A&A 413, 1045-1063 (2004)
DOI: 10.1051/0004-6361:20031582
T. Gehren1 - Y. C. Liang1,2 - J. R. Shi1,2 - H. W. Zhang1,2,3 - G. Zhao1,2
1 - Institut für Astronomie und Astrophysik der Universität München,
Scheinerstr. 1, 81679 München, Germany
2 - National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, PR China
3 -
Department of Astronomy, School of Physics, Peking University, Beijing 100871,
PR China
Received 17 June 2003 / Accepted 7 October 2003
Abstract
To determine the population membership of nearby stars we explore
abundance results obtained for the light neutron-rich elements 23Na and 27Al in
a small sample of moderately metal-poor stars. Spectroscopic observations are
limited to the solar neighbourhood so that gravities can be determined from
H IPPARCOS parallaxes, and the results are confronted with those for a separate
sample of more metal-poor typical halo stars. Following earlier investigations,
the abundances of Na, Mg and Al have been derived from NLTE statistical
equilibrium calculations used as input to line profile synthesis. Compared with
LTE the abundances require systematic corrections, with typical values of
+0.05 for [Mg/Fe], -0.1 for [Na/Fe] and +0.2 for [Al/Fe] in thick disk
stars where [Fe/H] .
In more metal-poor halo stars these values reach
+0.1, -0.4, and +0.5, respectively, differences that can no longer be
ignored.
After careful selection of a clean subsample free from suspected or known
binaries and peculiar stars, we find that [Na/Mg] and [Al/Mg], in combination
with [Mg/Fe], space velocities and stellar evolutionary ages, make possible an
individual discrimination between thick disk and halo stars. At present, this
evidence is limited by the small number of stars analyzed. We identify a gap at
[Al/Mg]
and [Fe/H]
that isolates stars of the thick
disk from those in the halo. A similar separation occurs at [Na/Mg]
.
We do not confirm the age gap between thin and thick disk found by Fuhrmann.
Instead we find an age boundary between halo and thick disk stars, however, with
an absolute value of 14 Gyr that must be considered as preliminary.
While the stellar sample is by no means complete, the resulting abundances
indicate the necessity to revise current models of chemical evolution and/or
stellar nucleosynthesis to allow for an adequate production of neutron-rich
species in early stellar generations.
Key words: line: formation - line: profiles - stars: fundamental parameters - stars: abundances - stars: late-type
Na, Mg and Al are in principle easy to observe in unevolved stars of spectral types F and G, because they have spectral lines of different strength and excitation in the visual wavelength region. Yet recent investigations of metal-poor stars have not converged toward a consistent picture (see Tomkin et al. 1985, 1992; Edvardsson et al. 1993; Timmes et al. 1995; McWilliam et al. 1995; Pilachowski et al. 1996; McWilliam 1997; Carretta et al. 2000; Chen et al. 2000; Fulbright 2000; Prochaska et al. 2000). Part of the problems connected with the interpretation of observed data seems to be owed to a large scatter of the resulting abundance ratios, and this can be due to a significant weakness in methodical approach, the assumption of local thermodynamic equilibrium (LTE) in the stellar atmospheres.
Large computers and even larger telescopes have made it relatively easy to obtain data for stars that could not be observed with high-resolution spectrographs only a decade ago. Since then most efforts have been focused on observing faint stars that are extremely metal-poor, mostly in hope to detect stars as near as possible to the oldest stellar population III. The results have opened a new empirical approach to the earliest Galactic nucleosynthesis, but they have not settled the basic questions: what type of stars were the first in the evolution of the Galaxy? What were the initial properties of the first stellar generations, and how did special elements and isotopes enter the chemical evolution of the Milky Way? Fuhrmann (1998, 2002) has repeatedly and almost convincingly argued that the local population of metal-poor dwarfs represents essentially the thick disk, and that halo stars constitute only an extremely tiny fraction of the local stellar mass which may or may not have been accreted from external star clusters or galaxies. Fuhrmann (2000) and Bernkopf et al. (2001) find that if there is any difference at all, the ages of both halo and thick disk stars must be very similar, i.e. beyond 12 Gyr. In fact, the identification and population membership of single stars has always remained somewhat arbitrary, in that kinematics or metal abundance or age alone never gave a conclusive answer. The most problematic identifier is the metal abundance because we learnt that there may exist low-abundance stars that kinematically would fit to thick disk velocities (Norris et al. 1985), and quite generally there may be a significant overlap of the populations in some, if not all, of the three parameters.
The status of the Galactic halo component is fairly uncertain. Its contribution
to Galactic mass and its origin are far from being understood. Before following
any more speculations it therefore seems worthwhile looking for further
empirical signs of population membership. While metal abundance in
general is admittedly a bad measure of the Galactic evolution time, metal
abundance ratios could be used instead, if the origin of certain elements
could be properly assigned to processes of nucleosynthesis and corresponding
stellar masses and time scales. Consequently, it takes a longer evolutionary
time to produce s-process elements than is necessary for r-process
nucleosynthesis because the involved stellar masses are different. Equally, the
production of -elements is followed on a shorter time scale than that of
iron, again due to the stellar masses involved in supernovae of type II and Ia,
respectively. The question arises: will there be a definite abundance ratio
which allows the individual discrimination of a halo star from a thick
disk star? There appears to be empirical evidence for [Eu/Mg] being such an
element ratio that allows a well-defined distinction between [Eu/Mg]
in stars of the disk populations (including the thick disk) and [Eu/Mg]
in halo stars (Mashonkina et al. 2003). This is particularly
interesting because both elements are thought to be produced in massive stars
shortly before or during the SN II explosion. Since europium is hard to observe
in metal-poor stars other elements might be preferred, and the abundances of
23Na, 24Mg and 27Al could provide such a discriminator.
Our investigation starts with a definition of our stellar samples and the reason for a separation between the mildly metal-poor sample and the extremely metal-poor halo stars. In the following section we will also discuss the observations and the extraction of echelle spectra, followed in Sect. 3 by a short presentation of the stellar parameters, temperature and gravity, microturbulence and iron abundance and the methods involved. Section 4 explains the conditions and models for which spectral line synthesis was obtained. NLTE line formation will emerge as the most important factor of the analyses. It will become evident, how much NLTE and LTE abundance results differ. This section will also show the element abundance results including a discussion of the abundance ratios and their comparison with recent analyses of other groups. The kinematic properties of the stars are described together with a very preliminary determination of stellar ages in Sect. 5. Although we are still far from a sound interpretation in terms of Galactic evolution and population membership, the final section adds a number of interesting conclusions.
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Figure 1:
Colour-magnitude diagram of 162 unevolved near-turnoff stars
fulfilling the selection criteria, based on H IPPARCOS parallaxes. Filled circles
refer to observed stars, open circles to the rest of the sample. Star-like
symbols document the metal-poor comparison sample. Symbol sizes increase with
the
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This sample contains but a very few extremely metal-poor stars. Thus we have extended our analysis to include as many metal-poor stars as possible that do not fit the above selection criteria. Spectra for these stars were taken from recent observations carried out mostly for different purposes. The results obtained for the second sample will serve as a control of the properties obtained for the less metal-poor stars of disk and halo components. The stellar samples are displayed in Fig. 1, which gives an account of the type of stars involved. The sample shown in Fig. 1 will not be complete in that it is not volume-limited. It is essentially flux-limited, with colour excess and availability of H IPPARCOS parallaxes providing the most important restrictions.
The FOCES observations of the stars in the northern hemisphere all cover a
spectral range from 3700 to 9800 Å on a total of 97 spectral orders. The
spectra were exposed on a 20482 CCD chip with m pixel size, providing a
spectral resolution of
per 2 pixel resolution element. Basic
observational data are given in the upper part of Table 1. For nearly
all stars the total exposure time was divided into 3 single exposures to allow
redundant data extraction. Unfortunately, due to substantial extinction from sky
haze and only medium seeing properties some of the exposures do not show the
expected signal. Yet most of the combined spectra display a
near
.
UVES observations cover a spectral range between 3300 and 6650 Å with
gaps of
100 Å around 4570 Å due to the beam splitter and near 5580 Å at the edge of the butted CCD. The spectral resolution is around
R =
60 000. These observations were originally intended to show a high signal in the
blue (see Mashonkina et al. 2003). Consequently the green/red spectra
have a S/N near 300 in most of the single exposures. Again, 3 exposures were
taken for each of the stars, for which the data are found in the bottom section
of Table 1.
Data extraction followed the standard automatic IDL program environment designed for the FOCES spectrograph (Pfeiffer et al. 1998), but with slight modifications also applicable to the UVES data. All echelle images including flatfield and ThAr were corrected for bias and scattered light background. Objects and ThAr exposures were extracted and corrected for flatfield response. Finally, bad pixels were detected and as far as possible removed by comparison of the 3 single exposures. This was particularly simple because most of the exposures were obtained in a time sequence. After wavelength calibration each of the spectra underwent a correlation with the solar spectrum to determine the radial velocity, before the single exposures were co-added without wavelength shift, thus ignoring any possible variations of radial velocity. This procedure could have been improved applying independent resampling of the single exposures before co-adding. However, it turned out that all velocity variations were well below a detectable level of 0.3 km s-1, whereas the subsequent spectrum analyses did not give any sign of kinematically broadened absorption lines. Thus we preferred to trade the disadvantage of a small velocity error against the advantage of co-adding single spectra without resampling.
Table 1:
Spectra obtained with the FOCES echelle spectrograph at the Calar Alto
2.2 m telescope in August 2001 (top section), and with the UVES echelle
spectrograph at the ESO VLT telescope in March 2001 (bottom section). Columns
are mostly self-explanatory. Parallaxes (mas), their errors ,
and proper
motions (mas/y) are taken from the H IPPARCOS catalogue (ESA 1997).
N and
refer to the number of spectra taken and the total exposure
time (s).
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Figure 2: Sample spectra of the velocity-corrected SB2 stars including HD 99383. The two components have been marked for the Cr I 201 line at 5254.98 Å. The flux axis is on an arbitrary scale, compressed to show approximately the same dynamic range for each star. The solar flux spectrum of Kurucz et al. (1984) is shown for comparison. |
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HD 133621, HD 134113: Orbital solutions by Latham et al. (2002) were recognized only after the observing runs.
HD 141335 (G224-81): Orbital solution by Goldberg et al. (2002) was
recognized only after the observing runs. Spectrum is double-lined with the
second component offset by 28 km s-1 to the red (see Fig. 2).
Mg I b lines show unusually strong wings with a clear asymmetry to the
red.
HD 142267: Noted by Hünsch et al. (1999) as a weak X-ray emitter. Nidever et al. (2002) list it as a possible radial velocity variable, but with questionable evidence from only 2 observations. Ca II H+K seem to be slightly filled in, but the absorption line spectra are unconspicuous.
BD+68 901: Core emission in the Ca II H+K lines (see Fig.
3). The spectrum looks normal.
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Figure 3:
Relatively
strong emission in the cores of the Ca II H+K lines is found in the
(normalized) spectra of BD+68![]() |
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HD 173084: Known as visual binary star with separation of 0.36 arcsec.
Double-lined spectrum with a velocity offset of
km s-1 (see Fig. 2).
HD 184855 (G92-15), HD 210631 (G18-35), G217-8, HD 10443 (G71-27): All four stars now have orbital solutions published by Latham et al. (2002).
G261-10: Double-lined spectroscopic binary with second component shifted
towards
km s-1 (see Fig. 2). This star is on the list of
Latham et al. (2002), but with no indication of RV variation.
HD 198300 (G230-49): Noted by Smith & Churchill (1998) because of faint Ca II H+K emission, which is not well documented on our spectra.
HD 200580: Only recently identified as a visual binary using speckle interferometry (Mason et al. 2001). The angular distance of the two stellar components is only 0.13 arcsec.
HD 221950: Cited by Clausen et al. (1997) as a photometric primary
star, the spectrum shows a clear double-lined structure with velocity separation
of
km s-1 and components of similar strength. Ca II H+K display
a filled-in core with significant asymmetry. As seen in Fig. 2, the
lines of both components are significantly broader than typical line widths of
other stars observed with the same instrument settings. Explained by rotation
alone, the additional line widths would fit to
km s-1.
HD 224930: Spectroscopic binary (Underhill 1963), secondary component is approximately 3 mag fainter. Ca II H+K lines with core emission (see Fig. 3).
HD 29907: Single-lined spectroscopic binary (Lindgren & Ardeberg 1996).
CD-52 2174: Double-lined spectroscopic binary with second component
shifted by
km s-1 (see Fig. 2).
BD-3 2525: Clearly double-lined spectroscopic binary with
km s-1 separation. Greenstein & Saha (1986) have published an orbit for a
single-lined binary. Latham et al. (1988) revised the binary orbit and
reported the double-lined structure of the spectra (see Fig. 2).
HD 99383: Nissen et al. (2002) list this star as a double-lined spectroscopic binary. As indicated in Fig. 2, there is no second line system found in our spectra.
Of all known binary stars only HD 200580, HD 224930, and HD 29907 have been
retained for spectroscopic analysis, and we will pay special attention to the
abundance results of these stars. The other spectroscopic binaries will be
analyzed separately. The second line system in double-lined binary spectra is
not always detected very easily. Using a correlation analysis with reference to
the solar flux spectrum of Kurucz et al. (1984) the spectral binary
structure becomes more evident. A small part of the SB2 spectra is displayed in
Fig. 2, where the two components have been marked for the
Cr I 201 line near 5255 Å. Note that the spectrum of HD 99383 is
clearly single-lined in our exposures. However, the lines appear slightly
broader than those of CD-522174 or BD-3
2525.
The existence of stellar chromospheres has been confirmed for solar-type members of the thin disk, but also for metal-poor subdwarfs (Smith & Churchill 1998). Our blue spectra seem to show chromospheric H+K line reversals for many of the stars analyzed here, all of them with B-V < 0.75. Although the relatively low signal does not allow to confirm this for individual stars, it is seen more clearly in the mean spectra of both samples.
Stellar atmospheric models have been calculated for the individual stars using
plane-parallel one-dimensional stratifications of temperature and pressure. Our
constraint of convective equilibrium is based on the mixing-length approach with
a mixing-length parameter
(Fuhrmann et al. 1993). Since
the spectrum analysis is carried out differentially with respect to the Sun, the
solar reference data are based on the same type of atmospheric model. Although a
differential analysis does not rule out systematic errors - in particular, when
large differences in basic parameters are encountered - this way of comparing
abundances for a large sample of stars seems to us more adequate than the use of
so-called absolute oscillator strengths.
Plane-parallel atmospheres are very unspecific in modeling kinematic
properties of the emitting gas flows. This is immediately evident from the solar
granulation, and the unsatisfactory situation must be improved with the
introduction of the microturbulence velocity. The concept of micro- and
macroturbulence is far from hydrodynamic reality but surprisingly successful in
describing the total integrated flux blocked by an absorption line, and also the
overall profile shape of the emerging line flux. However, it is limited to
symmetric profiles. The present status of horizontally homogeneous atmospheres
is also limited by the application of local thermodynamic equilibrium (LTE).
This introduces an inconsistency between the atmospheric structure and the
formation of absorption lines, which is not too important in stars near the main
sequence, but it may become more influential in slightly evolved stars of
reduced metal abundance. Except for these considerations our models include the
physics that provide the transport of energy using essentially 5 free parameters
to fit the observed line spectra. These are: effective temperature
,
surface gravity
,
microturbulence velocity
,
metal abundance
[Fe/H]
, and
-element abundance ratio [
/Fe],
where the last 3 parameters enter the opacity distribution functions (ODF),
which determine the optical depths. We use the ODF data provided by Kurucz
(1992), but rescaled by
to account for a
meteoritic iron abundance
(Anders & Grevesse
1989). Transfer and constraint equations are fully linearized with
respect to temperature and pressure. Our use of ODFs is restricted to the
calculation of the model atmosphere. The NLTE formation of absorption lines
instead uses a full opacity sampling (see Sect. 4).
Table 2: Metal lines used to determine element abundances. f values and damping constants have been determined from solar spectrum fits.
The most important problem is therefore the determination of the basic stellar parameters. Both
Two other stellar parameters, the "metal'' abundance [Fe/H] and the
microturbulence
,
are evaluated in close interaction. For this purpose a
list of selected Fe II lines is used to synthesize line profiles for
given
and
.
These lines are found, together with the line data
determined from the solar spectrum, at the end of Table 2. We mention
here that Fe II lines are synthesized under the assumption of LTE. The
choice of the line list follows the requirement of minimum blends and the need
to detect them (or at least a subset thereof) in extremely metal-poor stars.
These lines have been used in a closed loop iteration to determine [Fe/H] with
minimum scatter for a given
.
The final internal accuracy of the metal
abundance is around
([Fe/H])
and
km s-1.
Stellar atmospheres on the subgiant branch or near the bottom of the giant
branch depend on the specification of another abundance parameter, [/Fe], which
determines a significant fraction of the free electron pool at cooler surface
temperatures. [
/Fe] is dominated by all
elements with a low enough
ionization energy (which excludes oxygen and neon). This requires the knowledge
of abundance ratios, of which [Mg/Fe] is determined by NLTE analyses, whereas
[Si/Fe] or [Ca/Fe] were estimated assuming values lower than for [Mg/Fe].
Fortunately, the determination of [
/Fe] is not crucial for our analysis since
none of the stars in Table 3 is in the critical range of
temperatures and gravities. Therefore, all our atmospheric models are calculated
assuming [
/Fe] = [Mg/Fe]. The final atmospheric model parameters are iterated
whenever [Mg/Fe] or
deviate from the initial determination during the
subsequent abundance analysis.
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Figure 4:
Typical
example of collision-dominated Na I population departure coefficients
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Deviations from local thermodynamic equilibrium (LTE) in stars are generally expected in gaseous stratifications in which the kinetic equilibrium is determined by non-local interaction processes such as radiation transfer. Since collisions always act as sources of thermalization, the dominance of radiative processes is to be measured against collisional interactions. Typically, in hot stars the radiation field outperforms electronic collisions by an order of magnitude, whereas in cool stellar atmospheres the radiation field is much weaker, and collisions play a more important role. Therefore it is by no means evident why stars with atmospheres as cool as the Sun should be affected by NLTE conditions. However, as explored by Baumüller et al. (1997, 1998) and Zhao et al. (1998, 2000), two reasons for deviations from LTE are found in the atmospheres of metal-poor FG-type stars even near the main sequence. The collision rates are reduced due to a decrease of the electron density with metal abundance, because a large fraction of the electron donors at temperatures below 7000 K consists of metals instead of hydrogen. The radiation field is absorbed by a smaller fraction of metal atoms and ions (predominantly in the near UV), and as a result the photoionization rates tend to increase with falling metal abundance. Both effects are particularly important in such stars for ions far from their ionization maximum, such as most of the neutral metals with ionization energies below 8 eV. These elements are more than 90% ionized, and the neutral atoms become a minority that is sensitively reacting on particle interactions. On the other hand, the kinetic equilibrium of majority ions as Ca II or Ba II is often balanced by line transitions (see Mashonkina et al. 1999).
While collisions between atoms and electrons or heavier particles seem to be very similar for the different atomic species, photoionization rates not only depend on the radiation field (and the line blocking), but at least as much on the atomic properties. Whereas atoms such as Na I have very small photoabsorption cross-sections and hence photoionization rates, the situation is completely different for atoms of Mg I, Al I, and even Fe I. Their atomic structure provides them with particularly strong photoabsorption cross-sections, and their photoionization rates can be very large.
Table 3: Final stellar parameters and abundances of the program stars. For the description of stellar masses and ages see Sect. 5.
Quite generally, in cool metal-poor stellar atmospheres, atoms of the first kind cascade recombining electrons down to their ground state because there is no efficient ionization. Simultaneously, the reduced electronic collisions cannot maintain a thermal excitation equilibrium. Displayed in Fig. 4, the level populations of this type of atom are above those of thermal equilibrium in the region of line formation associated with that level. Consequently, the increased opacities in the line cores and inner wings produce a line optical depth scale that is significantly shifted to the outer layers, and the line source functions are larger than the Planck function. Both effects work in opposite directions, but the result is dominated by the increased optical depth scale, and the Na I D lines are intrinsically deeper in NLTE calculations than in LTE. Therefore a reduced NLTE element abundance is necessary to fit an observed line profile.
Although sodium is not the only example for such a downward cascade in a neutral
metal, the opposite case of enhanced photoabsorption cross-sections seems to be
more common. Examples for photoionization-dominated neutral metals have been
given above. At present, these atomic species present a considerable challenge
to the theory of atomic interactions and their kinetic equilibrium. The reason
for that is evident in the solar spectrum, where the strongest absorption edges
in the near UV are not produced by the hydrogen atom despite of its extremely
high abundance. It is the Al I ground state with its low ionization
energy of only 5.98 eV that produces the deepest cut in the solar flux at 2070 Å, followed by the Mg I P
ionization edge at 2510 Å.
While the other low-excitation levels of Mg I,
S and
P
have also strong photoabsorption cross-sections, all other bf opacity
sources in the near UV are less important. Fe I is, however, a notable
exception. According to Bautista's (1997) calculations its large number of
low-excitation levels each has a photoabsorption cross-section significantly in
excess of what is found in more simple atomic configurations. Here, the high
number of large photoionization rates up to levels of 4 or 5 eV excitation
energy produces an important sink to the population of neutral iron. A typical
example of photoionization-dominated departure coefficients is given in Fig. 5.
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Figure 5: Typical example of photoionization-dominated Mg I departure coefficients in the moderately metal-poor disk star HD 157089 (top) and the Al I departure coefficients in G69-8 (bottom). In spite of hydrogen collisions all neutral energy levels are strongly underpopulated. |
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Our present investigation is based on the atomic properties documented in a
number of earlier reports (Baumüller et al. 1996, 1997,
1998; Zhao et al. 1998, 2000), where most of the problems
have been described in more detail. It deviates from these results in that our
atomic models have been redesigned. The number of levels was extended to include
all known terms plus some hydrogenic terms near the ionization edge. All
bf radiative cross-sections have been included from close-coupling
calculations of Butler (1993) and Butler et al. (1993). At the same
time the opacity distribution function background has been replaced by an
opacity sampling method. The reason for this change lies in the extremely strong
UV blocking factors introduced by the ODF method, which do not fit to the
observed solar spectrum. The change from ODF to OS background opacities was also
performed to remove the corresponding background opacity dependence on the NLTE
elements themselves. Therefore, compared with earlier published data, the new
models produced slightly different population departure coefficients, which in
turn required a new empirical determination of the hydrogen collision factors
.
The full analysis of all observed spectra including the solar spectrum of Kurucz
et al. (1984) allows the choice of a reasonable compromise which,
iterated together with all basic stellar parameters, led to the values of
for Mg I and Na I, whereas Al I required a smaller
value of
.
These numbers differ from those used before because the
photoionization rates were considerably enhanced with reduced OS background
opacities. There is also no longer an advantage in assuming an
excitation-dependent collision factor as in Baumüller et al. (1996) or
Zhao et al. (1998). Stronger thermalization would not fit to a common
mean abundance scale including all lines analyzed here (see below). In
particular, LTE abundances always produce significant differences between
resonance or low-excitation and subordinate lines of all three elements.
All calculations have been carried out with a revised version of the DETAIL program (Butler & Giddings 1985) using accelerated lambda iteration following the extremely efficient method described by Rybicki & Hummer (1991, 1992). As described in Sect. 3, the analyses for all stars were iterated until basic stellar parameters and single line NLTE abundances were consistent. In spite of substantial changes in the atomic models, the basic properties of the Na, Mg and Al abundances are still the same as in our previous analyses.
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Figure 6: Abundance ratios [X/Fe] for both LTE (left) and NLTE (right) analyses. Filled circles refer to our moderately metal-poor sample, star-like symbols to the extremely metal-poor comparison stars. Some apparently peculiar stars are marked (see text). Differences between LTE and NLTE abundance ratios are displayed in Fig. 12. |
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Our analysis rests on between 6 and 8 lines for each of the elements considered. Since our results are based on a differential analysis with respect to the Sun, Table 2 lists the relevant line data with their final solar fit values. The Fe II lines have been synthesized under LTE conditions only, because they are not affected by NLTE departures. For Na I, Mg I and Al I, profile synthesis has been performed for both LTE and NLTE, however, the data in Table 2 refer only to the NLTE analysis. Our fit to the solar line spectrum requires some priority as far as the line data are concerned. Thus the primary choice of input data are the oscillator strengths, while the damping constants have been adjusted to fit the line wings. The resulting values of the van der Waals damping constants are mostly near to those calculated according to Anstee & O'Mara's (1991, 1995) tables, though they are systematically lower by 0.1 to 0.3 dex. This is similar to the damping constant fits for neutral iron lines (Gehren et al. 2001). We note here that, accepting Anstee & O'Mara's damping constants without change, some of the oscillator strengths would have to be modified substantially in order to produce a solar line profile fit.
Doppler broadening of the line profiles follows the usual assumption of random
atmospheric motions represented by microturbulent velocities. Clearly, this
representation by a single parameter,
,
is at its limit when
high-resolution spectra are analyzed. There can be no doubt that lines of the
Fe II multiplet 42 are formed under conditions quite different from those
of the other multiplets. Similar problems are found among the lines of the other
elements. As a result, we have fitted the observed metal line profiles with a
maximum of accuracy, however, being aware that there remains an uncertainty in
the profile shapes that is equivalent to some 0.05 dex in abundance results. In
fact, this may be the limiting accuracy obtainable in a plane-parallel
atmospheric model. We note that this uncertainty is significantly smaller than
most of the NLTE effects found in the metal-poor stars.
Spectrum synthesis is performed interactively using the IDL/Fortran-based SIU software package of Reetz (1993). It was designed to use large databases including relevant data for ion and molecular lines. Once that model atmosphere and NLTE calculations are available, all spectrum synthesis calculations, whether LTE or NLTE, are performed in real time and displayed on the screen for further evaluation. All parameters can be adjusted, which is particularly important for the external broadening of line profiles encountered by macroturbulence, rotation, or the spectrograph entrance slit. Interactive spectrum synthesis with SIU allows for a full consideration of profile blends, hyperfine structure and isotopic splitting, where abundances for each element can be adjusted individually. Consequently, all abundance results presented here are derived from profile fits using level populations, either calculated from Saha/Boltzmann distributions or from NLTE. For most of the lines in metal-poor stars, profile fits can be established with a simple Gaussian external broadening function.
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Figure 7: Abundance ratios [X/Y] for NLTE analyses as a function of [Mg/H] (left) and of [Mg/Fe] (right). Symbols are the same as in Fig. 6. |
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No dependence of abundances on excitation energy is observed for the individual
stars, because the final NLTE collision factors
have been optimized to
avoid this. Although this is no guarantee for consistency of the abundance
analysis, it is significantly better than if LTE were assumed. The basic results
for LTE and NLTE are displayed in Fig. 6 demonstrating the enormous
differences between LTE and NLTE line profile fit abundances.
Clearly, these differences are minor when [Mg/Fe] is concerned, with a limiting
adjustment of
dex for the extremely metal-deficient stars. Thus the
overall run of [Mg/Fe] abundance ratios is changed only gradually. This is
completely different for [Na/Fe] and [Al/Fe]. NLTE corrections for Na I
increase from
at moderate metal deficiencies
in the old thin disk to -0.4 in the extremely metal-poor stars. This confirms
the results of Baumüller et al. (1998) obtained with slightly different
atomic data and spectroscopic observations. By nature of the atomic interaction
processes, the NLTE corrections for Na are largest for the resonance line
doublet, which is the only spectral feature identified in the extremely
metal-poor subdwarfs. The NLTE corrections for Al I are even stronger
ranging from
for near-solar metal abundances to +0.5 in the extremely metal-poor subset which, formulated in the traditional
chemical evolution picture, would transform Al from a straight "secondary''
element to a near "primary'' one.
Figure 6 may be used to identify stars that are not following the
general abundance trends. As can be recognized from the comments in
Sect. 2, such stars are no longer a minority in the whole stellar sample.
The removal from spectral abundance analysis of roughly a third of the original
star list due to binarity highlights the problems that are encountered. The
remaining star list (Table 3) still includes the visual or
single-lined binaries HD 29907, HD 200580, and HD 224930. It also includes
metal-poor stars that do not give immediate evidence for a binary system but
show X-ray emission (HD 142267) or emission cores in the Ca II H+K
lines (BD+68 901 and, again, HD 224930). Moreover, it includes a
peculiar halo star, HD 74000, well known for its extreme nitrogen overabundance
(Carbon et al. 1987) and its peculiar [Ba/Fe] and [Eu/Fe] ratios
(Mashonkina et al. 2003), which together indicate that this star is
probably not representative of a standard evolutionary scenario. Some of these
stars, in particular HD 29907, HD 74000 and HD 200580, are immediately
identified in Fig. 6. Each of them carries an extreme [Na/Fe] and/or
[Al/Fe] ratio with no peculiarity found in [Mg/Fe]. G170-56 is a different case.
Radial velocity variations of this star are not confirmed in Latham et al.'s
(2002) list of observations (their Table 2), where any possible variation
(<1 km s-1 over a baseline of 15 yr) is hidden in the measurement error. With
a quite unusual solar Mg/Fe ratio, both Na and Al are extremely underabundant.
Most of the analyses of other groups found in the literature are based on the assumption that LTE is a sufficient approximation of the thermodynamic state of the stellar atmospheres.
Cowan et al. (2002) reanalyzed a typical halo giant, BD+17 3248,
with an overall metal deficiency of
.
They note a discrepancy
when fitting the Na I D lines and the excited doublets at 5685 and 8190
Å which is typical for LTE results, as is their strong underabundance of Al
derived from the resonance lines. The latter is found already in the work of
Ryan et al. (1996), who admit that their values may be affected by
departures from LTE.
Fulbright (2000, 2002), in his PhD thesis, analyzed a large number
of metal-poor stars, of which 15 stars are in common with our present sample.
His temperature scale differs from ours by
K, with
a systematic trend running from -65 K for stars with near-solar metal abundance
to -410 K for the extremely metal-poor star BD-4
3208. Part of this
large difference may be due to his use of overshooting model atmospheres which
tend to produce an enhanced temperature stratification. The adjustment of
effective temperatures until Fe I lines of different excitation energies
fit to a common abundance value, is unsuitable to determine
.
It is
similar to the assumption of an LTE Fe II/Fe I ionization
equilibrium to determine
.
Therefore, his abundances of neutral ions,
such as Na I and Al I should come out at values that are lower
than ours. However, various approximations or assumptions interfere, and his
abundance trends for 23Na and 27Al are buried in a scatter that our results do
not reproduce. His sample may contain much more binaries or peculiar stars than
ours, and his line data - collected from different sources - are responsible
for at least some of the scatter.
Chen et al. (2000) published abundance data for a large sample of old
(thin) disk stars with an admixture of thick disk members. Since their metal
abundances are mostly well above [Fe/H] = -1, their [Na/Fe] ratios are only
marginally affected by NLTE effects. They were aware of that [Al/Fe] is slightly
more affected by deviations from LTE, even at thin disk metal abundances. In
their Al/Fe ratios, calculated as all the other abundances under the assumption
of LTE, the NLTE abundance corrections of
are missing.
Thick disk stellar abundances have been investigated by Prochaska et al.
(2000). They present results for a preliminary subsample of 10 stars with
[Fe/H] between -0.43 and -1.09. Their admittedly small sample seems to present
a similar gap in [Fe/H] between -0.7 and -1.0, slightly broader than seen in
our data in Fig. 6. Their Na/Mg and Al/Mg ratios are derived under
the assumption of LTE using only the subordinate lines of these elements (which
are also part of our line list). The stars in their list all show a high [Al/Fe]
ratio of 0.4, which is very similar to our data for thick disk stars
derived under NLTE conditions. The [Na/Fe] data are around 0.1, also similar to
our thick disk results under NLTE. The striking similarity can be understood,
since both Na and Al results for excited lines differ only by a small amount
(-0.1 or +0.1) between LTE and NLTE (see Fig. 12). The analysis is
thus approximately on the same scale as ours, but includes only the thick disk.
Following Baumüller et al. (1998), Carretta et al. (2000; see also Gratton et al. 1999) seem to be the only group that applied NLTE calculations to sodium. Their results are very much in agreement with ours in that their NLTE corrections lead to a small metallicity-dependent increase of the [Na/Fe] ratios. Their scatter for medium-to-extreme metal-deficiency is, however, roughly two times the one we find. In view of the small sample we have analyzed here, it is not obvious whether that difference is significant.
Table 4:
Kinematic data of the program stars. Velocities are in km s-1. Radial velocities are own measurements, except for binaries. LSR
velocities are calculated for a solar motion of
=
(10.0, 5.2, 7.2) km s-1. Preliminary population membership assignments in the
rightmost column are for thin disk (D), thick disk (T) and halo (H). See last
section for an explanation.
Following the work of Roman (1950), the original classification of stellar populations was based much more on kinematics than on anything else. It is therefore of high priority to check if kinematic properties and abundance ratios have more in common than a coarse change of mean metal abundance itself. Although it was a general belief that metal-poor stars belong to the halo, and metal-rich stars to the disk, represented by populations II and I, respectively, the detection of the thick disk by Gilmore & Reid (1983) changed the simple picture of a spherical and a highly flattened stellar subsystem completely. Fuhrmann (2002) argues that in fact halo stars may constitute an extremely small minority of all Galactic stars, where most of the stars with intermediate kinematics instead belong to the thick disk.
The determination of Galactic kinematics, which are represented in the (U,V,W)
system with respect to the Local Standard of Rest (LSR), is therefore an
important aspect. In reference to the LSR we have adopted a solar motion of
km s-1 (see Binney & Merrifield
1998). For all stars except the known binaries we have entered our own
radial velocity measurements, combined with H IPPARCOS parallaxes and proper motions.
Table 4 provides the resulting space velocities including the total
velocities calculated with reference to the LSR. The last column comments the
population membership in a preliminary way which is based on a combination of
kinematic properties, stellar ages and metal abundance ratios (predominantly
[Al/Mg]) described in the discussion. For a number of stars the results are
highly uncertain, and we will come back to these below.
The first inspection of the table shows that quite a number of stars in the
sample seem to have kinematic properties that are far from the expected standard
of any population. This is seldom a problem with a discrimination between thin
and thick disk, because in particular the thin disk W velocities tend to be
significantly smaller than those of the other populations. However, the most
important discrimination between thick disk and halo cannot exclusively be based
on the W velocity component. There are stars like HD 29907 with extreme
high-velocity kinematics and very low metal abundance. However, the extreme
velocity consists mostly of a radial component, the W velocity could even fit a
thin disk star, and the metal deficiency is not extended to Al. Other stars such
as HD 148816 or G170-56 even have retrograde Galactic orbits, but their
metal abundance is near [Fe/H] = -0.8, their [Al/Mg] and [Na/Mg] ratios are far
from typical for halo stars, and the [Mg/Fe] ratio of G170-56 is that of a
typical solar-type thin disk star. HD 97320, in spite of its relatively low
metal abundance of [Fe/H] = -1.24 and its [Mg/Fe] ratio of 0.41, kinematically
fits more to thin disk properties. Thus, the remaining typical members of the
halo population, originally included in our high-proper motion sample, are
G188-22 and G242-4 (for which we unfortunately failed to determine the Al
abundance). Most kinematic stellar properties are only loosely correlated with
population membership. This holds at least for the U and W components, and for
the
velocity. All these space velocities do not discriminate
between thick disk and halo. The correlation between low V velocities and low
[Al/Mg] ratios, however, is slightly more instructive.
The kinematic status of our sample is shown in the Toomre diagram presented in Fig. 8. Whereas we already learn from Fuhrmann's (1998) work that thin and thick disk are not easily identified kinematically, this seems to hold even more for the kinematic differences between thick disk and halo, in particular, since some stars may not belong to any of these populations. Stars like G170-56 or HD 29907 seem to have followed some peculiar history and may have been accreted from dwarf galaxies or star clusters. Yet, even comparing kinematics and abundance ratios alone, Fig. 9 suggests that the difference between thick disk and halo is relatively well described by the [Al/Mg] ratio and, less accurately, by the [Na/Mg] ratio. In both diagrams, the thin disk is separated in the upper right edge. The difference between thick disk and halo could be marked by a horizontal line corresponding to roughly [Al/Mg] = -0.15, and [Na/Mg] = -0.4. Together with special kinematical properties such as the V velocity component, the subdivision of the latter two components is already quite convincing.
![]() |
Figure 8: Toomre diagram of space velocities. Symbols are the same as in Fig. 6. As compared with our results, the diagram does not seem to offer a very good discrimination between populations. |
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![]() |
Figure 9: Correlation between abundance ratios and orbital V velocity component. Symbols are the same as in Fig. 6. |
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Another step towards the identification of single star population membership
could be the age. Using the well-defined spectroscopic stellar effective
temperatures together with the absolute magnitudes based on H IPPARCOS parallaxes
(see Table 3) masses and approximate ages can be interpolated
according to [Fe/H] and [/Fe] from adequate tracks of stellar evolution.
Such calculations have recently become available through the work of VandenBerg
et al. (2000), who updated and extended earlier work including new
physics for the equation of state and opacities. There is no account for helium
or metal diffusion at this time, but our most interesting objects may still be
represented quite well differentially.
Using
as coordinate of the HR diagram, there remains no need to
introduce any empirical calibration to a colour index like B-V. Errors for most
of the absolute magnitudes are small (exceptions are easily identified in Table 1). A typical result in Fig. 10 reproduces the interpolated
stellar evolutionary track of HD 148816. The interpolation procedure is by no
means simple, because changes between effective temperatures and ages for tracks
of same composition but different masses are highly non-linear. Bergbusch &
VandenBerg (1992) and VandenBerg et al. (2000) have proposed a
suitable method that uses special morphological points, between which
neighbouring evolutionary tracks may be safely interpolated. A similar approach
was used here, so the interpolated final tracks always used
tracks representing pairs of
,
[Fe/H], and [
/Fe],
respectively.
We note that the knowledge of [/Fe] is as important as that of the other
stellar parameters. Table 5 gives an account of the error
propagation of the observed stellar parameters from which the unreliability of
absolute stellar ages is immediately evident. Therefore we will only
discuss comparative ages within the sample, where we have to remember that some
of our parameters may in fact lead to uncertainties of the range indicated in
the table. Compared with this the errors due to the interpolation method may be
negligible.
Table 5: Error propagation of basic stellar parameters when interpolating tracks of stellar evolution for HD 148816. Variations are given with respect to the final parameters in Table 3.
Masses and ages determined in this way are given in the last two columns of Table 3. Except for errors in the analysis of our stellar spectra we have to consider two important influences we cannot at present assess in an adequate way. The first one concerns the true composition of
![]() |
Figure 10:
Typical path of stellar evolution for HD 148816 (cross), linearly
interpolated for mass, metal abundance and ![]() |
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Lower effective temperatures would lead to an average increase of the stellar ages between 1 and 3 Gyr (see Table 5), which is at variance with the assumption that our ages are already too high.
This could lead to a statistical decrease of the stellar ages of
Gyr.
In Fig. 11 we finally show the distribution of abundance ratios with
stellar ages. Comparison with Figs. 7 and 9 confirms that
ages, kinematic parameters and [Mg/Fe] are equivalent variables against which
the stellar abundance ratios can be evaluated. This is not trivial, because
these three variables are not very well correlated. Therefore, only [Mg/Fe]
brings out the different systematic trend of [Na/Mg] and [Al/Mg] in Fig. 7.
![]() |
Figure 11: Correlation between abundance ratios and stellar ages. Symbols are the same as in Fig. 6. |
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The small number of stars analyzed here already allows a surprising variety of results to be noted. This depends on an a posteriori reselection of the "typical'' sample stars. Whereas our criteria for this process can be formulated clearly, we admit that such an iterative selection method may not be fully accepted. Our justification is the existence of a surprisingly large fraction of stars with peculiarities such as
As a result our original sample of 38+14 stars, already reduced to 27+11 stars in Sect. 2, will be reduced once more to exclude from the discussion of a clean sample the binaries HD 200580, HD 224930, HD 29907, and the peculiar star HD 74000. Figures 6 to 9, and 11 should therefore be corrected correspondingly.
The abundance analyses of all elements included here have once again clearly
indicated the importance of NLTE when analyzing the stellar spectra. This does
not only hold for the abundance analyses themselves, which often are performed
only as equivalent width analyses, irrespective of whether the synthesized
profiles have anything in common with the observed spectra. Taking care of
accurate profile fits, our abundance results are summarized in
Fig. 12, which outlines the increase of the differences between LTE and
NLTE analyses with decreasing metal abundance, as predicted. We emphasize here
again, that our NLTE calculations are conservative in that we have used the
typical free parameter, the collision factor
,
to force the abundances of
all lines analyzed to a common value (minimizing abundance scatter). Moreover,
the full system of equations and parameters simultaneously fits the solar
spectrum.
It is also important to realize that the "scatter'' seen in Fig. 12
is not an artefact of an uncertain abundance determination. Carefully
identifying the outliers such as G170-56 with an Al abundance difference of 0.6,
it becomes evident that the difference between LTE and NLTE abundances is
not only a matter of stellar parameters such as
,
,
and
perhaps [Fe/H], but it also depends on the Al abundance itself. Therefore any
table giving such correction values would have to be multi-dimensional.
![]() |
Figure 12: Difference of element abundance ratios calculated under NLTE and LTE assumptions and displayed in Fig. 6 (see text). |
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One of the more ambitious goals of this investigation is to identify single stars as members of a population. It is not clear at present, whether the very concept of populations tells us much about Galactic evolution or perhaps only summarizes a few mean properties acquired during various proto-Galactic or later merging processes. It has been noted repeatedly that kinematic data alone do not allow the unambiguous identification of a stellar population. In particular the overlap between halo and thick disk stars has not yet been fully evaluated, and Fig. 9 does not suggests a clear limit of V velocities either. The original classification of populations by their space velocities is therefore almost exclusively of statistical value. The work of Edvardsson et al. (1993), mainly confined to what we still address as the thin disk, has shown that the overall metal abundance represented by [Fe/H] is an insufficient indicator of most criteria of Galactic evolution. This is probably due to inhomogeneous star formation, which may be an order of magnitude more important for the short time intervals encountered during the evolution of thick disk or halo. Figure 6 therefore does not reveal too much of a population structure except that it helps to identify some outliers, here in particular G170-56.
Figure 7 instead allows to note some more important abundance trends. One is the gradual decline of [Na/Mg] with [Mg/Fe], irrespective of the stars being thin or thick disk members. It is not seen as clearly in Figs. 9 or 11. Thus it is more a global correlation that represents nucleosynthesis under quite different conditions. It has to be confronted with the striking difference in [Na/Mg] when comparing (thick) disk with halo stars at the same [Mg/Fe] ratio. There is no such trend with [Mg/Fe] among the [Al/Mg] abundance ratios of the (thick) disk stars, all of which seem to have within a very small scatter the same [Al/Mg] ratio. Then, if 23Na and 27Al are secondary elements that require the pre-existence of metals, why are they not built up in the same way under disk evolution conditions which may provide the continuous enrichment of Fe/Mg by SNe Ia and thus increase the overall metal abundance? The [Al/Na] ratio emphasizes this problem, with even halo stars showing a solar abundance ratio, but the disk stars (with many of them belonging to the thick disk) cluster around [Al/Mg] = 0.3, although with considerable scatter.
The comparison of [Na/Mg] and [Al/Mg] in Fig. 7 could tell us that
hydrostatic carbon and neon burning in (thick) disk stars result from two
different sites of nucleosynthesis or at least different masses of the SN II
progenitors. A similar scenario has been investigated by Tsujimoto et al.
(2002) who have tried to account for the strong scatter of [Na/Mg] and
[Al/Mg] ratios they found in the literature. Though that scatter is largely an
artefact from the LTE assumption underlying all those analyses or from the
various methods of determining stellar parameters, the overall trends seem to be
well reproduced assuming the local propagation of synthesized material through
adjacent SN shells, in which new stellar generations are initiated. While the
yields of both Na and Al in type II supernovae are under debate (Woosley &
Weaver 1995; Umeda et al. 2000), it seems that such a model
environment for chemical evolution is able to explain the different abundance
ratios of Na/Mg and Al/Mg for [Mg/H] > -2. Both [Na/Mg] and [Al/Mg] in the
(generally more metal-poor) halo stars represent a significant even-odd effect,
the amount of which is somewhere between the results of zero-metal supernovae of
type II (Woosley & Weaver 1995) and the production factors of very
massive population III stars (Heger & Woosley 2002). According to their
investigation population III stars between 40 and 150
end in a black
hole and do not contribute to chemical evolution. Whereas the synthesis of Na in
the disk populations can be explained by gradual enrichment including SN Ia,
the constant solar [Al/Mg] in both thin and thick disk remains a puzzle. Timmes
et al. (1995) predict an [Al/Mg] ratio gradually increasing from -0.4 in
the metal-rich halo to -0.3 in the early thick disk with a further increase to
+0.2 at solar metallicity. Our high level of Al abundance ratios in the thick
and thin disk, seen also in Edvardsson et al. (1993), is at variance with
that prediction. It may require some fine-tuning of preferably discontinuous
and/or time-dependent initial mass functions to produce our observed Al/Mg
ratio.
![]() |
Figure 13: Gap in metal abundance [Fe/H] and abundance ratio [Al/Mg], where no star is found. Such a plot may help to identify population membership. |
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There are, however, three stars out of our less metal-poor sample that have [Al/Mg] ratios near the upper edge of the typical halo sample (G188-22, HD 160693) or even somewhat above (G170-56). Whereas G188-22 seems to have halo properties when kinematics and age are inspected, HD 160693 is a different case. Its [Mg/Fe] ratio documents that this star must have been formed from matter recycled only in SN II, but its age is roughly 4 to 5 Gyrs less than that of typical halo stars separating it from the halo in terms of Galactic evolution. HD 160693 could therefore be a former member of an accreted dwarf galaxy or globular cluster. G170-56 on the other side has a typical thin disk [Mg/Fe] ratio which requires that its formation must have been based on matter recycled in SN Ia. We note that stars like G170-56 are not found in the thick disk either. Therefore we should rule out its membership in both thick disk and halo. Its exceedingly low Na and Al abundances are certainly surprising in view of the retrograde Galactic orbit. Quite probably, also G242-4 would have to be attributed to the halo, although its Al abundance could not be analyzed here.
A number of additional comments refer to population membership. Most of them
have to do with kinematical properties of the stars. A star like HD 97320 would
never be classified as a halo star from kinematics alone; both abundance ratios
and age clearly show that it must belong at least to a pre-disk origin.
HD 148816 is the kinematical counterpart of HD 97320. With a retrograde orbit
this star seems to have more in common with G170-56 than any other star in the
sample, although its [Al/Mg] and [Na/Mg] are atypical for the high [Mg/Fe]
ratio. At the same time HD 148816 has only a moderate metal deficiency of -0.78 which, for a halo star, would also be at an extreme end. In summary it
seems that roughly two thirds of the remaining stars on our two lists can be
attributed to the thick disk or halo populations, whereas some of the other
stars cannot be classified in such a simple way. The important question whether
our Galaxy can be composed of three homogeneous populations will have to await
more results from future spectral analyses. At present the identification of
population membership is based essentially in Fig. 13, where halo
stars and disk stars seem well separated at
,
.
A similar boundary is found at [Na/Mg]
.
The discrimination
between thin and thick disk stars follows Fuhrmann (1998) using [Mg/Fe] =
0.25 as a boundary. This classification scheme is reproduced in the last column
of Table 4.
Stellar ages displayed in Fig. 11, are not only very high and therefore unrealistic as absolute data but also do not exhibit a clear separation between thick and thin disk as was suggested by Fuhrmann (2000). Fuhrmann notes an age gap between roughly 9 and 12 Gyrs, which in fact refers to the age interval that is most densely populated in our sample. This could in part be due to the use of different evolutionary tracks, but to first order one would expect a shift of ages. Figure 11 shows that the region of the proposed age gap between thin and thick disk is populated by a mixture of stars identified both as thin or thick disk. The age overlap is near 2 Gyrs, and it is hard to think of any simple combination of analysis errors that could produce it. Unfortunately, Fuhrmann's and our work have only one star in common, HD 204155, which is outside the critical age range. Even accepting a misidentification for many stars in that range, there would be no age gap.
The question arises therefore, whether the age itself is a good population identifier. It may be that the stars filling in the gap that Fuhrmann claims to have detected are (a) no members of our standard populations, and/or (b) did not appear in the small local volume of Fuhrmann's sample. In the first case, quite a large fraction of our stars must come from separate evolutionary processes such as merging or tidal disruption of galaxies or clusters; stars like that could also belong to the intermediate population with [Mg/Fe] around 0.2, which stands out from the trends in thin and thick disk. In the second case, Fuhrmann's small local volume, although statistically complete, might not be representative for more extended parts of the Milky Way. Of course, it would be much more likely that our sample is not representative.
We note, however, that the age limit between halo and thick disk population seems to be much better defined in Fig. 11. In our analysis it lies near 14 Gyr, but that value could be reduced when evolutionary tracks with diffusion will become available.
The results presented in this paper are therefore not yet robust enough to warrant further speculations. Work extending the data set is under way.
Acknowledgements
This project was supported by the Deutsche Forschungsgemeinschaft (DFG) under grants Ge 490/26-1, 490/27-1, 490/28-1 and 490/30-1, and by the National Natural Science Foundation of China (NSFC) under grants 10173014 and NKBRSF G1999075406. We gratefully mention the help of Keith Butler and Andreas Korn. Don VandenBerg kindly supported us making available his most recent calculations of stellar evolution.