Mon. Not. R. Astron. Soc. 313, 99±111 (2000)
Chemical composition of eclipsing binaries: a new approach to the
helium-to-metal enrichment ratio
 lvaro GimeÂnez3,4
Ignasi Ribas,1w Carme Jordi,1,2 Jordi Torra1,2 and A
1
Departament d'Astronomia i Meteorologia, Universitat de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Spain
Institut d'Estudis Espacials de Catalunya, Edif. Nexus-104, Gran CapitaÁ, 2-4, E-08034 Barcelona, Spain
3
Laboratorio de AstrofõÂsica Espacial y FõÂsica Fundamental, Apartado 50727, 28080 Madrid, Spain
4
Inst. AstrofõÂsica de AndalucõÂa, Apartado 3004, 18080 Granada, Spain
2
Accepted 1999 October 21. Received 1999 October 18; in original form 1999 August 12
A B S T R AC T
The chemical enrichment law Y(Z) is studied by using detached double-lined eclipsing
binaries with accurate absolute dimensions and effective temperatures. A sample of 50
suitable systems was collected from the literature, and their effective temperatures were
carefully re-determined. The chemical composition of each of the systems was obtained by
comparison with stellar evolutionary models, under the assumption that they should fit an
isochrone to the observed properties of the components. Evolutionary models covering a
wide grid in Z and Y were adopted for our study. An algorithm was developed for searching
the best-fitting chemical composition (and the age) for the systems, based on the
minimization of a x 2 function. The errors (and biases) of these parameters were estimated by
means of Monte Carlo simulations, with special care put on the correlations existing
between the errors of both components. In order to check the physical consistency of the
results, we compared our metallicity values with empirical determinations, obtaining
excellent coherence. The independently derived Z and Y values yielded a determination of
the chemical enrichment law via weighted linear least-squares fit. Our value of the slope,
DY=DZ ˆ 2:2 ^ 0:8; is in good agreement with recent results, but it has a smaller formal
error and it is free of systematic effects. Linear extrapolation of the enrichment law to zero
metals leads to an estimation of the primordial helium abundance of Y p ˆ 0:225 ^ 0:013;
possibly affected by systematics in the effective temperature determination.
Key words: stars: abundances ± binaries: eclipsing ± stars: evolution ± ISM: abundances ±
galaxies: abundances.
1
INTRODUCTION
The slope of the chemical enrichment law Y(Z), namely DY/DZ, is
most frequently determined from observations of extragalactic H ii
regions. Considerable disagreements are found in the results of
various works. Indeed, Lequeux et al. (1979) found a value of
about 3 whereas Pagel et al. (1992) obtained a raw value of 6,
which was subsequently corrected to a final estimation of 4 ^ 1:
On the other hand, Izotov, Thuan & Lipovetsky (1997) determined
a lower value of the slope, close to 2. More recently, Izotov &
Thuan (1998) published a very detailed discussion based on a
sample of 45 low-metallicity H ii regions and derived a slope of
DY=DZ ˆ 2:3 ^ 1: This result is also in agreement with the
review of Peimbert (1995), which concluded that the observations
are compatible with the theoretical value of about 2.5. Along with
the works based on extragalactic H ii regions, other authors have
w
E-mail: [email protected]
q 2000 RAS
attempted a determination of DY/DZ through the analysis of the
fine structure of the ZAMS as a function of Y and Z. The first
results were very uncertain, as demonstrated by the value
DY=DZ ˆ 5 ^ 3 obtained by Perrin et al. (1977)and the lower
limit DY=DZ . 2 derived by Fernandes, Lebreton & Baglin
(1996). The situation is slightly improved in the recent work of
Pagel & Portinari (1998), who used the accurate Hipparcos
parallaxes to infer a global estimate of DY=DZ ˆ 3 ^ 2: The same
value was also found by Fernandes et al. (1998). A different
approach was employed by Renzini (1994), who considered star
counts in the Galactic Bulge to estimate the helium content of
metal-rich stars. He obtained a value of DY/DZ from 2 to 3, with 3
being a strict upper limit. The accurate knowledge of DY/DZ is
crucial for many astrophysical aspects, such as chemical evolution
of the Milky Way and other galaxies, and for stellar evolution in
general. We therefore attempted to undertake an estimation of the
enrichment law from a new point of view, devoting special
attention to evaluating the possible systematic effects. Our
100
I. Ribas et al.
approach is based on the study of eclipsing-binary data. It is well
known that double-lined eclipsing binaries are the source for the
most accurate and simultaneous determinations of masses and
radii (see e.g. Andersen 1991, hereafter A91). These determinations are possible through the analysis of radial velocity and light
curves. Furthermore, if detached double-lined eclipsing binaries
(hereafter DDLEBs) are considered, no significant mass transfer
has occurred between the components, since they are smaller than
their respective Roche lobes. In such cases, the mutual interaction
can be safely neglected and the components can be assumed to
evolve like single, individual stars. Therefore, DDLEBs provide
absolute dimensions for two single stars that can be assumed to
have a common origin both in time and chemical composition. In
this situation, evolutionary models should be able to predict the
same age for both components of the system. The comparison
between model predictions and observations determines the
chemical composition that best reproduces the fundamental
properties of the eclipsing-binary components. The initial helium
abundance (Y) and metallicity (Z) computed in this way are related
to the intrinsic composition, in contrast to spectroscopic
determinations, which provide the atmospheric abundances and
which might not be directly assimilated into the intrinsic
composition when atmospheric peculiarities are present. Therefore, these Z and Y determinations can be confidently used for
determining the helium-to-metal enrichment ratio, as we will show
in the subsequent sections. This paper is organized as follows: the
basic ingredients of our analysis, i.e. stellar evolutionary models
and eclipsing-binary sample, are the subject of Sections 2 and 3;
the fitting algorithm, including the numerical results and an
evaluation of the errors and biases, are described in Section 4; the
results of the metal content determination are compared with
Figure 1. (a) Isochrones and evolutionary tracks computed with Genevagroup models (dotted line) and CG models (solid line) for a chemical
composition of Z ˆ 0:02 and Y ˆ 0:30: The physical ingredients of these
two models are very similar, as deduced from the resemblance between
their tracks and isochrones. (b). Same as panel (a) but for Padova-group
models (dotted line) and CG models (solid line) using a chemical
composition of Z ˆ 0:02 and Y ˆ 0:28: In this case, most of the physical
ingredients are also very similar, except for the adopted amount of
convective overshooting, which shows up as a difference in the main
sequence hook region increasing with mass.
observational determinations in Section 5; Section 6 contains the
interpretation of the fitting algorithm output (Z and Y) in terms of
the chemical enrichment law (slope and primordial helium
abundance); and, finally, a brief summary of the main results is
provided in Section 7.
2
A D O P T E D E VO L U T I O N A RY M O D E L S
We adopted the recent set of models of Claret (1995), Claret &
GimeÂnez (1995), Claret (1997) and Claret & GimeÂnez (1998)
(altogether referred to as CG models). These models are especially
well-suited for our purposes because they cover a wide range, both
in metallicity (Z) and initial helium abundance (Y), which allows
us to consider Y as a free parameter. The prescriptions for dealing
with some physical ingredients adopted by CG (such as mass loss,
neutrino emissions, radiative opacities, equations of state or
convective energy transport) are very similar to those considered
by all current models. Our results will therefore not be influenced,
from a general point of view, by the particular election of CG
Figure 2. Left panel illustrates the way evolutionary tracks change as a
function of the initial helium abundance (Y) in the log g 2 log T eff plane,
for a given Z. In the right panel, the effect of a change in the initial metal
abundance (Z), for a fixed Y, is shown.
q 2000 RAS, MNRAS 313, 99±111
Eclipsing binaries: He/metal enrichment ratio
models. This fact is illustrated in Fig. 1, where evolutionary tracks
and isochrones computed with CG models, Geneva-group models
(Schaller et al. 1992) and Padova-group models (Bressan et al.
1993) are compared. However, it is true that there are still, even
for the main sequence region, open questions (convection
parameters, mass-loss rate, etc.) that might have some influence
on our results.
An illustration of how chemical composition affects the
evolution of a star is presented in Fig. 2, where evolutionary
tracks for several initial masses with different chemical compositions are compared. As can be seen, a change in Y essentially
influences the effective temperature of the model (Teff increases
with Y). On the other hand, Z has a small effect on Teff (i.e.
uncorrelated with Y) but a more noticeable effect on log g and on
the predicted age.
3
SAMPLE OF ECLIPSING BINARIES
A very careful selection of DDLEBs with accurate determinations
(1±2 per cent errors) of masses and radii was compiled in A91.
These systems were mainly observed and analysed by the
Copenhagen group, although Andersen included some systems
from Popper (1980) and other authors. The list comprises 45
systems, from which we discarded HS Aur, YY Gem and FL Lyr,
since they have components with M , 1 M( and the CG models
do not cover this range. A similar case is that of TZ For, whose
primary component is in an evolved stage (core He burning),
beyond the coverage of the CG models. EK Cep, whose secondary
component is still in the pre-main sequence (PMS) phase, was not
taken into account since only post-ZAMS models were considered
with sufficient detail. Additionally, two systems (V624 Her and
WW Aur) were rejected because the small errors quoted seem to
be underestimated (the fractional radii come from single-band
light curves and the systems are partially eclipsing). This makes a
total of 38 systems adopted from Andersen's compilation. We
have also considered the revisions and updatings for four systems
(V539 Ara, b Aur, EE Peg and DM Vir) that have appeared since
1991. A complementary search of the literature was done in order
to enlarge the list with recent publications. A careful evaluation of
the accuracy of the parameters led to a final number of 12
additional systems: AD Boo, IT Cas, Y Cyg, V380 Cyg, V909
Cyg, HS Hya, TV Nor, V3903 Sgr, V906 Sco, CD Tau, BH Vir
and HV 2274. The latter, which belongs to the LMC, was included
because of its low metal content, which will be of great help in
establishing the chemical enrichment law for a wide range of
metallicities. The physical properties of the components of the 50
DDLEBs in the sample are listed in Table 1, and a log M 2 log g
plot is shown in Fig. 3.
Unfortunately, the effective temperatures of the systems listed
in A91 were not so carefully evaluated as the absolute dimensions:
they were adopted directly from several publications, with no
further revision. These temperatures are thus highly non-uniform,
since different calibrations were used by different authors. Most of
the temperatures, coming from analysis dating down to 1975, were
based on old photometric calibrations. Since then, atmosphere
models (Kurucz 1979, 1991, 1994) have been very much improved
yielding new and more accurate photometric calibrations. This
fact drove us to undertake a revision of all effective temperatures
with the aim of using state-of-the-art calibrations and a procedure
as uniform as possible. We collected the available standard
photometry for all systems from the source references. Preference
was given to intermediate-band rather than wide-band measurements. Part of this work has already been done by ourselves and
published in a previous paper (Jordi et al. 1997b), which includes
individual StroÈmgren photometry for a large fraction (29) of the
systems in our current sample. The photometric grids of
Napiwotzki (1998, private communication), based on Kurucz
ATLAS9 atmosphere models (Kurucz 1991, 1994), were adopted
for temperature determination. These grids are an improved
version of those presented in Napiwotzi et al. (1993) by considering revised atmosphere models and including the metallicity
dependence on the photometry. For six of the remaining systems,
individual StroÈmgren photometry was also available. For five
systems we used joint StroÈmgren indices, whereas the temperatures of six binaries (the coolest ones) had to be based on
Johnson±Cousins photometric calibrations (Popper 1980). The
temperatures of the components of TV Nor were computed from
their joint Geneva photometric indices (intermediate-band). The
effective temperature of CD Tau is based on the `infrared flux
method' (IRFM) (Blackwell et al. 1990) applied to IR photometry.
Two systems (Y Cyg and EE Peg) have accurate spectroscopically-determined effective temperatures. They come from
NLTE model fits to Balmer lines in the case of Y Cyg, and
synthetic spectra fitting in a wide spectral range for EE Peg.
Finally, the temperatures of V380 Cyg and HV 2274 are based on
atmosphere model fits to UV spectrophotometry. The adopted Teff
values for all systems and their errors are also listed in Table 1.
4
Figure 3. log M 2 log g plot of the DDLEBs included in our sample. 1s
error bars are also shown, and are even smaller than the size of the symbol
in some cases. The presence of systems below the ZAMS line for Z ˆ 0:02
and Y ˆ 0:28 is related to chemical composition differences.
q 2000 RAS, MNRAS 313, 99±111
101
THE FITTING ALGORITHM
Stellar evolutionary models describe the interior structure and the
observable properties of a star (R, Teff, L, M) of a given initial
mass and chemical composition (X,Y,Z), as a function of time. On
the other hand, the analysis of DDLEBs yields high-accuracy
determinations of the mass and the radius (or log g) of the
components, and also their effective temperatures. Since only for a
very few systems is the atmospheric metallicity [Fe/H] accurately
known, we ignore this information at this point. Thus, the age of
the system and both Z and Y are the unknown parameters of our
DDLEBs. The knowledge of only the masses and radii of the
102
I. Ribas et al.
Table 1. Masses, surface gravities and effective temperatures for the 50
DDLEBs included in our sample.
Name
BW Aqr
V539 Ara
b Aur
AD Boo
GZ CMa
EM Car
QX Car
YZ Cas
IT Cas
PV Cas
SZ Cen
WX Cep
CW Cep
RS Cha
RZ Cha
Y Cyg
MY Cyg
V380 Cyg
V442 Cyg
V478 Cyg
V909 Cyg
V1143 Cyg
DI Her
AI Hya
HS Hya
KW Hya
x 2 Hya
GG Lup
TZ Men
UX Men
TV Nor
U Oph
V451 Oph
EW Ori
V1031 Ori
M/M(
log g (cgs)
log Teff (K)
1.488 ^ 0.022
1.386 0.021
6.24
0.07
5.31
0.06
2.381 0.020
2.306 0.020
1.438 0.016
1.237 0.013
2.206 0.019
2.005 0.018
22.89
0.32
21.43
0.33
9.267 0.122
8.480 0.122
2.314 0.021
1.350 0.010
1.330 0.009
1.328 0.008
2.815 0.050
2.756 0.054
2.317 0.026
2.277 0.021
2.539 0.050
2.329 0.045
13.52
0.39
12.08
0.29
1.858 0.016
1.821 0.018
1.518 0.021
1.509 0.027
17.57
0.27
17.04
0.26
1.811 0.025
1.786 0.030
11.1
0.5
7.02
0.25
1.564 0.024
1.410 0.023
16.67
0.45
16.31
0.35
1.98
0.03
1.75
0.03
1.391 0.016
1.347 0.013
5.185 0.108
4.534 0.066
2.145 0.038
1.978 0.036
1.255 0.008
1.219 0.007
1.978 0.036
1.488 0.017
3.613 0.079
2.638 0.050
4.116 0.040
2.509 0.024
2.487 0.025
1.504 0.010
1.238 0.006
1.198 0.007
2.053 0.022
1.665 0.018
5.198 0.113
4.683 0.090
2.776 0.063
2.356 0.052
1.194 0.014
1.158 0.014
2.473 0.018
2.286 0.016
3.981 ^ 0.020
4.075 0.022
3.926 0.017
4.096 0.022
3.930 0.010
3.962 0.010
4.180 0.011
4.364 0.019
3.989 0.012
4.083 0.016
3.857 0.017
3.928 0.016
4.140 0.020
4.151 0.021
3.995 0.011
4.309 0.010
4.158 0.009
4.175 0.020
4.165 0.025
4.171 0.024
3.486 0.008
3.677 0.007
3.640 0.011
3.939 0.011
4.059 0.024
4.092 0.024
4.047 0.023
3.961 0.021
3.909 0.009
3.907 0.010
4.136 0.012
4.145 0.012
4.007 0.023
4.014 0.023
3.166 0.027
4.132 0.038
3.999 0.016
4.146 0.019
3.919 0.015
3.909 0.013
4.403 0.012
4.288 0.017
4.323 0.016
4.324 0.016
4.297 0.018
4.307 0.017
3.584 0.011
3.850 0.010
4.326 0.006
4.354 0.006
4.079 0.013
4.270 0.010
3.712 0.015
4.188 0.019
4.301 0.012
4.364 0.010
4.225 0.010
4.303 0.009
4.272 0.009
4.306 0.009
4.221 0.010
4.278 0.012
4.081 0.015
4.153 0.018
4.038 0.014
4.196 0.015
4.401 0.010
4.426 0.010
3.560 0.008
3.850 0.019
3.800 ^ 0.007
3.807 0.007
4.268 0.012
4.248 0.012
3.971 0.009
3.964 0.009
3.805 0.006
3.775 0.007
3.927 0.017
3.914 0.017
4.531 0.026
4.531 0.026
4.395 0.009
4.376 0.010
3.959 0.014
3.821 0.016
3.811 0.007
3.811 0.007
4.032 0.012
4.027 0.012
3.878 0.015
3.893 0.015
3.912 0.012
3.944 0.012
4.449 0.011
4.439 0.011
3.883 0.010
3.859 0.010
3.816 0.010
3.816 0.010
4.538 0.008
4.534 0.008
3.850 0.010
3.846 0.010
4.309 0.007
4.299 0.013
3.839 0.006
3.833 0.006
4.484 0.015
4.485 0.015
3.987 0.021
3.944 0.021
3.820 0.008
3.816 0.008
4.241 0.020
4.185 0.020
3.851 0.009
3.869 0.009
3.813 0.003
3.806 0.003
3.900 0.006
3.836 0.007
4.068 0.008
4.043 0.008
4.170 0.014
4.041 0.024
4.022 0.010
3.857 0.012
3.785 0.007
3.781 0.007
3.960 0.007
3.892 0.005
4.211 0.015
4.188 0.015
4.037 0.016
4.006 0.017
3.776 0.007
3.762 0.007
3.889 0.028
3.923 0.026
Ref.
1,2
2,3
2,4
5
1,2
1
1,2
1,2
6
1,2,7
1,2
1,2
1,2
1,2
1,2
8
1,2
9
1
1
10
1,2
1,2
1,2
11
1,2
1,2
1,2
1,2
1,2
12
1,2
1,2
1
1,2
Table 1 ± continued
Name
EE Peg
IQ Per
AI Phe
z Phe
PV Pup
VV Pyx
V1647 Sgr
V3903 Sgr
V760 Sco
V906 Sco
CD Tau
CV Vel
BH Vir
DM Vir
HV 2274
(LMC)
M/M(
2.156
1.335
3.521
1.737
1.236
1.195
3.930
2.551
1.565
1.554
2.101
2.099
2.189
1.972
27.27
19.01
4.980
4.620
3.378
3.253
1.442
1.368
6.100
5.996
1.165
1.052
1.454
1.448
12.2
11.4
0.024
0.011
0.067
0.031
0.005
0.004
0.045
0.026
0.011
0.013
0.022
0.019
0.037
0.033
0.55
0.44
0.090
0.073
0.071
0.069
0.016
0.016
0.044
0.035
0.008
0.006
0.008
0.008
0.7
0.7
log g (cgs)
4.129
4.327
4.208
4.323
3.596
3.997
4.122
4.309
4.256
4.278
4.089
4.088
4.253
4.289
4.058
4.143
4.177
4.259
3.656
3.858
4.087
4.174
4.000
4.023
4.307
4.351
4.108
4.106
3.536
3.585
0.013
0.009
0.019
0.013
0.014
0.012
0.009
0.012
0.010
0.011
0.009
0.009
0.012
0.012
0.016
0.013
0.021
0.019
0.012
0.013
0.010
0.012
0.008
0.008
0.014
0.017
0.009
0.009
0.027
0.029
log Teff (K)
3.940
3.802
4.111
3.906
3.705
3.793
4.149
4.072
3.840
3.841
3.979
3.979
3.975
3.949
4.580
4.533
4.217
4.202
4.017
4.029
3.792
3.792
4.254
4.251
3.789
3.750
3.806
3.806
4.362
4.364
0.005
0.005
0.008
0.008
0.012
0.010
0.010
0.007
0.010
0.010
0.009
0.009
0.014
0.014
0.020
0.022
0.013
0.013
0.020
0.020
0.004
0.004
0.012
0.012
0.005
0.006
0.010
0.010
0.003
0.003
Ref.
13
1,2
1
1,2
1
1,2
1,2
14
1,2
15
16
1,2
17,18
2,19
20
References±1: A91, 2: This work, 3: Clausen (1996), 4: NordstroÈm &
Johansen (1994), 5: Lacy (1997a), 6: Lacy et al. (1997), 7: Popper (1987),
8: Simon, Sturm & Fiedler (1994), 9: Guinan et al. (in preparation), 10:
Lacy (1997b), 11: Torres et al. (1997), 12: North, Studer & KuÈnzli (1997),
13: Linnell, Hubeny & Lacy (1996), 14: Vaz et al. (1997), 15: Alencar, Vaz
& Helt (1997), 16: Ribas, Jordi & Torra (1999a), 17: Popper (1997), 18:
Clement et al. (1997), 19: Latham et al. (1996), 20: Ribas et al. (1999b).
components does not unequivocally determine these parameters.
Therefore, the effective temperatures of the components, although
we know them not to be so reliable as the absolute dimensions,
need to be included in the algorithm. This is because a change in
the value of Y is equivalent to changing the effective temperature
(see Fig. 2), or, in other words, the value of the effective
temperature mostly determines the initial helium abundance of the
system, whereas it has less effect on the initial metal abundance. If
a chemical enrichment law, i.e. Y(Z), is assumed, the knowledge of
the masses and radii is sufficient to unequivocally determine the
best fitting isochrone (see Ribas 1996; Jordi et al. 1997a; Pols
et al. 1997). However, if the isochrone is forced to fit all data,
including the observed effective temperature, any systematics
present in its determination may introduce a bias on the derived
ages and metallicities. When Y is left free, its determination may
be affected by any systematics in the temperature, but these
systematics would have little influence on the estimation of the
age and the metallicity. Therefore, we preferred to make use of all
the observational information, and not to impose any restriction on
the initial helium abundance with the aim of avoiding possibly
biased solutions. Another point that might introduce some bias is
the age origin. The evolutionary models of CG place their time
origin on the ZAMS. But there is a PMS phase whose duration
depends on the mass of the star. Several simulations for the ages
and mass differences of the stars in our sample showed that the
effect of not considering the time spent in the PMS phase is
negligible and that it does not introduce any bias. However, for the
q 2000 RAS, MNRAS 313, 99±111
Eclipsing binaries: He/metal enrichment ratio
103
sake of completeness, the ages derived from our post-ZAMS
models were corrected for a very small additive term, due to PMS
evolution, which we computed from the models of Iben (1965).
4.1
Foundations of the algorithm
When the mass and the surface gravity are known, each
combination (Z,Y) in the evolutionary models predicts an age
and an effective temperature for the star. We therefore constructed
a function (x 2) that depends on the difference of the predicted
ages for both components …log tA 2 log tB †; and on the difference
between the observed (superscript `obs') and model predicted
(superscript `mod') effective temperatures …log T obs
eff A;B 2
log T mod
eff A;B †: Our algorithm is based on finding the initial
chemical composition that provides the smallest possible value
of this function, i.e. the best description of the observed properties
of the system components. Therefore, two procedures are
involved: interpolation in the evolutionary models and minimization of a bidimensional function.
(i) The interpolation procedure uses as input data, on the one
hand, the grid of evolutionary tracks of CG computed for different
(Z, Y) values (Z ˆ 0:004±0:03; Y ˆ 0:18±0:42) and, on the other
hand, the following stellar parameters: the initial chemical
composition (Z*, Y*), the observed (instantaneous) mass (M*)
and the observed surface gravity (log g*). Three basic steps are
then followed. The first step consists in the calculation of a set of
evolutionary models (SEM) for the assumed chemical composition (Z*, Y*) by interpolating within the discrete grid of (Z, Y)
values. Taking into account the uniform distribution of the grid
points, the interpolation can be easily performed, first as a
function of Y and then as a function of Z. Some tests indicated that
the best results are obtained when using a linear interpolator in Y
and a logarithmic interpolator in Z. In the second step, complete
evolutionary tracks for the observed mass M* of each component
are interpolated between two contiguous tracks of the SEM at
(Z*, Y*). Logarithmic interpolation yields the best results as
indicated by several tests. Finally, in the last step, the evolutionary
track for M* is linearly interpolated at the observed log g*
yielding the age, effective temperature, luminosity, initial mass
and internal structure constants. An iterative scheme was used to
correct the interpolation mass (which was initially adopted as M*)
by taking into account mass loss during the evolution (CG adopted
the equation from Nieuwenhuijzen & de Jager 1990). Thus, the
initial mass (MZAMS) of the star is obtained, after a few iterations,
in such a way that a zero-age star with MZAMS evolves and yields a
star of M* …M* , M ZAMS † and log g* after losing a fraction of the
mass via stellar wind. We have to mention that this correction was
very small (below 1 per cent) for most systems in our sample. The
most critical system is probably EM Car, but even in that case the
amount of mass loss is insignificant (2 per cent). A flux diagram
that shows all the steps and the parameters involved in the
interpolation procedure is presented in Fig. 4.
An evaluation of the interpolation errors was done in two steps
(see Asiain, Torra & Figueras 1997 for a complete description of
this procedure). First, a whole SEM of a given (Z,Y) was
suppressed and interpolated from the remaining ones. The
resulting and the suppressed SEMs were then compared for
several different masses, which yielded the values listed in the first
three rows of Table 2. This is actually an overestimation of the
actual mean interpolation errors by about a factor of four, since the
density of the initial grid was degraded when suppressing a certain
q 2000 RAS, MNRAS 313, 99±111
Figure 4. Flux diagram showing the individual steps of the interpolation
procedure. Input data are Z*, Y*, M* and log g*. Notice the iterative
process designed to correct for mass loss.
Table 2. Standard deviations of the difference
between actual and interpolated evolutionary tracks
for several different masses. The figures listed here
are, approximately, a factor of four larger than the
estimated mean errors. The first three rows correspond
to the interpolation for a given (Z, Y ), and the
remaining rows correspond to the mass interpolation
alone.
s (D log Teff)
s (D log g)
s (Dt /t )
s (D log Teff)
s (D log g)
s (Dt /t )
15 M(
5 M(
2.5 M(
1.5 M(
0.002
0.017
0.045
0.002
0.010
0.006
0.001
0.009
0.035
0.002
0.007
0.015
0.007
0.018
0.038
0.002
0.010
0.019
0.010
0.028
0.035
0.005
0.010
0.009
(Z, Y). The error due to mass interpolation was evaluated in the
same way, i.e. by eliminating a whole track of a given mass from
the SEM and interpolating it from its neighbours. The results of
the comparison of the interpolated track with the real track are
shown in the last three rows of Table 2. These values are also an
overestimation of the mean interpolation errors by about a factor
of four, for the same reason as previously exposed. In summary,
the interpolation errors can be estimated to be below 1.5 per cent
in all the parameters for the evolutionary stages of the systems in
the sample.
(ii) The interpolation procedure previously described provides
the age and the effective temperatures, among other parameters, of
each component of the system for a given (Z*, Y*). We therefore
obtain two age determinations that can be mutually compared, and
two effective temperatures that can be compared to the
observational determination. Thus, the following function for
each DDLEB system was evaluated:
x2 …Z; Y† ˆ vlog t
…log tA 2 log tB †2
s2log tA ‡ s2log tB
‡ vlog T A
mod 2
…log T obs
eff A 2 log T eff A †
2
slog T A
‡ vlog T B
mod 2
…log T obs
eff B 2 log T eff B †
;
2
slog T B
…1†
104
I. Ribas et al.
where t is the age, s p is the standard error associated to the
parameter p and v p is the relative weight that is assigned to p. The
criteria for the weight system were the following: twice the weight
to the ages than to temperatures, since the former come from the
more reliable determinations (masses and radii) and the temperature of the primary was given twice the weight than that of the
secondary, since the latter is usually fainter and its photometric
indices have larger uncertainties. With this prescription, the
relative (non-normalized) weights were the following:
vlog t ˆ 6;
vlog T A ˆ 2;
vlog T B ˆ 1:
…2†
Nevertheless, tests made with different weight systems showed a
very weak influence on the final results. This can be explained by
the essentially uncorrelated behaviour of Z and Y, i.e. Z depends
mostly on the age and Y is determined by the value of the effective
temperature. Owing to the small effect of the weight system used,
we decided not to adopt any special criteria depending on the
characteristics of a given system, e.g. totally eclipsing cases.
Equation (1) makes use of all the information available for the
eclipsing binaries. Considering either TeffB/TeffA or T effB 2 T effA
instead of the actual values of the temperatures could be an
alternative functional form. The advantage of such an approach is
obvious, since the temperature ratio (or difference) can be
obtained from the light curve analysis and it is a scale-independent
quantity. Several tests were made in order to evaluate the
possibility of using this prescription. The results were negative
since the temperature ratio essentially does not depend on Z and it
varies very slowly as a function of Y, thus the temperature ratio is
not a discriminant parameter. Therefore, the actual value of the
effective temperature is needed to obtain an estimation of the
initial helium abundance of the system. There should exist a
certain chemical composition that yields the same ages for both
components and effective temperatures identical with the observed
ones, i.e. x2 ˆ 0: But, taking into account the observational errors
and the inaccuracies of the models, our algorithm searches for the
smallest value of x 2 that is assumed to represent the chemical
composition that best reproduces the observed parameters of the
components. x 2, as defined in Eq. (1), is a non-analytical
bidimensional function, and the `downhill simplex method'
(Nelder & Mead 1965) was used to find its minimum. The
computational implementation of this method, which requires
only function evaluations and no derivatives, is explained in Press
et al. (1992). In the regions where the function is multivalued, i.e.
main sequence hook (MS hook), the x 2 function was computed by
taking into account all the different value combinations (three or
nine, depending on the position of one or both components in the
multivalued region). In such a case, we adopted the combination
that yielded a smaller value for x 2.
4.2
Results
From its definition, the x 2 function becomes more sensitive to Z
and Y changes when both components of the DDLEB system have
Figure 5. 3-D and contour plots of the x 2 function (equation 1) for eight eclipsing-binary systems in the sample. Cases with well-defined minimum, shallow
and elongated minimum, and minimum outside the (Z,Y) range are shown for illustrative purposes. The discontinuity in the bottom panels is caused by the
location of the system in a multivalued region (MS hook).
q 2000 RAS, MNRAS 313, 99±111
Eclipsing binaries: He/metal enrichment ratio
different locations in the H±R diagram. This can be easily
understood in terms of an isochrone that has to fit two points in the
H±R diagram with given error bars: the closer the two points, the
larger the range of Z and Y that can fit both at the same time. Thus,
our analysis yields only reliable results for those eclipsing-binary
systems whose components are separate enough in the log M 2
log g diagram. For the remaining ones, no information about the
model predictions can be extracted from isochrone fitting. The
separation was defined as
s
…M A 2 M B †2 …log gA 2 log gB †2
Sˆ
…3†
‡
s2MA ‡ s2MB
s2log gA ‡ s2log gB
of the sample are shown in Fig. 5. Examples of systems with a
well-defined minimum or a shallow minimium were selected for
illustrative purposes. Also, some systems with components close
to the multivalued region or with a minimum outside the explored
(Z, Y) range are shown. In general, the depth of the minimum,
which is related to the reliability of the solution, increases when
both components are far from the ZAMS and when S is large. The
latter has already been justified, and the former can be easily
understood in terms of the age-degeneracy close to the ZAMS. In
that position, small variations in the input parameters lead to very
large variations in age, and so sensitivity of the function is greatly
decreased.
The results for those systems with a solution, i.e. those with a
minimum in the covered (Z, Y) range, are shown in Table 3. GZ
CMa, for which a small extrapolation was needed, is also
included. The output best-fitting parameters include the age and
the chemical composition (Z, Y) that best reproduces the system
properties, and the corresponding value of x2min : The latter is
and the minimum separation parameter was set to S ˆ 2; which
means that both components have to be separated by at least twice
the standard errors. With these restrictions, the number of
DDLEBs selected for further analysis was reduced from 50 to
41. A 3-D plot and a contour plot of x 2(Z, Y) for several systems
Table 3. Results of the fitting algorithm. Those systems with a minimum in the covered (Z, Y) range are listed
in the left part, except for GZ CMa for which a solution involving a small extrapolation was accepted. The
errors were computed through a Monte Carlo simulation (see text). Notice that V380 Cyg and HV 2274 are also
listed, although their analysis was not done through the fitting algorithm. The right part contains further
information for those systems that do not have a minimum in the covered (Z, Y) range.
Name
V539 Ara
AD Boo
GZ CMa
EM Car
YZ Cas
WX Cep
CW Cep
V380 Cyg
V442 Cyg
V1143 Cyg
AI Hya
HS Hya
KW Hya
x 2 Hya
TZ Men
TV Nor
U Oph
V451 Oph
V1031 Ori
EE Peg
AI Phe
z Phe
V1647 Sgr
V3903 Sgr
V906 Sco
CD Tau
CV Vel
HV 2274
Name
BW Aqr
b Aur
QX Car
SZ Cen
RS Cha
DI Her
GG Lup
UX Men
EW Ori
IQ Per
PV Pup
V760 Sco
BH Vir
q 2000 RAS, MNRAS 313, 99±111
105
log age (yr)
Z
Y
x2min
7.614 ^ 0.015
9.303 0.041
8.694 0.020
6.791 0.011
8.654 0.014
8.789 0.024
6.666 0.063
7.209 0.031
9.158 0.032
8.650 0.980
9.143 0.024
9.139 0.050
8.953 0.027
8.281 0.024
8.288 0.030
8.562 0.029
7.675 0.032
8.403 0.031
8.846 0.010
8.563 0.032
9.721 0.007
7.842 0.019
8.489 0.042
6.377 0.033
8.420 0.025
9.433 0.035
7.758 0.009
7.243 0.012
0.012 ^ 0.003
0.022 0.006
0.032 0.006
0.005 0.002
0.016 0.004
0.014 0.003
0.023 0.007
0.020
0.020 0.007
0.016 0.008
0.020 0.004
0.011 0.009
0.013 0.003
0.014 0.003
0.014 0.002
0.015 0.002
0.017 0.005
0.015 0.002
0.016 0.004
0.026 0.007
0.010 0.006
0.013 0.001
0.013 0.002
0.010 0.003
0.016 0.003
0.026 0.007
0.007 0.005
0.007
0.282 ^ 0.027
0.243 0.030
0.312 0.035
0.238 0.083
0.270 0.026
0.240 0.025
0.298 0.101
0.28
0.303 0.058
0.242 0.028
0.271 0.048
0.240 0.054
0.201 0.026
0.276 0.025
0.254 0.027
0.255 0.034
0.253 0.046
0.279 0.037
0.248 0.038
0.283 0.034
0.241 0.049
0.290 0.022
0.224 0.038
0.276 0.047
0.253 0.040
0.263 0.036
0.199 0.048
0.260
.03
0.039
0.205
0.079
0.001
0.156
0.081
0.252
0.086
0.385
0.044
0.060
0.121
0.180
0.113
0.217
0.209
0.054
0.045
0.134
0.106
0.725
0.043
0.017
0.500
0.133
0.032
Comments
Unreliable extrapolated solution at Z @ 0:03 and Y @ 0:4: However, x2 , 1 in some regions.
Unreliable extrapolated solution at Z @ 0:03 and Y @ 0:4: However, x2 , 1 in some regions.
Extrapolated solution at Z ˆ 0:035 and Y ˆ 0:368; with x2min ˆ 0:047:
Metal-poor solution, but very strongly dependent on the value of the overshooting parameter.
Solution at Z ˆ 0:018 and Y ˆ 0:264; with x2min ˆ 2:32: x2 . 1 in all the (Z,Y) range.
ZAMS system. No solution in the (Z, Y) range, but x2 , 1 in some regions.
ZAMS system. No solution in the (Z, Y) range, but x2 , 1 in some regions.
Extrapolated solution at Z ˆ 0:039 and Y ˆ 0:329; with x2min ˆ 0:018:
Extrapolated solution at Z ˆ 0:042 and Y ˆ 0:336; with x2min ˆ 0:017:
The solution places the B component between the first two points of the evolutionary track.
ZAMS system. No solution in the (Z, Y) range, but x2 , 1 in some regions.
ZAMS system. No solution in the (Z, Y) range, but x2 , 1 in some regions.
No solution in the (Z, Y) range, but x2 , 1 in some regions.
106
I. Ribas et al.
related to the quality of the best-fitting solution that the models are
able to provide. Notice the large error in the age of V1143 Cyg,
which is caused by the proximity of its components to the ZAMS.
Information about the remaining systems (systems with ZAMS
components, without a solution in the covered (Z, Y) range or with
extrapolated solutions) is also presented in Table 3. Some
comments on a handful of systems should be given at this time.
The age and chemical composition of HV 2274 and V380 Cyg,
both included in Table 3, were not computed using the fitting
algorithm. Careful, individual analysis of these systems have been
published separately (HV 2274: Ribas et al. 1999b; V380 Cyg:
Guinan et al. in preparation). In the case of HV 2274, both
components are too close in the log g 2 T eff diagram to be suitable
for applying the fitting algorithm. Therefore, a spectrophotometric
metallicity determination (checked against the known LMC metal
content) was adopted. Only the age and Y were considered as free
parameters, which were confidently determined in spite of the
similarity of the components. V380 Cyg contains the most evolved
massive star in the sample. At its evolutionary stage, the models
are critically influenced by the value of the overshooting
parameter. In Guinan et al. (in preparation), the analysis is done
with a set of specially-built evolutionary tracks with various
overshooting parameters, and by comparing with the spectrophotometric metallicity determination. The value derived from
this detailed study is listed in Table 3. Finally, SZ Cen is also a
moderately-evolved system but with intermediate-mass components. Similarly to V380 Cyg, the chemical composition obtained
from the fitting algorithm is strongly influenced by the value of
the overshooting parameter, owing to the critical location of the
components in the log g 2 log T eff diagram. A thorough discussion on this point is provided in Ribas et al. (in preparation).
Therefore, SZ Cen, with anomalously low metal and helium
abundances obtained from the algorithm, was not considered
for further work. Also in Ribas et al. (in preparation), other
moderately-evolved intermediate-mass systems in the current
sample (WX Cep, AI Hya and V1031 Ori) are shown not to
experience the same kind of behaviour, owing to the less-critical
location of their components in the log g 2 log T eff diagram.
Therefore, the chemical composition determinations for these
systems are reliable.
4.3
Errors
The errors of the output best-fitting parameters were computed
through a Monte Carlo simulation in order to achieve the best
possible estimation and to be able to analyse the possibility of
biases. We must point out that in both cases only the uncertainties
of the input parameters were considered, since the interpolation
errors were shown to be negligible. A very important remark must
be made concerning the correlation of the errors associated with
the parameters of both components. Actually, the relative values
RB =RA ; M B =M A and T effB =T effA are known with a significantly
better accuracy than the actual absolute values. The determination
of these ratios comes from certain specific characteristics of the
light and radial velocity curves that are accurately modelled.
However, when the standard errors of M, log g and Teff for both
components are provided, this correlation is not explicitly shown.
We should remark here that k ˆ RB =RA is quite often a poorlydefined parameter for systems with very similar components and
eclipses of equal depth. However, as we mentioned previously, we
did not consider for further study those systems with similar
components, therefore minimizing the possibility of such
degeneracy. The correlation between the errors of the components
has crucial importance when evaluating the uncertainties of our
algorithm, since the relative position of the components in the
H±R diagram, which fixes the slope of the isochrone, is better
defined than their absolute location. Unfortunately, the errors of
RB =RA ; M B =M A and T effB =T effA are not explicitly shown in most
of the publications. In order to evaluate them, we compared the
standard errors of the absolute parameters with the errors of the
parameter ratios for several systems for which both of them are
known. We found that, in all cases, the relative errors are a factor
of 2.5 smaller than the error estimations of the parameters
themselves. This error ratio was also adopted for all our systems.
A simulation considering 500 points was run for each of the
systems with a solution in the (Z, Y) range. This number ensures
an statistical error below 5 per cent, which is suitable for our
purposes. We generated a random set of parameters M, log g and
Teff for the primary component following a normal distribution
around the observed values with a standard deviation equal to the
observed error. Once a new set of parameters is generated for the
primary component, they are re-scaled to the secondary
component by means of the ratios RB =RA ; M B =M A and
T effB =T effA : Then, a new random set of parameters for the
secondary component is generated around the re-scaled values
with a standard deviation 2.5 times smaller than the observed
error. After running the simulation, the error of each parameter
was adopted as the standard deviation of all the solutions obtained.
The inverse value of x2min of each solution was used as the weight
for computing the standard deviation. Thus, we considered a
larger weight when the minimum of the solution was better
defined, i.e. the fundamental actual parameters could be more
accurately reproduced. For systems close to the limit of the
covered region in the (Z, Y) range and with shallow minima, only
the inner region of the Gaussian distribution was considered when
computing the standard deviation, in order to avoid possible
extrapolation biases. The errors adopted are listed, together with
the corresponding parameters, in Table 3. The smallest errors in Z
and Y are about 0.002 and 0.025, respectively, for the sharpest
minima, and they rise to about 0.010 and 0.100, respectively, for
the systems with the widest, more poorly-defined minima. The
mean errors are ksZ l ˆ 0:004 and ksY l ˆ 0:04; i.e. around 20 per
cent in both cases.
4.4
Biases
A biased determination of the best-fitting parameters may result if
the x 2 function presents a strong asymmetric shape in the
minimum region. In order to study this possibility, we used the
results of the Monte Carlo simulation described in the previous
section to compare the distribution of the solutions with a
Gaussian distribution. A simple way of evaluating the asymmetry
of the distribution consists in comparing the best-fitting parameters with the mean values of the 500 simulations. We computed
these mean values for the systems in Table 3. The differences
between these values and the best-fitting parameters are listed in
Table 4. As may be seen, the differences are small in all cases and
always smaller than the standard errors listed in Table 3.
Additionally, there does not exist any systematic trend either in
ˆ
the form of a zero point (the mean differences are kZ 2 Zl
20:001 ^ 0:002 and kY 2 Yl ˆ 0:000 ^ 0:012) or as a functional
dependence of Z or Y.
Thus, we conclude that no significant statistical bias is present
when considering a sample of systems. However, some individual
q 2000 RAS, MNRAS 313, 99±111
Eclipsing binaries: He/metal enrichment ratio
Table 4. Difference between the best-fitting chemical composition and
the mean value obtained from a Monte Carlo simulation …n ˆ 500†: GZ
CMa (extrapolated solution), V380 Cyg and HV 2274 (both have
chemical composition values computed without using the fitting
algorithm) are not included.
Name
Z 2 Z
Y 2 Y
Name
Z 2 Z
Y 2 Y
V539 Ara
AD Boo
EM Car
YZ Cas
WX Cep
CW Cep
V442 Cyg
V1143 Cyg
AI Hya
HS Hya
KW Hya
x 2 Hya
TZ Men
0.000
20.004
0.000
20.002
0.001
20.005
20.001
0.002
20.003
20.003
0.000
0.001
0.000
20.002
20.012
0.001
20.007
0.014
20.008
20.004
0.018
20.008
20.024
20.004
0.011
0.001
TV Nor
U Oph
V451 Oph
V1031 Ori
EE Peg
AI Phe
z Phe
V1647 Sgr
V3903 Sgr
V906 Sco
CD Tau
CV Vel
0.000
0.001
0.001
20.001
0.003
20.001
0.000
0.000
20.001
20.002
20.002
20.002
0.003
0.033
0.012
20.020
0.014
20.011
0.001
20.001
0.000
0.001
20.006
20.009
Figure 6. Histograms showing the distribution of 500 simulated solutions
for the chemical composition of the eclipsing binaries V539 Ara and HS
Hya. The vertical solid line represents the actual best-fitting solution listed
in Table 3, and the vertical dotted line is the mean value of the distribution.
Notice the asymmetric behaviour, especially in the case of HS Hya.
effects are not discarded, since they may depend on the actual
location of the system in the H±R diagram. This possibility is
statistically difficult to evaluate, owing to the fact that it is related
to a single system. Since the asymmetry that we observe is rather
small in all cases, no significant bias should be expected for any of
the systems. As an example, Fig. 6 shows histograms for Z and Y
computed for two extreme cases: V539 Ara and HS Hya.
V539 Ara is a system with a well-defined and deep minimum
that presents a very slightly asymmetric distribution with an
extended wing towards high Z. Nevertheless, the mean value of the
distribution is indistinguishable from the best-fitting values
(dotted and solid lines in the figure, respectively), indicating
that no significant bias is present. HS Hya is an example of a
system with a shallow minimum. In this case, the solution is not so
well defined and a clear asymmetric trend is observed in the figure
(towards high Z and high Y). Thus, the mean values of the
distribution are larger than the best-fitting parameters. Even in an
q 2000 RAS, MNRAS 313, 99±111
107
extreme case, such as HS Hya, a possible bias is a factor of two
smaller than the standard error listed in Table 3. Despite the
asymmetric distribution displayed by a small number of systems,
we adopted the criterion of quoting the standard deviation of a
normal distribution as the error of our determination. In our
examples, such a criterion includes 82 per cent and 57 per cent (Z
and Y) of the simulated solutions for V539 Ara, and 82 per cent
and 75 per cent (Z and Y) for HS Hya.
5
R E S U LT S O N M E TA L C O N T E N T
Although it has been clearly stated that the metal abundance that
we derive has an intrinsic character, a comparison with the
atmospheric determinations should provide compatible results in
most cases. Only systems with atmospheric anomalies (e.g.
metallic line spectra), are expected to exhibit large differences,
whereas the best agreement should be for stars with deep outer
convection zones, i.e. mid-F and cooler. Unfortunately, very few
systems have spectroscopic determinations of the atmospheric
metal content. The only systems with published spectroscopic
[Fe/H] are UX Men (Andersen, Clausen & Magain 1989), EE Peg
(Linnell et al. 1996), AI Phe (Andersen et al. 1988) and CD Tau
(Ribas et al. 1999a). A metallicity estimation can be also obtained
from the StroÈmgren photometric index d m8 for A3±G2 stars. For
those systems in this spectral type range, we computed the
metallicity following the calibrations of Smalley (1993) and
Nissen (1988). All the available empirical metal abundance
determinations and the comparison with the isochrone fitting
results are listed in Table 5. The extrapolated solution for UX
Men listed in Table 3 has been not included, since it was
considered to be unreliable. Some remarks must certainly be made
concerning the systems with atmospheric metallic anomalies: GZ
CMa, AI Hya, KW Hya and EE Peg. For three of them, the metal
abundance deduced from the atmospheric parameters is considerably larger than the intrinsic metallicity deduced from the models.
This is a clear indication of the presence of physical processes
(diffusion) that lead to an enhancement of the metal abundance in
the external layers of the star (Michaud 1980). The remaining
system, GZ CMa, also with a metallic line spectrum, presents an
atmospheric metallicity in good agreement with the results of the
isochrone fitting. The metallic character that leads to its classification is probably related to a real, intrinsic high metal abundance
rather than to atmospheric anomalies. Regarding those systems with
non-metallic spectra, the agreement between observational
metallicity determinations and the results of our fitting algorithm
is excellent. Indeed, all the differences in Z are within ^0:005:
An interesting possibility of the results is the analysis of the
metal abundance as a function of the mass or the age of the
system. The mean values of the chemical composition of all 27
galactic systems listed in Table 3 are
kZl ˆ 0:016 ^ 0:006
kYl ˆ 0:260 ^ 0:028:
However, it may be illustrative to split these systems into those
composed of O- and B-type stars and those composed of A- and Ftype stars. This means that, for the first group, which includes 11
OB-type systems,
kZl ˆ 0:014 ^ 0:005
kYl ˆ 0:265 ^ 0:029;
and for the remaining 15 systems with A- and F-type components
(AI Phe, with a K-type component, has not been included):
kZl ˆ 0:018 ^ 0:006
kYl ˆ 0:257 ^ 0:029:
108
I. Ribas et al.
Table 5. Observational metallicity determination, and comparison with the best-fitting solution from the
algorithm. Z( ˆ 0:0188 (Schaller et al. 1992).
Name
GZ CMa
YZ Cas
WX Cep
V380 Cyg
AI Hya
HS Hya
KW Hya
TZ Men
UX Men
V1031 Ori
EE Peg
AI Phe
CD Tau
HV 2274 (LMC)
[Fe/H]obs
Zobs
0.22
0.03
20.03
20.03
0.5
20.17
0.5
20.03
0:04 ^ 0:10
20.25
0.5
20:14 ^ 0:10
0:08 ^ 0:15
20:45 ^ 0:06
0.032
0.020
0.017
0.018
0.07
0.013
0.06
0.018
0:021 ^ 0:005
0.011
0.06
0:014 ^ 0:003
0:023 ^ 0:008
0:007 ^ 0:001
Therefore, the mean metallicity of the OB-type systems comes out
to be marginally smaller than that of the systems with AF-type
components. This result supports the indications of a low metal
content for the early-B type system GG Lup found by Andersen,
Clausen & GimeÂnez (1993). This system, which was not
considered in our analysis owing to its unevolved stage, actually
falls below the solar metallicity ZAMS in Fig. 3. Another
interesting study carried out by Kilian (1992), based on atmospheric abundance determinations, also revealed a substantially
sub-solar metallicity (by almost a factor of two) for a sample of
early-type stars in OB associations. A very thorough study on the
chemical evolution of the galactic disc was published by
Edvardsson et al. (1993). The authors collected high-accuracy
spectroscopic and photometric data for a sample of 189 field stars.
Their results show that, for the ages of our DDLEBs, no
correlation between the age and the metallicity is present. Indeed,
the age±metallicity relation is quite flat, even for ages close to
10 Gyr, and with a large scatter in metallicity. Studies based on
open-cluster data also confirm the presence of a large metallicity
scatter and do not support the existence of an age±metallicity
relation (Friel 1995, and references therein). Thus, our result is not
in contradiction with these studies. In addition, the age-span of our
DDLEBs is small (the oldest system, AI Phe, is only 5 Gyr old)
and we are only sampling a restricted spatial area around the Sun,
mostly within 1 kpc.
6 C H E M I C A L E N R I C H M E N T R AT I O A N D
PRIMORD IA L H ELIUM A BUN DAN CE
The initial metal and helium abundances (Z, Y) are independent
output parameters provided by our fitting algorithm. Thus, the
data listed in Table 3 allow us to establish an empirical
determination of the enrichment law and, as a consequence, an
estimation of the primordial helium abundance. Since the
chemical enrichment law is thought to be universal, the results
for the LMC system HV 2274 and for the remaining Galactic
systems can be treated together. The determined helium
abundance of these 28 systems is plotted versus the metallicity
in Fig. 7. Error bars have not been represented for each point, but
an illustration of the typical errors is provided in the bottom-right
part of the figure.
As can be seen, a quite well-defined linear dependence is
present in the figure, in spite of the rather large scatter and the
error bars associated with each point. Even though both axes are
Z obs 2 Z
0.000
0.004
0.003
0.05
0.002
0.05
0.004
20.005
0.03
0.004
20.003
Comments
Photometric, primary Am star
Photometric, based on the B comp.
Photometric
Spectrophotometric
Photometric, primary Fm star
Photometric
Photometric, primary Am star
Photometric, based on B comp.
Spectroscopic
Photometric
Spectroscopic, primary Am star
Spectroscopic
Spectroscopic
Spectrophotometric
Figure 7. The initial helium abundance (Y) versus the initial metallicity (Z)
for a sample of 28 DDLEBs (Table 3). Typical error bars are shown in the
bottom-right part of the panel. The solid line represents a weighted linear
regression, the result of which is in the top part of the panel.
affected by observational errors, we decided to compute a
weighted linear regression of Y as a function of Z, since the
mean relative error of the former is more than three times larger
than that of the latter. The adopted relationship was expressed in
the form
Y ˆ Y p ‡ …DY=DZ†Z;
…4†
where Yp can be assimilated to the value of the primordial helium
and DY/DZ is the enrichment ratio. We tested different weight
systems leading to very similar results, and we finally adopted
relative weights proportional to the inverse of the quadratic sum of
the errors in Y and Z. The resulting values of the least-squares fit
and their uncertainties are
Y p ˆ 0:225 ^ 0:013
DY=DZ ˆ 2:2 ^ 0:8;
with a correlation coefficient r ˆ 0:5 and an rms residual in Y of
0.024. The scatter of the relation is therefore rather large, but in
agreement with what is expected from the mean uncertainty of the
measurements …ksY l ˆ 0:04†: The large dispersion is probably
caused by the random errors of the fundamental parameters,
q 2000 RAS, MNRAS 313, 99±111
Eclipsing binaries: He/metal enrichment ratio
especially the effective temperature. However, the influence of
these errors is already included in the uncertainties quoted for the
fitted abundances. The main concern is not the effect of the
random errors but the presence of a systematic trend. No
systematic errors are expected in the mass and surface gravity
determination of the DDLEB components, since the analysis of
different systems are completely independent and possible sources
of systematic effects are always studied in detail. The most likely
source of systematic errors is the effective temperature determination, since it is a scale-dependent quantity and any zero-point
offset would have a direct influence on the determined initial
helium abundance. The atmospheric parameters of a star (the
effective temperature is included) are affected by its rotation rate
(Figueras & Blasi 1998). As we showed in Section 3, the effective
temperature of most of the systems in the sample was obtained
through photometric indices and calibrations, which can both be
influenced by rotation. This influence depends on two parameters:
the fractional angular velocity (the ratio between the observed and
the critical value) and the inclination of the rotation axis with
respect to the visual, which can be adopted to be close to 908 for
DDLEBs. A rotating star is therefore similar to a non-rotating one
with smaller effective temperature. We collected the linear
rotational velocities for most of the systems in our sample and
interpolated in the tables of Collins & Sonneborn (1977) to obtain
the critical angular velocity and the StroÈmgren indices for both the
rotating and the non-rotating cases. Then we computed a
correction to our photometry as the difference between the
StroÈmgren indices that include rotation and those of an identical
non-rotating star. All the components of the systems in our sample
have fractional angular velocities below 0.3 and most of them
(65 per cent) are below 0.2. Therefore, the rotation correction is
expected to be very small. Actually, we found that the temperature
difference was in all cases around or below 100 K. Moreover,
differences of at most 50 K were found between the components in
the same binary system. Therefore, the influence of stellar rotation
on the temperature difference between the system components can
be neglected, and, thus, the metallicity determination is not
affected, since it is only determined by the relative location of the
components. However, the derived initial helium abundance may
be systematically influenced by the change in the effective
temperature (see Fig. 2). Since rotation has only significant
influence on Y and not on Z, Yp and DY/DZ are not equally
affected. Actually, the slope of the relation remains unchanged
when an offset in the y-axis is considered, and so DY/DZ can be
considered free of systematics caused by the effective temperature. However, the zero point (Yp) is directly influenced by a
systematic trend on the helium abundance determination. As an
illustration of the order of magnitude of the effect, a systematic
increment of the effective temperature of only about 2 per cent
would produce a change of Yp from 0.225 to 0.24. Further
discussion on this point is provided at the end of this section.
6.1
Comparison with previous results
Once the presence of systematic errors in our DY/DZ determination has been ruled out, we can compare the result with the recent
values described in Section 1, either coming from the analysis of
H ii regions, ZAMS structure or Galactic Bulge stars. We find a
good general agreement, with our estimate lying around the
`mean' value deduced from all determinations. Also, the
agreement with the most accurate value of Izotov & Thuan
(1998) is very good. However, it has to be pointed out that the
q 2000 RAS, MNRAS 313, 99±111
109
uncertainty associated with our determination is the smallest one
when compared with all the previous results. The enrichment law
that we have obtained can also be compared with what several
authors adopt when building their stellar evolutionary models for a
range of metallicities. In contrast with the models of CG, some of
the most widely-used models adopt a fixed law Y(Z) to compute
the initial helium content as a function of the metallicity. In
summary, the Geneva group (Schaller et al. 1992; Shaerer et al.
1993a,b; Charbonnel et al. 1993) adopts Y ˆ 0:24 ‡ 3Z; the
Padova group (Bressan et al. 1993; Fagotto et al. 1994a,b) uses
Y ˆ 0:23 ‡ 2:5Z; and the Cambridge group (Pols et al. 1998)
employs Y ˆ 0:24 ‡ 2Z: We see that the slope of the enrichment
law considered by all these current models lies within the error
bars of our empirical determination, the Padova-group and
Cambridge-group adoptions being the closest to our estimate.
Regarding the primordial helium abundance, hundreds of papers
in the literature are devoted to the study of this important
parameter. Actually, most of the studies obtain primordial helium
abundances that fall in the range between 0.23 and 0.24.
Therefore, this is a quite well-established quantity and the main
concern is now the determination of the third decimal place. We
shall only review here some of the most recent results. A thorough
work on the observational determination of the primordial helium
abundance was published by Pagel et al. (1992). The authors
measured the abundance of helium, nitrogen and oxygen for a
sample of 33 extragalactic H ii regions and evaluated the possible
statistical and systematic errors. They obtained a value of Y p ˆ
0:228 ^ 0:005…stat:† ^ 0:005…syst:†; that was subsequently modified by Olive & Steigman (1995) to Y p ˆ 0:232 ^ 0:003 after
excluding some discrepant objects. Peimbert (1996) collected all
the available Yp determinations from 1979 to 1995, and obtained a
weighted average of Y p ˆ 0:234 ^ 0:005; very close to the
previously-mentioned value. An update of the primordial helium
determination was presented by Olive, Steigman & Skillman
(1997). They obtained and analysed new data on low-metallicity
extragalactic H ii regions and derived a primordial helium
abundance of Y p ˆ 0:234 ^ 0:002…stat:†; again in very good
agreement with the determinations of Olive & Steigman and
Peimbert. Olive et al. also obtain an upper bound to Yp of 0.237, at
the 95 per cent confidence level. This value seems to be in
contradiction to the recent result published by Izotov & Thuan
(1998), who obtain Y p ˆ 0:244 ^ 0:002 and claim that a value as
low as Y p ˆ 0:234 can be excluded. Thus, the determination of the
primordial helium abundance is still an open question, although in
the region of 3±4 per cent. We have to remark that all these works
are based on the observation and modelling of spectral features of
H ii regions. Therefore, the results are subject to possible
systematic errors related, e.g., to the values of the helium
emissivity adopted or to the amount of undetected neutral helium.
As we earlier pointed out, our primordial helium abundance
determination is not as reliable as the helium enrichment value,
since it may be affected by systematic errors in the determination
of the effective temperature. Thus, we cannot contribute to the
discussion about the accurate estimation of Yp. However, the
argument can be reversed and we can adopt a quite accurate
primordial helium abundance of Y p ˆ 0:24: Then, when comparing with our estimation Y p ˆ 0:225; we deduce that the possible
systematics present in the effective temperature determination can
be constrained to be around 2 per cent as a mean, although this
figure may change as a function of the photometric region. This is
the reflection of an increment of the helium abundance by 0.015.
In conclusion, large systematics or zero-point offsets in the
110
I. Ribas et al.
effective temperature determination can be ruled out, thus
reinforcing our confidence in the whole analysis and in the
resulting parameters.
7
CONCLUSIONS
The chemical composition of a sample of 50 eclipsing binaries
with accurate fundamental parameters was studied by comparison
of evolutionary-model predictions with observations. This was
done through an algorithm founded on the assumption that
evolutionary models should be able to fit an isochrone to both
components of the system for a certain chemical composition. We
made use of the evolutionary models of CG that constitute a grid
with several (Z, Y) values. Thus, both Z (metallicity) and Y (initial
helium abundance) were treated as free parameters. The fitting
algorithm is based on the interpolation of the evolutionary models
and the minimization of a x 2 function that depends on the
chemical composition and the fundamental parameters of the
components. The best-fitting model parameters are those that
predict the smallest value of x 2. The algorithm yields the bestfitting values for the age and the chemical composition (Z, Y) of
the system (among other parameters) for those cases where a
minimum is found. The errors associated with the chemical
composition determination were evaluated by means of Monte
Carlo simulations. We collected all the available empirical metal
abundance determinations (photometric, spectroscopic and spectrophotometric) for the systems in our sample and we compared them
with the resulting values of the fitting algorithm. All the
differences in Z were found to be within ^0.005, except for
those systems with atmospheric anomalies. When studying the
correlation between the metal abundance and the spectral type, we
found that the mean metallicity of systems with OB-type
components is marginally smaller than that of the systems with
AF-type components. Since both the values of Z and Y are output
parameters of our fitting algorithm, we were able to undertake a
determination of the chemical enrichment law. We obtained
DY=DZ ˆ 2:2 ^ 0:8; in good agreement with the most recent
results, but with a smaller formal error. Moreover, possible
systematic effects were discussed and ruled out. Also, an
estimation of the primordial helium abundance was derived by
extrapolation of the linear enrichment relationship. Our result,
Y p ˆ 0:225 ^ 0:013 is less accurate than that obtained through
other methods and appears to be slightly lower than current
estimations. This may be justified by the presence of small
systematic deviations of the effective temperature determination
(rotation effects or a zero-point offset). It should be emphasized
that this is the first time that the helium-to-metal enrichment
ratio and the primordial helium have been obtained through the
analysis of DDLEBs. Furthermore, our results are based on the
determination of the intrinsic values of Z and Y and not on
abundances deduced from atmosphere modelling and spectral
fitting.
AC K N O W L E D G M E N T S
J. V. Clausen is thanked for his comments and suggestions, which
have improved the quality of this paper. This work was supported
by the Spanish CICYT under Contract ESP97-1803. IR acknowledges the grant of the Beques predoctorals per a la formacio de
personal investigador by the CIRIT (Generalitat de Catalunya)
(ref. FI-PG/95-1111).
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