object	stellar parameters	(1)	(2)	(3)
	$\rm [Fe/H]$	$M_{\rm bol}$	$M_\odot$	$T_{\rm eff}$	age	$T_{\rm eff}$	age	$T_{\rm eff}$	age	$T_{\rm eff}$	$\triangle t_1$	$\triangle t_2$	$\triangle t_3$	note
HD 19445	-1.95	4.91	0.74	6016	10.8	6289	11.3	6243	13.4	6122	$\pm 0.2$	$\pm 1.0$	9.9-18.0
	0.07	0.11	0.037	80									12.5-14.3
HD 45282	-1.52	1.98	0.90	5282	8.9	5321	9.2	5218	10.2	5267	$\pm 0.2$	$\pm 0.1$	8.3-12.6	1
	0.06	0.31	0.045	80									9.8-10.6
HD 103095	-1.35	6.33	0.64	5110	8.9	5128	5.8	5195	9.7	5184	$\pm 1.5$	$\pm 1.6$	< 18.1	2
	0.10	0.05	0.032	80
HD 194598	-1.12	4.45	0.84	6058	9.3	6232	9.4	6230	11.1	6151	$\pm 0.4$	$\pm 1.0$	7.7-15.0
	0.07	0.16	0.042	80									10.3-11.9
HD 201891	-1.05	4.46	0.81	5943	11.9	--	12.4	--	14.4	--	+1.4	$\pm 0.5$	10.5-18.7	3
	0.08	0.09	0.041	80

The derived ages appear to be rather consistent except for the 5.8 Gyr for HD 103095 (step 2), which is the most unevolved and least massive object (see Fig. 2 of [Fuhrmann 1998]). The final ages (Y=0.235) range from 9.7-14.4 and are therefore in rough agreement with cluster ages ([Salaris & Weiss 1998]) computed with similar models. In the case of HD 45282, we can derive the age also directly from the evolutionary tracks for this mass ( $0.9\,M_\odot$ ), i.e. without employing our fit formulae. We then obtain for step 1 8.79 Gyr (compared to 8.9 from Eq. (2); Col. 6) and 10.43 Gyr compared to 10.2 Gyr (Col. 10) for the final mixture interpolation. This emphasizes the negligible error due to Eqs. (1) and (2) for these typical ages. In general, we find that ages obtained by linear interpolation between the tracks agree with the fitting formulae results to 5% or better.

Columns 12-14 of Table 8 list the age uncertainties resulting from the errors in the observational quantities, which are given in the second lines of Cols. 2-5. Obviously, the mass uncertainty of 5% (assumed for Col. 4) is by far too large to allow accurate age determinations. The resulting age range ( $\triangle t_3$ ) is of order 8 Gyr, especially for the lower masses. This emphasizes the need for evolved objects close to or after the turn-off. HD 45282 is such an object, beginning already its RGB ascent. For three objects we give $\triangle t_3$ under the assumption that the mass is accurate to 1% (second line of $\triangle t_3$ ), which would lead to acceptable uncertainties. This is also the achievable accuracy in detached eclipsing binary systems ([Paczynski 1997]). The age uncertainty due to metallicity ( $\triangle t_1$ ) is almost negligible and that due to brightness - i.e. distance - errors ( $\triangle t_2$ ) of order 1 Gyr or smaller. An exception is HD 103095, which, due to its low mass and unevolved state is of course most sensitive. In this case, the upper mass limit of $0.68\, M_\odot$ is actually inconsistent with the lowest (zero-age) brightness of our stellar models. A lower limit for $\triangle t_3$ is therefore missing. HD 201891 is outside the metallicity range of our models; its age of 14.4 Gyr, which is the highest of all objects, might be the result of applying Eq. (2) outside its definition range. The upper limit of the metallicity range ( ${\rm [Fe/H]}=-0.97$ ) was not explored. For HD 94598 the small extrapolation was allowed.

As a further consistency test we derived effective temperatures by interpolating between the tracks. These $T_{\rm eff}$ are always given in the column following that with the age. The agreement with the observed temperatures is, at least in the final case (Y=0.235) of order of the $T_{\rm eff}$ -error, with a tendency, however, that our temperatures are higher. This could be an indication that diffusion, which has been ignored here, is indeed active ([Salaris et al. 2000]). $T_{\rm eff}$ for HD 103095 was derived from the $0.6\,M_\odot$ tracks only for the reasons given in the previous paragraph.

We finally comment on the use of models including diffusion. Assuming that the typical metal depletion for a low-mass star of cosmological age is of order 0.3 dex ([Salaris et al. 2000]), we have applied the t(L)-relation of our D-models to HD 19445 and HD 194598. Then the final (step 3) ages turn out to be 13.6 and 10.7 Gyr, which is slightly older than in the C-case. Effective temperatures are reduced to 5931 resp. 5194 K. Both values are again within the observational uncertainties.

4.3 Conclusions

The comparison with other calculations and the application to (single) stars with determined absolute stellar parameters revealed that the largest errors in age determinations based on our stellar evolution tracks are (1) mass, which must be known to 1% accuracy and (2) systematic uncertainties/differences in and between theretical models.

A physical source of uncertainty concerns the effectiveness of diffusion. In our D-calculations, full diffusion of hydrogen and helium with coefficients calculated following [Thoul et al. 1994] was included. This leads in many cases, due to the extremly thin convective envelopes of metal-poor main-sequence stars, to an almost complete depletion of the models in helium, which accumulates below the convective layers. As soon as the star gets cooler, the convective envelope deepens and the helium is mixed back to the surface, as is reflected in the vanishing differences in the HRD in Fig. 3 . [Salaris et al. 2000] recently have investigated in detail the proper use of isochrones to be fitted to either GC or field halo subdwarf data, when diffusion is included. The main point to be stressed is that the present surface metallicity of an individual subdwarf is not the initial one, but is lower by 0.1-0.4 dex (depending on mass and age) due to diffusion. In a GC, however, [Fe/H] is usually determined from red giants, in which the original surface metallicity has been restored by convection. Here, an evolutionary track with this initial metallicity and diffusion included would be the correct one to be used for an individual star.

Arguments in favor of diffusion acting close to how it is calculated are the solar model ([Richard et al. 1996]; [Guenther & Demarque 1997]) and the temperatures of main-sequence subgiants with HIPPARCOS-distances, which, according to [Morel & Baglin 1999] and [Salaris et al. 2000] can be explained by the fact that diffusion leads to lower temperatures (see Fig. 3). Arguments against the full action of sedimentation (rather, arguments in favour of an additional mixing process counteracting diffusion) are the remaining discrepancy between solar models and the seismic Sun just below the convective envelope ([Richard et al. 1996]) and the presence of $^7{\rm Li}$ in old metal-poor stars ([Vauclair & Charbonnel 1998]). In addition, we remark that [Morel & Baglin 1999] tried to explain temperature differences of about 100 K, while in metal-poor low-mass stars the effect of diffusion might reduce $T_{\rm eff}$ by 200 K or more. This leads to colours so red that the comparison with the turn-off colour of some globular clusters would yield negative reddening (e.g. M 5, for which models without diffusion result in a reddening of only 0.02 mag). We added a few test calculations (for the case $M/M_\odot=0.8$ , Y=0.25, Z=0.001), in which either convective overshooting (as in [Schlattl & Weiss 1999]) or an enhanced stellar wind (following [Vauclair & Charbonnel 1995]) or both was employed to reduce the effect of gravitational settling. For pure diffusion a surface helium abundance at the end of the main sequence of $Y_{\rm s} = 10^{-4}$ results; the overshooting models retain up to $Y_{\rm s} = 0.04$ , those with a Reimers mass loss ( $\eta=0.4$ ) $Y_{\rm s} = 0.06$ , and those with both effects $Y_{\rm s} = 0.09$ . Note that the TO age of this star is only 9.8 Gyr. For a cosmological TO age of about 12 Gyr the mass would be lower and the sedimentation effect smaller due to the larger extent of the convective envelope (see also [Salaris et al. 2000]).

Without elaborating further on this discussion, the true effectiveness of diffusion might lead to main-sequence lifetimes somewhere between the extremes of no and full diffusion. All arguments brought forward here concern the photospheric properties of stars; however, the evolutionary speed is determined by the central evolution (diffusion leading to a faster aging by adding helium to the core). On the other hand, the processes counteracting diffusion near to the photosphere could do the same at the center (e.g. rotation-induced mixing). Therefore, the true main-sequence life-time might be in between the two limiting cases investigated here; the difference between them being of order 1 Gyr (see Tables A1-A24). A similar result was obtained by [Castellani & Degl'Innocenti 1999], who discussed the effect in the case of globular cluster isochrones.

To conclude, we have presented an extensive grid of metal-poor low-mass stellar models. The intention is that these data could be used for determining stellar ages, if global parameters such as mass, luminosity and composition of individual halo or globular cluster stars are known. The data can also be used for standard isochrone construction. To facilitate age derivation, we have presented fitting formulae, which reproduce the evolutionary results with an accuracy of 5% or better for the age range of interest ( $\approx 10$ Gyr). We consider the uncertainty of the evolutionary ages to be of order 1 Gyr (at cosmic ages) due to systematic uncertainties in the models and calculations and another 1 Gyr (at most) due to the unknown effectiveness of diffusion. In this respect, the accuracy of the fitting formulae is within these principal uncertainties.

Application of our fit formulae to five metal-poor (halo) field stars with accurately known metallicity and brightness and reasonably well-determined mass resulted in ages between 9.7 and 13.4 Gyr (except for one star with a metallicity outside our model grid). Such ages appear to be in reasonable agreement with recent globular cluster age determinations ([Salaris & Weiss 1998]) using similar stellar models. The uncertainties due to metallicity or distance errors are smaller than the model uncertainties, but the 5% mass uncertainty results in an age error of up to $\pm4$ Gyr. A 1% accuracy in mass must be achieved to make this error source comparable to all others. It appears that the use of the fitting formulae does not introduce an additional error source of relevance.

We have not discussed the importance of the effective temperatures, which in the stellar models depend on the convection theory or parameter used. This is because $T_{\rm eff}$ is a very insensitive discriminator between different masses; it should therefore not be used to select the mass of the evolutionary track to be compared. On the other hand, if the stellar mass is known (with some error), most likely the effective temperature of the corresponding track is within the error range. Finally, for known mass, errors in the models' effective temperature do not influence the t-L-relation. This is different from isochrone age determinations, where $T_{\rm eff}$ influences the morphology and therefore the luminosity of the turn-off, as illustrated by [Mazzitelli et al. 1995] by using two different convection theories. However, $T_{\rm eff}$ -values for stars with determined mass and luminosity will provide independent checks for the quality of the stellar models. Our test application results in effective temperatures, which are within the given uncertainty of the observationally determined ones. This we regard as an encouraging confirmation of our models.

4 Discussion

4.1 Comparison with other theoretical results

4.2 A test application

4.3 Conclusions