[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
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
but brighter by about 0.02 dex for
.
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
-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
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
and
,
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
is larger by
.
For Z=0.001, the differences
are very similar (
and a TO-mass higher
by
). 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.
Z | ![]() |
age |
![]() |
![]() |
age |
![]() |
![]() |
age |
![]() |
![]() |
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 |
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 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]),
Cas ([Lebreton et al. 1999]), or Gl570BC ([Forveille et al. 1999]),
are too metal-rich (
). 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 (
)
and
therefore are assumed to be
-enriched in agreement with our
model compositions. Errors in
and
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
is
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
(cf. Table 1). In Col. 8, the age
obtained from interpolation to the observed
(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).
object | stellar parameters | (1) | (2) | (3) | ||||||||||
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![]() |
age |
![]() |
age |
![]() |
age |
![]() |
![]() |
![]() |
![]() |
note | |
HD 19445 | -1.95 | 4.91 | 0.74 | 6016 | 10.8 | 6289 | 11.3 | 6243 | 13.4 | 6122 | ![]() |
![]() |
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 | ![]() |
![]() |
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 | ![]() |
![]() |
< 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 | ![]() |
![]() |
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 | ![]() |
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
(
), 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 (
)
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
under the assumption that the mass is accurate
to 1% (second line of
), 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 (
)
is almost negligible
and that due to brightness - i.e. distance - errors (
)
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
is actually
inconsistent with the lowest (zero-age) brightness of our stellar
models. A lower limit for
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 (
)
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
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
-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]).
for HD 103095 was derived from the
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.
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
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
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
,
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
results; the overshooting
models retain up to
,
those with a Reimers mass loss
(
)
,
and those with both effects
.
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 (
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 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
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
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,
-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.
Acknowledgements
The continuing support by B. Paczynski, who stimulated this work, is gratefully acknowledged. We are also indebted to S. Degl'Innocenti and M. Salaris for comparing their results with ours, and to D. Alexander and F. Rogers for providing us with their opacity tables calculated for our particular needs. H. Ritter very diligently read the paper and helped to improve it considerably.
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